Interferometric observations of pre-main sequence disks - Caltech ...

Università degli Studi di Milano
Facoltà di Scienze Matematiche, Fisiche e Naturali
Dottorato di Ricerca in Fisica, Astrofisica e
Fisica applicata - XIX Ciclo
FIS/05-Astronomia e Astrofisica – Matricola R05633
Interferometric observations of
pre-main sequence disks
Tutore: Prof.ssa Antonella Natta
– Andrea Isella –
Coordinatore: Prof. Gianpaolo Bellini
Anno Accademico 2005-2006

A Rossana

CONTENTS
I Introduction 1
1 Scientific overview 3
1.1 Star formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Disk structure in the pre-main sequence phase . . . . . . . . . . . . . . 8
1.3 Grain growth and planetary formation . . . . . . . . . . . . . . . . . . 15
1.4 Outline and aims of the thesis . . . . . . . . . . . . . . . . . . . . . . . 19
2 Introduction to interferometry 23
2.1 A bit of history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Basic principle of astronomical interferometry . . . . . . . . . . . . . . 26
2.3 Optical versus millimeter interferometry . . . . . . . . . . . . . . . . . 29
2.4 Introduction to modelling techniques . . . . . . . . . . . . . . . . . . . 34
II Dust in the inner disk 39
3 The shape of the inner rim in proto-planetary disks 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

iv CONTENTS
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 The rim shape . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 The rim SED . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Rim images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.4 Grain size distribution . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.1 The 3µm bump . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.2 The rim radius . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.3 LkHa101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.1 Evaluation of Eq. 3.1 for single grains . . . . . . . . . . . . . . 64
3.6.2 SED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Large dust grains in the inner region of circumstellar disks 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Target stars and observations . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 The dust evaporation and the “puffed-up” inner rim . . . . . . . 72
4.3.2 Visibility model . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Comparison with the observations . . . . . . . . . . . . . . . . . . . . 75
4.4.1 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.2 MWC 758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.3 VV Ser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.4 CQ Tau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.5 V1295 Aql . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4.6 MWC 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.7 AB Aur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS v
4.5 Comparison with previous analysis . . . . . . . . . . . . . . . . . . . . 79
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6.1 Presence of large grains . . . . . . . . . . . . . . . . . . . . . 81
4.6.2 Inclination and position angle . . . . . . . . . . . . . . . . . . 83
4.6.3 Improving the model constrains . . . . . . . . . . . . . . . . . 84
4.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 85
5 More on the “puffed–up” inner rim 95
5.1 Fuzzy and sharp rims . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 New shapes for the rim . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 New observational results . . . . . . . . . . . . . . . . . . . . . . . . . 101
III Gas in the inner disk 105
6 AMBER/VLTI spectroscopy of HD 104237 107
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Observations and data reduction . . . . . . . . . . . . . . . . . . . . . 110
6.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
IV The structure of the outer disk 117
7 The keplerian disk of HD 163296 119
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2 Observations and data reduction . . . . . . . . . . . . . . . . . . . . . 121
7.2.1 PBI observations . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2.2 SMA observations . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2.3 VLA observations . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3 Observational results . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3.1 Disk morphology and apparent size . . . . . . . . . . . . . . . 123

vi CONTENTS
7.3.2 Disk kinematic . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3.3 Free-free contribution and spectral index . . . . . . . . . . . . 128
7.3.4 Disk mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.4 Disk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.4.1 Continuum emission . . . . . . . . . . . . . . . . . . . . . . . 132
7.4.2 CO emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.5.1 Method of analysis . . . . . . . . . . . . . . . . . . . . . . . . 134
7.5.2 Continuum emission . . . . . . . . . . . . . . . . . . . . . . . 135
7.5.3 CO emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.6.1 The disk outer radius . . . . . . . . . . . . . . . . . . . . . . . 139
7.6.2 The gaseous disk . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 142
V Conclusions and future prospects 149
8 Summary and Conclusions 151
8.1 The inner disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2 The outer disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9 Future developments 157
9.1 Gas in the inner disk of Herbig Ae stars . . . . . . . . . . . . . . . . . 157
9.2 Probing planet formation through interferometric observations . . . . . 161
Published Papers 169
Bibliography 171

Part I
Introduction

CHAPTER 1
1.1 Star formation
Scientific overview
Star formation in our galaxy is a continuous process that occurs inside large concentra-
tions of cold gas and dust which contain more than 50% of the interstellar matter. Char-
acterized by temperatures of about 10 K, extents of tens of parsec and masses between
10 4 and 10 6 M⊙, the Giant Molecular Clouds are sustained against the gravitational
collapse by supersonic turbulent motions (Zuckerman and Evans, 1974) which lead to
strong density inhomogeneity inside the cloud. When the gravity on these smaller scale
condensations wins the gas pressure, the material collapses and the stellar formation
starts. During the contraction, clouds break in smaller parts following a fragmentation
process that ends with the formation of cores when the collapse becomes adiabatic, i.e.,
the gas compression corresponds to an increase in temperature. Before that, the col-
lapse has been isothermal since the gas density was too low to prevent the cooling of the
cloud.
Cores generally have extent of about 0.1 pc; the gas density is between 10 4 and
10 5 molecules cm −3 and the temperature is similar to the temperature of the original
molecular cloud. Given these properties, the typical core mass is of few M⊙, higher than

4 Scientific overview
the critical Jeans mass for the gravitational instability (∼1M⊙). The gravity overwhelms
the thermal pressure and the core collapses in a free-fall time scale of few 10 5 years.
While the core rotation has been found to be negligible compared to gravity (Ar-
quilla and Goldsmith, 1986; Caselli et al., 2002), magnetic field and turbulence could
play an important role in the core stability delaying or preventing the star formation.
Many authors think that molecular core are indeed supported against gravity by mag-
netic field (Ostriker et al., 1999; Li et al., 2004); they can collapse only after the mag-
netic field has been dissipated by ambipolar diffusion (Spitzer, 1978) on a time scale
of 10 6 yr, an order of magnitude higher than the typical free-fall time. Balancing grav-
ity with magnetic energy, critical masses comparable with core masses can be obtained
with a magnetic field of about 50µG. Recent measurements of magnetic fields in few
dense cores (Crutcher et al., 2006) suggest that most cores are closed to equilibrium,
but with large uncertainties. Moreover there are authors who strongly disagree with this
scheme, and believe that magnetic field has no role in the core collapse (Nakano, 1998).
Turbulent motions can also prevent cores from collapsing. In many cases, the velocities
measured from the width of molecular lines are indeed large enough to sustain the core
against the gravity. On the other hand even turbulent motions dissipate in a time com-
parable to the free-fall time (Nakano, 1998) and the core collapse can be only delayed.
If the core rotation can not prevent the collapse, the specific angular momentum
determines the stellar multiplicity of the final system and has a strong influence on the
structure of the circumstellar environment. Hydrodynamic models showed indeed that
the fragmentation of an isothermal core depends strongly on its initial angular momen-
tum (Matsumoto and Hanawa, 2003; Boss and Myhill, 1995) and that that process is
likely to be the dominant mode of binary formation (Bodenheimer et al., 2000). For
an exhaustive description of the stellar multiplicity see the review of Goodwin et al.
(2006).
In the case of a slowly rotating sphere of gas of constant angular velocityΩ, the
collapse is monolithic and leads to a structure composed by four concentric regions
(Terebey, Shu and Cassen, 1984):

1.1 Star formation 5
• in the center of the system an accreting core forms with radius R⋆ and increasing
mass M⋆= ˙Macct. Since the collapse is adiabatic the gas pressure re-establishes
itself trying to reach the hydrostatic equilibrium. Both the gravitational energy
produced during the contraction and the accretion luminosity Lacc= GM⋆ ˙Macc/R⋆
are released by the system that become a protostar;
• an accreting disk forms between R⋆ < R < Rc. The disk outer radius, Rc =
Ω 2 cst 3 /16, is the centrifugal radius defined as the place where the rotational ve-
locity vrot=ΩR 2
in f /R is equal to the free-fall velocity v f f∼ (GM⋆) 1/2 R −1/2 . The
centrifugal radius corresponds to the maximum distance at which the infalling
material impact the disk;
• a spherical envelope in free-fall between Rc

6 Scientific overview
Log νFν
Log νFν
Log νFν
Log νFν
-7
-8
-9
-7
-8
-9
-7
-8
-9
-7
-8
-9
Core
Active
disk
CLASS II
Passive
disk
CLASS III
Remnant
disk
CLASS 0
CLASS I
Protostar
Star
Star
11 12 13 14 15
Log ν (Hz)
zZ
5000 AU
500 AU
Figure 1.1: The figure shows the formation of a single star and the evolution of the circum-
stellar material. The left panels show the spectral energy distribution of the system at different
50 AU
evolutionary stages; the right panels show the corresponding system geometry.
5 AU

1.1 Star formation 7
the mid-infrared. Class II stars are characterized by a SED composed by the protostellar
emission and the flux arising from the circumstellar disk composed by gas and dust;
in this phase most of the circumstellar material is in the disk. The SED peaks in the
near infrared and decreases roughly as a power law from the mid infrared to millimeter
wavelengths. Finally, for the Class III objects the emission of the circumstellar material
becomes negligible and the SED has a pure photospheric shape.
The protostellar phase ends when the core has either accreted onto the star+disk or
dissipated by winds and jets. At this point the mass of the central star is practically
fixed and the mass accretion rate ˙M decreases to very low values (typically ˙M< ∼ 10 −7 M⊙
yr −1 ). The star evolves by slow contraction on the Kelvin-Helmholtz timescale (tKH∼
GM 2 ⋆ /R⋆L⋆) toward the ZAMS, where the stellar luminosity is due to the hydrogen
burning. During this pre-main sequence phase, the stellar luminosity is dominated by
the gravitational energy released in the contraction while the accretion luminosity is
negligible.
Pre-main sequence stars are located on the Herzsprung-Russell diagram in a region
to the right of the ZAMS and evolutionary tracks have been computed by several au-
thors (Stahler and Palla, 2005). By comparing the position of any individual star to the
evolutionary tracks, it is possible to determine its age and mass (a more direct way to
determine the mass of the central star will be discussed in Sec. 1.2). In this way it has
been found that the age of pre-main sequence stars range from 10 5 to 10 7 yr. It is also
generally accepted to call T Tauri stars (TTS) the low-mass (M⋆< 1.5M⊙) pre-main se-
quence stars, while Herbig Ae/Be stars (HAe/Be, since the work of George H. Herbig in
1960) are the intermediate mass pre-main sequence stars (1.5M⊙< M⋆< 8M⊙). Higher-
mass stars reach the ZAMS while still accreting and do not have a pre-main-sequence
phase (Beuther et al., 2006).
Both TTS and HAe/Be stars show a strong stellar activity in the form of continuum
excess emission in the UV and IR, emission lines, strong photometric and spectroscopic
variability, ejection of matter as wind and jets (see the review of Bally et al., 2006). In
many cases the activity is strictly related to the presence of circumstellar disks, which

8 Scientific overview
are the subject of this thesis.
1.2 Disk structure in the pre-main sequence phase
Circumstellar disks are an essential component of the process of stellar formation.
Nearly all stars are thought to be born with circumstellar disks (Beckwith and Sar-
gent, 1996; Hillenbrand et al., 1998) and direct evidence of disks exist in a increasing
number of pre-main sequence stars of all masses.
Even if spectroscopically and spatially resolved observations of TTS and HAe stars
have given a wealth of information on circumstellar disk structures, no theory or model
can simultaneously account for all the physical processes occurring in circumstellar
disks – viscous accretion onto the central star, mass loss due to outflow, irradiation by
the central star, interaction with the stellar wind and magnetic field, turbulent mixing of
material, dust grain growth, gradual settling of the dust towards the disk mid plane and
planetary formation – and many questions about the structure and evolution of circum-
stellar disks are still open.
The simple case of an accretion disk where the mass accretion rate ˙M is constant
and the viscous accretion is the only source of heating has been discussed by Shakura
and Sunyaev (1973), while Lynden-Bell and Pringle (1974) also discussed the time-
dependent case. In these models, viscous stresses within the disk transport material
towards the star and angular momentum to the disk outer regions in a viscous timescale
tν= r 2 /ν, depending on the viscosityν.
Observation of a large sample of pre-main sequence stars has shown that after the
infall phase is finished (in 10 5 years), the total disk mass decreases with time and most of
disk material is dispersed after about 10 7 years (Meyer et al., 2006). After Shu (1992),
it has been clear that molecular viscosity alone was too small to account for the disk
lifetime and an additional viscosity should be introduced to explain the observations.
To avoid having to solve the problem of viscosity in detail, Shakura and Sunyaev
introduced theαprescription in which the viscosity is expressed by the relationν=αhcs

1.2 Disk structure in the pre-main sequence phase 9
where h is the pressure scale height of the disk, cs is the sound speed and the parameter
α contains all the uncertainties on the real nature of the viscosity. Assuming that the
material orbits with a keplerian rotation around the star and that the disk is in hydrostatic
equilibrium in the vertical direction, the viscous time scale, which is the time to move
matter from r to the star, is
tν∼ 3×10 6
0.01 r
5/4 yr (1.1)
α 100AU
which, forα = 0.01, is roughly consistent with the fact that disks have in general
disappeared by the time stars reach the main-sequence.
A useful quantity to describe how the mass is distributed in a disk is the surface
densityΣ(r) defined as the integral of the mass densityρ(r, z) in the vertical direction
z. In the case of a steady disk, i.e. characterized by a constant mass accretion rate,
the equation of mass conservation into the disk implies thatΣ∝r −3/4 α −1 . As it will
be discussed in Chapter 7, high angular resolution millimetric observations of a large
number of HAe and TTS disks indicate that the surface density generally decreases with
the distance from the central star as the power low r −0.8 − r −1 , indicating thus an almost
constant value ofαthrough all the disk.
Among the many disk instabilities suggested to explain the observed value ofα(Pa-
paloizou and Lin, 1995; Balbus, 2003), the magnetorotational instability (MRI), present
in a weakly magnetized disk, is at the present the most accepted mechanism to drive
turbulent viscosity in disks and transport angular momentum outwards (Velikhov, 1959;
Chandrasekhar, 1960; Balbus and Hawley, 1991; Stone and Pringle, 2001). However,
many authors claim that neither thermal ionization, cosmic ray ionization, nor X-rays
are able to provide a sufficient number of free electron to have MRI operating in the
disk-mid plane where most of the disk mass and angular momentum are confined (Gam-
mie, 1996; Glassgold et al., 1997; Ilgner and Nelson, 2006). Indeed, in the case of TTS
disks, Gammie (1996) has shown that the ionization fraction is sufficient to couple the
magnetic field to the gas only very close to the central star (r< ∼ 0.1 AU) or in a superficial
layer of the disk. Between these active layers, there is a dead layer corresponding to the

10 Scientific overview
disk regions close to the mid plane where the magnetic field and the gas are not coupled
and the MRI can not operate.
A second efficient process of angular momentum transfer that may operate in pre-
main sequence disk is gravitational instability (Tomley et al., 1991; Laughlin and Bo-
denheimer, 1994; Bertin, 1997; Bertin and Lodato, 1999; Lodato and Bertin, 2001 and
2003; see also the recent review of Durisen et al., 2006). Gravitational instability may
dominate the angular momentum transfer in the earliest stages of the stellar evolution,
when the mass of the circumstellar disk may be comparable with that of the central
star. In the more evolved evolutionary phase typical of TTS and HAe stars, most of the
circumstellar mass is accreted in the central star with the result that the measured disk
masses are usually below the value required for gravitational instability. The existing
estimates of the mass of the pre-main sequence disks are in most cases obtained from
the measure of the optically thin dust emission at millimeter wavelength. This method
is however based on strong assumptions on the dust opacity and it can be affected by
large uncertainties (see also the discussion in Chapter 7). Thirty years after the work of
Shakura and Sunyaev, the nature of the disk viscosity is still matter of debate.
For anα-disk (a disk based on theαprescription), the temperature at each radius
r is Td ∝ r −3/4 , obtained by equating the heating rate, due to viscous dissipation, to
the cooling rate, computed assuming that the disk radiates as a black body at the local
temperature Td. However, for most disk around TTS and HAe stars the heating by
stellar radiation is dominant over viscous heating and the disk temperature depends on
the angle at which the stellar radiation strikes the disk surface. Since the circumstellar
material is composed by a large amount of dust and gas, all the stellar flux intercepted
by the disk is absorbed and re-emitted at longer wavelength, between the near infrared
and the millimeter. Disk where the stellar radiation is the dominant source of heating
are refereed to as reprocessing disks. Kenyon and Hartmann (1987) recognized that
reprocessing disks in hydrostatic equilibrium are flared, i.e., the disk opening angle
increases with r. The flaring geometry allows the disk to capture a significant portion
of the stellar radiation at large radii where the disk is cool, increasing the mid to far

1.2 Disk structure in the pre-main sequence phase 11
infrared emission and explaining the strong infrared excess observed around TTS and
HAe stars. The resulting temperature profile is flatter than in aα-disk and typically
Td∝ r −1/2 . At any given radius, the vertical dependence of the density on z is obtained
by integrating the equation of hydrostatic equilibrium, which gives
ρ(r, z)=ρ(r, 0) exp(−z 2 /2h 2 ), (1.2)
where the disk pressure scale is h∝r 9/7 . The density decreases rapidly with z, so that
the matter is concentrated on the disk mid plane.
The SED of a disk can be computed in first approximation assuming that each disk
annulus emits as a black body at the effective temperature Td, so that the observed flux
at any given frequencyνis given by the relation
Fν= cosθ
D 2
Rd
Rin
Bν(Td)(1−e −τν )2πrdr, (1.3)
where Rin and Rd are the inner and outer disk radius, D is the distance from the observer,
θ is the disk inclination with respect to the line of sight (θ=0means face-on disks), Bν
is the black body emission. The optical depthτν is
τν= 1
cosθ kνΣ(r) (1.4)
where kν is the total (gas+dust) opacity at the frequencyν. For conditions typical of
HAe and TTS, the main portion of the energy is emitted in the wavelength range from
∼1µm to 100µm, where the SED usually shows a slow dependence on the wavelength
(see Fig. 1.2). At longer wavelengths the SED is characterized by a power law profile
with a slope depending on the grain properties and optical depth as discussed in Sec. 1.3.
As direct consequence of the radial monotonic decrease of the disk temperature,
observations at different wavelengths map disk emission at distances from the central
star which increase with the observational wavelength. Between 1µm and 7µm, the
emission is dominated by the disk regions very close to the central star (r< ∼ 0.5 AU)
and characterized by temperatures higher than about 500–1000 K. The study of this
innermost region of the disk is particularly interesting because it is the locus where
the gas is supposed to accrete on the star, where the interaction between the stellar

12 Scientific overview
Figure 1.2: From Dullemond et al. (2006). Build-up of the SED of a flaring circumstellar disk
and the origin of various components: the near infrared bump is supposed to originate in the
puffed-up inner rim, the infrared dust features (as the silicate ones between 10µm and 20µm)
from the warm surface layer, and the underlying continuum from the deeper and cooler disk
regions. Typically the near and mid-infrared emission comes from small radii, while the far-
infrared and the millimeter emission come from the outer disk regions.
radiation and the disk is stronger and where the magnetic field is supposed to drive the
often observed outflows and jets that propagate perpendicularly to the disk plane. In the
innermost region of the disk, where the temperature is higher than∼1500 K, most of the
dust sublimates (Pollack et al., 1994) and the opacity undergoes a strong discontinuity
(Natta et al., 2001; Dullemond et al., 2001). If the gas inside the dust sublimation radius

1.2 Disk structure in the pre-main sequence phase 13
absorbs a negligible amount of stellar radiation, the dust inner rim is perpendicularly
illuminated by the central star and is expected to be hotter and higher, i.e. with an
higher pressure scale, than a standard reprocessing disk. The resulting puffed-up inner
rim dominates the near-infrared emission and naturally explains the strong near-infrared
excess observed in most of the SEDs of HAe and TTS stars. The presence of a puffed-up
inner rim can influence the structure of the outer regions of the disk shadowing the dust
from the direct stellar radiation. Meeus et al. (2001, 2003) divided the SEDs of HAe
stars into two groups: those with strong far-infrared flux (group I) and those with weak
infrared flux (group II), suggesting that the latter are characterized by a disk that lies
in the shadow of the puffed-up inner rim. Although the real existence of the puffed-up
inner rim is still under debate (Vinkovic et al., 2006), model predictions are in good
agreement with the spatially resolved near infrared interferometric observations of a
number of HAe and TTS stars (see the review of Millan-Gabet et al. 2006). The second
part of the thesis (Chapters 3, 4 and 5) is dedicated to the discussion of the structure of
the puffed-up inner rim and to the analysis of interferometric observations of the inner
disk.
The structure of the gaseous disk inside the dust sublimation radius is still poorly
known (see the review of Najita et al., 2006). Existing high resolution spectroscopic
observations of CO and water near infrared emission lines indicate that the gas extends
very close to the central star, probably until the distance at which the gaseous disk
is truncated by the stellar magnetic field lines (Shu et al., 1994). Inside this point,
typically located at few stellar radii, the gas moves along the magnetic field lines and
accretes on the central star emitting the UV radiation usually observed in HAe and TTS
spectra as UV veiling (Hartman et al., 1998; Bouvier et al., 2006). The accreting gas
is also supposed to be responsible for the hydrogen recombination lines (Hα, Brγ, Paβ,
etc) characteristic of most HAe and TTS. Both the amount of veiling and the hydrogen
line intensities depend on the accreting gas density and are used to measure the mass
accretion rate (Muzerolle et al., 2001; Natta et al., 2006). Chapter 6 is dedicated to
the spectrally resolved near infrared interferometric observations of the gas in the dust-

14 Scientific overview
depleted inner disk.
The mid-infrared part of the SED comes from disk regions at distances between
∼5 AU and∼50 AU, where the disk opacity and the thermal emission are dominated
by the dust. The dust density is generally high enough that the disk is optically thick
to its own emitted radiation from near to far infrared and the resulting SED is in first
approximation the sum of many black bodies with different effective temperatures. A
closer look at the physics of a reprocessing disk reveals however that the disk surface
temperature is generally higher than the effective temperature Td (Calvet et al., 1991;
Chiang and Goldreich, 1997). Dust grains in the surface layers are directly exposed
to the stellar radiation and are hotter than dust grains in the disk interior which are
heated only by the diffused infrared emission of the disk itself. As a consequence, the
thermal emission from this surface layer produces emission features in good agreement
with nearly all the TTS and HAe stars spectra (Meeus et al., 2001; Kessler-Silacci et
al., 2006). The analysis of spectroscopic features, mainly in mid-infrared, give a lot of
information of the dust composition and size and will be discussed in more detail in the
next section.
Finally, particularly interesting for the study of circumstellar disk is the range of long
wavelengths, from sub-millimeter to radio, where the dust opacity can be approximated
by a power law, kν∝λ −β (Beckwith and Sargent, 1991), and the disk is optically thin to
its own thermal radiation (τν < ∼ 1). In this case, the observed flux Fν is directly related
to the dust surface densityΣ(see Eq. 1.3 and Eq. 1.4) and both the disk mass and the
parameterβcan be derived from observations at different wavelengths (Mannings and
Sargent, 1997; Testi et al., 2001 and 2003; Natta et al., 2004). Millimeter interferome-
ters allow to resolve spatially the continuum disks observations and both spatially and
spectrally the molecular lines. Using CO vibrational transitions, it has been confirmed
that in most of the observed TTS and HAe stars the gas is orbiting the central star with
velocity patterns in good agreement with keplerian rotation. In this case, the gas veloc-
ity gives a direct measure of the mass of the central object, which is independent of the
position of the star on the HR diagram and of the evolutionary tracks. Taking advantage

1.3 Grain growth and planetary formation 15
of varying opacity of the different transitions of CO and its isotopes, is possible to probe
disk properties at different depth and calculate the relative CO abundance. For some
nearby stars, the continuum observations show asymmetry in the radial dust distribu-
tion which can be caused by dynamical perturbations of unknown companions or giant
planets (Pietú et al., 2005; Corder et al., 2005). All these points will be discussed in
more detail in Chapter 7 which describes new millimeter observations of the HAe star
HD 163296.
1.3 Grain growth and planetary formation
Circumstellar disks around TTS and HAe are composed for about 99% of gas and for
only 1% of solid dust grains. Nevertheless, as seen in the previous section, the dust
component is responsible for the observed disk thermal emission, which remains a key
diagnostic tool to detect disks and characterize their structure. Dust grains dominate the
disk opacity, control the disk temperature and geometry, regulate the chemical reactions
of molecular species as CO and H2O in the colder parts of disks, and are the building
block for the formation of planetesimals and planets within disks.
From the diffuse interstellar medium to Class I objects, dust grains are, and remain,
a mixture of amorphous silicates and carbons, with a size distribution from∼ 0.01µm
to∼ 0.2µm (Draine et al., 2003; Bianchi et al., 2003; Kessler-Salacci et al., 2005). The
situation changes drastically in circumstellar disks in which they grow many order of
magnitude in size forming kilometers-size body (planetesimals) and even planets.
While the growth from sub-micron to micron sizes is probably driven by collisional
aggregation in Brownian velocity fields (Blum, 2004), it is still under debate if the for-
mation of meter size objects occurs through collisional aggregation (see the review of
Dominik et al., 2006) or gravitational instabilities in the over-dense dust regions of the
disk (Youdin and Shu, 2002; Rice et al., 2004 and 2006). Modelling the grain growth
requires a complex analysis of the dust and gas coupling, dust sticking properties and
velocity fields within the disk. While growing in mass, particles start to decouple from

16 Scientific overview
the gas and sediment on the disk mid plane, increasing the collisional efficiency and
probably reaching a dynamical equilibrium in which large particles form smaller ones
through destructive collisions and vice versa. Given the extreme complexity of the
problem, different models gives often contradictory results and the formation of plan-
etesimals remain at the moment an open question.
As consequence of the grain growth and settling, circumstellar disks are predicted
to be vertically stratified with small grains in the uppermost layers of the disk and large
grains in the disk mid plane. Furthermore, grain growth should be much slower in
the outer regions of the disk than in the regions closer to the star, which should be
completely depleted of small grains. The situation becomes even more complicated if
we consider that inward radial circulations, due for example to the disk viscosity and/or
to the dust drag due to the gas, bring material close to the star where dust sublimates.
At the same time, opposite radial circulations, due, for example, to the stellar wind
or to the magnetorotational instability, move the inner material outward permitting the
ricondensation of grains and the formation of crystalline structures (Gail, 2004).
Despite all the theoretical difficulties in understanding the dust evolution, grain prop-
erties in disks can be explored with a large number of techniques and over large range
of wavelengths. It is worth to stress that since the disks around pre-main sequence stars
have high optical depth, observations at optical and infrared wavelengths allow to ob-
tain information only on grains located in the very superficial layer of the disk where
τ≃1. Such observations are thus sensitive only to a very small fraction of the dust mass
which is concentrated on the disk mid plane (see Eq. 1.2). An important exception to
this general rule is represented by near-infrared interferometric observations which are
able to spatially resolve the regions close to the star where the dusty disk is truncated
by the dust evaporation. We will show in Chapter 4 that in this case, the measure of the
location of the dust evaporation radius gives information on dust grains present in the
disk mid plane.
Particularly important to the study of the chemical composition, size and structure
of the dust grains is the mid-infrared spectral region (between∼ 3µm and∼ 100µm),

1.3 Grain growth and planetary formation 17
rich in vibrational resonances of many dust species. In disks around HAe and TTS,
this dust features are generally seen in emission and are thought to originate in the
optically thin disk surface layer, directly illuminated by the stellar radiation (Calvet et
al., 1991; Chiang and Goldreich, 1997). The more common features observed around
TTS and HAe stars are those of amorphous and crystalline silicates, located around
10µm and 20µm. Recent Spitzer data (e.g. Furlan et al., 2006) show a large variation
in the strength and shape of this features, confirming the correlation firstly identified by
van Boekel et al. (2005): weaker is the feature flatter is its shape. This is interpreted in
terms of the growth of silicate grains from sub-micron to micron sizes; if grains grow
further, the silicate emission disappears.
The presence of crystalline grains around HAe and TTS stars implies that the pris-
tine material underwent a drastic change in its structure due to some, not yet known,
energetic process. Using spectrally resolved mid-infrared interferometry, van Boekel et
al. (2004) showed for the first time that the disk surface inside 1-2 AU around three HAe
stars is much more crystalline than the outer disk surface, supporting the idea that crys-
tallization occurs in the inner disk (Gail, 2004). However crystalline silicates are also
present at larger distances (∼ 20 AU), and in the comets in our Solar System, suggesting
that the radial drift can have an effective role in the dust evolution within circumstellar
disks.
At millimetric and centimetric wavelengths disks become optically thin to their own
thermal radiation and the observations can probe the bulk of the dust mass concentrated
on the disk mid plane. The SED of the continuum emission at these wavelengths can be
directly related to the grain emissivity which, in turn, depends on the grain properties.
The millimetric dust opacity can be approximated by a power law, k∼λ −β , where the
exponentβdepends strongly on the grain size distribution and on its upper limit. In
a size distribution composed only by grains larger than the observing wavelength, i.e.
centimeter size grains, the dust opacity is grey andβ=0, whileβis close or larger than
unity only for sub-millimeter grains (see Fig. 1.3). If the disk emission is optically thin
(τν≪1) and the Reyleigh-Jeans approximation is verified (Bν(T)∝λ −2 ), the emitted

18 Scientific overview
Figure 1.3: From Natta et al. (2006). Opacity indexβfor grains with a size distribution n(a)∝
a −q between amin and amax. The indexβis computed between 1 and 7 mm for compact segregated
spheres of olivine, organic materials, water ice (Pollack et al., 1994), and plotted as function of
amax. Different curves correspond to different values of q, as labelled. In all cases, amin≪ 1 mm.
The dotted curve showsβfor grains with single size amax. The histogram on the right shows the
distribution of the values ofβderived from millimeter observations. About 60% of the objects
haveβ≤1, and amax > ∼ 1 cm.
flux is expressed by the power law Fν ∝ λ −α . Using Eq. 1.3 and Eq. 1.4 it can be
demonstrated using that in this caseα=β+2. This simple relation allow to translate
the observed value ofαinto the opacity power low indexβ. To verify the fundamental
assumption that the millimetric disk emission is optically thin, the emission itself must
be spatially resolved and the disk size measured (Testi et al., 2001 and 2003).
There are at present several disks whose millimeter emission has been resolved with
interferometers (Natta et al., 2004; Rodmann et al., 2006) and in Chapter 7 we will
present new multi-wavelengths observations of the HAe star HD 163296. The obtained

1.4 Outline and aims of the thesis 19
values ofβvaries from 1.6 (very similar to the interstellar medium and Class 0/I objects)
to 0.5, with most of the objects withβ< ∼ 1, corresponding to the presence of centimeter
size grains (Natta et al., 2004; Draine, 2006).
In order to constrain dust growth models, the next step is to check for variations of
the dust properties as a function of the radius. Our observation of HD 163296 are the
first attempt to obtain the multi wavelengths high angular resolution and high sensitivity
observations required for such analysis and we refer to the Chapter 7 for the results.
The major actual limitation to understand planetary formation is that none of the
discussed techniques is sensitive to bodies larger than few centimeters. In principle,
planetesimals of few kilometers in size could be detected through the dynamical pertur-
bations exerted on the surrounding material. Particularly interesting is the discovery of
clumps in the dust density distribution in the disk around the close-by HAe star AB Aur
(Corder et al., 2005; Pietú et al., 2005). On the other hand, the detection of meter-size
bodies is at the moment practically impossible.
1.4 Outline and aims of the thesis
The aim of this thesis is to study the structure of circumstellar disks around pre-main
sequence stars. Taking advantage of high spatial resolution interferometric observations
we have focused our attention on the gas and dust properties both in the inner disk, at
fractions of AU from the central star, and in the outer disk regions, at R>50 AU.
The thesis is divided into five parts: a general introduction and a general review of
astronomical interferometry (this Chapter and Chapter 2), a study of the dust in the inner
disk (Chapter 3, 4, 5), a study of the gas in the inner disk (Chapter 6), a study of gas and
dust in the outer disk (Chapter 7) and the final remarks (Chapter 8 and 9).
Chapter 2 describes the basis of interferometry, similarities and differences between
the near-infrared and millimeter interferometry and a presentation of the modelling tech-
niques used to analyse interferometric observations.
In Chapter 3 we describe the structure of the innermost part of circumstellar disks.

