Extrasolar Planets - Confronting the Theory of Planet Formation with Observations

INTERNATIONAL
SPACE
SCIENCE
INSTITUTE
SPATIUM
Published by the Association Pro ISSI No. 41, May 2018

Editorial
Extrasolar planets – why should we
care? Simple answer … first of all,
a planet in general is an interesting
place, we should best know, we inhabit
one. The Universe is filled
with planetary systems, and for
over two decades now increasingly
more of them are being discovered.
This is neither a surprise nor anything
to start being bored of, in
fact every new planet helps to improve
our knowledge and also to
push the limits of what we want to
learn. This exactly is the implication
given by the title of the talk:
“A Laboratory to Confront the
Theory of Planet Formation with
Observations”.
However, almost with each new
discovery, there is at least one new
question that is opened up as well.
The more we observe, the more
we realise that there is still much
more to learn, in fact, there are a
huge number of open questions in
planet formation.
Still, the search for extrasolar planets
is a demanding undertaking and
needs extremely precise instrumentation.
It is understandably
hard to detect a light signal of a
small object next to a million or
billion times brighter one, its host
star. Similarly, it is not easy to identify
a tiny dark spot on a bright object
or a minuscule shift in a star’s
spectrum due to the presence of
one or more companions.
As instrumentation techniques are
progressing, this fosters new models
to be developed to explain what
has been observed. Of course, the
ultimate test for all new theories
are then the observations again,
and a fruitful combination of models
and data can lead to better understanding
of our environment.
The sheer number of planets now
has made statistics an appropriate
tool to do exactly that.
The successive extension of our
laboratory space from the home lab
Earth to the neighbour lab Solar
System to remote parts is still ongoing,
and we are in for surprises.
Discovering new worlds far outside
our own and noting their vast
diversity does not only help in understanding
basic physical processes
but also in realising that the
very system we have the chance to
inhabit is special among all the
planetary systems discovered so far.
The present text is based on a lecture
by Prof. Christoph Mordasini
in the Pro ISSI seminar series. It
has been edited and arranged with
valuable input from Prof. M.C.E.
Huber and Dr. H. Schlaepfer.
Anuschka Pauluhn
Mönthal, February 2018
Impressum
ISSN 2297–5888 (Print)
ISSN 2297–590X (Online)
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Title Caption
Artist’s view on how common
planets are around the stars in the
Milky Way – the rule rather than
the exception. (Credit: ESA, C.
Carreau).
The inset shows the plot of a computed
and a measured planetary
mass distribution, from Benz et al.
(2014).
SPATIUM 41 2

Extrasolar Planets
Confronting the Theory of Planet Formation with Observations 1
by Prof. Christoph Mordasini, University of Bern
Introduction
Naturally, our understanding of
planets has so far been swayed by
the Solar System planets that have
been observed for ages. These have
shaped theories that can now be
tested against increasing numbers
of examples from systems that are
very different from ours. New systems,
some with new and unexpected
properties are forcing us to
adapt and re-think our theories, as
already anticipated some years ago,
just shortly after the first extrasolar
planetary systems had been discovered
2 . Every new discovery is
checked – does it fit into our models?
If so, fine, one more for the
current theories; if not, what is
wrong – faulty observation, flaws
in the measurement, or a completely
new feature that needs more
explanation and study. The goal is
not “to detect for the detection’s
sake”, but to improve our know l-
edge, our theory, our understanding
of the world we live in.
The exoplanet detection count
jumped up after 1995: Michel
Mayor and Didier Queloz had detected
the gas giant planet 51 Pegasi
b 3 , which orbits a rather ordinary
G-type main-sequence star, not
too much different from our Sun
and thus had found a rather big
planet of half the mass of Jupiter
with an orbital period of only 4.23
days. Following this discovery of a
gas giant in a short-period orbit
close to its host star, astronomers
now looked as a matter of course
for more unexpected types of
companions!
Several branches can be followed
in planetary research – from the
exoplanets’ orbital dynamics to
Figure 1: The Solar System planets, together with some dwarf planets located beyond Neptune’s orbit in the Kuiper belt.
Sizes to scale, not distances. Credit: The International Astronomical Union/Martin Kornmesser.
1
The current text is a summary of Prof. Mordasini’s talk for the Pro ISSI audience on 15 March 2017.
It was drafted by Dr. Anuschka Pauluhn and revised by Prof. Mordasini.
2
See Spatium 6 by W. Benz (2000).
3
The official name of an exoplanet is a combination of the parent star’s name and a lower-case letter, in the order of
detection. The first planet of the system is denoted “b”, the second “c”, etc.
SPATIUM 41 3

