Planetary population synthesis

Lecture 15
Lecture Universität Heidelberg WS 11/12
Dr. C. Mordasini
Planetary
population synthesis
Mentor Prof. T. Henning

Lecture 15 overview
1. Motivation for population synthesis
2. Method
3. Formation tracks and a-M distribution
4. Mass distribution
5. Semimajor axis distribution
6. Stopping distance of close-in planets
7. Radius distribution
8. Metallicity distribution

1. Motivation for planetary
population synthesis

Deal with (a lot of) statistical data
The last years of observational exoplanet research was characterized by a tremendous increase in
data, i.e. in large number of detections. Of particular importance are the discoveries by high precision
radial velocity observations, and by the KEPLER satellite (transit photometry). But also ground based
transit observations, microlensing and direct imaging observations contribute to the very large data
set.
More, even bigger data sets are expected to come in the future (e.g. the GAIA satellite, an astrometric
mission), or WFIRST (a microlensing satellite).
Due to the different observational techniques, the different data sets contain different constraints which can in
the end help to increase our understanding of the physics of planet formation. But it is often difficult to unite the
different constraints into one coherent picture.

surements
fraction of
derstandiondistriauer
et al.
.
error and
he largest
ncertainty
ly to 35%
ansit over
randomly
e duration
al transit.
ion (1) by
tio threshnservative
does not
are grazed
perfect
intervals.
erval (apransit)
by
ct the nuly
equally.
adius
agnitude
te the dered
planet
rginalizing
puted ocranges
of
50 days.
values in
rrence for
ays. The
) increases
rence with
(4)
anets havervalcenstant,
and
se param-
R⊕ bins
a lack of
ng a max-
. Each rat
fraction,
planet der
of target
by the bi-
(5)
in a spec-
Number of Planets per Star with P < 5
Number of Planets per Star with P < 50 days
0.100
0.010
0.001
0.12
0.10
0.08
0.06
0.04
0.02
Deal with (a lot of) statistical data
1.0 1.4 2.0 2.8 4.0 5.7 8.0 11.3 16.0 22.6
Planet Radius (RE) 0.00
1.0 1.4 2.0 2.8 4.0 5.7 8.0 11.3 16.0 22.6
Planet Radius (RE) Fig. 5.— Planet occurrence as a function of planet radius for
planets with P

2. Method

specialized
models
The essence of population synthesis
Ida & Lin 2004++
Thomes et al. 2008
Mordasini et al. 2009++
Miguel et al. 2008,2009
extraction process
Towards a “Standard model of planet formation and
evolution” or “Super-Montecarlo”....
- you need specialized models to
know what is important
- while you get the essence, you
have lost the subtlety of the original
- but what is left is a concentrate
of many effects
- and lets you see the big
picture (hopefully)
population
synthesis

Extended core accretion model
Formation model tested in the Solar
System (Alibert, Mordasini, Benz 2004)
rvational motivation for population synthesis 9
Mordasini et al. 2009a
Mordasini et al. 2009b
Observed
population
e 1.1: The red line shows the total number of known extrasolar planetary companions as a function
epoch of discovery. Note how the number has grown very quickly in the last years, approximately as
nce 1995 (the green fitting curve goes as t 2.5 ). The data is taken from J. Schneiders extrasolar planet
opedia at exoplanet.eu. Before 1995, four pulsar planets, and a low mass companion of HD 114762
m et al., 1989) were known. Even if the latter object has a projected mass in the planetary mass domain
are indications that this system is seen in a rather face-on configuration, so that the true mass of the
nion could be much larger, maybe even in the low mass M star regime (Hale, 1995).
cations of population synthesis (Popov & Prokhorov, 2004).
ost of the planets in the plot were discovered by the radial velocity (RV) method, where
easures the wobble of the star around the common center of mass of the star-planet
. A part of the thesis was also the observational search of extrasolar planets with the
ethod using the HARPS spectrometer (Pepe et al., 2004) which yielded a number of
sting discoveries No (Santos match: et al., 2004a; improve,
Lovis et al., 2005, 2006; Udry et al., 2006; Mayor
, 2008; Curto et al., 2006). This observational work within the HARPS consortium (PI:
el Mayor) allowed change to learn andparameters understand the most important observational technique to
t extrasolar planets. It also made possible a very direct access to the latest observational
s, a direct insight in the various biases involved in the detection process, and a direct
oration between the observational approach at the Obsérvatoire de Genève, and the
etical approach of our group in Bern. Therefore, the observational results obtained by
V method form the main observational comparison data that was used in the theoretical
Population Synthesis Principle
Draw and compute
synthetic
planet population
Apply observational
detection bias
Comparison:
Observable sub-population
- Distribution of semi-major axis
- Distribution of masses
- Fraction of hot/cold Jupiters
- Metallicity effect
Initial Conditions: Probability
distributions & parameters
Disk gas mass
Disk dust mass
Disk lifetime
Match
From
observations
Cross check
Couple to other detection
methods
Predictions
(going back to the full
synthetic population)
Model solution
found!

