Download Core formation in giant planets formed through

Core formation in giant planets
formed through gravitational
collapse
Diplomarbeit
der Philosophisch-naturwissenschaftlichen Fakultät
der Universität Bern
vorgelegt von
Frédéric Carron
2008
Leiter der Arbeit
Prof. Dr. W. Benz
Physikalisches Institut
Abteilung Weltraumforschung und Planetologie
Contents
1. Introduction
5
2. Formation of planets
7
2.1. Formation and structure of the protoplanetary disc
2.1.1. Structure . . . . . . . . . . . . . . . . . . .
2.1.2. The minimum-mass solar nebula (MMSN) .
2.2. Stability of discs and the Q-parameter . . . . . . .
2.3. Core-accretion model . . . . . . . . . . . . . . . . .
2.4. Giant gaseous protoplanet (GGPP) model . . . . .
2.5. Cross section . . . . . . . . . . . . . . . . . . . . .
2.5.1. Geometrical cross section . . . . . . . . . .
2.5.2. Gravitational cross section . . . . . . . . . .
2.5.3. Cross section of a gas ball . . . . . . . . . .
2.6. Hill radius . . . . . . . . . . . . . . . . . . . . . . .
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simulation
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3. Tools used
23
3.1. The Mercury integrator package . . . . . . . . . .
3.1.1. Principle of symplectic integrators . . . .
3.1.2. The utilization of Mercury . . . . . . . . .
3.1.3. Collisions and ejections . . . . . . . . . .
3.1.4. Test of Mercury . . . . . . . . . . . . . . .
3.2. Interaction of planetesimals with the atmosphere
3.2.1. Summary of the theoretical assumption .
3.2.2. Energy and mass deposition . . . . . . . .
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4. N-body simulation
23
23
25
26
26
28
28
31
32
4.1. Initial distribution of the planetesimals in the n-body
4.2. Impact statistics . . . . . . . . . . . . . . . . . . . .
4.2.1. Discussion of the results . . . . . . . . . . . .
4.2.2. Comparison with impacts in Jupiter . . . . .
5. Simulation of the interaction of the planetesimals with the growing clump
5.1. Structure of the clump . . .
5.1.1. Clump 1 . . . . . . .
5.1.2. Clump 2 . . . . . . .
5.2. Modication of CM module
5.3. Results of the simulations .
7
8
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13
16
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32
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Contents
5.4. Mass accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6. Conclusions and outlook
58
A. Generate a random variable following a given distribution using an uniform
distribution
60
A.1. A f (x) ∼ xp distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
A.2. A Rayleigh distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3
Abstract
In this thesis we investigate the core formation through accretion of planetesimals in
giant planets formed through gravitational collapse.
First, we make a n-body simulation with planetesimals and giant planets. This gives
us a statistic of the impacts of the planetesimals in the giant planet. Second, we simulate
the interaction of the planetesimals with the atmosphere of the planet. In that way we
look which part of the mass of the planetesimals reaches the centre of the giant planet and
which part is ablated in the atmosphere of the planet. Finally, the combination of both
results allows us to say how much mass is accreted by the giant planet, which fraction of
the mass reaches the centre of the planet and which fraction is ablated in the atmosphere.
We nd that with a typical initial mass distribution in the protoplanetary nebula we
can form cores of several earth masses if the giant protoplanet is dense enough.
1. Introduction
Before the rst discovery of an extrasolar planet in 1995 from Mayor and Queloz [1995],
there was no proof of the existence of extrasolar planetary systems. Since then, approximately 250 extrasolar planets were discovered. A large part of these planets are giant
gaseous planets. Unfortunately the formation of giant planets is still not well understood.
There coexist two dierent models of the formation of giant planets. The standard
model is the core-accretion model described by Pollack et al. [1996]. In this model the
formation of a giant planet occurs in two steps. In a rst step a rock-ice core is formed
by collisional accretion of planetesimals. When the core has reached a critical mass the
second step starts. In this step the core accretes the surrounding gas. Depending on
the initial protoplanetary disc mass, the simulations of Pollack et al. [1996] predict the
formation of planets like Jupiter in a timescale of about 1 × 106 − 5 × 107 years. Considering the typical lifetime of the disc of about 5 × 106 − 7 × 106 years [Haisch and Lada,
2001], the time to form a planets seems too long. Because once the disc is gone, no gas
accretion is possible anymore and therefore no giant planet can be formed.
This contradiction has motivated the second formation model. This model described
by Boss [1997] is the so called giant gaseous protoplanet (GGPP) model. According to
this model the formation occurs only in one step. Gravitational instabilities in the protoplanetary disc cause a local collapse of the gas and dust into a giant planet. This allows
to form planets in a few thousands of years. A main dierence of the GGPP model
compared with the core-accretion model is that it need no core to explain the formation
of a planet and that it does not predict the formation of a core. This means that the new
formed giant gaseous protoplanet has the same abundance of elements as the protoplanetary disc which has a solar composition. However, observations and theoretical models
show that the composition of giant planets in our solar system is enriched in heavy elements (i.e. elements with molecular weight greater than helium) compared with the sun.
Saumon and Guillot [2004] estimate the total amount of heavy elements in Jupiter to
8 - 38 M⊕ (M⊕ is the mass of the earth) which corresponds to an enrichment by a factor
of 1.5 - 6 compared with the abundance of heavy elements in the Sun. Furthermore
Saumon and Guillot [2004] estimate that in Jupiter between 0 M⊕ and 11 M⊕ of the
heavy elements are in the form of a solid core. In Saturn the mass of the core is between
9 M⊕ and 22 M⊕ and the total amount of heavy elements corresponds to an enrichment
by a factor of 6 - 14 compared with abundance of heavy elements in the Sun [Saumon
and Guillot, 2004].
5
1. Introduction
The fact the GGPP model do not explain the enrichment in heavy elements of giant planets compared to a solar elements composition can be perceived as an argument
against this model. But a possible explanation of the enrichment in heavy elements
could be the accretion of planetesimals by the giant planets. If a planet accretes enough
planetesimals, it could be enriched by a signicant amount of heavy elements. If enough
planetesimals reach the centre of the planet, the formation of a solid core is probable.
The goal of this work is to determine if a core can be formed after the creation of a
GGPP through accretion of planetesimals. For this we use a prole of a protoplanet computed by Mayer et al. [2004] and simulate impacts of planetesimals with a programme
developed by Christoph Mordasini [Mordasini, 2004] of the theoretical astrophysics research group of the university of Bern. This programme calculates the trajectory of
planetesimals in a planetary atmosphere. In particular this programme calculates the
gas drag, the ablation and mechanical destruction of the planetesimals. So we can look
if planetesimals can reach the centre of the GGPP. This part of the work is described in
section 5.
To determine with which velocities planetesimals collide with the GGPP we use the
programme Mercury of Chamber [1999]. This part of the work is describes in section 4.
Of course some notions of the formation of planetary systems are indispensable to
investigate the formation of cores in giant planets. So we recapitulate the formation of
planets in section 2. Especially we give a summary of the core-accretion model in section
2.3 and a summary of the giant gaseous protoplanet model in section 2.4.
6
2. Formation of planets
2.1. Formation and structure of the protoplanetary disc
1
This section gives a short overview of the formation of the protoplanetary disc. The
formation of planets always starts with the formation of a central star. Stars are formed
in giant molecular clouds (GMC). GMC are cold and dense regions in the galaxy. A
GMC is mainly composed of molecular hydrogen (approx. 71 % of the mass) and helium
(approx. 27 % of the mass). Only the two other per cent of the mass is composed of
heavy elements (i.e. elements with molecular weight greater than He). Approximately
half of the heavy elements is in the form of dust. The temperature of GMC is about 10
K and the average density is about 109 particles per m3 . The total mass of a GMC is
approximately 105 − 106 solar masses.
Spontaneous, or after a disturbance (for example created by a shockwave of a supernovae),
the GMC can become locally gravitationally unstable. A part of the GMC starts then to
fragment and to collapse into a protostar. In a rst approximation an isothermal cloud
of the temperature T and the mass m will collapse when its thermal energy ET is smaller
than the gravitational potential energy EG . The thermal energy is given by
E T = N kB T =
m
kB T
µmH
(2.1)
where N is the total number of molecules in the cloud, kB is the Boltzmann constant,
mH is the mass of one hydrogen atom, and µ is the mean molecular weight (in atomic
mass unit u).
For a spherical cloud an estimation of the gravitational potential is
EG =
Gm2
R
(2.2)
where G is the gravitational constant and R is the radius of the cloud.
Therefore the condition for collapse EG > ET can be written as
m
Gm2
>
kB T.
R
µmH
(2.3)
Assuming a cloud of constant density ρ, the radius of the cloud can be written as R =
1
Reference: [Irwin, 2003],[Benz, 2007]
7
2. Formation of planets
3m
4πρ
(1/3)
and the condition (2.3) leads to
r
m > mJ =
3 1
√
4π ρ
kB T
GµmH
3/2
1
≈√
ρ
kB T
GµmH
3/2
.
(2.4)
mJ is the so called Jean's mass. The Jean's mass gives a simple method to determine
the minimal mass of the cloud that will collapse at a given temperature and density.
Once the collapse of the cloud has begun, the denser part of this cloud will collapse
more quickly. This leads to a fragmentation of the cloud. Each fragment nally forms its
own star. During the collapse, the gravitational binding energy is converted into thermal
energy. First the cloud is optically thin, this means that the thermal radiation of the
nebula can escape from the gas. In a later phase the centre of the clump, which collapses
more quickly, becomes optically thick. At this point the temperature of the centre rises
very quickly. This slows down the velocity of the collapse but does not stop it. The
collapse stops only when the temperature is high enough to start the initial fusion of the
new born protostar.
Not the whole fragment of the cloud collapses in the protostar because the fragment
of the cloud has at the beginning of the collapse some net rotation due to the dierential
rotation of the galaxy. The entire angular momentum has to be conserved during the
collapse. So the inner part of the cloud will begin to rotate more rapidly. This considerably disturbs the ow to the centre of particles which are in the plane perpendicular
to the rotation axis (Laplacian plane). Once the centrifugal force, due to the rotation,
equals the gravitation force the migration will stop. This causes an accumulation of dust
and gas on the Laplacian plane, and leads to the formation of the protoplanetary disc.
In order to understand the process of planets formation it is important to have an idea
of the time needed to form a protoplanetary disc and its lifetime. The timescale of the
collapse and the formation of the protostar is about 105 years [Irwin, 2003]. At this time
the disc has a mass between 0.01 and 0.1 solar mass. Obvervations show that the disc
evolves within 106 − 107 years into a disc with a low mass and a low density [Haisch and
Lada, 2001]. After this time the planet formation is not longer possible. Taking into
consideration that our solar system exists since about 4.5 × 109 years the available time
to create a planet is relatively short.
2.1.1. Structure
The typical radius of a protoplanetary disc is approximately a few hundreds astronomical
units (AU). The thickness of the disc is approximately 10 % of the local distance to
the central star. In a good approximation the disc is axisymmetric so that cylindrical
coordinates (r, φ, z) are useful to describe the its structure. (The z -axis is the rotation
axis, r the distance from the rotation axis and φ the azimuth angle. But there is no
dependence of φ as long as the disc is axisymmetric. See gure 2.1.)
8
2. Formation of planets
Figure 2.1.: A schematic view of the protoplanetary disc. Gas and dust is drawn grey.
Vertical structure
The vertical structure of the disc is described by the hydrostatic equilibrium:
dp
= −ρgz = −ρg sin θ
dz
(2.5)
where p is the pressure and ρ the density of the gas, gz is the z-component of gravitational
acceleration and the angle θ is arctan zr .
?
The gravitational acceleration is given by g = rGM
2 +z 2 where M? is the mass of the central
z
star. For r << 1 and without self-gravitation of the gas and the dust, the gravitational
acceleration can be written in a good approximation as:
GM?
g∼
.
