CondensationLecture 8

Lecture Universität Heidelberg WS 11/12Dr. C. Mordasini

Based partially on script of Prof. W. Benz Mentor Prof. T. Henning

Bond et al. 2010

Lecture 8 overview

1. Condensation

1.1 Carbonaceous chondrites

1.2 The thermodynamic of condensation

1.3 Examples

1.4 The full sequence

1.5 Water ice condensation

1.6 Condensation in extrasolar systems

1.1 Carbonaceous chondrites

Carbonaceous chondritesThe condensation of dust grains out of the gas phase represents the very first phase of planet growth.

The most primitive sub-class are so called CI-chondrites. In their appearance, these mostly small, black, and very friable rocks remind more of a piece of tar or charcoal than of a stone. They contain a large fraction of water (bound in silicates) of 17-22%. The iron content (in form of iron oxides) is about 25% in mass. Carbon makes about 3-5%. Interestingly, amino acids are also present.

While the Earth, the Moon and many other planetary bodies show clear signs of chemical differentiation and fractionation, the most primitive meteorites, the so-called carbonaceous chondrites, seem to present an unaltered image of the chemical composition of the nebula as it was 4.6 billion years ago.

The chemical/mineralogical composition shows that the origin of CI-chondrites is in the outer part of the solar system (>4 AU) since they never have been heated above 50°C during their formation and subsequent evolution.

Except for some volatile elements like hydrogen or oxygen are the relative elemental abundances in CI-chondrites nearly identical to those measured in the photosphere of the sun. Lithium is depleted in the sun relative to CI-chondrites, as it is destroyed by nucleosynthesis.

Holweger

1.2 The thermodynamic of condensation

Thermodynamic equilibrium

In order to compute which elements condense where, we assume that changes in temperature and density occur on a relatively long timescale compared to the chemical reaction timescale. This is a reasonable assumption at least for the inner part of the disk where temperatures and gas densities are high. Under this assumption, we can presume that these changes always occur at constant temperature and pressure (which will however by different as function of the distance from the star) and in thermodynamical equilibrium .

The first model that computed the sequence of solid which emerge from the gas phase when we let the gas slowly cool was Grossman & Larimer 1974. Here we reproduce some aspects of this work as well as some later improvement to this simple minded approach.

After the disk has formed, it cools, and new dust grains condense out (the assumption that first a disk forms, and then condensation happens is clearly an idealization, In reality these processes would occur partially concurrently). The formation of the early dust grains proceeds therefore along a condensation sequence in which the most refractory elements condense in the inner regions of the nebula while volatile elements condense only at larger distance (outside the icelines).

The collapse of the interstellar gas cloud that leads to the formation of the protoplanetary nebula is a relatively violent process during which temperatures high enough to vaporize most (but not all) solids are reached. Therefore, the dust grains originally contained in the gas will mostly get vaporized. Solids which survived the collapse (so-called presolar grains) are tiny, very refractory grains like nano-diamonds, graphite particles or silicon carbide (SiC) grains.

Thermodynamic potentialsIn a thermodynamical system, processes will spontaneously continue until the relevant thermodynamical potential is minimized. Examples are:

1) In a isothermal-isochor system in equilibrium, the Helmoltz free energy F will be minimal.

F = U - TS

2) In a isothermal-isobar system in equilibrium, the Gibbs free energy (also called free enthalpy) G will be minimal.

G = F + pV

= U - TS + pV

= H - TS

H is the enthalpy U + pV. Note that the unit of these potentials is erg.

In the situation that the chemical reactions happen at constant temperature and pressure, the free enthalpy or the Gibbs energy G is the natural choice for the thermodynamical description of the changes.

Thermodynamic potentials II

G = H − TS → dG = dH − TdS − SdT with H the enthalpy

H = U + pV → dH = dU + pdV + V dp = δQ + V dp

We have made use of the first principle of thermodynamics: dU = δQ− pdV

For a reversible change we must therefore have

dG = dH − TdS − SdT = δQ + V dp− δQ− SdT = V dp− SdT

where we have used the definition of the entropy: dS =δQ

T

Clearly, for a process that takes place at constant temperature and constant pressure, we have dG=0 in the final state (equilibrium).

The free enthalpy just as the entropy are thermodynamic potentials defined to within a constant. It is therefore useful to define standard conditions to be used as reference point. The standard conditions are generally set to be T=298 K and p=1 atm.

To illustrate this concept, let us compute the change in free enthalpy for the reaction taking place at standard conditions:

Example

Let us define the free enthalpy change (final minus initial) at these standard conditions as

∆G00 = G00(H2O)−G00(H2)−12G00(O2)

where the double 0 subscript indicates standard p and T. The free enthalpy of the individual components can be interpreted as the free enthalpy of formation of the substance.

H2 +12O2 → H2O g

∆G00 = −258.8− 0.0− 12

· 0 = −258.8 k J /mole

Note that by convention the free enthalpy of the most stable form of a substance is taken to be zero. The change of the free enthalpy is negative, which means the reaction is exergonic and thus a favored reaction (spontaneous).

From tables we can the following values:

Condensation in the nebulaSince the changes in the nebula do not happen at reference temperature and pressure, we need to be able to compute the change in free enthalpy for other thermodynamical conditions.

1) changes at constant temperature

dG = V dp = nRTdp

p→ G(p, T )−G0(T ) = nRT ln

�p

p0

�From the definition of the free enthalpy change, we have in this case (dT=0) for an ideal gas

where G0(T) stands for G(p0,T) the free enthalpy at standard pressure but at temperature T. Clearly p0 is the reference pressure (1 atm).

In equilibrium, ΔG(p,T) = 0, and

(1)

where the Δ represents the difference operator ''after'' – ''before'' of the chemical reaction. The pi are the partial pressures.

We can apply this formalism to describe a chemical reaction that occurs at different pressures but constant temperature. If the reaction involves i components, i=1,...,N, each with different concentrations ni, we have

Condensation in the nebula IITo understand this formalism, we consider the simple example

aA + bB → cC + dD

In equilibrium, we must have:

�

i

ni ln�

pi

p0

�= ln

�pC

p0

�c �pD

p0

�d

�pA

p0

�a �pB

p0

�b

= −∆G0(T )RT

= lnKp

For the reaction constant Kp(T) we have

2) changes at constant pressure

From the definition of the free enthalpy change, we have in this case (dP=0)

dG = −SdT → G(p, T )−G0(p) = −� T

T0

S(T )dT

where T0 is the reference temperature 298 K. To compute S(T) we need to recall the definition of the entropy:

Condensation in the nebula III

dS =δQ

T→ S(T )− S0 =

� T

T0

δQ

T=

� T

T0

cpdT

T= cp ln

�T

T0

�

where S0 is the entropy at standard condition.

