L10 From Planetesimals to protoplanets

wn of constrained quantity. For this reason, the dust-to-gas ratio is
idered varied in our simulations, and
Mass
takes two
growth
values depending
rate
on
II Accretion of planetesimals
n pro- the mid-plane temperature of the disc: fD/G for temperatures
ration below 150 K and 1/4 fD/G for higher temperatures. In principle,
Formation of a core
aminar the position of the iceline should evolve because of the viscous
, type I evolution of the disc. However, since our treatment of the plan-
Lissauer 1993
nearly etesimals disc is very simple, we do not take into account this Accretion rate of gas very low
. evolution.
Notes:
type I We assume that due to the scattering effect of the planet, the Depletion of the feeding zone
- the velocity dispersion of planetesimals enters only in focusing factor, but is the key factor.
r than surface density of planetesimals is constant within the current
anaka - the growth feeding zone rate but is larger decreases in with disks time with proportionally larger planetesimal to the mass surface densities. MZ < critical mass
nowl- - since accreted generally (and/or ejected decrease from thewith disc) distance, by the planet. planets The feed- grow slower at large distances.
erived - Fg can ingbe zone much is assumed more to complex extend toin a distance the three-body of 4 RH oncase. either
al fac- side of the planetary orbit, where RH ≡
nDecrease that of planetesimal surface density
rvival,
As the protoplanet grows, the surface density of planetesimals
ut just
must decrease in proportion. For the assumption of accretion
tofrom be: a feeding zone with spatially constant planetesimal surface
density one finds for a planet with semimajor axis a
(16)
entum
(17)
xpresof
the
Mplanet
1/3
aplanet is the
3M∗
Hills radius of the planet. For the inclinations and eccentricities
of the planetesimals, we use the following prescription (P96):
1 2GMplanetesimal 1
i =
√ , (20)
aplanet rplanetesimal 3Ω
where Mplanetesimal and rplanetesimal are the mass and radius of
planetesimals, at the location of the planet, and
e = max 2i, 2 RH
· (21)
aplanet
Finally, we also take into account the ejection of planetesimals
due to the planet, using the ejection rate given by Ida & Lin
(2004):
accretion rate
ejection rate =
02; and Bate et al. 2003) the migration prowalk,
and the mean value of the migration
be highly reduced, compared to the laminar
shown by Menou & Goodman (2004), type I
ass planets can be slowed down by nearly
itude in regions of opacity transitions.
ations seem to indicate that the actual type I
le may in fact be considerably longer than
stimated by Ward (1997) or even by Tanaka
hese reasons, and for lack of better knowlse
for type I migration the formula derived
002) reduced by an arbitrary numerical faceen
1/10 and 1/100. Tests have shown that
r is small enough to allow planet survival,
not change the formation timescale but just
igration (see Sect. 3.1).
velocity for low mass planets is taken to be:
Γ
net , (16)
Lplanet
lanet(GM∗aplanet)
4 Vesc,disk
, (22)
Vsurf,planet
1/2 is the angular momentum
e total torque Γ is given by:
2 Mplanet rPΩp
αΣ,P)
ΣPr
M∗ Cs,P
4 PΩ2 D/G
below 150 K and 1/4 fD/G for higher temperatures. In principle,
the position of the iceline should evolve because of the viscous
evolution of the disc. However, since our treatment of the planetesimals
disc is very simple, we do not take into account this
evolution.
We assume that due to the scattering effect of the planet, the
surface density of planetesimals is constant within the current
feeding zone but decreases with time proportionally to the mass
accreted (and/or ejected from the disc) by the planet. The feeding
zone is assumed to extend to a distance of 4 RH on either
side of the planetary orbit, where RH ≡
p, (17)
dlog Σ
und velocity and αΣ ≡ dlog r . In this expres-
P refers to quantities at the location of the
ration, two cases have to be considered. For
when their mass is negligible compared that
Mplanet
1/3
aplanet is the
3M∗
Hills radius of the planet. For the inclinations and eccentricities
of the planetesimals, we use the following prescription (P96):
1 2GMplanetesimal 1
i =
√ , (20)
aplanet rplanetesimal 3Ω
where Mplanetesimal and rplanetesimal are the mass and radius of
planetesimals, at the location of the planet, and
e = max 2i, 2 RH
· (21)
aplanet
Finally, we also take into account the ejection of planetesimals
due to the planet, using the ejection rate given by Ida & Lin
(2004):
accretion rate
ejection rate =
4 Vesc,disk
, (22)
Vsurf,planet
where Vesc,disk = d, compared to the laminar
u & Goodman (2004), type I
be slowed down by nearly
f opacity transitions.
