Download L8 Condensation

Lecture 8
Condensation
Bond et al. 2010
Lecture Universität Heidelberg WS 11/12
Dr. C. Mordasini
Based partially on script of Prof. W. Benz
Mentor Prof. T. Henning
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 chondrites
The condensation of dust grains out of the gas phase represents the very first phase of
planet growth.
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 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.
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
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.
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).
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.
Thermodynamic potentials
In 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.
Thermodynamic potentials II
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.
G = H − T S → dG = dH − T dS − 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 − T dS − SdT = δQ + V dp − δQ − SdT = V dp − SdT
δQ
where we have used the definition of the entropy: dS =
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.
Example
To illustrate this concept, let us compute the change in free enthalpy for the reaction taking
place at standard conditions:
1
H2 + O2 → H2 O g
2
Let us define the free enthalpy change (final minus initial) at these standard conditions as
∆G00
1
= G00 (H2 O) − G00 (H2 ) − G00 (O2 )
2
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.
From tables we can the following values:
∆G00
1
= −258.8 − 0.0 − · 0 = −258.8 k J /mole
2
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).
Condensation in the nebula
Since 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
From the definition of the free enthalpy change, we have in this case (dT=0) for an ideal gas
dp
dG = V dp = nRT
→ G(p, T ) − G0 (T ) = nRT ln
p
�
p
p0
�
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).
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
where the Δ represents the difference operator ''after'' – ''before'' of the chemical reaction. The
pi are the partial pressures.
In equilibrium, ΔG(p,T) = 0, and
(1)
Condensation in the nebula II
To understand this formalism, we consider the simple example
aA + bB → cC + dD
In equilibrium, we must have:
�
i
ni ln
�
pi
p0
�
�

= ln  �
For the reaction constant Kp(T) we have
pC
p0
�c �
pD
p0
�d 
∆G0 (T )

= ln Kp
�a � �b  = −
RT
pA
pB
p0
p0
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
S(T )dT
T0
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
δQ
dS =
→ S(T ) − S0 =
T
�
T
T0
δQ
=
T
where S0 is the entropy at standard condition.
�
T
T0
dT
cp
= cp ln
T
�
T
T0
�
In case of a chemical reaction taking place at standard pressure but varying temperature, we
can write (integral of S over T):
�
� �
�
T
∆G(p, T ) − ∆G0 (p) = −∆S0 (T − T0 ) − ∆cp T ln
− (T − T0 )
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).
At equilibrium, we will again have ΔG(p,T) = 0, so that
�
∆G(p, T ) − ∆G0 (p) = −∆S0 (T − T0 ) − ∆cp T ln
�
T
T0
�
�
− (T − T0 )
(2)
Finally, in our isothermal-isobar situation, we can combine (1) and (2), taking into account the
enthalpy of formation of the substances:
(3)
1.3 Examples
Example 1: Dissociation of hydrogen
For 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 II
Solving 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
For the specific heats we assume an ideal gas, therefore
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):
Dissociation of hydrogen IV
1.0
0.1
0.01
ptot [atm]
0.001
10-8
10-6
10-4
10-2
10�4
10�5
1000
1500
2000
2500
Temperature [K]
3000
3500
Fraction α of dissociated H2
Fraction α of dissociated H2
1
0.8
0.6
ptot [atm]
0.4
10-8
10-6
10-4
10-2
0.2
0.0
1000
1500
2000
2500
3000
3500
4000
Temperature [K]
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 I
At 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
�
T
298
�
�
�
�
T
∆G0 (T ) = −3.698 × 105 + 153.42(T − 298) − 0.58 T ln
− (T − 298)
298
∆S(T ) = SFe (s) (T ) − SFe (g) (T ) = −153.42 − 0.58 ln
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 II
To 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(F e) = ptot
�(F e)
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 sequence
The 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 II
Grossman & 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.
In order for the equilibrium
condensation model to be correct,
the various timescales for the
chemical reactions (gas-gas, gassolids, 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.
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.
The full sequence III
The 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 IV
The 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
Water ice condensation
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.