their structure and composition,
and in particular also that of their
atmospheres 4 . Of course, all these
studies are closely related, and of
central interest is the way planets
are evolving – following their entire
life cycle from formation to
later stages in their career in a planetary
system like in ours, for
example.
However, observing them is not
simple in most cases. Stars, on the
other hand, are much easier to
observe; they outshine any planet
around them. Indeed, a large part
of their evolution has been reconstructed
a long time ago. Ordering
them according to their temperature
and luminosity, stars can be
classified into certain groups in the
famous Hertzsprung-Russell diagram
5 . Such a helpful diagram
would be nice to have for planets,
so that one could find suitable systematics
of classes and evolution.
With more and more exoplanets
found, this has now become possible.
An example of such a classification
of planets is the mass-distance
diagram, relating the planets’
masses (in multiples of Earth masses
M ) to their distances to the host
star (defined by the semi-major
axis of their orbits, and given in
multiples of the Earth’s distance
from the Sun, the so-called Astronomical
Unit, AU).
Figure 2: Two planetary disks observed with the SPHERE instrument mounted on ESO’s Very Large Telescope (VLT). The
central part of the image at the location of the bright host star has been blocked in order to reveal the fainter surrounding.
Credit: ESO.
4
See Spatium 36 by H. Lammer (2015).
5
H-R diagram: A scatter plot of absolute magnitude of stars and their effective temperature. It turns out that luminosity
and temperature are not random but related to physical and chemical properties of the stars and to their age. In this diagram,
most stars are found along a region, which is called the main sequence. A star remains on the main sequence while
it is fusing hydrogen in its core. Other regions in the H-R diagram define classes of stars with specific properties such as
giant or white dwarf stars.
SPATIUM 41 4

The structure of
the Solar System
Planetary studies are as old a discipline
as humans have been observing
the sky, and even the possibility
of habitable planets somewhere
in the Universe had already been
suggested by scholars like Immanuel
Kant 6 . Until the last decades,
however, the observation of planets
had been restricted to the Solar System,
and all of the early theories on
planet formation had been derived
from the conditions found there.
Figure 3: Protoplanetary disks, observed in the infrared by the Hubble Space Telescope.
Most of the nebulae represent the small dust particles around the stars, which
are seen because they are scattering the starlight. Credit: D. Padgett (IPAC/Caltech),
W. Brandner (IPAC), K. Stapelfeldt (JPL) and NASA/ESA.
Moreover, what had been the
source of knowledge on the formation
of planets a few years ago was
given by only two states in their
life cycle, namely their very beginning
– in nebular disks during star
formation processes, and the end
product – in a developed planetary
system.
The protoplanetary nebular disks
(various examples are shown in Figure
2 and Figure 3) are formed during
and almost immediately after
the collapse of a molecular cloud
into a protostar. As material further
from the protostar begins to
fall inward, the conservation of angular
momentum prevents it from
falling directly onto the protostar
and the material will flatten into a
disk that surrounds the protostar.
These disks can stretch directly
from the protostar to distances of
hundreds of astronomical units
(i. e., well over a hundred times the
distance between Sun and Earth).
Because the dust and gas of the disk
are heated by light from the newborn
star, the parts of the disk closest
to the star will be the hottest
and the parts farthest from the star
will be the coldest. This dust and
gas will emit as a black body, with
the hotter material emitting mainly
in the infrared, and the colder in
the (sub-) millimetre wavelength
bands. Consequently, these wavelengths
are a good target for observations,
and a combination of
space- and ground-based infrared
and (sub-) millimetre interferometric
observations have been providing
valuable data for decades.
At least one of the outcomes of such
a process of planet formation is
very well known – the Solar System
– and, of course, any model
for planet formation has to be consistent
with the constraints given
by the Solar System planets. The
basic structure and the conditions
in the Solar System are characterised
by a few remarkable properties
(cf., Figure 4 overleaf ).
“Orderly Orbits”: Our Solar System
features prograde, nearly coplanar
and nearly circular orbits. The orbital
motions and rotations of the
6
Already in 1755 Immanuel Kant, based on work by Thomas Wright (1750) and probably also by Emanuel Swedenborg
(1734), developed a theory on the formation and evolution of the Solar System (Allgemeine Naturgeschichte und Theorie
des Himmels, 1755). Independently, Pierre-Simon Laplace later (1796) developed a similar theory on the formation
of the Solar System, cf., Kant-Laplace theory. The assumption that the Solar System formed from nebular material, i. e.,
the “nebular hypothesis” is still the basis of modern cosmogony. In fact, Kant was also convinced of the existence of extraterrestrial
life forms.
SPATIUM 41 5