Formation model
Planet-Planet
1. Disk migration
2. Planet planet interaction
(n-body)
3. Planetesimal-envelope
interaction
4. Gaseous envelope structure
5. Protoplanetary gas disk
6. Disk of planetesimals
Standard components, but coupled

1 Metallicity
assume same in star
and disk
Stellar [Fe/H] from spectroscopy.
Gaussian distribution for [Fe/H]
with µ ~0.0, σ~ 0.2. (e.g. Santos
et al. 2003)
2 Disk (gas) masses
Thermal continuum emission from cold dust at mm
and submm wavelengths (Ophiuchus nebula).
β Pic 12 K6–M4 0 ZS04 JA06
NGC 2353 12 K0–M4 0
Initial conditions
N. C. Santos et al.: Statistical properties of exoplanets 367
+6 K05 this work
Collinder 65 25 K0–M5 0 +7 Tuc-Hor 27 K1–M3 0
K05 this work
+8 NGC 6664 46 K0–M1 0
ZS04 JA06
+4 S82 this work
References. Schmidt (1982, S82),Park et al. (2001, P01),Hartmann et al. (2005, Ha05),Kharchenko et al. (2005, K05),Mohanty et al
(2005, M05),Sicilia-Aguilar et al. (2005, SA05),Carpenter et al. (2006, C06),Lada et al. (2006, L06),Jayawardhana et al. (2006, JA06)
Sicilia-Aguilar et al. (2006, SA06),Dahm & Hillenbrand (2007, D07),Briceño et al. (2007, B07),Jeffries et al. (2007, JE07), Hernández et al
(2007, He07),Sana et al. (2007, S07),Caballero (2008, C08),Flaherty & Muzerolle (2008, FM08),Luhman et al. (2008, L08),Sicilia-Aguila
et al. (2009, S09),Zuckerman & Song (2004, ZS04).
Santos et al. 2003
3 Disk lifetime
L-band (3.4 μm)
photometry:
- excess caused by μsized
dust @ ~900K
... ok to < 10 AU
Fig. 2. Left: metallicity distribution of stars with planets making part of the CORALIE planet search sample (shaded histogram) compared
with the same distribution for the about 1000 non binary stars in the CORALIE volume-limited sample (see text for more details). Right: the
percentage of stars belonging to the CORALIE search sample that have been discovered to harbor planetary mass companions plotted as a
function of the metallicity. The vertical axis represents the percentage of planet hosts with respect to the total CORALIE sample.
Haisch et al. 2001, Fedele et al. 2010
NGC 2024
Trapezium
ANDREWS ET AL.
IC 348
NGC 2362
Fig. 4. facc (dots) versus fIRAC (squares) and exponential fit for facc (dot
suggests that we may be looking at the approximate limit on the
metallicity of the stars in the solar neighborhood.
Here we have repeated the analysis presented in Paper II,
but using only the planet host stars included in the well defined
CORALIE sample7 Fig. 3. Accreting stars-frequencysurvey as a function (Udry et of al. age. 2000). New data The(based metallicities for this latter sam-
on the VIMOS survey) are shownple as were (red) computed dots, literature fromdata a precise as (green) calibration of the CORALIE
squares. Colored version is available
Cross-Correlation
in the electronic
Function
form.
(see Santos et al. 2002a); since the
4 Initial semimajor axis of the seed embryo:
calibrators used were the stars presented in Paper I, Paper II,
. This sub-sample He i 5876 has Åa total in emission of 41 ob- (EW and= this −0.5 paper, Å,–0.6Årespectively).
the final results are in the very same scale.
jects, ∼60% of them having planets The Analytical discovered evidence of in large the work con- (Lissauer & Steward 1992) and
Hα10% together The knowledge with the He of ithe emission metallicity is distribution for stars in
text of the CORALIE survey itself. most Here likely we have due included to ongoing all mass the solar accretion, neighborhood and these (and twoincluded stars
numerical simulations (Kokubo & in Ida the CORALIE 2000): sam- spacing
