---
abstract: 'Planets open gaps in discs. Gap opening is typically modeled by considering the planetary Lindblad torque which repels disc gas away from the planet’s orbit. But gaps also clear because the planet consumes local material. We present a simple, easy-to-use, analytic framework for calculating how gaps deplete and how the disc’s structure as a whole changes by the combined action of Lindblad repulsion and planetary consumption. The final mass to which a gap-embedded gas giant grows is derived in tandem. The analytics are tested against 1D numerical experiments and calibrated using published multi-dimensional simulations. In viscous alpha discs, the planet, while clearing a gap, initially accretes practically all of the gas that tries to diffuse past, rapidly achieving super-Jupiter if not brown dwarf status. By contrast, in inviscid discs—that may still accrete onto their central stars by, say, magnetized winds—planets open deep, repulsion-dominated gaps. Then only a small fraction of the disc accretion flow is diverted onto the planet, which grows to a fraction of a Jupiter mass. Transitional disc cavities might be cleared by families of such low-mass objects opening inviscid, repulsion-dominated, overlapping gaps which allow most of the outer disc gas to flow unimpeded onto host stars.'
author:
- |
    M. M. Rosenthal$^{1\href{https://orcid.org/0000-0003-3938-3099}{\includegraphics[scale=0.4]{orcid.pdf}}}$[^1], E. I. Chiang$^{2,3\href{https://orcid.org/0000-0002-6246-2310}{\includegraphics[scale=0.4]{orcid.pdf}}}$, S. Ginzburg$^{2}$, R. A. Murray-Clay$^{1\href{https://orcid.org/0000-0001-5061-0462}{\includegraphics[scale=0.4]{orcid.pdf}}}$\
    $^{1}$ Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\
    $^{2}$ Department of Astronomy, University of California at Berkeley, CA 94720, USA\
    $^{3}$ Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA
bibliography:
- 'refs.bib'
date: Released 
title: How Consumption and Repulsion Set Planetary Gap Depths and the Final Masses of Gas Giants
---

\[firstpage\]

accretion, accretion discs – planets and satellites: formation – planets and satellites: gaseous planets – planets and satellites: physical evolution – planet-disc interactions – protoplanetary discs

Introduction {#sec:intro}
============

Annular gaps in protoplanetary discs are often attributed to embedded planets. The interpretation stems from the theory of satellite-disc interactions that successfully predicted the existence of shepherd moons in planetary rings (e.g., @gt_1982). Satellites in rings, and by analogy planets in discs, repel material away from their orbits as the waves they excite at Lindblad resonances dissipate and impart angular momentum to the ambient medium (see also @gr_2001; @gs_2018). The repulsive, gap-forming planetary Lindblad torque competes against the disc’s viscous torque which diffuses material back into the gap.

Most studies of protoplanetary disc gaps concentrate exclusively on the Lindblad and viscous torques (e.g., @fsc_2014; @kmt_2015; @zzh_2018) and neglect how gaps can also deepen because embedded planets consume local disc gas. Exceptions include, e.g., @znh_2011, @dk_2015 [@dk_2017], and @mfv_2019, whose numerical simulations of planet-disc interactions allow for both planetary accretion and planetary torques. Our aim here is to give an elementary and analytic accounting of both effects: to understand, for planets on fixed circular orbits, how Lindblad repulsion and planetary consumption combine to set gap depths. This is a two-way feedback problem—planetary accretion affects the gas density inside the gap, but the density inside the gap determines the rate of planetary accretion (@gc_2019a, [-@gc_2019b]). Accordingly we will calculate how gas giants grow in tandem with their deepening gaps. Much of our analytic framework is the same as that of @tt_2016 and @tmt_2020, who used it to study nascent planets in viscous discs; we will explore both viscous and inviscid discs. The problem of planetary accretion within disc gaps is also a global one insofar as a planet can accrete gas that is brought to it from afar, from regions outside the gap. Thus we will stage our calculations within circumstellar discs that transport mass across decades in radius. This opens up another form of feedback: in feeding the planet, the disc can have its entire surface density profile changed (e.g., @ld_2006; @znh_2011; @o_2016).

Our work is organized as follows. In section \[sec:consumption\] we describe how Lindblad repulsion and planetary accretion of disc gas (“consumption”) work together to determine gap depths and surface density profiles of viscous circumstellar accretion discs. Our largely analytic considerations are supplemented with simple numerical experiments modeling planet-disc interactions and disc evolution in 1D (orbital radius). In section \[sec:consumption\] we fix, for simplicity, the planet mass; in section \[sec:mass\], we allow the planet mass to grow freely and solve the full two-way feedback problem. In section \[sec:windy\], motivated by recent theoretical and observational developments, we consider discs that transport their mass not by viscous diffusion but rather by angular momentum losses from magnetized winds. For such inviscid, wind-driven discs, accretion is not diffusive but purely advective, and embedded planets carve out especially deep gaps in the absence of viscous backflow. We summarize and discuss the implications of our findings on gas giant masses and disc structure, including the structure of transitional discs, in section \[sec:sum\].

A simplified study such as ours will not capture important (and sometimes poorly understood) effects, among them planetary migration (e.g., @kn_2012; @dhm_2014; @dk_2015 [@dk_2017]; @fc_2017; @kts_2018; @mnp_2020), eccentricity evolution (both of the planet and the disc; e.g., @pnm_2001; @gs_2003; @kd_2006; @dc_2015; @mfv_2019), and the 3D dynamics of circumplanetary discs (e.g., @fzc_2019). Our goal is not so much to be realistic but to acquire some intuition about the interplay of Lindblad repulsion and planetary accretion, and to provide a baseline understanding that can guide the development and interpretation of more sophisticated models. Where possible, we place our results in context with state-of-the-art numerical experiments in the literature (see in particular section \[sec:sum\]).

Viscous discs: Surface Density Profiles at Fixed Planet Mass {#sec:consumption}
============================================================

We study how the surface densities of viscous accretion discs are shaped by repulsive planetary Lindblad torques in addition to planetary accretion of disc gas (“consumption”). Section \[sec:oom\] contains analytic considerations which are tested numerically in section \[sec:num\]. In these sections, while we allow the disc surface density to deplete by consumption, we do not simultaneously allow the planet’s mass to increase. This fixing of the planet’s mass is done for simplicity, to see how the planet affects the disc but not vice versa. In section \[sec:mass\], we free up the planet’s mass and allow two-way feedback between planet and disc.

Order-of-magnitude scalings {#sec:oom}
---------------------------

Consider an accreting planet embedded in a viscous disc. From Figure \[fig:sig\_cartoon\] we identify three disc surface densities: $\Sigma_{\rm p}$ at the orbital radius of the planet ($r = r_{\rm p}$), $\Sigma_+$ exterior to the planet, and $\Sigma_-$ interior to the planet. The planet depresses the local surface density because it is both consuming disc gas and repelling disc gas away by Lindblad torques. Our goal is to estimate the depth of the planet’s gap in relation to the inner and outer discs: $\Sigma_{\rm p}/\Sigma_-$ and $\Sigma_{\rm p}/\Sigma_+$. We assume a steady state where the disc has viscously relaxed: given a viscosity $\nu$, the system age $t$ is at least as long as the diffusion time $r^2/\nu$ across the disc. In addition, $t$ is at most the planet growth timescale $M_{\rm p}/\dot{M}_{\rm p}$, so that we may consider the planet mass fixed at any given moment.

Mass flows steadily inward at rate $\dot{M}_+$ from the outer disc. Part of this flow is accreted by the planet at rate $\dot{M}_{\rm p}$, with the rest feeding the inner disc which accretes onto the star at rate $\dot{M}_-$. Dropping numerical pre-factors (these will be restored in later sections), we have $$\begin{aligned}
\dot{M}_+ &= \dot{M}_- + \dot{M}_{\rm p} \nonumber \\
\Sigma_+ \nu &\sim \Sigma_- \nu + \dot{M}_{\rm p} \nonumber \\
&\sim \Sigma_- \nu + A \Sigma_{\rm p} \,. \label{eqn:mass}\end{aligned}$$ There are a number of assumptions embedded in these order-of-magnitude statements. For $\dot{M}_+$ and $\dot{M}_-$ we have substituted standard steady-state expressions for a disc of shear viscosity $\nu$ (e.g., @fkr_2002), valid asymptotically at locations far from any mass sink ($|r-r_{\rm p}| \gtrsim r_{\rm p}$). At the same time, the locations we are considering in the outer and inner discs are not so far from the planet that we need to account for spatial variations in $\nu$, which may change by order-unity factors over length scale $r$.

For the planet’s accretion rate, we have assumed in (\[eqn:mass\]) that it scales linearly with the local surface density $\Sigma_{\rm p}$ with proportionality constant $A$: $$\label{eqn:Adef}
\dot{M}_{\rm p} = A \Sigma_{\rm p} \,.$$ This assumption is satisfied, e.g., by a planet accreting at the Bondi rate (e.g., @fkr_2002): $$\begin{aligned}
 \label{eq:m_dot_bondi_simp}
    \dot{M}_{\rm p, Bondi} &\sim \rho_{\rm p} c_{\rm s} R_{\rm B}^2 \nonumber \\
    &\sim \frac{\Sigma_{\rm p}}{H} c_{\rm s} \left( \frac{GM_{\rm p}}{c_{\rm s}^2} \right)^2  \end{aligned}$$ where $\rho_{\rm p}$ is the disc midplane mass density near the planet, $c_{\rm s}$ is the disc sound speed, $R_{\rm B} = GM_{\rm p}/c_{\rm s}^2$ is the Bondi radius, $H = c_{\rm s}/\Omega$ is the disc scale height, $\Omega$ is the orbital angular frequency, and $G$ is the gravitational constant. Then $$\label{eqn:A}
A_{\rm Bondi} \sim \frac{m^2}{h^4} \Omega r^2$$ where $m \equiv M_{\rm p}/M_\star$ is the planet-to-star mass ratio, and $h\equiv H/r$ is the disc aspect ratio. Ginzburg & Chiang ([-@gc_2019a], their section 1.1) discusses how Bondi accretion may be valid for “sub-thermal” planets whose masses are less than $$\label{eqn:thermalmass_0}
M_{\rm thermal} \sim h^3 M_\star$$ the mass for which the Bondi radius $R_{\rm B}$, the Hill radius $R_{\rm H} \sim m^{1/3} r$, and the disc scale height $H$ are all equal. A sub-thermal planet has $R_{\rm B} < R_{\rm H} < H$—its gravitational sphere of influence has radius $R_{\rm B}$, set by gravity and thermal pressure—and should accrete at the Bondi rate, isotropically from the all-surrounding disc (@gc_2019a; see also fig. 1 of @tt_2016 for evidence supporting the Bondi $m^2$ scaling, taken from the 3D simulations of @dkh_2003). For a super-thermal planet having $M>M_{\rm thermal}$, the hierarchy of length scales reverses so that $R_{\rm B} > R_{\rm H} > H$—the planet’s sphere of influence, now set by gravitational tides at radius $R_{\rm H}$, “pops out” of the disc—and arguably the planet accretes in a more 2D fashion, presenting a cross-section of order $R_{\rm H} H$ to disc gas that shears by at a velocity $\Omega R_{\rm H}$. The corresponding “Hill rate” for consumption is then $$\label{eqn:hillsimple}
\dot{M}_{\rm p,Hill} \sim \rho_{\rm p} \times R_{\rm H}H \times \Omega R_{\rm H} \sim \Sigma_{\rm p} R_{\rm H}^2 \Omega$$ whence $$A_{\rm Hill} \sim m^{2/3} \Omega r^2 \,.$$ A Hill-based scaling for consumption is commonly used in 2D disc-planet hydrodynamical simulations (e.g., @znh_2011 [@dk_2015; @dk_2017; @mfv_2019]). We have assumed in writing the above that the planet masses are large enough for accretion to be hydrodynamically-limited as opposed to cooling-limited (@gc_2019a, cf. their fig. 1).

