A&A 458, 661-668 (2006)
K. Kornet1,2 - S. Wolf1 - M. Rózyczka2
1 - Max Planck Institute for Astronomy, Königstuhl 17, 69117 Heidelberg, Germany
2 - Nicolaus Copernicus Astronomical Center, Bartycka 18, Warsaw 00-716, Poland
Received 23 June 2005 / Accepted 7 March 2006
We examine the predictions of the core accretion - gas capture model concerning the efficiency of planet formation around stars with various masses. First, we follow the evolution of gas and solids from the moment when all solids are in the form of small grains to the stage when most of them are in the form of planetesimals. We show that the surface density of the planetesimal swarm tends to be higher around less massive stars. Then, we derive the minimum surface density of the planetesimal swarm required for the formation of a giant planet both in a numerical and in an approximate analytical approach. We combine these results by calculating a set of representative disk models characterized by different masses, sizes, and metallicities, and by estimating their capability of forming giant planets. Our results show that the set of protoplanetary disks capable of giant planet formation is larger for less massive stars. Provided that the distribution of initial disk parameters does not depend too strongly on the mass of the central star, we predict that the percentage of stars with giant planets should increase with decreasing stellar mass. Furthermore, we identify the radial redistribution of solids during the formation of planetesimal swarms as the key element in explaining these effects.
Key words: accretion, accretion disks - planetary systems: protoplanetary disks - planetary systems: formation
Radial velocity surveys led to the discovery of over 150 extrasolar planets around main sequence stars. Published descriptions of most of them can be found in the references given by Marcy et al. (2005) and Mayor et al. (2004). Those surveys have been the most successful in the case of G dwarf stars, because such stars have well-defined spectroscopic features and show only a little photospheric activity. Consequently, most of the known extrasolar planets orbit stars similar to our Sun. Due to the constant progress in the detection techniques, the observational programs recently started to also include stars with lower masses on a larger scale, namely M dwarfs. Moreover, some of these surveys are now particularly dedicated to lower-mass stars (e.g. Bonfils et al. 2004; Endl et al. 2003). So far, these efforts have led to the discovery of three planets around two M dwarf stars: Gliese 876b,c (Marcy et al. 2001,1998) and GJ436b (Butler et al. 2004).
From the theoretical point of view, the problem of giant planet formation around M dwarfs was studied recently by Laughlin et al. (2004). They addressed it within the core accretion - gas capture model (CAGCM) that provides the most widely accepted scenario explaining the formation of giant planets in both the Solar System and extrasolar planetary systems. This model predicts that first a solid planetary core is formed by collisional accumulation of planetesimals. When the core reaches a mass of a few Earth masses, it starts to accrete gas, and an extended hydrostatic envelope is built around it. As the accretion rate of gas is greater than the accretion rate of solids at this time, the envelope eventually becomes more massive than the core. When this happens, a runaway accretion of gas ensues, which is terminated either by tidal interactions of the planet with the protoplanetary disk or by the dissipation of the disk. CAGCM has found supporting evidence in the discovery that stars with planets have higher metallicities than field stars (Fischer & Valenti 2003; Santos et al. 2000). This is because the formation time of giant planets decreases with increasing surface density of the planetesimal swarm (Pollack et al. 1996), which in turn increases with the primordial metallicity of the protoplanetary disk. Thus, giant planets are expected to form more easily in disks with higher metallicity.
Laughlin et al. (2004) conclude that M dwarfs have a limited ability to form Jupiter-mass planets. This is a direct consequence of their assumption that the surface density of the planetesimal swarm out of which planetary cores are formed scales linearly with the mass of the central star. However, the solid component of the protoplanetary disk evolves in a different way than the gaseous component (Weidenschilling & Cuzzi 1993). Due to the gas drag, a significant redistribution of solids takes place, and in the inner disk their surface density can be substantially enhanced compared to the initial one (Weidenschilling 2003; Stepinski & Valageas 1997). In general, the efficiency of the processes responsible for the redistribution of dust depends on the mass of the central star. An obvious conclusion is that analysis of the formation of giant planets around stars with various masses should include the global evolution of solids in protoplanetary disks. A simple model of the evolution of solids was proposed by Kornet et al. (2005,2002). Applying it to solar-like central stars, these authors reproduced the observed correlation between stellar metallicity and the probability of a planet occurring (a similar result was independently obtained by Ida & Lin 2004b).
