Inside-Out Planet Formation. VIII. Onset of Planet Formation and the Transition Disk Phase

TL;DR

The study probes the onset of Inside-Out Planet Formation (IOPF) by examining pebble trapping at the dead zone inner boundary (DZIB) as the disk accretion rate $\dot{m}$ declines. Using a self-consistent, viscously heated inner-disk model with an $\alpha$-profile tied to the DZIB, the authors compute how the DZIB moves inward and how pebble drift and trapping respond to evolving disk conditions, deriving a minimum trapping size $a_{p,\rm trap}$ that scales as $\dot{m}^{1/3}$. They show that efficient trapping near $r_{\rm DZIB}\approx0.1\,\mathrm{au}$ requires pebbles of order $a_p\sim0.5\,\mathrm{mm}$ and that the trapped-pebble flux—and thus the mass of the first planet that can form (the Vulcan planet) via shallow-gap opening—depends strongly on the assumed pebble-size distribution and possible early pebble-flux boosts. The results provide a boundary condition for IOPF and offer a potential explanation for the transition-disk phase arising around $\dot{m}\sim10^{-9}\,M_\odot\,\mathrm{yr}^{-1}$, illustrating how disk evolution sets the inner planetary architecture. Remaining uncertainties include DZIB fluctuations and dust-growth processes, motivating future, more realistic simulations of DZIB-dust evolution coupling.

Abstract

Inside-Out Planet Formation (IOPF) is a theory of {\it in situ} formation via pebble accretion of close-in Earth to Super-Earth mass planets at the pressure maximum associated with the dead zone inner boundary (DZIB), whose location is set initially by thermal ionization of alkali metals at $\sim1,200\:$K. With midplane disk temperatures determined by viscous accretional heating, the radial location of the DZIB depends on the accretion rate of the disk. Here, we investigate the ability of pebbles to be trapped at the DZIB as a function of the accretion rate and pebble size. We discuss the conditions that are needed for pebble trapping to become efficient when the accretion rate drops to $\sim10^{-9}\:M_\odot\:{\rm yr}^{-1}$ and the resulting DZIB is at $\sim 0.1\:$au, which is the expected evolutionary phase of the disk at the onset of IOPF. This provides an important boundary condition for IOPF theory, i.e., the properties of pebbles when planet formation begins. We find for our fiducial model that typical pebble sizes of $\sim0.5\:$mm are needed for pebble trapping to first become efficient at DZIBs near 0.1~au. This model may also provide an explanation for the first emergence of the transition disk phase in protoplanetary disks with accretion rates of $\sim10^{-9}\:M_\odot\:{\rm yr}^{-1}$.

Figures (6)