Sun, the planets and their moons
are predominantly in one sense 7 ,
and the planets circle the Sun on
orbits with small inclinations. Mercury’s
inclination is rather large
with 7 °, and Pluto, also formerly
known as a planet, has an orbital
inclination of 17.1 ° from the ecliptic;
but note that Pluto is now classified
as a “dwarf planet” and not
considered to be a planet any more.
“Rocks, Gas and Ice”: There is a kind
of order or sequence of the planets
in our Solar System: with increasing
distance from the Sun, there
are the rocky planets (Mercury,
Venus, Earth, Mars), the gas giants
(Jupiter, Saturn), and finally the ice
planets (Uranus and Neptune).
“A distinguished inner circle”: In our
Solar System no planets are found
inside 0.4 AU and no planets reside
outside Neptune’s orbit, i. e., beyond
30 AU.
Figure 4: Mass-distance diagram with
the positions of the Solar System planets
Venus, Earth, Jupiter, Saturn, Uranus,
Neptune, showing the areas 1 and
2 where until the mid-nineties no planets
were thought to reside and the preferred
location 3, the “comfort zone”
for planets in the Solar System.
Figure 5: The planets of the Solar System, from left to right, Mercury, Venus,
Earth and Mars, Jupiter and Saturn, Uranus and Neptune. Sizes are shown approximately
to scale, distances not. Image credit: NASA.
Giant planets, gas and ice:
Giant planets, also referred to as Jovian planets, are usually mainly composed of
low-boiling-point materials (gases or ices), rather than rock or other solid matter,
but massive planets containing large amounts of solids also exist. The giant
planets consist primarily of high-pressure fluids above their critical points, where
distinct gas and liquid phases do not exist. The principal components are hydrogen
and helium in the case of Jupiter and Saturn (gas giants) and water, ammonia
and methane in the case of Uranus and Neptune (ice giants). A planet is called
“Hot Jupiter” when its mass is similar to Jupiter but its orbit lies much closer to
its host, with the orbital period being rather short, on the order of 10 days or
less, and with the surface temperature of its atmosphere being accordingly high.
At masses greater than roughly 13 Jupiter masses, the planets would start burning
deuterium and thus qualify as so-called brown dwarfs.
Rocky planets:
Rocky planets, also called terrestrial or telluric planets, are composed primarily
of silicate rocks or metals. Within the Solar System, the terrestrial planets are
the inner planets closest to the Sun, i. e., Mercury, Venus, Earth, and Mars. Terrestrial
planets have a solid planetary surface, making them substantially different
from the larger giant planets, which are composed mostly of some combination
of hydrogen, helium, and water existing in various physical states.
Planet formation as
suggested from the
Solar System
Surely, a satisfactory theory has to
explain all these conditions. The
central ingredient and fundamental
quantity to start with is the mass
present in the protoplanetary disk:
how much raw material is available?
Early models for planet formation
used a kind of reverse engineering:
the idea was to assume
that the planets had been formed
where they are now in the disk,
take the distribution of hydrogen
and helium gas and other elements
present in the Solar System and
7
However, the rotations of Venus and Uranus are retrograde.
SPATIUM 41 6

Figure 6a: Left: The surface density of the protoplanetary disk for the Solar System planets versus distance from the host star
in a doubly logarithmic plot, from Ruden (2000). The sudden increase of the density marks the position of the snow line.
Right: A schematic of the temperature structure of the MMSN, the minimum mass solar nebula model for our system.
Figure 6b: An artist’s impression of the water snow line around the young star V883 Orionis, as detected with the Atacama
Large Millimeter/Submillimeter Array ALMA. Credit: A. Angelich (NRAO/AUI/NSF)/ALMA (ESO/NAOJ/NRAO).
A nice terrestrial snow line is shown on the right.
spread it out to their nearest neighbours.
The distribution of gas and
dust and the temperature are the
initial conditions – and then a baby
planet is set to grow from a grain
to a planetesimal and further. An
important parameter in the disk is
thus the radial density and its variation.
By neglecting the vertical
variation one defines the so-called
surface density, i. e., the disk density
integrated over the vertical direction
of the disk; in principle this
describes how much mass per surface
shell is available. The surface
density provides a snapshot of the
mass distribution, governed by the
viscous and gravitational torques of
the gas and solids that determine
the accretion flows and the angular
momentum transport. The basic
assumptions for the simplistic
solar nebula (also called the “minimum
mass solar nebula”, MMSN)
modelled the gas surface density by
a so-called power law, decreasing
with distance as r –3/2 . This fundamental
quantity is shown in Figure
6a in a doubly logarithmic plot
(that bridges several orders of magnitude
and presents the strongly
decreasing function as a straight
line).
The abrupt rise in the plot of
Figure 6a marks the so-called snow
line (also called frost line or ice line):
This is the location in a protoplanetary
disk, in the case of the Solar
System between Mars and Jupiter,
where the surface density increases
due to the condensation of water.
The inner parts of the disk are
hotter due to the irradiation from
the host star. Of course, this snow
line can also be defined for other
volatile compounds like ammonia,
methane, carbon monoxide, carbon
dioxide, and its particular distance
depends on the substance and the
ambient pressure and temperature.
SPATIUM 41 7