In this paper we will calculate the growth of planets from sub-thermal to super-thermal masses, so will have occasion to use both $A_{\rm Bondi}$ and $A_{\rm Hill}$. We recognize that the 2D picture motivating our Hill scaling may not be correct; in 3D, meridional flows from gap walls can feed the planet along its poles [@smc_2014; @msc_2014; @fc_2016]. Relatedly, the disc density scales with height $z$ above the midplane as $\exp[-z^2/(2H^2)]$ (for an isothermal atmosphere), which implies that a considerable fraction of the disc mass resides between $|z|= H$ and $2H$; accordingly, the planet does not pop out of the disc until it is strongly super-thermal, i.e., until $m$ is a large multiple of $h^3$ (cf. equation \[eqn:thermalmass\_0\]). An isotropic version of super-thermal accretion controlled by the Hill sphere gives $\dot{M}_{\rm p,Hill,iso} \sim \rho_{\rm p} \times R_{\rm H}^2 \times \Omega R_{\rm H}$ or $A_{\rm Hill,iso} \sim m \Omega r^2 / h$. Yet another prescription for accretion is given by @tw_2002: $A_{\rm TW} \sim m^{4/3} \Omega r^2 / h^2$, an empirical relation based on their 2D numerical simulations (see also @tt_2016). To the extent that these alternative scalings increase with $m$ more steeply than our nominal $A_{\rm Hill} \propto m^{2/3}$, whatever final super-thermal planet masses we derive should be lower limits (see sections \[sec:proc\] and \[sec:sum\]).

![Sketch of the disc surface density and accretion flow in the vicinity of a planet. The planet is located at orbital radius $r_\mathrm{p}$, inside a gap having surface density $\Sigma_\mathrm{p}$. At $r > r_{\rm p}$, the disc surface density is $\Sigma_+$ and mass accretes inward at rate $\dot{M}_+$. Downstream of the planet, at $r < r_{\rm p}$, the corresponding surface density and accretion rate are $\Sigma_-$ and $\dot{M}_-$, respectively. The difference $\dot{M}_+ - \dot{M}_-$ is the accretion rate onto the planet $\dot{M}_{\rm p}$.[]{data-label="fig:sig_cartoon"}](Fig1_newA.pdf){width="\linewidth"}

Momentum conservation provides another relation between the surface densities. It is easiest to write down downstream of the planet in the accretion flow (in the inner disc), as the flow of momentum upstream (in the outer disc) is complicated by the mass sink presented by the planet. In the inner disc there are no sinks of mass or momentum, only a steady transmission of mass inward and angular momentum outward (assuming, as we do throughout this paper, a non-migrating planet; see section \[sec:sum\] for pointers to the migrating case). The rate at which angular momentum is carried viscously outward by the inner disc equals the viscous transport rate local to the planet, plus the repulsive Lindblad torque exerted by the planet on the disc:[^2] $$\begin{aligned}
\Sigma_- \nu \Omega r^2 \sim \Sigma_{\rm p} \nu \Omega r^2 + B \Sigma_{\rm p} \Omega r^2 
\label{eqn:momentum}\end{aligned}$$ with $$\label{eqn:B}
B \sim \frac{m^2}{h^3} \Omega r^2$$ given by the standard @gt_1980 linear Lindblad torque, integrating the effects of all Lindblad resonances up to the torque cutoff. A similar statement to (\[eqn:momentum\]), dropping the viscous term local to the planet, was made by @fsc_2014. Given $B$, (\[eqn:momentum\]) can be solved for the gap contrast with the inner disc: $$\begin{aligned}
\frac{\Sigma_{\rm p}}{\Sigma_-} \sim \frac{1}{1 + B/\nu} 
\label{eqn:oom1}\end{aligned}$$ (see also @dm_2013; @kmt_2015; @gs_2018). Combining mass conservation (\[eqn:mass\]) with momentum conservation (\[eqn:momentum\]) yields the gap contrast with the outer disc: $$\begin{aligned}
\frac{\Sigma_{\rm p}}{\Sigma_+} \sim \frac{1}{1 + (A + B)/\nu} \,.
\label{eqn:oom2}\end{aligned}$$ An equivalent equation is derived by @tt_2016 [their appendix B] and @tmt_2020 [their equation 26]. Equations (\[eqn:oom1\]) and (\[eqn:oom2\]) inform us that planetary consumption ($A \neq 0$) leads to asymmetric gap contrasts: a deeper gap relative to the outer disc than to the inner disc. The outer gap contrast is the more important insofar as the outer disc controls surface densities everywhere downstream; in other words, $\Sigma_+$ is the independent variable while $\Sigma_{\rm p}$ and $\Sigma_-$ are dependent variables. Equation (\[eqn:oom2\]) states that, given $\Sigma_+$, the effects of accretion ($A$) and repulsion ($B$) in setting the gap depth $\Sigma_{\rm p}$ are additive (not multiplicative). If $A > B$, then consumption dominates.

For $A=A_{\rm Bondi}$ and $B$ given by (\[eqn:B\]), $$\label{eqn:ab}
A_{\rm Bondi}/B \sim 1/h > 1$$ and consumption dominates repulsion in setting the gap depth, independent of planet mass in the sub-thermal regime. On the other hand, for $A=A_{\rm Hill}$, $$\label{eqn:ab_Hill}
A_{\rm Hill}/B \sim m^{-4/3} h^3 $$ which says that for super-thermal planets that are massive enough, repulsion dominates consumption ($A_{\rm Hill}/B < 1$).

We may also solve for the relative accretion rates: $$\begin{aligned}
\frac{\dot{M}_{\rm p}}{\dot{M}_+} &\sim \frac{A\Sigma_{\rm p}}{\Sigma_+\nu} \sim  \frac{A/\nu}{1+(A+B)/\nu} \label{eq:m_dot_ratio} \\
 \frac{\dot{M}_-}{\dot{M}_+} &\sim \frac{\Sigma_-}{\Sigma_+} \sim \frac{1+B/\nu}{1+(A+B)/\nu} \,. \label{eqn:mdot-0}\end{aligned}$$ A couple example limiting cases of (\[eq:m\_dot\_ratio\]) and (\[eqn:mdot-0\]) are as follows. If we take $A/B = A_{\rm Bondi}/B \sim 1/h > 1$ and further assume that $B/\nu > 1$ so that the inner gap contrast is significant (equation \[eqn:oom1\]), we find $$\begin{aligned}
    \frac{\dot{M}_{\rm p}}{\dot{M}_+} &\sim 1 - B/A_{\rm Bondi} \sim 1 - h \label{eqn:mdotp}\\
    \frac{\dot{M}_-}{\dot{M}_+} &\sim B/A_{\rm Bondi} \sim h \label{eqn:mdot-}\end{aligned}$$ which says that the planet consumes nearly all of the mass supplied to it by the outer disc, leaving behind a fraction $h$ to feed the inner disc. If instead we take $A/B = A_{\rm Hill}/B$ and further assume $B > A_{\rm Hill} > \nu$ (repulsion-limited and deep gap), then $$\begin{aligned}
    \frac{\dot{M}_{\rm p}}{\dot{M}_+} &\sim A_{\rm Hill}/B \sim m^{-4/3} h^3 < 1 \label{eqn:mdotp_Hill}\\
    \frac{\dot{M}_-}{\dot{M}_+} &\sim 1 - A_{\rm Hill}/B \sim 1 - m^{-4/3} h^3 \label{eqn:mdot-_Hill}\end{aligned}$$ and the planet diverts only a small fraction, $A_{\rm Hill}/B$, of the disc accretion flow onto itself.

The order-of-magnitude considerations presented here are firmed up in subsequent sections, including in Appendix \[sec:analy\_ss\], where we derive in greater analytic detail the surface density profile and mass accretion rates, drawing from @ld_2006.

Numerical simulations {#sec:num}
---------------------

### Procedure {#sec:proc}

We solve numerically for the 1D evolution of a viscously shearing disc [e.g., @fkr_2002] with a planetary mass sink. The governing equation for the surface density $\Sigma(r,t)$ in cylindrical radius $r$ and time $t$ reads $$\begin{aligned}
\pd{\Sigma}t=\frac{3}{r}\pd{}r\left[r^{1/2}\pd{}r\left(r^{1/2}\nu\Sigma\right)\right] - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p}) 
\label{eq:dsig_dt}\end{aligned}$$ where $\delta$ is the Dirac delta function and $r_{\rm p}$ is the radial position of the planet (held fixed). For the viscosity $\nu$ we employ the @ss_alpha $\alpha$-prescription: $$\label{eq:ss_alpha}
\nu = \alpha c_{\rm s}^2/\Omega = \alpha h^2 \Omega r^2$$ where $\Omega$ is the Keplerian orbital frequency around a $1 \, M_\odot$ star, $c_{\rm s} = \sqrt{k_{\rm B}T/\overline{m}}$, the disc temperature is $T = 200 \, {\rm K} (r/{\rm au})^{-1/2}$, $k_{\rm B}$ is Boltzmann’s constant, $\overline{m} = 2m_{\rm H}$ is the mean molecular mass, $m_{\rm H}$ is the mass of the hydrogen atom, and $$h \equiv H/r = c_{\rm s}/(\Omega r) \simeq 0.054 \left( \frac{r}{{\rm 10 \, au}} \right)^{1/4} \,.$$ We fix $\alpha = 10^{-3}$ for the results in this section. Given these inputs, $\nu = \nu(r) \propto r^1$.

