Disk-Fed Giant Planet Formation
Massive giant planets, such as the ones being discovered by direct imaging surveys, likely experience the majority of their growth through a circumplanetary disc. We argue that the entropy of accreted material is determined by boundary layer processes, unlike the “cold-” or “hot-start” hypotheses usually invoked in the core accretion and direct collapse scenarios. A simple planetary evolution model illustrates how a wide range of radius and luminosity tracks become possible, depending on details of the accretion process. Specifically, the proto-planet evolves towards “hot-start” tracks if the scale-height of the boundary layer is , a value not much larger than the scale-height of the circumplanetary disc. Understanding the luminosity and radii of young giant planets will thus require detailed models of circumplanetary accretion.
Subject headings:accretion, accretion disks — planets and satellites: formation — planets and satellites: gaseous planets — planets and satellites: physical evolution
We are now in an era where giant planets orbiting other stars can be directly imaged. The current sample is small, but in the coming years the next generation of direct detection instrumentation (e.g. SPHERE - Vigan et al. 2010, GPI - Macintosh et al. 2008, HiCIAO - Yamamoto et al. 2013) will grow this sample over the coming years, and these campaigns recently yielded an actively accreting proto-planet (Sallum et al., 2015). Since sub-stellar mass objects do not generate any internal luminosity from nuclear burning, they passively cool over time, such that there is a large degeneracy between luminosity, age and mass. This issue is particularly acute at young ages when the planets may not have cooled significantly from their formation conditions, so that even for a fixed age and mass a planet could admit a large range of luminosities, depending on its initial thermal content (e.g. Spiegel & Burrows, 2012).
One can turn this issue on its head and with dynamical constraints on a planet’s mass learn about its initial thermal content and perhaps its formation. The original approach assumed that giant planets started cooling from an arbitrary high entropy state (Stevenson, 1982; Burrows et al., 1997), with such “hot start” models now typically associated with the outcome of fragmentation in the protoplanetary disc (Boss, 1997). By contrast, Marley et al. (2007) used the standard core-accretion picture (Pollack et al., 1996; Bodenheimer et al., 2000) and argued that newly formed planets would be significantly cooler than the “hot-start” models, with initial cooling times typically years in these “cold-start” models. Observationally, the luminosity of a handful directly imaged exoplanets with mass constraints are inconsistent with the “cold-start” scenario (Marleau & Cumming, 2014) in possible tension with the core accretion model. On the other hand, it appears difficult to form giant planets ( M) through gravitational instability (Rafikov, 2005; Kratter et al., 2010).
While there have been attempts to blur these formation channels into “warm-start” scenarios (e.g. Spiegel & Burrows, 2012; Mordasini et al., 2012; Mordasini, 2013), all such models implicitly assume that accretion takes place in a spherically symmetric manner. Accreting material would then be processed by a shock at the planet’s surface, which plays a key role in determining the initial entropy of the planet (e.g. Marley et al., 2007). However, this is unlikely to be how giant planets accrete their mass in a realistic scenario. Excess angular momentum of the accreting material is likely to form a circumplanetary disc through which material can accrete and be thermally processed. In this work, we argue that disc accretion can drive the entropies of forming giant planets up to traditional “hot-start” values, even in the core accretion framework, and that this indeed likely to happen.
2. Overview of Accretion Paradigm
Most current models of planet formation within the core accretion scenario assume that the planet remains embedded in the disc until the disc disperses at some later time. However, it is well known that once a planet grows to become massive enough to perturb the disc then it can open a gap (e.g. Lin & Papaloizou, 1993). A rough estimate of the mass at which the planet can open a gap is given by the “thermal-mass”, when the planet’s Hill sphere - - exceeds the local scale height of the disc (). For a typical passively heated protoplanetary discs with temperature (Kenyon & Hartmann, 1987), this qualitatively requires:
Therefore, it is likely that a giant planet in the region 1-30 AU will accrete the majority of its mass after gap opening. Once the gap has opened, simulations suggest that the incoming accretion streams possess enough angular momentum to form a circumplanetary disc (e.g. D’Angelo et al., 2002; Ayliffe & Bate, 2009; Martin & Lubow, 2011; Szulágyi et al., 2014). Thus, the entropy of the accreting planetary material is no longer associated with the parent protoplanetary disc, but is rather processed by the circumplanetary disc and is associated with the accretion process that transfers material from the disc to the planet. Two such mechanisms exist: magnetospheric and boundary-layer accretion. Gas giant planets are hypothesised to have magnetic fields, and indeed Jupiter has a field strength of 5 Gauss. In order to determine whether accretion is controlled by magnetic fields (in the magneto-spherical accretion model) or whether the disc extends all the way to the planet’s surface we must determine the magnetospheric truncation radius.