=
r2
(2.6)
For small θ, gz can be approximated as:
GM? ∼ GM? ∼ GM? z
= Ω2K z
=θ 2 =
r2
r
r2 r
q
?
here we introduce the Keplerian rotation rate ΩK = GM
.
r3
(2.7)
gz = sin θ
With the relation p = c2 ρ, where c is the speed of sound in the gas, the hydrostatic
equilibrium (2.5) takes the form:
Ω2 z
dρ
= −ρ K2 .
dz
c
(2.8)
This dierential equation can be solved by separation of variables
2
2
ΩK z
Ω2 z
Ω2 z 2
dρ
= − K2 dz ⇒ ln ρ = − K 2 + const. ⇒ ρ = ρ0 e− 2c2 ,
ρ
c
2c
(2.9)
with ρ0 = ρ(z)|z=0 .
√
With the introduction of the vertical scale height H = 2 ΩcK the vertical density prole
is simply given by:
z2
ρ = ρ0 e− H 2 .
9
(2.10)
2. Formation of planets
And the surface density Σ is:
Z∞
Σ=
ρ(z)dz =
√
πρ0 H.
(2.11)
−∞
Radial structure
Assuming that the gas rotates in a circular orbit around the protostar, the radial structure
2
can be determined using the fact that the centripetal acceleration ac = vr (with v as the
orbital speed of the gas) is equal to the radial acceleration. The radial acceleration is the
?
sum of the gravitation force GM
and the acceleration due to the pressure gradient ρ1 dp
dr .
r2
Therefore the basic equation for the radial structure is
GM? 1 ∂p
v2
=
+
.
r
r2
ρ ∂r
(2.12)
2.1.2. The minimum-mass solar nebula (MMSN)
We know that planets are formed from a gas cloud with solar composition. Today the
planets in our solar system have clearly not a solar composition. The minimum-mass
solar nebula (MMSN) is the minimum mass of gas and dust with solar composition that
allows to create the planets of our solar system.
To nd the value of the MMSN, the dust-to-gas ratio ζ is needed:
ζ=
mdust
= f (T )
mgas
(2.13)
where mdust is the mass of dust (high Z elements, Z = atomic number), mgas the mass
of gas and f (T ) a function of the temperature. This function is approximately a step
function. For temperature below about 170 K rocks and ices (water, methane, ammonia,...) are in a form of condensates. For these cold regions Hayashi [1981] estimates the
1
. In warm regions between 170 K and 1500K only solids
dust-to-gas value to ζcold ≈ 60
with high condensation temperature are in the form of condensates. In particular ice
1
does no more exist. Hence the dust-to-gas ratio drops to ζcold ≈ 240
[Hayashi, 1981].
The limit between warm and cold region is called the ice-line. In our solar system the
giant planets (Jupiter, Saturn, Uranus, and Neptune) are beyond the ice-line.
If we assume that:
ˆ the planets were formed at their present location
ˆ the mass of high Z matter in giant planet is about 80M⊕ (M⊕ is an earth mass)
ˆ the mass of terrestrial planets is about 2M⊕ ,
we can make a simple approximation of the mass of a MMSN as following [Benz, 2007]:
MM M SN =
2
ζwarm
+
80
= 5280M⊕ = 0.016M .
ζcold
10
(2.14)
2. Formation of planets
− 3
2
With the assumption that the surface density follows the law Σ = Σ0 rr0
(where
Σ0 is the surface density at the distance r0 from the central star) and with a more
sophisticated calculation Hayashi [1981] gets a MMSN mass between 0.35 AU and 36 AU
of 0.013M .
Figure 2.2 shows the density of the disc like assumed by Hayashi [1981].
Figure 2.2.: The plot shows the surface density of the MMSN like assumed by Hayashi
[1981]. We see at 2.7 AU the ice-line. Beyond the ice-line it is cold enough
that ice can condense, this leads to a jump of the solid density of a factor
four. Apart from this jump the density of gas and solids decreases with the
3
distance of the sun r like r− 2 . The total mass of the MMSN between 0.35
AU and 36 AU is 0.013 M .
The gure is taken from [Ruden, 1999].
2.2. Stability of discs and the Q-parameter
The stability of the disc plays a central role in the direct collapse planet formation model
(GGPP model). Only if the protoplanetary disc becomes locally unstable, a local gravitational collapse of the disc is possible. A relative simple study of the stability can
be made for an isothermal uniformly rotating disc with zero thickness. This is done in
detail in Binney and Tremaine [1987]. This section gives a brief summary of the stability
considerations and mainly follows the steps explained in Binney and Tremaine [1987] and
in Benz [2007].
We start with the equations which determine the dynamics of the disc: the continuity
equation (2.15), the Euler equation (2.16) and the Poisson's equation (2.17). The equations are written for a rotating frame of reference which rotates with the same angular
11
2. Formation of planets
~ = (0, 0, Ωz ) as the unperturbed disc:
velocity Ω
∂Σ
+ ∇· (Σ~v ) = 0,
∂t
∂~v
1
~ × ~v ) + Ω
~ 2 (xe~x + y e~y ),
+ ~v · ∇~v = − ∇p − ∇φ − 2(Ω
∂t
Σ
∆φ = 4πGΣδ(z).
(2.15)
(2.16)
(2.17)
Σ is the surface density of the disc, t is the time, ~v is the velocity of the gas, p is the
~ is the angular velocity of the disc, φ is the gravitational potential
pressure of the gas, Ω
eld, ~x = (x, y, z) is the position vector, e~x and e~y are unit vectors that point in the same
direction as the x and y axes. The equations 2.15 - 2.17 have an equilibrium solution for:
(2.18)
Σ = Σ0 ,
(2.19)
~v = 0,
(2.20)
2
p = p0 = c Σ 0 ,
where c is the sound speed in the gas.
We introduce small perturbations of the form
f (x, y, t) = f0 (x, y, t) + f1 (x, y, t), 1
(2.21)
in the equilibrium quantities. Where f (x, y, t) is some physical quantity (for example the
surface density Σ(x, y, t)), f0 (x, y, t) is the solution of the equilibrium state, f1 (x, y, t)
is the perturbation. We now plug in the perturbed quantities in the equations 2.15 - 2.17
and keep only linear terms in . That leads to the linearized equations which describe
the evolution of the perturbations:
∂Σ1
+ Σ0 ∇· v~1 = 0,
∂t
∂ v~1
c2
~ × v~1 ),
= − ∇Σ1 − ∇φ1 − 2(Ω
∂t
Σ0
∆φ1 = 4πGΣ1 δ(z).
(2.22)
(2.23)
(2.24)
We make the ansatz
~
Σ1 = Σa e−i(k·~x−ωt) ,
(2.25)
−i(~k·~
x−ωt)
v1 = (vax e~x + vay e~y )e
~
φ = φ e−i(k·~x−ωt) ,
1
a
,
(2.26)
(2.27)
where ~k is the wave vector of the disturbance. Without loss of generality we set ~k = k e~x .
12
2. Formation of planets
It is straightforward to show that the ansatz satises the equations 2.22 - 2.24 when:
ω 2 = 4Ω2 − 2πGΣ0 |k| + k 2 c2
(2.28)
This is the dispersion relation for the uniformly rotating sheet. If ω 2 is smaller than
0 the perturbations will grow exponentially (see equation 2.25 - 2.27) and the disc will
become unstable.
Now we search the minimal value of ω 2 as a function of k . Setting the rst derivative of ω 2
2π 2 G2 Σ2
0
. Perturbations
to zero, we nd that ω 2 has a minimum of 4Ω2 − c2 0 when k = πGΣ
c2
πGΣ0
with a wave number around c2 are the most unstable ones.
The condition ω 2 < 0 can be written as:
1
cΩ
> .
πGΣ0
2
(2.29)
This calculation is only valid for a uniformly rotating disc. A planetary disc has a
dierential rotation but a similar calculation can be made [Binney and Tremaine, 1987],
[Toomre, 1964]. For dierentially rotating discs we introduce the Q-parameter
Q=
cκ
πGΣ0
(2.30)
2
2 . For a nearly Keplerian disc the
+
4Ω
where κ is the epicyclic frequency κ2 = r dΩ
dr
epicyclic frequency is approximately the Keplerian angular speed ΩK .
For axisymmetric disturbances the stability criterion condition can be written as Q > 1.
For nonaxisymmetric disturbances the condition for stability change to Q ? 1.5 [Durisen
et al., 2006].
2.3. Core-accretion model
As already said, the core-accretion model explains the formation of giant planets with
the creation of a rock-ice core and with a strong gas accretion of this core. In simulations
Pollack et al. [1996] has showed that the formation consists of three phases.
In the rst phase the core is formed by accretion of planetesimals. Once the planetesimals reach a size of the order of 10 km the gravitation force becomes signicant. The
gravitation force leads to an increase of the eective collision cross section in comparison
with the geometrical cross section (see section 2.5). Therefore big planetesimals grow
faster than small ones. Approximately 5 × 105 years after the formation of the star a
number of planetary cores (so-called embryos) is formed. The cores have approximately
90 % of the original mass in its local feeding zone [Irwin, 2003]. The feeding zone is the
zone from which a planetary core can accrete planetesimals. Planetesimals which are not
in the feeding zone are not accreted by the planetary core. For a planetary core which
13
2. Formation of planets
is orbiting on a circular orbit around the central star, the feeding zone has the form of
an annular strip and is extended to a distance of 3.7RH on either side of the orbit of the
planetary core, where RH is the Hill radius (see section 2.6) [Pollack et al., 1996]. The
rest of the mass stays in smaller planetesimals. The mass of the embryos varies with the
distance to the star. Typical values are about 10 M⊕ (M⊕ is the Earth mass) at 10 AU.
At 1 AU the mass of the embryos is about 0.1 M⊕ .
The second phase starts when the embryo has enough mass to accrete the surrounding
gas. This happens when the embryo reaches a mass of the order of 10 M⊕ . The gas
accretion happens because the envelope (the envelope is the bound gas to the core) cools
and contracts. The contraction allows the planetary cores to accrete some additional
gas. The augmentation of the mass of the planet enlarges the feeding zone what allows
the accretion of some other planetesimals. The deceleration of the planetesimals heats
up the envelope again. Therefore the cooling and the resulting gas accretion happen
slowly. The exact duration of this phase strongly depend on the surface density of the
disc. Pollack et al. [1996] predict a duration of this phase of about 1 × 106 − 5 × 107 years.
The third phase begins when the accreted gas reaches a critical mass (approx. the mass
of the solid part of the planet). At this point the envelope starts to contract quickly.
The surrounding gas ows in. This phase stops only when the planet has accreted the
whole surrounding gas. The duration of this process is very short in comparison with
the second phase (approx. 105 years) [Irwin, 2003]. Figure 2.3 shows a simulation of the
formation of Jupiter.
Hence the timescale for the formation of giant planet is dominated by the duration
of the second phase, which is between 1 × 106 − 5 × 107 years [Pollack et al., 1996].
Considering the typical lifetime of the disc of about 5 × 106 − 7 × 106 years [Haisch and
Lada, 2001], the time to form a planet seems too long. Because once the gas of the disc is
gone, no gas accretion is anymore possible. To form giant planets in a shorter time than
around 106 years a solid density about four times larger than in the MMSN is needed.
This contradiction was a motivation to develop the direct collapse formation model (see
section 2.4).
But the core-accretion model has several advantages with regard to other models.
First, it gives an explanation for the formation of all kinds of planets. Earth like planets
never reach phase 2. Giant ice planets like Uranus and Neptune never reach phase 3 or
reach it very late when the most part of the gas of the nebula has already gone. Gaseous
giant planets make the whole formation process before the protosolar nebula has gone.
Secondly it also explains why giant planets are enriched in heavy elements (heavier than
He) with regard to the solar element composition. And it naturally explains how the
cores of planets are built. We remember here that giant planets of your solar system
are enriched in heavy elements with regard to the solar element composition and they
probably have a solid core [Saumon and Guillot, 2004] at the centre (see section 1).
14
2. Formation of planets
Figure 2.3.: Simulation of the formation of Jupiter, taken from Pollack et al. [1996]. It
shows the planet's mass as a function of time. Mp is the total mass of the
planet, MXY is the mass of the envelope and MZ is the mass of the core. We
see clearly the three formation phases. In the rst 5 × 105 years only the core
is built. Then we see the slow accretion of gas and solid until ≈ 7.5 × 106
years when the envelope mass is equal to the core mass. In the third phase
the mass of Jupiter increases very quickly.