20 Scientific overview
We propose a model in which the dusty component of the disk is truncated by the dust
evaporation forming a puffed-up inner rim. Introducing the dependence of the dust
evaporation temperature on the gas density, we predict that the bright side of the rim
is curved, rather than vertical, as expected when a constant evaporation temperature
is assumed. We demonstrate that both the radial position of the rim and the emitted
flux depend on the dust grain size. We finally compute synthetic images of the curved
rim seen under different inclinations and show how images become more and more
asymmetric for increasing inclinations.
Chapter 4 describes the analysis of interferometric observations of six HAe stars in
the framework of our puffed-up inner rim model. Assuming that astronomical silicates
are the most abundant dust component, we found that in four cases dust grains larger
than few microns are required by, or consistent with the observations. We constrain the
inclination of the inner disk and, where possible, compare our results with the values of
the outer disk existing in the literature.
In Chapter 5 we discuss new developments of the puffed-up inner rim model that
followed the publication of the results presented in Chapter 3.
In Chapter 6 we discuss the first results of spectrally resolved interferometric obser-
vation (VLTI/AMBER) of the Brγ emission of the HAe star HD 104237. We point out
that the line emission and the continuum infrared emission, arising from the puffed-up
inner rim, have similar size scales. We conclude that the line emission is not compatible
with standard magnetospheric accretion models and most likely originate in a compact
wind, starting from a region at 0.2-0.5 AU from the central star.
Chapter 7 describes recent multi wavelengths interferometric observations of the
HAe star HD 163296, performed with the Sub-Millimeter Array, the Plateau de Bure
Interferometer and the Vary Large Array, between 0.87 mm and 7 mm. Molecular CO
emissions show a very large circumstellar disk with a velocity pattern in good agree-
ment with Keplerian rotation. The mass of the central star and the disk parameters are
determined. Comparing the CO and continuum emission, we suggest that the dust prop-
erties undergo a sharp discontinuity at about 200 AU from the star and discuss possible

1.4 Outline and aims of the thesis 21
explanations.
Chapter 8 summarizes the main results discussed in the thesis while in the Chapter 9
we discuss possible future new observations and developments.

CHAPTER 2
2.1 A bit of history
Introduction to interferometry
Interferometry is based on wave properties of light as first observed by Thomas Young
in 1803 through his famous two-slit experiment: if monochromatic light from a distant
point source propagates through two-slits, see Fig. 2.1, the resulting illumination pattern
is composed by bright and dark fringes, due respectively to the constructive and destruc-
tive interference between the secondary waves produced along the slits. The condition
for constructive interference implies that the fringe spacing∆Θ of the intensity distri-
bution is inversely proportional to the projected separation between the two slits, called
baseline B, in units of the radiation wavelengthλ:
∆Θ= λ
. (2.1)
B
The corresponding fringe spatial frequency (fringes per unit angle) is thus
u= B
. (2.2)
λ
Imagine now another point-source of light (of equal brightness, but incoherent with the
first) located at an angle ofθ=λ/2B from the first source (see right panel of Fig. 2.1).

24 Introduction to interferometry
baseline=b
Point source
at infinity
Incoming plane waves
∆θ =
2 slits
Fringe spacing
λ /b radians
Interference pattern
(Visibility = 1)
λ
= Wavelength
Point sources
at infinity
separated by
1/2 the fringe
spacing
2 sine waves
destructively
interfere
(Visibility = 0)
Figure 2.1: From Monnier (2003). Young’s two-slit interference experiment (monochromatic
light) is presented to illustrate the basic principle behind stellar interferometry. On the left is the
case for a single point-source, on the right a double source with an angular separation half of the
fringes spacing. The interference patterns shown represent the intensity distribution.
In this case, the two illumination patters are out of phase of 180 ◦ , cancelling each other
out and presenting an uniformly illuminated screen. This simple experiment shows a
fundamental property of interference, namely that the contrast of fringes is a function
of the geometry of the source.
The application of interferometry in astronomy has been firstly proposed by Fizeau
in 1868, who suggested to measure the fringes pattern produced by the stellar light to

2.1 A bit of history 25
measure stellar diameters. The practical implementation of this technique arrived many
years later with Michelson who, in 1891, measured the angular diameter of Jupiter’s
satellites. For the following thirty years astronomical interferometry stalled; only in
1921, Michelson and Pease successfully measured the diameter of the super-giant star
Betelgeuse using an interferometer at the Mount Wilson Observatory. While techni-
cal difficulties delayed the growth of optical interferometry until the mid-1970’s, radio
interferometry had a fast development after the second world war. In 1946, Ryle and
Vonderg constructed a radio analogue of the Michelson interferometer and soon discov-
ered a number of new cosmic radio sources. During 1950s and 60s, Ryle and Hewish
developed the aperture synthesis method at radio wavelengths which allows the source
image reconstruction combining the signals from many different pairs of antennas. In
the 70s, they applied this method to reproduce a one-mile effective aperture using an ar-
ray of three antennas with a diameter of 60 feet each. A common technique in aperture
synthesis imaging is to use the rotation of the Earth to increase the number of different
antenna pairs included in an observation. In the last forty years, aperture synthesis has
been widely applied in radio interferometry with the result that astronomical images of
good quality can be now obtained (see Chapter 7)
Despite the improvement in the radio domain, optical interferometry has only been
developed from the mid-1970’s with the advent of new technology: detectors, actuators,
servo-control, etc. Labeyrie (1975) was the first to directly combine the light from two
separated small telescopes at the Nice observatory. In the 1980s, the aperture synthe-
sis technique was extended to visible and infrared astronomy by the Cavendish Astro-
physics Group, providing the first very high resolution images of nearby stars. These
development allowed the planning and construction in the mid-90s of several optical in-
terferometers. In particular, with the ESO Very Large Telescope Interferometer (VLTI),
the Keck Interferometer (KI) and the CHARA array, optical interferometry is becoming
a familiar technique for an increasing number of astronomers.

26 Introduction to interferometry
2.2 Basic principle of astronomical interferometry
A fundamental property of interferometers is the angular resolution. Classical diffrac-
tion theory has established the Reyleigh Criterion for defining the (diffraction-limited)
resolution∆θ of a filled circular aperture of diameter D, at the observational wavelength
λ:
∆θ=1.22 λ
D
rad. (2.3)
Considering a binary star system, this criterion corresponds to the angular separation on
the sky where one stellar component is centered on the first null of the diffraction pattern
of the other; the binary is then said resolved. A similar criterion can be defined for an
interferometer: the same binary system is resolved by an interferometer if the fringe
contrast goes to zero at the longest baseline. As shown in Fig. 2.1, this correspond to
the angular separation∆θ=λ/2B.
The fringe contrast is historically called visibility and, for the simple two-slit inter-
ferometer considered here, can be written as
|V|= Imax− Imin
Imax+ Imin
= Fringe amplitude
, (2.4)
Average Intensity
where Imin and Imax denote the minimum and maximum intensity of the fringes. From
this definition, it follows that the left and the right fringe patterns of Fig. 2.1 have vis-
ibility of 1 and 0, corresponding respectively to non resolved and resolved sources.
Moreover, if the source is shifted with respect to the optical axis of the system (see the
right panel of Fig. 2.1), the corresponding fringe patter is also shifted by the same an-
gleφ, called phase. Visibility and phase are usually expressed together as the complex
visibility V=|V|e iφ , which completely defines a pattern of interference fringes.
The importance of measuring complex visibilities derives from the van Cittert–
Zernike theorem (see Thompson et al., 2001 for a complete discussion) which relates
the fringe contrast to the Fourier components of the impinging brightness distribution.
In particular, the theorem states that for spatially incoherent sources in the far field,
the value of the visibility V, measured with a baseline B, is equal to the normalized

2.2 Basic principle of astronomical interferometry 27
Figure 2.2: Schematic representation of an interferometer composed by the telescopes or anten-
nas. If the source brightness distribution has a small angular extension, it can be approximated
with a function I(l,m) in a plane tangential to the celestial sphere. The corresponding spatial
frequency are measured in the uv plane as described in the text.
Fourier transform of the sky brightness distribution corresponding to the fringe spatial
frequency u=B/λ rad −1 ; the phase of the fringe pattern is also equal to the Fourier
phase of the same spatial frequency component.
In the real case of a bi-dimensional astronomical source whose radiation comes from
a small portion of the sky, it is possible to define a system of Cartesian coordinates (l, m)
on the tangent plane to the celestial sphere at the point where the object is located, and
express the surface brightness as a function I(l, m). In this case, the van Cittert–Zernike

28 Introduction to interferometry
theorem can be mathematically expressed in the form
−i2π(ul+vm) P(l, m)I(l, m)e dl dm
V(u, v)=
I(l, m) dl dm
(2.5)
where u=Bu/λ and v=Bv/λ are the spatial frequencies corresponding to the projec-
tions of the baseline B along the orthogonal directions (u, v) (see Fig. 2.2), and P(l, m)
is a function that describes the sensitivity of each element of the interferometer to the
direction of arrival of the radiation, called primary beam.
The previous equation can be inverted in the form
P(l, m)I(l, m)∝ V(u, v)e +i2π(ul+vm) du dv (2.6)
which stat that, known the primary beam P(l, m), it is possible to recover the sky bright-
ness distribution I(l, m) of whatever source if a suitable number of measurements of
V(u, v) are obtained; the meaning of “suitable number” will be discussed in the follow-
ing.
Using the synthesis aperture technique, we can measure the visibility only in a finite
number of places in the uv-plane. We can thus define a sampling function S (u, v), which
is zero where no data have been taken. In this case, the image that we can recover is the
the so-called dirty image
I d
(l, m)∝ S (u, v)×V(u, v)e +i2π(ul+vm) du dv∝ B d (l, m)∗(P(l, m)I(l, m)), (2.7)
where B d (l, m) is the Fourier transform of the sampling function, commonly known as
dirty beam. The dirty beam is the response on the interferometer to a point source and
it corresponds to the Airy pattern for a single-dish circular telescope. The dirty image,
obtained from the Fourier transform of the observed visibilities, is thus the convolu-
tion of the brightness P(l, m)I(l, m) and the dirty beam B d (l, m). The sampling function
is a linear filter that cuts all the not sampled spatial frequencies and the correspond-
ing angular scales. Its choice should be therefore done with attention on the basis of
the presumed size and morphology of the source. In general, for a good image recon-
struction, the sampling function should be as uniform as possible. On the contrary, if

2.3 Optical versus millimeter interferometry 29
the sampling of the uv-plane is sparse and non uniform then the dirty beam is charac-
terized by strong and numerous sidelobes; in these cases the brightness profile of the
source in dirty image results distort and hard to identify. Nevertheless, the dirty beam
is determined by the known sampling of the uv-plane and the dirty image can be par-
tially corrected through complex processes of deconvolution which assume “a priori”
information about the source (see Cornwell et al., 1999, for a detailed discussion on the
existing deconvolution techniques).
Together with the sampling of the uv-plane, the sensitivity of an interferometer is
the crucial quantity that define the range of source types that can be observed. For a
single dish telescope, sensitivity and resolution are strongly coupled, depending both
on the telescope diameter D, respectively through D 2 andλ/D. This is not true for
an interferometer where the spatial resolution depends on the maximum baseline Bmax
while the sensitivity depends mainly on the total collecting area, namely the number of
telescopes×the telescope’s diameters. The major limits to the sensitivity are due to the
atmosphere, to the optical transmission and to the detector/background noise. All these
problems depend strongly on the observational wavelength; they affects in a different
way optical and radio observations and will be discussed in the following section.
2.3 Optical versus millimeter interferometry
Optical and millimeter interferometry are essentially the same technique used in two
different wavelength domains. Although they share the same fundamental properties
their technical implementation exhibits some significant differences.
Both optical and millimeter interferometers measure the complex visibility of the in-
terference fringe pattern produced by separated apertures, called telescopes, siderostats
or antennas. The principle of these two techniques are based on the van Citter-Zerniche
theorem, i.e. the visibility is directly related to the Fourier transform of the spatial
distribution of the brightness of the observed object (see Eq. 2.5). By using separated
apertures, both techniques achieve high angular resolution observations, with a resolu-

30 Introduction to interferometry
tion tens to hundreds of time larger than single aperture in the same wavelength domain.
They both use arrays of telescopes and, as described in the previous section, have to find
the most suitable telescope configuration to reach their objective. Moreover, both optical
and radio interferometry require to calibrate the measured complex visibilities.
Optical and millimeter interferometry differ, first of all, for the angular resolution
that can be achieved, defined by the fringe spacingλ/2B. Fixed the maximum baseline
B, the wavelength dependence implies that an interferometer observing in the near-
infrared may achieve an angular resolution three orders of magnitude higher than at
millimeter wavelengths. In practise, the existing millimeter interferometers are charac-
terized by antennas configurations more extended that the optical array and the differ-
ence is generally lower. For examples, while the VLTI has a maximum baseline of∼200
m, corresponding to a resolution of few milli-arcsec in the near-infrared, the Plateau the
Bure Interferometer can operate with a baseline of∼ 700 m and can achieve a resolution
of∼0.3 ′′ at 1 mm; for the Very Large Array the resolution is∼0.2 ′′ at 3 cm (B=36000
m). In future, with the Atacama Large Millimeter Array it will be possible to achieve a
resolution of∼0.01 ′′ at 1 mm with a baseline of 15000 m.
The atmosphere is the source of a second important difference. Due to the diver-
sity of the temperature between the ground and the upper layers of the atmosphere,
convection occurs and creates turbulent eddies, characterized by different refractive in-
dices. When looking to objects through the atmosphere, the rays of light are therefore
randomly deviated. The atmospheric turbulence is characterized by the spatial Fried’s
parameter r0, which corresponds to the spatial scale of the atmospheric turbulence, and
the coherence time t0, which define the temporal variation of the turbulence. At opti-
cal wavelengths, typical values of r0 and t0 are in the range 0.1–1 m and 10-100 ms
respectively; moreover, the physical analysis of the turbulence shows that both r0 and
t0 depend on the wavelength asλ 6/5 . At optical wavelengths, the dominant effect of
atmospheric turbulence is the corrugation of the wavefront both at the level of each in-
dividual aperture, seeing, and at the level of the interferometer, atmospheric piston. As
in the case of single optical telescopes, interferometers use adaptive optics systems to

2.3 Optical versus millimeter interferometry 31
correct for the seeing and stabilize the light on each aperture. Nevertheless, the optical
path difference between two beams fluctuates in time and the fringes move back and
forth on the plane of the detector, blending after the coherence time t0∼ 10−100 ms.
This effect, called piston, can be corrected using a fringe tracker that measures the op-
tical path difference at a frequency higher than t0 and sends the appropriate correction
to a delay lines system the correct the incoming beams. The fringe tracker allows thus
to stabilize the phase of the signal for integration times longer than t0 and to increase
the signal-to-noise ratio on the detector; the absence of a fringe tracker on most of the
available optical interferometers is one of the major causes of their low sensitivity.
In the millimeter domain the situation is slightly easier. The spatial Fried’s param-
eter, r0, is larger than the antenna sizes. Observations of each antenna are therefore
diffraction limited and no seeing correction is required. However, millimeter observa-
tions may be strongly affected by variable ionospheric and tropospheric conditions, e.g
the time dependent water vapour column density which can introduce variations in the
pathlength, and hence variations in the interferometric phase. This causes the loss of
visibility amplitude over the integration time and reduces the spatial resolution. Since
at millimeter wavelengths the atmosphere coherence time t0 is of several minutes, it is
possible to calibrate the visibility phase using the phase referencing technique which
consists on short observations of a calibrator source with known phase within a few
degrees of the target source every 5-10 minutes.
The application of the phase referencing technique in the optical is very challenging,
with the result that the absolute phase calibration can be hardly achieved even with the
fringe tracker. Indirect and partial information on the absolute phase can however be
obtained using the closure phase technique. The basic idea of this method, introduced
by Jennison (1958), is founded on the principle that the sum of the observed phases
around a close loop of at least three baselines, is equal to the sum of the intrinsic phases
due only to the source.
To understand this statement, we show in Fig. 2.3 a configuration of three telescopes

32 Introduction to interferometry
Figure 2.3: Modified from Monnier (2003). The figure explain the principle behind the closure
phase analysis. As described in the text, phase errors introduced at any telescopes causes equal
but opposite phase shifts, cancelling out if the closure phase is calculated on a close loop of three
or more telescopes.
where, for each pair (i, j), the observed phase can be expressed by the relation
φi j=φ0,i j+θi−θ j, (2.8)
whereφ0,i j is the intrinsic source phase andθi,θ j are the phase shifts introduced by
atmospheric variations and/or instrumental errors at each antenna. Defining the closure
phase as the sum of the observed phases,β123=φ12+φ23+φ31, the Eq. 2.8 implies that
the termsθi cancel each other out with the result that the closure phase is equal to the
sum of the intrinsic phases:
β123=φ12+φ23+φ31=φ0,12+φ0,23+φ0,31. (2.9)
Closure phase measurements can be used to obtain information on the single Fourier
phasesφ0,i j, which are required to reconstruct the source image, with complex methods

2.3 Optical versus millimeter interferometry 33
whose description is out of the aim of this thesis (see Monnier, 2003). Nevertheless,
closure phase has an important property that gives information on the morphology of
the source: if the source has a centro-symmetric brightness distribution, the closure
phases is 0 ◦ for every choice of the telescopes configuration. It is easy to prove this by
imaging the centro-symmetric brightness exactly at the center of the image: in this case
every single phaseφ0,i j is indeed zero. Fixed the three telescopes array configuration,
the closure phase increases with the asymmetry of the surface brightness of the source;
a practical application of this method to the study of the structure of the inner regions of
circumstellar disks is shown in Chapters 5 and 9.
A third difference between radio and optical interferometry comes from the type
of detection and beam combination. In the radio domain, the signal detection occurs
at the antenna level thanks to the heterodyne technique: the signal is coupled with a
reference signal of high coherence and therefore one records the amplitude and the phase
of the coming electric field. The signals from each antenna are then digitized and the
combination takes place in the correlator. An electronic phase delay is applied to take
into account the difference in path length between the two arms. In the optical domain,
the heterodyne technique can not be applied and the interferometry is obtained by direct
detection of interference fringes. The light beams are propagated to a central lab where
the optical paths are equalized using delay lines, and are combined to form interference
fringes which are recorded. Sensitivity of optical interferometers is strongly limited by
the low throughput due to the large number of reflections between the telescopes and the
final detector. Taking into account also the light loss in filters, dichroics and along the
beam transport, the visible-light transmission results to be generally between 1% and
10%.
The direct consequence of all these differences is that while images are routinely
produced by radio interferometers, this is not the case in the optical domain. Although
optical interferometers have made enormous progress in the last few years, in most of
cases we are still limited to the interpretation of visibility measurements.

34 Introduction to interferometry
2.4 Introduction to modelling techniques
Radio and optical interferometric observations can by analysed at two different levels.
Where the quality of optical observations do not allow the image reconstruction, the data
analysis has to be performed in the Fourier space, comparing the observed visibilities
with synthetic values obtained starting from theoretical models. On the contrary, radio
observations generally produce images and the analysis can thus be directly performed.
Nevertheless, even if this latter method appears of easier application, it suffers strong
limitation when the signal-to-noise ratio of the observations is low and/or the sampling
of the uv-plane is sparse and not uniform. In this case, the dirty image is characterized
by strong sidelobes that have to be removed with the deconvolution. However, decon-
volution is a non-linear process which increases the noise level, specially in the case of
weak extended sources. This is often the case of observations of circumstellar disks,
where the surface brightness decreases with the angular distance from the central star
and the outermost disk regions are lost in the observational noise. The consequence is
that the comparison between disk models and cleaned images is often inaccurate.
As for the optical interferometry, the best way to analyze the data is therefore to
compare the observed visibilities with those predicted by our models. Weighting each
visibility data point by its known thermal noise, it is possible to define how well the
model fits the data adopting the usualχ 2 notation:
χ 2 =Σi[(Remod,i− Reobs,i) 2 + (Immod,i−Imobs,i) 2 ]·Wi, (2.10)
where Re∗,i and Im∗,i are the real and imaginary parts of the i-th observed (obs) or model-
predicted (mod) visibility. The weight Wi, provided in the observed uv-table (the table
containing the baselines Bu and Bv, the corresponding visibility and the weight), is de-
rived from the system temperature Tsys, the spectral resolution∆ν, the integration time
τ, the effective collecting area of one antenna Ae f f and the loss of efficiency introduced
by the correlatorη:
Wi= 1
withσi=
σ2 √ 2kTsys
Ae f fη √ . (2.11)
τ∆ν

2.4 Introduction to modelling techniques 35
Models of possible circumstellar disk images can by schematically divided in two
category: centro-symmetric models, for which the visibility can be expressed analyti-
cally introducing the Bessel functions, and more complex asymmetric models, for which
a numerical solution of the image Fourier transform is required.
To the first class belong all the geometrical models such as point sources, uniform
or Gaussian disks and rings, either circular or elliptical, and all their possible combina-
tions, i.e, multiple systems composed by the sum of two or more of theme. When the
model has a circular symmetry it is easier to switch to polar coordinates and express the
brightness of the image I(l, m) as a function I(ρ) of the radial coordinateρ= √ l 2 + m 2 .
In this case, the corresponding visibility depends also only on the radial coordinate
r= √ u 2 + v 2 , through the relation
V(r)=2π
∞
0
I(ρ)J0(2πρr)ρ dρ, (2.12)
where J0 is the zeroth-order Bessel function of the first kind:
J0(x)= 1
2π
2π
0
exp (−ix cosθ) dθ. (2.13)
For a circular ring of radiusρ and infinitesimal thickness, the brightness can be ex-
pressed through theδfunction as
I(ρ)= 1
δ(ρ−ρ0), (2.14)
2πρ0
and the corresponding normalized visibility is then:
V(r)= J0(2πρ0r). (2.15)
Any circularly symmetric function can be described as the integral of infinitesimally
thickness rings with varying radius and intensity and, since the Fourier transform is
addictive, its visibility is the integral of the corresponding visibility curves. In this
way, the visibility of a uniform disk of angular diameterθand normalized integrated
brightness is given by
V(r)=2 J1(πθr)
, (2.16)
πθr

36 Introduction to interferometry
where J1 is the first-order Bessel function of the first kind; while for a normalized Gaus-
sian disk (remember that the Fourier transform of a Gaussian function is also a Gaussian
function)
V(r)=exp
− (πθr)2
, (2.17)
4 ln 2
whereθis the full width half maximum of the disk. Finally, the visibility of a uniform
ring characterized by angular inner diametersθ and width f is
V(r)=
2
πθr(2 f+f 2 ) {(1+ f )J1[(1+ f )πθr]− J1(πθr)}. (2.18)
Circular symmetric models can be considered as a first approximation of face-on
disk images and can be used to estimate the angular diameter of the emission. Ring
models, in particular, have been widely applied in near-infrared interferometry to cal-
culate the disk inner radius (Monnier and Millan-Gabet, 2002; Eisner et al., 2004; see
the discussion in Chapter 3). In most cases, the disk inclination is however not known
and the face-on assumption is not justified. Inclining a circular ring the resulting im-
age is an ellipse. Through a transformation of coordinates, and taking advantage of the
similarity properties of the Fourier transform, it is possible to re-conduce the elliptical
case to the circular one and express the visibility trough the Bessel functions. For an
exhaustive description of analytical calculation of visibility we remand to Perrin and
Malbet (2003).
If the model image has a complex morphology, a numerical bi-dimensional compu-
tation of the visibility is required. Thanks to the fast improvement of the technology,
Fourier transform of images composed by millions of pixels can by today performed
on a scale time of seconds or minutes, on a normal personal computer. This time is
incredible fast if compared to the hours, or even days, required not more than 20 years
ago. All the popular commercial software for numerical analysis, i.e., IDL and Mat-
Lab, contain dedicated routines for multi dimensional Fourier transform calculations.
Another large number of codes is freely available on the web. In particular, all the data
analyses discussed in this thesis have been performed using a free C subroutine library
for computing discrete Fourier transform called FFTW (Fast Fourier Transform in the

2.4 Introduction to modelling techniques 37
West); documentation, features and benchmarks can be found on the web site 1 . The
input of this numerical recipe is a matrix of arbitrary size, containing the brightness
values I(i, j) for each pixel of the model image, defined by discrete coordinates i and j
(corresponding to the continuum coordinate l and m of Fig. 2.2). The output consists in
a table containing the real and imaginary part of the visibility at the corresponding spa-
tial frequencies. The comparison with the observed visibility can be finally preformed
interpolating the theoretical visibilities, calculated on a regular grid, in the points of
the uv-plane sampled during the observation and calculating theχ 2 as discussed at the
beginning of this Section.
If the observations are characterized by an uniform sampling of the uv-plane, with
baselines in the interval Bu,v,min-Bu,v,max, we can expect to observe source features on an-
gular scale sizes ranging from∼λ/Bu,v,max up to∼λ/Bu,v,min. To perform the comparison
between models and observations, we must therefore create a model image able to re-
produces such features. In particular, the grid spacing, i.e. the pixel sizes∆i and∆ j, and
the number of pixels on each axis, Ni and N j, must allow the representation of all these
scales. In terms of range of uv points sampled, the requirements are∆i≤λ/2Bu,max,
∆ j≤λ/2Bv,max and Ni∆i≥λ/Bu,min, N j∆ j≥λ/Bv,min.
1 http://www.fftw.org

Part II
Dust in the inner disk

CHAPTER 3
The shape of the inner rim in
proto-planetary disks
This Chapter has been published in Astronomy & Astrophysics:
A. Isella 1,2 and A. Natta 2 , 2005 A&A, 438, 899
“The shape of the inner rim in proto-planetary disks”
1 Dipartimento di Fisica, Univeristá di Milano, via Celoria 16, 20133 Milano, Italy
2 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy
Abstract: This Chapter discusses the properties of the puffed-up inner rim that forms in
circumstellar disks when dust evaporates. We argue that the rim shape is controlled by a funda-
mental property of circumstellar disks, namely their very large vertical density gradient, through
the dependence of grain evaporation temperature on gas density. As a result, the bright side of
the rim is curved, rather than vertical, as expected when a constant evaporation temperature
is assumed. We have computed a number of rim models that take into account this effect in a
self-consistent way. The results show that the curved rim (as the vertical rim) emits most of its