Lodders 2003
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.
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).
e formation model for each set of initial con- the computational variable required by our model, and the corretemporal evolution of the planet (formation sponding observable, the stellar metallicity [Fe/H], we assume:
well as its final properties (mass, semimajor (1) the stellar content in heavy elements is a good measure of the
tc., Under
Sect. 5.2).
abundance
of heavy
elements in out
the of
disk
during
the simplistic assumption that theoverall
fraction
of material
that condenses
the
gas isformaynthetic
planets
wouldfor
remain
undetected
tion time.
for this
assumption
comes
from the
small difconstant
except
the increase
at by
the iceline,
weSupport
can write
for the
initial solid
surface
density
al techniques. So, to be able to compare the ferences between solar photospheric and meteoritic abundances
profilewith the observed one, we apply in (Asplund et al. 2005); (2) a scaled solar composition and (3) a
pulation
ailed synthetic detection bias (Paper II). In negligibly small influence of the change of the relative heavy
n the sub-population of observable synthetic element content on the relative hydrogen content in the comparin Here,
the sixth
step, we
performed
quantitative
Σ(r,t=0)
is the
gas surface
density at
t=0 (which
is obviously
fD/G isformation
the dust in
atively
small [Fe/H]
domainillofdefined)
interest and
for planet
per to
II)gas
to compare
thethe
properties
of assumed
this ob- that
the solar
neighborhood
(−0.5
[Fe/H]
0.5). Then,
Then, similar
to
ratio. For
later, it is
it is the
same in the
disk≤as
in the≤ star.
we
exoplanet
sub-population
with a comparison
Murray
et al. (2001), we can write
can relate
it to the so-called
stellar metallicity
[Fe/H].
solar planets.
fD/G
= 10[Fe/H]
(6)
fD/G,%
ariables
where
is thetracer
dust toofgas
ratio
corresponding
to [Fe/H]of= 0.
This formula implies that we assume that
iron fisD/G,%
a good
the
relevant
overall amount
Carlo
variables
to describe
the varying
initiali.e. a
This
formula
implies
that we assume
that iron is
is defined
a good tracer
solids
available
for planet
formation
scaled
solar
composition.
The metallicity
planetary formation process. Three describe of the relevant overall amount of solids available for planet foras
disk and one the seed embryo.
mation. Robinson et al. (2006) have found that at a given iron
abundance, planet host stars are enriched in silicon and nickel
s ratio in the protoplanetary disk fD/G de- over stars without planets, indicating that the above relation is a
ther with Σ0 ) the solid surface
simplification.
This density.
means that
a star with the same Fe content as the sun has
computed.
/G between 0.013 and 0.13 were
Measurements
of thespectroscopically.
heavy element abundance
in like
the Sun
[Fe/H]=0.
[Fe/H] can
be determined
For solar
the domain of Σ0 , this corresponds to ini- yield the amount (for complete condensation) of high Z material
FGK stars in the solar neighborhood, one finds a Gaussian
e densities at a0 = 5.2 AU of between 0.65 that existed initially in the form of uniformly mixed fine dust
distribution
[Fe/H] with a mean µ ~0.0, and a dispersion σ~ 0.2
For comparison, the MMSN has
a value of forgrains.
However, what is relevant for our simulations is the con(e.g.
Santos
et
al.
2003).
2
2.5 g/cm (Hayashi 1981).
centration of solids in the innermost 20 AU of the disk at a later
urface density Σ0 at 5.2 AU givesThe
the amount
stage, namely when the dust has evolved into the 100 km planspread
2 in [Fe/H] by about 1 dex shows that initial dust-to-gas
. Values between between 50 and ratios
1000 g/cm
used inone
our model.
in disksetesimals
vary by about
order or magnitude.
Mordasini et
2009
MMSN
isal.estimated
to have had a value of
As has been shown by Kornet et al. (2001), the transition
Initial solid surface density profile
vant for low mass planets (see Tanaka et al. 2002), is reduced
ovfor more massive central stars.