stars known to have companions with areminimum classifiedmasses as accreting lowerstars.
ple)Weestimateafractionofaccret- permits us to determine the percentage of planet host stars
than ∼18 MJup; changing this limiting to stars e.g. 10 in NGC MJup 6231 does of not 11/75 peror metallicity 15 (±5%). bin. WeThe warn result the reader
between bodies Δ ∝ a
is seen in Fig. 2 (right panel). As
change any of the results presented below. that this might be a lower limit we can to the perfectly actual see, fraction the probability of accret- of finding a planet host is
The fact that planets seem to orbit ingthe stars; most further metal-rich investigation stars a strong is needed function to disentagle of its metallicity. the nature This result confirms former
in the solar neighborhood has led some (accretion groupsvs tobinarity/rapid build planetda
rotation) analysis done ofda the in Paper systems II and with bylarge Reid (2002). For example, here
search samples based on the high metal Hα10% p(a)da content (>300 of kmtheir s host we can see that about = 7% dlog(a) of the stars in the CORALIE const. sample
stars. Examples of these are the stars BD-10 3166 (Butler et al. having metallicity a between 0.3 and 0.4 dex have been discov-
2000), HD 4203 (Vogt et al. 2002), and HD 73526, HD 76700, ered to harbor a planet. On the other hand, less than 1% of the
HD 30177, and HD 2039 (Tinney et al. 2002). Although clearly stars having solar metallicity seem to have a planet. This result
increasing the planet detection rate, these kind of metallicity bi- is thus probably telling us that the probability of forming a giant
ased samples completely spoil any statistical study. Using only planet, or at least a planet of the kind we are finding now, de-
stars being surveyed for planets in the context of the CORALIE pends strongly on the metallicity of the gas that gave origin to
survey (none of these 6 stars is included), a survey that has the star and planetary system. This might be simple explained if
never used the metallicity as a favoring quantity for looking for we consider that the higher the metallicity (i.e. dust density of
planets, has thus the advantage of minimizing this bias. the disk) the higher might be the probability of forming a core
−1 0.8
0.6
0.4
0.2
(a)
ted line) and for fIRAC (dashed line).
Andrews et al. 2010 (b) (c) (d)
NGC 6531
We identified 26 cluster members in NGC 6531 based on the
presence of Hα emission and presence of Li. 13 other sources
show presence of Li 6708 Å, but have Hα in absorption. As in
the case of NGC 6231, these might be cluster members with no
or a reduced chromospheric activity. We measured the EW of L
6708 Å of these 13 sources and compared them with the typi
) but low EW [Hα].
cal EW of the 26 stars in NGC 6531 showing also Hα emission
Page 5 of 7
0.0
4 3 2 1 0
Log M⇤M ⇥
5 Stellar mass

3. Formation tracks
and
mass-distance distribution

Edge of
comp. disk
Formation of the a-M diagram
Dittkrist et al. in prep
One model,
very different
outcomes.
isothermal type I
adiabatic type I
saturated type I
type II
Mstar=1M⊙, alpha=7x10-3 Starting mass
Irradiated disk
2% interstellar grain opacity

Inner boundary comp. disk
0.1 AU
Type II Migration
Limit
Synthetic a-M diagram
Giants
Planetary Desert
Failed cores
(Proto-terrestrial planets)
Timescale Limit
Iceline
upturn
Mainly icy core
Mainly rocky core
Menv / Mheavy > 10
1< Menv / Mheavy < 10
Menv / Mheavy < 1
Mstar=1 M⊙, alpha=7x10 -3
Irradiated disk. Σ(0.1)=0
non-isothermal migration
0.3% interstellar grain opacity