42 The shape of the inner rim in proto-planetary disks
radiation in the near and mid-IR, and provides a simple explanation for the observed values
of the near-IR excess (the “3µm bump” of Herbig Ae stars). Contrary to the vertical rim, for
curved rims the near-IR excess does not depend much on the inclination, being maximum for
face-on objects.We then computed synthetic images of the curved rim seen under different incli-
nations; face-on rims are seen as bright, centrally symmetric rings on the sky; increasing the
inclination, the rim takes an elliptical shape, with one side brighter than the other.
3.1 Introduction
The structure of the inner regions of circumstellar disks associated with pre-main se-
quence stars is the subject of intense research. Interferometers working in the near
infrared are providing the first direct information on the morphology of disks on scales
of fractions of AU. They show that in the majority of cases the observed visibility curves
are not well reproduced by flared disk models; rather, they are consistent with the emis-
sion of a ring of uniform brightness, of radius similar to the dust evaporation distance
from the star (Millan-Gabet et al., 2001; Tuthill et al., 2001).
The interferometric results provide strong support for the idea that the inner disk
structure deviates substantially from that of a flared disk because dust evaporation in-
troduces a strong discontinuity in the opacity, which results in a puffed-up rim at the
dust destruction radius, where dust is exposed directly to the heating stellar radiation.
The idea of a puffed-up inner rim was proposed by Natta et al. (2001) and developed
further by Dullemond et al. (2001, hereafter DDN01) for Herbig Ae stars, to account for
the shape of the near-infrared excess of these stars (the “3-µm bump”). These authors
pointed out that the rim also had the right properties to explain the early interferomet-
ric results of Millan-Gabet et al. (2001). Recent theoretical work by Muzerolle et al.
(2004) has shown that the condition required to produce a puffed-up inner rim are in-
deed likely to exist in most Herbig Ae and T Tauri stars. The concept of such an inner
rim has been widely used to interpret near-IR interferometric data for Herbig and T

3.1 Introduction 43
Tauri stars (Eisner et al., 2004; Muzerolle et al., 2003; Colavita et al., 2003; Eisner et
al., 2003; Monnier and Millan-Gabet, 2002; Millan-Gabet et al., 2001). Its effects on
the disk structure and emission at larger radii have been discussed by Dullemond and
Dominik (2004), who propose that the classification of Herbig Ae stars in two groups,
based on the shape of the far-infrared excess (Meeus et al., 2001), can be interpreted as
differences between objects where the outer disk emerges from the shadow of the inner
rim and objects where this never happens.
In spite of its success in accounting for a variety of observations, the actual structure
of the rim has not been much discussed. DDN01 adopted for their models a very crude
approximation, namely that the illuminated side of the rim is “vertical”, and that its
photospheric height is controlled by radial heat diffusion behind the rim. Such a model,
taken at face value, has the obvious disadvantage that the rim emission vanishes for
objects seen face-on, for which the projection on the line of sight of the rim surface
is null, and for objects seen edge-on, where the rim obscures its own emission. This
is clearly inconsistent with observations of the SED, which show that all the Herbig
Ae stars with disks have similar near-IR excess, regardless of their inferred inclination
(Natta et al., 2001; Dominik et al., 2003).
The vertical shape of the illuminated face of the rim is clearly not physical, as
pointed out already by DDN01. Several effects are likely to “bend” the rim: among
them, one can expect that radiation pressure on dust grains or dynamical instability,
due to self-shadowing effects, could modify the illuminated face of the rim (Dulle-
mond, 2000; DDN01). None of these suggestions, however, has been explored further.
In this Chapter, we will discuss in detail a different process, not mentioned so far,
which depends exclusively on the basic physics of dust evaporation, i.e., on the depen-
dence of the evaporation temperature on gas density. Circumstellar disks are charac-
terized by a very large variation of the density in the vertical direction, so that the dust
evaporation temperature varies by several hundred degrees in a few scale heights; mov-
ing vertically away from the disk mid plane along the rim, dust will evaporate at lower
and lower temperatures, i.e., further away from the central star. This very simple effect

44 The shape of the inner rim in proto-planetary disks
curves significantly the inner face of the rim, as we will describe in the following.
The Chapter is organized as follows. In Sec. 3.2 we describe the model we use to
compute the rim shape and its observational properties. The results are presented in
Sec. 3.3, where we discuss also how the rim depends on dust properties. A discussion
of the results, in view of the existing observations, follows in Sec. 3.4, and a summary
is given in Sec. 3.5.
3.2 Model description
Our model of the inner rim of passive-irradiated flaring disks joins the two different
analytical methods to solve the structure of a circumstellar disk proposed respectively
by Calvet et al. (1991; 1992, hereafter C92) and Chiang and Goldreich (1997, hereafter
CG97). The temperature in the rim atmosphere is determined using the analytical solu-
tion of the problem of the radiation transfer as in C92, neglecting the heating term due to
the mass accretion. The vertical structure of the rim is then computed in a way derived
from CG97 and DDN01, adding a relation between the dust vaporization temperature
and the gas density as proposed in Pollack et al. (1994). As a result we obtain a curved
model for the inner rim whose features are described in the next paragraph.
Although the expressions for the dust temperature derive from a first order solution
of the radiation transfer equation, we found (see Appendix) good agreement with the
correct numerical result in most cases. To zero order, we can compute the rim structure
avoiding proper radiation transfer calculations.
In the limit where the incident angleαof the stellar radiation onto the disk surface
isα≪1, the equations of the temperature forτd = 0 andτd ≫ 1 (whereτd is the
optical depth for the emitted radiation) are formally equal to the optically thin Ts and
mid plane Ti temperatures introduced by CG97 (two-layer approximation). Therefore,
while the two layer approximation is useful to study the structure and the emission
features (e.g. silicate features at 10µm) of the flaring part of the disk (as in DDN01),
it must be abandoned in modeling the inner rim, sinceα≃1. Nevertheless we can

3.2 Model description 45
adapt the CG97 treatment of the vertical structure of the disk to the inner rim using the
appropriate expressions for the dust temperature.
In order to clarify this concept and to introduce the relation between the vaporization
temperature of dust and the gas density, the basic equations are briefly summarized. We
refer to the cited works for a physical discussion of the equations.
We suppose that the disk is heated only by the stellar radiation and we callαthe
incident angle between the radiation and the disk surface. The incident beam is absorbed
exponentially as it penetrates the dusts and ifτd is the optical depth for the emitted
radiation, the dust temperature T(τd) is given by the relation (C92)
T 4 (τd)=T 4 2
R⋆
1
⋆ · µ(2+3µǫ)+ − 3ǫµ2 e
2r ǫ (−τd/µǫ)
(3.1)
whereµ=sinα andǫ is an efficiency factor that characterizes the grain opacity, defined
as the ratio of the Planck mean opacity of the grains at the local temperature T(τd) and
at the stellar temperature T⋆, respectively. In writing the previous equation we have
assumed that the scattering of the dust grain is negligible and that the Planck mean
opacity is defined as:
∞ 0
KP(T)=
Bν(T)kνdν
∞ . (3.2)
Bν(T)dν
0
Following Muzerolle et al. (2003), the continuum emission of the disk is assumed to
originate from the surface characterized by the optical depthτd= 2/3.
In the flaring part of the disk, for whichµ≪1, Eq. 3.1 can be rewritten in terms
of the two layer approximation, proposed in CG97, in which the interior of the disk
(withτd≫ 1) is heated to the temperature Ti by half of the stellar flux, while the other
half of the stellar flux is re-emitted by the superficial layer heated at the optically thin
temperature Ts:
T 4 (τd≫ 1)≃T 4 i = µ
2
T 4 (τd= 0)≃T 4 s
= 1
ǫ
2
R⋆
T
r
4 ⋆
2
R⋆
T
2r
4 ⋆
(3.3)
(3.4)

46 The shape of the inner rim in proto-planetary disks
Following DDN01 we assume that the dust component of the disk is truncated on the
inside by dust evaporation, forming an inner hole of radius Rin. Inside this hole, only
gas may exist but as long as the gas is optically thin to the stellar radiation (Muzerolle
et al., 2004) we can neglect its absorbing effect. Near the inner radius Rin, since the rim
surface is nearly perpendicularly exposed to the stellar radiation (µ≃1), Eq. 3.3 and
3.4 must be replaced by the more general relations
T 4 (τd≫ 1)≡T 4 ∞ =µ(2+3µǫ)
T 4 (τd= 0)≡T 4
0 =
2µ+ 1
ǫ
derived as limiting cases from Eq. 3.1.
2
R⋆
T
2r
4 ⋆ , (3.5)
2
R⋆
T
2r
4 ⋆, (3.6)
Since the dust in the rim is by definition close to the evaporation temperature, the
value ofǫ is fixed and depends only on the grain absorbing cross section. Moreover, to
first order, since the difference between T(τd= 0) and T(τd=∞) is never very large, at
the inner radius (µ=1) the ratio between these two temperature depends only onǫand
is given by:
T0
T∞
=
1/4 2ǫ+ 1
. (3.7)
ǫ(2+3ǫ)
The rim is defined by the condition that the incoming stellar radiation is entirely
absorbed by the intervening dust and in the following we will refer to the surface where
τs= 1 to define its shape and location. Once the grain evaporation temperature Tevp is
fixed, one can see from Eq. 3.1 that the rim has a sharp surface whenǫǫcr, the dust temperature increases with the optical depth and
the transition to optically thick regimes is controlled by the geometrical dilution of the
stellar radiation; in this case the transition fromτs= 0 to very high values occurs in a

3.2 Model description 47
log(δR/R in )
0
-1
-2
-3
-4
-5
-6
0.2 0.4 0.6 0.8 1
Figure 3.1: Width of the region over which 99% of the stellar radiation is absorbed as function
ofǫ.
relatively broader region (see Fig. 3.1); however, also in these cases the rim is located
by theτs= 1 surface with an accuracy better than 10–20%.
ε
With the assumption that the disk is in hydrostatic equilibrium in the gravitational
field of the central star and that is isothermal in the vertical direction z, the gas density
distribution is expressed by the Gaussian relation
where
ρg(r, z)=ρg,0(r) exp(−z 2 /2h 2 ), (3.8)
h
r =
T∞
Tc
r
R⋆
1/2
(3.9)
defines the relation between the pressure scale h and the interior temperature T∞ at a
distance r from the central star. The temperature Tc is a measure of the gravitational
field of the central star, expressed by
Tc= GM⋆µg
kR⋆
whereµg is the mean molecular weight of the gas.
(3.10)

48 The shape of the inner rim in proto-planetary disks
Note that the interior temperature of the disk T∞ depends on the incident angle
α between the stellar radiation and the disk surface, through Eq. 3.5. The quantity
µ=sinα accounts for the projection of a disk annulus on a plane perpendicular to the
incident radiation and is given by the relation
r
dz(r)
µ=sin arctan + arctan −
z(r)
dr
π
, (3.11)
2
where z(r) is the axisymmetric equation of the surface of the disk that we want to deter-
mine. For the standard flaring model, the incident angleα is given by the relation
sinα≡µ≃ 0.4R⋆
r
+ r d(H/r)
dr
, (3.12)
where H is the photospheric height of the disk, defined as the height to which the optical
depth of the disk to the stellar radiation isτs= 1, on a radially directed ray. For the inner
rim we can retain this definition for H and place z(r)=H(r) in Eq. 3.11, where H is
thus given by the relation
KP(T⋆)
µ
∞
H(r)
ρd(z, r)dz=1. (3.13)
As shown in DDN01, the ratioχ=H/h, between the photospheric and pressure
height, is a dimensionless number of the order of 4-6, depending weakly on the angle
α, the densityρd and the Planck mean opacity of dust at the stellar temperature T⋆.
Finally, to obtain the dust density distribution from Eq. 3.8 we assume a constant ratio
ρd/ρg= 0.01.
We can follow CG97 and DDN01 to obtain a self-consistent solution of Eq. 3.2, 3.5,
3.6, 3.8, 3.9, 3.11, 3.13 to determine the structure of the disk, as long as dust evaporation
can be neglected. When dust evaporation is important, DDN01 have developed an ap-
proximate solution of the rim/disk structure under the assumption of constant Tevp. The
inner rim has a vertical surface toward the star located at the dust evaporation distance
Rin. The vertical photospherical height H depends on the dust density behind the rim;
since the rim is higher than the flaring disk, it casts a shadows over the disk. In this
shadowed region, assuming that there is no external heating except the stellar radiation,

3.2 Model description 49
the pressure height h depends only on the radial heating diffusion. The exact determi-
nation of the structure of the rim in this diffusive region would thus require to solve
the problem of radiation transport in two dimensions (see Dullemond, 2002). Since this
goes well beyond our aims, we adopt the approximated relation used in DDN01
d(rT 4 ∞ )
dr
≃− rT 4 ∞
. (3.14)
h
Since h ∝ (T∞R 3 ) 1/2 (from Eq. 3.9), the temperature T∞ can be eliminated and for
h≪R we obtain the relation for h behind the rim
d(h/r)
dr
1
=− . (3.15)
8R
The solution of Eq. 3.15 can therefore be used to determine the densityρd(r, z) behind
the rim through Eq. 3.8.
We now introduce the relation between the evaporation temperature of the dust and
the gas density. The physical reasons for this effect can be easily understood thinking
of evaporation as the process by which equilibrium between the gas pressure and the
surface tension of the dust grains is reached: the higher the gas density, the higher will
be the evaporation temperature. We adopt for the dust in the disk the model proposed by
Pollack et al. (1994). In this model, the grains with the higher evaporation temperature
are the silicates, which will therefore determine the location of the rim. Their evapora-
tion temperature (see Table 3 in Pollack et al.) varies with the gas density roughly as a
power law, of the kind:
Tevp= Gρ γ g (r, zevp) (3.16)
where G=2000 andγ=1.95·10 −2 . Sinceρg varies exponentially with z (see Eq. 3.8),
for zevp= h the evaporation temperature of dust is only 1% smaller than that on the mid
plane, but for zevp=H∼ 5h (for a reasonable value ofχ) the difference is about 20%
and dust evaporation takes place at a larger distance from the star, curving the rim.
To obtain a self-consistent determination of the structure of the curved rim, we im-
plemented a numerical method able to solve Eq. 3.16 together with the set of equa-
tions 3.2, 3.5, 3.6, 3.8, 3.9, 3.11, 3.13. The distance of dust evaporation in the mid plane

50 The shape of the inner rim in proto-planetary disks
Rin is computed for z=0 in the set of equations and is taken as the starting point of
the radial grid on which the rim structure is computed. As a result of the calculations,
we obtain the location in the (R, z) plane of the rim surfaces characterized by a constant
value of the optical depth. The surface forτs= 0 is thus the evaporation surface of dust
grains, forτs=1 we obtain the surface relative to the photospheric height H, defined
through the Eq. 3.13, while forτd=ǫ·τs= 2/3 we obtain the emitting surface of the
rim.
Moving on the rim surface away from the star, the incident angleαdecreases and,
when it approaches zero, the determination of the rim shape becomes very difficult. This
is mainly due to the fact that the described solution for the radiation transfer neglects
the heat diffusion between contiguous annulus of the rim. Therefore both the mid plane
temperature T∞ and the pressure height h of the rim goes unphysically to zero forα=0,
according to Eq. 3.5 and 3.9. To avoid this unrealistic behaviour, we use the approxi-
mated relation discussed previously (see Eq. 3.15) to determine the pressure height h in
the region where the diffusion is the dominant heating source. The transition distance
between the region of the rim heated by the star and those heated by the diffusion is de-
termined imposing continuity of dh/dr. The photospherical height H is then determined
as for the vertical rim.
3.3 Results
Using the model described in the previous section, we compute the structure of the inner
rim for a disk heated by a star with temperature T⋆= 10000K, mass M⋆= 2.5M⊙ and
luminosity L⋆= 47L⊙. We take a disk surface densityΣ(r)=2·10 3 (R/AU) −1.5 g cm −2 ,
and a dust-to-gas mass ratio dust/gas=0.01. In our models, this value ofΣcorresponds
to a mid plane gas density of about 10 −8 g/cm 3 at 0.5AU from the star. Note that the
results are not very sensitive to the exact value ofΣ, as long as the inner disk remains
very optically thick.
The dust properties are those of the astronomical silicates of Weingartner and Draine

3.3 Results 51
Photospherical height, H (AU)
0.12
0.08
0.04
ε = 0.92
ε = 0.58
ε = 0.25
ε = 0.14
0
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
R (AU)
ε = 0.08
Figure 3.2: Photospherical height H (τs= 1) of the inner rim for different values ofǫ, as labelled.
The curves have been computed for a star with T⋆= 10000K, M⋆= 2.5M⊙, L⋆= 47L⊙. Each
curve ends where the rim becomes optically thin to the stellar radiation.
(2001). In our models, we consider that all the grains have the same size and charac-
terize their properties with the quantityǫ, the ratio of the mean Planck opacity at the
evaporation temperature to that at the stellar temperature (see Eq. 3.2). The evaporation
temperature of silicate grains vary from 1600K, for gas density ofρg= 10 −6 g/cm 3 , to
1000K forρg= 10 −16 g/cm 3 (see Eq. 3.16). For the Weingartner silicates and a vapor-
ization temperature Tevp∼ 1400 K,ǫ∼ 0.08 for grains of radius a=0.1µm, and grows
to values of about unity for grain radii>5µm.
3.3.1 The rim shape
The shape of the rim is shown in Fig. 3.2, which plots the locus ofτs = 1 (i.e., the
photospheric height H) as function of R for different values ofǫ. The ending point of
the rim (Rout, Hout) is when the rim becomes optically thin at the stellar radiation.
Forǫ ≥ǫcr ∼ 0.58, corresponding to silicate grains bigger than 1.3µm, the inner
radius Rin, the outer radius Rout and the maximum photospheric height of the rim Hout,
all vary very little, with values Rin≃0.50 AU, Rout≃ 0.65 AU and Hout≃ 0.09 AU. For
smaller values ofǫ, the rim becomes steeper and the inner radius increases. Forǫ= 0.08,
corresponding to grains with radius 0.1µm, the rim has Rin= 1.16 AU, Rout= 1.34 AU
and Hout= 0.14 AU.

52 The shape of the inner rim in proto-planetary disks
Effective Temperature, T eff (K)
1400
1350
1300
1250
1200
1150
1100
1050
1000
ε = 0.25
ε = 0.58
ε = 0.92
950
1 1.05 1.1 1.15 1.2 1.25
R/Rin Figure 3.3: Effective temperature (i.e., T(τd= 2/3)) along the rim surface for different values
ofǫ, as labelled. Each curve is plotted as function of R/Rin, where Rin is the distance of the rim
from the star on the disk mid plane. Note that, in fact, not only R but also z changes along each
curve, as shown in Fig. 3.2. Forǫ>ǫcr≃ 0.58 Te f f is almost independent of the grain opacity.
The shape of the rim can be roughly characterized by the ratio of its maximum height
Hout over width∆R=(Rout−Rin). This quantity, which is nominally infinity in a vertical
rim, becomes in the curved rim models∼ 0.6 forǫ> ∼ ǫcr and is∼ 0.8 forǫ∼ 0.14−0.08.
In other words, as grains grow, the inner rim approaches the star but the bending of the
surface varies very little.
For any given value ofǫ, the dust temperature along the rim is not constant (as in
the vertical rim) but decreases from values of about 1400 K, typical of silicate evap-
oration temperatures at density of 10 −8 g/cm 3 , to 1200 K for density of 10 −11 g/cm 3 .
Fig. 3.3 plots the effective temperature of the rim (i.e., T(τd= 2/3)) along its surface
for different values ofǫ. Forǫ> ∼ ǫcr the effective temperature is equal to the vaporiza-
tion temperature; forǫ

3.3 Results 53
ever, is sufficiently small that most of the rim emission occurs in the near-IR, as for the
vertical rim.
3.3.2 The rim SED
The fraction of stellar luminosity intercepted by the rim, given by the ratio Hout/Rout,
varies from 10%, for small values ofǫ, to 14%, forǫ≥ǫcr. Assuming that the rim is
in thermal equilibrium, the intercepted radiation is equal to the total emitted flux FIR.
For eachǫ, we have computed the spectral energy distribution of the rim emission for
different inclination angles. As discussed in Sec. 3.2, the rim emission is computed
as that of a blackbody at the local temperature along the rimτd = 2/3 surface. We
will come back in the Appendix to this assumption, which, in any case, gives a very
good approximation to the global properties of the rim emission (see also Muzerolle
et al., 2003; C92). Most of the emission, as expected from the range of temperatures,
occurs in the near-IR. We have computed the fraction of the stellar luminosity re-emitted
by the rim in the wavelength range 1.25-7.0µm for different values ofǫ as function of
the inclination (Fig. 3.4). This near-IR excess peaks at zero inclination, where has values
between∼ 10% (smaller grains) and∼ 20% (larger grains). As the inclination increases,
the near-IR excess decreases slowly, reaching values between 5% and 8%, depending
onǫ. For inclination higher then∼80 ◦ the rim emission is self-absorbed. Note that for
the large grains, withǫ> ∼ ǫcr, the near-IR excess becomes almost independent ofǫ.
The behaviour described in Fig. 3.4 is very different from what one obtains in the
case of the vertical rim. As described in the previous section, neglecting the dependence
of the evaporation temperature of grains from gas density results in the inner face of the
rim to be vertical (as in DDN01). Withǫ=ǫcr and Tevp≃ 1400 K (equal to the evapo-
ration temperature at the inner radius of the curved rim), the inner radius Rin is the same
as for the curved rim; the photospheric height, evaluated using the approximation de-
scribed by Eq. 3.15, is also the same (Hout= 0.09 AU); the fraction of stellar luminosity
intercepted by the vertical rim is 17%. However, the value of FNIR observed for different
inclination angles is very different, with a very strong (and opposite) dependence on i

54 The shape of the inner rim in proto-planetary disks
L NIR /L *
0.2
0.16
0.12
0.08
0.04
0 0 10 20 30 40 50 60 70 80
Inclination angle (deg)
Figure 3.4: Near infrared emission of the curved rim (integrated between 1.25−7µm and
normalized to L⋆) for different inclination angles and different values ofǫ. Starting from the
bottom, the curves refer toǫ= 0.08 (long-dashed), 0.14 (short-dashed), 0.25 (dotted), 0.58=
ǫcr (short-dash-dotted), 0.92 (long-dash-dotted). For the last two values ofǫ the emission is
almost the same and the lines overlap. The continuum line plots the emission of the vertical rim,
computed forǫ=ǫcr.
(Fig. 3.4). In particular, FNIR vanishes for face-on disks, and is maximum (of the order
of 20%) for very inclined systems, just before self-absorption sets in.
There are also differences between the curved and vertical rim models concerning
the shape of the predicted 3µm bump. While the vertical rim has a constant temperature
of 1400K over all its surface, in the curved rim the temperature varies from 1400K on
the mid plane to about 1200K at the outer edge (see Fig. 3.3) and the SED is broader
than a single-temperature black body. In practice, however, this is only a minor effect.
3.3.3 Rim images
Fig. 3.5 shows synthetic images of the curved rim at 2.2µm for grains withǫ= 0.08
(radius of about 0.1µm) andǫ=ǫcr= 0.58 (radius of about 1.3µm). As described in

3.3 Results 55
Sec. 3.3.1 (see Fig. 3.2 and Fig. 3.3), the inner radius of the rim is larger for smaller
grains and the surface brightness is lower, due to the lower effective temperature of the
emitting surface.
For the comparison with the vertical model of the rim, the left panels of Fig. 3.5
show the images of the vertical rim, calculated forǫ= 0.58. The largest difference is
at low inclinations: for i=0 ◦ (face-on disks) the vertical rim vanishes, as the projected
emitting surface along the line of sight is zero, while the curved rim has a centrally
symmetric ring shape. For a distance of 144pc, the inner radius of the ring, forǫ=ǫcr,
is about 4 milliarcsecond (mas); its brightness peaks practically at Rin (Rpeak/Rin∼ 1.05)
and decreases slowly (by about a factor of two) outward, until at Rout it drops to very
low values. At this distance, the width of the rim (roughly its FWHM) is about 0.8 mas.
For higher inclination the central symmetry is lost and the projected image is an
ellipse with one edge brighter than the other. In general, however, the brightness distri-
bution of the rounded rim is much more symmetric than that of the vertical rim.
3.3.4 Grain size distribution
The results shown so far have been computed assuming that all the grains have the same
composition and size. This is in practice unrealistic, and one wonders how the results
will change if grains with different properties are present. Different grains absorb differ-
ently the stellar radiation and reach very different equilibrium temperatures. The degree
of complexity of the radiation transfer problem increases remarkably (see Wolfire and
Cassinelli, 1986, 1987) and there is no approximate solution such as Eq. 3.1. In fact, to
the best of our knowledge even numerical solutions are not available to describe accu-
rately the transition region where some grains are cooler than evaporation and survive
while others do not.
We can however estimate the effect of a grain mixture in the following way. Let us
consider for simplicity a classical MRN grain size distribution, characterized by a power
law a −q with a the radius of the dust grains, q=3.5, and a varying between a minimum
value amin= 0.01µm and a maximum value amax≫amin. From what we have discussed

56 The shape of the inner rim in proto-planetary disks
Figure 3.5: Synthetic images of the curved rim for different values ofǫ and inclination angle
(i = 0 ◦ is face-on); comparison with the vertical rim. The left and central panels show the
images of the curved rim calculated forǫ= 0.08 (silicate grains with radius a=0.1µm) and
ǫ=ǫcr= 0.58 (a=1.3µm). The right panels show the images for the vertical rim calculated
forǫ= 0.58. The surface brightness of the rim, plotted in colors, is computed for a wavelength
of 2.2µm. The stellar parameters are as in Sec. 3.3, the distance is d=144pc. The vertical rim
for i=0 ◦ has zero brightness. Note that as discussed in Sec. 3.3 forǫ>ǫcr the images of the
curved rim remain the same.
in Sec. 3.2, if amax is smaller than 1.3µm (corresponding toǫ = ǫcr for the adopted
silicate grains), all the grains have a maximum temperature atτd= 0 and the higher

3.4 Discussion 57
the value ofǫ, the smaller the evaporation distance from the star. As soon as the largest
grains can survive, the stellar radiation is rapidly absorbed and all the other grains will
also survive. In this case, the shape of the inner rim is controlled by the largest grains in
the distribution. The shapes in Fig. 3.2 and the emitted fluxes shown in Fig. 3.4 should
not change significantly as long as one interpretsǫas relative to the largest grains.
The situation is more complex if amax> 1.3µm, since the grains withǫ>ǫcr can
survive near the star only in an optically thin regime. However, our single-grain models
show that there is very little difference in the location and shape of the rim as soon as
ǫ>ǫcr. This suggests that varying amax above the “transition” value (i.e., the value of
ǫ at which the dependence of T onτchanges from decreasing to increasing) will not
change the rim properties any further, at least to zero order.
It is also likely that grains have not just different size but also different chemical
composition. In this case, the rim location and properties are determined by the dust
species with the highest evaporation temperature, as long as its contribution to the disk
opacity is sufficient to make the disk optically thick. As discussed in Sec. 3.2, in the
Pollack et al. (1994) dust model silicates have the highest evaporation temperature. This
is why we have performed all our calculations for silicate grains. However, one should
keep in mind that, in some dust models, graphite contributes most of the opacity at
short wavelengths; since graphite has a much higher evaporation temperature (Tevp∼
2000 K) than silicates, its presence would move the rim much closer to the central star.
Although the details of the rim shape may change, its curving, which is caused only by
the dependence of Tevp on the density, will not disappear.
3.4 Discussion
3.4.1 The 3µm bump
The curvature of the inner side of the rim has important consequences on the observable
near-IR excess, as shown in Fig. 3.4, which shows that the predicted near-IR excess

58 The shape of the inner rim in proto-planetary disks
Figure 3.6: Histogram of the observed values of LNIR/L⋆ for a sample of 16 Herbig Ae stars
from Natta et al. (2001) and Dominik et al. (2003). Note that the Dominik et al. values
are computed between 2 and 7µm, while those from Natta et al. between 1.25 and 7µm. The
uncertainties are in most cases rather large, due to the subtraction of the photospheric flux, which
contributes significantly to the observed ones at short wavelengths, and to variability (see Natta
et al. for some examples).
ranges between about 10 and 20%, and that there is no significant dependence on the
inclination of the disk with respect to the observer. These results compare very well
with the existing observations. Fig. 3.6 shows the observed values for a well-studied
sample of Herbig Ae stars from Natta et al. (2001) and Dominik et al. (2003); of a total
of 16 objects, only one (HD 142527) has LNIR/L⋆> 0.25, and one (HD 169142)< 0.09.
The observations of the near-IR SEDs of Herbig Ae stars have also shown that there
is no systematic variation of the near-IR excess with the inclination of the disk. The

3.4 Discussion 59
sample of Fig. 3.6 includes some rather face-on objects such as AB Aur (LNIR/L⋆∼
0.20, i∼20 ◦ − 40 ◦ ; Fukagawa et al., 2004) and HD 163296 (LNIR/L⋆∼ 0.21, i∼30 ◦ ;
Grady et al., 2000) and some UXORS variable, such as UX Ori, WW Vul and CQ Tau,
which are generally considered to be close to edge-on, and have values of LNIR/L⋆∼
0.12−0.25. This, which has been a puzzle for the vertical rim, finds a natural physical
explanation in our models, which predict that the near-IR excess depends little on the
inclination and that, in particular, does not vanish for face-on objects. Actually, our
models predict that the largest excesses should be seen in face-on objects, and it would
be interesting to explore in detail if this is indeed the case. However, for such a study to
be significant, one would need a large sample of objects with known inclinations, which
is at present not available.
Another interesting aspect of our results is the dependence of LNIR/L⋆ on grain prop-
erties. If taken at face-value, one should expect that only objects with relatively large
grains (greater than about 1µm) can have large values of LNIR/L⋆, and that low val-
ues of the near-IR excess can only occur in face-on objects with small grains. These
properties of the rim emission are potentially important for a better understanding of
grain properties in the inner disk and should be pursued further in the future, combining
observations of disks with well known inclinations with models that explore a broader
range of grain properties than we considered in this Chapter.
3.4.2 The rim radius
One side product of our models is Rin, the distance from the star to the rim on the disk
mid plane. For a face-on disk, Rin practically coincide with the position of the peak of
the near-IR brightness. For fixed values of the stellar parameters, Rin depends on the
dust properties; its value is shown in Fig. 3.7 for different values ofǫ. Larger grains
can survive closer to the star than smaller grains and, because of the inversion in the
temperature gradient, Rin does not change much with the grain size once this exceeds
the value for whichǫ=ǫcr, so that for L⋆= 47L⊙, Rin > ∼ 0.5AU. Rin scales roughly as
L 0.5
⋆ , so that one can expect Rin > ∼ 0.1 AU for L⋆∼ 2L⊙. These values, especially for

60 The shape of the inner rim in proto-planetary disks
small grain sizes, are larger than the predictions of simple, optically thin calculations,
because Eq. 3.1 correctly includes the effect of the diffuse radiation field. Note that if
backward scattering cannot be neglected, the temperature atτ=0 will be even higher,
increasing further the size of the inner hole (see Appendix).
Values of Rin have been derived for a number of objects from near-IR interfero-
metric observations (see, for example, Akeson et al., 2000 and 2002; Millan-Gabet et
al., 2001; Monnier and Millan-Gabet, 2002; Colavita et al., 2003; Eisner et al., 2003,
2004 and 2005). The results depend not only on the assumed disk model, but also on
other quantities such as the stellar SED, the disk inclination etc., so that a direct com-
parison with our values of Rin can be misleading. We note, however, that Rin ∼ 0.5
AU for intermediate-mass objects is roughly consistent with the observations, suggest-
ing that in many objects grains have grown to sizes a> ∼ 1.3µm (see also Monnier and
Millan-Gabet, 2002).
This conclusion, of course, needs to be taken with great caution. There are a number
of effects that can change the model-predicted Rin, in addition to different grain sizes.
For example, on one hand smaller values of Rin can be due to some low-density gas
in the inner disk hole, able to absorb the UV continuum from the star (Monnier and
Millan-Gabet, 2002; Akeson at al., 2005). Also, the presence of graphite, which has
an evaporation temperature of about 2000 K, can decrease Rin. On the other hand, the
presence of accretion at a significant rate can increase Rin by heating grains in the inner
disk to temperatures higher than those produced by the photospheric radiation alone
(Muzerolle et al., 2004).
In practice, one needs to compare in detail the model predictions of the quantities
measured with interferometers (visibility curves, their dependence on baseline and hour
angle, phase measurements) for specific, well-known objects. We have performed such
a study with the aim of assessing the impact of curved rim models on the interpretation
of current (and future) interferometric observations. In particular, we have included a
discussion of the effect of the asymmetries of the rim projected images when not face-
on. The results are presented in the next Chapter.

