HEAVY-ELEMENT ENRICHMENT OF A JUPITER-MASS PROTOPLANET AS A FUNCTION OF ORBITAL LOCATION

Jupiter contains mostly hydrogen and helium with a smaller fraction of heavy elements. Constraints on the mass of Jupiter's heavy elements can be found by fitting its measured gravitational field using theoretical equations of state. Theoretical models of Jupiter's interior predict that the total mass of heavy elements ranges from 20 to 40 M_⊕ (Saumon & Guillot 2004; Nettelmann et al. 2008; Militzer et al. 2008). In addition, measurements by the Galileo probe mass spectrometer suggest that Jupiter's atmosphere is enriched in heavy elements by a factor of up to 3 compared with the Sun (Young 2003). Observations of transiting giant planets suggest that the composition of gas giant planets varies over a wide range. While some extrasolar planets are found to contain mainly hydrogen and helium, others possess large amounts of heavy elements (Guillot 2008).

Giant planet formation models must be able to explain the abundances of heavy elements in Jupiter and other giant planets. The standard model for giant planet formation is "core accretion" (Pollack et al. 1996; Hubickyj et al. 2005; Lissauer & Stevenson 2007) in which formation starts with the buildup of a solid core. Once the core reaches a critical mass of roughly 10 M_⊕, runaway gas accretion occurs leading to the accumulation of a massive gaseous envelope. Until recently, the formation timescale in this model was uncomfortably long compared with observationally derived lifetimes of protoplanetary disks (Haisch et al. 2001). However, recent work offers several ways to overcome the "timescale problem" in the core accretion model (e.g., Hubickyj et al. 2005; Dodson-Robinson et al. 2008a, 2008b; Lissauer et al. 2009); one of them is planetary migration (Alibert et al. 2005; Dodson-Robinson et al. 2008a, 2008b). Observations of "hot Jupiters" have supported the paradigm in which planets are formed at large radial distances and then migrate inward to their present locations. Alibert et al. (2005) have shown that Jupiter can form within ∼1 Myr starting with a planetary embryo at 8 AU.

An alternative model for giant planet formation is disk instability (Boss 1997; Mayer et al. 2002). In this model, giant planets are formed as a result of gravitational instabilities in the protoplanetary disk surrounding the young star. If the formed "clumps" cool fast enough to stay gravitationally bound (Rafikov 2007; Boss 2009) they can evolve to become gaseous protoplanets (Durisen et al. 2007). Gravitational instabilities can occur at very large radial distances of tens of astronomical units. In fact, the formation of a clump farther from the protostar in this model is favored due to lower temperatures and faster cooling in the outer regions. Therefore, if giant planets are formed via disk instability, it is certainly possible that they are formed farther out and migrate inward to their present locations.

In the disk instability model, protoplanets are initially formed with a stellar composition; they can be enriched in heavy elements after their formation as a result of planetesimal accretion. Helled et al. (2006), hereafter Paper I, presented a calculation of heavy-element enrichment in a Jupiter-mass clump, taking into account different planetesimal sizes, compositions, and velocities. The location of the protoplanet, however, was limited to 5.2 AU, Jupiter's current location. Observations of close-orbiting extrasolar giant planets suggest that planetary migration may be a common phenomenon. In addition, the recent direct imaging of extrasolar planets has revealed that giant planets can be present at much larger radial distances than that of Jupiter in the solar system (Marois et al. 2008).

In this paper, we investigate possible heavy-element enrichment of a Jupiter-mass clump formed at different radial distances and disk environments (masses and density profiles). The implications of the results for Jupiter and extrasolar giant planets are discussed.

The amount of solids available for accretion by a protoplanet depend on the disk mass and its radial density profile. The total (gas and dust) surface density Σ profile can be given by

$$\begin{equation} \Sigma (a)=\Sigma _0\left(\frac{a}{5\, {\rm AU}}\right)^{-\alpha }, \end{equation} \tag{ 1 }$$

where Σ₀, the surface density at 5 AU, is determined by the disk mass, a is the radial distance, and α is a parameter which defines the steepness of the density function. We take the disk's inner and outer boundaries to be 0.1 and 30 AU, respectively. The value for the inner boundary is typically taken as 0.1 AU (Andrews & Williams 2007) since disks are expected to be truncated by the stellar magnetosphere in the innermost regions (Shu et al. 1994). Indeed, observations suggest truncation radii of about 0.1 AU (Eisner et al. 2005). The value of the inner boundary radius, however, is unimportant as long as it is significantly smaller than the outer boundary radius (Ruden & Lin 1986). A disk which ranges up to 30 AU, as considered here, covers the region in which giant planets are likely to form, and allows surface densities that are consistent with planet formation. Disks are expected to expand with time, and observations of disks suggest that they can be as large as hundreds of AU (Andrews & Williams 2007); however, our calculation concentrates on the first 10⁵ years, before the disk expands to such large distances (Dodson-Robinson et al. 2008a, 2008b).