Apart from the mass sink, equation (\[eq:dsig\_dt\]), which combines the 1D mass and momentum equations, is identical to the diffusion equation governing an isolated viscous disc as derived by @lp_1974. What is missing is an explicit accounting for the repulsive Lindblad torque exerted by the planet. Many studies include the planetary torque by introducing, into the momentum equation, a term for the torque per unit radius that scales as ${\rm sgn}(x)/x^4$, where $x \equiv r - r_{\rm p}$ (e.g., @lp_1986; @ld_2006). Compared against 2D hydrodynamical simulations, this $1/x^4$ prescription has been shown in 1D studies to reproduce the azimuthally averaged surface density profiles of repulsive gaps near their peripheries (at $x \gtrsim 4 H$) but not near gap centers (at $x \lesssim 4 H$; @fsc_2014, their section 4.3). In particular the 1D torque density prescription, which assumes angular momentum is deposited locally and neglects wave propagation, fails to recover the flat bottoms of gaps and the surface densities there (cf. @gs_2018 who use the @gr_2001 wave steepening theory to lift these assumptions). This shortcoming of the $1/x^4$ prescription means that it cannot be used to compute the planetary accretion rate $\dot{M}_{\rm p}$, which depends on knowing the gas density in the planet’s immediate vicinity.

What we do instead to include the repulsive Lindblad torque when calculating planetary accretion is as follows. Within the radial grid cell at $r = r_{\rm p}$ of width $\Delta r_{\rm p}$, the surface density is reduced after every timestep $\Delta t$ according to $$\begin{aligned}
 \label{eqn:euler}
\Sigma (r_{\rm p}, t+\Delta t) = \Sigma (r_{\rm p}, t) - \frac{\dot{M}_{\rm p} (t) \Delta t}{2 \pi r_{\rm p} \Delta r_{\rm p}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, ({\rm simulation})\end{aligned}$$ where the label “simulation” reminds us that this equation applies to the numerical simulation only and should not be used outside of that context. It is in evaluating $\dot{M}_{\rm p}$ that we include, in a “sub-grid” manner, the repulsive Lindblad gap: $$\begin{aligned}
 \label{eqn:subgrid}
\dot{M}_{\rm p}(t) &= A \times \frac{\Sigma (r_{\rm p},t)}{1+B/\nu} \,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \, \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,({\rm simulation}).\end{aligned}$$ What equation (\[eqn:subgrid\]) says is that the disc surface density the planet actually “sees” when consuming local gas is lower than the numerically computed “grid-level” surface density $\Sigma (r_{\rm p},t)$—lower by the Lindblad reduction factor $1/(1+B/\nu)$ (equation \[eqn:oom1\]). In other words, repulsion is encoded/enforced at a sub-grid level. We stress that equation (\[eqn:subgrid\]) is used only in our numerical simulation to capture repulsion and should not be used outside of it; contrast (\[eqn:subgrid\]) with, e.g., (\[eq:m\_dot\_ratio\]), and note that $\Sigma(r_{\rm p},t)$ is notation specific to the simulation and should not be confused with $\Sigma_{\rm p}$, the actual surface density at the planet’s position.

Our numerical procedure captures the gap depth but not the gap width, as the sub-grid modification is restricted (for simplicity) to the grid cell containing the planet. We consider this crude scheme acceptable insofar as we are more interested in the gross magnitudes for $\Sigma_{\rm p}/\Sigma_+$ and $\Sigma_{\rm p}/\Sigma_-$ and less interested in the precise surface density gradients. An untested assumption underlying our numerical procedure—and in our steady-state analytics—is that material flows radially through the gap at whatever velocity $u_r$ is needed to maintain continuity, i.e., to enforce $\dot{M}_- = \dot{M}_+ - \dot{M}_{\rm p} = -2\pi \Sigma_{\rm p} r_{\rm p} u_r$ (where $u_r < 0$ for accretion toward the star). We cannot test this assumption as we do not resolve the flow dynamics inside the gap. We will call out this assumption in the results to follow (sections \[sec:nc10au\] and \[sec:numwind\]). Also, as a reminder, we note that while the surface density changes as a result of consumption, in this subsection we fix $M_{\rm p}$, i.e., we do not update $M_{\rm p}$ using $\dot{M}_{\rm p}$ (this assumption is relaxed in section \[sec:mass\]).

In evaluating the consumption and repulsion coefficients $A$ and $B$, we make choices similar to those in our earlier order-of-magnitude analysis (section \[sec:oom\]), except that now we include numerical pre-factors for greater precision: $$\begin{aligned}
A_{\rm Bondi} &= 
0.5 \, \Omega r^2 \frac{m^2}{h^4}  && {\rm for} \, \operatorname{sub-thermal}\,\, m \leq 3 h^3  \label{eqn:bondi}\\
A_{\rm Hill} &= 2.2 \, \Omega r^2 m^{2/3} && {\rm for} \, \operatorname{super-thermal}\,\, m > 3 h^3 \label{eqn:hill}\\
B &= 0.04 \, \Omega r^2 \frac{m^2}{h^3}
\label{eqn:kanagawa}\end{aligned}$$ where all quantities are evaluated at $r_{\rm p}$. The pre-factor of $0.5$ in equation (\[eqn:bondi\]) is calibrated using 3D simulation results for $\dot{M}_{\rm p}$ from D’Angelo et al. ([-@dkh_2003]; these are re-printed in fig. 1 of @tt_2016). The coefficient of 2.2 in equation (\[eqn:hill\]) follows from requiring that (\[eqn:bondi\]) match (\[eqn:hill\]) at the thermal mass $$\label{eqn:thermalmass}
M_{\rm thermal} \equiv 3 h^3 M_\star \simeq 0.5 \left( \frac{h}{0.054} \right)^3 M_{\rm J}$$ defined by equating $H$ with $R_{\rm H} = (m/3)^{1/3} r$, with $M_{\rm J}$ the mass of Jupiter. Equation (\[eqn:kanagawa\]) is taken from the numerical 2D simulations of @kmt_2015 [see also @dm_2013 and @d_2015 who report similar results].

Note further that the expressions we used in section \[sec:oom\] for the steady disc accretion rates $\dot{M}_+$ and $\dot{M}_-$ should be amended with the numerical pre-factor $3\pi$, i.e., $\dot{M}_+ = 3\pi \Sigma_+ \nu$ and similarly for $\dot{M}_-$ (e.g., @fkr_2002). This correction is already embedded in the diffusion equation (\[eq:dsig\_dt\]). Including this pre-factor in equation (\[eqn:mass\]) implies that $A$ should be replaced with $A/(3\pi)$ in equations (\[eqn:oom2\])–(\[eqn:mdot-\_Hill\]). Putting it all together, we have $$\begin{aligned}
 \label{eq:A_B_visc}
    \frac{A_{\rm Bondi}}{3 \pi B} \simeq \frac{1.3}{h} > 1\end{aligned}$$ implying that consumption always dominates for sub-thermal masses. Furthermore, $$\begin{aligned}
 \label{eq:A_B_visc_Hill}
    \frac{A_{\rm Hill}}{3 \pi B} \simeq 5.5 m^{-4/3} h^3 \simeq 1.0 \left( \frac{m}{5 \times 10^{-3}} \right)^{-4/3} \left( \frac{h}{0.054} \right)^3\end{aligned}$$ implying that repulsion dominates for super-thermal masses exceeding a “repulsion mass”

$$\begin{aligned}
 \label{eqn:repulsionmass}
M_{\rm repulsion,visc} &\simeq 3.6 h^{9/4} M_\star \nonumber \\
& \simeq 5.3 M_{\rm J} \left( \frac{h}{0.054} \right)^{9/4} \simeq 5.3 M_{\rm J} \left( \frac{r}{10 \, {\rm au} } \right)^{9/16} \nonumber \\
&\simeq 1.2 h^{-3/4} M_{\rm thermal} \simeq 11 \left( \frac{0.054}{h} \right)^{3/4} M_{\rm thermal}  \,.\end{aligned}$$

For $M<M_{\rm repulsion,visc}$, consumption dominates and the planet accretes nearly all the disc gas that tries to diffuse past; for $M > M_{\rm repulsion,visc}$, repulsion dominates and the planet’s accretion rate falls below the disc accretion rate. The above expression for $M_{\rm repulsion, visc}$ depends on our assumption that planetary accretion follows our Hill scaling $A_{\rm Hill} \propto m^{2/3}$ for super-thermal masses. As discussed in section \[sec:oom\], this assumption might not be correct. If instead of $A_{\rm Hill}$ we use $A_{\rm TW} = 0.29 \,\Omega r^2 \,m^{4/3}/h^2$ as found from the 2D numerical simulations of @tw_2002, we would find $A_{\rm TW}/(3\pi B) \simeq 4 \,( M_\mathrm{p}/M_\mathrm{J} )^{-2/3} ( h/0.054)$, in which case the mass above which repulsion dominates would change to $M_{\rm repulsion,visc,TW} \simeq 9 \, M_{\rm J} \,[r / (10 \, {\rm au})]^{3/8}$. This is nearly twice the value of $M_{\rm repulsion,visc}$ given by (\[eqn:repulsionmass\]), and would imply a more extended consumption-dominated growth phase. Insofar as our nominal model adopts $A_{\rm Hill}$ which leads to a more limited consumption-dominated growth phase, the planet masses we compute for our viscous disc model are lower limits.

So far we have described how we compute the mass sink term, which includes the sub-grid Lindblad torque, in equation (\[eq:dsig\_dt\]). The remaining diffusive term is solved in a standard way. We first change variables to $z \equiv r^{1/2} \nu \Sigma$ and $y \equiv 2 r^{1/2}$ so that the diffusive portion of equation reads $$\begin{aligned}
 \label{eq:kep_simp}
\pd{z}t &= \frac{12 \nu}{y^2} \pd{^2 z}{y^2}\end{aligned}$$ with non-constant diffusion coefficient $12 \nu/y^2$. We solve equation as an initial value problem using an implicit scheme (e.g., @ptv_2007). Our computation grid extends from an inner boundary of $r_\mathrm{in} = 0.01 \, \mathrm{au}$ to an outer boundary of $r_\mathrm{out} = 500 \, \mathrm{au}$, and is divided into 300 cells that are uniform in $\Delta y$. We fix the timestep $\Delta t = 10^{-4} t_{\nu,{\rm p}}$, where $t_{\nu,{\rm p}} \equiv r_{\rm p}^2 / \nu (r_{\rm p}) \simeq 1.7$ Myr is the viscous diffusion timescale at the planet’s orbital radius of $r_{\rm p} = 10$ au (where $h \simeq 0.054$). Recognizing that our transformed variable $z$ is proportional to the viscous torque $2 \pi \nu \Sigma r^3  d \Omega / dr \propto r^{1/2} \nu \Sigma$, we use a torque-free inner boundary condition, $z (r_\mathrm{in}) = 0$, as would be the case if the disc were truncated by a co-rotating stellar magnetosphere (shearless boundary layer). At the outer boundary we assume the torque gradient $\partial z / \partial r \,(r_{\rm out}) = 0$. Neither boundary condition is critical as we are interested in the flow near the planet, away from either boundary.