Finally, in our isothermal-isobar situation, we can combine (1) and (2), taking into account the enthalpy of formation of the substances:

(3)

∆G(p, T )−∆G0(p) = −∆S0(T − T0)−∆cp

�T ln

�T

T0

�− (T − T0)

�

where ΔS0 is the difference of standard entropy of the reaction (''after'' – ''before'') and Δcp is the difference in specific heat at constant pressure taking place as a consequence of the reaction (cp is assumed to be independent of temperature).

In case of a chemical reaction taking place at standard pressure but varying temperature, we can write (integral of S over T):

∆G(p, T )−∆G0(p) = −∆S0(T − T0)−∆cp

�T ln

�T

T0

�− (T − T0)

�(2)

At equilibrium, we will again have ΔG(p,T) = 0, so that

1.3 Examples

Example 1: Dissociation of hydrogenFor the reaction H2 ⇒ H + H, at equilibrium, we must have from equation (1)

We have taken into account that one H2 becomes two H. In order to deal with the partial pressures, it is convenient to define the dissociated fraction α, so that α=0 means H2 only, while α=1 means complete dissociation. We assume that we start with n moles of undissociated H2. We can then write the following table:

H2 H total

nb of moles (1-α)n 2 αn (1+α)n

molar fraction (1-α)/(1+α) 2α/(1+α) 1

partial pressure (1-α)ptot/(1+α) 2αptot/(1+α) ptot

Inserting these partial pressures in the expression above yields (ptot will be given by our nebula model while p0 is the reference pressure of 1 atm):

Dissociation of hydrogen IISolving for α yields

To determine Kp(T), we can use the results for the change at constant pressure. The entropy change is (end minus beginning):

Finally we write with eq. 3 for the change of the free enthalpy

From lookup tables we find the following numerical data:

so

Dissociation of hydrogen III

so

Grouping all terms yields:

With this equation, we can calculate and thus finally α(T,ptot) which is

the quantity in which we are interested.

Numerically we find for four different nebular pressures ptot (typical will be 10-4 atm):

For the specific heats we assume an ideal gas, therefore

Dissociation of hydrogen IV

1000 1500 2000 2500 3000 350010�5

10�4

0.001

0.01

0.1

1

1000 1500 2000 2500 3000 3500 40000.0

0.2

0.4

0.6

0.8

1.0

Frac

tion α

of d

isso

ciat

ed H

2

Frac

tion α

of d

isso

ciat

ed H

2

Temperature [K] Temperature [K]

ptot [atm]

10-8

10-6

10-4

10-2

ptot [atm]

10-8

10-6

10-4

10-2

Notes:

•A high total pressure inhibits dissociation.•One dissociation begins, it is a very steep function of temperature especially at low pressure.•In the temperature range where dissociation occurs, the fraction of molecular hydrogen is also a strong function of pressure.•Even at relatively high pressures, essentially all hydrogen is dissociated by 3500 K. According to the nebula models we studied earlier, such a high temperature is reached very close to the star only, at least after the accretion rate of gas is no more very high.•For T<1000 K, all gas is molecular. From this we conclude that in most of the disk, we have H2, not H.•In our calculation, we have made a number of assumptions which may not always be justified: ideal gas law, constant specific heat, etc. For more accurate calculations, it is important to include all these effects.

Example II: Condensation of iron IAt equilibrium, for Fe (g) ⇒ Fe (s), we can write from equation (1) for the reaction constant (activity for pure solids is set to unity)

To compute the reference free enthalpy as a function of the temperature, we proceed as in the example before. Looking up appropriate tables, we obtain:

SFe(s)(T ) = 27.06 + 25.10 ln�

T

298

�; SFe(g)(T ) = 180.49 + 25.68 ln

�T

298

�

∆S(T ) = SFe(s)(T )− SFe(g)(T ) = −153.42− 0.58 ln�

T

298

�

∆G0(T ) = −3.698× 105 + 153.42(T − 298)− 0.58�

T ln�

T

298

�− (T − 298)

�

The first term on the RHS is the enthalpy of vaporization at standard conditions. With these equation, we can calculate the partial pressure of Fe vapor as a function of temperature.

We note that for T≈T0=298 K, we have so that the vapor pressure is

This corresponds to the classical form of the vapor pressure law p(T)=p0 e-C/T.

Condensation of iron IITo actually compute the condensation temperature of iron in the solar nebula we must have a suitable model of the solar nebula. In this simple example, let us assume that we have a constant total pressure (the increase of the total pressure due to vaporized Fe is neglected). In a good approximation this total pressure is equal to the hydrogen and helium partial pressures and the iron partial pressure follows from abundance considerations:

ptot � p(H2) + p(He)

For an element i we can write (Xi =mole fraction)

On the cosmochemical scale, atomic abundances are normalized to the number of silicon atoms of log(ε(Si))=6. Therefore,

p(Fe) = ptot

��(Fe)

0.5�(H) + �(He)

�= 5.31× 10−5

ptot

Here we have assumed hydrogen to be in molecular form. The standard abundances for the solar nebula are: log(ε(Fe))=5.95; log(ε(H))=10.45; log(ε(He))=9.45. This partial pressure plots as a horizontal line in the diagram. The intersection between the two curves yields the condensation temperature of iron as condensation occurs when the vapor pressure is equal the partial pressure. We find about 1350 K for ptot=10-4 atm i.e. p(Fe)=5.31 x 10-9 atm.

1.4 The full sequence

The full sequenceThe computation of the full condensation sequence is a complicated task (Grossman & Larimer 1974, Rev. Geophys., 12, 71). We present here some of their results.

Condensation of two refractory solids: 1) Corundum: 2 Al (g) + 3O (g) → Al2O3 (s) 2) Spinel: Mg (g) + 2 Al (g) + 4 O (g) → MgAl2O4 (s)

Note that the track for the vapor phase is not a horizontal line as in the previous example. Here Grossman & Larimer assumed a solar nebula model, including relative abundance, which means that pressure, temperature have to be considered.

For corundum, the condensation temperature is found as before at the intersection of the two lines and gives T=1758 K. For spinel, the situation is somewhat more complicated. If corundum would not condense first, Spinel would condense at T=1685 K. However, the condensation of corundum removes aluminum and oxygen and thus changes the slope of the partial pressure curve below T=1758 K (arrow). According to Grossman & Larimer, spinel only condenses at about T=1500 K.

The full sequence IIGrossman & Larimer (1974) computed the full sequence of condensation for a number of elements. The abundance of the different elements were taken to be solar and the total pressure was set to 10-4 atm.

Individual bodies in the solar system do not match exactly this condensation sequence.•For example, Mercury's bulk uncompressed density from the condensation model is 4.3 g/cm3 as opposed to the 5.5 g/cm3 observed.•Mercury contains about 70% iron, Venus 30%. This large difference is in contrast to the close proximity of the condensation curves of Fe and Mg2SiO4 (“rock”).•Finally, simulations of the last stages of planetary accumulation have shown that planets are not made from materials originating from narrow feeding zones but rather are collected over sizable areas of the solar nebula implying considerable mixing.