The relation between the disk mass and the surface density at 5 AU, Σ₀, is then given by

$$\begin{equation} \Sigma _0 = \frac{M_{\rm disk}(2-\alpha)}{2 \pi 5^{\alpha } {\rm \; AU}^2} \left(30^{2-\alpha } -0.1^{2-\alpha } \right)^{-1}, \end{equation} \tag{ 2 }$$

where the disk mass M_disk is given by M_disk = ∫^30 AU_0.1 AU2πaΣ(a)da. The solid surface density σ is taken to be Σ/70, as commonly taken for the solar nebula (Weidenschilling 1977). Since the masses and density distributions of protoplanetary disks have wide ranges (Eisner et al. 2008; Eisner & Carpenter 2006; Andrews & Williams 2007; Andrews et al. 2008), we consider three different disk masses, 0.1, 0.05, and 0.01 M_☉, with M_☉ being the Sun's mass, and take α values of 1/2, 1, and 3/2. Figure 1 shows the solid surface density for the cases considered. In all cases the surface density decreases with radial distance; however, the density decreases more slowly with radius for smaller α values. Table 1 summarizes the solid surface density at 5 AU (σ₀) values for the different disk masses and density profiles. Since we take the solid surface density as a constant fraction of the gas density (constant metallicity), it increases with increasing disk mass. However, high solid surface densities can also be the result of high metallicity. It is therefore possible to have low-mass disks with relatively high solid surface densities, or low solid surface densities for massive disks, for high-metallicity and low-metallicity disks, respectively. The implications of high solid surface density as a result of stellar metallicity are discussed in Section 3.2.

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The mass of solids accreted by a protoplanet depends on the solid surface density (the available mass at a certain location) and the protoplanet's efficiency in capturing solid planetesimals. To compute the planetesimal capture rate, we model the evolution of a Jupiter-mass protoplanet, with an initial state as expected in the disk instability model, similar to that used in Paper I. The evolution is followed assuming that the protoplanet is spherical, hydrostatic, and isolated (see Paper I and Helled et al. 2008 for further details). The effect of planetesimal accretion is not included in the evolution calculation. Figure 2 presents the physical parameters of the protoplanet as a function of time for the first 10⁵ years.

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As in Paper I, we follow planetesimal trajectories with increasing impact parameters in order to determine the largest impact parameter for which planetesimal capture is possible, b_capture. The computed trajectory accounts for gas drag and gravitational forces, assuming that the gravitational interaction is two body (Podolak et al. 1987; Paper I). The protoplanet's capture radius R_capture is defined as the planetesimal's closest distance of approach to the protoplanet's center for the critical impact parameter (see Paper I for details). Both the protoplanet radius and capture radius decrease as the protoplanet evolves. The difference between the protoplanet radius and capture radius as a function of time can be seen in Figure 2. The capture radius is smaller than the actual radius of the planet especially at the early stages of the evolution, when the density in the outer envelope is low, and the planetesimal must pass through more dense inner regions to lose kinetic energy by gas drag and be captured by the protoplanet. The capture radius is typically smaller by a factor of a few than the largest impact parameter for capture (Paper I).

Once the capture radius is known the capture cross section can be computed. A two-body gravitational cross section is given by πb²_capture; however, this cross section accounts only for the planetesimal and the protoplanet. To account for the influence of the Sun, we follow Paper I and take the capture cross section to be πR²_captureF_g, as expected for a three-body interaction (Greenzweig & Lissauer 1990). The gravitational enhancement parameter F_g is taken as unity, providing a conservative estimate for the capture cross section. The planetesimals are taken to be 1 km radius bodies composed of a mixture of ice, rock, and organics with an average density of 2 g cm⁻³, and a random velocity of 1 km s⁻¹. Once the capture cross section is known, the accretion rate can be computed.