6).
ou- 3. the disk structure. High central mass stars result in
higher gravity in the vertical direction. On the other
erhand, viscosity
dissipation
onthe
the Keplerian
ro- The factor
fR/I represents
thedepends
effect of
iceline: frequency, and disks around high mass stars are hotter.
he
Numerical calculations show that disks around high mass
on
stars are generally thinner, the first effect being more imon
C. Mordasini et al.: Extrasolar planet population synthesis. I.
1141
portant.
le,
among
ng 4. the Keplerian frequency which governs,
2.2. Migration
rate other things,
For an actively accreting disk, the iceline position depends on
the accretion rate of solids.
ity
The migration of the protoplanet occurs in two main regimes
depending
its mass.(viscous
Low mass planets
undergo type I midiskupon
mass
heating).
The plot shows the position of
cal
gration (Ward 1997; Tanaka et al. 2002) which depends linearly
Moreover, the disk model takes into
account
theaice
effect
offunction
on the
body’s
mass. The prevalence
planets hasof
led the initial gas surface density
tic
the
iceline
asofaextrasolar
to suspect
the actual type I migration
than
thethatgravitational
ra- rate is probably sigal. photoevaporation, at distances larger us
nificantly
lower
than
currently
estimated
(Menou
& Goodman
Σ
at
5.2
AU
(upper
three
lines). It corresponds to an initial
0
dius
R
(see
Veras
and
Armitage,
2004),
which
depends
linearly
g
2003; Nelson & Papaloizou 2004). For this reason, we allow for
opdisketreduction
midplane
Tmid
a arbitrary
of the type
Itemperature
migration rate as calculated
in of 170 K. The iceline is
on
the
mass
of
the
central
star
(see
Adams
al.
2004).
Finally,
he
Tanaka et al. (2002) by a constant efficiency factor fI .
plotted
forchanges
three
values
thetheviscosity parameter α: 0.01
the
location
of
the
iceline
depends
on
the
andfrom
presThetemperature
migration type
type I to typeof
II when
vel
becomes massive enough to open a gap in the disk. We
sure in the disk. All other parametersplanet
being
equal,
the
iceline
(dashed
line),
(solid
line)
assume that this happens
when0.007
the Hill radius
of the planet
be- and 0.001 (dotted line). The
greater
than thestars,
density scale
&
is located at larger distances around comes
higher
mass
the height
ef- H̃ of the disk (Linto
lower
three
lines
correspond
Papaloizou 1986). Planetary masses where the migration regime an initial Tmid of 1600 K,
fect begin of the order of 1 to 2 AUchanges
(depending
on such
thea thermal
disk’scriterion only, as found
can be low with
roughly
the
evaporation
temperature
of rock.
also
by
Papaloizou
&
Terquem
(1999)
who
use
a
similar
condimass),
going
from
M
=
0.5M
to
M
=
2.0M
.
Analytical
pastar
⊙
star
⊙
tion.−3
This is especially the case as due to disk evolution, the disk
10 height
, our
nominal
value
H̃ decreases
with time,
so that the minimal mass
ent fit of our disk models for α = 7 × scale
to open a gap decreases. This effect is emphasized by the
parameter
(Shakura
&not include irradiation,
ss, of the Shakura & Sunyaev viscosityneeded
fact that The
our disk figure
model currently
on does
the
so that especially
the end
of disk
Sunyaev,
1973), shows that the position
of the towards
iceline
can
beevolution, H̃ becomes
tarFig. 1. Position
of the iceline a as a function of the initial gas surface smaller than
in
a
disk
including
it,
and
smaller
planets can open
right
shows
the
initial
Fit
for
α=0.007
(Alibert
et
al.
2011)
density
Σ
at
5.2
AU
(upper
three
lines).
It
corresponds
to
an
initial
T
approximated
as
of
a gap (Edgar et al. 2007). The order of magnitude we obtain is
of 170 K. The iceline is plotted for three values of α: 0.01 (dashed line), however solid
consistent surface
with the one derived
from Armitage & Rice
density
ry.0.007 (solid line) and� 0.001 (dotted line).