For field strengths similar to gas giants in our solar system this indicates that the magnetic field cannot truncate the disc and accretion will proceed all the way to the star. Alternatively, one would need a planetary field strength in excess of gauss for the truncation radius to exceed the planetary radius (see also Quillen & Trilling, 1998; Fendt, 2003; Lovelace et al., 2011; Zhu, 2015). The expected magnetic field strengths of young massive giant planets is thought to be times Jupiter’s field strength (Christensen et al. 2009). Thus, it is highly likely that the circumplanetary disc will accrete onto the planet through a boundary layer (e.g. Lynden-Bell & Pringle, 1974).
Since the gas disc lifetime limits the accretion timescale for gas giants to Myr we know the accretion rates onto the protoplanets must be high with values of order M yr expected. These accretion rates can lead to extremely large accretion luminosities (e.g. Rafikov, 2008b; Owen, 2014; Zhu, 2015) of order L. Approximately, half of this is released in the disc and the remaining fraction in the boundary layer. Such accretion luminosities are many orders of magnitude larger than the internal luminosities of planets in “cold-start” scenarios ( L) and higher than the majority of “hot-start” models ( L; e.g. Marley et al., 2007). The ratio of accretion luminosity to internal cooling luminosity is:
where & are the internal flux and luminosity of a passively cooling planet with mass and radius . Therefore, for a planet forming through disc accretion, the disc and boundary layer will strongly irradiate the surface of the planet, not unlike the earliest stages of star formation (e.g. Adams & Shu, 1986; Rafikov, 2008a). External irradiation of a gas giant with an internal luminosity many orders of magnitude smaller is also similar to the hot Jupiter problem. The irradiation pushes the radiative-convective boundary deeper into the planetary interior, preventing the radiation from escaping as easily, thus suppressing cooling and contraction (Guillot et al., 1996; Burrows et al., 2000; Arras & Bildsten, 2006). Therefore, by accreting through a disc, the planet cooling will be suppressed and it will retain a higher entropy than if it were cooling passively.
3. Simple Disc-Fed Planetary Formation Model
We assume that the bulk of the planet’s mass is contained in a convective envelope surrounding the core. We can evaluate the binding energy of this envelope by assuming that the envelope mass exceeds the core mass and that its structure is described by a polytrope with , so that the total binding energy () is:
where & are the planet’s mass and radius, respectively. As the planet accretes, its total binding energy evolves. Following Hartmann et al. (1997), who studied the evolution of accreting low-mass stars, we assume the bulk of the planet remains convective and describe the evolution as:
The accretion efficiency parameter, , represents the internal energy of the accreted matter, is the accretion rate and describes the radiative losses from the planetary surface. The boundary layer can be described using a “slim-disc” model (e.g. Abramowicz et al., 1988; Popham & Narayan, 1991). Global integration of the slim-disc energy equation (Popham, 1997) shows that111Assuming the discs luminosity is large compared to :
where is the luminosity radiated away by the disc surface layers, is the heat capacity, is the angular velocity of the planet, is the Keplerian angular velocity at the radius of the planet and is the angular momentum flux normalised to . Here we do not attempt to model the evolution of the planetary rotation rate, assuming it is not close to break-up, but note that given an explicit boundary layer model the evolution of the planetâs angular velocity could be computed as well. The second term of the left-hand side represents the energy advected into the planet and the right hand side represents the energy dissipated by the disc and boundary layer, where we neglect the role of the term. Therefore, the fraction of the accretion luminosity advected into the star is (Popham, 1997):
where is the scale height of the boundary layer at the planetary radius.