4. The planetary
mass distribution

Mordasini
et al. 2009b
Model incompleteness
Planetary Initial Mass Function PIMF
Peak at low masses
Neptunian Bump
Minimum: Planetary desert
Flat Giant’s Plateau
Superjupiter Tail
• Complex structure, dominated by low mass planets
• Consistent w. non-detection of Jupiters around 90% stars.
• Maxima at masses similar to Solar System planets.
MStar=1 MSun
Nominal Model
Type Mass (MEarth) %
(Super)-Earth < 7 58
Neptunian 7-30 17
Intermediate 30-100 6
Jovian 100-1000 14
Super-Jupiter > 1000 4
Model predicts that planets with
M < 30 MEarth account for over
75% of all planets

Mordasini et al. 2012
PIMF: Dependence on disk properties
Metallicity Disk mass Disk lifetime
Fe/H mainly just scales
PIMF for giant planets:
Fe/H: threshold, but final
mass not given by Fe/H
•higher number of giants
•but not more massive
Inverted metallicity
effect for low mass
Weak metallicity
effect for Neptunes
Disk mass changes the MF
shape for giant planets.
High disk mass: giant
planets of higher mass, but
less of lower mass.
Disk lifetime changes both.
Long living disks: giants
• more numerous and
• higher mass
-Correlation with MD

We derive the statistical properties of planets from the 1330 FGKM target stars
for which we have uniform precision of 3 m s −1 and at least 6 years duration of
observations. Detected exoplanets have minimum masses, Msin i, between 6 MEarth
Mass distribution: low RV precision
and ∼15 MJup, with an upper mass limit corresponding to the (vanishing) tail of
the mass distribution. The planet mass distribution is shown in Fig. 1 and follows
a power law, dN/dM ∝ M −1.05 54), 55) affected very little by the unknown sin i. 41)
The paucity of companions with Msin i greater than 12 MJup confirms the presence
of a “brown dwarf desert” 54) for companions with orbital periods up to a decade.
Using a synthetic observational bias for a RV survey of 10 years at 10 m/s precision, find sub-population
of detectable synthetic planets.
Number of Planets
20
15
10
5
Planet Mass Distribution
dN/dM ∝ M −1.05
104 Planets
Keck, Lick, AAT
Marcy et al. 2005
0
0 2 4 6 8 10 12 14
M sin i (MJUP) Fig. 1. The histogram of 104 planet masses (Msin i) found in the uniform 3 m s −1 Doppler survey
of 1330 stars at Lick, Keck, and the AAT telescopes. The bin size is 0.5 MJup. The distribution
of planet masses rises as M −1.05 from 10 MJup down to Saturn masses, with incompleteness at
lower masses.
The mass distribution is very well reproduced.
Forming really massive planets is hard (insufficient mass & time)

Observation
?
Towards the underlying mass distribution
Udry & Santos 2007
All instruments
HARPS (1 m/s)
Observational
bias
Hints of the Neptunian bump and
the minimum at 30 ME?
Synthetic
Mordasini et al. 2009
10 m/s
1m/s
0.1 m/s
Full Population
Very strong sign of core accretion
(all models - Ida & Lin, Mordasini et al, Miguel et al.)

Depth of the minimum
rch for southern extra-solar planets. XXIII. 9
e
-
to
so
n
a;
en)
er
sa
ly
ns
re
ts
yrre
st
el
Dependence on gas accretion rate in runaway
in Mordasini Fig. 7. Theoretical et al. 2011
mass distribution from Neptunian to Jovian
Planetary gas accretion rate
limited to slow viscous disk
accretion rate.
Shallow minimum.
Planetary gas accretion rate not
limited to disk accretion rate for
gas already in the planet’s hill
sphere (fast accretion).
Deep minimum.
Detections important for directly
constraining formation theory.