The planetesimal accretion rate is given by (Safronov 1969)

$$\begin{equation} \frac{dm(a)}{dt}=\pi R^2_{\rm capture}(t) \sigma (a,t) \Omega (a), \end{equation} \tag{ 3 }$$

where σ is the surface density of solid material and Ω is the protoplanet's orbital frequency.

The total available mass of solids in the planet's feeding zone is given by

$$\begin{equation} M_{\rm av}(a)\,{=}\,\pi \big(a_{\rm out}^2\,{-}\,a_{\rm in}^2\big)\sigma (a) \,{=}\, 16 \pi a^2 \sigma _0 \left(\frac{a}{5 {\rm \; AU}}\right)^{-\alpha } \!\!\left(\frac{M_p}{3\,M_{\odot }}\right)^{1/3}\!\!, \end{equation} \tag{ 4 }$$

where a_out and a_in are the outer and inner radii of the feeding zone, respectively. The planetesimals are assumed to be uniformly spread on either side of the orbit to a distance a_f, which depends on the eccentricity and inclination of the planetesimal orbits, as well as the Hill sphere radius of the protoplanet R_H. We take a_f ∼ 4R_H, with $R_H=a\left(\frac{M_p}{3\,M_{\odot }}\right)^{1/3}$ and M_p is the planet's mass. The inner and outer boundaries of the feeding zone are then taken to be a_in = a − a_f and a_out = a + a_f, respectively. We assume that planetesimals do not get in or out of the planetary feeding zone. A detailed discussion of how the feeding zone is defined can be found in Pollack et al. (1996).

The size of the feeding zone is proportional to a² while the solid surface density is proportional to a^−α. As a result, the available mass of solids M_av in the disk is proportional to a^2−α (Equation (4)). Since we use α values smaller than 2, the available mass increases with radial distance. Figure 3 shows the available mass for accretion as a function of distance from the star for the three disk masses and density profiles considered. Although the available mass of heavy elements for accretion by a Jupiter-mass clump is found to increase with radial distance, it does not imply that this available mass can actually be accreted. The accretion rate is proportional to Ω, the protoplanet's orbital frequency, which goes as a^−3/2, and to σ, the solid surface density (∝a^−α); both decrease with radial distance. As a result, the accretion timescale is significantly longer at the disk's outer regions.

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The protoplanet can be enriched with heavy elements as long as the accretion process is efficient. While the protoplanet is extended (a few tenths of AU) it fills most of the area of its feeding zone, planetesimals are slowed down due to gas drag, and are absorbed by the protoplanet. In fact, both accretion and ejection occur all the time, but as long as the body is extended ejection is negligible. The accretion efficiency decreases with decreasing size of the protoplanet, and once the body is sufficiently small planetesimals no longer pass through its envelope, and instead of being accreted they get ejected from the protoplanet's vicinity. The radius of the protoplanet changes significantly once molecular hydrogen begins to dissociate and a dynamical collapse occurs. For a Jupiter-mass clump the dynamical collapse takes place after a few 10⁵ years (DeCampli & Cameron 1979; Bodenheimer et al. 1980; Paper I). After the collapse the body has much larger internal densities and temperatures, with a dimension of a couple of Jupiter radii. In addition, the protoplanet is expected to open a gap in the gas disk and undergo type II migration (Lin & Papaloizou 1985; Nelson et al. 2000). Once a gap is opened and inward migration occurs, planetesimals are mostly ejected by the migrating planet and accretion no longer dominates. In addition, it is unclear how planetesimals are distributed within the gap and how solids from the disk (or in the gap's vicinity if it exists) interact with the protoplanet (Goldreich et al. 2004). The timescale for type II migration depends on the disk's viscosity and on the protoplanet's radial distance (D'Angelo et al. 2003). The migration timescale τ_mig decreases with increasing disk mass (massive disks are expected to be more viscous) and with decreasing radial distance (τ_mig ∝ a^1/2). For simplicity, for all cases considered here we take the maximum time for accretion as 10⁵ years. The accretion timescales for low-mass disks and/or large radial distances are not expected to be more than a few 10⁵ years. For these cases, the accreted mass calculated here can be taken as a lower bound.