The temperature and mass distributions
provide the relevant building
blocks for planets: The lack of
close-in planets in the Solar System
can be explained by the fact that
there is simply not enough material
found close to the star in order
to form planetesimals. There are
less solids close to the star, and giant
planets can form only outside
the snow line in annuli of sufficient
volume for them to grow.
Core-accretion
The standard model of planet formation,
the core-accretion theory,
is a “bottom-up” process: first the
planetesimals form solid cores,
some of which later accrete massive
gaseous envelopes and become
giant planets. The remaining cores
collide to form both ice giants and
terrestrial, rocky planets. The
planet grows by accretion. If the
planet does not change its distance
from the host star, the stellar gravity
limits the area from where it can
accrete mass to an annulus around
the planet’s orbit, whose width is a
few times the Hill sphere radius 8 ,
the so-called feeding zone. Consequently,
the mass of a planet can
grow locally only to a limiting mass
M iso , the isolation mass. Following
the standard accretion theory, five
to ten Earth masses are required for
Jovian-mass planet growth, but at
close distances to the star, the mass
availability is smaller. Thus, the
classical models predict giant planets
only at larger distances.
Similarly, the “pure” core-accretion
model explained why the positions
of the planets in the Solar
System do not exceed about 30 AU.
The accretion by collisional growth
is governed by the so-called Safronov
relation 9 , which states that
the planetesimal’s change of mass
is proportional to the surrounding
density, the effective cross-section
of its interaction with its environment
and its velocity – and both,
density as well as velocity decrease
with distance from the host. The
further away from the star, the
longer it thus takes to grow to a
certain mass, and the planets have
to form within the lifetime of the
protoplanetary disk! So, the model
implies that the gas giants form
earlier, typically on time scales of
the lifetime of the protoplanetary
disk, which have been observed to
be not longer than 10 million
years 10 , and the rocky planets take
more time, around hundred million
years. That would nicely explain
the lack of far-out planets in
the Solar System: the time scales
just do not fit, there is not enough
material left at the boundary of the
disk by then. The possibility that
the giant planets could form fast
and directly from a gravitational
instability in the protoplanetary
gas disk was not much supported
in most classical theories.
Figure 7: Mass-distance diagram, showing the areas 1 and 2 where no planets were
thought to form, this time with positions of the exoplanets (counted until 2017)
included. It clearly shows that planets in general are by far less discriminating in
choosing their preferred locations than our Solar System ones. Not only all kinds
of close-in planets are found but also far-out ones.
8
The Hill sphere approximates the gravitational sphere of influence of a smaller body in the presence of perturbations
from a more massive body.
9
In 1969 Viktor Safronov quantitatively described the states of accretion and terrestrial planet formation.
10
The lifetimes of protoplanetary disks have been determined by observations in the infrared (from the warmer core
region of 100 K to 1500 K) and millimetre (from the outer regions of colder dust of around 10 K) wavelength bands
to be widely in the range from 10 6 years to 1.5 × 10 7 years, and not longer. While the protostar accretes, the mass of its
envelope/disk decreases significantly and the formation of planetary systems cannot exceed the lifetime of a sufficiently
massive disk. Thus, the time available for planet formation is limited.
SPATIUM 41 8