15
2. Formation of planets
2.4. Giant gaseous protoplanet (GGPP) model
As said above the giant gaseous protoplanet (GGPP) model is an interesting alternative
to the core-accretion model. Boss [1997] proposes that in the early period of the protoplanetary disc giant planets, due to gravitational instability, may be created from the
collapse of the gas disc.
Mayer et al. [2004] study the evolution of gravitationally unstable protoplanetary discs
and the possible giant planet formation with three-dimensional smoothed particle hydrodynamics (SPH) simulations. They use disc masses out to 20 AU between 0.075 M and
0.125 M and central stars usually with a mass of 1M . The minimum Q-parameter
(see: section 2.2) is at the beginning of their simulation between 0.8 and 2 (mostly about
1.4). For their calculation Mayer et al. [2004] use either a locally isothermal equation
or an adiabatic equation. Within a few hundreds of years the Q-parameter can locally
decrease below 1, and parts of the disc become gravitationally unstable. This can lead to
the formation of clumps which are gravitationally bound regions of the disc and stable
to tidal disruption. The clumps contract quickly to a density several orders of magnitude higher than the initial disc densities. And nally this leads to the formation of
protoplanets. After approx. 103 years these planets have masses between 1 MX and 7
MX (MX is the Jupiter mass). Figure 2.4 shows an example of the evolution of the Q
prole in the simulation of Durisen et al. [2006]. And gure 2.5 shows an example of the
density of a gravitationally unstable protoplanetary disc after an evolution of 350 years
(simulation also from Durisen et al. [2006]).
The GGPP model has no problem to explain the formation of planets in the relative
short life time of the gas nebula. The giant planets are formed in approx. a thousandth of
the life time of the gas nebula. No core is needed in this model to simulate the formation
of planets. The formation of giant planets with very small cores is so possible. That is
not the case with the core-accretion model.
According to observation and theoretical models Jupiter has a solid core with a mass
between 0 and 11 M⊕ and Saturn has a core with a mass between 9 M⊕ and 22 M⊕
[Saumon and Guillot, 2004]. Hence is seems probable that all giant planets have a core.
But it is not clear how a core in a giant gaseous protoplanet is formed. According to
Boss [1997] a 1 MX GGPP contains 6 M⊕ of heavier elements than H and He. It is
possible that the heavy elements, if they are in the form of dust grain, grow by collisional
coagulation and move to the centre of the planets. A core could be formed in this way.
But this scenario does not explain why planets are enriched in heavy elements compared
with the sun. A possible explanation could be that giant gaseous protoplanets accrete
planetesimals after their formation.
The GGPP model explains only the formation of giant planets. It is not develop to
explain the formation of other planets like Uranus and Neptune which are be formed by
slow accretion of planetesimals like described in section 2.3.
16
2. Formation of planets
Figure 2.4.: Q as a function of the distance from the central star (R) of the disc at three
dierent times t: the thin solid line shows t = 0 a, the dashed line shows
t = 160 a and the thick solid line show t = 240 a. At the beginning the
minimum of Q is about 1.4. With the time the parameter Q locally decreases
far below 1. A fragmentation occurs between t = 160 a and t = 240 a at
approx. 12 AU.
The gure is taken from [Durisen et al., 2006].
Figure 2.5.: This gure shows the density of the same disc as in gure 2.4 after 350 years.
On the left side a locally isothermal equation of state is used for the whole
simulation. On the right side the locally isothermal equation of state is also
used but once the density exceed a critical value the simulation switches to
an adiabatic equation of state.
Brighter shades are for higher densities. The densities vary between 10−11
kg
and 10−3 m3 . The radius of the plotted disc is 20 AU.
The gure is taken from [Durisen et al., 2006].
17
2. Formation of planets
2.5. Cross section
The accretion of planetesimals by growing planets is an important aspect of the formation mechanism. Through accretion, planets increase their mass. The accretion is also
a heating mechanism of the gaseous envelope because of the deceleration of the planetesimals in the atmosphere. This can play an important role for the evolution of the
atmosphere's structure.
The interaction with the atmosphere is relatively complex to describe, but simulations
programmes exist (see section 3.2). In a rst step it is important to know how much
planetesimals are accreted. This can be determined with the concept of the cross section.
And as long as we do not calculate the interaction of planetesimals with the atmosphere
and make a two body model, it can be calculate analytically.
2.5.1. Geometrical cross section
In a rst step we ignore the gravitation force. Further we idealize the planet as a solid
sphere of radius r1 and the planetesimals as spheres of radius r2 . If the planetesimals y
on a straight line, all planetesimals which get closer than rc = r1 + r2 to the centre of
the planet will collide. In other words a planetesimal ying in one direction will collide
with the planet if it goes through an area of σ = πrc2 . σ is called the cross section.
2.5.2. Gravitational cross section
The geometrical cross section is increased if there is a gravitational interaction between
the planet and the planetesimal. This is simply due to the fact that a planetesimal will
not y on a straight line but will be attracted by the planet.
The calculation of the cross section with gravitation in a 2 body system is relatively
simple. Again we consider two solid spheres with radii of r1 and r2 and with masses of
m1 and m2 . The positions of the spheres are ~x1 and ~x2 . Therefore the position of the
~ cm is
centre of mass X
~ cm = ~x1 m1 + ~x2 m2 .
X
m1 + m2
(2.31)
Now we dene the relative position x of the spheres and the reduced mass µ,
~x = ~x1 − ~x2 ,
µ=
m1 m2
.
m1 + m2
(2.32)
(2.33)
The energy of the system can now be written as
E=
m1 + m2 ~˙ 2
µ
Gm1 m2
Xcm + ~x˙ 2 −
.
2
2
|x|
18
(2.34)
2. Formation of planets
Without loss of generality, we set the velocity of the centre of mass to zero. Now the
energy is
Gm1 m2
µ
.
(2.35)
E = ~x˙ 2 −
2
|x|
And the angular momentum ~l is
~l = µ~x × ~x˙ .
(2.36)
Now we use the conservation of energy and of angular momentum to derive the cross
section. First we consider the situation when the two bodies are innitely far from each
other. We set the initial velocity to ~x˙ 2∞ = (−ẋ2∞ , 0, 0). Therefore only the third component of the angular momentum vector will not be zero ~l = (0, 0, x2,∞ ẋ∞ ) = (0, 0, sẋ∞ ),
where s is the impact parameter at innity and is equal to the second component of ~x
at innity. So the energy and the angular momentum are
µ ˙2
~x ,
2 ∞
= µs~x˙ ∞ .
E∞ =
l∞
(2.37)
(2.38)
Figure 2.6.: A schematic view of the situation at innity. We see the two bodies with a
radius of r1 respectively r2 , the vector ~x = ~x1 − ~x2 , the velocity at innity
ẋ2∞ , and the initial impact parameter s.
In a second step we consider the situation at closest approach and set the closest
distance between the two spheres to r. If the distance is smaller, the spheres will collide.
The energy and the angular momentum are
µ ˙2
Gm1 m2
~xca −
2
| xca |
2
˙
= µxca ~xca .
Eca =
lca
(2.39)
(2.40)
If xca is smaller than rc = r1 + r2 the planesimals will collide with the planet. We dene
scrit as the maximum s that leads to a collision between the two bodies. And we use the
equations (2.37 -2.40) to express the square of the critical impact parameter s2crit as a
19
2. Formation of planets
function of rc and ~x˙ 2∞ . We get
s2crit
2Gm1 m2 rc2
= rc2
= rc2 +
| rc | ~x2∞
1+
vesc
~x˙ ∞
2 !
.
(2.41)
q
In the last step we use the fact that the escape velocity is given by vesc = 2Gmrc1 m2 . If
the initial impact parameter is smaller than scrit , the bodies will collide. Therefore we
can write the gravitational cross section σgrav as
!
vesc 2
2
.
(2.42)
σgrav = πrc 1 +
~x˙ ∞
2.5.3. Cross section of a gas ball
If the planet is not constituted of solids but of gases, the cross section will change with
regard to the gravitational cross section. In the extreme case it is possible that a solid
planetesimal ies through the whole planet without loosing enough energy to be captured
by the internal planet. The exact cross section depends not only on its mass and radius
but also on the structure of the atmosphere. But capture cross section will always be
equal or smaller than that of a solid planet with same mass and radius.
We can express the real cross section σ as the following product
σ = σgrav f ρ(r), p(r), T (r), ~x˙ 2∞ ,
(2.43)
where f ρ(r), p(r), T (r), ~x˙ 2∞ is a function depending on the density, the pressure and
the temperature of the planet as a function of distance to the centre r, and f also depends
on the initial velocity ~x˙ 2∞ . f is always smaller or equal to 1.
In order to determine the cross section we use the programme of Mordasini [2004] (see
section 3.2). The programme computes the trajectories of planetesimals, which start
from a distance of 100 Hill radii (numerical realization of innity) of the planet and
have a chosen velocity, for dierent initial impact parameter s. Then it searches the
maximal impact parameter where the planetesimal is no more captured by the planet.
Planetesimals are considered to be captured if they can not escape from the Hill sphere.
Figure (2.7) taken from the diploma work of Christoph Mordasini [Mordasini, 2004] shows
the situation.
2.6. Hill radius
The Hill radius gives an approximation of the range of the gravitational inuence of a
planet which is orbiting around a central star. If a body is closer to a planet than the
Hills radius, it will mainly feel the gravitational acceleration of the planet otherwise it
will mainly fell the gravitation of the sun.
20
2. Formation of planets
Figure 2.7.: This gure shows the principle to nd the critical impact parameter s. For
the critical impact parameter a planetesimal can just not come out of the
Hill sphere. In the gure a planet (with a solid core) is set at the position
x = 0 and y = 0. The impact parameter s is in this case scaled with the
impact parameter of the core sc,d .
This gure is taken from Mordasini [2004].
Between the star and the planet at a distance of one Hill radius from the planet, a test
particle is under equilibrium considering the three following force: the gravitational forces
of the planet Fg,pl. , the gravitational force of the star Fg,? , and the centrifugal force Fz
of a body orbiting with the same angular frequency as the planet. (Figure 2.8 shows the
situation.)
This condition leads to the equation Fz + Fg,pl. − Fg,? = 0 or:
ΩK (a)2 a +
Gm
GM?
−
= 0,
2
RH
(a − RH )2
(2.44)
q
GM
where ΩK (a) =
is the Keplerian angular velocity at a distance a of the star, RH
a3
is the searched Hill radius, m is the mass of the planet and M? is the mass of the star.
The insertion of ΩK (a) in equation (2.44), the division through G and the multiplication
with the common denominator lead to the equation:
2
2
M? R H
(a − RH )3 + ma3 (a − RH )2 − M? a3 RH
= 0.
Now we substitute
with x and assume in a second step that x 1.
⇒ M? a5 x2 (1 − x)3 − 1 + ma5 (1 − x)2 = 0.
(2.45)
RH
a
21
(2.46)
2. Formation of planets
Figure 2.8.: A schematic view of the star and the orbiting planet. The Hill sphere is
drawn with the color gray. The positions of the two Lagrangian points are
also drawn. A particle at L1 or L2 is under equilibrium with the centrifugal
force and the gravitational forces if the test particle rotate with the same
angular velocity as the planet.
And for small x it follows
and nally x =≈
m
3M?
(1/3)
⇒ −3M? a5 x2 + ma5 ' 0,
(2.47)
r
(2.48)
or
RH ≈
3
m
a.
3M?
As we see it in this formula the gravitational inuence of a planet depens in a strong way
on the distance of the planet from the star.
The two points on the axes planet-star at a distance of RH of the planet correspond to
the two Lagrangian points L1 and L2 . We remember than the Hill sphere gives only
an approximation of the range of the gravitational inuence of a planets. In reality
this range is not spherical. A detailed derivation of the forms of Hill surfaces an of the
Lagrangian points is given in Beutler [2005].
22
3. Tools used
As said in the introdudtion, we want to investigate the core formation through accretion
of planetesimals in giant planets formed through gravitational collapse. For this we make
an n-body simulation with planetesimals and giant planets. This gives us a statistic of
the impacts of the planetesimals in the giant planet. And then we simulate the interaction of the planetesimals with the atmosphere of the planet. The combination of both
results permits to nd the wanted results.
