Annual Report 2011 Max Planck Institute for Astronomy

44 III. Selected Research Areas
Number of planets
100
50
0
10
Msin i [MEarth ]
the fact that always exactly the same formation model is
used. This is a basic outcome similar to the observational
result. In Figure III.1.3, one can for example find tracks
that lead to the formation of hot Jupiters. Most embryos
however remain at low masses, since they cannot accrete
a sufficient amount of planetesimals to start rapid gas accretion
and become a giant planet.
Comparison with observations
100
Fig. III.1.4: Comparison of the observed and the synthetic
planetary mass distribution. The left panel shows the distribution
of planetary masses as found with high precision radial
velocity observations (Mayor et al. 2011). The blue line gives
the raw count, while the red line corrects for the observational
bias against the detection of low-mass planets. The right panel
Among the many outputs that can be compared with observations,
one of the most fundamental results of population
synthesis is a prediction for the distribution of
planetary masses. It is obvious that the planetary mass
function has many important implications, including the
question about the frequency of habitable extrasolar planets.
In the left panel of Fig. III.1.4, the planetary mass
function is shown as derived recently from high precision
radial velocity observations of FGK dwarfs (Mayor
et al. 2011). It makes clear that below a mass of approximately
30 Earth masses, there is a strong increase in the
frequency. Very low-mass planets of a few Earth masses
are very frequent. The right panel shows the predicted
mass function from population synthesis calculations of
planets around a 1 M 0 star. Also in the theoretical curve,
there is a strong change in the frequency at a similar mass
as in the observations. This is explained by the fact that
in this mass domain, planets start to accrete nebular gas
in a rapid, runaway process. They then quickly grow to
masses of M 100 M Earth . It is unlikely that the proto-
Normalized Fraction
0.08
0.06
0.04
0.02
0
1 10
10 2
M sin i [M Earth ]
10 3 10 4
shows the planetary mass function as found in a population
synthesis calculation. The black line gives the full underlying
population, while the blue, red, and green lines are the detectable
synthetic planets at a low, high, and very high radial velocity
precision.
planetary disk disappears exactly during the short time
during which the planet is transformed from a Neptunian
into a Jovian planet. This makes that planets with intermediate
masses 30 M Earth are less frequent (“planetary
desert”, cf. Ida & Lin 2004). The “dryness” of the desert
is directly given by the rate at which planets can accrete
gas, while the mass where the frequency drops represents
the mass where runaway gas accretion starts. We thus see
how the comparison of synthetic and actual mass function
constrains the theory of planet formation. We further
note that model and observation agree in the result that
low-mass planets are very frequent.
A typical example how planet population synthesis
can be used to study the global effects of a given physical
mechanism is shown in Fig. III.1.5. The plot compares
the observed distribution of radii of planets inside
of 0.27 AU as found by the Kepler satellite (Howard et
al. 2011) with the radius distribution in three different
population synthesis calculations. The three calculations
are identical except for the value of f κ . This parameter
describes the reduction factor of the opacity due to grains
relative to the interstellar value. A value of f κ 1 means
that the full interstellar opacity is used, while f κ 0
means that we are dealing with a grain-free gas where
only molecular and atomic opacities contribute. These
opacities are used when calculating the internal structure
of the planets. There, the strength of the opacity is important
for the rate at which planets can accrete primordial
H/He envelopes. At low opacities, the liberated gravitational
potential energy of the accreted gas can be easily
radiated away, allowing the envelope to contract, so
Credit: C. Mordasini