In order for the equilibrium condensation model to be correct, the various timescales for the chemical reactions (gas-gas, gas-solids, solids-solids) must all be significantly shorter than the cooling time of the nebula.This was not the case at all times and non-equilibrium models should be considered.

The full sequence IIIThe condensation calculations have been further improved by e.g. Lodders 2003 or Ebel 2006.

Equilibrium stability relations of vapor, minerals and silicate liquid as a function of temperature (T) and total pressure (P) in a system with solar bulk composition.

The full sequence IVThe result of such calculations are the condensation temperatures of important minerals as listed in the table.

Lodders 2003, total pressure 10-4 bars

Jones, total pressure 10-3 bars

We note that hydrogen and helium do not condense for temperatures expected in the nebula. Methane only condenses at large distances.

1.5 Water ice condensation

It is a long standing, classical (and plausible) assumption to associate the large change of the surface density due to ice condensation with the global structure of the Solar System, in particular the division between terrestrial planets inside, and giant planets outside. We will see later how the increase of the surface density affects planetary growth (specifically the so called isolation mass, and the growth timescale).

The condensation of water ice sets the representative temperature for the appearance of volatile ices (e.g. methane ice). Note that some oxygen is removed from the gas by the formation of silicates and oxides. Namely ~23% of all oxygen is incorporated into rocky elements (Al, Ca, Mg, Si, and Ti) before water ice condenses.

Water ice condensation

As we can understand from the table, the clearly most important condensation temperature is the one of water ice at about 180 K at a pressure of 10−4 bar (some other calculations indicate lower temperatures of T ≈ 150 K). The reason is that for a solar composition, the surface density of condensible materials rises dramatically once water ice forms (by about a factor 4). Thus, Σ(ices + rock) ≃ 4 Σ(rock). The exact ratio is uncertain. Classical calculations (Weidenschilling 1977) found 4.2. Recent calculations (Min et al. 2011) indicate a smaller jump factor of about 2.8.

Lodders 2003

Initial solid surface density profileUnder the simplistic assumption that the fraction of material that condenses out of the gas is constant except for the increase at the iceline, we can write for the initial solid surface density profile

1144 C. Mordasini et al.: Extrasolar planet population synthesis. I.

defined as the semimajor axis where the inner boundary of theplanet’s feeding zone touches the inner boundary of our com-putational disk at amin = 0.1 AU, (“the feeding limit”) i.e. atatouch = amin/(1!4(Mplanet/(3 M"))1/3). If a planet has migratedto atouch, all we can state is that its final semimajor axis wouldbe #atouch (it is also possible that it eventually would have falleninto the host star), and what its mass at atouch was.

3. Monte Carlo method

The basic idea of using a Monte Carlo method to synthesizeplanetary populations is to sample all possible combinationsof initial conditions (protoplanetary disk mass, metallicity, etc.)with a realistic probability of occurrence. This leads to all pos-sible final outcomes of the formation process (i.e. planets) alsooccurring with their relative probabilities. We first explain thegeneral six step procedure that we used.

In the first step, we identified four crucial initial condi-tions, and studied the domain of possible values they can take(Sect. 3.1). Some other initial conditions had to be kept constantduring the synthesis of one population, for simplicity or compu-tational time restrictions (Sect. 3.2). In the second step, we de-rived probability distributions for each of the four Monte Carlovariables (Sect. 4). In the third step, we draw in a Monte Carlofashion large numbers of sets of initial conditions. The forth stepconsists of using the formation model for each set of initial con-ditions, giving the temporal evolution of the planet (formationtracks, Sect. 5.1) as well as its final properties (mass, semimajoraxis, composition etc., Sect. 5.2).

Many of these synthetic planets would remain undetected bycurrent observational techniques. So, to be able to compare thesynthetic planet population with the observed one, we apply inthe fifth step a detailed synthetic detection bias (Paper II). Inthis way, we obtain the sub-population of observable syntheticplanets. Ultimately, in the sixth step, we performed quantitativestatistical tests (Paper II) to compare the properties of this ob-servable synthetic exoplanet sub-population with a comparisonsample of real extrasolar planets.

3.1. Monte Carlo variables

We use four Monte Carlo variables to describe the varying initialconditions for the planetary formation process. Three describethe protoplanetary disk and one the seed embryo.

1. The dust-to-gas ratio in the protoplanetary disk fD/G de-termines (together with !0) the solid surface density.Models with fD/G between 0.013 and 0.13 were computed.Combined with the domain of !0, this corresponds to ini-tial solid surface densities at a0 = 5.2 AU of between 0.65and 130 g/cm2. For comparison, the MMSN has a value ofapproximately 2.5 g/cm2 (Hayashi 1981).

2. The initial gas surface density !0 at 5.2 AU gives the amountof gas available. Values between between 50 and 1000 g/cm2

were used. The MMSN is estimated to have had a value ofabout 100!200 g/cm2 (Hayashi 1981).

3. The last variable that characterizes a disk is the rate at whichit loses mass due to photoevaporation Mw. For the popula-tion presented below, it was allowed to vary between 5 $10!10 M%/yr and 3 $ 10!8 M%/yr.

4. The initial semimajor axis of the seed embryo within thedisk, astart, is the fourth variable. It can take values of 0.1 #astart # 20 AU.

3.2. Parameters

Some other initial conditions of the model were kept constantfor all planets of a given population. We mention only the mostimportant parameters here. More details can be found in Alibertet al. (2005a). For the nominal population discussed in Sect. 5,we use a viscosity parameter ! for the disk model of 0.007 andan e"ciency factor for type I migration fI of 0.001. The influ-ence of these two important parameters is briefly discussed inSect. 5.3.3, and will be further considered in forthcoming publi-cations. In this and the companion paper the mass of the centralstar M" is kept constant at 1 M%.

4. Probability distributions

In the next step we determine the probability of occurrence ofa certain combination of initial conditions. In the ideal case, theprobability distributions for all our variables would be deriveddirectly from observations. Unfortunately, in reality, this is notpossible either because in some cases observations do not existor, even if they exist, a certain amount of modeling is necessaryto extract the distributions from the observations.