The surface density of the disc is initialized with the similarity solution for an isolated viscous accretion disc with $\nu \propto r^1$ (@lp_1974; @hcg_1998): $$\begin{aligned}
 \label{eqn:sim}
    \Sigma(r,0) = \frac{M_{\rm disc}}{2 \pi r_1^2} \frac{r_1}{r}  e^{-r/r_1} \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, 
    ({\rm simulation})\end{aligned}$$ where $M_{\rm disc}  = 15.5 \, M_{\rm J} \simeq 0.015  \, M_\odot$ is the initial mass of the disc and $r_1 = 30$ au is a characteristic disc radius (where the diffusion time is $r_1^2/\nu \simeq 5$ Myr). We consider two fixed planet masses, $M_{\rm p} = 0.3 \,M_{\rm J} < M_{\rm thermal}$ and $M_{\rm p} = 10 \,M_{\rm J} > M_{\rm thermal}$. Planet masses that freely grow are modeled in section \[sec:mass\].

At every timestep, we first advance $\Sigma(r,t)\rightarrow \Sigma(r,t+\Delta t)$ for all $r$ according to (\[eq:kep\_simp\]) using the implicit solver, and then we advance $\Sigma(r_{\rm p},t)\rightarrow \Sigma(r_{\rm p},t+\Delta t)$ using (\[eqn:euler\]) and (\[eqn:subgrid\]). This procedure is repeated until the disc is evolved for several $t_{\nu,{\rm p}}$, long enough for the disc near the planet to achieve a quasi-steady state.

![How the surface density profile of a viscous disc responds to a planet that both consumes disc gas, and repels gas away by Lindblad torques. Surface densities are calculated from our 1D numerical simulation of a planet of fixed mass, either $M_{\rm p} = 0.3 \,M_{\rm J}$ (top panel) or $M_{\rm p} = 10\, M_{\rm J}$ (bottom panel), at $t = 3 t_{\nu,\mathrm{p}}$ when the disc near the planet at $r_{\rm p} = 10$ au has viscously relaxed. When computing the planetary accretion rate $\dot{M}_{\rm p}$, the gap is modeled as a single cell whose “true” surface density equals the grid-level $\Sigma$ lowered by a factor of $(1+B/\nu) \simeq B/\nu$; plotted here are the true sub-grid values $\Sigma_{\rm p}$. Accordingly, the planet’s gap is not spatially resolved and its width should not be taken literally from this figure. Red double-tipped arrows have lengths equal to their associated variables in dex, and demonstrate good agreement between numerics and analytics. The planet of mass $M_{\rm p} = 0.3 \,M_{\rm J}$, accreting at the Bondi rate, creates an asymmetric gap, with the inner disc surface density $\Sigma_-$ lower than the outer $\Sigma_+$ by $A_{\rm Bondi}/(3\pi B) > 1$; conditions are always consumption-dominated for Bondi accretion and $B$ as given by (\[eqn:kanagawa\]). The planet of mass $M_{\rm p} = 10\, M_{\rm J}$, accreting at the Hill rate, creates a symmetric gap where $\Sigma_-/\Sigma_+ \sim 1$; conditions here are repulsion-dominated as $M_{\rm p} > M_{\rm repulsion,visc}$ (equation \[eqn:repulsionmass\]).[]{data-label="fig:sigma_detailed"}](Fig2_newA.pdf){width="\linewidth"}

### Results {#sec:res222}

Figure \[fig:sigma\_detailed\] shows, for $M_{\rm p} = \{0.3, 10\} M_{\rm J}$, the numerically computed surface density profiles $\Sigma(r)$ at $t = 3 t_{\nu,{\rm p}} \simeq 5$ Myr. Overlaid for comparison is our numerical solution without a planet, which we have verified matches the analytic time-dependent similarity solution of @lp_1974. For the case with a planet, rather than plot at face value the numerically computed (grid-level) $\Sigma (r_{\rm p},t)$, we plot that value multiplied by the sub-grid reduction factor $1/(1+B/\nu)$—this is the “true” value for $\Sigma_{\rm p}$ that incorporates the repulsive Lindblad torque. Since this sub-grid correction factor is applied to only a single grid point, we cannot resolve gap widths; our focus instead is on the gross gap contrasts $\Sigma_{\rm p}/\Sigma_+$ and $\Sigma_{\rm p}/\Sigma_-$.

The surface density profiles shown in Figure \[fig:sigma\_detailed\] conform to the analytic considerations of section \[sec:oom\]. For $M_{\rm p} = 0.3 \, M_{\rm J}$ (top panel), conditions are consumption-limited: $\Sigma_+/\Sigma_{\rm p} \sim A_{\rm Bondi}/(3\pi\nu)$ (equation \[eqn:oom2\] in the limit $A_{\rm Bondi}/(3\pi) > B > \nu$) and the surface density of the entire interior disc is depressed relative to the same disc without a planet by a factor of $\Sigma_+ / \Sigma_- \sim \dot{M}_+ / \dot{M}_- \simeq A_{\rm Bondi}/(3\pi B)$ (equations \[eqn:mdot-0\] and \[eqn:mdot-\]). By comparison, for $M_{\rm p} = 10 \, M_{\rm J}$ (bottom panel), the gap is more nearly symmetric, $\Sigma_+/\Sigma_- \sim 1$ (equations \[eqn:mdot-0\] and \[eqn:mdot-\_Hill\]), and deep and repulsion-dominated, $\Sigma_+/\Sigma_{\rm p} \sim B/\nu$ (equation \[eqn:oom2\] in the limit $B > A_{\rm Hill}/(3\pi) > \nu$).

So long as consumption is stronger than repulsion in the sense that $A/(3\pi) > B$—a condition that we have shown always obtains for sub-thermal masses accreting at the Bondi rate, and for sufficiently low-mass super-thermal masses accreting at the Hill rate ($M < M_{\rm repulsion,visc}$)—repulsion does not much affect the gap surface density $\Sigma_{\rm p}$. Figure \[fig:m\_final\_visc\] demonstrates that different choices for the repulsion coefficient $B = \{10^{-2}, 10^{-3}, 10^{-4}\}\times A_{\rm Bondi}$ all yield practically the same $\Sigma_{\rm p}$ (when corrected to the true sub-grid value) relative to $\Sigma_+$. What repulsion, in combination with consumption, affects instead is how much gas leaks past the planet into the inner disc: the three different values for $B$ in Figure \[fig:m\_final\_visc\] yield three inner disc surface densities that, from equation (\[eqn:mdot-0\]), scale as $\Sigma_-/\Sigma_+ \simeq (1+B/\nu)/[1 + A_{\rm Bondi}/(3\pi \nu)]$. This factor scales as $3\pi B/A_{\rm Bondi}$ when $A_{\rm Bondi}/(3\pi) > B > \nu$ (dot-dashed and dotted lines), and as $1/[1 + A_{\rm Bondi}/(3\pi\nu)]$ when $B < \nu$ (solid line; in this limit repulsion has no effect).

![Same as Figure \[fig:sigma\_detailed\] for the case $M_{\rm p} = 0.3 M_{\rm J}$, but for different choices of $B$ scaled to $A_{\rm Bondi}$. As long as $A_{\rm Bondi}/(3 \pi) > B$, the planet’s gap is consumption-dominated and its surface density $\Sigma_\mathrm{p}$ is independent of the repulsion coefficient $B$. The depression of the inner disc relative to the outer disc is, however, sensitive to $B$ for $B > \nu$; $\Sigma_-/\Sigma_+ \simeq (1+B/\nu)/[A_{\rm Bondi}/(3\pi\nu)]$. []{data-label="fig:m_final_visc"}](Fig3_newA.pdf){width="\linewidth"}

Viscous discs: Gas Giant Growth {#sec:mass}
===============================

Numerical calculation at $r_{\rm p} = 10$ au {#sec:nc10au}
--------------------------------------------

We now relax the assumption that the planet mass remains fixed, and at every timestep update $M_{\rm p}$ according to $\dot{M}_{\rm p}$ computed using equation (\[eqn:subgrid\]). Our numerical procedure is unchanged from section \[sec:num\] except that we initialize the planet mass at $M_{\rm p}(0) = 0.1 \, M_{\rm J}$ and allow it to grow. For our nominal disc parameters ($\alpha = 10^{-3}$, $h = 0.054$ at $r_{\rm p} = 10$ au), a starting planet mass of $0.1 M_{\rm J}$ ($m \simeq 0.95 \times 10^{-4}$) implies that, initially, $A = A_{\rm Bondi}$, $A_{\rm Bondi}/(3\pi B) \simeq 1.3/h \simeq 24$ (a consumption-dominated gap), $A_{\rm Bondi}/(3\pi \nu) \simeq 19$ (a strong outer gap contrast), and $B/\nu \simeq 0.79$ (a weak inner gap contrast).

Figure \[fig:sigma\_mdot\_M0\_10\_MJ\] shows two snapshots in time of $\Sigma(r)$ and the disc mass flow rate $\dot{M}_{\rm disc}(r) = -2 \pi \Sigma r u_r$, where $$\begin{aligned}
    u_r = -\frac{3}{\Sigma r^{1/2}} \frac{\partial}{\partial r} \left(\nu \Sigma r^{1/2} \right)\end{aligned}$$ is the gas radial velocity (e.g., @fkr_2002) evaluated numerically from our solution for $\Sigma$ (omitting the single-point discontinuity at $r = r_{\rm p}$). Note that $\dot{M}_{\rm disc} > 0$ indicates inward mass transport, toward the star. The planet accretes predominantly from the outer disc, notwithstanding a small contribution from the inner disc before the disc has viscously relaxed; this early-time contribution can be seen at $t = 0.3\,t_{\nu,{\rm p}}$ when $\dot{M}_{\rm disc} < 0$ from $r \sim 3$ au to the planet’s orbit. The behaviour of $\dot{M}_{\rm disc}$ at $r \sim 100$ au is characteristic of a viscous disc near its turn-around “transition radius” (@lp_1974; @hcg_1998), outside of which the disc has not yet viscously relaxed; this outermost disc behaviour is not caused by the planet.

Embedded in Figure \[fig:sigma\_mdot\_M0\_10\_MJ\] is our assumption, first mentioned in section \[sec:proc\], that the disc flow inside the gap maintains continuity. At $t = 3 t_{\nu,{\rm p}}$, $\dot{M}_{\rm disc} (r > r_{\rm p})$ is, to within a factor of 2, the same as $\dot{M}_{\rm disc} (r < r_{\rm p})$. Because the gap surface density $\Sigma_{\rm p}$ at this time is about 4 orders of magnitude smaller than the surface densities $\Sigma_+$ and $\Sigma_-$ outside the gap, the radial velocity $|u_r|$ within the gap must be 4 orders of magnitude larger than the radial velocities outside, to maintain the near-constancy of $\dot{M}_{\rm disc}$ across $r_{\rm p}$. Since the radial accretion velocities away from the gap are of order $r/t_\nu \sim \nu/r \sim \alpha h c_{\rm s} \sim 2$ cm/s, we must have $|u_r| \sim 0.2$ km/s within the gap. How such a radial velocity is achieved is not specified by our model, which does not resolve the gap spatially.