The rapid progress in observational techniques opens up the possibility of testing the correctness and predictive power of the model proposed by Kornet et al. (2005). To that end, we extend their analysis and calculate probabilities of planet occurrence around stars with different masses (both smaller and larger than ), which may be compared to future observational data. In Sect. 2 we explain our approach to the evolution of protoplanetary disks and planet formation. The results of our calculations are presented in Sect. 3 and discussed in Sect. 4.
We model the protoplanetary disk as a two-component fluid,
consisting of gas and solids. The gaseous component is described
by the analytical model of Stepinski (1998), which gives the surface
density of gas,
as a function of distance afrom the star and time t, in terms of a selfsimilar solution to
the viscous diffusion equation. The viscosity coefficient is given
by the standard prescription:
The evolution of solids is governed by two equations. The first of them is the continuity equation for the surface density of solid material, . The second one, describing the evolution of grain sizes, can be interpreted as the continuity equation for size-weighted surface density of solids, , where s(a)is the radius of solid particles at a distance a from the star. The equations are solved numerically on a moving grid whose outer edge follows the outer edge of the dust disk. The details of the method can be found in Kornet et al. (2001).
|Figure 1: The initial surface density ( left panel) and temperature ( right panel) as a function of distance from the star in the protoplanetary disk with an initial mass and outer radius . Different curves correspond to different values of the mass of the central star in solar masses as described by the labels.|
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We model the formation of a giant planet in situ, so the
orbital parameters of the planet do not vary in time. Our
procedure for the evolution of the protoplanetary cores is based
on the following assumptions: (1) core accretion starts when
solids at a given distance a from the star reach radii of
(2) at each time, the planetesimals are mixed well through the feeding zone of the planet, so their surface
is always uniform in space, but
usually decreasing with time as planetesimals accrete onto
the planet; (3) the planetesimals do not migrate into the feeding
zone from outside or inside and vice versa, but they can be
overtaken by the boundary of the feeding zone as it expands due to
the growing mass of the planet. Under these assumptions the growth
of protoplanetary core mass
can be described by
the formula given by Papaloizou & Terquem (1999),
To calculate the rate of gas accretion onto the protoplanet, one should
solve the equations of mass conservation,
hydrostatic equilibrium, energy generation from accretion of
planetesimals and quasi-static contraction, and radiative or
convective energy transport, as given e.g. in Bodenheimer & Pollack (1986).
Following Ida & Lin (2004a), we use a simplified approach based
on fits to the numerical results. We assume that the accretion of
gas starts when the core reaches critical mass of
In the CAGCM the process of planet formation is naturally split into two main phases. In the first one, dominated by collisional accumulation of dust grains, a planetesimal swarm is formed in the protoplanetary disk. In the second phase, dominated by gravitational interactions, planetary cores are assembled and subsequently accrete planetesimals and gas from the disk. In the following subsections we investigate how each phase is influenced by the mass of the central star.
To illustrate how the mass of the central star influences the formation of the planetesimal swarm, we follow the evolution of a protoplanetary disk with an initial mass and initial outer radius for three values of (0.5,1 and 4 ). Figure 1 shows the initial distributions of and T in the midplane of the disk. In all three cases the gas is distributed very similarly, with dropping monotonically from at 0.01 AU from the star to at the outer edge of the disk. The changes of slopes in the distribution of correspond to transitions between different powerlaws describing the opacity of the disk matter in different temperature ranges (Ruden & Pollack 1991). The distribution of temperature is qualitatively very similar to the distribution of . In the disk around a star T drops from at to a few Kelvins at the outer edge of the disk. The evaporation temperature assumed for the dust grains in our models ( ) is reached at . Note that at a given radius T increases as the mass of the central star is increased. It is the result of the increasing vertical component stellar gravity, due to which the scale height of the disk is reduced.
Initially, the dust is well mixed with the gas, with the ratio
constant everywhere in the disk. As the disk evolves, surface
density and temperature of gas slowly decrease due to accretion
and viscous spreading; however, the dust component evolves in
quite a different way. The grains grow in size due to mutual
collisions and gain inward radial velocities due to the gas drag.
If they cross the evaporation radius, they sublimate and are
accreted onto the star on the viscous timescale as a vapour.
However, if their growth time is shorter than the timescale of
inward migration, they manage to reach sizes of a few km before reaching the evaporation radius. Their radial motions are
then stopped and the planetesimal swarm attains its final form.
Figure 2 shows the distribution of planetesimals in our
model after 106 yr from the beginning of its evolution. In all
three cases, the outer radius of the planetesimal swarm is much
smaller than the initial outer radius of the disk. The difference
is larger for models with smaller .