4.1. Dust to gas ratio fD/G – [Fe/H]

To establish a link between the dust-to-gas ratio fD/G, which isthe computational variable required by our model, and the corre-sponding observable, the stellar metallicity [Fe/H], we assume:(1) the stellar content in heavy elements is a good measure of theoverall abundance of heavy elements in the disk during forma-tion time. Support for this assumption comes from the small dif-ferences between solar photospheric and meteoritic abundances(Asplund et al. 2005); (2) a scaled solar composition and (3) anegligibly small influence of the change of the relative heavyelement content on the relative hydrogen content in the compar-atively small [Fe/H] domain of interest for planet formation inthe solar neighborhood (!0.5 # [Fe/H] # 0.5). Then, similar toMurray et al. (2001), we can write

fD/GfD/G,%

= 10[Fe/H] (6)

where fD/G,% is the dust to gas ratio corresponding to [Fe/H] = 0.This formula implies that we assume that iron is a good tracerof the relevant overall amount of solids available for planet for-mation. Robinson et al. (2006) have found that at a given ironabundance, planet host stars are enriched in silicon and nickelover stars without planets, indicating that the above relation is asimplification.

Measurements of the heavy element abundance in the Sunyield the amount (for complete condensation) of high Z materialthat existed initially in the form of uniformly mixed fine dustgrains. However, what is relevant for our simulations is the con-centration of solids in the innermost 20 AU of the disk at a laterstage, namely when the dust has evolved into the 100 km plan-etesimals used in our model.

As has been shown by Kornet et al. (2001), the transitionfrom the very early dust phase to the later planetesimal phase in-volves a number of coupled mechanisms of dust-dust and dust-gas interactions like dust settling to the midplane, dust growthby coagulation and radial drift. This leads to a redistribution ofthe solids within the disk, which can in turn have important ef-fects on planetary formation (Kornet et al. 2005). The key pointis that these processes lead to an increase of the solid to gas ra-tio in the inner (<&10!20 AU) planet forming regions of the disk

Here, Σ(r,t=0) is the gas surface density at t=0 (which is obviously ill defined) and fD/G is the dust to gas ratio. For the later, it is assumed that it is the same in the disk as in the star. Then, we can relate it to the so-called stellar metallicity [Fe/H].

The spread in [Fe/H] by about 1 dex shows that initial dust-to-gas ratios in disks vary by about one order or magnitude.

This formula implies that we assume that iron is a good tracer of the relevant overall amount of solids available for planet formation i.e. a scaled solar composition. The metallicity is defined as

This means that a star with the same Fe content as the sun has [Fe/H]=0. [Fe/H] can be determined spectroscopically. For solar like FGK stars in the solar neighborhood, one finds a Gaussian distribution for [Fe/H] with a mean µ ~0.0, and a dispersion σ~ 0.2 (e.g. Santos et al. 2003).

Mordasini et al. 2009

Iceline positionThe factor fR/I represents the effect of the iceline:

For a disk similar as the MMSN, we find an iceline position of about 3.7 AU.

2 Y. Alibert et al.: Extrasolar planet population synthesis. III

Finally, Kornet et al. (2006) have calculated the influence

of the central star mass on the evolution of the protoplanetary

disk, by focusing in particular on the redistribution of solids in

the disk. Using simple model to relate the resulting structure of

the protoplanetary disk and the probability of presence of a gi-

ant planet, they have concluded that, globally, the probability to

harbor a Jupiter mass planet increases it the mass of the primary

decreases. Interestingly enough, their conclusions are very dif-

ferent from the ones of Laughlin et al. (2004) and Ida & Lin

(2005). Indeed, in these two latter cases, the surface density of

planetesimals is assumed to scale positively with the mass of the

primary star, whereas in Kornet et al. (2006), the redistribution

of solids being more efficient around low mass stars, leads to

higher surface densities of solids.

In a recent paper, Mordasini et al. (2009a, paper I) have pre-

sented planet population synthesis calculations based on the ex-

tended core-accretion formation model of Alibert et al. (2005a)

which takes into account migration and disk evolution. This

model has been shown to reproduce some of the bulk proper-

ties of the giant planets in our own Solar System (Alibert et al.

2005b), and of the three Neptune mass planet system discov-

ered around HD69830 (Lovis et al. 2006, Alibert et al. 2006).

Population synthesis calculations rely on the calculation of thou-

sands of planet formation models, each one assuming a differ-

ent set of initial conditions. In particular, the mass of the pro-

toplanetary disk, its lifetime, its metallicity (more exactly the

planetesimals-to-gas ratio in the disk), and the starting location

of the planet’s embryo have to be specified. The distribution

functions for these initial conditions are taken, as far as possible,

directly from astronomical observations. Furthermore, by taking

into account the observational bias introduced by radial velocity

surveys (Naef, 2004), it was possible to compare in a statistical

way the observed population of giant planets with the synthetic

ones. Using Kolmogorov-Smirnov (KS) tests, Mordasini et al.

(2009b, paper II) showed that this approach yields a planet pop-

ulation whose mass and semi-major axis distributions match the

properties of the actually observed planets to a confidence level

of close to 90 %.

In this paper, we extend the calculations presented in pa-

pers I and II to the formation of planets around stars of different

masses. In order to isolate the effects of varying the stellar mass,

we use the same formation model varying only the central star

mass. Note that we also consider cases in which the properties of

the protoplanetary disk are functions of the mass of the primary.

In these cases, the characteristics of the protoplanetary disk were

varied as well.

Since we not only use the exact same procedure as explained

in detail in paper I and II but also all the same value for all nu-

merical parameters, we will not discuss our approach in any de-

tail here but refer the reader to these papers. The present paper

is organized as follows: in Sect. 2, we discuss the influence of

the mass of the primary on the formation process itself as well

as on the corresponding initial conditions. In Sect. 3, the out-

come of model in which the mass of the central star is varied

from 0.5 M⊙ to 2.0 M⊙ are presented. In particular, we discuss

the changes in the properties of the resulting planets (mass and

semi-major axis, composition) as well as the influence of metal-

licity in the formation process. Finally, Sect. 4 will be devoted to

the discussion and conclusions.

2. Formation model and initial conditions

2.1. Influence of the central star mass

The planet formation model we use is a simplified version of

the one presented in Alibert et al. (2005a). In this model, we

consider a planetary embryo initially located at a given distance

of the central star, in a disk consisting of gas and solids. The

complete model is described in paper I and we will not present

it again here.

The actual mass of the central star (also referred as ”pri-

mary”) enters in many of the physical processes involved during

planet formation. Explicitly, it enters in:

1. the Hill’s radius (RH = aplanet

�Mplanet

3Mstar

�1/3, where Mplanet and

Mstar are the planet and star masses, and aplanet is the planet

semi-major axis) of the planet. This radius is a measure of

the size of the planet’s feeding zone, and (indirectly) of its

envelope mass. Planets forming around more massive stars

can accrete planetesimals originating from a smaller region

of the protoplanetary disk.

2. the type I migration rate. Type I migration rate, which is rele-

vant for low mass planets (see Tanaka et al. 2002), is reduced

for more massive central stars.

3. the disk structure. High central mass stars result inhigher gravity in the vertical direction. On the otherhand, viscosity dissipation depends on the Keplerian fre-quency, and disks around high mass stars are hotter.Numerical calculations show that disks around high massstars are generally thinner, the first effect being more im-portant.