Figure \[fig:Mp\_vs\_t\_visc\] displays the planet’s mass as a function of time. We identify a consumption-dominated phase during which the planet grows from 0.1 to $5 \,M_{\rm J}$ ($M < M_{\rm repulsion,visc}$; equation \[eqn:repulsionmass\]) and a slower repulsion-dominated phase between 5 and $8 \,M_{\rm J}$ ($M > M_{\rm repulsion,visc}$). During the first phase, accretion starts at the Bondi rate and switches to the Hill rate once $M_{\rm p} > M_{\rm thermal} \simeq 0.5 \,M_{\rm J}$ (equations \[eqn:thermalmass\] and \[eqn:bondi\]–\[eqn:hill\]). A consumption-dominated ($A/(3\pi) > B$) and deep ($A/(3\pi) > \nu$) gap implies from (\[eq:m\_dot\_ratio\]) that $\dot{M}_{\rm p} \simeq \dot{M}_+$, i.e., the planet’s accretion rate is about as large as it can be. During the final repulsion-limited phase, when $M_{\rm p} > 5 \,M_{\rm J}$ and $A_{\rm Hill}/(3\pi) > B$, consumption slows and the planet undergoes a last near-doubling in mass as the remainder of the disc diffuses away, onto the star.

Analytic estimates of the final planet mass
-------------------------------------------

We can compare our numerical result for the final mass at $r_{\rm p} = 10$ au to the following analytic estimates, derived by neglecting the initial short-lived Bondi accretion phase and assuming that at all times the planet accretes at the Hill rate ($A=A_{\rm Hill}$) and has a large inner gap contrast ($B >\nu$): $$\begin{aligned}
\dot{m} &= \frac{A_{\rm Hill} \Sigma_{\rm p}}{M_\star} \nonumber \\
&= \frac{A_{\rm Hill}}{M_\star} \frac{\Sigma_+ \nu}{A_{\rm Hill}/(3\pi) + B}\end{aligned}$$ where we have used (\[eqn:oom2\]). At small orbital distances, final planet masses exceed $M_{\rm repulsion,visc}$ and so their final growth phase is repulsion-limited: $$\begin{aligned}
\label{eqn:viscmdot}
\dot{m} & = \frac{A_{\rm Hill}}{B} \frac{\Sigma_+ \nu}{M_\star} \nonumber \\
    & = 55 \alpha h^5  m^{-4/3} \frac{\Sigma_+ r_{\rm p}^2}{M_\star} \Omega \,.\end{aligned}$$ We approximate $\Sigma_+$ using the similarity solution for an isolated viscous disc with no planet and $\nu \propto r^1$: $$\begin{aligned}
 \label{eqn:simtime}
    \Sigma_+ \sim \frac{M_{\rm disc}}{2\pi r_1^2} \left(\frac{r_1}{r_{\rm p}}\right) T^{-3/2} e^{ -(r_{\rm p}/r_1)/T}\end{aligned}$$ where $T \equiv 1 + t/t_1$, $t_1 \equiv r_1^2/[3 \nu(r_1)]$, and $M_{\rm disc}$ is the initial disc mass (@lp_1974; @hcg_1998). Integrating equation (\[eqn:viscmdot\]) from $t=0$ to $t$ gives $$\begin{aligned}
 \label{eqn:fullsolvisc}
    m(t) \sim & \left(\frac{385}{18\sqrt{\pi}}   \frac{M_{\rm disc}}{M_\star} \frac{h^5}{h_1^2}  \frac{r_1}{r_{\rm p}} \right)^{3/7} \nonumber \\   &\times  \left[\text{Erf}\left(\sqrt{\frac{r_{\rm p}}{r_1}}\right)-\text{Erf}\left(\sqrt{\frac{r_{\rm p} t_1}{r_1 (t + t_1)}}\right)\right]^{3/7} (\operatorname{repulsion-limited})\end{aligned}$$ where $h_1$ is the disc aspect ratio at $r_1$. As $t \rightarrow \infty$, equation (\[eqn:fullsolvisc\]) simplifies to $$\begin{aligned}
 \label{eqn:asymsolvisc}
    m_\mathrm{final,visc} \sim& \left[\frac{385}{18\sqrt{\pi}}   \frac{M_{\rm disc}}{M_\star} \frac{h^5}{h_1^2}  \frac{r_1}{r_{\rm p}}
    \text{Erf}\left(\sqrt{\frac{r_{\rm p}}{r_1}}\right)
    \right]^{3/7} (\operatorname{repulsion-limited})
    \end{aligned}$$ which further simplifies in the limit $r_{\rm p} \ll r_1$ (away from the initial disc outer edge) to $$\begin{aligned}
\label{eqn:solviscsimple}
    M_{\mathrm{final,visc}} &\sim 10 \, M_\mathrm{J}  \, \left( \frac{M_{\rm disc}}{15.5 \,M_{\rm J}} \right)^{3/7} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{9/28} \,\, (\operatorname{repulsion-limited})\end{aligned}$$ for our fiducial parameters. Note that $M_{\rm final,visc}$ in these limits is independent of $\alpha$ and $r_1$. Equation (\[eqn:solviscsimple\]) may be reproduced to order-of-magnitude by multiplying $\dot{m}$ (evaluated at $t = t_1$) by $t_1$. In Figure \[fig:Mp\_vs\_t\_visc\] we plot equation (\[eqn:asymsolvisc\]) as the uppermost horizontal dashed line, labeled $M_{\rm final,visc}$.

At the largest orbital distances, conditions tend to remain consumption-limited as $M_{\rm p}$ stays below $M_{\rm repulsion,visc}$. Then the planet accretes nearly all of the disc gas that tries to diffuse past the planet—and diffusion can be in the outward direction ($\dot{M}_{\rm disc} < 0$) if the planet is located near or beyond the disc’s turn-around radius. Accordingly we estimate the planet mass as $$\begin{aligned}
M_{\rm p}(t) \sim \int^t_0 |\dot{M}_{\rm disc}| dt \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\operatorname{consumption-limited}) \end{aligned}$$ where $\dot{M}_{\rm disc}$ is approximated by the no-planet similarity solution (equation 35 of @hcg_1998). For $r_{\rm p}\leq r_1/2$, $$\begin{aligned}
 \label{eqn:consume1}
M_{\rm p}(t) &\sim M_{\rm disc} \left( e^{-r_{\rm p}/r_1} - \frac{e^{-(r_{\rm p}/r_1)/T}}{\sqrt{T}} \right) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\operatorname{consumption-limited})\end{aligned}$$ and for $r_{\rm p}>r_1/2$, $$\begin{aligned}
 \label{eqn:consume2}
M_{\rm p}(t) \sim M_{\rm disc} \left( \sqrt{\frac{2r_1}{r_{\rm p}}} e^{-1/2} - e^{-r_{\rm p}/r_1} - \frac{e^{-(r_{\rm p}/r_1)/T}}{\sqrt{T}} \right) \nonumber \\
(\operatorname{consumption-limited}).\end{aligned}$$ We will make use of equations (\[eqn:fullsolvisc\]), (\[eqn:consume1\]), and (\[eqn:consume2\]) in section \[sec:sum\] when we discuss, in the context of observations, how the final planet mass depends on disc mass and orbital distance.

![Snapshots of the surface density profile $\Sigma(r)$ and disc accretion rate $\dot{M}_{\rm disc}(r) = -2 \pi \Sigma u_r r$ ($> 0$ for accretion toward the star) for a planet embedded at $r_{\rm p} = 10$ au in a viscous $\alpha = 10^{-3}$ disc. The planet mass is allowed to freely grow starting from $M_{\rm p}(0) = 0.1 M_{\rm J}$. At $t = 0.3 t_{\nu,{\rm p}}$, the planet resides in a consumption-dominated, asymmetric gap (top panel, dashed curve) and accretes from regions both exterior and interior to its orbit which have not yet viscously relaxed (bottom panel, dashed curve). At the later time $t = 3 t_{\nu,{\rm p}}$, the planet has grown sufficiently (see also Figure \[fig:Mp\_vs\_t\_visc\]) that its gap is now repulsion-dominated and more symmetric (top panel, solid curve); the planet now accretes only from the outer disc, reducing the flow of mass into the inner disc by less than a factor of 2 (bottom panel, solid curve. At this time we have multiplied $\dot{M}_{\rm disc}$ by a factor of 5 for easier viewing).[]{data-label="fig:sigma_mdot_M0_10_MJ"}](Fig4_newA_v3.pdf){width="\linewidth"}

![Accretion history of a planet of initial mass $M_{\rm p}(0) = 0.1 M_{\rm J}$ embedded at $r_{\rm p} = 10$ au (where $h = 0.054$) in a viscous disc of initial mass $M_{\rm disc} = 15.5 M_{\rm J}$. Transitions from Bondi accretion to Hill accretion ($M_{\rm thermal}$, equations \[eqn:bondi\]–\[eqn:hill\] and \[eqn:thermalmass\]), and from consumption to repulsion-dominated gaps ($M_{\rm repulsion,visc}$, equation \[eqn:repulsionmass\]), are indicated. An analytic estimate of the final planet mass is plotted as $M_{\rm final,visc}$ (equation \[eqn:asymsolvisc\]), computed assuming repulsion-dominated conditions (at $r_{\rm p} = 10$ au for this disc mass, conditions are actually intermediate between the repulsion and consumption limits, and so plotting equation \[eqn:consume1\] which assumes consumption-dominated conditions would give a similar result as equation \[eqn:asymsolvisc\]; see also Figure \[fig:m\_final\_r\_visc\]).[]{data-label="fig:Mp_vs_t_visc"}](Fig5_loglog_3lines.pdf){width="\linewidth"}

Planets in Inviscid Wind-Driven Discs {#sec:windy}
=====================================

Motivated by recent ALMA observations that point to little or no turbulence in protoplanetary discs (e.g., @p_etal_2016; @f_etal_2017), and by theoretical work arguing that discs are, for the most part, laminar because they are too cold and dusty to support magnetorotational turbulence (e.g., @g_1996; @pc_2011; @b_2011), we here turn away from the $\alpha$-based picture of turbulent and diffusive discs, and consider instead inviscid (zero viscosity) discs that accrete by virtue of magnetized winds (e.g., @b_etal_2016; @b_2016). We review how wind-driven discs work in section \[sec:wind\] and how planets open repulsive gaps in inviscid discs in section \[sec:inv\]. We then study how repulsion combines with consumption to set gap depths and planetary accretion rates, analytically in section \[sec:invoom\] and numerically in section \[sec:numwind\].