dicate that the actual type I
e considerably longer than
d (1997) or even by Tanaka
d for lack of better knowlgration
the formula derived
an arbitrary numerical fac-
100. Tests have shown that
h to allow planet survival,
formation timescale but just
t. 3.1).
mass planets is taken to be:
(16)
/2 is the angular momentum
is given by:
2 Ωp
ΣPr
s,P
2 GM⊙/aplanet is the escape velocity form
the central star, at the location of the planet, Vsurf,planet
=
4 PΩ2 the position of the iceline should evolve because of the viscous
evolution of the disc. However, since our treatment of the planetesimals
disc is very simple, we do not take into account this
evolution.
We assume that due to the scattering effect of the planet, the
surface density of planetesimals is constant within the current
feeding zone but decreases with time proportionally to the mass
accreted (and/or ejected from the disc) by the planet. The feeding
zone is assumed to extend to a distance of 4 RH on either
side of the planetary orbit, where RH ≡
p, (17)
dlog Σ
αΣ ≡ dlog r . In this exprestities
at the location of the
s have to be considered. For
is negligible compared that
iven by the viscosity of the
Mplanet
1/3
aplanet is the
3M∗
Hills radius of the planet. For the inclinations and eccentricities
of the planetesimals, we use the following prescription (P96):
1 2GMplanetesimal 1
i =
√ , (20)
aplanet rplanetesimal 3Ω
where Mplanetesimal and rplanetesimal are the mass and radius of
planetesimals, at the location of the planet, and
e = max 2i, 2 RH
· (21)
aplanet
Finally, we also take into account the ejection of planetesimals
due to the planet, using the ejection rate given by Ida & Lin
(2004):
accretion rate
ejection rate =
4 Vesc,disk
, (22)
Vsurf,planet
where Vesc,disk = case. Moreover, as shown by Menou & Goodman (2004), type I
migration of low-mass planets can be slowed down by nearly
one order of magnitude in regions of opacity transitions.
These considerations seem to indicate that the actual type I
migration timescale may in fact be considerably longer than
the one originally estimated by Ward (1997) or even by Tanaka
et al. (2002). For these reasons, and for lack of better knowledge,
we actually use for type I migration the formula derived
by Tanaka et al. (2002) reduced by an arbitrary numerical factor
fI chosen between 1/10 and 1/100. Tests have shown that
provided this factor is small enough to allow planet survival,
its exact value does not change the formation timescale but just
the extent of the migration (see Sect. 3.1).
The migration velocity for low mass planets is taken to be:
daplanet
Γ
= −2 fIaplanet , (16)
dt
Lplanet
where Lplanet ≡ Mplanet(GM∗aplanet)
2 GM⊙/aplanet is the escape velocity form
the central star, at the location of the planet, Vsurf,planet
=
GMplanet/Rc is the planet’s characteristic surface speed, and
1/2 is the angular momentum
of the planet and the total torque Γ is given by:
2 Mplanet rPΩp
Γ = (1.364 + 0.541αΣ,P)
ΣPr
M∗ Cs,P
4 PΩ2 evolution of the disc. Howev
etesimals disc is very simple
evolution.
We assume that due to th
surface density of planetesim
feeding zone but decreases w
accreted (and/or ejected from
ing zone is assumed to exte
side of the planetary orbit, w
Hills radius of the planet. Fo
of the planetesimals, we use
1 2GMplanetesimal
i =
aplanet rplanetesimal
where Mplanetesimal and rplane
planetesimals, at the location
e = max 2i, 2
p, (17)
dlog Σ
where Cs is the sound velocity and αΣ ≡ dlog r . In this expression,
the subscript P refers to quantities at the location of the
planet.
For type II migration, two cases have to be considered. For
low mass planets (when their mass is negligible compared that
of the disc) the inward velocity is given by the viscosity of the
disc. As the mass of the planet grows and becomes comparable
RH
·
aplanet
Finally, we also take into ac
due to the planet, using the
(2004):
accretion rate
ejection rate =
Vesc,disk
Vsurf,planet
where Vesc,disk = Ejection
As the mass of the protoplanet increases, it becomes at some point massive enough to also
eject planetesimals form the nebula.
2 GM⊙/
the
central star, at the loc
GMplanet/Rc is the planet’s
Rc is the planet’s capture rad