Figure 4(a) presents the captured mass in the first 10⁵ years of the planetary evolution for all the cases considered. The black, blue, and red curves are for disk masses of 0.1, 0.05, and 0.01 M_☉, respectively. The solid, dashed, and dotted curves refer to density distributions proportional to a^−1/2, a⁻¹, and a^−3/2, respectively. As expected, higher solid surface density (more massive disks) results in larger enrichment in heavy elements. Lower α values allow heavy-element capture at larger radial distances due to a more moderate decrease in density. Although the available mass for accretion increases with radial distance, the location at which the maximum mass can be accreted does not exceed 15 AU. This is because the accretion time is significantly longer in the outer regions and the protoplanet cannot capture the entire available mass within 10⁵ years. The location of the maximum shifts slightly outward with decreasing α since more mass is available at distant locations.

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3.1. Jupiter

3.2. Extrasolar Planets

We present a calculation of heavy-element enrichment via planetesimal capture by a Jupiter-mass clump created by disk instability. The accreted mass is computed considering different formation radial distances for the protoplanet (5–30 AU), and various disk environments. We consider three disk masses (0.1, 0.05, and 0.01 M_☉) and solid surface density distributions (∝a^−1/2, a⁻¹, and a^−3/2). Massive disks have high surface densities allowing for a significant enrichment of heavy elements. Increasing values of α (increasing the density function's steepness) lead to lower heavy-element enrichment in the outer part of the disk.

The available mass for accretion increases with radial distance. The accretion rate, however, decreases with radial distance due to its dependence on both the surface density and orbital frequency, both of which decrease with radial distance. The maximum time for accretion is taken to be 10⁵ years, the period in which planetesimal accretion is efficient. The exact amount of heavy elements accreted will depend on both the disk properties (mass, density distribution, viscosity, etc.) and the planetary contraction rate. The contraction rate can be slower when the effects of planetesimal accretion and stellar radiation are included in the evolution calculation. At small radial distances these effects are unimportant because the accretion timescale is shorter or comparable to the 10⁵ years considered. At large radial distances the stellar radiation effect is small but since the accretion time is significantly longer a slower contraction provides more time for accretion, and therefore increases the captured mass. As a result, the accreted heavy-element mass for large radial distance presented here is a lower bound.

For the disk parameters considered here we find that heavy-element enrichment in a Jupiter-mass protoplanet can range from about 1 to 110 M_⊕. The result shows that protoplanets with similar initial masses (one Jupiter mass in our case) can end up having significantly different compositions, as well as final masses. Since both the planetary contraction rate and accretion rate change with planetary mass, protoplanets with varying initial masses will have different enrichments. The planetary enrichment also depends on the solid surface density which changes with the disk's mass, size (outer boundary), and its density profile. The enrichments presented here are therefore consequences of the values chosen to define the disk properties. Once disk configurations can be described more accurately, the planet's enrichment could be better constrained. Finally, a planet's enrichment also changes with the planetesimal properties (size, composition, velocities), as shown in Paper I. The capture efficiency (cross section) decreases with increasing planetesimal size, density, and random velocity. Our calculations can be repeated using different disk environments, planetary masses, and planetesimal properties.

We conclude that Jupiter could have formed at larger radial distances than its current location (the exact radial distance depends on the assumed disk parameters) and still accrete between 20 and 40 M_⊕ of heavy elements, the heavy-element mass predicted by interior models (Saumon & Guillot 2004). Since the disk's temperature decreases with increasing radial distance, at larger distances planetesimals can consist of amorphous ice in which volatiles can be trapped. Formation at more distant regions explains the enrichment of heavy elements and volatiles measured in Jupiter's atmosphere (Owen et al. 1999). However, since the disk's temperature can increase with increasing surface density, a more detailed analysis is needed to evaluate the temperature profile, and the radial distances required to allow the existence of amorphous ice and efficient trapping of volatiles. In such an investigation larger radial distances than considered here may be required.

Our work suggests that the final composition of the protoplanet can change considerably depending on the planetary "birth environment." The large variation we derive in heavy-element enrichment explains the different compositions of observed extrasolar gas giant planets (Guillot 2008). Since higher surface density leads to larger heavy-element enrichments, our model results are found to be consistent with the correlation between heavy-element enrichment and stellar metallicity (Guillot et al. 2006).

We conclude that the disk instability model can lead to both giant planets with nearly solar compositions and planets which are significantly enriched with heavy elements. Our findings support the disk instability model of gas giant planet formation.

R.H. thanks Alan Boss, Nadar Haghighipour, Brad Hansen, and Morris Podolak for fruitful discussions and helpful suggestions. R.H. acknowledges support from NASA through the Southwest Research Institute. G.S. acknowledges support from the NASA PGG and PA programs.