3.5 Summary and conclusions 61
R in (AU)
1.3
1.1
0.9
0.7
0.5
0.3
0.2 0.4 0.6 0.8 1
Figure 3.7: Behaviour of the inner radius of the rim for different values ofǫ, evaluated for the
star and disk parameters described in Sec. 3.3. Note that forǫ> ∼ ǫcr∼ 0.58 (corresponding to
silicate grains of 1.3µm) the radius of the inner rim is almost constant around 0.5 AU.
3.4.3 LkHa101
To the best of our knowledge, the only image of the inner region of a circumstellar disk
is that obtained by Tuthill et al. (2001), for LkHa101 using Keck in the H and K bands.
The results are reminiscent of our images, showing an elliptical ring with a side much
brighter than the other. The similarity is very interesting but one should keep in mind
that LkHa101 is an early B star with a luminosity between 500 and 50000 L⊙, depending
on its distance. Our model may not apply to such an object (e.g., Monnier et al., 2005).
3.5 Summary and conclusions
In this Chapter, we discuss the properties of the inner puffed-up rim that forms in cir-
cumstellar disks when dust evaporates. The existence of a rim has been claimed starting
from the work of Natta et al. (2001), both on theoretical and observational grounds.
Here, we investigate the shape of the illuminated face of the rim. We argue that this
ε

62 The shape of the inner rim in proto-planetary disks
shape is controlled by a fundamental property of circumstellar disks, namely their very
large vertical density gradient, through the dependence of grain evaporation temperature
on density. As a result, the bright side of the rim is naturally curved, rather than vertical,
as expected when a constant evaporation temperature is assumed.
We have computed a number of rim models that take into account this effect in a
self-consistent way. A number of approximations have been necessary to perform the
calculations, and we discuss their validity. The basic result (i.e., the curved shape of the
rim illuminated face) appears to be quite robust.
For a given star, the rim properties depend mostly on the properties of the grains, and
very little on those of the disk itself, for example the exact value of the surface density.
The distance of the rim from the star is determined by the evaporation temperature (at
the density of the disk mid plane) of the dust species that has the highest evaporation
temperature, as long as its opacity is sufficient to make the disk very optically thick;
in the model of Pollack et al. (1994) of the dust in accretion disks, silicates have the
highest evaporation temperature. Therefore, we have assumed in our models dust made
of astronomical silicates, and varied their size over a large range of values. We find that
the rim properties do not depend on size as soon as a> ∼ 1.3µm; the values of the rim radii
observed with interferometers suggest that in many pre-main–sequence disk grains have
grown to sizes of 1–fewµm at least.
The curved rim (as the vertical rim) emits most of its radiation in the near and mid-
IR, and provides a simple explanation for the observed values of the near-IR excess
(the “3µm bump” of Herbig Ae stars). Unlike the vertical rim, for curved rims the
near-IR excess does not depend much on the inclination, being maximum for face-on
objects and only somewhat smaller for highly inclined ones. This is in agreement with
the apparent similarity of the observed near-IR SED between objects seen face-on and
close to edge-on.
We have computed synthetic images of the curved rim seen under different inclina-
tions. Face-on rims are seen as bright, centrally symmetric rings on the sky; increasing
the inclination, the rim takes an elliptical shape, with one side brighter than the other.

3.6 Appendix 63
However, the brightness distribution of curved rims remains at any inclination much
more centrally symmetric than that of vertical ones. In the next Chapter we will discuss
the application of the curved rim models to the interpretation of near-IR interferometric
observations of disks.
3.6 Appendix
In this appendix we discuss in detail some of the assumptions on radiation transfer made
in building our rim models. We use as templates the results of a radiation transfer code
developed by E. Krügel, which is described in Habart et al. (2004). The code considers
a plane-parallel slab of dust, illuminated on one side by a star with properties as in Sec.
3.3.
Figure 3.8: Temperature as function of the optical depth to the stellar radiationτs for small
grains (ǫ= 0.08) and very large grains (ǫ= 1). The dashed curves are the results of the radiation
transfer code, the solid lines the temperature obtained from Eq. 3.1.

64 The shape of the inner rim in proto-planetary disks
3.6.1 Evaluation of Eq. 3.1 for single grains
We show in Fig. 3.8 the comparison of the temperature derived from Eq. 3.1 with the
results of the radiation transfer code for two cases, one withǫ= 0.08 (small grains) and
one withǫ= 1 (very large grains). In both cases scattering is neglected. The temperature
is plotted as function of the optical depth to the stellar radiationτs. One can see that Eq.
3.1 gives the correct values of T forτs=0 and forτs=∞. For intermediate values of
τs, the results are in both cases accurate within 10%. This is more than adequate for the
purposes of this Thesis.
3.6.2 SED
We compare now the SED of a black-body at temperature T(τd= 2/3) from Eq. 3.1, to
the results of the radiation transfer code (dashed curves) for the same grains of Fig. 3.8.
The approximation is very good for large grains. For small values ofǫ, the discrepancy
is larger, and, in particular, one cannot reproduce any dust feature. However, also in the
caseǫ= 0.08, the difference in LNIR/L⋆ is only of 4%.
3.6.3 Scattering
Scattering has the obvious effect of increasing the grain temperature at the slab surface
and decreasing it at large optical depth. We compare in Fig. 3.9 the temperature profile
of grains withǫ= 0.08 when scattering is properly included in the radiation transfer
(dashed line) rather than suppressed (Qsc= 0.). The curves are labelled with the val-
ues of ˜ω, defined as Qsc(1−g)/Qsc(1−g)+Qabs, where g measures the asymmetry of
the scattering phase function; ˜ω=0 if Qabs= 0 or if the scattering is forward peaked
(g=1). All quantities are averaged over the stellar radiation field and the correspond-
ing SEDs show that the value of LNIR/L⋆ decreases by about 25% when scattering is
included.

3.6 Appendix 65
Figure 3.9: Left panel: comparison of the SED computed with the radiation transfer code (dotted
lines) and as a black-body at T(τd= 2/3) from Eq. 3.1 (solid line) forǫ= 0.08 andǫ= 1.0, as
labelled. Right panel: Temperature profile for grains withǫ= 0.08, when scattering is included
Acknowledgements
( ˜ω=0.7; dashed line) or suppressed ( ˜ω=0.0; solid line).
We would like to thank Endrik Krügel for having allowed us to make use of his radiation
transfer code, Leonardo Testi, Kees Dullemond, Carsten Dominik, Giuseppe Lodato
and Giuseppe Bertin for useful discussions. The authors acknowledge partial support
by MIUR COFIN grant 2003/027003-001.

CHAPTER 4
Large dust grains in the inner region of
circumstellar disks
This Chapter has been published in Astronomy & Astrophysics:
A. Isella 1,2 , L. Testi 2 and A. Natta 2 , 2006 A&A, 451, 951
“Large dust grain in the inner region of circumstellar disks”
1 Dipartimento di Fisica, Univeristá di Milano, via Celoria 16, 20133 Milano, Italy
2 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy
Abstract: Simple geometrical ring models account well for near-infrared interferometric
observations of dusty disks surrounding pre-main sequence stars of intermediate mass. Such
models demonstrate that the dust distribution in these disks has an inner hole and puffed-up
inner edge consistent with theoretical expectations. In this Chapter, we reanalyze the available
interferometric observations of six intermediate mass pre-main sequence stars (CQ Tau, VV Ser,
MWC 480, MWC 758, V1295 Aql and AB Aur) in the framework of a more detailed physical
model of the inner region of the dusty disk. Our aim is to verify whether the model will allow us
to constrain the disk and dust properties. Observed visibilities from the literature are compared

68 Large dust grains in the inner region of circumstellar disks
with theoretical visibilities from our model. With the assumption that silicates are the most
refractory dust species, our model computes self-consistently the shape and emission of the
inner edge of the dusty disk (and hence its visibilities for given interferometer configurations).
The only free parameters in our model are the inner disk orientation and the size of the dust
grains. In all objects with the exception of AB Aur, our self-consistent models reproduce both
the interferometric results and the near-infrared spectral energy distribution. In four cases,
grains larger than∼1.2µm, and possibly much larger are either required by or consistent with
the observations. The inclination of the inner disk is found to be always larger than∼30 ◦ , and
in at least two objects much larger.
4.1 Introduction
Understanding the properties and evolution of the dust grains contained in proto-planetary
disks around pre-main sequence stars is important because they are the seeds from which
planets may form. We have now strong evidence that grains in disks are very different
from the grains in the diffuse interstellar medium and in the molecular clouds from
which disks form, as reviewed, e.g., by Natta et al. (2006). In many objects, observa-
tions with millimeter interferometers have provided strong evidence that the grains in
the outer and cooler regions of the disk (further than 50 AU from the star) have been
hugely processed, and have grown from sub-micron sizes to millimeter and centimeter
ones. Closer to the star, however, in the regions were planets are more likely to form,
observational evidence has been confined to grains close to the disk surface. For these,
which however account for a tiny fraction of the total dust mass, emission in the silicate
features has shown a correlation between the shape of the feature and its strength that is
interpreted as due to growth of the grains from size a∼0.1µm to a∼1µm (van Boekel
et al., 2003 and 2004; Meeus et al., 2003). In this inner disk, the properties of the grains
in the disk mid plane are still unknown.
In the last few years, due to the new long baseline near-infrared interferometers,

4.2 Target stars and observations 69
many important steps forward in the study of the internal regions of circumstellar disks
have occurred. The available near-infrared interferometric observations of T Tauri
(TTS) and Herbig Ae (HAe) stars (Eisner et al., 2003 and 2004; Millan-Gabet et al., 2001;
Tuthill et al., 2001; Monnier et al., 2005) confirm the idea that the inner disk properties
are controlled by the dust evaporation process which produce a “puffed-up” inner rim at
the dust destruction radius (Natta et al., 2001; Dullemond, Dominick and Natta, 2001,
hereafter DDN01). In these models, the location and shape of the rim depends on the
properties of grains located not on the disk surface but on its mid plane.
As discussed in the previous Chapter, we have recently proposed models of the
“puffed-up”inner rim which include a self-consistent description of the grain evapora-
tion and its dependence on the gas density (Isella and Natta, 2005; hereafter IN05). IN05
have explored a large range of grain properties, and discussed how the location of the
rim depends on grain properties. In this Chapter, we will use the IN05 models to analyze
the existing interferometric data of the best observed HAe stars to explore, in practice,
the constraints on grain properties provided by this technique and their uncertainties.
As a byproduct of the modeling process, one obtains also the orientation of the
inner disk (i.e. its inclination with respect to the line of sight and its position angle);
this can be compared with the orientation of the outer disk, obtained from millimeter
observations of the molecular gas and dust emission and/or scattered light in the optical.
The Chapter is organized as follows. In Sec. 4.2 we describe the available interfer-
ometric observations of the target stars. The IN05 model for the inner rim is briefly
summarized in Sec. 4.3 and used to fit the observations of the individual objects in
Sec. 4.4. A comparison of the results with previous analysis of the same data is pre-
sented in Sec. 4.5. Our results are discussed in Sec. 4.6. Conclusions follow in Sec. 4.7.
4.2 Target stars and observations
Our sample is composed of six HAe stars (AB Aur, CQ Tau, VV Ser, MWC 480,
MWC 758 and V1295 Aql), for which near-infrared interferometric observations ex-

70 Large dust grains in the inner region of circumstellar disks
Table 4.1: Stellar parameters. Stellar parameters are from Hillenbrand et al. (1992),
van den Ancker et al. (1998), Chiang et al. (2001), Straizys et al. (1996), Mannings et
al. (2000) and references therein.
Source Spectral Type d T L M Av
(pc) (K) (L⊙) (M⊙)
AB Aur A0pe 144 9772 47 2.4 0.5
MWC 480 A2/3ep+sh 140 8700 25 2.2 0.25
MWC 758 A5IVe 230 8128 22 2.0 0.22
CQ Tau A8 V/F2 IVea 100 8000 5 1.5 1.00
VV Ser B9/A0 Vevp 260 10200 49 3.0 3.6
V1295 Aql B9/A0 Vp+sh 290 8912 83 4.3 0.19
ist in the literature. Tab. 4.1 summarizes the physical properties of the target stars. All
the stars are classified as young stellar objects with masses ranging from 1.5 to 4.3 solar
mass and a spectral type between A0/B9 and A8/F2. CQ Tau and VV Ser belong to the
family of UXORs and are characterized by large and irregular variability.
We use visibility measurements of the target stars from the literature, obtained with
interferometric observations carried out with PTI (Palomar Testbed Interferometer) in
K band (λ0 = 2.2µm,∆λ=0.4µm) described in Eisner et al. (2004). For AB Aur
and V1295Aql, IOTA observations are also available (Millan-Gabet et al., 2001) for the
K’(λ0= 2.16µm,∆λ=0.32µm) and H (λ0= 1.65µm,∆λ=0.30µm) bands.
4.3 Model description
We use a model based on the assumption that the near-infrared emission of HAe stars
originates in the “puffed up” inner rim which forms in the circumstellar disk at the dust
evaporation radius (Natta et al., 2001; DDN01).
In IN05 we revised the concept of the “puffed up”inner rim, introducing the depen-

4.3 Model description 71
Figure 4.1: Sketch of the structure of the inner part of a proto-planetary disk. Panel (a) shows
a disk with an inner puffed up rim that casts a shadow over the outer part of the disk. Farther
than 5-10AU from the central star, the flaring disk emerges from the rim shadow. Panel (b)
presents the image of the inner rim, computed with the IN05 models for a star with an effective
temperature of 10000K and disk inclination of 30 ◦ . Panel (c) shows a vertical (edge on) section
of the inner rim. The curvature of the surface of the rim is caused by the variation of the dust
evaporation temperature with the height above the disk mid plane (see text).
dence of the dust evaporation temperature on the local gas density, following the dust
model of Pollack et al. (1994). The main result is that the surface of the rim presents a
curved shape (see Fig. 4.1), whose features are summarized in the following.

72 Large dust grains in the inner region of circumstellar disks
4.3.1 The dust evaporation and the “puffed-up” inner rim
Following the suggestion of Natta et al. (2001), we assume that the dust component
of a circumstellar accreting disk is internally truncated by the dust evaporation process,
forming an inner hole of radius Revp inside of which only gas can survive. If the radiation
absorption due to this inner gas is negligible (as is often the case; see, e.g., Muzerolle et
al., 2004), dust evaporation occurs where the equilibrium temperature Td of grains em-
bedded in the unattenuated stellar radiation field, equals their evaporation temperature
Tevp. In IN05 we used an analytical solution of the radiation transfer problem (Calvet
et al., 1991 and 1992) to calculate the grain temperature inside the disk and we showed
that evaporation occurs at a distance from the star that can be expressed as:
2
1500
Revp[AU]=0.034·
Tevp
L⋆
L⊙
2+ 1
, (4.1)
ǫ
where L⋆ is the stellar luminosity andǫ is the ratio of the Planck mean opacity at Tevp
to that at the stellar effective temperature T⋆,ǫ = κP(Tevp)/κP(T⋆). The quantityǫ
measures the cooling efficiency of the grains; it depends on the wavelength dependence
of the absorption efficiency of the grains and varies with grain composition and size.
If the dust in the proto-planetary disk is composed of different types of grains, the
location and structure of the inner rim depends on the properties of the grains with the
highest evaporation temperature. In the dust model proposed by Pollack et al. (1994),
the most refractory grains are silicates for which Tevp is given by the relation
Tevp(r, z)=2000·[ρg(r, z)] 0.02 , (4.2)
valid for the gas densityρg in the range between 10 −18 g/cm 3 and 10 −5 g/cm 3 . In the
following analysis, we will therefore assume that the inner disk dust is made of silicates,
with optical properties given by Weingartner and Draine (2001); thus,ǫ is uniquely
defined by the grain radius a, and we will use a, rather thanǫ, as a model parameter.
Assuming that the proto-planetary disk is in hydrostatic equilibrium in the gravita-
tional field of the central star and that it is isothermal in the vertical direction z, the gas

4.3 Model description 73
densityρg(r, z) has its maximum value on the mid plane and decreases with z as
ρg(r, z)=ρg,0(r) exp(−z 2 /2h(r) 2 ), (4.3)
where h is the pressure scale height of the disk. The mid plane density can be expressed
as a power-law of rρg,0(r)=ρg,0(r0)(r0/r) γ , withγof the order of 2–3 (see, e.g., Chiang
and Goldreich, 1997).
The decrease ofρg with z, combined with Eq. 4.2, implies that the silicate evapora-
tion temperature varies from, i.e.,∼1500 K on the mid plane (assuming a typical gas
density of∼ 10 −7 g/cm 3 ) to∼ 1000 K at z/h=6.4 and∼ 800 K at z/h=8. Since Tevp
decreases with z, it is immediately clear from Eq. 4.1 that the distance from the star at
which dust evaporates increases with z, describing a curved surface as shown in Fig. 4.1.
The dependence of Tevp on the gas density is an important factor when computing
the shape of the rim in the vertical direction, where the gas density varies by many
orders of magnitude while the distance from the star is practically unchanged. In the
radial direction, we expect relatively small variations ofρg,0, for any reasonable value of
the disk mass, so that the distance of the rim from the star, measured in the mid plane,
is practically independent of the gas density.
The emission of the rim is computed assuming that it originates from the surface
characterized by an effective temperature Te f f = T(τd = 2/3), whereτd is the opti-
cal depth for the emitted radiation. The Te f f surface, therefore, defines the observed
location and shape of the rim. In IN05, we discussed how the Tevp and the Te f f sur-
faces behave for small and large silicate grains, and showed that the Te f f surface moves
closer to the star for increasing grain size until a critical value, which for silicates is
about 1.2µm. Larger grains produce rims with Te f f surfaces practically independent of
a. Therefore, for a fixed stellar luminosity, silicates with a∼1.2µm give the minimum
value of the distance of the rim from the star. Conversely, if the measured rim distance
is equal to this minimum value, one can derive from it only a lower limit (∼ 1.2µm) to
the grain size.
The rim emission peaks at near-infrared wavelengths. Atλ< ∼ 5−7µm, one can assume
that the observed flux is the sum of the stellar+rim emission, with only negligible

74 Large dust grains in the inner region of circumstellar disks
contribution from the outer disk (see, e.g., DDN01). We model the stellar photospheric
flux using standard Kurucz model atmospheres.
4.3.2 Visibility model
Due to the limited coverage of the uv-plane of the existing near-infrared interferometers,
it is not possible at present to recover full images from the available data, and one has
to resort to the analysis of the visibilities on given interferometric baselines.
Starting from the synthetic images of the inner rim (see Fig. 3.5 at pg. 56), we
compute the predicted visibility values using a Fast Fourier Transform recipe. For face-
on inclination, due to the circular symmetry of the rim image, the visibility depends only
on the length of the baseline B. For inclination greater than zero, the image of the rim
has an “elliptical” shape: the minor axis decreases with increasing inclination and the
upper half of the rim becomes brighter than the lower part. For baselines oriented along
the direction of the minor axis of the rim image, the visibility decreases more slowly
than for those oriented along the major axis. For all other orientations of the baseline,
the visibility will have values intermediate between these two (see Fig. 4.3). Moreover,
due to the Earth rotation during the observation, the baseline corresponding to a fixed
telescope pair moves in the uv plane describing an ellipse. Along this ellipse, each point
is related to the hour angle HA of the target object in the sky. In the next section we use
the V 2 -B plot and V 2 -HA plot, to show how the models fit the observations.
The visibility model takes into account the emission of the central star, modeled as
a uniform disk of radius R⋆. If F⋆ and Fd are respectively the stellar and the inner rim
flux at the wavelength of the observation, the total visibility is given by the relation:
V 2 2 F⋆V⋆+ FdVd
= , (4.4)
F⋆+ Fd
where V⋆ and Vd are the visibility values of the star and of the disk. Note that for the
PTI configuration (baselines between 84m and 100m) V⋆ is in all cases very closed to 1.

4.4 Comparison with the observations 75
4.4 Comparison with the observations
4.4.1 Model parameters
Once the stellar and dust properties are fixed, the model-predicted visibilities depend on
the dust grain radius a, which completely defines the rim structure, and two parameters
(observational parameters in the following) that describe the orientation of the disk,
namely the inclinationιand position angle PA.
For each star, we firstly compute the predicted rim structure varying the size of the
grains from very small to very large values. As discussed in Sec. 4.3, we assume that sil-
icates are the most refractory component; we take the optical properties of astronomical
silicates defined by Weingartner and Draine (2001). Other disk parameters (i.e., mass
and density radial profile) can be neglected in this analysis. We fixρg,0(Rrim)∼10 −7
g/cm 3 which gives a total disk mass of about 0.1M⊙, for a nominal value ofγ=2.5 and
an outer disk radius of the disk of 200 AU.
Once the structure of the “puffed up” inner rim is calculated, the predicted visibility
depends on the orientation of the disk in the sky, defined by the inclinationιof the mid
plane of the disk with respect to the line of sight and its position angle PA, measured
from north to east and relative to the major axis of the projected image of the disk on
the sky. The inclination is defined so thatι=0 ◦ identifies a face-on disk whileι=90 ◦
corresponds to an edge-on disk. For inclinations higher than 80 ◦ the rim emission is
likely absorbed by the outer regions of the disk and the IN05 model can not be applied.
In practice, we compute visibility models for each object varying a,ιand PA in-
dependently. We then select the best models calculating the reducedχ 2 between the
visibility data and the theoretical values calculated at the same points in the uv plane.
The observed near-infrared fluxes, and the IOTA data when available, are then “a poste-
riori” used to check the quality of the fit and, when possible, to reduce the degeneracy
due to the small number of visibility points. For some stars, the existing data do not
constrain the parameters, but still define a range, outside of which the fit to the data is
very poor.

76 Large dust grains in the inner region of circumstellar disks
Tab. 4.2 shows in column 2 the best values of the astronomical silicate radius a and,
in column 3, the corresponding values of the radius of the inner rim Rrim. The two
observational parameters i and PA are given in columns 4 and 5, respectively. Note that
the free parameters are in boldface; Rrim is a derived quantity.
4.4.2 MWC 758
PTI visibilities are fitted by a family of models, with parameters varying between the
two extreme cases shown in Fig. 4.4. In one case, the disk has small grains of radius
a=0.17µm,ι=48 ◦ and PA=134 ◦ ; in the other, big grains with a≥1.2µm,ι=40 ◦
and PA=145 ◦ . Models with a values within this range will fit the observed visibilities
equally well, provided that we varyιand PA in an appropriately way.
However, if we consider also the constraints set by the SED at near-infrared wave-
lengths, we find that only models with big grains fit it reasonably well (see the right
panel of Fig. 4.4). The best-fitting model (χ 2 r
= 2.0) has then a≥1.2µm, inner rim
radius is Rrim= 0.32AU, rim effective temperature (at z=0) is 1460K. The near-infrared
flux, LNIR, integrated between 2µm and 7µm is 25% of the total stellar luminosity, sim-
ilar to the observed value. Once we fix a, the formal uncertainties onι, estimated from
the surface where the reducedχ 2 equalsχ 2 min + 1, are quite small,±3◦ . More realistic
uncertainties are of the order of 10 ◦ for bothιand PA.
Note that in MWC 758 the PTI visibilities define quite well the orientation of the
disk, even when the SED is not used to constrain the grain size. In particular, the
inclination cannot be lower than about 30 ◦ .
4.4.3 VV Ser
The results for the star VV Ser are shown in Fig. 4.5. As for MWC 758, the inter-
ferometric observations allow different sets of model parameters. Namely, we obtain
similar values of the reducedχ 2 (∼ 1.2) for all grain sizes a> ∼ 0.4µm. Over this range
of a, inclination and position angle vary in the intervals 45 ◦ − 80 ◦ and 60 ◦ − 120 ◦ , re-

4.4 Comparison with the observations 77
spectively, with lower inclinations for larger grains. The correlation betweenιand PA
is very strong, and the uncertainties in these two parameters remain very large even for
fixed a.
Although the fit is never very good, the VV Ser SED is better accounted for by large
grains (see the right panel of Fig. 4.5) and in Table 2 we show the best values of the
parameters for a≥1.2µm. The rim effective temperature is 1400 K, the near-infrared
excess is 21% of L⋆.
4.4.4 CQ Tau
The limited number of visibility points does not allow us to constrain all the parameters
of the disk. Fig. 4.6 shows two models, with the same level of confidence (χ 2 r
∼ 1); the
two models have similar orientations (inclinations of 52 ◦ and 46 ◦ with position angles
of 168 ◦ and 164 ◦ respectively) but very different grain sizes (a=0.3µm and a≥1.2µm)
and radii of the inner rim (0.25AU and 0.16AU, respectively). For a ≥ 1.2µm, the
effective temperature of the rim (at z=0) is Te f f = 1480K, while Te f f = 1050 for
a=0.3µm. The SEDs of the two models are compatible with the observed fluxes, with
LNIR= 13%−18% L⋆. All the models with a within this range, and similarιand PA,
have similarχ 2 values. Outside this range, models give a much poorer fit to the data.
As for MWC 758, the visibility data constrain well the orientation of the disk on the
sky. The inclination, in particular, has to be quite large, 40 ◦< ∼ ι< ∼ 55 ◦ .
4.4.5 V1295 Aql
The PTI observations of V1295 Aql are characterized by a very small number of visi-
bility points and the disk parameters are hardly constrained. Even adding the IOTA data
does not help due to the big errors that affect these observations. Note also that PTI and
IOTA observations are performed at different wavelengths, K and H respectively.
As for CQ Tau, we show the models for the two extreme sets of parameters that give
an equally good fit (Fig. 4.7) withχ 2 r
∼ 1. All the intermediate combinations of a,ι

78 Large dust grains in the inner region of circumstellar disks
and PA can explain the observations as well. In order to fit the values of visibility, an
inclination ranging between 40 ◦ and 65 ◦ is required. More face-on systems can not in
general reproduce the visibility spread in the IOTA data and the V 2 − HA behaviour of
the PTI points. The grain radius varies from a≥1.2µm (Rrim= 0.7 AU, Te f f= 1400 K)
to a=0.3µm (Rrim= 1.2 AU, Te f f= 970 K) while the position angle can not be defined
at all.
The right panel of Fig. 4.7 shows the comparison between the observed and the
predicted SED. More inclined disks can in general reproduce better the photometric
measurements around 1.5µm. The near-infrared emission of disk with an inclination of
less than 50 ◦ is peaked at about 4µm and can not reproduce the infrared excess between
1µm and 2µm; in both models, the integrated near-infrared flux is LNIR∼ 20% L⋆.
4.4.6 MWC 480
Also in this case, due to the narrow range of available baselines, the PTI visibilities of
MWC 480 are consistent with different sets of parameters at the same level of confidence
(χ 2 r
∼ 2). Fig. 4.8 shows the two extreme disk configurations characterized by quite
similar parameters for the dust grain size (a=0.2µm−0.3µm; Rrim=0.63–0.53 AU and
Te f f≃ 1250 K), but very different values of the inclination (ι=35 ◦ andι=60 ◦ ) and the
position angle (ψ=60 ◦ andψ=168 ◦ ). Several intermediate configurations reproduce
the observed data as well. The degeneracy can not be removed even using the SED
(Fig. 4.8) which is similar in the two models (LNIR/L⋆= 0.18%−0.14%) and it is only
roughly consistent with the photometric values.
4.4.7 AB Aur
AB Aur is the only HAe star that cannot be fitted with the IN05 models. The PTI
visibilities require a face-on rim, consistent with the inclination derived from large-scale
images in scattered light (Grady et al., 1999; Fukagawa et al., 2004) and at millimeter
wavelengths (Corder et al., 2005; Piétu et al., 2005). For these inclinations, the V 2 data