4. the Keplerian frequency which governs, among other things,

the accretion rate of solids.

Moreover, the disk model takes into account the effect of

photoevaporation, at distances larger than the gravitational ra-

dius Rg (see Veras and Armitage, 2004), which depends linearly

on the mass of the central star (see Adams et al. 2004). Finally,

the location of the iceline depends on the temperature and pres-

sure in the disk. All other parameters being equal, the iceline

is located at larger distances around higher mass stars, the ef-

fect begin of the order of 1 to 2 AU (depending on the disk’s

mass), going from Mstar = 0.5M⊙ to Mstar = 2.0M⊙. Analyticalfit of our disk models for α = 7 × 10

−3, our nominal valueof the Shakura & Sunyaev viscosity parameter (Shakura &Sunyaev, 1973), shows that the position of the iceline can beapproximated as

rice

AU=

�Σ5.2AU

10g/cm2

�0.44

�

Mstar

M⊙

�0.1(1)

where Σ5.2AU is the initial gas surface density at 5.2 AU2, andwe assume that the gas surface density follows a power lawwith a −3/2 slope. We recall here that, as mentioned in Alibert

et al. (2005), the structure of the disk is calculated without tak-

ing into account the irradiation of the central star, which itself

depends on the mass of the primary. However, higher mass stars

tend to have higher luminosities, which also translates in an ice-

line located at larger distances.

Finally, the mass of the central star also enters in the determi-

nation of the initial conditions used for the population synthesis

2the location of the iceline does not vary as long as the gas surface

density evolves, and is only governed by the initial gas surface density,

see papier I

Fit for α=0.007 (Alibert et al. 2011)

The figure on the right shows the initial solid surface density (black line). The initial gas surface density is derived from the similarity solutions of the viscous accretion disk problem.

C. Mordasini et al.: Extrasolar planet population synthesis. I. 1141

Fig. 1. Position of the iceline aice as a function of the initial gas surfacedensity !0 at 5.2 AU (upper three lines). It corresponds to an initial Tmidof 170 K. The iceline is plotted for three values of !: 0.01 (dashed line),0.007 (solid line) and 0.001 (dotted line). The lower three lines corre-spond to an initial Tmid of 1600 K, roughly the evaporation temperatureof rock. The rockline arock is however not taken into account in the nom-inal model, due to the di"culty in defining its relevant location, as diskevolution is very rapid close-in and irradiation e#ects might be impor-tant (cf. Paper II).

where Rg is taken to be 5 AU, amax is the size of the disk, and thetotal mass loss Mw due to photo-evaporation is an input parame-ter which together with the ! parameter determines the lifetimeof the disk.

For simplicity, we adopt an initial profile of the gas disksurface density according to the phenomenological model ofHayashi (1981), !(a, t = 0) = !0 (a/a0)!3/2 where !0 is thesurface density at our reference distance (a0 = 5.2 AU), and thecomputational disk extends from amin = 0.1 AU to amax = 30 AU.The initial total gas disk mass in the computational disk is then4"!0a3/2

0 (a1/2max ! a1/2

min). For the initial profile with ! " a!3/2 theaccretion rate decreases from the inner to the outer parts of thedisk. As shown in e.g. Papaloizou & Terquem (1999), the in-ner parts of the disk evolve rapidly toward a state of constantaccretion rate M#. Therefore, the inner initial gas disk profile istruncated in order to obtain an accretion rate lower than a con-stant value of order 3 $ 10!7 M%/yr. This allows us to speed upthe calculation of the disk evolution.

The initial solid surface density is given by !D =fD/G fR/I!0 (a/a0)!3/2 (Hayashi 1981; Weidenschilling et al.1997) where fD/G is the dust-to-gas ratio of the disk, and fR/Iis a factor describing the degree of condensation of ices. Itsvalue is set to 1/4 in the regions of the disk for which the ini-tial mid-plane temperature exceeds the sublimation of water ice(Tmid > 170 K), and 1 otherwise. The semimajor axes aice wherethis temperature is reached as a function of initial gas surfacedensity !0 is plotted in Fig. 1. For a minimum mass solar neb-ula (MMSN) like !0 (100!200 g/cm2), the iceline is as expectedfound between 2 and 4 AU (Hayashi 1981). Note that in the ac-tive disk model we use, the e#ect of stellar irradiation on thetemperature structure of the disk is not included.

2.2. Migration rate

The migration of the protoplanet occurs in two main regimesdepending upon its mass. Low mass planets undergo type I mi-gration (Ward 1997; Tanaka et al. 2002) which depends linearlyon the body’s mass. The prevalence of extrasolar planets has ledus to suspect that the actual type I migration rate is probably sig-nificantly lower than currently estimated (Menou & Goodman2003; Nelson & Papaloizou 2004). For this reason, we allow fora arbitrary reduction of the type I migration rate as calculated inTanaka et al. (2002) by a constant e"ciency factor fI.

The migration type changes from type I to type II when theplanet becomes massive enough to open a gap in the disk. Weassume that this happens when the Hill radius of the planet be-comes greater than the density scale height H of the disk (Lin &Papaloizou 1986). Planetary masses where the migration regimechanges can be low with such a thermal criterion only, as foundalso by Papaloizou & Terquem (1999) who use a similar condi-tion. This is especially the case as due to disk evolution, the diskscale height H decreases with time, so that the minimal massneeded to open a gap decreases. This e#ect is emphasized by thefact that our disk model currently does not include irradiation,so that especially towards the end of disk evolution, H becomessmaller than in a disk including it, and smaller planets can opena gap (Edgar et al. 2007). The order of magnitude we obtain ishowever consistent with the one derived from Armitage & Rice(2005), since they give a gap opening condition (including thee#ect of viscosity) of Mplanet/M# >& !1/2(H/aplanet)2. In our sim-ulation, the transition typically occurs when the aspect ratio ofthe disk has become tiny, between 2 and 3%, meaning a tran-sition at tens of Earth masses. We note that Crida et al. (2006)have derived a new criterion for gap opening which depends onboth the disk aspect ratio and the Reynolds number. Using sucha modified transition mass has some influence on the planetaryformation tracks (see Sect. 5.3.5).

Type II migration (Ward 1997) itself comes in two forms:As long as the local disk mass is large compared to the planet’smass Mplanet (called “disk dominated” migration in Armitage2007), the planet is coupled to the viscous evolution of the diskand its migration rate is independent of its mass. The plane-tary migration timescale is then the same as the gas viscoustimescale (e.g. Ida & Lin 2004a). Once the local disk mass andthe planet’s mass become comparable, migration slows down(Lin & Papaloizou 1986) and eventually stops. Due to the inertiaof the planet the disk can no longer deliver the amount of angularmomentum necessary to force the planet to migrate at the gas’radial speed (e.g. Trilling et al. 1998; called “planet dominated”migration in Armitage 2007).