Mass distribution II: high RV precision
uncorrected
for obs. bias
corrected for obs. bias
Mayor et al. 2011
•RV: thanks to 1 m/s precision observations, a new huge population of low
mass planets has emerged in the last few years.
•bi (tri?) modal distribution: minimum at about 30 Earth masses, too.
•neptunian bump: strong increase between 30-15 ME.
•overall maximum at small masses.

Synthetic PIMF and rv precision
Observational RV precision 10, 1 and 0.1 m/s
•Bimodal at 1 m/s (Jovian, Neptunian), and trimodal at 0.1 m/s (Jov., Nept., proto-terrestrial)
•At least qualitatively similar as observations.
•At low masses, model incomplete, so direct comparison very difficult,
•but distinction between Jovian (w. gas runaway) and Neptunian (w/o gas runaway)
•and distinction between objects which migrated significantly (Neptunian), and those
formed in situ (proto-terrestrial), could be real.

5. The semimajor axis
distribution

where they have to stop before falling onto the star. Several stopping mechanisms
have been proposed, invoking, e.g., a magnetospheric central cavity of the accretion
disk, tidal interactions with the host stars, Roche-lobe overflow by the young inflated
giant planet, or photoevaporation. The question is, however, still debated. Alternative
points of view invoke in situ formation (Bodenheimer, Hubickyj & Lissauer
2000), possibly triggered through disk instabilities (Boss 1997, Durisen et al. 2007).
Note however that, even in such cases, subsequent disk-planet interactions leading to
N
20
15
10
Semimajor axis distribution: giant planets
5
0
1 2 3
log(Period) (days)
uncorrected for obs. bias
The semimajor axis distribution of giant planets found by RV consists of a
•pile up at a period of about 3 days (0.04 AU). Stopping mechanism for migration?
•maybe this pile up is only tiny, when properly correcting for obs. bias...
•a period valley (10 d < P < 100 d). Timescale effect?
•an upturn at about 1 AU. Reservoir at large distances? Original formation region?
Udry· Santos
Udry & Santos 2007
Msini>0.75 MJ
Msini

Synth. semimajor axis distribution
ig. 8. Starting position relative to the iceline of embryos growing
Preferred starting
ventually to planets with a final mass M ≥ 300M⊕ as a function
f [Fe/H]. Symbols show the disk gas mass: Red filled squares are
ow [MD/MSN] < −0.1. Black triangles are intermediate [MD/MSN]
location (6-7 AU)
-0.1 to 0.4), and green open circles finally are those disk with a large
MD/MSN] > 0.4.
he minimal necessary planetesimal disk mass is reduced at
mall distances, in the second case at large distances while in
he other parts of the plot, the points lie on top of each other.
his shows that at small distances (where the core accretion
imescales are short) the small isolation masses (and the associted
long Kelvin-Helmholtz timescales for gas accretion) limit
iant planet growth. Isolation masses decrease with decreasing
emimajor axis (at least for the chosen solid surface profile),
hich must be canceled out by an increasing ΣS. Faster type
C. Mordasini et al.: Extrasolar planet population synthesis IV 13
-embryos of giant-planetsto-be
come from outside the
iceline (Ida & Lin 2004). More
solids=>more massive cores.
-high [Fe/H]: start also inside.
Planets inside 0.1 AU excluded Mp>300 MEarth Mstar= 1 Msun
formation, and influences in this way the birth place of giant
planets to be. In the left panel of Fig. 8 we have directly plotted
the position where embryos start to grow astart which later become
planets larger than 300 M⊕, relative to the position of the
iceline aice in the corresponding disk, as a function of [Fe/H].
Additionally, the disk lifetime was color coded. Note that in our
simulation, the location of the iceline is independent of [Fe/H]
(but dependent on Σ0). Possible effects of a modification of the
disk temperature structure and thus the iceline location via a
metallicity dependent disk opacity are thus not included.
The figure shows that at low metallicities [Fe/H] −0.3,
planets can only start to form in a small region with a width
of 2-3 AU just outside the iceline. In this low [Fe/H] disks,
only in this region massive cores can form quickly enough.
With increasing [Fe/H], the zone expands to larger radii, and
Distance migrated [AU]
for [Fe/H] −0.15, giant planets can also form inside the ice-
line. This corresponds, with the highest Σ0 of the distribution
(which is necessary, see next section) to a solid surface density
about 3.5 times as high as ΣS,SN. These results are qualitatively
similar to those of Ida & Lin (2004a), except for a somewhat
Log(Mdisk/MMSN)
lower numerical value (3.5 instead of 5). The absence of red
squares at negative values of astart − aice shows that the disk
masses must always be rather large for the formation of giant
planets Typical inside the iceline, migration
even if [Fe/H] is large. At higher
[Fe/H], giant planets can form at almost all semimajor axes if
concurrently [MD/MSN] is high indicated by green open circles
distance
(about 1 AU to 20 AU), but the preferred formation location is
still beyond the iceline. Finally, at the highest [Fe/H], the loca-
-about 3-5 AU. Not so much...
tion of the iceline becomes less important.
One therefore sees from the figure, that while at low metallicities,
pathways to giant planets are very restricted and in particular
only possible in a small part of the disk, this is not the
case for higher metallicities, where giant planets can form all
over the disk.
There is another important effect for the starting positions,
which cannot be directly seen in Fig. 8 which was already
quickly mentioned in sect. 3.2.3. For giant planet formation
at low [Fe/H], high [MD/MSN] Mordasini et areal. needed 2012
for compensation
18 C. Mordasini et al.: Extrasolar p
Fig. 12. Distribution of final semimajor axes of synthetic planets larger th
for [Fe/H]< −0.2, blue solid lines for −0.2