4.5 Comparison with previous analysis 79
imply a very small inner radius, about two times smaller that the smallest Rrim obtained
using the IN05 model (see Fig. 4.2). If, to put the discrepancy in a more physical
context, we take Tevp as a free parameter, we find good agreement with the PTI data for
Tevp∼ 2800 K (dashed line), a value by far too high not only for silicates but also for
any other type of grains (e.g., Pollack et al. 1994).
The situation becomes even less clear if we consider also the IOTA observations
(squared points), since they seem to indicate the presence of a more inclined disk with an
inner radius between 0.26 AU and 0.51 AU. No additional information can be obtained
from the analysis of the spectral energy distribution, since all the inner rim models
with an effective temperature between 1500 K and 2500 K are compatible with the
photometric data. We will come back to AB Aur in Sec. 4.6.
4.5 Comparison with previous analysis
Fits to the same interferometric data analyzed in Sec. 4.4 have been obtained by Eisner
et al. (2004, hereafter E04) assuming a toroidal shape for the puffed-up inner rim, based
on the simplified DDN01 model. In these fits, the free parameters are the location of the
rim Rrim and the two observational parameters,ι and PA. The E04 results are shown in
Tab. 4.2.
We note that for three objects (MWC 758, VV Ser and CQ Tau) the E04 inclinations
are in agreement within the errors with the values obtained with the IN05 model, while
for the other two (V1295 Aql and MWC 480) the E04ιestimates are consistent with
the lowest value of the range derived in this Chapter.
The largest differences are in the derived values of Rrim: the IN05 inner radii are
always larger than E04 results, with a maximum difference of a factor∼ 3 if we consider
our maximum Rrim in the MWC 480 system. While for CQ Tau the two values are almost
the same, for all the other stars the difference is a factor 1.5 and 2. This discrepancy is
mainly due to the difference between IN05 and E04 models. In particular, in IN05,
the curved shape of the emitting surface is self-consistently calculated, allowing a more

80 Large dust grains in the inner region of circumstellar disks
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Baseline (m)
PTI
IOTA
IN05 - R in =0.51
E04 - R in =0.26
Figure 4.2: V 2 data for AB Aur plotted in function of the baseline. The square and circle points
refer respectively to the IOTA and PTI observations in K band. The solid line is relative to the
best prediction of the self-consistent model of inner rim (a≥1.2µm, Rrim=0.51 AU,ι=20 ◦ ).
The dashed line is obtained with the IN05 model using a≥1.2µm with an ad hoc Tevp= 2800
K (Rrim= 0.26 AU).
correct determination of the dependence of the rim emission on the inclination of the
disk.
Moreover, the IN05 model takes into account the effect of the radiation transport
within the disk, even if in an approximate way (see Appendix of the previous Chapter).
This supplementary heating is neglected by E04, who calculate the dust temperature
taking into account only the direct stellar radiation. The ratio between the two values of
Rrim is given by the relation
Rrim
ˆRrim
= ˆǫ (2+1/ǫ), (4.5)
where the ˆǫ, ˆRrim and ˆT 2 evp are the values used by E04 and the inner radius ˆRrim is given

4.6 Discussion 81
by the relation
ˆRrim= 1
ˆT 2
L⋆
4πσ
evp
1
. (4.6)
ˆǫ
Assuming the same value of the dust emissivity (ǫ=ˆǫ), the ratio Rrim/ ˆRin is∼ 1 for
ǫ

82 Large dust grains in the inner region of circumstellar disks
lar medium (a=0.01−0.1µm, Weingartner and Draine, 2001), confirming that grain
growth has taken place in the innermost disk regions (van Boekel et al., 2004).
Even if the predicted near-infrared excess agrees well with the photometric observa-
tions, some interesting differences exist between the theoretical and the observed spec-
tral energy distributions. With the exception of CQ Tau, the predicted SEDs always
peak at a wavelength slightly longer than found in the observations: the flux at short
wavelengths (between 1.5µm and 2.2µm) is thus generally underestimated while the
flux between 2.2µm and 7µm is overestimated. This may be due to the fact that in our
models the SED is computed assuming that each point on the surface of the rim emits
as a black body at the local effective temperature. This approximation is energetically
correct but may not reproduce the exact wavelength dependence of the emitted radiation
(see Appendix in Chapter 3).
The rim models fail only in the case of AB Aur, where silicates, of whatever size,
produce rims that are too distant from the star to be consistent with the observations. As
shown in Sec. 4.4.7, PTI and IOTA data give somewhat contradictory results, and more
interferometric data are clearly required. However, unless further observations drasti-
cally change the present picture, the discrepancy between the rim model predictions and
the data is highly significant, and some of the basic underlying assumptions need to be
changed. It is possible that in AB Aur grains more refractory than silicates dominate the
dust population in the inner disk; however, the Tevp required (∼ 2800 K) is too high for
any dust species likely present in disks (e.g., Pollack et al., 1994).
It is more likely that gas in the dust-depleted inner region absorbs a significant frac-
tion of the stellar radiation, shielding the dust grains which are therefore cooler than in
our models. This requires a high gas density in the inner disk, as expected if the accre-
tion rate is high; in general, the accretion rates of HAe stars (including AB Aur) are low
enough to ensure that the gaseous disk remains optically thin (Muzerolle et al. 2004).
However, our knowledge of the accretion properties and gas disks of HAe stars is very
poor, and should be investigated further.
AB Aur may be more than just an oddity. The presence of optically thick gas inside

4.6 Discussion 83
the inner rim has been proposed to explain the near-infrared interferometric observations
of some very bright Herbig Be stars, for which the visibility data suggest inner disk
radii many times too small to be consistent with the puffed-up rim models (Malbet et
al., 2006; Monnier et al., 2005). If this is the case, AB Aur could be the low-luminosity
tail of the same phenomenon, which, given its small distance and large brightness, could
be used to understand a whole class of objects.
4.6.2 Inclination and position angle
Inclination and PA are well constrained by the existing data only in one case (MWC 758).
We want to stress, however, that values of the parameters outside the ranges given in
Tab. 4.2 do not fit the data at all. In particular, there are no objects consistent with
face-on disks, or in general with a centro-symmetric brightness distribution. This rules
out models, such as those of Vinkovic et al. (2006), where most of the near-infrared
flux is contributed by a spherically symmetric shell around the star, rather than by a
circumstellar disk.
Two stars (CQ Tau and VV Ser) belong to the group of UXOR variables, which
are interpreted as objects with disks seen close to edge-on (Grinin et al., 2001; Natta
and Whitney, 2000; Dullemond et al., 2003). We derive for them large inclinations, in
agreement with this interpretation.
For some of our targets, there are in the literature estimates of the orientation of
the outer disk on the plane of the sky obtained with millimeter interferometers (Testi el
al., 2001 and 2003; Manning and Sargent, 1997; Piétu et al., 2005; Corder et al., 2005).
These determinations refer to the outer disk, i.e., to spatial scales of 50–100 AU at
least. The comparison with the values derived in the near-infrared for the inner disk (on
scales of less than 1 AU) can provide information on possible distortions in the disk, e.g.
variations of the inclination with radius. For the three disks with millimeter data (MWC
758, CQ Tau and MWC 480), there is agreement (within the uncertainties) between the
inclination obtained by infrared and millimeter observations. However, it is certainly
premature to exclude the existence of disk distortions, given the large uncertainties that

84 Large dust grains in the inner region of circumstellar disks
affect both the millimeter and the near-infrared estimates. More accurate interferometric
observations in the two wavelength ranges and self-consistent models of the disk at all
physical scales are required.
An interesting case is that of VV Ser, whose disk has recently been imaged as a
shadow seen against the background emission in the 11.3 PAH feature (Pontoppidan et
al., 2006); These authors derive an inclination (of the outer disk) of about 70 ◦ and a
position angle of 13 ◦ ± 5 ◦ . While the inclination is consistent with the upper limit of
the range we obtain for the inner disk, the position angle is off by almost 90 ◦ . This
discrepancy is intriguing, and deserves further investigation.
4.6.3 Improving the model constrains
Near-infrared interferometric observations of disks around pre-main sequence stars are
still few and sparse. It is clear from our analysis that even in the most favorable cases
more visibility data at different baselines are necessary to narrow the range of possible
disk inclinations and grain properties.
Given the huge demands of telescope time that interferometric observations require,
it is useful to make use of model predictions in preparing the observations and in choos-
ing the baseline configurations that can constrain the disk structure.
CQ Tau represents an example of how the IN05 model can be used in this context. To
better constrain the inner rim structure, one will need observations with baselines longer
than 130m (available with the VLT interferometer), for which the predicted values of the
squared visibility parameters are very different for different disk models (see Fig. 4.6).
On the other hand, observations with baseline shorter than 60m could better constrain
the inner radius of MWC 480, since a degeneracy in the models is present at longer
baselines.
V1295 Aql represents a still different case, in which the degeneracy in the values of
the predicted squared visibility can be removed observing at distant hour angles for the
same baseline configurations, in order to determine the visibility variations at the same
baseline, due to the inclination of the disk.

4.7 Summary and conclusions 85
4.7 Summary and conclusions
In this Chapter we have analyzed the near-infrared interferometric observations of the
six best observed HAe stars using the rim models developed by Isella and Natta (2005),
discussed in Chapter 3. Our aim was to explore the potential of near-infrared interfer-
ometry to constrain the properties of the grains in the inner disks of these stars.
The basic assumptions of the IN05 rim models are that the inner disk structure is
controlled by the evaporation of dust in the unattenuated stellar radiation field, as ex-
pected if the gaseous disks have low optical depth, and that the most refractory grains
are silicates. The IN05 self-consistent models for the puffed-up inner rim reproduce
both the interferometric observations and the near-infrared spectral energy distribution
of all the objects we have studied, with the exception of AB Aur, which we have briefly
discussed.
For the five stars where we are able to obtain a good fit to the data, we can estimate
the grain sizes in the rim, i.e., in the mid plane of the inner disk. We find that in four
cases grains larger than∼1.2µm are either required by or consistent with the data. Only
in one case do we find that the existing data require a∼0.2−0.3µm. Note that this
value of a=1.2µm is a lower limit to the grain size: grains can be much larger, since
the rim location and shape do not change significantly if the grains grow further.
As a result of the model-fitting, one derives also the inclination and position angle of
the disk on the plane of the sky. We find that, in general, these parameters are not well
constrained by the existing data. However, in all cases we can fit, inclinations lower
than 30 ◦ are not consistent with the observations and the surface brightness distribution
can not be circularly symmetric. This rules out a spherical envelope as the dominant
source of the near-infrared emission.
For some objects, estimates of the inclination of the outer disk have been obtained
from millimeter interferometric observations; within the uncertainties, they agree with
the values obtained for the inner disk.
Our analysis shows that near-infrared interferometry is a very powerful tool for un-

86 Large dust grains in the inner region of circumstellar disks
derstanding the properties of the inner disks, in particular when combined with physical
models of these regions. However, at present the existing data are for many objects still
too sparse in their coverage of the uv plane to allow an accurate determination of the
disk parameters. We expect that this will be improved in the future. In this context,
since near-infrared interferometry is and will remain a very time demanding technique,
we stress the importance of using physical models of the inner region of the disk in
planning future observations.
Acknowledgements
We are indebted to Josh Eisner and Rafael Millan-Gabet for providing us with the PTI
and IOTA data. The authors acknowledge partial support for this project by MIUR PRIN
grant 2003/027003-001.

IN05 model Eisner et al. (2004) outer disk
Source a Rrim ι PA Rrim ι PA ι PA
(µm) (AU) (deg) (deg) (AU) (deg) (deg) (deg) (deg)
MWC 758 ≥ 1.2 0.32 40 145 0.21 36 +3
−2 127 +4
−3 46 116
VV Ser ≥ 1.2 0.54 50−70 60−120 0.47 42 +6
−2 166 +17
−6 72±5 13±5 (b)
CQ Tau 0.3−≥1.2 0.16−0.25 40−55 145−190 0.23 48 +3
−4 106 +4
−5 63 +10
−15 2±13 (c)
V1295 Aql 0.3−≥1.2 0.7−1.2 40−65 0.55 23 +15
−23
MWC 480 0.2−0.3 0.53−0.63 30−65 0.23 28 +2
−1 145 +9
−6 20−40 147−180 (a,d)
AB Aur impossible to fit 0.25 8 +7
−8 15−35 (e, f,g,h)
50−110
Table 4.2: Best fitting model parameters. From column 2 to 5 are reported the best fit parameters for the “puffed-up” inner rim, obtained
with the IN05 model: the grain radius a, the radius of the inner rim Rrim, the inclinationι and the position angle PA. The free parameters
of the model are presented in bold face. Columns 6,7 and 8 show the values of the radius of the inner rim, the inclination and the position
angle, obtained by Eisner et al. (2004). Finally, the last two columns show the available estimates of inclination and position angle for
the external region of the disk: (a) Mannings and Sargent (1997); (b) Pontoppidan et al. (2006); (c) Testi et al. (2001 and 2003); (d)
Simon et al. (2000); (e) Fukagawa et al. (2004); ( f ) Grady et al. (1999); (g) Corder et al. (2005); (h) Piétu et al. (2005).
+6 (a)
−5
4.7 Summary and conclusions 87

Face-on disk image
Baseline (m)
V 2
1
0.5
0
Inclined disk image
Baseline (m)
Figure 4.3: Predicted visibilities for two different inclinations of the inner rim. The two panels on the left show the predicted image for
a face-on rim (ι=0 ◦ ) and the relative visibility squared V 2 : since the image is circularly symmetric, the visibility squared values are the
same for every baseline orientation. The two panels on the right show the predicted image for an inclined rim (ι=60 ◦ ). In this case, the
values of V 2 depend on the baseline orientations: V 2 decreases rapidly for baselines oriented along the major axis of the image (solid
line) while V 2 decreases slowly for a baseline oriented along the minor axis (large and small dashes). For the intermediate orientation
(dashed line), V 2 is in between the two extreme values.
V 2
1
0.5
0
88 Large dust grains in the inner region of circumstellar disks

V 2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Baseline (m)
MWC758
V 2
0.40
0.20
0.40
0.20
0.40
0.20
NS
NW
SW
-3 -2 -1 0 1
hour angle
Figure 4.4: The left and central panels show K-band V 2 data for MWC 758 respectively plotted as function of the baseline and of the
hour angle, for the three different PTI baseline orientations (NS, baseline length of 110 m in direction North-South; NW, 86 m direction
North-West; SW, 87 m direction South-West). PTI measurements (Eisner et al., 2004) are shown by dots. The right panel shows the
spectral energy distribution of MWC 758. The de-reddened photometric fluxes are from Eisner et al. (2004, filled circles), Malfait et
al. (1998, empty circles), van den Ancker et al. (1998, filled squares) and Cutri et al. (2003, empty squares); the dotted line shows
the photospheric stellar flux. The solid lines show the predictions of the best-fitting model with small grains (a=0.17µm,ι=48 ◦ ,
PA=134 ◦ ); the dashed lines the best-fitting model with big grains (a≥1.2µm,ι=40 ◦ , PA=145 ◦ ). As described in Sec. 4.4.2, all
the intermediate disk configuration can reproduce the observations at almost the same level of confidence. Each curve on the left panel
corresponds to a different orientation of the baseline on the plane of the sky (see Fig. 4.3). The stellar parameters are given in Tab. 4.1.
λ F λ (erg cm 2 s -1 )
1e-07
1e-08
1e-09
0.5
1
λ (μm)
5
4.7 Summary and conclusions 89

V 2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Baseline (m)
VVSer
V 2
0.40
0.20
0
0.60
0.40
0.20
0.40
0.20
NS
NW
SW
0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
hour angle
Figure 4.5: Same as Fig. 4.4 for VV Ser. The solid curves show the model predicted values of the K-band V 2 for a rim with a=0.52µm,
ι=65 ◦ and PA=80 ◦ ; the dashed curves for a rim model with a≥1.2µm,ι=55 ◦ and PA=115 ◦ . The photometric data are from
Eisner et al. (2004, filled circles), Cutri et al. (2003, empty squares), Hillenbrand et al. (1992, empty squares), Rostopchina et al. (2001,
filled squares). Stellar parameters in Tab. 4.1.
λ F λ (erg cm 2 s -1 )
1e-07
1e-08
1e-09
0.5
1
λ (μm)
5
90 Large dust grains in the inner region of circumstellar disks

V 2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Baseline (m)
CQTau
V 2
0.40
0.20
0.40
0.20
0.40
0.20
NS
NW
SW
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
hour angle
Figure 4.6: Same as Fig. 4.4 for CQ Tau. The solid curves show the predictions of the model with a=0.3µm (Rrim= 0.25 AU),ι=52 ◦
and PA=168 ◦ ; the dashed curves the model with a≥1.2µm (Rrim=0.16 AU),ι=46 ◦ and PA=164 ◦ . The photometric data are
from Eisner et al. (2004, empty squared), Glass and Penston (1974, filled squares), ISO catalogue (empty circles) and Natta et al. (2001,
filled triangles) and references therein.
λ F λ (erg cm 2 s -1 )
1e-08
1e-09
0.5
1
λ (μm)
5
4.7 Summary and conclusions 91

V 2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Baseline (m)
V1295Aql
V 2
0.30
0.20
0.10
0.30
0.20
NW
SW
0.10
-0.5 0 0.5 1 1.5
hour angle
Figure 4.7: Same as Fig. 4.4 for V1295 Aql. The solid lines plots the results of a model with a≥1.2µm (Rrim= 0.7 AU),ι=63 ◦ and
PA=162 ◦ . The dashed lines a model with a=0.3µm (Rrim= 1.2 AU),ι=40 ◦ and PA=80 ◦ . The photometric measures are from
Eisner et al. (2004, filled circles), Malfait et al. (1998, open circles), Kilkenny et al. (1985, filled squared), Glass and Penston (1974,
open squared). The short baseline points in the left panel (empty squares) are from IOTA, while the central panel shows only the PTI
data.
λ F λ (erg cm 2 s -1 )
1e-07
1e-08
1e-09
0.5
1
λ (μm)
5
92 Large dust grains in the inner region of circumstellar disks

V 2
1
0.8
0.6
0.4
0.2
0
0 50 100 150 200
Baseline (m)
MWC480
V 2
0.40
0.20
0.40
0.20
NW
SW
-3 -2 -1 0 1
hour angle
Figure 4.8: Same as Fig. 4.4 for MWC 480. The solid lines are for the disk model characterized by a=0.2µm (Rrim= 0.63 AU),ι=35 ◦
and PA=30 ◦ . The dashed lines are relative to a disk model characterized by a=0.3µm (Rrim=0.53 AU),ι=60 ◦ and PA=102 ◦ . The
photometric data are from Eisner et al. (2004, filled circles), Malfait et al. (1998, open circles), Cutri et al. (2003, empty squares) and
van den Ancker et al. (1998, filled squares).
λ F λ (erg cm 2 s -1 )
1e-07
1e-08
1e-09
0.5
1
λ (μm)
5
4.7 Summary and conclusions 93

CHAPTER 5
More on the “puffed–up” inner rim
After that Natta et al. (2001) and Tuthill et al. (2001) independently suggested that
the dusty component of pre-main sequence circumstellar disks have to be internally
truncated at the dust evaporation radius, many efforts have been made to understand the
possible implications of this effect on the circumstellar disk structure.
Dullemond, Dominik and Natta (2001, hereafter DDN) were the firsts to include the
dust evaporation in a self-consistent numerical solution of the disk structure, obtaining
the puffed-up inner rim model summarized in Fig. 5.1. Even if the model was based on
some very crude approximations (i.e, negligible viscous and self–diffused disk heating,
negligible gas absorption inside the dust evaporation radius, black-body dust grains,
constant dust evaporation temperature), its predictions were in good agreement not only
with the observed spectral energy distributions of many HAe and TTS stars, but also
with the first interferometric measurements of the disk inner radii of Millan-Gabet et
al. (2001). Moreover, the presence of disk shadowed regions, shielded from the stellar
radiation by the rim, has been suggested to explain the different mid-infrared spectral
slopes observed in a large sample of HAe stars (Meeus et al. 2001).
The major problem of the DDN model was the predicted vertical rim surface and
the consequent strong dependence of the rim emission on the disk inclination angle (see

96 More on the “puffed–up” inner rim
rim radiation
star
inner hole shadowed region flaring disk
inner rim
CG97
flaring disk
R in
H rim
diffusion
projection of
the wall
diffusion
R fl
H cg
disk surface
disk atmosphere
Figure 5.1: Sketch of the “vertical puffed-up inner rim” model from Dullemond, Dominick
and Natta (2001). Tacking into account the dust evaporation that occurs close to the central
star, the dusty component of the disk is truncated at radius Rin. At this distance, the dust is
perpendicularly heated by the stellar radiation and is hotter than in the classical flaring disk
model labeled with CG97 (from Chiang and Goldreich, 1997). The pressure scale height of the
disk at Rin is thus higher than in CG97, giving rise to a “puffed-up” inner rim that appears as a
“vertical wall”. The disk regions behind the rim lay in the rim shadow with a vertical structure
driven only by the heating radial diffusion into the disk. At larger distances and out of the rim
shadow, the disk has the same properties of the flaring disk model of CG97.
Fig.3.4). It required that all the observed HAe and TTS stars had an inclination of∼60 o ,
which was both physically improbable and in contrast with other observational evidence
which indicated a face-on disk around AB Aur.
The inner rim model published in Isella and Natta (2005, IN05) removed some of
the approximation of the DDN model. In particular, we (i) introduced a realistic dust
R

5.1 Fuzzy and sharp rims 97
opacity to calculate the evaporation distance, (ii) considered the additional heating term
due to the radiation diffused by the disk itself, (iii) introduced the dependence of the dust
evaporation temperature on the surrounding gas density to compute a more realistic rim
shape.
As a consequence of the first point, we showed that it is possible to calculate the size
of the dust present in the inner disk from the observed inner disk radius; the results have
been discussed in Chapter 3 and 4. The latter two points have been the subject of follow-
up works recently appeared in the literature. The main results are briefly summarized in
the following.
5.1 Fuzzy and sharp rims
Introducing the additional heating term due to the radiation field diffused by the disk–
itself, we found that the rim atmosphere, defined as the region in which 99% of the
stellar radiation is absorbed, can be either very sharp or fuzzy, depending on the size
of the dust grains present into the disk. Assuming a nominal star of L⋆= 50L⊙ and
Teff=10000 K, our calculations show that silicate grains smaller than about 1µm are
mainly heated by the stellar radiation field and that their temperature decreases along the
rim atmosphere, starting from the evaporation temperature atτ=0. In this case, given
the high density of the dust, all the stellar radiation is absorbed in a very narrow region if
compared with the radius of the inner rim (see Fig. 3.1 at pg. 47). On the contrary, larger
grains are more efficiently heated by the diffused infrared radiation originating from the
disk interior and their temperature tends to increase in the rim atmosphere, starting from
the value atτ=0at which the grains reach the evaporation temperature (see Fig. 3.8 at
pg. 63). Since the grain temperature cannot be higher than the evaporation temperature,
large grain can thus survive only in an optically thin atmosphere (τ∼0). In this case,
the rim atmosphere can be very extended and comparable with the inner rim radius (Fig.
3.1).
Using a different formalism, the dependence of the rim atmosphere width on the dust

98 More on the “puffed–up” inner rim
Figure 5.2: From Vinkovic (2006). Dust dynamics in protoplanetary disks brings dust closer to
the star than the inner edge of optically thick dusty disk. Such a disk structure is possible if this
zone is populated only by big grains and it is vertically optically thin. These conditions enable
disk thermal cooling and dust survival. The resulting sublimation zone spans from the optically
thin to the optically thick inner disk radius. Its physical size is∼0.2 AU in Herbig Ae stars and
∼0.03 AU in T Tau stars.

5.2 New shapes for the rim 99
grain size has been recently confirmed by Vinkovic (2006) who extended the calculation
to the case of a mixture of grain sizes. He showed that dust grains of different size have
different temperature at the surface of the rim (whereτ=0), while their temperature is
the same in the disk interior. The exact grain size composition in the rim atmosphere
depends on the amount of the external flux because different grain sizes sublimate at
different distance from the star. Moreover, he confirmed that a large optically thin sub-
limation zone (∼ 0.2 AU for Teff=10000 K) may exist in presence of micron size grains
(see Fig. 5.2). Note that this value is almost the same obtained with the single-size
approximation adopted in IN05 (see Fig. 3.1 at pg. 47).
The gas present inside the rim atmosphere may be enriched by metals coming from
the sublimated dust, which can favour a further grain growth by recondensation on the
surface of existing grains. The author also suggest that the sublimation zone may play an
important role in dusty wind processes, since its small optical depth and small distance
from the central star should result in gas properties that are more susceptible to non-
gravitational forces capable of launching a dusty wind (such as magnetic fields) than
the rest of the dusty disk. This conclusion is particularly interesting if combined with
the indication that a disk wind launched just in front of the inner rim may be responsible
for the Brγ emission detected in HD 104237 (see Chapter 6).
5.2 New shapes for the rim
The shape of the puffed-up inner rim in the IN05 model is driven by the dependence
of the dust evaporation temperature on the surrounding disk density. Close to the mid-
plane, where the density are high, the dust evaporation temperature is also high; moving
away from the mid plane, the gas density rapidly decreases and the corresponding evap-
oration temperature also drops. For fixed stellar parameters and dust properties, the
resulting dust evaporation surface is therefore curved rather than vertical, as predicted
by the DDN model.
Recently, Tannirkulam et al. (2007; hereafter T07) have shown that dust growth

100 More on the “puffed–up” inner rim
Rim height[A.U]
0.15
0.10
0.05
IN05 values for density dependent dust sublimation model
TORUS values for density dependent dust sublimation model
TORUS values for dust-segregation model (this work)
_ _ _ _ Analytic estimate for dust-segregation model
large
grains
dust
segregation
small
grains
0.00
0.0 0.5 Radius[A.U] 1.0 1.5
Figure 5.3: Courtesy of A. Tannirkulam, to be published in Tannirkulam at al. (2007). The
figure shows the height of the inner rim above the disk mid plane. The rim is defined as the
τ=1surface (forλ=5500Å) computed along radial lines from the central star. The solid lines
labelled with “small grains” and “large grains” indicate the IN05 model calculated for single
grain sizes of 0.1µm and 1.2µm respectively. The diamond indicate the rim calculated with the
TORUS code for the same grain sizes and without the dust settling (only the density-dependent
sublimation temperature is present). The stars indicate the rim surface calculated with TORUS
introducing the dust settling (the dust segregation model). Finally, the dashed line is an analytical
estimate of the dust evaporation front for the dust segregation model.
and settling can curve the rim surface on larger scales than the ones produced by the
density-dependent sublimation temperature. A number of observational (i.e. Retting et
al., 2006) and theoretical (Dullemond and Dominik, 2004; Tanaka et al., 2005) results
have shown that during the disk evolution, small dust grains tend to coagulate in larger
grains and settle towards the disk mid plane. In absence of strong vertical turbulence the
disk should therefore be characterized by a vertical grain size stratification dominated
by large grains in the mid plane and small ones in the disk upper layers. Since the dust
sublimation distance depends on the grain size (i.e., larger grains can survive closer to
the central star), dust settling may thus lead to a significant rim curvature. Fig. 5.3 shows
the results obtained by T07 using a Monte Carlo approach to calculate the structure of
an inner rim, where the vertical dust size stratification has been parametrized following
the predictions of theoretical grain evolution models (Dullemond and Dominick, 2004).
In absence of dust settling, the rim shape is in good agreement with the IN05 models

5.3 New observational results 101
(see the solid lines and the empty diamonds in the figure). On the contrary, when the
settling operates, the rim shape is strongly curved (see the stars in the figure). The
rim dominates the near-infrared disk emission and it appears as a thick ring, larger
than the IN05 model, when seen at face-on inclinations. At higher inclination the T07
model predicts more symmetric images than those showed in Fig. 3.5, corresponding
to smaller closure phase values when observed with near infrared interferometers. One
should note that the dust settling parametrization used by T07 is valid in absence of
vertical turbulent motions which mix the circumstellar material and destroy the grain
size stratification. If a significant degree of vertical turbulence exists (see i.e. Ilgner and
Nelson, 2006), the IN05 rim shapes would be more correct. In general, the inner rim
should have a shape in between the to extreme cases predicted by IN05 and T07. In any
case, the dusty disk inner rim will have a fuzzy and significantly curved surface, very
different from the “vertical wall” predicted by the first models of Dullemond, Dominick
and Natta (2001).
5.3 New observational results
Given the spatial resolution required to observed the inner regions of nearby pre-main
sequence disk, only near-infrared interferometry allow to clarify the real structure of
the dusty disk inner rim. As we have discussed in Chapters 2 and 4, all the current
results are based on measuring the module of fringe visibility, while the visibility phase
is required to unambiguously detect deviation from simple symmetries. While atmo-
spheric turbulence corrupts the direct measurement of the fringe phase, by combining
three or more telescopes it is possible to measure the closure-phase, a phase quantity
that depends only on the source intrinsic phase. Today, only three facilities (the Palo-
mar Testbed Interferometer, PTI; the Infrared Optical Telescope Array, IOTA; and the
VLT Interferometer ,VLTI) allow closure phase measurements and IOTA is the only one
that has produced useful observations of young stellar objects so far. In particular, Mon-
nier et al. (2006) have recently performed a closure-phase survey of 8 nearby HAe stars

102 More on the “puffed–up” inner rim
Milliarcseconds
Milliarcseconds
15
10
5
0
-5
-10
-15
15
10
5
0
-5
-10
-15
HD 45677
Elliptical Ring Model
15 10 5 0 -5 -10 -15
Milliarcseconds
HD 45677
Skewed Ring Model
15 10 5 0 -5 -10 -15
Milliarcseconds
Data Vis2
Data Vis2
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4
Model Vis2
0.6 0.8
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4
Model Vis2
0.6 0.8
Data Closure Phases (degs)
Data Closure Phases (degs)
10
0
-10
-20
-30
-40
-40 -30 -20 -10 0 10
Model Closure Phases (degs)
10
0
-10
-20
-30
-40
-40 -30 -20 -10 0 10
Model Closure Phases (degs)
Figure 5.4: From Monnier at al. (2006). Top left panel shows the best-fit elliptical ring model to
the IOTA data for the star HD 45677 (north is up and east is left). Top middle panels compares
the observed visibility data to the model data. Top right panel compares the observed closure
phases with the model closure phases. This kind of models represents spherical or elliptical
envelopes as predicted by Vinkovic et al. (2006). Since this model is constrained to be cen-
trosymmetric, all closure phases are identically zero and cannot fit the observations. The bottom
panels present the best-fit “skewed ring model”. The closure phases are fairly well explained by
a strong northwest skew. Some data points are not well explained by these models, suggesting
a patchier dust distribution. This model is similar to what predicted by Isella and Natta (2005)
and Tannirkulam et al. (2007).