The exoplanet
revolution
Fine! That’s it – all done and understood.
However, now enter the
1990ies and the successive “exoplanet
revolution”, starting with
the first Hot Jupiter in 1995. But
not enough – the more planetary
systems had (and have) been discovered,
the more was (and is being)
understood that things are really
different out there – and that
our Solar System is in fact a very
special instance of a planetary system.
Ordering the new findings in a
mass-distance diagram like in Figure
4, gives a slightly different impression
of planets’ favored locations,
see Figure 7.
Of course, it is difficult to understand
the structure of an iceberg
seeing only its tip. That this guideline
applies to the theory of planet
formation as well became obvious
Figure 8: Cumulative number of detections
grouped by the detection
method, generated by the NASA Exoplanet
Archive operated by the California
Institute of Technology. The planets
are grouped by detection method.
See also on https://exoplanetarchive.
ipac.caltech.edu/exoplanetplots/.
with the increasing number of exoplanet
detections: most of the
planets found did not fit into the
well-ordered scheme known from
the Solar System. Soon it became
clear that a lot of different and possibly
new physical processes had to
be included in the models in order
to represent the findings. Figure 8
shows the count of detections as of
March 2018. The current count
can be followed on the webpages
www.exoplanet.eu and www.exoplanets.org.
New theories and before-neglected
approaches had to be invoked to
explain the observations of Hot
Jupiters and far-out planets. The
model of core accretion alone did
not seem to be enough and needed
to be extended; for example, the
theory of gravitational instability
was revived, and also the idea of
“static” planetary orbits was partially
abandoned in favour of more
“dynamic” ones and the possibility
of orbital migration.
Detection techniques
for exoplanets
All signals from exoplanets are extremely
faint, and most methods so
far are indirect – the planet is detected,
for example, via its influence
on the motion of its host star
or via its influence on the path of
light emitted by another, distant
star.
The most important methods for
finding exoplanets are
1. The radial velocity method (RV)
This is a spectroscopic method that
measures the net Doppler shift in
as many as possible lines of a star’s
spectrum. The result is a velocity
at which the star is moving in direction
of the observer’s line of
sight. Removing all other known
motions, like, e.g., that of the telescope
relative to the barycentre of
the Solar System, the resulting motion
is due to the influence of the
planetary orbits. The signals are
tiny – of the order of metres per
second. Due to the viewing geometry,
the measurements yield the
product of the mass of the planet
and the sine of the unknown inclination
angle i between the orbital
plane and the plane of the sky,
M P sin i, which is a lower limit to
the mass only. The method is most
sensitive to massive planets and to
those in short-period orbits. Additionally,
the number of spectral
lines emitted by a star is crucial for
the precision. One of the most precise
instruments for this technique
is the HARPS (High Accuracy
Radial velocity Planet Searcher)
spectrograph, attached to the ESO
3.6 m telescope in La Silla, Chile.
It has been developed and built
mainly by the universities of Geneva
and Bern and can detect velocity
shifts down to 0.6 m/s.
2. Transit techniques
The theory is simple, the measurements
are hard. As the previous
method, it is an indirect detection
via the light curve of the host star.
Transit photometry identifies
planet candidates by the periodic
drops in observed stellar brightness
that are caused by the planet obscuring
a portion of the stellar disk
once per orbit. Transit photometry
can only detect planets whose or-
SPATIUM 41 9

Figure 9: Directly imaged planet in the disk of Beta Pictoris. This “Super-Jupiter” with a mass of roughly 7 Jupiter masses
was first measured with the VLT in November 2003 and identified as a planet many years later. Credit: Lagrange/ESO.
bits are viewed nearly edge-on, and
it measures size, not mass. To get
good signals, it is necessary to go
to space. After the first interesting
transit detections with the CoRoT
spacecraft, the NASA Kepler mission
has provided thousands of candidates
that need to be further investigated
via the RV method. In
order to precisely measure planetary
radii and even test for the presence
of atmospheres, the CHEOPS
(CHaracterizing ExOPlanet Satellite)
mission of the University of
Bern will provide high-precision
photometry of transiting planets.
Its launch is planned for 2018, like
that of NASA’s Transiting Exoplanet
Survey Satellite TESS. Both
missions can also help to prepare a
target catalogue for the Hubble
Space Telescope (HST) follow-on
mission, the James Webb Space
Telescope (JWST).
3. Direct imaging
Relative to their host stars, planets
are very faint light sources. Coronagraphs,
i. e., telescopes with an occulting
central disk, are thus used
to block the light of the star, and
most observations have been made
in the infrared where the planets
are brighter than in the visible part
of the spectrum. (The ratio of the
spectral radiances of Jupiter to the
Sun is 10 –9 in the visible range,
however 10 –6 and larger in the
infrared and longer wavelengths.)
The demand on angular resolution
is high as well: to resolve an angular
diameter of 1 AU at a distance
of 1 pc (which spans 1 arcsec, by
definition of the parsec 11 ) corresponds
to resolving the diameter of
a human hair at 20 m distance. To
overcome disturbances by the atmospheric
seeing (that are caused
by the irregular variation of the
local index of refraction), use of
adaptive optics is mandatory. The
great advantage of direct signals
from the planet is the possibility for
spectroscopic analysis and thus to
learn about the atmospheres. This
method favours planets orbiting
less luminous stars. A spectacular
directly-imaged planet around the
host star Beta Pictoris located 63
light years from Earth is shown in
Figure 9: Although the first images
had been taken in 2003, only with
improved data reduction methods
in 2008 the faint source could be
identified as a planet. Follow-up
observations then showed the
planet re-appear on the other side
of the star.
4. Gravitational microlensing
Gravitational microlensing occurs
when the gravitational field of a
star acts like a lens, magnifying the
light of a distant background star.
This happens only when the two
stars are almost exactly aligned.
Lensing events are brief, lasting for
weeks or days, as the two stars and
Earth are all moving relative to
each other. If the foreground lensing
star has a planet, then that planet’s
own gravitational field can
make a detectable contribution to
the lensing effect. As the alignment
11
One parsec (pc) is the distance at which the Sun-Earth distance 1 AU subtends one second of arc, 1 arcsec. 1 pc =
3.26 light years.
SPATIUM 41 10