• Figure 1: Structural profiles of Inside-Out Planet Formation (IOPF) disk models used in this paper, assuming dead zone $\alpha=10^{-4}$ for accretion rates of $10^{-8}$, $10^{-9}$ and $10^{-10}\:M_\odot\:{\rm yr}^{-1}$. All models are for a one solar mass central star. From top to bottom, the rows show: gas mass surface density ($\Sigma_g$); midplane temperature ($T$); midplane opacity ($\kappa$) (assumed to be vertically constant); disk aspect ratio ($h/r$); enclosed mass in solids ($M_d(<r)$), i.e., initially assumed to be dust, summed in the disk from $r_{\rm 1200K}$ out to radius $r$ for a solid to gas mass ratio of $f_s=0.01$; disk viscous $\alpha$ profile. Note the difference in location and width of the $\alpha$ transition zones.
• Figure 2: (a) Top row: Map of pebble radial drift velocity plotted in parameter space of disk radius, $r$, and pebble radial size, $a_p$. Red color is used for inward drift velocities; blue for outward velocities. From left to right, the panels show results for disks with $\dot{m}=10^{-8}\:M_\odot\:{\rm yr}^{-1}$, $\dot{m}=10^{-9}\:M_\odot\:{\rm yr}^{-1}$ and $\dot{m}=10^{-10}\:M_\odot\:{\rm yr}^{-1}$. All disks have the same dead zone $\alpha=10^{-4}$. The minimum size of pebbles that experience a zone of outward migration (blue) and thus can be trapped near the DZIB becomes smaller as $\dot{m}$ decreases, i.e., 0.10 cm at $\dot{m}=10^{-8}\:M_\odot\:{\rm yr}^{-1}$, 0.063 cm at $\dot{m}=10^{-9}\:M_\odot\:{\rm yr}^{-1}$ and 0.035 cm at $\dot{m}=10^{-10}\:M_\odot\:{\rm yr}^{-1}$. (b) Middle row: As (a), but now showing radial velocity of fixed-size pebbles as a function of radial location. (c) Bottom row: As (a), but now showing pressure gradient ($d \ln{P} / d \ln{r} \equiv - k_P$) profile near DZIB. All models have similar values $k_P \simeq -1$ in the region just interior to the pressure maximum.
• Figure 3: (a) Left: Evolution of disk and planet properties as a function of accretion rate. Note, the evolution during planet formation is expected to be from a high accretion rate state (on the right) to a low accretion rate state (on the left), as indicated by the arrow at the bottom of the figure. These results assume constant opacity $\kappa=10\:{\rm cm^2\: g}^{-1}$ and $\alpha=10^{-4}$. Upper panel: Minimum size for a pebble to be trapped at the DZIB versus accretion rate (white solid line). Pebbles in the blue region are trapped, while those in the red region are dragged inward through the DZIB along with gas accretion. The black solid line is the critical size to enter the Stokes drag regime (above the line) from the Epstein drag regime (below the line). Middle panel: Location of pressure maximum predicted by Eq. \ref{['eq:r_dzib']}. Lower panel: Mass of the first planet to form in the IOPF model due to shallow gap opening leading to pebble isolation (Eq. \ref{['eq:mg1']}). (b) Right: As (a), but now shown as a function of radial location of the DZIB, which evolves from large to small distances, and with the middle panel showing accretion rate versus $r_{\rm DZIB}$.
• Figure 4: (a) Top panel: Minimum trapping size at the DZIB location (red dashed line), shown as a function of disk accretion rate, with evolution expected to occur from right to left. The peak pebble sizes of three assumed log-normal distributions (peaking at 0.5, 1, and 2 mm and with dispersion $\sigma_p=0.2$ dex) are indicated by the horizontal lines. (b) Bottom panel: Evolution of DZIB pebble trapping fraction, $f_{\rm trap}$, with disk accretion rate for the three pebble size distributions described in (a).
• Figure 5: (a) Upper panel: Effective pebble flux that is trapped at the DZIB as a function of disk accretion rate for the models with $a_{p,{\rm peak}}=0.5, 1, 2\:$mm (green, orange, blue lines, respectively) and for $f_{p,{\rm boost}}=1$ and 10 (solid and dashed lines, respectively). For the fiducial evolving disk model in which accretion rate declines by a factor of 10 every 1 Myr, the point at which the time-integrated trapped pebble mass equals the gap opening mass is indicated by solid and open circles for cases with $f_{p,{\rm boost}}=1$ and 10, respectively. (b) Middle panel: Vulcan planet formation timescales, $t_{\rm form,1}\equiv M_{p,1}/(f_{\rm trap}\dot{m}_p)$, i.e., the time to accumulate enough material for a planet to have enough mass to open a shallow gap in the disk, i.e., with $M_{p,1}=M_{\rm G,D}$, for the same cases shown in (a). Note, these times are based on instantaneous rates. The solid and open circles mark the stages at which the time-integrated trapped pebble mass equals the gap opening mass, as described in (a). (c) Bottom panel: Evolution of gap opening mass, $M_{\rm G,D}$, at the DZIB, i.e., $M_{p,1}$, with disk accretion rate.