This is because small
solid bodies evolving in disks around less massive stars gain
higher inward velocities and tend to migrate to smaller radius
before reaching km-sizes. The maximum velocity of the inward drift
can be estimated as
Due to the inward migration of solids and their confinement to much smaller radii, the final surface density of planetesimals is increased locally within a factor of a few in comparison with the initial value of . As this effect is larger in disks around less massive stars, their final planetesimal swarms tend to be more favourable to the formation of giant planets.
|Figure 2: The surface density of solids as a function distance from the star in the protoplanetary disk with the same parameters as in Fig. 1 after from the beginning of its evolution. Different curves correspond to different values of the mass of the central star in solar masses as described by the labels.|
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To quantify the influence of on the effectiveness of giant planet formation from a planetesimal swarm, we introduce the concept of the minimum surface density . We define it as the minimum value of the initial surface density of planetesimals needed to form a Jupiter-mass (1 ) planet in less than the lifetime of the protoplanetary disk . For we adopt a value of 3 .
First, by solving the set of Eqs. (7), (9), and (12) with different values of
a function of distance from the star. The results are shown in Fig. 3 for the same values of
as before. Close to the star (
is a decreasing function a. In this regime, the accreting
protoplanetary core rapidly accumulates all planetesimals in its
feeding zone and reaches the isolation mass
Afterwards, the accretion of planetesimals is negligible and the
planet grows mainly due to the accretion of gas. Upon integrating
Eq. (12), we obtain the minimum isolation mass
needed to form a 1
|Figure 3: The minimum surface density of a planetesimal swarm needed to from a 1 planet in less than 3 106 yr as a function of distance from the central star. Different curves are obtained for different masses of the central star, as labeled in units of solar masses in the upper right corner.|
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For sufficiently large radii
changes its slope and becomes an increasing function of a. In this regime, the time scale of accretion of the planetesimals onto the core is larger than the
lifetime of the disk and becomes the main factor to determine
Consequently, the surface density of
planetesimals in the feeding zone never drops much below its
initial value. To describe the formation of a planet analytically
under these conditions, we divide the whole process into two phases. During the first phase the planet exclusively due to the accretion of planetesimals grows. We assume that the surface density
of planetesimal swarm
does not change in time,
because the core only accumulates a negligible fraction of solids
present in the feeding zone. From Eq. (7) we obtain
|Figure 4: The minimum surface density of the planetesimal swarm needed to form a 1 planet in less than 3 106 yr as a function of distance from the central star, calculated with a constant value of . The solid line represents the results of exact values obtained numerically, while dotted lines represent analytical approximations given by Eqs. (16) and (22). The mass of the central star is .|
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As we see, the minimum surface density of the planetesimal swarm required for the formation of a gas giant planet in a time shorter than the lifetime of protoplanetary disk is a complicated function of . In a given planetesimal swarm such a planet forms more easily around less massive star if its orbital radius is smaller than 10 AU, and around a more massive star if its orbital radius is larger than 10 AU.
The results of the last two subsections allow us to investigate the influence of on the whole process of giant planet formation for a broad set of models of protoplanetary disks. We calculate the grid of models similar to the one described in Sect. 3.1, but with different values of the initial disk mass M0 and outer radius R0. To cover the range of masses and sizes of disks observed in nature, we choose M0in the range of 0.02 to 0.2 . The range of R0 is adjusted for every metallicity so that all models in which formation of giant planets is possible could be accounted for. For the viscosity coefficient we chose a value of 0.001 (Papaloizou & Nelson 2003). We follow each model until all solids are in the form of planetesimals or are accreted onto the star. Then, we evaluate every model with planetesimals to determine whether the surface density of planetesimals exceeds the minimum surface density anywhere in the disk. Models with this property are labeled as planet bearing. For each such model we determine the minimum and maximum distances from the star at which the surface density of the planetesimal swarm is larger than . The results obtained for different values of the stellar mass are shown in Figs. 5-7.
|Figure 5: The plane of initial parameters of protoplanetary disk models [M0, ]. Minimum and maximum distance from a star at which the formation of a 1 planet is possible within 3 106 yr is indicated by contours and a grey scale, respectively. White circles indicate disk models in which the surface density of the planetesimal swarm is everywhere lower than the critical value for planet formation. Black circles indicate disks in which all solids are accreted onto the star.|
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The area of the region occupied on the [M0,R0] plane by the planet bearing disks is clearly anticorrelated with the mass of the central star. As we have shown in Sect. 3.1, solid grains gain higher inward velocities in disks around less massive stars, and the resulting planetesimal swarms have higher surface densities. We see that reducing the mass of the central star increases the maximum R0 for which planet formation is possible in disks with the same initial mass M0.