Finally, assuming that the boundary layer obscures an area of the planetary surface, is given by:
where is the flux emerging from the planetary surface at an angle to the pole. As discussed above, the accretion luminosity often exceeds the internal luminosity by several orders of magnitude. Rafikov (2008a) showed that the integral in Equation 8 can be written as:
Calculation of can be performed assuming emission from a standard -disc (e.g. Adams & Shu, 1986; Popham, 1997; Rafikov, 2008a). The function is not analytic; however, given an opacity law of the form , can be calculated following Rafikov (2008a). A power-law fit to the opacity law in the planetary regime is given by Rogers & Seager (2010) with & . Adopting this fit, the function is shown in Figure 1, indicating that irradiation from the disc can lead to cooling luminosities several tens of percent lower than for a passively cooling planet.
Therefore, we write as:
where is the luminosity of an isolated planet.
In order to fully evolve our accreting planet-disc system we need to know the scale height of the boundary layer at the planetary surface. Boundary layers are poorly understood; since they are stable to the MRI and an -viscosity prescription may lead to unphysical solutions (Pringle, 1977; Popham & Narayan, 1992). Recently Belyaev et al. (2013a, b) showed that angular momentum transport can occur through waves arising from the sonic instability in boundary layers; however, a simple boundary layer model including this mechanism does not exist. We can still make some reasonable estimate of the scale height by using the disc scale height just outside the boundary layer. Since one would expect enhanced dissipation in the boundary layer due to the sharper angular momentum gradients present, we might expect the scale height to be larger in the boundary layer than in the disc. For an actively heated disc, where one equates radiative cooling with local viscous dissipation in a Keplerain disc, the scale height in the disc, , is given by:
where is the mean particle weight in amu. In obtaining Equation 11 we have neglected the “standard” factor. This factor arises if one applies a zero-torque boundary condition at the boundary-layer/disc interface, resulting in a decrease in the surface density towards the planet’s surface. This specific boundary condition is unlikely to be appropriate for the boundary-layer/disc interface which is more likely to have an angular velocity and thus associated torque of a full Keplerian disc (e.g. Popham & Narayan, 1991). Since Equation 11 is weakly sensitive to the input parameters, we assume that the scale height of the boundary layer222Since we only evaluated the scale height in the disc not the boundary layer we cannot verify this assumption. remains constant for the entire evolution and choose values . Combining Equations 5 & 10, we obtain the evolution equation for the radius of an accreting convective planet:
To integrate Equation 12, one needs to know the passive luminosity of the planet as a function of planet mass and radius. To calculate this we use the mesa code (Paxton et al., 2011) to produce a series of hydrostatic models as a function of planet mass and radius, assuming a 10 M rocky core and the Freedman et al. (2008) opacities. The resulting luminosities (left-panel) and Kelvin-Helmoltz time-scales (, right-panel) are shown in Figure 2.
We can now understand how the proto-planet will evolve from Equation 12. The first term on the RHS is , where is the mass evolution time-scale, while the second term is approximately . Therefore, if the proto-planet will cool and shrink. If then the evolution of the planet depends on the internal energy of the incoming material, and as such there is a critical above which the planet increases in radius as it accretes, otherwise it shrinks. This critical boundary layer height is , which is for and for . Given typical temperatures in the boundary layer, we expect the gas to be monotonic, so that for the proto-planet’s radius will increase if . This critical value of the boundary layer scale height is not much larger than the estimated disc scale height (Equation 11), thus we may expect the planet’s radius to increase rather than decrease during circumplanetary disc accretion. The left panel of Figure 2 shows that, in the giant planet regime g, luminosity increases with increasing radius as the mass increases. Therefore, disc accretion through a boundary layer can potentially drive the planet to high luminosities, comparable or even larger than typical values in the “hot-start” scenarios.
We consider the evolution of a proto-planet accreting through a boundary layer. The planet is taken to be an initially 50 M planet with a 10 M core and we assume the core doesn’t grow in mass during the evolution. We assume that the accretion rate is constant and the radius of the planet evolves according to Equation 12, where the relative scale height of the boundary layer to the planet’s radius remains fixed for the entire accreting period. We consider a variety of formation times, final masses, initial cooling times for the 50 M proto-planet and values.
The radius evolution of a proto-planet with final mass 1 M (top) and 10 M (bottom), accreting at a constant rate with a formation time of 1 Myr (left) and 10 Myr (right), are shown in Figure 3. Three initial cooling times of years, years & years are shown each with values of 0.15, 0.25, 0.35, 0.45.