Mainly two programmes are used for this work. First we need a programme to perform
the n-body simulation with a large number of planetesimals. For this part we use the
Mercury integrator package (see section 3.1). We also need a programme which can
simulate impacts of planetesimals in atmospheres of giant planets. For this part the
programme developed by Christoph Mordasini is used (see section 3.2). This chapter
gives a short overview of this two programmes.
3.1. The Mercury integrator package
The Mercury integrator package (hereafter Mercury ) is a software package, written by
Chamber [1999], for simulations of n-body systems. The programme is written in Fortran
77 and is freely available on the web site of Chamber [1999].
Mercury is optimized for the calculation of objects which are moving in the gravitational
eld of a large central body. This is typically the case in a planetary system where
the motion of the planets is mainly due to the gravitational force of the central star.
Smaller bodies like planets cause mostly only small disturbances in the gravitational eld.
This fact is used in symplectic integrators, which are faster than conventional n-body
algorithm. They also have the advantage that they do not cause long term accumulation
of energy errors. But symplectic integrators become inaccurate if the disturbances in the
gravitational force become bigger than the attraction force of the central body. That is
the case if two planets are close to each other. For this reason Chamber [1999] uses in
Mercury a hybrid system. Mercury uses a symplectic integration method as long as there
are no close encounters between planets. If there are, Mercury switches to a conventional
integrator.
3.1.1. Principle of symplectic integrators
This section gives a short description of the principle of symplectic integrators and mainly
follows the steps explained in Chamber [1999].
23
3. Tools used
The Hamiltonian H , which is the sum of kinetic and potential energy, has in a n-body
system the following form
H=
N X
N
N
X
X
mi mj
p2i
−G
,
2mi
|ri − rj |
(3.1)
i=1 j=i+1
i=1
where pi is the momentum of the body i, mi is the mass of the body i, ri is the position
of the body i, N is the number of bodies, and G is the gravitational constant.
To calculate the evolution of any quantity q , which is a function of x and p and in
particular does not explicitly depend on the time ( ∂q
∂t = 0), we can use Hamilton's
equations of motion
∂H
∂pi
∂H
= −
.
∂xi
dxi
dt
dpi
dt
=
(3.2)
(3.3)
And we can write the time derivative of q as
N
dq X
=
dt
i=1
∂q dxi
∂q dpi
+
∂xi dt
∂pi dt
N X
∂q ∂H
∂q ∂H .
=
−
= F q.
∂xi ∂pi
∂pi ∂xi
(3.4)
i=1
In the last step we dene the operator
N X
∂ ∂H
∂ ∂H
F =
−
.
∂xi ∂pi
∂pi ∂xi
(3.5)
i=1
The resulting dierential equation can easily be solved
dq
= Fq ⇒
dt
Zq(t)
q(t−τ )
1
dq =
q
Zt
F dt0 ⇒ q(t) = eτ F q(t − τ ),
(3.6)
t−τ
where t is the time argument and τ the time step. The exponential term can be written
as a Taylor series, and we get
τ 2F 2
dq
= 1 + τF +
+ . . . q(t − τ ).
(3.7)
dt
2
Here we can split the Hamiltonian into two parts H = HA + HB . And we dene the
operators A and B like F in equation (3.5) but for the Hamiltonian HA respectively HB
instead of H . So we can express the time evolution of q with
dq
= eτ (A+B) q(t − τ ).
dt
24
(3.8)
3. Tools used
Now the strategy is to split up the exponential term eτ (A+B) in a product of exponential
functions which depend only on A or on B . It is important to notice that the operator
A and B do not commute. This means in particular that generally eA+B 6= eA eB .
They are dierent possibilities to split the exponential term. For example, if we want
second-order integration as it is used in Mercury, we get
τB
τB
dq
= e 2 eτ A e 2 q(t − τ ).
dt
(3.9)
It is relatively easy to show that this equation corresponds with the exact equation (3.8)
to the order O(τ 3 ). For that we only need to write the exponential term in equation
(3.9) as a Taylor series and compare the result with the Taylor series of equation (3.8).
An important characteristic of symplectic integrators is that they produce no long
term energy error. This is due to the fact that the integrator exactly solve a problem
which is close to the real one.
Chamber [1999] has shown that the smaller the ratio A/B is, the smaller the error term
O(τ 3 ) will be. For this reason it makes sense to split up the Hamiltonian H into a
dominant part HA and a small part HB . For a planetary system a good separation
could be made as following. HA describes the motion of the planets around the central
star but without gravitational forces between the planets. And HB describes only the
motion of the planets due to the attractive forces of the other planets. In general HA
is signicantly bigger than HB , but this is no more the case if the distance between
two planets becomes small enough. Therefore close encounters between planets causes a
decrease of the accuracy of the integrator. A solution to this problem could be to reduce
the time step τ . But a change of the time step leads to a shift of the energy of the whole
system. And this could destroy the symplectic characteristic of the integrator.
For these reasons Mercury switches during close encounters of bodies to a conventional
n-body integrator.
3.1.2. The utilization of
Mercury
The utilization of Mercury is relatively simple. The software package Mercury contains
a good manual, written by Chamber [1999], which explains in detail how to use Mercury.
This section gives only a brief summary of the important aspects of the utilization of
Mercury.
The programme package contains the following three programmes.
Mercury6_1.for
problem.
is the main programme. It does the whole integration of the n-body
converts the results of mercury6_1.for to a Keplerian orbital element.
This allows to follow the evolutions of the orbits of all bodies
Element6.for
Close6.for
is used to get detailed information about close encounters between bodies.
25
3. Tools used
There are dierent les to set the integration parameters like the start and ending time
or the output time interval.
We can dene two types of objects in Mercury
are bodies that feel the gravitational forces of all other bodies. They can
also collide with all other bodies.
Big bodies
are bodies that do not feel the gravitational force of other small bodies.
They do not necessarily have zero mass. If they have a non zero mass, they inuence
the motion of big bodies. Otherwise they are test particles. Furthermore they
cannot collide with other small bodies.
Small bodies
The initial position and velocity of each body can be dened in two les, one for big
bodies and one for small ones. The initialization can be made directly in Keplerian
orbital elements or in Cartesian coordinates.
3.1.3. Collisions and ejections
It is possible that a body collides with an other during the integration. In this case the
more massive body will accrete the less massive one. This means that Mercury stops the
integration of the less massive body and adds the mass of it to the more massive one.
Bodies could also be ejected out of the planetary system. If a body reach a critical
distance of the central star Mercury will simply stop the integration for this body. The
critical distance can be dened and is normally of the order of 100 AU.
Finally, if the distance between a body and the central star becomes smaller than a
minimal value, the integration will also stops. In this case Mercury assume that the
body fall into the central star.
3.1.4. Test of
Mercury
To assure that Mercury works well, we make an integration of the outer planets of
the solar system (Jupiter, Saturn, Uranus, Neptune, Pluto) over one million years, and
compare the results with known results from Beutler [2005]. Figures 3.1(a)-3.1(b) show
the evolution of the eccentricities of the outer planets over one million years. Figure
3.1(a) is obtained with Mercury. Figure 3.1(b) is taken from Beutler [2005].
26
3. Tools used
0.12
Jupiter
Staturn
Uranus
Neptune
Pluto - 0.14
0.1
Eccentricity
0.08
0.06
0.04
0.02
0
0
200
400
600
800
1000
kYears
(a) Simulation with
Mercury
(b) Reference from Beutler [2005]
Figure 3.1.: Evolution of the eccentricities of the outer planets over one million years.
Figure a) is obtained with Mercury. Figure b) is taken from Beutler [2005].
For Pluto e − 0.14 is plotted where e is the eccentricity.
We see a strong correlation between the two plots. Plots for the other Keplerian orbital
elements show a similar correlation. The results of Beutler [2005] are smoother because
the eccentricity was averaged over a time intervals of ve revolutions of Jupiter in order
to avoid short period eects.
We can conclude that Mercury works well.
27
3. Tools used
3.2. Interaction of planetesimals with the atmosphere
Christoph Mordasini [Mordasini, 2004] has written a simulation programme to calculate
the interaction of planetesimals with an atmosphere. This simulation module (hereafter
CM module ) was written as a part of a larger project whose goal is to simulate the
formation of giant planets with migration and disc evolution [Alibert et al., 2005].
For given pressure p, density ρ and temperature T of an atmosphere, CM module calculate
the trajectory of a planetesimals in the atmosphere.
Apart of the gravitational force, CM module takes also into consideration drag force,
thermal ablation and mechanical destruction.
A very detailed description of CM module is given in Mordasini [2004], and a short
description of the physical aspects is given in Alibert et al. [2005, chap. 2.3.2]. This
section will only give a short summary of CM module. A summary is needed because
CM module is with Mercury (3.1) one of the two most important tools for this work.
3.2.1. Summary of the theoretical assumption
Structure of the planet
The structure of the planets is assumed to be central symmetric, i.e. pressure p, density ρ
and temperature T depend only on r, where r is the distance of the centre of the planets.
The structure has to be given to the programme as a table of r, p(r), ρ(r), T (r) and
M (r), where M (r) is the mass of the planets within the sphere of radius r,
Zr
M (r) =
4πr2 ρ(r0 )dr0 .
0
CM module was written in order to study planets which have a core. So normally there
is also a solid core in the middle of the planets. For this work some modication has
been made to treat a planet without core.
Structure of the planetesimals
Planetesimals are dened with their mass and with the material properties (density, melting temperature, etc.). For the calculation of the gravitational force the planetesimals
are treated as point masses, for the drag force the radius is of course used. The form of
the planetesimal is assumed to be spherical. If fragmentation of a planetesimal occurs,
the planetesimal is treated as a number N of smaller and identical planetesimals.
Gravity
The gravitational acceleration of the planetesimals ~agrav is calculated as
~agrav = −
GM (r)
· ~r.
r3
28
3. Tools used
This means CM module considers the gravitational acceleration of a two bodies system
(the planet and the planetesimals), i.e. the attraction of the central star is ignored. This
assumption makes sense because in the atmosphere of the planets the main attraction is
caused from the planet.
The acceleration on the planet due to the planetesimals is also ignored because the mass
of the planet is much bigger than the mass of the planetesimals.
Drag force
Once a planetesimal enters into the atmosphere of the planet, it feels an aerodynamic drag
force. The calculation of the drag force is not trivial. The main diculty is that on the one
side the density of an atmosphere can easily change by 10 orders of magnitude. And on
the other side the properties of in-falling planetesimals, like velocity and radius, are very
variable. So the calculation of the gas drag has to cover a wide range of hydrodynamical
regimes.
CM module calculates the drag force with the formula for turbulent regime of Landau
and Lifshitz [1959]:
~r
1
F~D = CD ~r˙ 2 S ,
2
r
(3.10)
where F~D is the drag force, CD the drag coecient for a sphere which is a function of
the gas properties and the velocity and radius of the planetesimal, ρ(r) the density of
the gas, and S the cross section of the planetesimal. This function is calculated with
the equation of Henderson [1976], which calculate CD in a way that formula 3.10 gives a
realistic drag force value for a very wide range of ow regimes. So that we can also use
formula 3.10 when we do not have a turbulent regime.
Thermal ablation
When a planetesimal is braked through the drag force, it looses kinetic energy. This
energy is converted into thermal energy. A fraction of the energy heats the planetesimal
the other part heats the surrounding gas. If the heat ux is high enough, this eect can
lead to melting or vaporization of the surface of the planetesimal.
A simple way to describe the change of the planetesimal's mass is given as following. The
work dWD due to the drag force is:
dWD = F~D · d~r = FD dr,
(3.11)
where FD =| F~D |. If we assume that the energy Qabl is needed to melt or vaporize a
mass unit and that not the whole dissipated energy heats the surface of the planetesimals
but only CH times the dissipated energy, the change of the mass of the planetesimals per
time unit is:
dM
dWD 1
1
1
=−
= − CH CD ~r˙ 3
S.
(3.12)
dt
dt Qabl
2
Qabl
29
3. Tools used
The main diculty of using this formula is to determine the value of CH . Like CD , CH
is a function of the gas properties and the velocity and radius of the planetesimals. It
can vary by about ve orders of magnitude.
This expression for ablation neglects: the thermal emission of the surface of the planetesimals, the absorption of the ambient thermal radiation, the convective energy exchange
with the gas and also the heat transport to the inner part of the planetesimal. The last
point is not treated in CM module, the others are.