We can also see that in a given disk the inner edge of the planet-bearing region moves inward as we decrease the mass of the star (for its radius is on the average equal to 0.55 that for ). This effect is mainly caused by the larger surface density of planetesimal swarms produced by disks around less massive stars. Differences in , while appreciable, are much less important (when is reduced by a factor of 2, drops by 20% only; see Eq. (16)). The outer edge of the planet-bearing region also moves closer to the star (for its radius is on the average equal to 0.75 that for ). This is because in most cases it coincides with the outer edge of the planetesimal swarm, which is more compact around less massive stars. The difference in the minimum surface density also tends to decrease the outer radius of the planet-bearing region, but it is again a second-order factor.
Generally, our model predicts that giant planets tend to form at tighter orbits around less massive stars, and wider orbits around more massive stars. However, at least in some cases their locations may be influenced by the effects of migration. We return to this point in Sect. 4.
One of the main results of extrasolar planet searches is the discovery that planet-bearing stars tend to have higher metallicities than field stars (Fischer & Valenti 2003; Santos et al. 2000). That correlation can be easily explained within CAGCM. In this scenario, the formation time of giant planets decreases with increasing surface density of the planetesimal swarm (see Eqs. (16) and (22)), which in turn is an increasing function of the original metal content of the protoplanetary disk. Consequently, the giant planets form more easily in disks with higher metallicities. Kornet et al. (2005) calculated the rates of giant-planet occurrence in disks with different metallicities around stars with mass . Their approach to the evolution of solids and to the formation of giant planets was the same as the one used in this paper. They were able to reproduce the observational correlation for disk models with viscosity parameter . Herein we extend their calculations onto disks around stars with various masses.
The change in the disk metallicity influences the processes
leading to the formation of planets in two ways. First, it
changes the structure of the gaseous disk by changing the opacity
in the disk. In our models we scaled the opacity by a constant
factor Z equal to the metallicity of the disk expressed in solar
units. This approach is justified by the fact that the opacity in
protoplanetary disks is mainly due to dust grains and molecules.
Second, the primordial metallicity of the disk determines the
initial ratio of dust-to-gas surface densities. In our models this
ratio is initially independent of the distance from the star and
is equal to
Following the procedure described by Kornet et al. (2005), for every Zwe calculate the area
of the region occupied on
plane by disks that form planets at
distances smaller than 5 AU from the central star. The last
restriction comes from the fact that currently we know only one extrasolar planet on a larger orbit - 55 Cnc d (Marcy et al. 2002). A measure of the rate of planet occurrence can be defined as
|Figure 8: The rate of planet occurrence as a function of the primordial metallicity of protoplanetary disks. Different lines are obtained for models with different masses of the central star, as labeled in solar units in the upper left corner. The histogram shows the observational data compiled by Fischer & Valenti (2003).|
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The results are presented in Fig. 8. It shows the rate of planet occurrence as a function of disk metallicity for three values of . As expected, is an increasing function of Z. The minimum value of Z below which no giant planets are formed at orbits smaller than 5 AU decreases with the mass of the central from for , to for .
We see that for Z smaller than 0.2, is an increasing function of . This is because most disks in which planets formation would be possible around less massive stars have outer parts that are gravitationally unstable, and the amount of solids present in their stable parts is too low to enable subsequent formation of giant planets according to CAGCM. However, as Z increases, smaller and smaller disks become planet-bearing for every M0, and the percentage of gravitationally unstable disks in which formation of giant planets is not possible decreases. Consequently, the factors promoting the formation of a giant planet around less massive stars as described in previous sections become important, and changes into a decreasing function of .
Based on a simple approach to the evolution of solids in protoplanetary disks, we investigated the influence of the mass of the central star on the formation of giant planets. We showed that due to the more efficient redistribution of solids the planetesimal swarms around less massive stars tend to have higher surface densities. Next, we derived the minimum surface density of the planetesimal swarm needed to enable formation of a giant planet within the lifetime of the protoplanetary disk, and we found that at distances from the star smaller than 10 AU it increases with the stellar mass. Farther away from the star the minimum density becomes anticorrelated with the mass of the star. However this effect is offset by the anticorrelation mentioned already between the mass of the star and the surface density of the planetesimals.