5.3 New observational results 103
with an angular resolution of about 5×12 milli-arcsec at 1.65µm. The data show the
presence of a resolved continuum emission in agreement with dust depleted inner holes
with radius between∼0.2–0.5 AU. In the case of AB Aur, RY Tau and HD 163296, the
measured visibilities show that a small fraction of the observed emission (

Part III
Gas in the inner disk

CHAPTER 6
Constraining the wind launching region
in Herbig Ae stars: AMBER/VLTI
spectroscopy of HD 104237
This chapter will appear in Astronomy & Astrophysics
and can be found at www.arxiv.org/abs/astro-ph/0606684.
E. Tatulli 2 , A. Isella 1,2 , A. Natta 2 , L. Testi 2 and the AMBER consortium 3 , A&A in press
“Constraining the wind launching region in Herbig Ae stars:
AMBER/VLTI spectroscopy of HD 104237”
1 Dipartimento di Fisica, Univeristá di Milano, via Celoria 16, 20133 Milano, Italy
2 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy
3 http://amber.obs.ujf-grenoble.fr
Abstract: We investigate the origin of the Brγ emission of the Herbig Ae star HD 104237
on Astronomical Unit (AU) scales. Using AMBER/VLTI at a spectral resolutionR=1500 we
spatially resolve the emission in both the Brγ line and the adjacent continuum. The visibility

108 AMBER/VLTI spectroscopy of HD 104237
does not vary between the continuum and the Brγ line, even though the line is strongly detected
in the spectrum, with a peak intensity 35% above the continuum. This demonstrates that the line
and continuum emission have similar size scales. We assume that the K-band continuum excess
originates in a “puffed-up” inner rim of the circumstellar disk, and discuss the likely origin
of Brγ. We conclude that this emission most likely arises from a compact disk wind, launched
from a region 0.2-0.5 AU from the star, with a spatial extent similar to that of the near infrared
continuum emission region, i.e, very close to the inner rim location.
6.1 Introduction
The spectra of pre-main sequence stars of all masses show prominent strong and broad
emission lines of both hydrogen and metals. These lines trace the complex circumstellar
environment that characterizes this evolutionary phase, and are very likely powered
by the associated accretion disks. The emission lines are used to infer the physical
properties of the gas, and to constrain its geometry and dynamics. Their exact origin,
however, is not known. The hydrogen lines, in particular, may originate either in the
gas that accretes onto the star from the disk, as in magnetospheric accretion models
(Hartmann et al., 1994), or in winds and jets, driven by the interaction of the accreting
disk with a stellar (Shu et al., 1994) or disk (Casse et al., 2000) magnetic field. For
Herbig Ae stars, it is additionally possible that they form in the inner disk (Tambovtseva
et al., 1999).
For all models, emission in the hydrogen lines is predicted to occur over very small
spatial scales, a few AUs at most. To understand the physical processes that occur at
these scales, one needs to combine very high spatial resolution with enough spectral res-
olution to resolve the line profile. AMBER, the three-beam near-IR recombiner of the
VLTI (Petrov et al., 2003), simultaneously offers high spatial and high spectral resolu-
tion, with the sensitivity required to observe pre-main sequence stars. These capabilities
were recently demonstrated by Malbet et al. (2006), who successfully resolved the lu-

6.1 Introduction 109
λ Fλ (erg cm 2 s -1 λ Fλ (erg cm
)
2 s -1 )
1e-07
1e-07
1e-08
1e-08
1e-09
1e-09
0.4 0.4 0.6 0.60.80.81
1 1.5 1.52
2 3 3 4 45
56
67
7
λ λ (μm)
HD104237 ISAAC/AMBER spectrum
1.4
1.2
1.0
0.8
?64
2.164
?66
2.166
x0= 2.1??E+3
λ (μm)
?68
2.168
Visibility
1.0
0.5
0.0
2.140 2140 2.160 2160 2.180 2180
Figure 6.1: Left: SED predicted by the “puffed-up” rim model used to normalise the K-band
continuum visibility. The contributions of the A star (long-dashed line), the K3 star (dotted line)
and the rim (short dashed line) are shown (see the text for the stellar parameters). The obser-
vations are from 2MASS (filled circles), Hipparcos (filled squares), and Malfait et al. (1998,
empty circles). An extinction of Av= 0.31 (van den Ancker et al., 1998) has been used to cor-
rect the data for interstellar absorption. Center: Comparison of Brγ observed with AMBER in
the photometric channels (solid line) and ISAAC (dotted line); the dashed line shows the ISAAC
spectrum smoothed to the spectral resolution of AMBER. Right: Visibility of HD 104237 as a
function of wavelength. The continuum has been normalised using the star+rim model, as
described in the text. The vertical line shows the Brγ wavelength.
minous Herbig Be system MWC 297 in both the continuum and the Brγ line. They
showed that the line emission originates from an extended wind, while the continuum
infrared excess traces a dusty accretion disk.
In this Chapter we concentrate on a lower mass, less active system, the Herbig Ae
system HD 104237. The central emission line star, of spectral type between A4V and
A8, is surrounded by a circumstellar disk, which causes the infrared excess emission
(Meeus et al., 2001) and drives a jet seen in Ly-α images (Grady et al., 2004). The
optical spectrum shows a rather narrow Hα emission with a P-Cygni profile (Feigelson
et al., 2003). The disk is seen almost pole-on (i=18 ◦ +14
−11 ; Grady et al., 2004), consistent
with the low value of v sin i (12 km s −1 ; Donati et al., 1997). Donati et al. (1997)
detected a stellar magnetic field of 50 G. Böhm et al. (2004) have revealed the presence
of a very close companion of spectral type K3, orbiting with a period of∼ 20 days, and
λ (μm)

110 AMBER/VLTI spectroscopy of HD 104237
whose bolometric luminosity is 10 times fainter than the one of the central star. In the
near infrared domain, spatially unresolved ISAAC observations (Garcia Lopez et al.,
2006) show a strong Brγ emission line, with a peak flux 35% above the continuum.
6.2 Observations and data reduction
AMBER observed HD 104237 on 26 February 2004 on the UT2-UT3 baseline of the
VLTI, which corresponds to a projected length of B=35 m. The instrument was set
up to cover the [2.121, 2, 211]µm spectral range with a spectral resolution ofR=1500,
which resolved the profile of the Brγ emission line at 2.165µm. The data consist of
10 exposures of 500 frames, with an integration time of 100ms for each frame. The
integration time is a trade-off between gathering enough flux in the photometric channel
and preventing excessive contrast loss due to fringe motion during the integration. At the
time of these early observations the star magnitude of K=4.6 was close to the sensitivity
limit of the medium spectral resolution mode, and as a result fringes are only visible in
≈ 25% of the frames. In the photometric channels the Brγ line is under the noise level of
the individual frames, but it is well detected when averaging them and it contributes 35%
of the total flux (Fig. 6.1, central panel). For comparison we overlay a higher spectral
resolution spectrum (R=8900, taken with ISAAC approximately one year before the
VLTI observations), as well as its smoothing to the AMBER spectral resolution. The
two spectra are consistent with each other, within the combined typical line variability
of Herbig Ae stars and calibration uncertainties.
Data reduction followed standard AMBER procedures (Tatulli et al. 2006). To
optimize the signal to noise ratio (SNR) of the visibility derived from these relatively
poor-quality data, we pre-selected frames with individual SNR greater than 1.5, ensuring
that fringes are present in all short exposures that enter the visibility.
For these observations the visibility amplitude could not be calibrated to an absolute
scale using an astronomical calibrator (unresolved star), due to non-stationary vibrations
in the optical train of the UT telescopes. With a 100ms integration time these vibrations

6.2 Observations and data reduction 111
randomly degrade the fringe contrast of the individual frames by a large factor, and the
statistics of that attenuation depends on uncontrolled factors (exact telescope and delay
line pointing, recent environmental history, etc) that do not similarly affect the target
star and its calibrator. Fortunately the contrast loss from vibrations is achromatic across
our small relative bandpass, and as a consequence the differential visibility, that is, the
relative visibility between the spectral channels, is unaffected. Our dataset therefore
allows us to investigate the visibility across the Brγ emission line compared to that in
the adjacent continuum. The differential visibilities are accurate to approximately 5%.
The excess near infrared continuum emission in Herbig Ae stars most likely origi-
nates in the innermost regions of their dusty circumstellar disks, at the dust sublimation
radius (Eisner et al., 2005; Isella et al., 2006). To approximately normalise the visi-
bilities, we thus scaled the observed continuum value to the predictions of appropriate
theoretical models of the disk inner rim emission (Isella and Natta, 2005). We computed
the structure of the “puffed-up” inner rim as described in Isella et al. (2006). We used
for the two stars the following parameters: Te f f= 8000K, L⋆=30L⊙ for the primary and
Te f f = 4700K, L⋆=3L⊙ for the secondary star. We adopted a distance of D=115 pc,
and a disk mass surface density ofΣ(r)=2·10 3 · r −3/2 g/cm 2 (with r in AU).
Fig. 6.1 (left) shows that the model using micron size astronomical silicates pro-
duces a very good fit of the SED. The star fluxes contribute approximately 30% (respec-
tively 20% and 10% for the primary and secondary star, which gives a binary flux ratio
of∼ 0.5 in the K band) of the total 2.15µm flux, and the inner rim appears as a bright
ring with radius Rrim= 0.45AU (3.8 mas at D=115 pc). The resulting model visibility
on the 35m baseline is V = 0.38 (Fig 6.1, right). Note that the orbital period of the
spectroscopic binary is 20 days. This leads to an average separation ofδs=0.15 AU
which corresponds to an angular separation of 1.2 mas. The central system (A star+K3
companion) is therefore completely unresolved on the current baseline, and both stars
are located inside the dust evaporation radius.

112 AMBER/VLTI spectroscopy of HD 104237
6.3 Results and discussions
Within the fairly small error bars, the visibility does not change across the Brγ emission
line. This result is robust and puts strong constraints on the relative spatial extent of the
line and continuum emission regions, demonstrating that they have very similar apparent
sizes. We use this constraint to probe the processes responsible for the the Brγ emission
in this star, and consider in turn the three main mechanisms usually invoked to interpret
the hydrogen line emission in pre-main-sequence stars. We translate each mechanism
to simple geometrical models of specific spatial extension, with the line strength fixed
at the observational value, and evaluate the resulting visibility across the line 1 .
Magnetospheric accretion: in such a model matter falls on the star along magnetic
field lines, and the base of this accreting flow is (approximately) inside the corota-
tion radius (Muzerolle et al., 2004). For HD 104237 we find Rcorot= 0.07AU, using
v sin i = 12km/s (Donati et al., 1997) and an inclination of i = 18 ◦
(Grady et al.,
2004). Note that this value is, at our spatial resolution level, similar to the separation
of the binary. Adopting Rstar and Rcorot as the inner and outer limits of the magne-
tospheric accreting region, Fig. 6.2 (upper panel) demonstrates that Brγ emission is
then confined much closer to the star than the dusty rim. The predicted visibility there-
fore increases significantly in the line, contrary to the observational result. Explaining
the observed visibility with magnetospheric accretion requires the corotation radius to
approach the inner rim, which would need an unrealistically lower stellar rotational ve-
1 Note that this analysis is valid as long as the continuum emission is resolved. In our case the contin-
uum is calibrated by a model and not by an unresolved star, and therefore there might be a chance that the
visibility of the continuum is close to 1. However, this peculiar scenario appears very unlikely since the
puffed-up rim model has been shown to be well representative of the very close environment of Herbig Ae
stars over a large luminosity range (Monnier et al., 2005). Furthermore, the hypothesis of a resolved con-
tinuum emission for HD 104237 is strongly supported by its measured accretion rate of 10 −8 M⊙ (Grady
et al., 2004). Indeed, Muzerolle et al. (2004) showed that for weak accretion rates ( ˙M

6.3 Results and discussions 113
V
V
V
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
0.6
0.5
0.4
0.3
•
•
•
2.14 2.15 2.16 2.17
wavelength (μm)
Figure 6.2: Comparison between the observed visibilities (empty square with error bars) and the
predictions (solid curves) of the simple geometrical models for the Brγ emission (sketched in the
same panels). The observed visibilities are scaled to match the continuum value predicted by the
“puffed-up” inner rim model (Isella and Natta, 2005) as described in Sec. 6.2. The continuum
emission arises both from the stellar photosphere (≈20%) and from the dusty disk inner rim,
located at the dust evaporation distance Rrim=0.45 AU and which appears as the bright gray
scale ring. The Brγ emission regions are shown as grid surfaces. The three panels illustrate the
different models discussed in Sec. 6.3: the upper panel represents the magnetospheric accretion
model in which the Brγ emission originates very close to the star, inside the corotational radius
Rcorot=0.07 AU; in the middle panel the Brγ emission originates between Rcorot and the rim
radius Rrim=0.45 AU, representing the gas within the disk model; the bottom panel shows the
outflowing wind model, in which the emission is confined close to the inner rim, between∼ 0.2
AU and∼0.5 AU.

114 AMBER/VLTI spectroscopy of HD 104237
locity (v

6.4 Conclusions 115
note however that the launching region of an X-wind, driven by the stellar magnetic field
(Shu et al., 1994), is close to the corotation radius and too small to be consistent with the
present data. Making the X-wind acceptable needs one to relax the assumption that most
of the line emission originates near the launching point, and instead have the brightest
Brγ region a factor 5−8 further away from the star. The disk-wind scenario in contrast
has its launching point a few tenths of an AU from the star and needs no adjusting.
This preference for a disk-wind is consistent with the size of the jet launching regions
inferred from the rotation of TTauri jets (Coffey et al., 2004).
Our analysis assumes that the continuum originates in a dusty “puffed-up” ring. That
model has been shown to match Herbig Ae and T Tauri stars over a large luminosity
range (Monnier et al., 2005), but it obviously needs to be verified for the specific case of
HD 104237, by obtaining calibrated interferometric observations over a few baselines.
6.4 Conclusions
We have presented interferometric observations of the Herbig Ae star HD 104237, ob-
tained with the AMBER/VLTI instrument withR=1500 high spectral resolution. The
visibility is identical in the Brγ line and in the continuum, even though the line repre-
sents 35% of the total 2.165µm flux. This implies that the line and continuum emission
regions have the same apparent size.
Scaling the continuum visibility with a “puffed-up” inner rim model, and using sim-
ple models to describe the Brγ emission, we have shown that the line emission is un-
likely to originate in either magnetospheric accreting columns of gas or in the gaseous
disk. It is much more likely to come from a compact outflowing disk wind launched
in the vicinity of the rim, about 0.5 AU from the star. This does not preclude accretion
from occurring along the stellar magnetic field detected by Donati et al. (1997), and
accreting matter might even dominate the optical hydrogen line emission, but our ob-
servations show that the bulk of the Brγ emission in HD 104237 is unlikely to originate
in magnetospheric accreting matter.

116 AMBER/VLTI spectroscopy of HD 104237
Our results show that AMBER/VLTI is a powerful diagnostic of the origin of the
line emission in young stellar objects. Observations of a consistent sample of objects
will strongly constrain the wind launching mechanism.
Acknowledgements
This work has been partly supported by the MIUR COFIN grant 2003/027003-001 and
025227/2004 to the INAF-Osservatorio Astrofisico di Arcetri. This project has benefited
from funding from the French Centre National de la Recherche Scientifique (CNRS)
through the Institut National des Sciences de l’Univers (INSU) and its Programmes Na-
tionaux (ASHRA, PNPS). The authors from the French laboratories would like to thank
the successive directors of the INSU/CNRS. C. Gil work was supported in part by the
Fundação para a Ciência e a Tecnologia through project POCTI/CTE-AST/55691/2004
from POCTI,with funds from the European program FEDER.
This work is based on observations made with the European Southern Observatory
telescopes. We made use of the ASPRO observation preparation tool from the Jean-
Marie Mariotti Center in France, the SIMBAD database at CDS, Strasbourg (France)
and the Smithsonian/NASA Astrophysics Data System (ADS). The data reduction soft-
wareamdlib is freely available on the AMBER site http://amber.obs.ujf-grenoble.fr.
It has been linked to the free software Yorick 2 to provide the user friendly interface
ammyorick.
2 ftp://ftp-icf.llnl.gov/pub/Yorick

Part IV
The structure of the outer disk

CHAPTER 7
The keplerian disk of HD 163296
This chapter describes the analysis of recent millimeter interferometric observations of
the HAe star HD 163296. While the data reduction and the implementation of disk
models required to analyse the data are complete, part of the analisys of the results is
still in progress. The results described in the following will appear in a forthcoming
paper (Isella A., Testi L., Natta A., Neri R., Wilner D, Qi C., 2007).
7.1 Introduction
Millimeter and submillimeter interferometers are providing an increasingly detailed de-
scription of disks around pre-main sequence stars of solar (T Tauri stars; TTS) and in-
termediate mass (Herbig Ae; HAe). Both the dust continuum emission and the emission
in molecular lines are observed and spatially resolved in a number of disks, yielding
information on the disk density and temperature, the dust properties and the gas chem-
istry and dynamics (e.g., Natta et al., 2006; Dutrey et al., 2006 and references therein).
The number of well-studied disks is still very small, practically restricted to the most
massive and luminous ones; still, it is clear that disks differ from one another. Recently,

120 The keplerian disk of HD 163296
it has been reported evidence of spiral structures in AB Aur, a 2-3 Myr old intermedi-
ate mass star, and of deviations from keplerian rotation (Piétu et al., 2005; Corder et
al., 2005); the classical TTS LkCa15 has a large inner hole of size∼ 50 AU depleted
of dust, while the HAe star MWC 480 has a smooth disk with an optically thick (at
millimeter wavelengths) inner region of radius∼35 AU (Piétu et al., 2006). Both spiral
structures and large gaps are evidence of dynamical perturbations, possibly due to the
presence of large planets or/and to the effects of the disk self-gravity (see the case of
IRAS 16293-2422B, Rodriguez et al., 2005) which however may favour the planetary
formation (see the review of Durisen et al., 2006).
The existence of unperturbed and distorted disks among pre-main sequence stars
suggests that the planet formation is actively occurring during this evolutionary stage,
leaving detectable marks on their parent disks. It is therefore important to study in detail
as many disks as possible, in order to characterize their basic properties and to detect
deviations from the simple patterns of homogeneous disks in keplerian rotation.
We report in this Chapter a detailed study of the disk associated to the HAe star
HD 163296, using observations in the continuum and CO lines obtained with three
different interferometers, namely the VLA at 7 mm, the Plateau the Bure Interferometer
at 1.3 and 2.6 mm and SMA at 0.85 mm. HD 163296 is a star of spectral type A1,
mass of roughly 2.3 M⊙, distance 122 pc. Early OVRO observations (Mannings and
Sargent, 1997) have shown the presence of a disk with a minimum mass∼ 0.03 M⊙ and
evidence of rotation from the CO lines. The disk is seen in scattered light by Grady
et al. (2000), with radius of∼ 500 AU; it has an associated jet seen in Ly-α with HST,
extending on both sides of the disk orthogonally to it. Natta et al. (2004) found evidence
of evolved dust in the outer disk of HD 163296 by comparing the VLA 7 mm flux to the
OVRO observations. The results we present here have much higher spatial resolution
and wavelength coverage than what previously known. They will allow us to measure
accurately the dynamics of the disk as well as the disk and dust properties and to test
the capability of disk models to account for the observations. As we will show, they
suggest that in HD 163296 it is not the inner disk which is dust-depleted but rather the

7.2 Observations and data reduction 121
outer disk, outside about 200 AU.
The structure of the Chapter is as follows. Sec. 7.2 will describe the observations.
The results will be presented in Sec. 7.3, where we will derive some of the disk param-
eters. A more detailed analysis, using self-consistent disk models of the dust and CO
line emission will be presented in Sec. 7.4; Sec. 7.5 contains the results, which will be
further discussed in Sec. 7.6. Summary and conclusions follow in Sec. 7.7.
7.2 Observations and data reduction
HD 163296 was observed with the Plateau de Bure Interferometer (PBI), the NRAO
Vary Large Array (VLA) and the Sub-Millimeter Array (SMA).
7.2.1 PBI observations
The IRAM/PBI observations were carried over the 2003/2004 winter season. The six 15
m dishes were used in the most extended configuration providing a baseline coverage
between 25 and 400 m. The corresponding angular resolutions are reported in Tab. 7.1.
The receivers were tuned to observe the 12 CO J=2–1 line and the nearby continuum at
1.3 mm, while at 2.7 mm the 13 CO J=1–0, and C 18 O J=1–0 lines were observed along
with the continuum. Bandpass, complex gain and amplitude calibrations were ensured
by observations of standard IRAM calibrators. The phase stability was excellent dur-
ing our observations and only a minimal amount of editing of the data was necessary.
All calibrations were performed using the standard CLIC suite of programmes within
the GILDAS software package. The calibrated uv data were then exported for the sub-
sequent analysis. The accuracy of the flux density scale calibration is expected to be
within 20% at these wavelengths.

122 The keplerian disk of HD 163296
7.2.2 SMA observations
The SMA 1 observations of HD 163296 was made on August 23rd, 2005 using the
Compact Configuration of seven of the 6 meter diameter antennas, which provided 21
independent baselines ranging in length from 8 to 80 meters. The SMA digital correlator
was configured with a narrow band of 512 channels over 104 MHz, which provided 0.2
MHz frequency resolution, or 0.18 km s −1 velocity resolution at 0.87 mm (345 GHz),
and the full correlator bandwith was 2 GHz. The weather was good withτ(1.3 mm)
around 0.06 and the double-sideband (DSB) system temperature were between 200 and
500 K. The source HD 163296 was observed from HA -3 to 4.5. Calibration of the
visibility phases and amplitudes was achieved with observations of the quasar 1921-
293, typically at intervals of 25 minutes. Observations of Uranus provided the absolute
scale for the flux density calibration and the uncertainties in the flux scale are estimated
to be 20%. The data were calibrated using the MIR software package 2 .
7.2.3 VLA observations
HD 163296 was observed at the VLA as part of a larger survey for 7 mm disk emission
around Herbig Ae stars (see Natta et al., 2004). Data were obtained with the array in
the C and D configurations, in several occasions from Dec 2001 through May 2003.
Accurate pointing was ensured by hourly pointing sessions at 3.6cm on a bright extra-
galactic object. The array offered baselines from the shadowing limit through∼ 3.4 km,
although some data was obtained in a hybrid DnA configuration, all the data from the
longer baselines had to be rejected due to large phase fluctuations that could not be
corrected. The resulting uv-plane coverage offered an angular resolution of 0.5 ′′ and a
maximum recoverable size of the order of∼40 ′′ . All the data was edited and calibrated
using standard recipes in AIPS. The short term complex gain variations were corrected
1 The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and
the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institu-
tion and the Academia Sinica
2 http://cfa-www.harvard.edu/cqi/mircook.html

7.3 Observational results 123
λ (mm) beam FWHM PA flux (mJy)
0.87 3 ′′ .14×2 ′′ .21 161 o 1910±20
1.3 1 ′′ .95×0 ′′ .42 7 o 705±12
2.8 3 ′′ .3×0 ′′ .94 8 o 77.0±2.2
7 1 ′′ .71×0 ′′ .81 172 o 4.5±0.5
Table 7.1: Integrated flux and beam size of the observations of HD 163296 performed with the
SMA (at 0.87 mm), PBI (at 1.3mm and 2.7 mm) and VLA (at 7 mm).
using frequent (few minutes cycle) observations of the quasar 1820−254, while the flux
scale was set observing the VLA calibrator 1331+305. This procedure is expected to be
accurate within∼ 15% at 7 mm.
7.3 Observational results
The continuum maps obtained at 0.87 mm, 1.3 mm, 2.8 mm and 7 mm are shown in
Fig. 7.1, the emission of the 12 CO J=2–1 transition is shown in Fig. 7.2, the emission
of the 12 CO J=3–2 transition is shown in Fig. 7.3 and the emission of the 13 CO J=2–1
transition in Fig. 7.4. At the wavelength corresponding to the C 18 O J=1–0 transition we
did not detected any emission; the continuum integrated fluxes are reported in Tab. 7.1.
With simple physical assumptions, these observations allow to determine some fun-
damental parameters of the star+disk system as the stellar and the disk masses, the
contribution to the observed fluxes of free-free gas emission, the wavelength depen-
dence of the observed flux at millimeter wavelengths and the related dust grain opacity.
7.3.1 Disk morphology and apparent size
The 12 ′′ ×12 ′′ continuum maps of HD 163296 are shown in Fig. 7.1. At all wavelengths,
the peak is coincident with the position of the optical star as measured from Hipparcos
and, given the respective synthesized beams FWHM (see Tab. 7.1), the emission is

124 The keplerian disk of HD 163296
Figure 7.1: Continuum maps of HD 163296 at 0.87 mm (SMA observations), 1.3 mm and 2.8
mm (PBI) and 7 mm (VLA), starting form the left. In the 0.87 mm map, in order to highlight the
extended morphology of the disk, the first contour level corresponds to 30 mJy (3σ), the second
to 6σ and the inner contour levels are spaced by 10σ. At longer wavelength the contour levels
are all spaced by 3σ, corresponding to 16 mJy at 1.3 mm, 3.3 mJy at 2.8 mm and 0.75 mJy at
7 mm. The small boxes show the relative synthesized beams. The integrated fluxes, the beam
dimensions and orientations are summarized in Tab.7.1.
resolved and elongated approximately in the east-west direction. Approximating the
source with a circularly symmetric geometrically thin disk and taking into account the
beam shape, the observed aspect ratio of the level contours implies an inclination of the
disk plane from the line of sight of about 45 o ± 20 o and a position angle of 120 o ± 30 o ,
roughly in agreement with the values obtained by Mannings and Sargent (1997) using
marginally resolved OVRO observations.
In the continuum map, the maximum elongations of the 3σ contours along the more
resolved direction varies between 4 ′′ at 0.87 mm to 0.9 ′′ at 7 mm. The fainter contour
levels are not centrally symmetric, showing a flux excess in the east half of the image,
better visible in the 0.87 and 1.3 mm maps. Both the disk size and the morphology will
be discussed in more detail in Sec. 7.6.1.
7.3.2 Disk kinematic
The emission maps of the 12 CO J=2-1 (at 1.3 mm) and J=3-2 (at 0.87 mm), and the
13 CO J=1-0 (at 2.8 mm) transitions are shown in Fig. 7.2, 7.3 and 7.4 respectively; the

7.3 Observational results 125
Figure 7.2: Velocity channel maps of the 12 CO J=2–1 line emission around HD 163296. The
LSR velocity (km s −1 ) is indicated in the upper left corner of each panel. The angular resolution
(synthesized beam), indicated in the small boxes, is 1.92 ′′ × 0.42 ′′ at PA 7 o ; the contour spacing
is 0.23 Jy/beam corresponding to 3σ. The last two panels show the integrated intensity (contour
levels spaced by 0.4 Jy/beam km s −1 ) and the velocity field (contour levels from 3 km s −1 to 9
km s −1 spaced by 0.5 km s −1 ) respectively.

126 The keplerian disk of HD 163296
Figure 7.3: Velocity channel maps of the 12 CO J=3-2 line emission around HD 163296. The
LSR velocity (km s −1 ) is indicated in the upper left corner of each panel. The angular resolution
(synthesized beam), indicated in the small boxes, is 3.1 ′′ ×2.03 ′′ at PA 161 o ; the contour spacing
is 1.5 Jy/beam corresponding to 3σ. The last two panels show the integrated intensity (contour
levels spaced by 5 Jy/beam km s −1 ) and the velocity field (contour levels from 3 km s −1 to 9 km
s −1 spaced by 0.5 km s −1 ) respectively.