At supersonic velocity the expression 3.12 is no more valid. Then the heat input is
proportional to the thermal emission of the shock front. In this case CM module computes
the post-shock temperature which is a function of the velocity of the planetesimal and
of the gas properties. Then the energy input Ein can be expressed as:
Ein = CH,rad (T )σT 4 ,
(3.13)
where T is the post-shock temperature, σ is the Stefan-Boltzmann constant and CH,rad is
a function depending on T , which describes how much energy of σT 4 reaches the surface
of the planetesimals.
Mechanical destruction
A planetesimal, which is ying with a high velocity through a dense region of an atmosphere, can be exposed to a very high aerodynamic pressure. The pressure at the front
pf ront of the planetesimal is higher than the pressure on the lateral side pside of the planetesimal. This pressure dierence ∆p = pf ront − pside can cause mechanical destruction.
For a rst approximation of ∆p we can use the Bernoulli's equation for incompressible
ow (and a constant gravitational potential) :
p = pamb −
ρv 2
,
2
(3.14)
where p is the static pressure, pamb is the ambient pressure, and ρv 2 /2 is the dynamic
pressure. The pressure at the front of the body is the same as the ambient pressure. On
the lateral side of the body the static pressure is now given by equation 3.14. And ∆p is
given by the dynamic pressure ρv 2 /2.
Obviously the assumption of incompressible ow is not really correct when the y velocities of the planetesimals are high, and they are high. For this reason CM module
uses a more sophisticated model for the calculation of ∆p. Essentially CM module use
an adiabatic deceleration and the correct properties of shock waves. That leads to the
following result.
 γ

p 1 + γ−1 Ma 2 γ−1 − p,
if Ma < 1,
2
∆p
(3.15)
γps

pps 1 + γps −1 Ma 2 γps −1 − p, if Ma ≥ 1,
2
30
3. Tools used
Where Ma is the Mach number (Ma = v/cs , where cs is the speed of sound), γ is the
ratio between the specic heats, the value with the subscript 'ps' are calculated with the
shock waves jump condition and the equation of state of Saumon et al. [1995]. A detailed
explanation of equation (3.15) is given in Mordasini [2004, chap. 3.3].
Once the pressure dierence ∆p overcomes a critical value ∆pcrit which coresponds to
the tensile strength or the self gravity tensile strength of the body, the body will start
to deform itself. How the deformation exactly occurs depend on if the body can react
sucient quickly to change of pressure. If ∆pcrit is built up in a timescale larger than
the time needed for a sound wave to travel through the body, the body will stay in a
pressure equilibrium. This is the so called static regime. Because the body always stays
in a pressure equilibrium and that the pressure on the side of the body is smaller than
at the front, the body will rst stretch itself in the lateral direction. Then it becomes
instable and fragments.
The other regime is the dynamical regime. Here the body is not in a pressure equilibrium
and the mechanical ablation occurs at the front of the body.
3.2.2. Energy and mass deposition
After CM module calculated the whole trajectory of a planetesimal through an atmosphere, a subroutine calculated the mass and energy deposition in each layers of the
atmophere. This is particularly important if the evolution of the planetary atmosphere
should be calculate.
31
4. N-body simulation
4.1. Initial distribution of the planetesimals in the n-body
simulation
The orbit of a planetesimal is dened by the six Keplerian orbital elements which are
the semimajor axis a, the eccentricity e, the inclination i, the longitude of the ascending
node, the argument of periapsis and the mean anomaly at a given epoch. This section
shows how the distribution of each element is chosen.
The distribution of the semimajor axis is made in such a way that it corresponds to
the surface density used by Alibert et al. [2005]. The surface density of the mass of the
planetesimals as a function of the distance from the star is dened as
Σ(r) =
dm(r)
dm(r)
=
,
dA
πrdr
(4.1)
where r is the distance from the star, m(r) is the mass of all planetesimals which are
closer to the star than r, dA = πrdr is an innitesimal surface element.
The initial surface density is set to
Σ(r) = Σ(r0 )
r
r0
−3/2
,
(4.2)
where Σ(r0) is the surface density of planetesimals at the distance r0 .
Within one n-body simulation run the mass of all planetesimals are the same. Therefore
the mass can be expressed as m(r) = N m0 where N is the number of planetesimals,
which are closer to the star than r, and m0 is the mass of one planetesimal. Now we can
calculate the density function of the planetesimals dN (r)/dr as a function of r,
dm(r)
Σ(r) =
πrdr
=
dN (r)
dr
=
⇒
−3/2
m0 dN (r)
r
= Σ(r0 )
πrdr
r0
−1/2
πr0 Σ(r0 ) r
.
m0
r0
(4.3)
(4.4)
For all planetesimals a semimajor axis between aplanet − 6RH and aplanet + 6RH is generated, where aplanet is the semimajor axis of the planet and RH the Hill radius of
the planet. Figure (4.1) shows an example of the distribution of the semimajor axis of
32
4. N-body simulation
planetesimals.
0.3
0.25
Probability
0.2
0.15
0.1
0.05
0
6
8
10
12
Semimajor axis [AU]
14
Figure 4.1.: Distribution of the semimajor axis of planetesimals. The planet has a mass
of 2.4 MX and a semimajor axis of 10 AU.
Appendix A shows how to generate a dN (r)/dr ∼ rp distribution for p ∈ R.
For the density function of the inclination fi (sin(i), sin(i0 )) and for the density function
of the eccentricity fe (e, e0 ) we assume, like Greenzweig and Lissauer [1992], a Rayleigh
distribution,
fe (e, e0 ) =
fi (sin(i), sin(i0 )) =
2
e
2e
exp − 2
2
e0
e0
sin2 (i)
2i
exp − 2
,
sin2 (i0 )
sin (i0 )
(4.5)
(4.6)
where e0 and sin(i0 ) are the root mean square of the corresponding distribution,
Z∞
fe (e, e0 )e2 de = e20
(4.7)
0
Z∞
fi (sin(i), sin(i0 )) sin2 (i)d (sin(i)) = sin(i0 )2 .
0
33
(4.8)
4. N-body simulation
The parameter e0 and sin(i0 ) are set by the prescription of Pollack et al. [1996].
s
2Gmplanetesimal 1
1
√
(4.9)
i0 =
aplanet
rplanetesimal
3ΩK
RH
,
e0 = 2max i0 ,
(4.10)
aplanet
where aplanet is the semimajor axis of the planet, mplanetesimal is the mass of a planetesimal, ΩK is the Keplerian rotation rate at the location of the planet and RH is the Hill
radius.
0.45
0.45
0.4
0.4
0.35
0.35
0.3
Probability
Probability
0.3
0.25
0.2
0.15
0.25
0.2
0.15
0.1
0.1
0.05
0.05
0
0
0
0.2
0.4
0.6
0.8
1
0
0.1
0.2
Inclination [degrees]
0.3
0.4
0.5
Eccentricity
(a) Inclination
(b) Eccentricity
Figure 4.2.: Distribution of the inclination and the eccentricity for ice planetesimals (ρ =
1000 kg/m). The planet has a mass of 2.4 MX and a semimajor axis of 10
AU.
The other three Keplerian elements are uniformly distributed.
In order to get a signicant statistic of the impacts we use, independently of the radius
of the planetesimals, for each run 10000 planetesimals. Unfortunately it is not possible
to set the whole mass of solids in form of planetesimals because the total number of
planetesimals would be too high to compute a run in a reasonable time. For example, if
we assume a typical surface density of solids of 75 kg/m2 at 5.2 AU, the mass of the solid
between 5 and 15 AU corresponds to the mass of approximately 108 ice planetesimals
with a radius of 100 km.
For the n-body simulation the mass of the planetesimals is set to zero. This means
that planetesimals are considered as test particles. Therefore the planet will conserve
during the whole simulation time exactly the same orbit. If the planetesimals would
have a mass the planet could migrate inward due to the ejection of planetesimals. But
the migration will not exceed a fraction of an AU. This is due to the relative low mass
34
4. N-body simulation
of the planetesimals compared to the mass of a planet. For a rough approximation we
assume that the whole mass of the planetesimals is concentrated in one body at 10 AU.
If we take the whole mass of solids between 5 AU and 15 AU for a surface density of 75
kg/m2 , this body would have a mass of 0.2 MX . The planet is also at 10 AU and has a
mass of 2.4 MX . Now we calculate the change of the semimajor axis of the planet if all
planetesimals are ejected. The energy of a body, with a semimajor axis a and a mass m,
is given by (two body case)
E=−
1 GM m
.
2
a
(4.11)
Therefore the conservation of energy leads to
−
1 GM mplanet 1 GM mplanetesimals
1 GM mplanet
−
=−
.
2
aplanet
2
aplanetesimals
2
ãplanet
⇒ ãplanet =
mplanet aplanet
.
mplanet + mplanetesimals
(4.12)
(4.13)
With the values of above we get a new semimajor axis of the planet of ã = 9.1 AU. This
means that the planet has reduced its semimajor axis by 10 % when we assume 100 %
ejection.
35
4. N-body simulation
4.2. Impact statistics
The n-body simulation programme Mercury (see chapter 3.1) treats planets like solid
bodies. If a planetesimal comes nearer to the centre of the planet than the planet's
radius, Mercury assumes that the planetesimal hits the planet and removes it. Some
little modications have been done in the programme Close6.for, which is used to Figure
4.3 illustrates the situation.
Figure 4.3.: For each impact between a planetesimal and the planet the velocity v =| ~v |
and the angle α are computed.
Our simulation integrates the evolution of the rst 105 years of the n-body problem.
After this time the rate of impact is considerably smaller as at the beginning. Figures
4.4(a) and 4.4(b) show the number of impacts as a function of time for the two dierent
protoplanet (the protoplanets are called: clump 1 and clump 2, the structure of these
protoplanets is described in section 5.1 ).
Now we look how much of the initial 10000 planetesimals have collided with the planets
and how much have been ejected out of the solar system (are farer away than 100 AU of
the central star) after a time of 105 years. Table 4.1 shows the results for all runs.
For the same clump the results are mainly the same for all types of planetesimals. But
the results are clearly dierent between clump 1 and clump 2. Because clump 2 is less
massive than clump 1 and also larger, the escape velocity from the surface of the clump
2 is smaller than the escape velocity of clump 1. This leads to a smaller ejection-rate.
The following histograms show the impact velocities and angles in the smaller clump
1 for dierent initial conditions.
36
4. N-body simulation
Distribution of the time of impacts
4
3.5
3.5
log10 (# of impacts in 1000 a)
log10 (# of impacts in 1000 a)
Distribution of the time of impacts
4
3
2.5
2
1.5
1
3
2.5
2
1.5
1
0.5
0.5
0
0
2
4
6
8
Time [a]
(a)
0
0
10
2
4
6
8
Time [a]
4
x 10
Impacts in Clump1
(b)
10
4
x 10
Impacts in Clump 2
Figure 4.4.: Distribution of the times of the impacts. These plots show the result for 100
km ice planetesimals. The results of the other planetesimals are similar.
Clump
Planetesimals
Size [km]
1
1
1
1
1
1
Ice
Ice
Ice
Stone
Stone
Stone
10
100
1000
10
100
1000
2
2
2
2
2
2
Ice
Ice
Ice
Stone
Stone
Stone
10
100
1000
10
100
1000
fhit [%]
57
55
50
55
55
55
56
56
55
55
55
52
fejected [%]
8
9
14
8
8
8
2
2
2
2
2
2
fstillorbiting [%]
35
36
36
36
36
36
42
42
43
43
43
45
Table 4.1.: fhit indicates the fraction of the initial 10000 planetesimals which collided
with the planets, fejected indicates the fraction of the ejected planetesimals and
fstillorbiting indicates the fraction of the planetesimals which are still orbiting
around the central star.
37
4. N-body simulation
Distribution of the impact angles
Distribution of the impact velocities
600
250
500
200
Number
Number
400
300
150
100
200
50
100
0
0
1000
2000
3000
4000
Impact velocity [m/s]
5000
0
0
6000
10
20
(a) Velocity
30
40
50
60
Impact angle [degree]
(b) Angle
70
80
90
α
Figure 4.5.: Distribution of the impact velocities and impact angles in clump 1 for ice
planetesimals with a radius of 10 km. The dashed line shows the escape
velocity of the planetesimals.