The lower
three lines
corre�
�
�
(2005),
since
they
give
a
gap
opening
condition
(including the
0.44
0.1
spond to an initial T of 1600 K, roughly
the evaporation temperature
1/2
>
(black
line).
The
initial
Mstarin the nom- effect of viscosity) of Mplanet /M∗ ∼ α (H̃/aplanet)2. In our simereof rock. rThe
icerockline a Σis5.2AU
however not taken into account
× location, as disk ulation, the transition typically occurs(1)
when the aspect ratio of
inal model, due =
to the difficulty in defining
its relevant
2
gas
surface
density
AU
evolution
is very rapid10g/cm
close-in and irradiation effects M
might
⊙ be impor- the disk has become tiny, between 2 and 3%, meaning a tranMordasini et al. 2009
Iceline position
ice
0
mid
mid
rock
is derived from the
where Σ5.2AU is the initial gas surface density
at 5.2 AU
, and of
similarity
solutions
For
disk
as
R isataken
to be 5similar
AU, a is the
sizethe
of the MMSN,
disk, and the
edwhere
we assume that the gas surface density follows a power law
total mass loss Ṁ due to photo-evaporation is an input paramethe viscous accretion
nu-ter we
finda −3/2
an
iceline
position
of
which
together
with the
α parameter
determines
lifetimethat, as mentioned in Alibert
with
slope.
We
recallthehere
disk.
de-of the
disk problem.
al. (2005),
structure
disk is calculated
without takabout
3.7
Foretsimplicity,
we AU.
adoptthe
an initial
profile ofof
thethe
gas disk
tant (cf. Paper II).
max
g
w
persurface density according to the phenomenological model of
sition at tens of Earth masses. We note that Crida et al. (2006)
have derived a new criterion for gap opening which depends on
2
both the disk aspect ratio and the Reynolds number. Using such
a modified transition mass has some influence on the planetary
formation 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’s
mass Mplanet (called “disk dominated” migration in Armitage
2007), the planet is coupled to the viscous evolution of the disk
Iceline position II
:7;&'<=/.1)#>/--#?)/807,@#
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).
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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.
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
Condensation in extrasolar systems
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.
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.
No. 2, 2010
COMPOSITIONAL DIVERSITY OF EXTRASOLAR TERRESTRIAL PLANETS. I.
Table 1
Statistical Analysis of the Host and Non-host Star Distributions of
Mg/Si and C/O
2.0
The plot shows the measured photospheric Mg/
Si vs. C/O for known planetary host stars. Solar
Elemental Ratio
Standard
values
are shown byMean
the black Median
star (Asplund
et
Deviation
al.Mg/Si
2005). The dashed line indicates a C/O value
Host stars
0.83 ± 0.04
0.80
0.22
of 0.8
and
marks
the
transitions
between
a
Non-host stars
0.80 ± 0.03
0.79
0.16
silicate-dominated
composition and a carbideC/O
Host stars
0.67 ± 0.03at 10−4 0.68
0.23
dominated
composition
bar.
1.5
C/O
1051
1.0
Non-host stars
0.5
0.67 ± 0.03
0.69
0.23
Notes. All values are based on the abundances determined in Bond et al. (2008).
0.0
0.5
Bond et al. 2010
1.0
1.5
2.0
2.5
Mg/Si
Figure 1. Mg/Si vs. C/O for known planetary host stars with reliable stellar
abundances. Filled circles represent those systems selected for this study. Stellar
photospheric values were taken from Gilli et al. (2006; Si, Mg), Beirão et al.
Here
weuncertainty
discuss
results
of All
Bond
al. 2010
The quoted
is thesome
standard error
in the mean.
ratios areet
elemental
number
ratios, not solar normalized
logarithmic values.
who
combined
condensation
sequence
calculations
for extrasolar planet host stars with
Ecuvillon et al. (2006; O). A conservative approach was taken
planet
accretion
simulations.
in determining
the average
error shown in Figure 1. The errors
published for each elemental abundance were taken as being
Condensation in extrasolar systems II
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.
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.
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.