7.3 Observational results 127
Figure 7.4: Velocity channel maps of the 13 CO J=1–0 line emission around HD 163296. The
LSR velocity (km s −1 ) is indicated in the upper left corner of each panel. The angular resolution
(synthesized beam), indicated in the small boxes, is 3.28 ′′ × 0.94 ′′ at PA 7 o ; the contour spacing
is 0.09 Jy/beam corresponding to 3σ. The last two panels show the integrated intensity (contour
levels spaced by 0.12 Jy/beam km s −1 ) and the velocity field (contour levels from 3 km s −1 to 9
km s −1 spaced by 0.5 km s −1 ) respectively.

128 The keplerian disk of HD 163296
velocity-position diagrams along the line of nodes are shown in Fig. 7.5. For all the
molecular transitions, the emission is resolved, showing a velocity pattern typical of an
inclined rotating disk characterized by a position angle of about 130 o (a more precise
estimate of the position angle will be presented in Sec. 7.5). The velocity-position
diagram calculated along this direction (see Fig. 7.5) shows a well defined “butterfly
shape” typical of keplerian rotation. A first estimate of the mass of the central object
and of the dimension of the disk can be obtained by comparing the observed velocities
with the keplerian law:
vǫ= C·ǫ −1/2 , (7.1)
whereǫ is the angular distance from the central star and vǫ is the component of the disk
rotational velocity along the line of sight. If the stellar mass M⋆ is in solar unit, the
stellar distance d in parsec andθis the disk inclination (θ=0 means pole-on disk),
the constant C≃30 √ M⋆/d sinθ is the component along the line of sight of the disk
rotational velocity (in km/sec) atǫ= 1 ′′ . As shown in Fig. 7.5 the envelopes of both
the 12 CO and 13 CO emissions are in agreement with C≃ 2.7±0.4 km/sec. This value
corresponds to a stellar mass of 2.0±0.5 M⊙ for an inclination of 45 o (obtained in the
previous section) and d=122 pc; the systemic velocity is 5.8 Km/sec. Note finally that
the disk rotation is clearly observed at least up to a distance of about 4 ′′ , corresponding
to a minimum disk outer radius of about 500 AU. We will discuss in more detail the
determination of the stellar mass and the disk outer radius in Sec. 7.5, using a detailed
model for the continuum and CO molecular emission.
7.3.3 Free-free contribution and spectral index
The continuum spatially integrated fluxes of HD 163296 and the corresponding spectral
energy distribution are given in Tab. 7.1 and shown in Fig. 7.6, respectively. In addition
to our measurements (full squares), observations at 7, 13, 36 and 61 mm are from Natta
et al. (2004 and references therein, empty circles) while observations at 0.75, 0.8, 0.85,
1.1 and 1.3 mm are from Mannings (1994, empty squares).

7.3 Observational results 129
Figure 7.5: Velocity-position plots along the plane of the disk for the C 12 O J=2-1 (upper panel),
the C 12 O J=3-2 (middle panel) and the C 13 O J=1-0 (lower panel) transitions. The angular offset
is measured with respect to the phase center of the observations corresponding to the position of
the central star. The contour levels are spaced by 2σ corresponding to 0.14 Jy/beam, 1 Jy/beam
and 0.06 Jy/beam respectively. The cross in the lower left of each panel gives the angular and
spectral resolution of the corresponding map. The thick solid lines marks the border where
emission is expected for a Keplerian disk inclined by 45 o and rotating around a 2.0 M⊙ point
source; the external and internal dashed lines correspond to stellar masses of 2.5 M⊙ and 1.5 M⊙
respectively. The horizontal and vertical straight dashed lines mark the systemic velocity (5.8
km/sec) and the position of the continuum peak.

130 The keplerian disk of HD 163296
F λ (mJy)
10000
1000
100
10
1
0.1
Free-free emission
1 10 100
λ (mm)
Figure 7.6: Spectral energy distribution at mm-cm wavelengths of HD 163296. Our new ob-
servations are shown with full squares, data from Mannings (1994) with empty squares and data
from Natta et al. (2004 and references therein) with empty circles. Observed fluxes at 36 mm
and 61 mm have been used to calculate the free-free contribution (dashed line) to the observed
flux. At 7 mm, the free-free subtracted flux are shown with crosses, while at shorter wave-
length the the free-free contribution is negligible. The resulting millimetric spectral indexαmm
(Fλ∝λ −αmm , see the solid line) is 3.0±0.1.
Assuming that the observed flux at 36 and 61 mm is dominated by free-free emission
from a wind, the free-free contribution at 7 mm corresponds to 1.2 mJy, while it is
negligible at shorter wavelengths.
After the subtraction of the free-free component, the millimeter spectral indexαmm
(Fλ∝λ −αmm ) calculated between 1 mm and 7 mm is 3.0±0.1, slightly higher that the
valueαmm= 2.6±0.2 obtained by Natta et al. (2004) using VLA and OVRO fluxes only.
7.3.4 Disk mass
Assuming that the dust emission is optically thin at millimeter wavelength, the measured
flux can be used to estimate the product of the disk total mass M times the dust opacity

7.4 Disk Models 131
kν, through the relation:
M· kν=
2 Fνd
, (7.2)
Bν(T)
where d is the source distance and T is the typical temperature of the emitting dust. At
millimeter wavelength, the dust opacity can by parametrized by a power low
kν= 0.01·(ν/229GHz) β cm 2 g −1 , (7.3)
where the normalization at 229 GHz (corresponding toλ=1.3 mm) assumes a dust/gas
mass ratio of 0.01 (Beckwith et al., 1990, 1991). Taking a characteristic gas temperature
for the outer disk of an early A star of 30K (Natta et al. 2000), the circumstellar material
has mass of 0.12 M⊙. This rough mass determination, that does not take into account the
presence of optically thick emission from the inner part of the disk, nor of temperature
gradients, will be discussed in more detail in Sec. 7.5.2.
With the assumption of optically thin emission, the slope of the dust opacity isβ=
α−2∼1, whereαis the spectral index obtained in the previous section.
7.4 Disk Models
The disk parameters derived in Sec. 7.3 under a number of very crude assumptions
can only provide order-of-magnitude estimates. A more quantitative analysis requires
to compare the observations to model predictions. We chose to perform the compari-
son using the observed visibilities (rather than reconstructed images), as discussed in
Sec. 2.4
This methodology requires to decide “a priori” which family of models is likely to
describe the observed object. In view of the results described in Sec. 7.3, we model
the millimeter emission of HD 163296 (continuum and CO lines) as coming from a cir-
cumstellar disk. We assume that the disk is heated by the stellar radiation only, and that
any viscous contribution can be neglected. This is very likely a good approximation,
given the relatively low accretion rate measured in HD 163296 (∼ 10 −7 M⊙/yr; Garcia

132 The keplerian disk of HD 163296
Lopez et al., 2006) and that the millimeter emission is dominated by the emission of
the outer disk, where the stellar heating is always dominant. The disk is in hydrostatic
equilibrium and flared; to start with, we consider that dust and gas are fully mixed, and
that no decoupling occurs. The mid plane temperature and disk scale height are com-
puted using the 2-layer approximation of Chiang and Goldreich (1997), as developed
by Dullemond et al. (2001). Similar models have been used in Testi et al. (2003) and
Natta et al. (2004), to analyze the (sub)millimeter emission of a number of Herbig Ae
stars. We refer to these papers for a more detailed description.
Once the stellar properties are known, the disk structure is completely characterized
by the following parameters: the disk mass (Md), the disk inner and outer radii (Rin and
Rd), the dependence of the surface density on radius (Σ∝r −p ) and the properties of
dust on the disk surface and mid plane. Note that some of these parameters are not well
constrained by the emission at millimeter wavelengths, which depends very little, e.g.,
on Rin and the surface dust properties. In addition, the observed emission depends on
the orientation of the disk with respect to the observer, which is characterized by the
inclinationι of the disk with respect to the line of sight (ι=0 for face-on disks) and the
position angle PA.
7.4.1 Continuum emission
The continuum emission at millimeter and submillimeter wavelengths is computed by
ray integration as in Dullemond et al. (2001). We describe the midplane dust opacity
at long wavelengths as a power-law of indexβ, withβafree parameter, as in Eq. 7.3.
The inner disk radius is the dust sublimation radius, as in the rim models of Isella et al.
(2006) for large (∼ 1µm) grains, and it is equal to 0.45 AU; the dust on the disk surface
is as in Natta et al. (2004). Neither of these two quantities is relevant for the following
analysis.

7.4 Disk Models 133
7.4.2 CO emission
The observed CO emission originates in the outer surface layers of the disk, at heights
that depend on the optical depth of the specific transition. Once the disk structure is
specified, as described above, one needs to compute, at each radius, the gas temperature
profile in the vertical direction. This is a complex problem, whose results depend on
a number of not well known properties, among them the X-ray field and the role of
very small grains in heating the gas (e.g., Dullemond et al. 2006). Therefore, we use a
parametric description assuming that for each CO transition the excitation temperature
is the same at all z and can be described as a power-law of r in the form:
Tline= Tline(r0)(r/r0) −q . (7.4)
The assumption of a constant excitation temperature along the vertical direction is cor-
rect for very optically thick lines, and/or if the velocity gradient along the line of sight
is very large. In both cases, the contribution at each wavelength from the line of sight
under consideration comes from a small region only, where the optical depth in the line
is of order unity. For any CO line, the height to which Tline(r) refers is therefore differ-
ent, depending on the density and velocity structure, as well as on the disk inclination
angle. The values of Tline(r0) and q for each CO transition are free parameters. A similar
procedure has been used by Dutrey et al. (1994) and Dartois et al. (2003) in their study
of the CO emission of T Tauri disks.
In the analysis of CO observations we have assumed 12 CO/H=7.0e-5 and 13 CO/H
= 1.0e-6, which correspond to an isotopomer ratio 12 CO/ 13 CO=70 (Beckwith and
Sargent, 1990; Dutrey et al., 1996; and reference therein)
As described in detail in the Appendix, we have developed a code which computes,
for each CO transition, the line intensity and profile as function of the disk parameters,
namely the inclination and PA, the surface density profile, the disk outer radius. In
addition, we assume that the disk is in keplerian rotation around the central star and
vary the stellar mass independently for each line.

134 The keplerian disk of HD 163296
Parameters Continuum 12 CO J=2-1 13 CO J=1-0
M⋆ (M⊙) 2.6 a 2.6 +0.3
−0.5 2.6±0.6
PA 120 o ± 20 o 128 o ± 5 o 130 o ± 8 o
Incl 40 o ± 12 o 45 o ± 5 o 50 o ± 8 o
Rout (AU) 200±15 550±50 500±80
p 0.81±0.01 0.6 +0.3
−0.1 1.0±0.5
Σ0 (10AU) (g/cm 2 ) 46±4 90±70 4 +12
−3
β 1.0±0.1 1.0 a 1.0 a
χ 2 r 2.6 1.17 1.15
Table 7.2: Parameter of the disk structure relative to the best fit models for the continuum and the
CO molecular emissions as described in Sec. 7.5. For each parameters uncertainties are given at
7.5 Results
7.5.1 Method of analysis
1σ level. a Fixed parameter.
The observations of HD 163296 have been analyzed by comparing the observed uv-
tables with synthetic uv-tables obtained from the models for the continuum and CO
emission described in Sec. 7.4. At each point in the synthetic uv-plane we have associ-
ated the corresponding weight in the real uv-plane. The analysis is thus independent of
image reconstruction and cleaning procedures, and takes into account the true sensitivity
of the three different interferometers.
We have in total seven sets of data corresponding to four continuum wavelengths
and three CO lines. For each set aχ 2 minimization has been performed exploring a
wide region of the space of model parameters. In all cases except the 12 CO J=3-2 line,
for which the analysis is still in progress, we find a single minimum ofχ 2 . For each

7.5 Results 135
Figure 7.7: Maps of the residuals relative to the best fit model for the continuum emission (see
Fig. 7.1 for the observations). The contour level are spaced by 3σ corresponding to 30 mJy at
0.87mm, 16 mJy at 1.3 mm, 3.3 mJy at 2.8 mm and 0.75mJy at 7 mm. The small boxes show
the relative synthesized beams.
parameter, the 1σ uncertainties are estimated asχ 2 1σ =χ2 m+ √ 2n, where n is the number
of degrees of freedom andχ 2 m is theχ2 value relative to the best fit model.
7.5.2 Continuum emission
The continuum data at the four observed wavelengths (0.85, 1.3, 2.7, 7 mm) have been
analyzed together, to estimate the best disk parameters and their uncertainty. We have
varied the two parameters which define the position of the disk on the plane of the sky,
namely the inclination and position angle, as well as the three physical parameters which
affect the continuum dust emission, namely the disk outer radius Rout, the slope p of the
surface density profile and the its valueΣ0 at 10 AU. Since the continuum emission
has a very weak dependence on the mass of the central star, we fixed M⋆=2.6 M⊙ (see
Sec. 7.5.3).
Tab. 7.2 shows the parameter values of the best fitting model; the residuals corre-
sponding to the four different wavelength are shown in Fig. 7.7. They have been recon-
structed from the residuals in the uv-plane with the same procedure used to obtain the
images in Fig. 7.1. Residual contours are generally lower than 3σ with the exception of
the 0.87 and 1.3 mm maps, where a flux asymmetry in the east half of the map (see also

136 The keplerian disk of HD 163296
Sec. 7.3.1) is clearly visible. This structure, not detected at longer wavelengths, requires
more resolved observations to be investigated in more details.
The position angle (120 o ±20 o ) and inclination (40 o ±12 o ) are in good agreement with
the rough estimate of Sec. 7.3.1. The position angle is not very well constrained by the
continuum data alone, due to the spatial resolution generally larger than 1 ′′ and to the
very elongated beam at 1.3 mm; within the uncertainty, it is in agreement with previous
estimates in the literature. On the contrary, the inclination is significantly lower than the
∼60 o obtained by Mannings and Sargent (1997) using less resolved OVRO observations,
and by Grady et al. (2000), analysing HST images in scattered light. In both cases the
uncertainties may be very large and it is not clear how significant is the discrepancy.
The surface density radial profile has a slope p=0.81±0.01 and the disk outer
radius is Rout= 200±15 AU. The very good constraints on both parameters is due to
the favourable orientation of the beam at 1.3 mm with the maximum resolution (0.42 ′′ )
in almost the same direction of the major axis of the disk. Note that larger values of Rout
are not consistent with the data, irregardless of the value of p.
The slope of the dust opacity derived from the model fit isβ=1.0±0.1. The derived
value ofβconfirms the presence of very large grains in the HD 163296 circumstellar
disk (Natta et al. 2004).
The disk total mass (for the adopted opacity given in Eq. 7.3) is 0.05±0.01 M⊙,
equal to the value obtained by Natta et al. (2004) and Mannings & Sargent (1997), but
more than twice lower than the crude estimate of Sec. 7.3.4. Note that the disk mass is
computed assuming that the surface density profile derived from the millimeter data at
large radii extends smoothly inward to the dust sublimation radius. Given the low value
of p, this is not a strong assumption (most of the mass is at large radii), but we cannot
rule out that more mass than we estimate is contained in the optically thick inner disk
regions at r< ∼ 10 AU.

7.5 Results 137
Figure 7.8: Comparison between the observed and the model predicted 12 CO J=2-1 emis-
sion. The upper panel shows the position-velocity diagram for the 12 CO J=2-1 transition (as in
Fig. 7.5). The lower panel shows the residuals relative to the best fit model parameters reported
7.5.3 CO emission
in Tab. 7.2. The contour levels are at 2σ as in Fig. 7.5.
The analysis of the CO emission has been carried out separately for the different CO
transitions and the corresponding best fit parameters are given in Tab. 7.2. At the mo-
ment, the observations of the 12 CO J=3-2 emission are still under analysis. However, we
think that these data will not contribute to a better constraint of the disk structure, given
the low spatial resolution of the SMA observations As for the continuum, the solutions
of the model fitting are unique with the reducedχ 2 close to 1.
The observed 12 CO J=2-1 emission is well fitted by a keplerian disk orbiting a cen-
tral star with mass M⋆ = 2.6 +0.3
−0.5 M⊙. Fig. 7.8 shows the position-velocity residuals
obtained subtracting the best fit model from the observed uv–table: no evidence of non-

138 The keplerian disk of HD 163296
keplerian rotation or stellar outflow is detected, within the actual instrumental sensitiv-
ity. Both the position angle (128 o ±5 o ) and the inclination (45 o ± 5 o ) are in agreement
with the values obtained from the continuum. Since the line is optically thick, the con-
straint on the gas surface density is poor with p=0.6 +0.3
0.1 andΣ0= 90±70 g/cm 2 . The
inferred outer radius of the disk is 550± 50 AU, more than three times larger than the
value obtained by the continuum and similar to the result obtained by Thi et al. (2004).
The disk outer radius will be discussed in detail in Sec. 7.6.1.
The model fit to the 13 CO J=1-0 line gives results consistent with those obtained
from the 12 CO J=2-1 line with exception of the value ofΣ0 which is significantly
smaller. This discrepancy may be due to a depletion of the 13 CO and will be discussed
in Sec. 7.6.2. In general, the parameters constraint obtained from the 13 CO is worse, as
expected given the lower resolution of the observations.
The radial temperature profile of the 12CO has a slope q=0.5 +0.2
−0.1 and a value of
40 +2
−5 K at 100AU. The radial temperature profile of the 13CO has a slope q=0.8±0.4
and a value of 30±10 K at 100AU. We will comment on the gas physical conditions in
Sec. 7.6.
7.6 Discussion
The results presented in Sec. 7.5 show that all the observations are consistent with the
emission of a circumstellar disk in keplerian rotation around a star of 2.6 +0.2
−0.4 M⊙, assuming
the Hipparcos distance of 122 pc. Within the error, the stellar mass is in agreement
with the value of 2.3 M⊙ (Natta et al., 2004) obtained from the location of the star on
the HR diagram, using Palla and Stahler (1993) evolutionary tracks; the corresponding
stellar age is of about 5 Myr. The good keplerian rotation pattern and the low disk mass
seem also to indicate that the disk self-gravity may have small effects on the HD 163296
disk structure and evolution, while deviations from keplerian rotation can be present in
more massive disks (Lodato and Bertin, 2003; Cesaroni et al., 2005)
The disk has a moderate inclination with respect to the line of sight (45 o ±4 o ) with a

7.6 Discussion 139
Rout
700
600
500
400
300
200
CO
Continuum
0.5 0.6 0.7 0.8 0.9 1
Figure 7.9:χ 2 contours in the plane p–Rout relative to the best fit models for the continuum and
the 12 CO J=2–1 observations, as labelled in figure. The contours correspond to uncertainties of
1, 3 and 10σ, whereχ 2 1σ =χ2 m +√ 2n andχ 2 m
p
is the minimun value corresponding to the disk
parameters reported in Tab. 7.2.
position angle of 128 o ±4 o . Both the dust continuum emission and the CO lines indicate
a rather shallow surface density profile,Σ∝r −0.81±0.01 . The discrepancies observed in
the disk outer radius andΣ0 will be discussed in the following.
7.6.1 The disk outer radius
The model fitting (Tab. 7.2) shows that the value of disk outer radius inferred from
the continuum dust emission is three times smaller than the value obtained from the

140 The keplerian disk of HD 163296
CO analysis. Since our method takes into account the sensitivity limits of the different
interferometric observations, this discrepancy can not be explained by the fact that the
disk outer regions have a continuum surface brightness below the sensitivity limit. In
other words, if we extend the disk model that fit the continuum to the outer radius of the
CO, we predict a continuum emission between 200 AU and 550 AU which should be
easily detected in our observations. This point is illustrated in Fig. 7.6.1, which shows
theχ 2 contours relative to 1, 3 and 10σ in the p-Rout plane for the 1.3 mm continuum
and the 12 CO J=2-1 line. The two regions are clearly separated even at the 10σ level
and no variation of the other model parameters can reduce the discrepancy.
To reconcile CO and dust observations, we find that it is necessary to introduce a
drop in the the continuum flux of a factor>30 at a distance of about 200 AU. With this
drop, the millimeter fluxes at larger r will be below the actual instrumental sensitivity
and will be lost in the observational noise. What can be the origin of such a drop?
In the optically thin regime, the continuum flux emitted at distance r from the star
depends on the mass surface densityΣ(r), the dust/gas ratioΠ(r), the dust opacity Kν(r)
and the midplane dust temperature T(r) through the relation
Fν(r)∝Σ(r)·Π(r)· Kν(r)·T(r). (7.5)
It is possible that a different disk geometry (e.g., a lower flaring angle) gives a mid-
plane dust temperature lower than the values predicted by our disk model. A lower
temperature limit, of∼10 K, is however imposed by the equilibrium with the interstellar
radiation field. Given that our models predict temperatures of 20-30 K in the millimeter
emitting regions, one can reduce Fν by a factor 3 at most. Moreover, it is difficult to
see how the temperature can have a large discontinuity at 200 AU. In any case, we are
exploring the effects of varying the degree of flaring on the model predictions.
A second possibility to explain the observed flux depletion is that the dust opacity
Kν(r) at distance larger than 200 AU is much lower because most of the grains have
grown into bodies large than about 1 m. This, however, contradicts theoretical calcu-
lations which predict that meter size objects, wherever they form, should drift inwards
with a velocity of about 1 AU/century (see the review of Dominik et al., 2006)

7.6 Discussion 141
Finally, a more likely possibility is that the dust is depleted in the outer disk due to
dynamical perturbation between the disk and an unknown companion orbiting the star
at large distance. Such a body may also account for the asymmetric shape detected in
the continuum emission (Fig. 7.7). The presence of a giant planet, or a brown dwarf,
orbiting in the outer disk of HD 163296 has also be suggested by Grady et al. (2000),
in order to explain the dark line observed in the scattered light HST images between
300 AU and 350 AU from the central star. While planets are invoked to explain the
large inner gaps observed in the dust distribution in the so called “transitional disks”
(i.e., Calvet et al. 2005), HD 163296 will be, if confirmed by future observations, the
first case in which a sub-stellar mass companion is found in the outer disk of a pre-main
sequence star.
7.6.2 The gaseous disk
The values ofΣ0 reported in Tab. 7.2 indicate that the 13 CO J=1–0 emission is consistent
with a ratio 13 CO/H∼ 10 −7 , about a factor 10 lower than what found in molecular
clouds. We note that a similar depletion in the 13 CO abundance seems to be common in
a number of observed TTS (see Dutrey et al., 1994, 1996). Since the 13 CO J=1–0 line
is more optically thin than the 12 CO J=2–1, it probes the gas conditions deeper into the
disk. Therefore, a possible explanation may be that part of the 13 CO is condensated onto
dust grains in the colder disk regions close to the disk midplane. A second possibility,
recently discussed by Jonkheid et al. (2006), is that the ratio 12 CO/ 13 CO may depend on
on the dust settling. More quantitative tests about these hypotesys are in progress.
Our models show that both the 12 CO and 13 CO lines are optically thick in the
HD 163296 disk. The surface density is therefore poorly constrained and the mass
of the gaseous disk is uncertain by more than one order of magnitude, in the interval
0.01-0.8 M⊙. Extrapolating the gas surface density obtained from the continuum dust
emission to the CO radius, we obtain a disk mass of 0.2 M⊙, about 8% of the stellar
mass and four times higher that the disk mass obtained from the dust emission. If this
is correct, HD 163296 has a rather large and massive disk and self-gravity may produce

142 The keplerian disk of HD 163296
significant effects in the outher and colder regions of the disk. Note that this estimate
relies on the assumed 1.3 mm opacity value, which sets the normalization of the surface
density profile at r∼10 AU, i.e., where the 1.3 millimeter dust continuum emission
becomes optically thin. Also in this case, a more detailed analisys of the results is still
in progress.
7.7 Summary and conclusions
This Chapter presents new observations of the disk of HD 163296 in the dust continuum
and CO lines obtained with the VLA (7 mm continuum), PBI (1.3 mm and 2.8 mm
continuum, 12 CO J=2-1 and 13 CO J=1-0 lines) and SMA (0.87 mm continuum and
12 CO J=3-2 line). The disk is well resolved in both lines and continuum.
We have compared the observations to the predictions of self-consistent disk models.
We find that the disk, as seen in CO lines, is very large (R = 540±40 AU), with
a keplerian rotation pattern consistent with a central mass of 2.6 +0.2
−0.4 M⊙. Within the
observational errors, there is no evidence of non-keplerian motions and/or significant
turbulent broadening. We obtain a disk inclination of 45 o ±4 o , lower than the value
of∼60 o found in literature, while the position angle of 128 o ±4 o is in agreement with
previous results.
The dust opacity has a power law dependence on wavelengthκ ∝λ −β withβ=
1.0±0.1 in the interval 0.87-7 mm. This value is similar to what has been measured in
a number of spatially resolved disks of HAe and TTS (e.g., Natta et al. 2004, Rodmann
et al. 2006), and is very likely an indication that the bulk of the solid material in these
disks has coagulated into very large bodies, of millimeter and centimeter size (Natta et
al. 2006). Within the accuracy of our data, we find no significant variation ofβwith r.
The continuum observations constrain the surface density profile (Σ∝r 0.81±0.01 ) for
r

7.7 Summary and conclusions 143
A comparison of the disk properties derived from the dust continuum and the CO
lines at r

144 The keplerian disk of HD 163296
Appendix
CO emission
In order to analyze the CO emission we developed a numerical code that solve the
general formulation of the radiation transfer equation along each direction between the
observer and the emitting source. If s is the linear coordinate along the line of sight,
increasing from the observer (s≡0) towards the source, the observed emission in each
direction is given by the relation
Iν=
∞
0
Sν(s)e −τν (s)Kν(s)ds, (7.6)
where the optical depth is defined in each point s through
τν(s)=
s
0
Kν(s ′ )ds ′
(7.7)
and Kν is the absorbing coefficient of the interstellar medium. Given the high gas densi-
ties in the pre-main sequence circumstellar disks, we can assume that all the CO levels
corresponding to the rotational transitions under investigation are thermalized. In this
case the source function can be approximated by the Planck function
Sν(s)= Bν(TCO(s))= 2hν3
c 2
1
, (7.8)
exp(hν/kTCO(s))−1
depending only on the local temperature of the gas TCO (c, h and k are respectively the
light speed, the Plank and Boltzman constant).
The absorbing coefficient of the circumstellar medium is due both to gaseous CO
and dust: Kν(s)=K CO
ν (s)+K d ν (s). For the dust Kd ν (s)=ρ(s)·kν, whereρ(s) is the local
density of the circumstellar material (gas+dust) and kν is the dust absorbing coefficient
for total mass unit given in Eq. 7.3. The CO absorbing coefficient is given by the relation
K CO
ν (s)=nl(s)·σν(s) (7.9)
where nl(s) is the total number of CO molecules at the lower level l of the transition and
σν(s) is the CO absorbing cross section.

7.7 Summary and conclusions 145
Figure 7.10: Schematic representation of the frame of reference adopted to calculate the CO
emission arising from a rotating disk. The disk mid plane and the observer lie respectively on
the (x, y) and (y, z) planes;θ is the disk inclination; d is the distance between the observer and
the central star; s is the linear coordinate along the line of sight increasing from the observer
(s≡0) towards the emitting source. Assuming that the material within the disk is subject to the
keplerian rotation around the central star, we call vφ the velocity of a mass element at distance r
and vk the projection of vφ along the line of sight.
Calling m0 andχCO the mean molecular weight of the gas and the fraction of CO
present in the gas respectively, the number of molecules nl(s) is given by the Boltzman
equation
ρ(s)
nl(s)=χCO
m0
· gl e−El/kTCO(s) , (7.10)
Z(TCO(s))
where gl= 2l+1 is the statistical weight of the lower level l of the transition, El=
(1/2)l(l+1)kT1 is the level energy, T1 is the temperature equivalent to the transition
energy and Z(TCO(s)) is the partition function at the gas temperature TCO(s). Following
Beckwith and Sargent (1993, and references therein), the absorbing cross sectionσν(s)
can be expressed in term of the integrated cross section of the transitionσ0 through the
relation
σν(s)=σ0·φν(s)·(1−e −hν/kTCO(s) ), (7.11)

146 The keplerian disk of HD 163296
whereφν(s) is the intrinsic line profile
φν(s)= c
1/2
mCO
mCO
· exp
2πkTCO(s) 2kTCO(s) ·∆2
v,
(7.12)
and
ν0
σ0= 8π3 kT1
h 2 c
(l+1) 2
2l+1 µ2 . (7.13)
In the previous equations, mCO is the CO molecular weight,ν0 is the rest frequency of
the molecular transition,∆v is the difference between the velocity vobs= (c/ν0)(ν−ν0),
corresponding to the frequencyν, and the component of the gas velocity along the line
of sight, vk(s),µis the dipole moment of the CO molecule. Note that writing Eq. 7.12,
we assumed that the intrinsic line width depends only on the thermal velocity dispersion
in the gas and that turbulent motion are negligible.
As shown in Fig. 7.5 and described in Sec. 7.3.2, the observed velocity patterns in
the CO transitions are in very good agreement with the keplerian rotation of the disk.
We can thus assume that gas moves on circular orbits around the star characterized by a
tangential velocity
GM⋆
vφ(r)= , (7.14)
r
where r is the radius of the orbit and M⋆ is the stellar mass. To calculate the velocity
vk(s), component of the tangential velocity vφ(r) along the line of sight, it is useful to
define a coordinates system centered on the star as shown in Fig. 7.10: the (x, y) plane
corresponds to the disk mid plane; the observer lies in the (y, z) plane and its position
it is defined by the inclinationθand the distance d from the star; each point of the
circumstellar space can be defined through the cylindrical coordinates r= x 2 + y 2 ,
φ=arctan (y/x) and z. Since d≫r, we can write
vk(s)vφ(r)·cosφ·sinθ. (7.15)
Finally, the velocity vk(s) can be calculated for each direction, knowing the geometrical
transformations between the coordinate s along the line of sight and the cylindrical
coordinates r andφ.