[Fe/H] - semimajor axis
Observations: No clear correlation of [Fe/H] & semi. axis (giants). Udry & Santos 2007
Somewhat surprising: High [Fe/H] allow giant planet formation inside aice..
•High [Fe/H]: Small starting positions
➡Small disk gas mass sufficient.
➡Drives slow migration.
➡Planet quickly more massive than local gas disk mass: slows down.
➡Migration only over small distance.
•Low [Fe/H]: Large starting positions
➡Mean necessary gas mass high.
➡Drives rapid migration
➡Planets need to migrate over large extent until collected enough planetesimal for runaway
and to eventually slow down.
➡Migration over large distance.
[AU] high [Fe/H] mean [Fe/H] low [Fe/H]
Effects compensate!
Complex interaction of
various processes.
Mfinal>300 Mearth
mean astart 6.2 6.8 7.2
mean a-astart -3.5 -4.0 -4.7
mean a 2.7 2.8 2.6

6. Stopping distances

m Kepler 13
Number of Planets per Star
0.1000
0.0100
0.0010
P 0 = 1.7 days
P 0 = 2.2 days
0.0001
0.68 1.2 2.0 3.4 5.9 10 17 29 50
Orbital Period (days)
Howard et al. 2011
P 0 = 7.0 days
Close-in low mass/radius planets
2−4 R E
4−8 R E
8−32 R E
ig. 7.— Measured planet occurrence (filled circles) as a funcn
of orbital period with best-fit models (solid curves) overlaid.
ese models are power laws with exponential cutoffs belowachareristic
period, P0 (see text and equation 8). P0 increases with
reasing planet radius, suggesting that the migration and parkmechanism
that deposits planets close-in depends on planet
ius. Colors correspond to the same ranges of radii as in Figure
The occurrence measurements (filled circles) are the same as in
ure 6, however for clarity the 2–32 R⊕ measurements and fit
excluded here. As before, only stars in the solar subset (Table
and planets with Rp > 2 R⊕ were used to compute occurrence.
e integrated occurrence to P = 50 days is given in
ble 4.
4. STELLAR EFFECTIVE TEMPERATURE
4.1. Planet Occurrence
n the previous section we considered only GK stars
th properties consistent with those listed in Table 1.
Mayor et al. 2011
particular, only stars with Teff =4100–6100Kwere
Cut off below P0:
-small radii 2-4 Re: P0 = 7 days
-large radii >4 Re : P0 = 2 days.
Neptunian and smaller sized further out than giant planets.
No pile up at 3 days. Different stopping mechanism?
Msini

Gas surface density [g/cm 2 ]
MMSN, irrad., Σ0 ∝ a -0.9,
,alpha=7x10 -3
Gas surface density [g/cm 2 ]

7. Radius distribution

,-$.&//$01$23&,-$
Formation of M-R diagram
Fraction Z of heavy elements
Z 95 %
•Rapid collapse at about 0.2 MJ
•when Z≈ 0.5
•Two groups for R (pre/post
collapse) during formation.
•After disk dispersal (T>10 Myrs),
slow contraction.
•At late times, characteristic
maximum at about 4 MJ (degeneracy).
•Z decreasing with total mass:
imprint of formation (TKH)!