of the bodies in space does not occur
twice, the measurement cannot
be repeated.
5. Timing methods, e.g., Pulsar
timing
Pulsars are rapidly rotating,
strongly magnetised, radio-wave
emitting neutron stars, which are
the remnants of supernova explosions.
From variations of their extremely
regular signal, the presence
of companions can relatively easily
be deduced. However, pulsars are
not too frequently found, and their
planetary systems have to overcome
extreme conditions. Timing
methods can also be applied to
some other classes of pulsating variable
stars. Their signals have to be
regular enough that radial velocities
of companions can be detected
by the Doppler shift in the pulsating
frequency.
6. Astrometry
Astrometric detection is based on
precise measurements of the position
of objects and the change of
position over time. If a star has a
planet, then the gravitational influence
of the planet will cause the
star itself to move in a tiny circular
or elliptical orbit. Effectively,
star and planet each orbit around
their mutual centre of mass. One
potential advantage of the astrometric
method is that it is most sensitive
to planets with large orbits.
This makes it complementary to
other methods that are most sensitive
to planets with small orbits.
However, very long observation
times will be required – years, and
possibly decades, as planets far
enough from their star to allow detection
via astrometry also take a
long time to complete an orbit. No
detections have been confirmed
yet, however, the ESA mission
Gaia (Global Astrometric Interferometer
for Astrophysics, launched
2013) is expected to detect thousands
of planets via this method.
The two first methods have been
the most productive as of March
2018. Depending on the method,
certain systems are more probable
to be found than others – this is the
observational bias. In general, given
the faint signals and large distances,
all techniques require utmost precision
of the measurements.
With all these methods a multitude
and enormous diversity of planets
has been discovered – and their features
and properties cannot any
more be explained by the models
that have been successful in describing
the conditions found in
the Solar System. In particular, we
find Hot Jupiters, as well as far-out
planets! Can we now find theories
and models to explain and reproduce
our findings?
The sheer number of exoplanets
suggests the employment of statistical
methods. It now is feasible to
use sample sets of exoplanets and
extract statistical constraints that
can be applied to theoretical models.
Furthermore, the populationwide
approach can be used to investigate
distributions of properties,
like their masses, semi major axes,
radii, and eccentricities.
Figure 10: There is an enormous diversity in protoplanetary disks. Such disks vary
in their lifetimes, gas masses, dust masses, and thus in the initial conditions for planetary
evolution. The image shows a collection of 30 protoplanetary disks in the
Orion nebula, observed by the Hubble Space Telescope in the visual wavelength
range. Credit: NASA, ESA, and L. Ricci (ESO).
SPATIUM 41 11

Simulating the
diversity
Through observations, we know
already a lot about protoplanetary
disks, i. e., the beginning of planetary
systems and also about their
fully developed state. Already these
disks show an enormous diversity
(see Figure 10). However, the knowledge
about the intermediate states,
i. e., how a planetary system becomes
one, is still rather limited today.
What is desired is a theory that
can bridge the gap of (so far) missing
information between the states
of the early disks and the observed
planetary systems. The task is clear:
given a distribution of initial conditions
– can the distribution of exoplanets
be generated by a physical
model?
Modelling – Population
synthesis
The important physics is hidden in
the term “formation model”. As
usual with modelling, one has to
break down the intrinsically complex
and interacting processes into
smaller bits and lower-dimensional
parts, in order to be able to handle
them mathematically and computationally.
If, given all the necessary
simplifications, a result is in
reasonable agreement with observation,
this could mean that our
understanding of what happens is
not too far off. Another important
application of course is to make
predictions – from the output of a
simulation, tell the observers what
to search for. This latter is in fact a
valid application of the method –
it helps space agencies designing
missions by defining goals for future
instruments.
A global formation model builds
on many detailed theories and
models that each addresses one specific
physical mechanism. Most
models are based on the core-accretion
paradigm; recent studies
employ also the gravitational instability
model. In general, the start
is a protoplanetary disk and at the
end of a simulation a fully developed
planetary system at an age of
several billion years should be the
result. The modular outline of such
a model (here the one from the
University of Bern) is shown in the
box.
A test bed for the theories
All models are complex composites
of several sub-models; every
single part is critically dependent
on the output of the step before,
and interrelated to the other parts.
There are lots of parameters to
tune, lots of assumptions and simplifications
have to be made and
new approaches have to be tested.
A lot of progress has been made in
the last years, but modelling always
has room for more improvements,
be it through new theoretical ideas,
better algorithms or sheer computing
power. Using such a modular
The method of population synthesis
uses a statistical approach in order
to model and understand the
planet formation process; it generates
an output of planetary systems
from a sample distribution of initial
conditions. The result is subsequently
filtered with the appropriate
restrictions applying for a
specific observation method, i. e.,
considering the observation bias,
in order to compare with the respective
measurements. For example,
gas and dust masses as given by
a number of observations are used
as input for a simulation scheme as
is displayed in Figure 11.
Figure 11: Flow diagram that shows the outline of population synthesis (Mordasini
et al 2009).
SPATIUM 41 12