Wind-driven accretion discs {#sec:wind}
---------------------------

Inviscid, wind-driven accretion discs do not behave diffusively. Instead they are governed by simple advection: at every radius $r$, material moves inward with a vertically-averaged radial speed $u_r$ because it has lost angular momentum to a magnetized wind. The mass carried away by the wind itself is small compared to the mass advected inward through the disc (see Appendix \[sec:winds\], in particular the discussion below equation \[eqn:lambda\]). Then from continuity, including our planetary mass sink, $$\label{eqn:cont2}
\frac{\partial \Sigma}{\partial t} = \frac{1}{ r}  \frac{\partial}{\partial r} \left( \Sigma r u_r \right) - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p}) \,.$$ In Appendix \[sec:winds\] we show how a wind-driven disc inspired by @b_etal_2016 and @b_2016 can have $u_r$ approximately constant ($<0$ for accretion). We utilize here, for simplicity, a constant $u_r \equiv c < 0$ model: $$\label{eqn:cont3}
\frac{\partial \Sigma}{\partial t} = \frac{c}{r} \frac{\partial}{\partial r} (\Sigma r) - \frac{\dot{M}_{\rm p}(t)}{2\pi r} \delta (r - r_{\rm p}) \,.$$

It is instructive to examine the solution to (\[eqn:cont3\]) when $\dot{M}_{\rm p}=0$. The no-planet solution is separable: $$\label{eqn:sol}
\Sigma(r,t) = f(r) g(t) = \frac{M_{\rm disc}}{2\pi (ct_{\rm adv})^2} \frac{|c|t_{\rm adv}}{r} e^{-r/(|c|t_{\rm adv})} e^{-t/t_{\rm adv}}$$ for constants $M_{\rm disc}$ (the initial disc mass) and $t_{\rm adv}$, which we interpret as a disc radial advection time or drain-out time. For $c = -4$ cm/s (a value we relate to magnetic field parameters in Appendix \[sec:winds\]) and $t_{\rm adv} = 3$ Myr, the characteristic disc size is $|c|t_{\rm adv} \simeq 25$ AU, which seems reasonable. Equation (\[eqn:sol\]) resembles the @lp_1974 solution for a viscous disc which gives, for $\nu \propto r^1$, a surface density profile that scales as $r^{-1} \exp(-r/r_1)$ at fixed $t$ (equation \[eqn:simtime\]). This spatial resemblance is not surprising, as our viscous disc happens also to have an accretion velocity that is constant with radius: $|u_r| \sim r/t_\nu \sim \nu/r =$ constant. However, the solutions differ in their time behaviours; at fixed $r$, the wind-driven surface density decays exponentially as $\exp(-t/t_{\rm adv})$, whereas our viscous disc decays as a power law $t^{-3/2}$ (within viscously relaxed regions at small radii; @lp_1974 [@hcg_1998]). Viscous discs evolve more slowly because they conserve their total angular momentum; they can only drain away on the inside by redistributing their angular momentum to the outside in a kind of zero-sum game. Wind-driven discs are not so constrained; they lose their angular momentum wholesale to a wind, and so can dissipate more quickly.

We emphasize that $u_r = c$ is a vertically averaged, mass weighted, radial accretion velocity. In simulations by @bs_2013 of discs whose magneto-thermal winds are anchored at their electrically conductive surfaces, accretion actually occurs in a vertically thin, rarefied layer several scale heights above the midplane. The radial accretion velocity in this high-altitude layer is fast, on the order of the sound speed $c_{\rm s}$. The bulk of the mass of the disc, below this layer, is inert (see fig. 10 of @bs_2013). It is with this static and inviscid gas, extending from the midplane to a couple scale heights above and below, that the planet interacts, as we now describe.

Repulsion in inviscid discs {#sec:inv}
---------------------------

Without viscosity, disc gas in the vicinity of the planet depletes indefinitely, as it is repelled by the planetary Lindblad torque but cannot diffuse back. Under these conditions, @gc_2019a derived how the gas density at the center of the planet’s gap scales with elapsed time $t$, for a given planet-to-star mass ratio $m = M_{\rm p}/M_\star$ and disc aspect ratio $h = H/r$ (see the inviscid branch of their equation 17, and also their appendix): $$\begin{aligned}
 \label{eqn:inv}
\frac{\Sigma_{\rm p}}{\Sigma_-} \sim h^{549/49} m^{-4} (\Omega t)^{-39/49} \equiv \widetilde{B}_{\rm inv}^{-1}\end{aligned}$$ where $\Omega$ is the orbital frequency of the planet, $\Sigma_{\rm p}$ is the surface density within the gap, and $\Sigma_-$ is the surface density downstream of the planet in the accretion flow (see Figure \[fig:sig\_cartoon\]). By construction, $t$ is the time over which the planet’s mass is close to its given value $m$ (say within a factor of 2). In practice, for inviscid discs where gaps deepen dramatically with increasing planet mass, the mass doubling time of a planet lengthens with each doubling, so $t$ is of order the system age.

Equation (\[eqn:inv\]) does not apply when $\widetilde{B}_{\rm inv} < 1$, i.e., when a repulsive gap has not yet been opened because not enough time has elapsed for a given planet mass. To account for this possibility, we generalize (\[eqn:inv\]) using $$\label{eqn:invgen}
\frac{\Sigma_{\rm p}}{\Sigma_-} \sim \frac{1}{1+\widetilde{B}_{\rm inv}}$$ by analogy with equation (\[eqn:oom1\]) for the viscous case. Note that $\widetilde{B}_{\rm inv}$ is dimensionless while its viscous counterpart $B$ has dimensions of viscosity.

Consumption and repulsion combined {#sec:invoom}
----------------------------------

We now assemble the physical ingredients laid out in sections \[sec:wind\] and \[sec:inv\] into a sketch of how consumption and repulsion combine in an inviscid, wind-driven disc. Following by analogy our analysis in section \[sec:oom\] for a viscous disc, we first write down mass conservation (see equation \[eqn:mass\] and Figure \[fig:sig\_cartoon\]): $$\begin{aligned}
 \label{eqn:massinv}
\dot{M}_+ &= \dot{M}_- + \dot{M}_{\rm p} \nonumber \\
2\pi \Sigma_+ r |c| &= 2\pi \Sigma_- r |c| + \dot{M}_{\rm p} \nonumber \\
&= 2\pi \Sigma_- r |c| + A \Sigma_{\rm p}  \end{aligned}$$ where in lieu of the viscosity we now have $r |c|$. After replacing $\Sigma_-$ in (\[eqn:massinv\]) using our momentum relation (\[eqn:invgen\]), we have $$\begin{aligned}
\Sigma_+  \sim (1+\widetilde{B}_{\rm inv})\Sigma_{\rm p} + \frac{A}{2\pi r|c|} \Sigma_{\rm p}\end{aligned}$$ which implies the outer gap contrast $$\begin{aligned}
 \label{eqn:outer2}
\frac{\Sigma_{\rm p}}{\Sigma_+} \sim \frac{1}{1+A/(2\pi r|c|) + \widetilde{B}_{\rm inv}} \,.\end{aligned}$$ As in the viscous case (equation \[eqn:oom2\]), we see here that consumption ($A/(2\pi r|c|)$) and repulsion ($\widetilde{B}_{\rm inv}$) add. Taking $A$ to be the Bondi value (equation \[eqn:bondi\]) gives the ratio $$\begin{aligned}
\label{eqn:sensitive}
\frac{A_{\rm Bondi}/(2\pi r|c|)}{\widetilde{B}_{\rm inv}} \sim &  \,\frac{0.5}{2\pi} \frac{h^{353/49}}{m^{2} (\Omega t)^{39/49}} \frac{\Omega r_{\rm p}}{|c|} \nonumber \\
\sim & \,0.04 \left( \frac{M_{\rm p}}{0.1 \, M_{\rm J}} \right)^{-2} \left( \frac{t}{3 \, {\rm Myr}}  \right)^{-39/49} \times \nonumber \\
 &  \left( \frac{|c|}{4 \, {\rm cm/s}} \right)^{-1} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{489/196}\end{aligned}$$ which informs us that repulsion dominates consumption ($\widetilde{B}_{\rm inv} > A_{\rm Bondi}/(2\pi r|c|)$) when $$\begin{aligned}
M_{\rm p} > M_{\rm repulsion,inv} \sim & \, 0.02 \,M_{\rm J} \left( \frac{t}{3 \, {\rm Myr}}  \right)^{-39/98} \times \nonumber \\
 &  \left( \frac{|c|}{4 \, {\rm cm/s}} \right)^{-1/2} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{489/392} \,.\end{aligned}$$ That repulsion dominates consumption even for small masses is in contrast to the viscous case (see equation \[eqn:repulsionmass\] for $M_{\rm repulsion,visc}$). Repulsion-dominated gaps are symmetric between the inner and outer discs (equations \[eqn:invgen\] and \[eqn:outer2\]): $$\begin{aligned}
\Sigma_{\rm p}/\Sigma_-  & \sim   \Sigma_{\rm p}/\Sigma_+ \sim  1/(1+\widetilde{B}_{\rm inv})  \nonumber \\
& \sim  2 \times 10^{-3} \left( \frac{h}{0.054} \right)^{549/49} \left( \frac{10^{-4}}{m} \right)^4 \left( \frac{3 \, {\rm Myr}}{t} \right)^{39/49}  \label{eqn:symmetric}\end{aligned}$$ where for the last equality we have assumed that the gaps are deep ($\widetilde{B}_{\rm inv} > 1$). Under these conditions, we may estimate a final accreted planet mass by time-integrating $$\begin{aligned}
\dot{M}_{\rm p} &= A_{\rm Bondi} \Sigma_{\rm p} \nonumber \\
&\sim A_{\rm Bondi} \frac{\Sigma_+}{\widetilde{B}_{\rm inv}} \nonumber \\
& \sim \frac{A_{\rm Bondi}}{\widetilde{B}_{\rm inv}} \frac{M_{\rm disc}}{2\pi (ct_{\rm adv})^2} \frac{|c|t_{\rm adv}}{r_{\rm p}} e^{-r_{\rm p}/(|c|t_{\rm adv})} e^{-t/t_{\rm adv}} \label{eqn:int}\end{aligned}$$ from $t=0$ to $\infty$, where for $\Sigma_+$ we have employed the no-planet solution (\[eqn:sol\]). This last approximation is analogous to the one we made in (\[eqn:simtime\]) for a viscous disc. Equation (\[eqn:int\]) integrates to yield $$\begin{aligned}
 \label{eqn:m_final_wind_full}
M_\mathrm{final,inv} & \sim  \left[\frac{1.5}{2\pi} \, \Gamma\left(\frac{10}{49}\right) \left(\frac{M_{\rm disc}}{M_\star} \right) \left(\frac{r_{\rm p}}{ |c|t_{\rm adv}} \right)  \right. \nonumber \\ &  \times \left. h^{353/49}(\Omega t_{\rm adv})^{10/49}   e^{-r_{\rm p}/(|c|t_{\rm adv})} \right]^{1/3} M_\star \nonumber \\
& \sim  0.3 \, M_{\rm J} \left( \frac{M_{\rm disc}}{15.5 \,M_{\rm J}} \right)^{1/3} \left( \frac{r_{\rm p}}{10 \, {\rm au}} \right)^{163/196} e^{-r_{\rm p}/(3|c|t_{\rm adv})}\nonumber \\
& (\operatorname{repulsion-limited})\end{aligned}$$ where $\Gamma$ is the gamma function, and the numerical evaluation uses our fiducial parameters including $|c| = 4 $ cm/s, $t_{\rm adv} = 3$ Myr, and $M_\star = 1 M_\odot$. Our estimated final mass of $0.3 \, M_{\rm J}$ at $r_{\rm p} = 10$ au remains smaller than $M_{\rm thermal} \simeq 0.5 M_{\rm J}$ and so our use of $A_{\rm Bondi}$ is self-consistent.