Mordasini et al in prep.
M-R diagram
Synthetic population and
all known planets with
known M-R outside 0.1
AU (Solar System &
Exoplanets)
Z = Mcore/M
Orange: Z ≤ 1%
Red: 1 < Z ≤ 5%
Green: 5 < Z ≤ 20%
Blue: 20 < Z ≤ 40%
Cyan: 40 < Z ≤ 60%
Magenta: 60 < Z ≤ 80%
Yellow: 80 < Z ≤ 95%
Brown: 95 < Z ≤ 99%
Black: Z > 99%

M-R diagram: effect of opacity
Zero grain opacity Full interstellar grain opacity
Mordasini et al in prep.
The lower the grain opacity in the envelope, the easier for the planet to accrete gas (reduced
pressure support of the envelope). During evolution: The smaller the opacity, the faster the
cooling (smaller radii).

Planet Radius, R p (R E)
32
16
0.001 0.002 0.004 0.0079 0.016 0.032 0.063 0.13 0.25 0.50 1.0
0.000035 0.00007 0.00014 0.00028 0.00056 0.0011 0.0022 0.0044 0.0088 0.018 0.035
8
4
2
1 (2)
58018
1 (4) 0.0021
4 (45) 0.022 2 (18) 0.0087 4 (60) 0.030 5 (153) 0.076 6 (208) 0.10 5 (198) 0.099
57907over
0.00007 all planets 57808 0.00078 with 57749P
0.00030 < 57653 50 days. 0.0010 57538 We 0.0027 57429 computed 0.0036 57240 oc- 0.0035
1 (5)
52618
It is worth identifying additional sources of error and
simplifying assumptions in our methods. The largest
source of error stems directly from 35% rms uncertainty
Planet Occurrence from Kepler 9
Planet Occurrence − d 2 in R⋆ from the KIC, which propagates directly to 35%
uncertainty in Rp. We assumed a central transit over
the full stellar diameter in equation (2). For randomly
distributed transiting orientations, the average duration
is reduced to π/4 times the duration f/dlogP/dlogR of a central transit.
p
Thus, this correction reduces our SNR in equation (1) by
afactorof 0.001
Howard et al. 2011
0.12
π/4, i.e. a true signal-to-noise ratio thresh-
old of 8.8 instead of 10.0. This is still a very conservative
2 (11) 0.0054 4 (39) 0.019 6 (69) 0.034 1 (15) 0.0071 1 (28) 0.014 1 (25) 0.012 3 (168) 0.082
58031 0.00019 58028 0.00067 58022 0.0012 58017 0.00025 58009 0.00049 58004 0.00043 57997 0.0029
merator and denominator of equation (2) nearly equally.
0.0010 1 (6)
0.00004 58009
0.0029 4 (34)
0.00010 58004
Planet Occurrence − f cell
detection threshold. Additionally, our method does not
account for the small fraction of transits that are grazing
√ and have reduced significance. We assumed perfect
t scaling for σCDPP values computed for 3 hr intervals.
This may underestimate 1 (9) 0.0042 σCDPP for 1 (15) a 60.0075 hr interval 1 (52) (ap- 0.026
58036 0.00015
58030 0.00026
58020 0.00090
proximately the duration of a P = 50 day transit) by
∼10%. These are minor corrections and affect the nu-
3.1. Occurrence as a Function of Planet Radius
0.017 3 (25)
0.00058 57998
Comparison: KEPLER data
0.012 1 (15)
0.00043 57988
0.0076 3 (70)
0.00027 57981
0.034 4 (154)
0.0012 57963
3 (21) 0.011 7 (64) 0.032 21 (269) 0.15 31 (521) 0.30 36 (893) 0.53 34 (1101) 0.79 18 (749) 0.61
56665 0.00037 55966 0.0011 54585 0.0051 52260 0.010 48639 0.019 43318 0.028 36296 0.021
all planets in a radius interval with P < 50 days. The
3 (10)
1 30446
0.68
0.0075 1 (10)
0.00026 22540
1.2
0.011 4 (50)
0.00040 15445
2.0
0.067 6 (59) 0.22 1 (18)
0.0023 9764 0.0077 5784
3.4 3.4df(R) 5.9
0.062 3 (85)
0.0022 3170
10
0.81 2 (41)
0.028 1605
17
0.95
0.033
29 50
Orbital Period, P (days)
Fig. 4.— Planet occurrence as a function of planet radius and orbital period for P 10 are shown Here as blackdf(R)/d dots. The phaselog space is Rdivided is into the a grid mean of logarithmically number spaced cells ofwithin planets which planethav occurrence
is computed. Only stars in the “solar subset” (see selection criteria in Table 1) were used to compute occurrence. Cell color indicates
planet occurrenceing with the P