model, the suitable statistics and
the corresponding observations the
extended theories can now be
tested, with emphasis on the aforementioned
open questions.
Hot Jupiters/close massive
planets:
The sub-models of the Bern model
0) Observed distributions of initial
conditions, i. e., properties of
protoplanetary disks, such as
metallicity, lifetimes, mass
1) Structure and evolution of the
gaseous protoplanetary disk
2) Gas surface density evolution:
viscosity/mass loss by photoevaporation/accretion
3) Structure and evolution of the
disk of small bodies
(planetesimals)
4) Core growth of protoplanet by
accretion of planetesimals and
collision
5) Radial structure model of the
gaseous envelope of the
protoplanet
6) Atmosphere of the protoplanet
7) Interaction of planetesimals and
the gaseous envelope of the
protoplanet
8) Radius of the solid core as a
function of its mass, bulk composition
and external pressure
due to the surrounding gas
envelope
9) Mass loss due to atmospheric
escape of the primordial H/He
envelope
10) Orbital migration due to tidal
interaction
11) Gravitational interaction of
young planets
Figure 12: An illustration of the Hot Jupiter K2-33b, which (at an age estimated
to be between five and ten million years) is one of the youngest exoplanets detected
to date. Its orbital period is about five days. Credits: NASA/JPL-Caltech.
The finding of massive planets
close to their host star (cf., Figure
12) revived some older theory to be
included in the models – the possibility
that planets are mobile: orbital
migration. In a protoplanetary
gas disk, embedded planets
and gas interact gravitationally
which can lead to an exchange of
angular momentum. The planet
reacts by adjusting its semi major
axis. Both in- and outward migration
are possible, depending on
planet mass and the properties of
the disk. Including the orbital migration
part in the models, close
massive planets can be successfully
reproduced. However, choosing
the parameters is not simple – in a
number of simulations, the planets
just end up falling into the star too
fast instead of orbiting peacefully
around it for a while. Quantitative
predictions are hard: it is to date
one of the most debated subjects in
formation theory, other mechanisms
are also possible and under
study.
Figure 13: HR 8799 in Columba where at least four massive planets are far out
ones. Credit: J. Wang, C. Marois.
SPATIUM 41 13

Far-out planets/timescales for
planet growth:
Direct imaging of the system HR
8799 in the Columba moving
group showed some surprisingly
large companions farther out than
30 AU (Figure 13 on page 13). Still,
a mechanism is needed to reproduce
these planets. The alternative,
or rather additional model to pure
core accretion that can be tested in
population synthesis is gravitational
instability. It proposes a selfgravitative
formation: gas in the
protoplanetary disks collapses under
its own gravity and directly
forms a large, gravitationally bound
clump with a mass of several Jovian
masses. Other possible explanations
for far-out planets that can be
tested in the models include various
accretion mechanisms to simulate
the still not well-understood
growth stages of particles in the
metre-size range.
Figure 14: Comparison of observed and
synthetic planetary masses as found with
high-precision radial velocity observations.
The black line shows the raw
count, while the red line is corrected for
the observational bias (that misses the
detection of low-mass planets). The
panel on the right shows the planetary
mass function (i. e., the number distributions
of planets as a function of their
mass) as found in a population-synthesis
calculation. The black line gives the
full underlying population, while the
blue, red, and green lines are the detectable
synthetic planetary mass distributions
using, respectively, low (10 m/s),
high (1 m/s), and very high (0.1 m/s) radial
velocity precision. Figure from
Mordasini et al. (2015).
Different planet types
(rocky, icy, gaseous):
In extrasolar planetary systems, a
large variety of both planets and
planetary structure has been observed
with no simple partition
into gas giants, icy or rocky planets.
Instead, many planets with
masses between the Earth and the
ice giants have been detected; there
is no large gap and no clear type
difference as in the Solar System.
Are these planets Super-Earths or
Mini-Neptunes? The key to this
question is in the composition
when it comes to modelling the atmospheres,
starting from the microphysics
of the grains suspended
there and defining the opacity
Figure 15: Simulated planetary tracks,
the relation of planets’ masses to their
distance from the host star, and how
they have arrived there – formation processes
as generated with the Bern model
(Mordasini et al 2009). The colours
mark different types of orbital migration,
which are determined by the
planet and disk mass, and the viscosity
of the disk gas.
SPATIUM 41 14