Distribution of the impact angles
Distribution of the impact velocities
600
250
500
200
Number
Number
400
300
150
100
200
50
100
0
0
1000
2000
3000
4000
Impact velocity [m/s]
5000
0
0
6000
(a) Velocity
10
20
30
40
50
60
Impact angle [degree]
(b) Angle
70
80
90
α
Figure 4.6.: Distribution of the impact velocities and impact angles in clump 1 for ice
planetesimals with a radius of 100 km. The dashed line shows the escape
velocity of the planetesimals.
38
4. N-body simulation
Distribution of the impact angles
Distribution of the impact velocities
600
250
500
200
Number
Number
400
300
150
100
200
50
100
0
0
1000
2000
3000
4000
Impact velocity [m/s]
5000
0
0
6000
10
20
(a) Velocity
30
40
50
60
Impact angle [degree]
(b) Angle
70
80
90
α
Figure 4.7.: Distribution of the impact velocities and impact angles in clump 1 for ice
planetesimals with a radius of 1000 km. The dashed line shows the escape
velocity of the planetesimals.
Distribution of the impact angles
Distribution of the impact velocities
600
250
500
200
Number
Number
400
300
150
100
200
50
100
0
0
1000
2000
3000
4000
Impact velocity [m/s]
5000
0
0
6000
(a) Velocity
10
20
30
40
50
60
Impact angle [degree]
(b) Angle
70
80
90
α
Figure 4.8.: Distribution of the impact velocities and impact angles in clump 2 for ice
planetesimals with a radius of 10 km. The dashed line shows the escape
velocity of the planetesimals.
39
4. N-body simulation
Distribution of the impact angles
Distribution of the impact velocities
600
250
500
200
Number
Number
400
300
150
100
200
50
100
0
0
1000
2000
3000
4000
Impact velocity [m/s]
5000
0
0
6000
10
20
(a) Velocity
30
40
50
60
Impact angle [degree]
(b) Angle
70
80
90
α
Figure 4.9.: Distribution of the impact velocities and impact angles in clump 2 for ice
planetesimals with a radius of 100 km. The dashed line shows the escape
velocity of the planetesimals.
Distribution of the impact angles
Distribution of the impact velocities
600
250
500
200
Number
Number
400
300
150
100
200
50
100
0
0
1000
2000
3000
4000
Impact velocity [m/s]
5000
0
0
6000
(a) Velocity
10
20
30
40
50
60
Impact angle [degree]
(b) Angle
70
80
90
α
Figure 4.10.: Distribution of the impact velocities and impact angles in clump 2 for ice
planetesimals with a radius of 1000 km. The dashed line shows the escape
velocity of the planetesimals.
40
4. N-body simulation
4.2.1. Discussion of the results
As described in section 4.1, dierent massive planetesimals have a dierent initial distribution of the inclination and of the eccentricity. Small planetesimals have all an initial
inclination near to zero (see equation 4.9). Bigger planetesimals have a larger mean inclination and have also a larger variation of the inclination. This dierence can be observed
in the impact statistics. An initial inclination near to zero leads to an accumulation of
the impact angles near to zero (central impact). The impact angles for bigger initial
inclination are distributed around 45°. The impact velocities are distributed around the
escape velocity. Impacts in the larger clump 2 (see section 5.1.2) are similar with the
dierence that the impact velocity is smaller due to the smaller escape velocity.
The clear dierences between the distributions of the impact angles can be understood. For this we consider the theoretical distribution of two borderline cases. First we
calculate the distribution of the impact angles if all planetesimals would come from the
same direction neglecting gravitation. Small planetesimals come mostly from the same
direction because they have a very low inclination. Therefore they are in the same plane
as the planet. Figure 4.11 shows the situation. The impact angle α is given by α = sin(a)
da
= cos(α). Assuming that the
where the radius of the sphere is set to 1. Therefore is dα
da
impact parameter a is distributed uniformly in (0,1), the distribution dN
dα ∼ dα = cos(α)
where N is the number of impacts. For high inclinations the planetesimals can come
from all directions. In the extrem case the directions of the planetesimals are uniformly
distributed. Nyenegger [2005] has shown that for random collisions the impact angles
follows the distribution dN
dα ∼ cos(α) sin(α).
Futhermore the impacts velocity in the larger clump 2 (see section 5.1.2) are smaller
than the ones in clump 1 (see section 5.1.1, this is due to the smaller escape velocity of
clump 2.
Figure 4.11.: Situation if all planetesimals come from the same direction. The impact
parameter a is uniformly distributed
41
4. N-body simulation
1
cos(alpha)
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Probability
Probability
1
0.9
0.5
0.4
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
cos(alpha) * sin(alpha)
0
0
10
20
30
dN
(a) dα
40
50
Angle [degrees]
60
70
80
90
0
10
20
dN
(b) dα
∼ cos(α)
30
40
50
Angle [degrees]
60
70
80
90
∼ cos(α) sin(α)
Figure 4.12.: Theoretical distribution of the impact angles for the two discussed cases.
4.2.2. Comparison with impacts in Jupiter
Jupiter has a mass of the same order of magnitude that clump 1 and clump 2 but its
radius is 650 times smaller than the one of clump 1. This leads to an escape velocity
about ten times higher for Jupiter than for clump 1. Therefore the impact velocities are
a factor 10 higher than for clump 1.
An other signicant dierence concerns the fraction fhit , which is the fraction of planetesimals which collided with the clump, and the fraction fejected which is the fraction of
planetesimals which are ejected. fhit for Jupiter is about 6 % and fejected is about 32 %.
fhit
Therefore the ratio fejected
is about 0.19. However, for the clump 1 this ratio is about
6.9. This large dierence is due to the higher escape velocity for Jupiter.
42
5. Simulation of the interaction of the
planetesimals with the growing clump
In section 4.2 we have produced a statistic of the impacts in the giant gaseous protoplanets (GGPP). The goal is now to investigate what happens to the planetesimals that
hit a GGPP. In particular, we want to know which ones reach the centre, which ones are
destroyed in the other parts, or which ones simply go through the GGPP. For this, we
generate a number of simulations of planetesimal-clump interactions with dierent start
velocities and start angles (see gure 4.3). The simulation is made from the point where
Mercury stops the n-body calculation by using the programme of Mordasini [2004] (CM
module ).
In the rst part of this chapter (section 5.1) we describe the structure of the prole of
the giant gaseous protoplanet which are used. The second part (section 5.2) describes the
modication that we made in CM module. Finally in section 5.3 we discuss the results
of the simulation.
5.1. Structure of the clump
Lucio Mayer [Mayer, 2007] gave us the structures of two giant gaseous protoplanet (socalled clumps) which are resulting from his SPH simulation of the gravitational instability
of the protoplanetary disc (see section 2.4). For their calculation Mayer [2007] use a adiabatic equation of state with γ = 1.4 (γ is the ratio between specic heats) and radiative
transfer is included using a ux-limited diusion (for details about the ux-limited diffusion see Mayer et al. [2007]).
Notice that [Mayer, 2007] calculated clump 1 with a better resolution than clump 2.
The two clumps have not the same structure. In order to get results for a broader range
of possible giant gaseous protoplanet (GGPP), we perform our calculations with the two
clumps. Table 5.1 gives an overview of the radius and the mass of the two clumps. The
dierence in the structure of the two clumps is mainly due to the fact that [Mayer, 2007]
use a 10 times better spatial resolution for clump 1 than for clump 2.
5.1.1. Clump 1
The plots (gure 5.1, 5.2) show the density, the pressure, the temperature and the mass
of the clump as a function of the distance to the centre r. As we see the density and
43
5. Simulation of the interaction of the planetesimals with the growing clump
r[AU ]
m[MX ] RH [AU ]
−1
Clump 1 3.13 × 10
3.77
1.06
Clump 2 7.49 × 10−1 1.34
0.75
Table 5.1.: Overview of the two clumps: r is the radius of the planet, m the mass and
RH the Hill radius.
the pressure rise quickly near the centre. The consequence is that a large fraction of the
mass is in the inner regions of the clump (see gure 5.2(b)).
The markers in form of a cross in the plots show the value of the original data of Mayer
[2007]. In order to get a prole with more data points to prevent big steps in the prole
we make a cubic interpolation of the logarithm value (for the mass no interpolation is
done, the mass is calculated with the interpolated density). Between the centre and the
most inside data point the density, the pressure and the temperature is assumed to be
constant.
6
0
10
−1
10
−2
10
10
5
10
4
Pressure [Pa]
Density [kg/m3]
10
−3
10
−4
10
3
10
2
10
1
−5
10
−6
10
10
0
10
−1
−7
10
0
1
2
3
r [m]
4
10
5
0
1
2
3
r [m]
10
x 10
(a) Density
4
5
10
x 10
(b) Pressure
Figure 5.1.: Structure of clump 1, gure a) shows the density as a function of the distance
to the centre r. Figure b) shows the pressure as a function of r. The solid
line shows the values used and the crosses show the values of the original
data points of Mayer [2007].
5.1.2. Clump 2
The second clump (see gure 5.3, 5.4) has a much smoother prole than the rst one.
Only the most outside part of the clump has a relative strong noise. To get a smoother
prole with more points, we make a polynomial t the density, the pressure and the
44
5. Simulation of the interaction of the planetesimals with the growing clump
30
8
800
x 10
7
700
600
5
Mass [kg]
Temperature [K]
6
500
4
3
400
2
300
200
0
1
1
2
3
r [m]
4
0
0
5
1
2
3
r [m]
10
x 10
(a) Temperature
4
5
10
x 10
(b) Mass
Figure 5.2.: Structure of clump 1, gure a) shows the temperature as a function of the
distance to the centre r. Figure b) shows the mass as a function of r. The
solid line shows the values used and the crosses markers show the values of
the original data points of Mayer [2007].
temperature data.
5.2. Modication of CM module
In order to produce a number of simulations of the planetesimal-clump interaction with
dierent start velocities and start angles, we have made some modications in the CM
module (see section 3.2) because it was originally written for planets with a core.
In the original form of the programme, every planetesimal stops if it comes nearer to the
planet's centre than the minimal tabulated radius in the atmosphere prole which gives
the solid core boundary.
The modication was meanly performed by changing this stopping condition. The criterion is set so that the calculation is stopped if the planetesimal is nearer to the centre
than an arbitrary critical value (set to 1 % of the radius of the planet) and if its velocity
is smaller than an arbitrary critical value (set to 10 % of the local escape velocity). If
we are interested in the mass which reaches the centre of the planet, we can look at the
mass of the planetesimals that fullle the above-mentioned condition.
If the temperature at the centre is higher than the melting point of the planetesimal's
material, the critical value of the velocity has an important eect on the end mass of
the planetisimals. The reason is that the smaller this critical value of the velocity is the
longer it takes to brake the planetesimals to the critical velocity. Therefore CM module
will simulate the evolution of the planetesimals over a longer time. During this time the
45
5. Simulation of the interaction of the planetesimals with the growing clump
2
−4
10
−5
10
10
1
Pressure [Pa]
Density [kg/m3]
10
−6
10
−1
−7
10
10
−2
−8
10
0
0
10
2
4
6
r [m]
8
10
10
12
0
2
4
10
x 10
(a) Density
6
r [m]
8
10
12
10
x 10
(b) Pressure
Figure 5.3.: Structure of clump 2, gure a) shows the density as a function of the distance
to the centre r. Figure b) shows the pressure as a function of r. The solid
line shows the used values and the crosses show the values of the original
data points of Mayer [2007].
3
200
2.5
180
2
Mass [kg]
Temperature [K]
30
220
160
1.5
140
1
120
0.5
100
0
2
4
6
r [m]
8
10
x 10
0
0
12
10
x 10
(a) Temperature
2
4
6
r [m]
8
10
12
10
x 10
(b) Mass
Figure 5.4.: Structure of clump 2, gure a) shows the temperature as a function of the
distance to the centre r. Figure b) shows the mass as a function of r. The
solid line shows the used values and the crosses show the values of the original
data points of Mayer [2007].