As Armitage (2007) and Thommes et al. (2008), we havefound that this braking phase plays a key role in determining thefinal semi-major axis of massive planets. The reason for this canbe seen in Fig. 2 where 2!a2 is plotted as a function of timeand semimajor axis for an example disk evolving under the in-fluence of viscosity and photo-evaporation. The quantity 2!a2

serves as the measure of the local disk mass to which the planet’smass is compared (Lin & Papaloizou 1986; Syer & Clarke 1995;Armitage 2007).

The plot shows that except at the very end, 2!a2 always in-creases with a. Initially, in the region where most giant plan-ets begin their formation (&5!10 AU), a mass of at least afew Jupiter masses is needed to enter into the braking phase.However, after 1!2 Myr, which is the typical timescale to buildprotoplanets that have a su"cient mass to migrate in type II mi-gration, a mass of the order of &10 M' at &1 AU is already

For an actively accreting disk, the iceline position depends on disk mass (viscous heating). The plot shows the position of the iceline aice as a function of the initial gas surface density Σ0 at 5.2 AU (upper three lines). It corresponds to an initial disk midplane temperature Tmid of 170 K. The iceline is plotted for three values of the viscosity parameter α: 0.01 (dashed line), 0.007 (solid line) and 0.001 (dotted line). The lower three lines correspond to an initial Tmid of 1600 K, roughly the evaporation temperature of rock.

Mor

dasi

ni e

t al.

2009

Passively irradiated disks are found to have the ice line at a smaller radius, sometimes closer than 1 AU (Garaud & Lin, 2007; Lecar et al., 2006).

Iceline position II

!"#

!$#

!%#

!&#

'#

()*# +,-./0.1#

2)34,/)*#5/)67,/8179-#

:7; &

'<=/.1)#>

/--#?)/807,

@#

#####'A$#################&AB####%A'####%AB#####$A'############"A'##(4-./,81#?)7>#.C1#D9,E#FG#

+/).CH-#I/.1)#87,.1,.#

J9./0K1#L/)1,.#673*#<LA6A@#M78/07,-#/,3#I/.1)#87,.1,.#/.#.C1#0>1#7?#.C1#+/).CH-#;)7I.C#

In the Solar System, the composition of the putative parent bodies of different classes of meteorites indicates that water-rich asteroids exist in the outer asteroid belt (Morbidelli et al., 2000). This suggests that the ice line in the Solar Nebula was located at about 3 AU.

Saas Fee Course 2011

One should however understand that the iceline position was dynamic and evolved in time. It is an active subject of research (e.g. Min et al. 2011).

1.6 Condensation in extrasolar systems

No. 2, 2010 COMPOSITIONAL DIVERSITY OF EXTRASOLAR TERRESTRIAL PLANETS. I. 1051

Mg/Si0.5 1.0 1.5 2.0 2.5

C/O

0.0

0.5

1.0

1.5

2.0

Figure 1. Mg/Si vs. C/O for known planetary host stars with reliable stellarabundances. Filled circles represent those systems selected for this study. Stellarphotospheric values were taken from Gilli et al. (2006; Si, Mg), Beirao et al.(2005; Mg), Ecuvillon et al. (2004; C), and Ecuvillon et al. (2006; O). Solarvalues are shown by the black star and were taken from Asplund et al. (2005).The dashed line indicates a C/O value of 0.8 and marks the transitions betweena silicate-dominated composition and a carbide-dominated composition at10!4 bar. Average 2! error bars shown in upper right. All ratios are elementalnumber ratios, not solar normalized logarithmic values.

issue by simulating late-stage in situ terrestrial planet formationwithin 10 extrasolar planetary systems while simultaneouslydetermining the bulk elemental compositions of the planets pro-duced. This is the first such study to consider both the dynamicaland chemical nature of potential extrasolar terrestrial planets andit represents a significant step toward understanding the diversityof potential extrasolar terrestrial planets.

2. SYSTEM COMPOSITION

The two most important elemental ratios for determining themineralogy of extrasolar terrestrial planets are C/O and Mg/Si. Note that throughout this paper, Mg/Si and C/O refer tothe elemental number ratios, not solar normalized logarithmicvalues often quoted in stellar spectroscopy (usually shownas [X/H] for the solar normalized logarithmic abundance ofelement X compared to H). The ratio of C/O controls thedistribution of Si among carbide and oxide species. Under theassumption of equilibrium, if the C/O ratio is greater than 0.8(for a pressure of 10!4 bar), Si exists in solid form primarilyas SiC. Additionally, a significant amount of solid C is alsopresent as a planet-building material. For C/O values below0.8, Si is present in rock-forming minerals as SiO4

4! (or SiO2),allowing for the formation of silicates. The silicate mineralogy iscontrolled by the Mg/Si value. For Mg/Si values less than 1, Mgis in pyroxene (MgSiO3) and the excess Si is present as othersilicate species such as feldspars. For Mg/Si values rangingfrom 1 to 2, Mg is distributed between olivine (Mg2SiO4) andpyroxene. For Mg/Si values extending beyond 2, all availableSi is consumed to form olivine with excess Mg available to bondwith other elements as MgO or MgS.

Just as stellar C/O values are known to vary within thesolar neighborhood (Gustafsson et al. 1999), the C/O valuesof extrasolar planetary systems also deviate from the solarvalue. The photospheric C/O versus Mg/Si values for knownextrasolar planetary host stars are shown in Figure 1, basedon stellar abundances taken from Gilli et al. (2006; Si andMg), Beirao et al. (2005; Mg), Ecuvillon et al. (2004; C), and

Table 1Statistical Analysis of the Host and Non-host Star Distributions of

Mg/Si and C/O

Elemental Ratio Mean Median StandardDeviation

Mg/SiHost stars 0.83 ± 0.04 0.80 0.22Non-host stars 0.80 ± 0.03 0.79 0.16

C/OHost stars 0.67 ± 0.03 0.68 0.23Non-host stars 0.67 ± 0.03 0.69 0.23

Notes. All values are based on the abundances determined in Bond et al. (2008).The quoted uncertainty is the standard error in the mean. All ratios are elementalnumber ratios, not solar normalized logarithmic values.

Ecuvillon et al. (2006; O). A conservative approach was takenin determining the average error shown in Figure 1. The errorspublished for each elemental abundance were taken as beingthe 2! errors (as the method used to determine them naturallyprovides the 2! error range) and were used to determine themaximum and minimum abundance values possible with 2!confidence for each system. The elemental ratios produced bythese extremum abundances were thus taken as the 2! range inratio values and are shown as errors in Figure 1.