(i. e., the attenuation coefficient,
describing how radiation is transported
in a medium). Here, as in
many other parts of the planet-formation
processes, various theories
are under debate, and a better understanding
of the grain dynamics
and how the growth from tiny
particles to planetesimals really
happens is desired. Within existing
models, various potential
mechanisms and scenarios can be
simulated. Figure 14 places side by
side an important result of the Bern
model and observations made by
using the radial velocity method.
Planetary life cycles:
The ultimate goal is to illustrate
the life cycle of planets, from their
birth to late stages, as is possible for
stars in the HR diagram. For example,
the formation process is best
visualised in planetary formation
tracks in the mass-semi-major axis
diagram, where different phases of
concurrent growth and migration
can be identified (Figure 15). These
tracks can be generated from the
models in population synthesis calculations.
The different phases in
the formation process lead to the
emergence of distinguishable subpopulations
of planets; there is a
vast group of low-mass planets, a
“horizontal branch”, a sub-population
of Neptune-mass planets extending
out to 6 AU, and the “main
clump”, a concentration of giant
gaseous planets at around 0.3 AU
to 2 AU.
Outlook
The Universe is filled with planetary
systems, and the study of these
systems is a good example of a
vastly developing theory, not insignificantly
being driven by progress
in observations. In fact, the
field of planetary studies has become
a sound example of a very
successful mutual exchange between
theory and measurement.
The combination of modelling and
statistics is an adequate and practical
tool for dealing with the increasingly
large number of samples
and advanced observations.
Further reading/Literature:
Spatium No 6, Benz, W., From Dust to Planets, Oct. 2000.
Models could possibly be used to
make predictions about the habitability
of a planet based on its formation
and evolution. This will be
supported by future high-precision
observational data, as not only
photometric but also spectroscopic
missions are foreseen to provide
more information about the geophysical
properties of planets.
Maybe in the not too far-away future
we can obtain more insight
into possibilities how life has been
evolving in the Universe.
All this is contributing to a better
understanding of our closer environment
and also a bit further out.
Surely, it has added to the comprehension
that the Solar System is indeed
special among the planetary
systems.
Spatium No 36, Lammer, H., Origin and Evolution of Planetary Atmospheres,
Nov. 2015.
Benz, W., Ida, S., Alibert, Y., Lin, D., Mordasini, C., Planet Population Synthesis,
in: Protostars and Planets IV. Eds: H. Beuther, R.S. Klessen, C.P. Dullemons,
T. Henning, Univ. Arizona Press, Tucson, 2014, p. 691.
Laughlin, G., Lissauer, J.J, Exoplanetary Geophysics – An Emerging Discipline,
2016, eprint arXiv:1501.05685.
Mordasini, C., Alibert, Y., Benz, W., Extrasolar planet population synthesis.
I. Method, formation tracks, and mass-distance distribution. Astronomy and
Astrophysics, 2009, vol. 501, p. 1139.
Mordasini, C., Mollière, P., Dittkrist, K.-M., Jin, S., Alibert, Y., Global models
of planet formation and evolution. International Journal of Astrobiology, 2015,
vol. 14, p. 201, eprint arXiv:1406.5604.
Ruden, S.P., The Formation of Planets, in: The Origin of Stars and Planetary
Systems. Eds: C.J. Lada and N.D. Kylafis, Kluwer Academic Press, 1999, p. 643,
eprint arXiv:astro-ph/9910331.
SPATIUM 41 15

SPATIUM
The Author
After his study of Physics and
Mathematics at the University of
Bern Christoph Mordasini very
soon specialized in Astronomy, not
only theoretically oriented but also
dedicated to observations.
In particular the topic of extrasolar
planets found his interest, and
after his master thesis on “Planetesimal
impacts into forming giant
planets” under the supervision of
Prof. Willy Benz had been awarded
the best thesis of the year in 2004
at the physics department of the
University of Bern, he continued
his research with a PhD thesis on
“Extrasolar planet population synthesis”
and a summa cum laude
graduation in 2008.
He spent several years at the Max-
Planck Institut for Astronomy in
Heidelberg as a post-doctoral fellow.
During this time he was
awarded an Alexander von Humboldt
fellowship, as well as a Reimar
Lüst fellowship.
In 2013 he finished his Habilitation
and became Privatdozent at
the University of Heidelberg, and
since 2015 has held a professorship
at the University of Bern where he
leads the research group “Planets-
InTime”. He is a member of the
HARPS and SPHERE consortia
that search for extrasolar planets
using the radial velocity and direct
imaging technique, respectively.