46
5. Simulation of the interaction of the planetesimals with the growing clump
planetesimal will permanently lose mass if the temperature of the atmosphere is higher
than its melting temperature. In the clump, used this problem is important for icy planetesimals, stony planetesimals are not aected because the temperature of the gas never
reaches high enough temperatures.
There are two other stopping conditions:
ˆ if the planetesimal ies away from the planet and its distance to the centre of the
planet is bigger than 2 Hill radius of the centre
ˆ if the planetesimal is completely ablated or destroyed.
These two conditions are not changed from stopping condidition of the original CM
module.
5.3. Results of the simulations
In a rst step it is interesting to know for which conditions planetesimals are bound
and for which they are not. This is shown in gures 5.5, 5.6. In black regions the
planetesimals are bound, in white they are not. There are no big dierences between
stony and icy planetesimals. The small dierences are due to the smaller density of ice
compared to stone. Therefore, for the same size and velocity, icy planetesimals have a
smaller kinetic energy than stone planetesimals. This leads to a more eective capture
rate of icy planetesimals than for stony planetesimals.
There are larger dierences between clump 1 and clump 2. Due to the much higher
density in the central part of clump 1 compared with the second clump, clump 1 can
more slowing down planetesimals. For velocity below 104 m
s no planetesimals can go
through the centre of clump 1. On the other hand in clump 2 planetesimals which are
larger than about 10 km and faster than about 1500 m
s are not stopped by the clump.
For each run we also look which part of the initial planetesimal's mass reaches the
centre of the clump and which part is ablated in the atmosphere. Some planetesimals
only lose some mass in the atmosphere but are not decelerated enough to be captured
by the planet. They escape from the Hill sphere of the planet and y away.
Figures (5.7 - 5.10) show the results in the form of a color map.
47
5. Simulation of the interaction of the planetesimals with the growing clump
Planetesimal’s radius = 100km
10
9
9
8
8
Impact velocity [km/s]
Impact velocity [km/s]
Planetesimal’s radius = 10km
10
7
6
5
4
7
6
5
4
3
3
2
2
1
0
1
10
20
30
40
50
60
Impact angle [degrees]
70
80
90
0
10
20
(a)
9
9
8
8
7
6
5
4
90
70
80
90
80
90
7
6
5
4
3
3
2
2
1
1
10
20
30
40
50
60
Impact angle [degrees]
70
80
90
0
10
20
(c)
30
40
50
60
Impact angle [degrees]
(d)
Planetesimal’s radius = 100km
Planetesimal’s radius = 1000km
10
10
9
9
8
8
Impact velocity [km/s]
Impact velocity [km/s]
80
Planetesimal’s radius = 10km
10
Impact velocity [km/s]
Impact velocity [km/s]
Planetesimal’s radius = 1000km
7
6
5
4
7
6
5
4
3
3
2
2
1
0
70
(b)
10
0
30
40
50
60
Impact angle [degrees]
1
10
20
30
40
50
60
Impact angle [degrees]
70
80
90
(e)
0
10
20
30
40
50
60
Impact angle [degrees]
70
(f)
Figure 5.5.: These gures show if a planetesimal is bound to the planet (black regions) or
if it ies away (white regions). Figures a), b) and c) show stony planetesimals
in clump 1. Figures d), e) and f) show icy planetesimals in clump 1.
48
5. Simulation of the interaction of the planetesimals with the growing clump
Planetesimal’s radius = 100km
10
9
9
8
8
Impact velocity [km/s]
Impact velocity [km/s]
Planetesimal’s radius = 10km
10
7
6
5
4
7
6
5
4
3
3
2
2
1
0
1
10
20
30
40
50
60
Impact angle [degrees]
70
80
90
0
10
20
(a)
9
9
8
8
7
6
5
4
90
70
80
90
80
90
7
6
5
4
3
3
2
2
1
1
10
20
30
40
50
60
Impact angle [degrees]
70
80
90
0
10
20
(c)
30
40
50
60
Impact angle [degrees]
(d)
Planetesimal’s radius = 100km
Planetesimal’s radius = 1000km
10
10
9
9
8
8
Impact velocity [km/s]
Impact velocity [km/s]
80
Planetesimal’s radius = 10km
10
Impact velocity [km/s]
Impact velocity [km/s]
Planetesimal’s radius = 1000km
7
6
5
4
7
6
5
4
3
3
2
2
1
0
70
(b)
10
0
30
40
50
60
Impact angle [degrees]
1
10
20
30
40
50
60
Impact angle [degrees]
70
80
90
(e)
0
10
20
30
40
50
60
Impact angle [degrees]
70
(f)
Figure 5.6.: These gures show if a planetesimal is bound to the planet (black regions) or
if it ies away (white regions). Figures a), b) and c) show stony planetesimals
in clump 2. Figures d), e) and f) show icy planetesimals in clump 2.
49
5. Simulation of the interaction of the planetesimals with the growing clump
Planetesimal’s radius = 10km
Planetesimal’s radius = 100km
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
10
0.4
4
0.3
3
0.2
2
0.5
5
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(a)
80
Planetesimal’s radius = 1000km
Planetesimal’s radius = 10km
1
10
1
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
Impact velocity [km/s]
9
3
0.5
5
0.4
4
0.3
3
0.2
2
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(c)
40
60
Impact angle [degrees]
80
0
(d)
Planetesimal’s radius = 100km
Planetesimal’s radius = 1000km
10
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
0
(b)
10
Impact velocity [km/s]
40
60
Impact angle [degrees]
0.2
2
0.5
5
0.4
4
0.3
3
0.2
2
0.1
1
0
0.1
1
20
40
60
Impact angle [degrees]
80
0
0
(e)
20
40
60
Impact angle [degrees]
80
0
(f)
Figure 5.7.: Figures a), b) and c) show the fraction of the initial mass which reaches the
centre of the clump in a solid form. Figures d), e) and f) show the fraction
of the initial mass which is ablated in the atmosphere. The calculation is
made for stony planetesimals colliding with clump 1.
50
5. Simulation of the interaction of the planetesimals with the growing clump
Planetesimal’s radius = 10km
Planetesimal’s radius = 100km
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
10
0.4
4
0.3
3
0.2
2
0.5
5
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(a)
80
Planetesimal’s radius = 1000km
Planetesimal’s radius = 10km
1
10
1
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
Impact velocity [km/s]
9
3
0.5
5
0.4
4
0.3
3
0.2
2
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(c)
40
60
Impact angle [degrees]
80
0
(d)
Planetesimal’s radius = 100km
Planetesimal’s radius = 1000km
10
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
0
(b)
10
Impact velocity [km/s]
40
60
Impact angle [degrees]
0.2
2
0.5
5
0.4
4
0.3
3
0.2
2
0.1
1
0
0.1
1
20
40
60
Impact angle [degrees]
80
0
0
(e)
20
40
60
Impact angle [degrees]
80
0
(f)
Figure 5.8.: Figures a), b) and c) show the fraction of the initial mass which reaches the
centre of the clump in a solid form. Figures d), e) and f) show the fraction
of the initial mass which is ablated in the atmosphere. The calculation is
made for stony planetesimals colliding with clump 2.
51
5. Simulation of the interaction of the planetesimals with the growing clump
Planetesimal’s radius = 10km
Planetesimal’s radius = 100km
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
10
0.4
4
0.3
3
0.2
2
0.5
5
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(a)
80
Planetesimal’s radius = 1000km
Planetesimal’s radius = 10km
1
10
1
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
Impact velocity [km/s]
9
3
0.5
5
0.4
4
0.3
3
0.2
2
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(c)
40
60
Impact angle [degrees]
80
0
(d)
Planetesimal’s radius = 100km
Planetesimal’s radius = 1000km
10
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
0
(b)
10
Impact velocity [km/s]
40
60
Impact angle [degrees]
0.2
2
0.5
5
0.4
4
0.3
3
0.2
2
0.1
1
0
0.1
1
20
40
60
Impact angle [degrees]
80
0
0
(e)
20
40
60
Impact angle [degrees]
80
0
(f)
Figure 5.9.: Figures a), b) and c) show the fraction of the initial mass which reaches the
centre of the clump in a solid form. Figures d), e) and f) show the fraction
of the initial mass which is ablated in the atmosphere. The calculation is
made for icy planetesimals colliding with clump 1.
52
5. Simulation of the interaction of the planetesimals with the growing clump
Planetesimal’s radius = 10km
Planetesimal’s radius = 100km
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
Impact velocity [km/s]
Impact velocity [km/s]
10
0.5
5
0.4
4
0.3
3
0.4
4
0.3
3
0.2
2
0.5
5
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(a)
Planetesimal’s radius = 1000km
Planetesimal’s radius = 10km
1
10
1
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
Impact velocity [km/s]
9
3
0.5
5
0.4
4
0.3
3
0.2
2
0.2
2
0.1
0.1
1
0
1
20
40
60
Impact angle [degrees]
80
0
0
20
(c)
40
60
Impact angle [degrees]
80
0
(d)
Planetesimal’s radius = 100km
Planetesimal’s radius = 1000km
10
1
10
1
9
0.9
9
0.9
8
0.8
8
0.8
7
0.7
7
0.7
6
0.6
6
0.6
0.5
5
0.4
4
0.3
3
Impact velocity [km/s]
Impact velocity [km/s]
0
80
(b)
10
Impact velocity [km/s]
40
60
Impact angle [degrees]
0.2
2
0.5
5
0.4
4
0.3
3
0.2
2
0.1
1
0
0.1
1
20
40
60
Impact angle [degrees]
80
0
0
(e)
20
40
60
Impact angle [degrees]
80
0
(f)
Figure 5.10.: Figures a), b) and c) show the fraction of the initial mass which reaches the
centre of the clump in a solid form. Figures d), e) and f) show the fraction
of the initial mass which is ablated in the atmosphere. The calculation is
made for icy planetesimals colliding with clump 2.
53
5. Simulation of the interaction of the planetesimals with the growing clump
5.4. Mass accretion
Now we have the impact statistic, and we know the characteristic of various impacts.
In particular we know for each impact the fraction of the mass which is accreted by the
planet. This allows us to calculate the total amount of the accreted mass. We take the
impact parameters from the n-body simulation (section 4.2) and compare them with the
results of section 5.3. Because the impact angles and impact velocities have a continuous
distribution we made a two dimensional (angle and velocity) bi-linear interpolation in
the array of results from section 5.3.
The results are shown in table 5.2. The table indicates which fraction of the mass of
the colliding planetesimals (according to the impact statistic) goes to the centre (nearer
than 1 % of the radius) of the clump.
Clump
Planetesimals
Radius [km]
1
1
1
1
1
1
Stone
Stone
Stone
Ice
Ice
Ice
10
100
1000
10
100
1000
2
2
2
2
2
2
Stone
Stone
Stone
Ice
Ice
Ice
10
100
1000
10
100
1000
fcentre [%]
39.3
33.4
9.3
1.9
0.0
4.0
17.5
0.4
0.0
16.8
9.0
0.0
fatmo [%]
24.6
16.3
5.2
70.6
55.0
18.6
0.0
0.0
0.0
1.8
0.8
0.1
ftotal [%]
63.9
49.6
14.5
72.5
55.0
22.6
17.5
0.4
0.0
18.6
9.8
0.1
Table 5.2.: fcentre indicates which fraction of the mass of the planetesimals can reaches
the centre of the clump in a solid form, fatmo indicates which fraction of the
mass is ablated in the atmosphere and ftotal is the sum of both and indicates
which fraction of the mass is accreted by the clump (ftotal = fcentre + fatmo ).
The values are averaged over all impact velocities and angles obtained in
section 4.2
.
We see that in clump 1 a relative large fraction of the mass of the colliding planetesimals is accreted by the clump. For stony planetesimals, the main part of the mass comes
to the centre in solid form. Only a small part is ablated in the atmosphere. For icy
planetesimals the situation is dierent, because the temperature of the clump is higher
than the melting temperature of ice. After some time a whole icy planetesimal would be
melted. This would probably create a liquid core.
The bigger the planetesimals are the more dicult is it to stop them. This reason leads
to a decrease of the accreted mass for large planetesimals.
54
5. Simulation of the interaction of the planetesimals with the growing clump
In clump 2 less planetesimals are bound because of the smaller density of the clump.
Therefore a large part of the planetesimals y through the clump without being stopped.