Number planets per star
is reduced to π/4 times the duration of a central transit.
Thus, this correction reduces our SNR in equation (1) by
afactorof
Radius distribution of KEPLER planets
π/4, i.e. a true signal-to-noise ratio threshold
of 8.8 instead of 10.0. This is still a very conservative
detection threshold. Additionally, our method does not
account for the small fraction of transits that are grazing
√ and have reduced significance. We assumed perfect
t scaling for σCDPP values computed for 3 hr intervals.
This may underestimate σCDPP for a 6 hr interval (ap-
Nominal Model: all a
T=5x10 9 yrs
Nominal Model: P

is reduced to π/4 times the duration of a central transit.
Thus, this correction reduces our SNR in equation (1) by
afactorof
Radius distribution of KEPLER planets II
π/4, i.e. a true signal-to-noise ratio threshold
of 8.8 instead of 10.0. This is still a very conservative
detection threshold. Additionally, our method does not
account for the small fraction of transits that are grazing
√ and have reduced significance. We assumed perfect
t scaling for σCDPP values computed for 3 hr intervals.
Number planets per star
Mordasini et al in prep.
Nearly no type I: all a
T=5x10 9 yrs
Nearly no type I: P

8. The metallicity
distribution

Correlation stellar [Fe/H] & giant planet frequency
N. Santos et al. (2005)
search sample (all stars)
stars with giant planets
•the detection probability for giant planets is a strongly increasing function of
the host star metallicity.
•No hot Jupiters found in globular cluster 47 Tuc ([Fe/H]=-0.76). Expected for
solar neighborhood frequency (~0.5%): seven discoveries.
•Best known star-planet correlation for exoplanets. Important constraint for
formation.

Mordasini et al. 2009b
Synthetic population observable at 10 m/s
[Fe/H]
Blue: Observation (Fischer & Valenti 2005)
Red: Observation (Udry & Santos 2007)
Black: Observable synthetic planets
cf. also Santos et al. 2004, 2005
∝ Z 2
∝ Z
Reproduced by the synthetic population.
•Dependence not strong enough: Additional
mechanisms? Planetesimal formation
(Johansen et al. 2010) ?
Large metallicity effect on RV detections
• Metal rich systems tend to produce more
massive planets
• Radial velocity method favors massive
objects
Does not mean there are no planets
around low [Fe/H] stars... we just can’t
detect them at 10 m/s...

Mayor et al. 2011
No metallicity effect for low mass planets
•HARPS high precision sample: [Fe/H] distribution for giant planet hosts (black), for hosts with
planets less massive than 30 ME (red), and for the global combined sample (blue).
•No metallicity effect for low mass planets. The dividing line lies at about 30 ME. Same as the
minimum in the mass distribution.
•Even absence of low mass planets at high [Fe/H]? And of Super Jupiters at low [Fe/H]?
?
?

Mordasini et al 2012
Metallicity of synthetic hot planets
C. Mordasini et al.: Extrasolar planet population synthesis IV 21
C. Mordasini et al.: Extrasolar planet population synthesis IV 21
Mstar=1 M⊙
Slow type I
migration
Fast type I
migration
a

Questions?