Our expression (\[eqn:m\_final\_wind\_full\]) for $M_{\rm final,inv}$ resembles equation (19) of @gc_2019a; ours is an improvement as we have accounted explicitly for the transport properties of the disc through the radial velocity $c$ (see the discussion of transport-limited accretion in their section 4.1).

Numerical simulations {#sec:numwind}
---------------------

We test the ideas in section \[sec:invoom\] by numerically solving the continuity equation (\[eqn:cont3\]) and the momentum equation (\[eqn:invgen\]). To model the planetary mass sink in equation (\[eqn:cont3\]), we utilize the same sub-grid procedure of section \[sec:num\], replacing equation (\[eqn:subgrid\]) with $$\begin{aligned}
 \label{eqn:m_dot_win}
\dot{M}_{\rm p}(t) = A \times \frac{\Sigma(r_{\rm p},t)}{1+ \widetilde{B}_{\rm inv}} \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, \,\,\,\, 
({\rm simulation})\end{aligned}$$ where $\Sigma(r_{\rm p},t)$ is the grid-level surface density in the bin containing the planet, and $A$ and $\widetilde{B}_{\rm inv}$ are given by equations (\[eqn:bondi\])–(\[eqn:hill\]) and (\[eqn:inv\]), respectively. The initial mass of the planet is set to $M_{\rm p}(0) = 0.1 \, M_{\rm J}$ (we will see that using smaller initial masses hardly changes the outcome). We solve the advective portion of equation (\[eqn:cont3\]) with a first-order upwind scheme (e.g., @ptv_2007) applied to a grid that extends from $r_{\rm in} = 0.01$ au to $r_{\rm out} = 500$ au across 300 cells uniformly spaced in $\log r$. We fix $c = -4$ cm/s and initialize the grid using (\[eqn:sol\]), with $t_{\rm adv} = 3$ Myr and $M_{\rm disc} = 15.5 M_{\rm J} = 0.015 M_\odot$, the same value chosen for our viscous disc calculations. Our timestep is set to $\Delta t = 0.2 \Delta r_{\mathrm{min}}/|c|$, where $\Delta r_\mathrm{min} = 3 \times 10^{-3} \, \mathrm{au}$ is our smallest bin width. Other disc properties such as $h(r)$ and $\Omega(r)$ are the same as before. For the outer boundary condition we impose a ghost cell just outside $r_{\rm out}$ where the surface density is fixed at 0.

Figure \[fig:m\_final\_wind\_tdisc\_a\] (the inviscid counterpart to Figure \[fig:sigma\_detailed\]) shows $\Sigma(r)$ at $t = t_{\rm adv}$ when $M_{\rm p}$ has grown to $0.3 \, M_{\rm J}$, illustrating many of the features anticipated from our analytic treatment. Without a planet, the surface density profile follows $r^{-1} \exp[-r/(|c|t_{\rm adv})]$ as expected from equation (\[eqn:sol\]). With a planet, a gap is created that is nearly symmetric between the inner and outer discs, and whose depth is dominated by Lindblad repulsion (enforced by our sub-grid scheme), not consumption (equation \[eqn:symmetric\]). The inviscid gap is deep (scaling as $m^{-4}$; @gs_2018; @gc_2019a; see also @d_2020). Figure \[fig:m\_final\_wind\_tdisc\_b\] (analogous to Figure \[fig:sigma\_mdot\_M0\_10\_MJ\]) provides snapshots of $\Sigma(r)$ and $\dot{M}_{\rm disc}(r)$ taken at different times, and Figure \[fig:sigma\_mdot\_M0\_wind\] (analogous to Figure \[fig:Mp\_vs\_t\_visc\]) plots $M_{\rm p}(t)$. Unlike in a viscous disc, our example planet in an inviscid disc does not consume most of the disc mass exterior to its orbit; the disc accretion rate profile $\dot{M}_{\rm disc}(r)$ is not much affected by the planet except during an initial transient phase at $t < t_{\rm adv}$. We see a need for a high radial accretion velocity $|u_r|$ within the gap (see also section \[sec:nc10au\]): to ensure that $\dot{M}_{\rm disc}$ grades smoothly across the gap as shown in Figure \[fig:m\_final\_wind\_tdisc\_b\], $|u_r|$ must increase in proportion to the gap contrast $\Sigma/\Sigma_{\rm p}$. Inviscid gap contrasts are on the order of $10^5$, and so $|u_r| \sim 10^5 |c| \sim 4$ km/s, comparable to the orbital velocity. Note that simulations of planets in inviscid discs have not reproduced the deep gaps expected from our analytics, finding gap contrasts only up to a factor of $\sim$10 (e.g., @fc_2017 [@mnp_2019; @mnp_2020]). On the one hand the simulations are of limited duration and so their gaps may not have fully developed; on the other hand, the simulations allow for orbital migration and hydrodynamical instabilities, effects which may prevent gaps from becoming too deep in reality.

That the disc accretion flow proceeds largely unimpeded from outside to inside the planet’s orbit is a consequence of the gap being repulsion-dominated (equation \[eqn:mdot-0\], with $\nu$ replaced by $r|c|$). The planet diverts such a small fraction of the disc flow that it grows from $0.1 \, M_{\rm J}$ to only $0.3 \, M_{\rm J}$; most of the original $15.5 \, M_{\rm J}$ contained in the disc drains onto the star. Figure \[fig:sigma\_mdot\_M0\_wind\] also shows that reducing the initial seed mass to $M_{\rm p}(0) = 0.01 M_{\rm J}$ hardly affect the final mass.

![How the surface density profile of an inviscid disc responds to a planet that consumes disc gas and repels gas away by Lindblad torques. The planet, located at $r_{\rm p} = 10$ au, freely accretes starting from a seed mass of $0.1 M_{\rm J}$; the $\Sigma$ profile shown here is taken at a time $t = t_{\rm adv} = 3$ Myr, when the planet has grown to $\sim$$0.3 M_{\rm J}$ (see also Figure \[fig:sigma\_mdot\_M0\_wind\]). As is the case throughout this paper, the planet’s gap is not spatially resolved, but is modeled as a single cell. The “true” surface density inside this cell equals the grid-level $\Sigma$ lowered by a factor of $\widetilde{B}_{\rm inv}$, whose magnitude is given by the red double-tipped arrow. The gap is repulsion and not consumption dominated ($\widetilde{B}_{\rm inv} > A/(2\pi r|c|)$, equation \[eqn:sensitive\]); as such, the gap is symmetric in the sense that the surface density contrast with the outer disc is practically the same as with the inner disc. This figure is the inviscid counterpart to Figure \[fig:sigma\_detailed\] which was made for a viscous disc.[]{data-label="fig:m_final_wind_tdisc_a"}](Fig6_newA.pdf){width="\linewidth"}

![Snapshots of the surface density profile $\Sigma(r)$ and disc accretion rate $\dot{M}_{\rm disc}(r) = -2 \pi \Sigma u_r r$ ($> 0$ for accretion toward the star) for a planet embedded in an inviscid, wind-driven disc. The planet mass is allowed to freely grow starting from $M_{\rm p}(0) = 0.1 \,M_{\rm J}$; the masses corresponding to the plotted times are $0.27\, M_{\rm J}$ ($t = 0.1\, t_{\rm adv} = 0.3$ Myr) and $0.34 \,M_{\rm J}$ ($t = 3 \,t_{\rm adv} = 9$ Myr; see also Figure \[fig:sigma\_mdot\_M0\_wind\]). At $t=3\,t_{\rm adv}$, the disc has relaxed into a quasi-steady state in the presence of the planetary mass sink, and $\dot{M}_{\rm disc}(r)$ looks essentially the same as it would without the planet; the accretion rate onto the planet is negligible compared to the disc accretion rate—the gap is repulsion-dominated—and so the disc is not materially affected. Even at $t = 0.1\,t_{\rm adv}$, the interior surface density $\Sigma_-$ and $\dot{M}_{\rm disc}$ depress by only $\sim$15% because of consumption.[]{data-label="fig:m_final_wind_tdisc_b"}](Fig7_newA.pdf){width="\linewidth"}

![Mass evolution of a planet embedded at $r_{\rm p} = 10$ au in an inviscid but still accreting disc of initial mass $M_{\rm disc} = 15.5 \,M_{\rm J}$. Within $\sim$1 disc advection time $t_{\rm adv}$, the planet, whose gap is repulsion-dominated ($\widetilde{B}_{\rm inv} > A_{\rm Bondi}/(2\pi r_{\rm p}|c|)$), grows to a mass of $\sim$$0.35 \, M_{\rm J}$. The final planet mass varies by only $\sim$10% when the initial seed mass $M_{\rm p}(0)$ varies by a factor of 10. This figure is the inviscid counterpart to Figure \[fig:Mp\_vs\_t\_visc\] which was made for a viscous disc.[]{data-label="fig:sigma_mdot_M0_wind"}](Fig8_newA.pdf){width="\linewidth"}

Summary and Discussion {#sec:sum}
======================

Planets open gaps in circumstellar discs in two ways: by repelling material away via Lindblad torques, and by consuming local disc gas. Measured relative to the disc outside the planet’s orbit, the two effects are additive: both repulsion and consumption add to deepen the planet’s gap relative to the outer disc (see equation \[eqn:oom2\] or \[eqn:outer2\]). Relative to the inner disc, downstream of the mass sink presented by the planet, the gap surface density contrast is set by repulsion only (see equation \[eqn:oom1\] or \[eqn:invgen\]).

Many planet formation studies (e.g., @tt_2016; @lee_2019) take the planet’s hydrodynamically-limited accretion rate $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc})$, where $\dot{M}_{\rm hydro}$ is the planetary accretion rate computed according to the hydrodynamics of flows in the immediate vicinity of the planet, and $\dot{M}_{\rm disc}$ is the local disc accretion rate (the mass crossing the planet’s orbital radius, per time). Prescribing the planet’s accretion rate in this way is equivalent to comparing consumption, as measured by the “consumption coefficient” $A \equiv \dot{M}_{\rm p}/\Sigma_{\rm p}$, where $\Sigma_{\rm p}$ is the surface density inside the gap, and repulsion, as measured by the “repulsion coefficient” $B \equiv T/(\Sigma_{\rm p} \Omega r^2)$, where $T$ is the repulsive planetary torque and $\Omega r^2$ is the angular momentum per unit mass (see also @tt_2016 and @tmt_2020 who use the same framework). Under consumption-limited conditions ($A/(3\pi) > B$), the planet’s accretion rate saturates to nearly the disc’s accretion rate: $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc}) = \dot{M}_{\rm disc}$. Otherwise, under repulsion-limited conditions ($A/(3\pi) < B$), $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc}) = \dot{M}_{\rm hydro}$.