With the results of section 4.2 we know the fraction (fhit ) of the initial planetesimals
which have collided with the clump. With the initial mass distribution of solids in the
nebula, we can also calculate the mass m0 of the solid that, at the beginning of the n-body
simulation, is between aplanet + 6RH and aplanet − 6RH (the planetesimals were initially
distributed in this range), where aplanet is the semimajor axis of the clump and RH is the
Hill radius of the clump. We also know the fraction fcentre , fatmo and ftotal (see table 5.2).
Therefore we can calculate the total mass that would accrete a clump in 105 years,
which corresponds to the simulated time of the n-body simulations, simply with:
(5.1)
Mtotal = m0 fhit ftotal ,
where m0 is the mass of all planetesimals between aplanet + 6RH and aplanet − 6RH ,
aplanet
Z +6RH
2πrΣ(r0 )
m0 =
r
r0
−3/2
dr.
(5.2)
aplanet −6RH
This is the simplest estimation of the mass of all planetesimals that we can do. In particular we neglect migration of the planetesimals.
kg
For the results of table (5.3) Σ(r0 ) is set to a value of 10 m2 for r0 = 10 AU. This
corresponds to an initial solid density like assumed by Hayashi [1981] for a minimummass solar nebula (MMSN) (see 2.1.2). The results scale linearly with the chosen Σ(r0 ).
For a disc with a mass 10 times higher than a MMSN, the accreted mass would be 10
times bigger. Table 5.3 also shows the values Mcentre and Matmo which are calculated
like Mtotal in equation 5.1 but with fcentre and fatmo instead of ftotal .
We clearly see a dierence between the two clumps. Clump 1 can accrete more mass,
especially if the radius of the planetesimals is large. It seems relatively easy to build a
core in Clump 1 even with a disc mass corresponding to a MMSN. In clump 2 the core
formation is not so eective but it is still possible to make a core, if we have part of the
planetesimals which are smaller than (1000 -100) km, what probably is the case.
For very small planetesimals with a radius in the order of meters the factor ftotal will go
to one, because the planetesimals are quickly decelerated to a terminal fall velocity (the
velocity where the gravitation force is equal to the drag force) when they collide with the
clump. The deceleration is relatively smooth and no strong ablation occurs. So all stony
planetesimals of about 1 m size will reach the centre without that they lose a signicant
amount of mass. Ice planetesimals of the same size would all melt in the atmosphere. If
we assume ftotal = 1 the value of Mtotal would go to about 15 M⊕ for clump 1 and to
about 12 M⊕ for clump 2.
55
5. Simulation of the interaction of the planetesimals with the growing clump
Clump
Planetesimals
Radius [km]
1
1
1
1
1
1
Stone
Stone
Stone
Ice
Ice
Ice
10
100
1000
10
100
1000
2
2
2
2
2
2
Stone
Stone
Stone
Ice
Ice
Ice
10
100
1000
10
100
1000
Mcentre [M⊕ ]
5.82
4.95
1.38
0.29
0
0.54
2.08
0.05
0
2.03
1.09
0
Matmo [M⊕ ]
3.64
2.41
0.77
10.84
8.14
2.5
0
0
0
0.22
0.1
0.01
Mtotal [M⊕ ]
9.46
7.35
2.15
11.13
8.14
3.04
2.08
0.05
0
2.25
1.19
0.01
Z[Z ]
1.42
1.32
1.09
1.49
1.36
1.13
1.26
1.01
1
1.28
1.15
1
Table 5.3.: Mcentre indicates the mass which reaches the centre of the clump in a solid
state after 105 years assuming an initial mass distribution in the disc like
kg
described in section 4.1 and with Σ(r0 ) = 10 m2 for r0 = 10 AU that corresponds to an 1 MMSN massive disc. Matmo indicates the mass which is
ablated in the atmosphere and Mtotal corresponds to the total accreted mass
by the planet. Z is the total mass of heavy elements over the total mass in
units of Z . Z corresponds to the Z value of the sun. For the Z value we
assume that the clump has an initial Z of one.
56
5. Simulation of the interaction of the planetesimals with the growing clump
We also calculate the enrichment Z in heavy elements, due to the accretion of planetesimals, compared with the sun. For this, we assume that the clumps have an initial
enrichment of Zinit = 1Z (Z corresponds to the Z value of the sun). If the clumps
would have an initial Zinit = 0, Z would be smaller by one. We remember that the results
are obtained for a MMSN. For a disc with a mass of four MMSN the enrichment would be
about 2.4 Z if we use stony planetesimals with a radius of 100 km and clump 1 . This
value corresponds approximately to the enrichment of Jupiter, which is between 1.5 and
6 [Saumon and Guillot, 2004]. To reach Z of approximately 10 Z , which corresponds
to the enrichment in heavy elements of Saturn, we would need a disc with a mass of 25
MMSN.
An other interesting comparison with the data of Saumon and Guillot [2004] is the
). We
ratio of heavy elements in the core over the total amount of heavy elements ( MMcentre
total
assume that the whole mass of the planetesimals, which reach the center, would build a
core and that the mass ablated in the atmosphere stays in the atmosphere. Again for 100
km stony planetesimals and clump 1 we get MMcentre
' 2. This is close to the estimation
total
is dicult to
of Saumon and Guillot [2004] for Saturn. For Jupiter the ratio MMcentre
total
determine because the mass of the core of Jupiter is poorly known (between 0 and 11
M⊕ [Saumon and Guillot, 2004])
For icy planetesimals no comparison can be made because the value Mcentre is very sensitive to the setting of the critical stopping condition (see section 5.2).
Furthermore we remember that in our simulations all planetesimals have the same size.
For better results we should use a distribution of the size of the planetesimals.
57
6. Conclusions and outlook
The goal of this work was to look if the formation of a core in a giant planet, formed
through gravitational collapse, is possible. Our results show that the formation seems
possible.
The exact mass of the core formed depends on the initial mass density, on the structure
of the giant planets and on the size of the planetesimals. Regarding the initial mass of
the nebula, we see that even with a disc mass corresponding to a MMSN we can build
cores of several earth masses. The dependence on the size of the planetesimals and of the
structure of the clump is mainly due to the fact that dense planet's atmospheres can better stop large planetesimals than low density atmospheres. In a low pressure atmosphere
planetesimals can y through the whole atmosphere without losing a signicant amount
of kinetic energy. The planetesimals will then escape from the Hill sphere of the planet.
Smaller planetesimals are much more easily stopped by an atmosphere. The deceleration
of such planetesimals is relatively smooth so that no signicant ablation or mechanical
destruction takes place. Therefore they can reach the centre of the planet without losing
much mass.
We have to keep in mind that this model is relatively simple. We have made various
simplications. Especially we did not consider the evolution of the clump. It is clear that
once a core is built the structure of the clump will evolve. And especially the assumption
that the planet consists only of gas would become incorrect. The core formed would stop
planetesimals which are ying through the centre of the planet. But the core will stay
very small compared with the radius of the clumps. A 10 M⊕ core with a density of 3000
kg/m3 has a radius a factor 4000 smaller than the radius of clump 1. On the other hand
the atmosphere would probably become much denser due to the core formed and so the
accretion of planetesimals would become more dicult.
An other simplication is the clear separation of the n-body simulation and the simulation of interaction between the atmosphere and the planetesimals. Planetesimals, which
hit the clump, are deleted from the n-body simulation. Therefore the trajectories of these
planetesimals are no more calculated even if the simulation of the interaction later shows
that these planetesimals are not bound to the planet. Therefore we never know what
really happens with such planetesimals.
An implementation that would consider the evolution of the clump and would link the
n-body simulation with the interaction of the clump is probably the best way to improve
the model of the core formation of giant planets formed through gravitational collapse.
58
6. Conclusions and outlook
Futhermore using a realistic distribution of the planetesimals size would also improve the
results.
59
A. Generate a random variable following
a given distribution using an uniform
distribution
Almost all programming languages have a procedure to generate an uniformly distributed
random number. With the help of a uniform distribution other distributions can be
generated. One method to generate such distributions is the inverse transform sampling.
For a given distribution f (x) the cumulative distribution function F (x) is dened as
Zx
F (x) =
f (x̃)dx̃.
(A.1)
−∞
If an inverse function F −1 (u) of F (x) exists, so that F (F −1 (u)) = x, and if F (x) is a
strictly increasing function, then F −1 (U ) follows the distribution of F if U is an uniform
random variable on (0, 1). This can be easily proven. Let be P (F −1 (U ) ≤ x) the
probability that F −1 (U ) is smaller than x. It follows that because F (x) is a strictly
increasing function and because U is uniformly distributed.
P (F −1 (U ) ≤ x) = P (F (F −1 (U )) ≤ F (x)) = P (U ≤ F (x)) = F (x).
(A.2)
In other words, if we know the inverse function F −1 (u) of the cumulative distribution
function of a distibution f (x), we generate this distribution f simply with F −1 (U ) if U
is a random variable in (1, 0).
This fact is used to generate a f (x) ∼ xn and a Rayleigh distribution.
A.1. A f (x) ∼ xp distribution
The goal is to generate a random variable x between x1 > 0 and x2 > x1 with a density
function f (x) ∼ xp , with x ∈ R+ and p ∈ R.
We have to distinguish between p 6= −1 and p = −1.
60
A. Generate a random variable following a given distribution using an uniform distribution
Case p 6= −1
We normalize F (x) so that F (x2 ) = xp+1
− xp+1
= 1. Therefore the probability density
2
1
function f (x) is
(
1
p if x ≤ x ≤ x ,
1
2
p+1
p+1 x
f (x) = x2 −x1
(A.3)
0
if x < x1 or x > x2 .
And the cumulative distribution function is


0

 p+1 p+1
x
−x1
F (x) =
p+1
p+1
x
−x

1

 2
1
if x ≤ x1 ,
if x1 ≤ x ≤ x2 ,
(A.4)
if x > x2 .
The inverse function of F −1 is
1
p+1
p+1 p+1
.
F −1 (u) = −uxp+1
+
ux
+
x
1
2
1
(A.5)
With U ∈ [0..1] as an uniformly distributed random variable the prescription
1
p+1
p+1 p+1
x = −U xp+1
+
U
x
+
x
1
2
1
(A.6)
produces a random variable x with the wanted properties.
Case p = −1
In the case where p = −1 we can proceed in an analog way and get
x=
xU
2
.
U −1
x1
(A.7)
Again is U a random variable uniformly distributed between 0 and 1.
A.2. A Rayleigh distribution
A Rayleigh distribution is given for x ∈ R+ by
2
x
2x
exp − 2 .
f (x, σ) =
σ
σ
61
(A.8)
A. Generate a random variable following a given distribution using an uniform distribution
σ is the standard deviation of the f (x)
v
uZ∞
2
u
x
u 2x
σ=t
exp − 2 x2 dx.
σ
σ
(A.9)
0
The cumulative distribution function is
2
x
F (x) = 1 − exp − 2 .
σ
Therefore the inverse function F −1 (u) is
p
F −1 (u) = − ln (−u + 1)σ.
(A.10)
(A.11)
Again we introduce U which is uniformly distributed between 0 and 1. So that the
prescription
p
x = − ln (−U + 1)σ
(A.12)
produces a Rayleigh distributed random variable x.
62
Acknowledgements
I would like to thank Prof. Dr. Willy Benz for giving me the opportunity to make
my diploma thesis at the research group of theoretical astrophysics and planetary science
(TAPS) and for his continuous support during this work. He always took time to give me
hints and explanations about dierent problems. I would also like to thank all the TAPSteam members who helped me in dierent ways. I am especially grateful to Christoph
Mordasini. He helped me a lot during the whole diploma thesis with explanations about
all kinds of problems.
I would also like to thank Prof. L. Mayer from the University of Zürich for the interesting discussions in Zürich and for answering a lot of questions about his data proles via
e-mails.
Many thanks to Matthias Baumgartner. He took time to read my whole diploma thesis
and helped me with the English. Furthermore I would like to thank Urs Gerber from
institute of theoretical physics for the good team work that we had during the whole
physics studies. Be it in solving quatum physics exercises or in performing laboratory
experiments, we always had a comfortable work together.
Finally I would like to express my gratitude towards my parents who always supported
me in my studies. And I would like to thank Isabel Lehmann for the great time that we
spend together.
63
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