The mean values of Mg/Si and C/O for all extrasolarplanetary systems for which reliable abundances are availableare 1.32 and 0.77, respectively, which are above solar values(Mg/Si" = 1.05 and C/O" = 0.54; Asplund et al. 2005).This non-solar average and observed variation implies that awide variety of materials would be available to build terrestrialplanets in those systems, and not all planets that form can beexpected to be similar to that of Earth. Of the 60 systems shown,21 have C/O values above 0.8, implying that carbide mineralsare important planet-building materials in potentially more than30% of known planetary systems. This implies that a similarfraction of protoplanetary disks should contain high abundancesof carbonaceous grains. Furthermore, infrared spectral featuresat 3.43 and 3.53 µm observed in 4% of protoplanetary diskshave been identified as being produced by nano-diamonds (Acke& van den Ancker 2006). Such high abundances of carbon-rich grains in nascent planetary systems are inconceivable ifthey have primary mineralogy similar to our solar system, thusimplying that C-rich planetary systems may be more commonthan previously thought. The idea of C-rich planets is not new(Kuchner & Seager 2005) but the potential prevalence of thesebodies has not been previously recognized, nor have specificsystems been identified as likely C-rich planetary hosts. Thesedata clearly demonstrate that there are a significant number ofsystems in which terrestrial planets could have compositionsvastly different to any body observed in our solar system.

Both host and non-host stars4 display the same distributionsin C/O and Mg/Si values (see Figure 2). The mean, median,and standard deviation for both the host and non-host stars areshown in Table 1. The values listed in Table 1 are based onthe stellar abundances determined in Bond et al. (2008) as Gilliet al. (2006), Beirao et al. (2005), Ecuvillon et al. (2006), andEcuvillon et al. (2004) provide abundances for all four elementsfor just three non-host stars, thus preventing host and non-hostcomparisons. It is essential to point out here that the values

4 Throughout this paper, non-host stars refer to stars observed as part of aplanet search program that are not currently known to harbor a planetarycompanion.

The plot shows the measured photospheric Mg/Si vs. C/O for known planetary host stars. Solar values are shown by the black star (Asplund et al. 2005). The dashed line indicates a C/O value of 0.8 and marks the transitions between a silicate-dominated composition and a carbide-dominated composition at 10−4 bar.

In the last section, we have assumed a scaled solar composition for other stars. Detailed spectroscopic observations however show that some other stars have a different elemental composition in their photosphere. This implies that in the nebula there was a different condensation sequence leading to planetary building blocks consisting of minerals different than in the Solar system.

Condensation in extrasolar systems

This could in turn lead to a different composition of the planets forming around such stars (even though we have seen that in Solar System it is not straightforward to go from the condensation sequence to the composition of the final planets). In the end, this possibly affects the ability of extrasolar terrestrial planets around other stars to be habitable.

Here we discuss some results of Bond et al. 2010 who combined condensation sequence calculations for extrasolar planet host stars with planet accretion simulations.

Bond et al. 2010

Condensation in extrasolar systems II

The exact composition of silicates that form is controlled by the Mg/Si value. The minerals vary from pyroxene (MgSiO3) and various feldspars (for Mg/Si<1), to a combination of pyroxene and olivine (Mg2SiO4) (for 1<Mg/Si<2) and finally to olivine with other MgO or MgS species (for Mg/Si>2). The solar Mg/Si value is 1.05 while the bulk Earth Mg/Si value is 1.02.

The C/O ratio controls the occurrence of C in form of graphite and other carbide phases like SiC, TiC. At high C/O>0.8, SiC becomes the dominant form of Si instead of silicates which are Si-O compounds (as found in the solar system). Additionally, a significant amount of solid C is also present as a planet-building material. Therefore, at C/O>0.8, so-called “carbon planets” form (Seager et al. 2007). The sun has C/O=0.54.

Planets with Mg/Si<1 i.e. lower than Solar System will form species will have melts with a felsic composition. Such magma is very viscous, so extrusive volcanism could be very explosive on such planets.1064 BOND, O’BRIEN, & LAURETTA Vol. 715

Final Composition - HD19994 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - HD108874 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - HD4203 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.1 0.2 0.3 0.4 0.5

Sim.1

Sim.2

Sim.3

Sim.4

O

Fe

Mg

Si

C

S

Al

Ca

Other

Figure 14. Schematic of the bulk elemental planetary composition for thehigh-C-enrichment systems HD19994 (top), HD108874 (middle), and HD4203(bottom). All values are wt% of the final simulated planet. Values are shownfor the terrestrial planets produced in each of the four simulations run for thesystem. Size of bodies is not to scale. Earth values taken from Kargel & Lewis(1993) are shown in the upper right of each panel for comparison.

(An extended version of this figure is available in the online journal.)

However, for planetesimals initially forming under diskconditions at later times the planetary composition for allsimulated planets changes to more closely resemble a C-enriched Earth-like planet, with planets dominated by O, Fe,Mg, and Si and a significant amount of C. Up to 4.37 wt%C is predicted to exist in the planets for the disk conditionsat 3 ! 106 yr. These planets are essentially C-enriched Earths,containing the same major elements in geochemical ratios withinlimits to be considered Earth like, but also an enhanced inventoryof C, primarily accreted as solid graphite. As for 55Cnc, it is

r (AU)0 1 2 3 4 5

So

lar

No

rmal

ized

Mas

s

1

10

100

1000HD 4203 HD27442 HD177830 HD 72659

Figure 15. Solid mass distribution within the disk for four known extrasolarplanetary systems. All distributions are normalized to the solar distribution.Mass distributions are shown for HD4203 (solid; Mg/Si = 1.17, C/O =1.86), HD27442 (dash-dotted; Mg/Si = 1.17, C/O = 0.63), HD177830 (longdash; Mg/Si = 1.91, C/O = 0.83), and HD72659 (short dash; Mg/Si = 1.23,C/O = 0.40).

Figure 16. Schematic of notional interior models is based on calculations ofbulk planetary compositions for disk conditions at t = 5 ! 105 yr resultingfrom three different planetary systems: Gl777 (HD190360; top), HD177830(middle), and HD108874 (bottom). Figures are to scale for planet and layersizes and planet location.

expected that if were we to incorporate time-varying equilibriumcompositions into our models that we would see C occurring inthe terrestrial planets for all simulation times.