& LAURETTA
Vol. 715
Solar Normalized Mass
1000
HD 4203
HD27442
HD177830
HD 72659
100
10
1
0
1
2
r (AU)
3
4
5
Bond et al. 2010
Figure 15. Solid mass distribution within the disk for four known extrasolar
planetary systems. All distributions are normalized to the solar distribution.
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).
Sim.2
Condensation in extrasolar systems III
Sim.3
Sim.3
Sim.3
Sim.4
Sim.4
Starting with the planetesimal surface density and composition
obtained from the condensation
Sim.4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
sequence, Bond et al. run accretion simulations of the final phaseSemimajor
of terrestrial
planet
formation.
Axis (AU)
0.0
0.2
0.4
0.6
0.8
1.0
Semimajor Axis (AU)
Final Composition - HD72659 (0.5 Myr)
0
Final Composition - HD4203 (0.5 Myr)
1
2
3
4
Semimajor Axis (AU)
Sim.1
Sim.1
Final Composition - Gl777 (0.5 Myr)O
Fe
Sim.2
Mg
Sim.2
Si
C
Sim.1
S
Sim.3
Sim.3
Al
Ca
Sim.2
Sim.4
Sim.4
Other
Sim.3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0
1.4
Bond et al. 2010
Semimajor Axis (AU)
0.1
0.2
0.3
0.4
0.5
Semimajor Axis (AU)
Sim.4
Figure 14. Schematic of the bulk elemental planetary composition for the
Composition - HD213240 (0.5 Myr)
Schematic Final
of the
bulk elemental planetary composition
for the Earth-like planetary systems
high-C-enrichment systems HD19994 (top), HD108874 (middle), and HD4203
All values
are HD4203
wt% of the final(Mg/Si
simulated planet.
Values are
shown =
found to form around HD72659 (Mg/Si = 1.23, C/O(bottom).
=
0.40)
and
=
1.17,
C/O
for the terrestrial planets produced in each of the four simulations run for the
0.0
0.2
0.4
0.6
0.8
system.
Size of
bodies
is not
to scale. Earth
values
taken
from Kargel
& Lewis1.0
1.86). All values are wt% of the final simulated planet.
Values
are
shown
for
the
terrestrial
(1993) are shown in the upper rightSemimajor
of each panel
for (AU)
comparison.
Axis
planets produced in each of the four simulation run.(AnSize
ofversion
bodies
is not
to scale.
Earth
extended
of this figure
is available
in the online
journal.) values
Final Composition - HD17051 (0.5 Myr)
are shown in the upper right of each panel for comparison.
O
Sim.1
Fe
Mg
Si
Sim.2
C
S
Sim.3
Al
However, for planetesimals initially forming under disk
Note that planets forming around HD72659 at 1 AUconditions
are Sim.1
roughly
like.planetary
Planets
closer in
at laterEarth
times the
composition
for all
simulated condensation
planets changes to temperature).
more closely resemble
contain more Al and Ca, which are more refractory (higher
Thea Cenriched Earth-like planet, with planets dominated by O, Fe,
Sim.2
model planets forming at about 0.3 AU around HD4203
are
however
veryamount
different.
They
should
Mg, and
Si and
a significant
of C. Up
to 4.37
wt%
C is predicted
exist in the
the planets
for the disk
conditionsin
have a crust made
of graphite.
The different composition
also to
affects
geological
evolution
Semimajor
Axis (AU)
6
at 3fields.
× Sim.3
10 yr.Both
These are
planets
are essentially
C-enriched
Earths,
terms10.of
plateoftectonics,
or planetary
the existence
of the
magnetic
thought
to
be
important
Figure
Schematic
the bulk elemental
composition for
Earthcontaining the same major elements in geochemical ratios within
like
planetary
systems
HD27442
(top),
HD72659
(middle),
and
HD213240
for the emergence of life.
(bottom). All values are wt% of the final simulated planet. Values are shown limits to be considered Earth like, but also an enhanced inventory
Ca
Other
Sim.4
0.0
0.2
0.4
0.6
0.8
Questions?