The accretion of material by a proto-planet via a circumplanetary disc and boundary layer can result in a large range of initial planet properties, from “hot start” to “cold start” conditions. For reference, “hot start” and the “cold start” radii from Spiegel & Burrows (2012) are shown as the points. In all cases initially and the proto-planet follows an evolutionary path determined by . Therefore, above the critical value of the proto-planet grows in radius and below it shrinks in radius, as expected. For the highest values considered, the proto-planet is driven to high enough luminosities that radiative cooling dominates and contraction ensues. This decrease in radius causes the planets to follow a convergent evolutionary path and the final properties of planets with are insensitive to their initial properties. However proto-planets forming with do not cool appreciably and final properties post-accretion bear the signature of the initial Kelvin-Helmholtz time-scales. For massive planets (10 M final masses), the radius starts to increase again once they reach about 2 . This can be understood from Figure 2, which shows that, at high masses, the cooling time increases as one increases mass at fixed radius. So a planet whose evolution was dominated by cooling at lower masses can evolve into a region where cooling becomes subdominant and the thermal content of the accreted material drives the evolution instead. Unsurprisingly, faster accreting proto-planets experience stronger evolution, as mass accretion dominates over cooling more prominently.
Finally, we can compute the luminosity evolution of our planets, including at late times after accretion stopped. We do this by continuing to evolve the planet according to Equation 12, however, we set and allow the planet to cool over its entire surface. The comparison of model tracks with the handful of directly imaged giant planets for which their luminosity and age has been measured is shown in Figure 4. Planetary formation through boundary layer accretion can drive planets onto “hot-start”-like evolutionary paths , which can successfully match the luminosity of young directly imaged exoplanets.
5. Discussion and Summary
A proto-planet embedded in a proto-planetary disc is likely to open a gap at modest masses, thus a giant planet is likely to accrete the majority of its mass through a circumplanetary disc. This scenario is unlike the direct collapse (gravitational fragmentation) framework where the high entropy disc material is turned into a planetary mass object on a short-timescale, in an essentially adiabatic fashion. It is also unlike the final stages of the standard core accretion framework, which assumes that the proto-planet accretes spherically, shocks and then radiate away the disc material’s entropy, accreting material with the same temperature as the proto-planet’s atmosphere, leading to comparatively small, low luminosity planets (e.g. Marley et al., 2007). In the disc accretion scenario considered here, we argue that the entropy of the accreting material depends on how the material is transported from the disc to the planet. Unless the proto-planet has a very strong magnetic fields ( gauss) this will occur through a boundary layer and the fraction of heat added to the proto-planet directly depends on the thermal structure of the boundary layer, with puffier boundary layers advecting more heat into the planet. There is a critical value of the height of the boundary layer above which the proto-planet becomes inflated by accretion and driven to high luminosities, , which is only slightly larger than the scale height of the circumplanetary disc feeding the boundary layer. To the extent that boundary layers are hotter than their circumplanetary disc, one may expect in the majority of cases that circumplanetary disc accretion will inflate the proto-planet, driving it to high luminosities in line with those expected from “hot-start” or direct collapse scenarios.
Before concluding, we note that our models of disc-fed planet formation are highly idealized. We have assumed an polytrope. For the largest boundary layers we are adding extremely high entropy gas on-top of lower entropy gas, so that the envelopeâs outer layers may become stably stratified. Accretion over a long period of time could result in a complicated layered structure of convective and radiative zones, meaning that an polytrope may no longer be a good description of the proto-planetary structure. If such a stably stratified region were very thin it may cool quickly, resulting in low luminosity planets fairly soon after accretion ceases. Simple models – a constant opacity, hydrostatic radiative envelope (c.f. Stevenson, 1982) – suggest a thick radiative layer with large thermal inertia, so that the inflated radii should remain that way for a long time after accretion stops. More detailed models are needed to address this reliably. One obvious avenue for progress is thus to couple boundary-layer accretion models with detailed structure calculations for the planetary interior.
In conclusion, disc-fed accretion may be the dominant mechanism shaping the properties of young giant planets, where the thermal structure of the boundary layer controls the amount of heat advected into the planet during late-stage growth. Such a process will naturally arise in the core-accretion scenario once the planet becomes massive enough to open a gap in the protoplanetary disc and angular momentum conservation leads to the formation of a disc around the proto-planet. It therefore seems that further development of boundary layer accretion theory is warranted to understand giant planet formation and to interpret the wealth of direct imaging data expected to become available over the next few years.
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