7.7 Summary and conclusions 147
In order to solve the described set of equations, we thus need an expression for the
circumstellar mass densityρ(s) and the temperature of the emitting gas TCO(s). In both
cases we can assume the cylindrical symmetry and writeρ(s)≡ρ(r, z) and TCO(s)≡
TCO(r).
For the emitting gas temperature, we choose the parametrization
TCO(r)=TCO(r0)(r/r0) −q , (7.16)
while the mass density is calculated assuming that the disk is in hydrostatic equilibrium
and vertically isothermal in the inner region at the mid plane temperature Tm(r), through
the relation
ρ(r, z)=ρ0(r)·e −z2 /2h 2 (r) , (7.17)
valid between the disk inner and outer radii Rin and Rout. The density on the disk mid
planeρ0(r) can by expressed as function of the surface mass densityΣ(r)=Σ0(r/r0) −p
through the relation
ρ0(r)=
Σ(r)
√ 2πh(r) . (7.18)
Finally, the pressure scale h(r) is given by the relation
2r
h(r)=
3kTm(r) , (7.19)
GM⋆m0
where Tm(r) is obtained solving the structure of a stellar-irradiated passive disk as de-
scribed in the previous section.
Note that the emitting gas temperature TCO has been parametrized independently of
the disk mid-plane temperature Tm, which governs the density structure of the disk. As
pointed out by Dartois et al. (2003), the 12 CO J=2-1 transition is in fact a good tracer
of the “CO disk surface” where the gas temperature is very different from the disk mid
plane.
The resulting model, produces brightness maps for each frequency (velocity) that
can be compared with the observations presented in Fig. 7.2 and 7.4.

Part V
Conclusions and future prospects

CHAPTER 8
Summary and Conclusions
This thesis discusses a study of circumstellar disks based on high spatial resolution in-
terferometric observations of intermediate (Herbig Ae; HAe) and low mass (T Tauri;
TTS) pre-main sequence stars. The aim was to improve our knowledge of the structure
and evolution of the circumstellar material by determining the composition, the size,
the density distribution, the temperature and the kinematic of gas and dust. The thesis
has been developed following two complementary lines: the characterization of disk re-
gions at fractions of AU from the central star through near-infrared interferometry (Part
II and III) and the study of the outer disk regions through millimeter interferometry (Part
IV). In order to analyse the observations, I developed physical models of the disk struc-
ture and emission, realized their numerical implementations and compared the model
predictions with the data. The main results are summarized in the following.
8.1 The inner disk
The physical definition and the numerical implementation of models for the near-infrared
emission arising from the circumstellar dust is described in Chapter 3. Following the
idea of Natta et al. (2001), we confirm that the spectral energy distribution between

152 Summary and Conclusions
∼1 and∼5µm of most of the observed HAe and TTS is dominated by the puffed-up
inner rim, which forms at the dust evaporation radius. We argue that the rim shape
is controlled by the large vertical density gradient through the dependence of the dust
evaporation temperature on the surrounding gas density. The dust density (and the cor-
responding evaporation temperature) is maximum on the disk mid-plane and decreases
in the vertical direction. As a result, the bright side of the rim is curved, rather than
vertical, as expected when a constant evaporation temperature is assumed. Contrary to
the vertical rim model proposed by Dullemond et al. (2001), the near-infrared excess of
our model does not depend much on the disk inclination, being maximum for face-on
objects. For a given star, the location of the inner rim depends mostly on the grain size
of the dust species with the highest evaporation temperature. For astronomical silicates,
the rim radius is maximum for grains of∼0.01µm and it reaches a minimum value for
grains of∼ 1µm (assuming a typical HAe star with L⋆= 50L⊙ and Teff=10000 K). In
this range of grain sizes the dust temperature is maximum on the evaporation surface
and decreases rapidly moving inside the disk, i.e., with increasing optical depth. As a
consequence, the rim has a very narrow optically thin atmosphere, which, for a typi-
cal Herbig Ae star, has width∼ 10 −6 AU. On the contrary, grains larger than∼ 1µm
are mainly heated by the radiation field diffused by the disk its-self; their temperature
increases with increasing optical depth and the dust evaporation occurs in a relatively
wide optically thin atmosphere (∆r∼10 −2 AU). The consequences of this effect have
been discussed further in Chapter 5.
For a large number of models, we have computed synthetic images of the curved rim
seen under different inclinations. Face-on rims are always seen as bright centrally sym-
metric rings, while the images become more skewed as the inclination increases. The
comparison with the interferometric observations of six pre-main sequence stars (CQ
Tau, VV Ser, MWC 480, MWC 758, V1295 Aql and AB Aur) is discussed in Chap-
ter 4. For each star, we performed aχ 2 test between the measured and the theoretical
visibilities varying independently the model parameters, namely the dust size, the disk
inclination and position angle. In all objects with the exception of AB Aur, our self-

8.1 The inner disk 153
consistent model reproduce both the interferometric results and the near-infrared spec-
tral energy distribution. In four cases we found that silicate grains larger than∼ 1µm
are either required by or consistent with the data. Only in the case of MWC 480 grains
of about 0.2–0.3µm are required to explain the large disk inner radius. Our analysis
shows that the dust present in the disk mid plane within 1 AU from the star has grown
from the ISM size (∼0.01µm) to micron, or larger, radii.
In the case of AB Aur, our model can not reproduce the available observations,
which require an inner disk radius too small to be due to dust evaporation in the unatten-
uated stellar radiation field. As a possible explanation, we suggest that the gas present in
the dust-depleted inner region absorbs a significant fraction of the stellar radiation, pre-
venting the dust grain from the evaporation. One should also keep in mind that AB Aur
is a very complex object. Recently, it has been the target of a large number of interfero-
metric observations with the Infrared Optical Telescope Array (IOTA) by Millan-Gabet
et al. (2006). The measured visibilities and closure phases indicate the presence of an
asymmetry in the circumstellar environment, probably due to a compact thermal emis-
sion at spatial scales of 1-4 AU. Moreover, both near-infrared coronographic images
(Fukagawa et al. 2004) and millimeter interferometric observations (Piétu et al., 2005)
have shown the existence of spiral density waves on scales of few tens of AU from the
central star. All these results indicate a complex morphology of the material close to
AB Aur which requires to be investigated further.
The third part of the thesis (Chapter 6) is dedicated to the analysis of spectrally
resolved AMBER/VLTI interferometric observations of the Brγ emission line of the
HAe star HD 104237. The line is quite strong in this object, with a peak intensity of 35%
of the continuum. The data show that the visibility does not vary between the continuum
and the line. This implies that the line and the continuum emission arise at roughly the
same distance from the central star. We assume that the near-infrared continuum excess
originates from the puffed-up inner rim of the circumstellar disk and discuss the origin
of the Brγ line. We conclude that this emission most likely comes from the base of a
compact disk wind localized between 0.2–0.5 AU from the star, just in front of the dusty

154 Summary and Conclusions
disk inner rim. Our conclusions are in contrast with the magnetospherical accretion
model (Hartmann et al. 1998), which predicts that the hydrogen lines originate in the
gas accretion flow at few stellar radii from the central star. Our results, if confirmed
by future observations, will thus bring a new insight into the processes that govern the
accretion of material on the central star during the HAe disks evolution.
8.2 The outer disk
The last part of the thesis (Chapter 7) is dedicated to the analysis of the observations
of the HAe star HD 163296, obtained with different interferometers (Plateau de Bure,
SMA and VLA) in the wavelength interval between 0.87 and 7 mm. The observations
show the presence of a spatially resolved disk both in the continuum (0.87, 1.3 and
2.8 mm) and in the CO lines ( 12 CO J=3–2 and J=2–1, 13 CO J=1-0). Comparing the
measured visibilities with the predictions of self-consistent disk models we obtain a
disk inclination of 45 o ±4 o and a position angle of 128 o ±4 o . The exponentβof the dust
opacity (κ∝λ −β ) between 0.87 and 7 mm isβ=1.0±0.1 and indicates the presence of
very large dust grains, of millimeter and centimeter size (Natta et al. 2006).
Seen in the CO lines, the disk has an outer radius of 540±40 AU and is characterized
by a keplerian rotation pattern consistent with a central star of 2.6 +0.2
−0.4 M⊙, in agreement
with the value of 2.3 M⊙ obtained by the stellar position on the HR diagram. Within the
observational errors, there is no evidence of non-keplerian motions and/or significant
turbulent broadening.
The continuum observations constrain the surface density profile (Σ∝r 0.81±0.01 ) for
r

8.3 Conclusions 155
to explain scattered light HST observations. HD 163296 is the first disk with evidence
of a big discontinuity in the millimetric dust emission at large radii, and it is clearly
worth it to investigate it further (see Sec. 9.2).
8.3 Conclusions
In this thesis I have shown that astronomical interferometry is a powerful technique
to investigate the properties of circumstellar disks around nearby pre-main sequence
stars. Near-infrared interferometric observations, albeit still challenging and telescope–
time consuming, probe the gas and dust distribution at fractions of AU from the central
star and allow the investigation of important physical processes responsible for the disk
evolution and the planet formation, namely the grain growth, the gas accretion on the
central star and the launching mechanism of the observed outflows. On the other hand,
long baseline millimeter interferometry remains a fundamental technique to study the
gas kinematic and the dust properties in the outer disk regions, on scales of tens of AU.
Due mainly to the limitations of the available near-infrared interferometers, direct
observations of the inner disk have been limited so far to the few brightest objects and
it is not known if the inferred properties are common to most of the pre-main sequence
stars. In Chapter 9 I will discuss some ideas for future investigations.
Finally, I would like to note that the self-consistent disk models, developed and dis-
cussed in my thesis, have explored the importance of physical processes in circumstellar
disk not considered before. Some follow-up works, which have appeared in the litera-
ture after the publication of the Isella and Natta (2005) paper, have been briefly reviewed
in Chapter 5.

CHAPTER 9
Future developments
As my thesis project developed, there were more open questions that attracted my atten-
tion. In some cases, they become proposals for observing time, which, however, have
not been implemented yet. I describe briefly these future developments in the following.
9.1 Gas in the inner disk of Herbig Ae stars
Even if the dust evaporation is responsible for the dusty disk truncation at fraction of
AU from the central star, about 99% of the circumstellar material is settled in a gaseous
disk that may extend inward until the stellar radius. If the central star has a magnetic
field, the gaseous disk may be truncated at few stellar radii and the gas captured by the
magnetic field lines, giving rise to magnetospheric accretion columns. The accreting
gas eventually hits the star, producing accretion shocks on the stellar surface. This
magnetospherical accretion model has been successfully applied to classical TTS, where
strong magnetic fields have been measured (see the review of Bouvier et al., 2006 at
PPV).
The situation is more confused in the case of HAe stars since the strength of the stel-
lar magnetic field is very uncertain (Hubrig et al., 2006) and stellar winds can strongly

158 Future developments
interact with the accreting gas. In the relatively large dust depleted inner region that
characterizes HAe stars (R∼0.5 AU), the gas density and temperature are correlated
with the mass accretion rate ˙M: the higher is ˙M, the warmer and denser is the gas. For
˙M∼ 10 −7 M⊙/yr−10 −8 M⊙/yr, the gas temperature in the dust depleted inner disk ranges
between∼1500 K, at the dust evaporation distance, and∼4000 K very close to the star
(Muzerolle et al., 2004). In this range of temperatures, CO and water should exist. Wa-
ter vapor in particular should be abundant since it is the second molecular reservoir of
oxygen after CO in the gas phase at temperatures below the dissociation temperature
(∼ 2500 K) and above the water-ice sublimation temperature (∼ 150 K). Between 1000
K and 2500 K, H2O is a strong molecular coolant, as well as the dominant source of
infrared opacity in the case in which dust grains are not present (Najita et al., 2000).
The presence of warm molecular gas in the circumstellar environment has been es-
tablished both for low and high mass young stellar objects. The fundamental CO emis-
sion at 4.6µm has been frequently detected in TTS (Najita et al., 2003), where it has
been modelled as coming from the warm gas (∼1000 K) on the dusty disk surface. The
CO overtone emission (bandhead at 2.3µm) requires on the other hand the presence
of gas at 2000 K–4000 K, it is weaker than the fundamental one and it appears more
commonly among higher luminosity objects. In TTS, it has been detected only in few
percent (∼ 6%) of the observed stars (Najita et al., 2003). High spectral resolution ob-
servations of the CO overtone emission show in most cases symmetric double-peaked
profiles, strongly supporting the idea that the lines originate in the inner part (∼0.3AU)
of a rotating disk (Najita et al., 2006 and references therein). The absence of red or
blue-shifted absorption similar to what seen in the Balmer line of TTS, seems to rule
out the possibility that the lines are emitted in winds or funnel flows. High resolution
spectra provide a strong evidence of the rotation of the inner disk and are crucial to un-
derstand how the gas accretes on the central star. Like the CO overtone transitions, the
ro-vibrational transitions of water are also expected to probe the high density conditions
in disks. Emission from hot water (see Fig. 9.1) has been detected in the near-infrared in
a few stars (both low and high mass) that also show CO overtone emission (Carr et al.,

9.1 Gas in the inner disk of Herbig Ae stars 159
Figure 9.1: Adapted from Thi & Bik (2005). ISAAC medium resolution (R=8900) spectrum of
08576nr292. The lower panel shows the observed spectrum. The box marks the region of the
spectrum where the steam features are found. The top panel shows the blow-up of this region.
The upper histogram is the continuum subtracted observed flux, normalized to the maximum
strength at∼ 2.280µm. The middle one shows the best fit discussed in the paper; the lower curve
is the synthetic spectrum before degradation to the resolution of the observed spectrum. The
strong detected lines are labeled 1 to 10. Solid bars in the lower part of each panel show the
spectral intervals that can be covered with CRIRES observations.

160 Future developments
2004; Najita et al., 2000; Thi and Bik, 2005). Emission from steam (2.280µm-2.293µm)
has been reported in disks around the low-mass young star SVS 13 (Carr et al., 2004),
and the strongly accreting TTS DG Tau (Najita et al., 2000). In velocity–resolved spec-
tra the water lines are much narrower than the CO lines: this is consistent with both
water and CO emission originating in a keplerian rotating disk with a decreasing radial
temperature profile. Given the lower dissociation temperature of water (2500 K) com-
pared to CO (4000 K), CO is expected to extend inward to smaller radii than water, i.e.,
to higher velocities and temperatures. Spectral synthesis modeling of the detected CO
and water steam features shows, however, that the water abundances relative to the CO
is a factor 2-10 below the predictions of chemical equilibrium (Carr et al., 2004; Thi
and Bik, 2005). This has at the moment no clear explanation but suggests that the UV
field and/or X rays may play an important role in the gas chemistry.
Since these results have been obtained on a very small sample of objects, we have
recently proposed a study of a sample of 6 HAe stars using the high resolution spectro-
graph CRIRES (R=30000) at the VLT in order to detect both the ro-vibrational steam
emission and the overtone CO emission.
For a typical HAe star (L⋆= 50L⊙) the dust depleted inner disk has a radius about
10 times larger than for a TTS (L⋆=0.5⊙). The corresponding gas emitting area is thus a
factor 100 larger than for TTS. The total mass of the emitting gas is also correspondingly
higher. Both effects should make the observation of CO and steam emission easier in
HAe than in TTS, making HAe stars the best choice to study the gas in the inner disk.
Our 6 target stars are part of a large sample of 35 HAe stars for which the mass
accretion rate has been derived from the Brγ luminosity (Garcia Lopez et al., 2006).
They are characterized by similar mass accretion rate, i.e. similar density of the gaseous
inner disk, but different value of the UV radiation field (see Tab. 9.1). For all the stars
the UV flux inside the dust-depleted inner disk is due only to the stellar radiation (for
˙M∼10 −7 the accretion UV flux is negligible) and increases of more than three orders
of magnitude from the later to the earlier stellar spectral type; the X ray radiation is also
likely negligible compared to the UV one. In this way we hope to study the chemistry of

9.2 Probing planet formation through interferometric observations 161
Name SpT L⋆/L⊙ lg ˙M (a) Revp(AU) (b) G(Revp) (c) G(0.5AU) (c)
HD 142527 F6 70 -7.2 0.5 2.2e+7 2.2e+7
HD 144432 F0 23 -7.1 0.4 7.1e+8 4.5e+8
HD 169142 A5 15 -7.4 0.2 2.7e+9 4.3e+8
HD 179218 B9 500 -6.7 1.2 9.6e+9 5.5e+10
HD 163296 A1 36 -7.1 0.3 1.5e+10 5.e+9
VV Ser A2 63 -6.5 0.4 3.1e+10 2.0e+10
Table 9.1: (a) The mass accretion rates are calculated from the Brγ line intensity (Garcia Lopez
et al., 2006); (b) dust evaporation distance calculated as described in Isella et al. (2006); (c)
integrated UV-flux (912Å

162 Future developments
tions of the dust emission have shown that the dust undergoes drastic changes during
the disk evolution. In particular, starting from a dimension of∼ 0.01µm, typical of the
interstellar medium, dust grows many orders of magnitude to form the centimeter grains
observed around low (T Tauri; TTS) and intermediate (Herbig Ae; HAe) pre-main se-
quence stars (see the review of Natta et al. (2006) and the discussion in Chapter 7).
While growing in mass and size, particles start to decouple from the gas and sediment
on the disk mid-plane. For bodies larger than about 1 km, self-gravity dominates and
the orbital dynamic is no more affected by the gas drag. Called planetesimals, these
particles move on keplerian orbits around the central star, do not participate to the in-
ward drift of the smaller material and can attract each other through their mutual gravity,
aiding further growth into planet-size objects. While this core-accretion scenario (see
the review of Lissauer and Stevenson, 2006) is a widely accepted theory of planet for-
mation, no clear observational confirmation exists so far.
Information about planet formation can be obtained through the analysis of the
known exoplanets. Starting from 1995, more than 170 planetary systems have been
discovered around close-by main sequence stars 1 ; the corresponding fraction of stars
with planets is about 6% (Udry et al., 2006). Most of them, detected through radial
velocity measurements, host giant planets (M> 0.02M jup) orbiting inside∼ 4 AU from
stars of late F, G and early K spectral types. Only in four cases (Chauvin et al., 2005;
Guenther et al., 2005; Biller et al., 2006), all discovered through direct images, plan-
ets have been found to orbit at distances larger than 100 AU. Due to the strong biases
that affect the detection techniques, these number are poorly representative of the initial
phase of planetary formation. Nevertheless, they raise many questions: is the small frac-
tion of discovered planetary systems real? If yes, how can it be related with the common
presence of circumstellar disks around pre-main sequence stars? Are the circumstellar
disks all proto-planetary disks? And again, where do planets form in disks? How do
they interact with the surrounding material? Is the planetary formation responsible for
the disk disappearance? Albeit challenging, investigation of these questions through
1 from “the Extrasolar Planets Encyclopedia” at http://exoplanet.eu/

9.2 Probing planet formation through interferometric observations 163
observations of pre-main sequence circumstellar disks is extremely important.
At infrared and millimeter wavelengths, planetesimals and proto-planets behave like
black bodies characterized by a temperature between tens to hundreds degrees, depend-
ing on the distance from the central star. Their total emitting area is so small that the
corresponding thermal emission is well below the actual instrumental sensitivity and
overwhelmed by the surrounding gas and dust. As a result, the direct detection of plan-
ets around pre-main stars is actually impossible. However, we can infer their presence
through the observation of the dynamical perturbations that massive bodies induce both
on the material surrounding them and on the global disk structure.
Examples of global perturbations are constituted by the so called transitional disks in
which a planet may be responsible for the clearing out of the inner disk. Planets may also
be responsible for circular gaps in the gas distribution observed in few HAe and TTS
(Calvet et al., 2005). Moreover, the presence of a planetary companion is suggested in
the case of HD 163296 by our millimetric observations (Chapter 7) and by HST images
(Grady et al., 2000). Local perturbations induced by a forming planet can be observed
in the form of small scale enhancements of the millimetric flux, corresponding to a local
increase of the dust density. Such structures have been observed in the HAe star AB Aur
(Corder et al., 2005; Piétu et al. 2005). For HD 163296 we also detected asymmetry in
the millimeter flux compatible with dynamical perturbations by an unknown companion
(Chapter 7).
Due to the spatial resolution of existing millimeter interferometers and adaptive op-
tics systems, all these observations refer to disk regions at distances larger than about
30 AU from the central star. In apparent contradiction with the discussed observations,
planetary formation theories suggests that planets should form inside about∼20 AU,
where the gas density is larger, and then eventually migrate outward (Levison et al.,
2006). At the distance of the nearby star forming regions (∼150 pc), this correspond
to an angular size smaller than 0.02 arcsec and can be achieved only through near- and
mid-infrared interferometry.
In the next future it will be therefore fundamental to increase the few available near-

164 Future developments
Figure 9.2: Analysis of the existing observations. The left and right panels show K-band V 2
data for MWC 758 respectively plotted as function of the baseline and of the hour angle, for the
three different PTI baseline orientations (NS, baseline length of 110m in direction North-South;
NW, 86m direction North-West; SW, 87m direction South-West). PTI measurements (Eisner et
al. 2004) are shown by dots. The solid lines show the predictions of the best-fitting model with
small grains (a=0.17µm); the dashed lines the best-fitting model with big grains (a≥1.2µm);
grain sizes in this range can reproduce the observations at almost the same level of confidence.
Each curve on the left panel corresponds to a different orientation of the baseline on the plane of
the sky (see Chapter 4 for more details).
infrared observations of pre-main sequence disks and combine these data with interfer-
ometric observations at longer wavelengths. In this way, infrared and millimeter ob-
servations will allow to characterize the disk structure at all distances from the central
star, from the dust evaporation radius, at fraction of AU, until the disk outer radius, at
hundreds of AUs.
As part of an AMBER/VLTI Guarantee Time Observation program led by the Astro-
physical Observatory of Arcetri (PI: Antonella Natta), a sample of more than 30 HAe/Be
stars will be observed in the following two years; the observations of a first subset of 8
objects have been scheduled for the ongoing ESO observing period. Using a triplet of
VLT Auxiliary Telescopes, we will obtain three visibility values, spanning on a baseline
range between∼30 and∼80 m, and one closure phase measurement. Even if these
observations alone will be not sufficient to constrain the structure of the inner disk, they

9.2 Probing planet formation through interferometric observations 165
Figure 9.3: Simulated images of the circumstellar disk puffed-up inner rims and relative values
of the closure phase for the VLTI configuration UT1-UT3-UT4. The left panel shows the pre-
dicted image of the puffed-up inner rim, obtained assuming the presence of micron size grains
for an inclination of 40 ◦ and a position angle of 145 ◦ ; the relative visibility is shown with dashed
line in Fig. 9.2 and 9.4. The central panel shows the predicted image for the disk model with
small grains (a=0.17µm) and the same inclination and position angle; the relative visibility is
shown with solid lines in Fig. 9.2 and 9.4. The right panel shows the closure phase for the two
models as a function of the hour angle. Due to the different size of the rim, the closure phase
of the model with small grains (solid line) is much larger than the value for the big grain model
(dashed line).
will constitute a solid data-base that we plan to extend with future observations.
The first object we have selected for a detailed follow-up is the HAe MWC 758, al-
ready discussed in Chapter 4. This well studied pre-main sequence star (spectral index
A5IVe, luminosity 22L⊙, mass 2.0M⊙, distance 230pc) has a strong IR excess (Malfait
et al., 1998), modelled as due to a flared disk with an inner rim supposed to be respon-
sible for the observed [OI] 6300Å emission (Acke et al., 2005). The circumstellar disk
has also been resolved using millimetric interferometric observations by Manning et al.
(1997), who inferred a disk inclination of 46 ◦ and a position angle of the major axis of
the disk of about 116 ◦ . As discussed in Sec. 4.4.2, the available PTI visibilities can be
fitted by a family of inner rim models, with parameters varying between the two extreme

166 Future developments
cases shown in Fig. 9.2. While the inferred inclination and position angle of the disk
are very similar to the values obtained from the millimetric observations, models with
grain sizes varying by a factor 10 (between 0.1µm and 1µm) fit the observations equally
well. Our model shows that it is however possible to solve the existing ambiguity and
constrain the grain size using new AMBER-VLTI observations. In particular, we predict
that the closure phase measurements will be a powerful diagnostic to constrain the grain
size, being strongly dependent on the disk geometrical structure, as shown in Fig. 9.3.
The observations of MWC 758 have been proposed to ESO/VLTI and are scheduled for
the ongoing ESO observing period.
Given the huge amount of time required to obtain useful near-infrared interferomet-
ric observations, it will be fundamental to combine data obtained with different interfer-
ometers (e.g. the Keck Interferometer and the Palomar Testbed Interferometer) in order
to improve the uv-plane coverage and obtain a good constrain of different theoretical
models.
Important information on the disk structure can also be obtained from mid-infrared
interferometric observations around 10µm. At this wavelength, the disk emission is
dominated by the dust inside 5-10 AU from the central star, where most planets are
supposed to form. The measured visibility can be compared with classical disk models
(e.g. passive irradiated disk models) to determine the disk structure and orientation.
Puffed-up inner rim models predict that disk regions between 1 and 10 AU lay in the
rim shadow (see Fig. 7.10); disk shadowed regions, being cooler and denser than the
ones exposed to stellar radiation, are supposed to favour planet formation. Shadowed
disks differ from full-flared disks by a lower mid-infrared continuum flux and have
been invoked to explain different shapes of the spectral energy distribution of HAe stars
(Meeus et al. 2001). Using VLTI or/and KI, it will be possible to confirm or rule out
this theory (Leinert et al., 2004; van Boekel et al., 2004 and 2005).
Finally, new millimeter interferometric observations will be fundamental to perform
a multi-scale study of pre-main sequence disks. Thanks to the new extended configura-

9.2 Probing planet formation through interferometric observations 167
tion of Plateau de Bure and the new CARMA array 2 , angular resolutions of about 0.3
and 0.1 arcsec (at 1mm) can be respectively achieved. This will allows us to resolve
the nearby circumstellar disks on scale sizes of∼20-40 AU (at distances of 150 pc) and
eventually detect planet forming regions.
2 https://www.mmarray.org/

Figure 9.4: Simulated observations of two different inner disk models, using different telescope configurations. The left panel shows
the predicted squared visibilities as a function of the baseline length: solid and dashed lines refer respectively to the model with small
(a=0.17µm) and large (a≥1.2µm) dust grains. Solid and empty circles indicate the visibilities for the VLT UT1-UT3-UT4 telescopes
configuration at four different hour angles. The central panel shows the variation of the visibility with the hour angle for each baseline.
Finally, the panel on the right shows the uv-plane coverage (dots) obtained with the chosen configuration. The predicted closure phases
are plotted in Fig. 9.3
168 Future developments

Published papers
• Isella, A.; Testi, L.; Natta, A., “Large dust grains in the inner region of circum-
stellar disks”, 2006, A&A, 451, 951
• Tatulli, E.; Isella, A.; Natta, A.; Testi, L.; Marconi, A.; and the AMBER consor-
tium, “Constraining the wind launching region in Herbig Ae stars: AMBER/VLTI
spectroscopy of HD104237”, to appear in A&A (astro-ph/0606684)
• Isella, A.; Natta, A., “The shape of the inner rim in proto-planetary disks”, 2005,
A&A, 438, 899
• Malbet, F.; Benisty, M.; De Wit, W. J.; Kraus, S.; Meilland, A.; Millour, F.;
Tatulli, E.; Berger, J. -P.; Chesneau, O.; and the AMBER consortium, “Disk and
wind interaction in the young stellar object MWC 297 spatially resolved with
VLTI/AMBER”, to appear in A&A (astro-ph/0510350)
In preparation
• Isella, A; Testi, L; Natta, A.; Neri, R.; Wilner, D.; and Qi C., “The keplerian disk
of HD 163296”

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Ringraziamenti
Al termine di questi tre anni di Dottorato i miei piú sinceri ringraziamenti vanno ad
Antonella Natta, Leonardo Testi e Giuseppe Bertin per aver reso materialmente possible
questo lavoro, per le sempre stimolanti discussioni e l’inestimabile aiuto.
Un grazie speciale a Malcolm Walmsley per la cordialitá, la simpatia e gli utili con-
sigli; ad Eric Tatulli per la preziosa collaborazione, sperando che possa continuare a
lungo; a Riccardo Cesaroni per la pazienza e la disponibilitá nel rispondere alle mie nu-
merose richieste di aiuto; a tutti i compagni di pranzo per i piacevoli momenti trascorsi
insieme.
Ringrazio e saluto con piacere tutto il personale dell’Osservatorio di Arcetri, a par-
tire dai nostri direttori Francesco Palla e Marco Salvati, tutto il personale ricercatore,
tecnico ed amministrativo per la cordialitá e la disponibilitá sempre dimostrata.
Infine, saluto con affetto tutti gli amici distribuiti tra Milano e Firenze, la mia famiglia
e sopratutto mia moglie Rossana per non avermi (ancora) buttato fuori di casa!