These two effects determine the set of initial parameters characterising protoplanetary disks that are capable of giant planet formation within the core accretion - gas capture scenario. We showed that this set is larger for less massive stars. This means that the percentage of stars with massive planets should increase with decreasing stellar mass (at least in the range 0.5-4 ). However, as discussed below, in the currently accessible range of orbital radii (<5 AU), the situation is not all that clear.
Based on the sets obtained for different metallicities, we determined the occurrence rate of planets with orbits smaller than 5 AU as a function of the mass and metallicity of the star. We took into account the fact that the outer region of the disk is gravitationally unstable in some models. Such regions are located farther than 5 AU from the central star, and planets formed there by disk fragmentation are not included in our occurrence rate. However, their presence reduces the amount of solid material available for the formation of planetesimals. For less massive stars this effect is so strong that it overcomes factors promoting planet formation, so that for metal-poor disks the rate of planet occurrence decreases with the mass of the central star. As a result, the minimum metallicity at which giant planets can form at orbits smaller than decreases from 0.6 for stars with masses of to 0.2 for .
In the metal-rich regime the percentage of entirely stable disks in which formation of giant planets is possible is larger, and stable regions of partly unstable disks contain enough solids to produce planetesimal swarms capable of giant planet formation. Consequently, both factors promoting planet formation around less massive stars are in play, and a clear anticorrelation between the stellar mass and planet occurrence rate is observed. At the same time, our model does not account for the presence of giant planets around metal-poor stars. This may be due to the fact that we do not include planets that have formed beyond 5 AU and later migrated inward. Such an assumption is valid as long as the number of these planets is small compared to the number of planets that have formed within 5 AU. However, as we move to lower metallicities, the percentage of giant planets with silicate cores decreases (the silicates simply become too scarce), while the percentage of planets forming from ice grains increases. Thus, in metal-poor systems the number of planets with ice cores that migrated from large orbits can become a large fraction of planets at orbits smaller than 5 AU.
|Figure 9: The rate of planet occurrence as a function of the primordial metallicity of protoplanetary disks. Disks around stars with masses of are represented by a solid line. Disks around stars with masses of are represented by dotted and dashed lines. In the first case the range of initial masses of the disks is the same as in the case of solar type stars, while in the second it was scaled according to the mass of the central star.|
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Obviously, our description of the evolution of solids is very simplified. The basic underlying assumptions like the single-size distribution of solid grains or the neglect of planet migration already have been discussed by Kornet at al. (2004, 2005). The main additional assumption introduced in the present paper is the independence of the initial parameters of protoplanetary disks on the mass of the central star. While admittedly ad hoc, it seems to be better than the one adopted by Laughlin et al. (2004), who scaled their initial surface density of planetesimals linearly with the mass of the star. They did not take into account the antecedent evolution of solids leading to the formation of planetesimal swarms, and concluded that the formation of giant planets around low-mass stars is difficult. Recent observations suggest that masses of protoplanetary disks do not strongly depend on masses of the central stars (Guilloteau 2005). Nevertheless, to investigate the influence of our assumption, we performed additional set of calculations with a mass of the central star of and with the initial masses of disks scaled by factor of 0.5. The results are shown in Fig. 9. In this case the probability of finding a planet does not seem to depend strongly on the mass of the central star, which is true for the whole range of metallicities we have considered. Still, our models show that the evolution of solids leading to the formation of planetesimal swarms is a vital factor facilitating the formation of giant planets, whose role should be particularly clear for low-mass stars.
Our models of gaseous disks do not reproduce recent observations by (Muzerolle et al. 2005), which show that the accretion rate in protoplanetary disks increases with the mass of the central star. However, in the mass range considered here this dependence is very weak, and for a given value of stellar mass the spread in accretion rates reaches two orders of magnitude. In our opinion these data do not invalidate our basic assumption that initial disk parameters do not depend on the mass of the star. We also assumed that heating by stellar radiation is negligible, whereas at least in some cases it can be a dominant source of energy in the outer regions of the disk (more efficient than the turbulent dissipation). As such, it may substantially change the structure of the disk and the radial velocities of solids. Currently we are working on models that will take these effects into account.
This project was supported by the German Research Foundation (DFG) through the Emmy Noether grant WO 857/2-1 and the European Community's Human Potential Programme through the contract HPRN-CT-2002-00308, PLANETS. K.K. and M.R. acknowledge support from the grant No. 1 P03D 026 26 from the Polish Ministry of Science.