HD177830. HD177830 has the highest Mg/Si (and Al/Si)ratio of any system simulated. This enrichment alters thecompositions of major silicate species present within the disk.While the solar system should have condensed both olivine andpyroxene between 0.35 and 2.5 AU, HD177830 is dominated byolivine beyond 0.3 AU and contains only a small region wherepyroxene is predicted to coexist. This unusual composition isreflected in the final planetary abundances as the planets containlarge portions of Mg (up to 22.33 wt%; see Figure 12) and

The plot shows the solid mass distribution obtained from the condensation sequence within the disk for four known extrasolar planetary systems. All distributions are normalized to the solar distribution. Mass distributions are shown for -HD4203 (solid; Mg/Si = 1.17, C/O = 1.86)-HD27442 (dash-dotted; Mg/Si = 1.17, C/O = 0.63) -HD177830 (long dash; Mg/Si = 1.91, C/O = 0.83) -HD72659 (short dash; Mg/Si = 1.23, C/O = 0.40).Bond et al. 2010

Condensation in extrasolar systems III

1062 BOND, O’BRIEN, & LAURETTA Vol. 715

Final Composition - HD213240 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8

Sim.1

Sim.2

Sim.3

Sim.4

O

Fe

Mg

Si

C

S

Al

Ca

Other

Final Composition - HD27442 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8 1.0

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - HD72659 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Sim.1

Sim.2

Sim.3

Sim.4

Figure 10. Schematic of the bulk elemental planetary composition for the Earth-like planetary systems HD27442 (top), HD72659 (middle), and HD213240(bottom). All values are wt% of the final simulated planet. Values are shownfor the terrestrial planets produced in each of the four simulations run for thesystem. Size of bodies is not to scale. Earth values taken from Kargel & Lewis(1993) are shown in the upper right of each panel for comparison.

Martian fractionation

line

Al/Si (weight ratio)

0.0 0.2 0.4 0.6 0.8

Mg

/Si (

wei

gh

t ra

tio

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

MarsEarth VenusHD213240HD72659HD27442

Earth fractionation line

Figure 11. Al/Si vs. Mg/Si for the planets of HD27442 (triangles), HD72659(squares), and HD213240 (circles). Values are for disk conditions at 5 ! 105 yr.Earth values are shown as filled circles and are taken from Kargel & Lewis (1993)and McDonough & Sun (1995). Martian values are shown as filled diamondsand are taken from Lodders & Fegley (1997). Venus values are shown as filledsquares and are taken from Morgan & Anders (1980). Note that the values forHD27442, HD72659, and HD213240 all extend off to the right, reaching Mg/Sivalues of up to 3.5.

Final Composition - HD17051 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - HD177830 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - 55 Cnc (0.5 Myr)

Semimajor Axis (AU)

0 1 2 3 4

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - Gl777 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8 1.0

Sim.1

Sim.2

Sim.3

Sim.4

O

Fe

Mg

Si

C

S

Al

Ca

Other

Figure 12. Schematic of the bulk elemental planetary composition for the low-C-enrichment systems 55Cnc (first from the top), Gl777 (second from the top),HD17051 (third from the top), and HD177830 (bottom). All values are wt% ofthe final simulated planet. Values are shown for the terrestrial planets producedin each of the four simulations run for the system. Size of bodies is not to scale.Earth values taken from Kargel & Lewis (1993) are shown in the upper right ofeach panel for comparison.

Schematic of the bulk elemental planetary composition for the Earth-like planetary systems found to form around HD72659 (Mg/Si = 1.23, C/O = 0.40) and HD4203 (Mg/Si = 1.17, C/O = 1.86). All values are wt% of the final simulated planet. Values are shown for the terrestrial planets produced in each of the four simulation run. Size of bodies is not to scale. Earth values are shown in the upper right of each panel for comparison.

1064 BOND, O’BRIEN, & LAURETTA Vol. 715

Final Composition - HD19994 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.2 0.4 0.6 0.8

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - HD108874 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Sim.1

Sim.2

Sim.3

Sim.4

Final Composition - HD4203 (0.5 Myr)

Semimajor Axis (AU)

0.0 0.1 0.2 0.3 0.4 0.5

Sim.1

Sim.2

Sim.3

Sim.4

O

Fe

Mg

Si

C

S

Al

Ca

Other

Figure 14. Schematic of the bulk elemental planetary composition for thehigh-C-enrichment systems HD19994 (top), HD108874 (middle), and HD4203(bottom). All values are wt% of the final simulated planet. Values are shownfor the terrestrial planets produced in each of the four simulations run for thesystem. Size of bodies is not to scale. Earth values taken from Kargel & Lewis(1993) are shown in the upper right of each panel for comparison.

(An extended version of this figure is available in the online journal.)

However, for planetesimals initially forming under diskconditions at later times the planetary composition for allsimulated planets changes to more closely resemble a C-enriched Earth-like planet, with planets dominated by O, Fe,Mg, and Si and a significant amount of C. Up to 4.37 wt%C is predicted to exist in the planets for the disk conditionsat 3 ! 106 yr. These planets are essentially C-enriched Earths,containing the same major elements in geochemical ratios withinlimits to be considered Earth like, but also an enhanced inventoryof C, primarily accreted as solid graphite. As for 55Cnc, it is

r (AU)0 1 2 3 4 5

So

lar

No

rmal

ized

Mas

s

1

10

100

1000HD 4203 HD27442 HD177830 HD 72659

Figure 15. Solid mass distribution within the disk for four known extrasolarplanetary systems. All distributions are normalized to the solar distribution.Mass distributions are shown for HD4203 (solid; Mg/Si = 1.17, C/O =1.86), HD27442 (dash-dotted; Mg/Si = 1.17, C/O = 0.63), HD177830 (longdash; Mg/Si = 1.91, C/O = 0.83), and HD72659 (short dash; Mg/Si = 1.23,C/O = 0.40).

Figure 16. Schematic of notional interior models is based on calculations ofbulk planetary compositions for disk conditions at t = 5 ! 105 yr resultingfrom three different planetary systems: Gl777 (HD190360; top), HD177830(middle), and HD108874 (bottom). Figures are to scale for planet and layersizes and planet location.

expected that if were we to incorporate time-varying equilibriumcompositions into our models that we would see C occurring inthe terrestrial planets for all simulation times.

HD177830. HD177830 has the highest Mg/Si (and Al/Si)ratio of any system simulated. This enrichment alters thecompositions of major silicate species present within the disk.While the solar system should have condensed both olivine andpyroxene between 0.35 and 2.5 AU, HD177830 is dominated byolivine beyond 0.3 AU and contains only a small region wherepyroxene is predicted to coexist. This unusual composition isreflected in the final planetary abundances as the planets containlarge portions of Mg (up to 22.33 wt%; see Figure 12) and

Note that planets forming around HD72659 at 1 AU are roughly Earth like. Planets closer in contain more Al and Ca, which are more refractory (higher condensation temperature). The model planets forming at about 0.3 AU around HD4203 are however very different. They should have a crust made of graphite. The different composition also affects the geological evolution in terms of plate tectonics, or the existence of magnetic fields. Both are thought to be important for the emergence of life.

Starting with the planetesimal surface density and composition obtained from the condensation sequence, Bond et al. run accretion simulations of the final phase of terrestrial planet formation.

Bond et al. 2010

Questions?