Final planet masses {#sec:final}
-------------------

In conventional viscous discs with large enough $\alpha$-diffusivities[^3] and our assumed parameters, planets begin their growth under consumption-dominated conditions and possibly continue their growth under repulsion-dominated conditions, arriving at final masses well in excess of a Jupiter. We show in Figure \[fig:m\_final\_r\_visc\] the final mass of a planet embedded in an $\alpha = 10^{-3}$ disc, as a function of the planet’s orbital distance $r_{\rm p}$, computed using our numerical code of sections \[sec:consumption\]–\[sec:mass\]. Final planet masses increase gradually from $4 \, M_{\rm J}$ at 1 au, to $8\,M_{\rm J}$ at 30 au, in a disc of initial mass $M_{\rm disc} = 15.5 M_{\rm J}= 0.015 M_\odot $. In a disc $5\times$ more massive, the corresponding range of planet masses is 9–$20 \, M_{\rm J}$. The final masses are not sensitive to $\alpha$ insofar as $\alpha$ controls only the timescale over which the disc evolves (modulo disc dispersal by some other means, e.g., photoevaporation; see @tmt_2020). Final masses do depend on the initial mass of the disc, scaling as $M_{\rm disc}^{3/7}$ under repulsion-dominated conditions (equation \[eqn:asymsolvisc\]) and $M_{\rm disc}^1$ under consumption-dominated conditions (equation \[eqn:consume1\] or \[eqn:consume2\]). The trend of final planet mass with distance shown in Figure \[fig:m\_final\_r\_visc\] follows, for the most part, the trend predicted for repulsion-limited conditions, except at large $r_{\rm p}$ where consumption dominates. The final mass profiles in Figure \[fig:m\_final\_r\_visc\] recall those of the super-Jupiters in the HR 8799 system; the four planets, located between 15 and 70 AU of their host star, have practically the same mass, about 6–7 $M_{\rm J}$ (@wgd_2018).

![Final planet masses grown from viscous discs having $\alpha =
10^{-3}$ and varying total mass (top vs. bottom panels). Planet masses are initialized at $0.1 \,M_{\rm J}$ and grown using the 1D numerical code of section \[sec:mass\], which utilizes the repulsive gap contrast of Kanagawa et al. ([-@kmt_2015]; see also @dm_2013 and @fsc_2014) and gas accretion that switches from Bondi to Hill at the thermal mass. Points are plotted at $t = 50\,t_1 = 85$ Myr, where $t_{\rm 1} = r_1^2/[3\nu(r_1)]$ is the viscous diffusion time at $r_1 = 30$ au. Analytic curves are given by equation (\[eqn:fullsolvisc\]) for the repulsion limit (dashed blue), and equations (\[eqn:consume1\])–(\[eqn:consume2\]) for the consumption limit (dotted orange), also evaluated at $t = 50 t_1$. At most orbital distances, planet mass growth is limited by repulsion-dominated gaps; only at the largest distances, where the disc aspect ratio is large, are gaps relatively harder to open and conditions remain consumption-limited. The analytics, which are derived assuming the planet mass is small compared to the disc mass, are a better guide for the more massive disc in the bottom panel.[]{data-label="fig:m_final_r_visc"}](Fig9_two_panel_50_ts_v2.pdf){width="\linewidth"}

Initially and everywhere in a viscous disc, a planet, despite opening a gap, consumes practically all of the disc gas that tries to diffuse past its orbit (equation \[eq:m\_dot\_ratio\] with $A/(3\pi) > B > \nu$, where $\nu$ is the disc viscosity). This consumption-limited behaviour persists up to a repulsion mass $M_{\rm repulsion,visc}
\simeq 5 \, M_{\rm J} \,[r_{\rm p}/(10 \, {\rm au})]^{9/16}$ (equation \[eqn:repulsionmass\]), above which repulsion dominates. The repulsion mass is not the thermal mass $M_{\rm thermal}$ (equation \[eqn:thermalmass\]), but exceeds it by a factor of $\sim$$h^{-3/4}$, where $h$ is the disc aspect ratio. Growth continues more slowly at $M_{\rm p} > M_{\rm repulsion,visc}$, with the planet mass increasing beyond $M_{\rm repulsion,visc}$ by up to a factor of $\sim$4 for our parameter choices.

Equation (\[eqn:fullsolvisc\]) gives an approximate analytic expression for the planet mass vs. time during this final repulsion-limited stage. It predicts that planet masses are of order $10 \,M_{\rm J}$ by the time the disc dissipates. This result is derived by assuming the planet accretes at a rate that scales as $A_{\rm Hill} = 2.2 \, \Omega r^2 \, m^{2/3}$, where $m$ is the planet-to-star mass ratio; this prescription is commonly adopted by hydrodynamical simulations of planet-disc interactions, and might be appropriate for super-thermal masses. If instead of $A_{\rm Hill}$ we use the empirical formula $A_{\rm TW} = 0.29 \, \Omega r^2 m^{4/3}/h^2$ drawn from 2D numerical simulations by @tw_2002, then the mass above which repulsion dominates changes to $M_{\rm repulsion,visc,TW} \simeq 9 \, M_{\rm J} \,[r / (10 \, {\rm au})]^{3/8}$, nearly twice the value of $M_{\rm repulsion,visc}$ derived using the Hill scaling. Using $A_{\rm TW}$ leads to a more extended consumption-dominated growth phase, and final planet masses larger by order-unity factors compared to those of the solid curves in Figure \[fig:m\_final\_r\_visc\]. Overall, it appears that in viscous discs, planets accrete a not-small fraction of the disc mass, which can be many tens of Jupiter masses (@tab_2017, their fig. 10; see also @pmp_2019). This is in agreement with @tmt_2020, who limit giant planet growth by incorporating photoevaporative mass loss from the disc.

In inviscid discs, conditions tend to be repulsion-dominated even at low planet masses. Without viscosity or turbulent transport to compete against, planetary Lindblad torques carve deep gaps that are repulsion-dominated even for sub-thermal planets accreting at the Bondi rate (equation \[eqn:sensitive\]). Repulsion-dominated gaps are symmetric in the sense that gap contrasts between the outer and inner discs are the same; accordingly, disc accretion rates are nearly continuous across the gap (e.g., Figure \[fig:m\_final\_wind\_tdisc\_b\]), which means that most of the disc mass is not diverted onto the planet (in the language of @tt_2016, $\dot{M}_{\rm p} = \min (\dot{M}_{\rm hydro}, \dot{M}_{\rm disc}) = \dot{M}_{\rm hydro}$). Maintaining the disc accretion rate across a gap demands that the radial accretion velocity within the gap be as large as the gap is deep. Whether such fast inflows are possible, and whether inviscid gaps can be as deep as expected from our analytics (cf. numerical simulations that find only shallow gaps; @fc_2017 [@mnp_2019; @mnp_2020]), are unresolved issues.

Figure \[fig:m\_final\_r\_wind\], analogous to Figure \[fig:m\_final\_r\_visc\], shows that final planet masses in our model inviscid discs range between $\sim$0.05 and $1\, M_{\rm J}$, more than an order-of-magnitude smaller than their viscous disc counterparts. For the most part, the masses computed for inviscid discs using our numerical 1D code are well reproduced by equation (\[eqn:m\_final\_wind\_full\]), derived in the repulsion limit. This formula, which predicts that final planet masses scale as $M_{\rm disc}^{1/3}$ and $r_{\rm p}^{163/196} \simeq r_{\rm p}^{0.83}$, is similar to that derived by Ginzburg & Chiang ([-@gc_2019a], their equation 19),[^4] and improves upon it by accounting for the structure and transport properties of the parent disc—specifically how the disc may accrete by shedding angular momentum through a magnetized surface wind (e.g., @b_2016).

Orbital migration in viscous discs has been shown in numerical simulations to enhance $\dot{M}_{\rm p}$ relative to the migration-free case [e.g., @dk_2017]. Including migration would only amplify our finding that final planet masses in viscous discs are large, approaching if not well within the regime of brown dwarfs. Accretion rates should also increase for planets migrating in inviscid, wind-driven discs;in 3D, strongly sub-thermal planets have been shown to migrate inward (@mnp_2020). We may need such enhancements in $\dot{M}_{\rm p}$ to explain, within an inviscid scenario, giant planets like our own Jupiter, i.e., to bring planet masses up to $1 M_{\rm J}$ at distances of 1–10 au (Figure \[fig:m\_final\_r\_wind\]). On the other hand, sub-Jupiter masses, down to $\sim$0.1 $M_{\rm J}$ in many cases, are inferred from ALMA observations of disc gaps (@zzh_2018), and suggest that planets there are strongly repelling inviscid gas.

The asymmetric gap we computed for the viscous disc model in Figure \[fig:sigma\_detailed\] suggests a strong, mostly one-sided migration torque forcing the planet inward. However, this is misleading because our numerical procedure does not spatially resolve the gap, whose true radial width lies between $H$ (the pressure scale height) and $r_{\rm p}$ (@gs_2018). Most of the migration torque is exerted by disc gas on the bottoms of gaps, displaced radially from the planet by $\sim$$\pm H$, and here the actual surface density gradients, and of course the surface density itself, are small (see also @kts_2018).

![Final planet masses in an inviscid, wind-driven disc of varying mass (top vs. bottom panels). Planet masses are initialized at $0.01 M_{\rm J}$ and grown using the 1D numerical code of section \[sec:numwind\], which uses the time-dependent gap contrast of @gc_2019a to model repulsion, in a purely advective disc whose height-averaged radial accretion velocity is $c = -4$ cm/s and exponential drain-out time is $t_{\rm adv} = 3$ Myr. Points are plotted after $5 t_{\rm adv} = 15$ Myr. They mostly respect equation (\[eqn:m\_final\_wind\_full\]), which gives final planet masses grown in repulsion-limited and deep ($\widetilde{B}_{\rm inv}$ &gt; 1) gaps (dashed curve not including the drop-off at the largest distances). At $r_{\rm p} \sim 100$ au, the disc has such low density that the planet’s initial growth timescale $M_\mathrm{p}/\dot{M}_\mathrm{p}$ is comparable to $t_\mathrm{adv}$; here there are not many doublings before the disc drains away. In this regime the planet does not open a substantial gap ($\widetilde{B}_\mathrm{inv} < 1$) and its final mass can be estimated analytically by integrating $\dot{M}_\mathrm{p} = A_\mathrm{Bondi} \Sigma$ with $\Sigma$ given by the no-planet solution (\[eqn:sol\]); the dashed curve is the minimum of the resulting expression (not shown) and . []{data-label="fig:m_final_r_wind"}](Fig10_newA.pdf){width="\linewidth"}

Transitional discs {#sec:transitional}
------------------

We have shown how a planet accreting from its parent disc can change the disc’s entire complexion. This make-over is most evident for a planet that siphons away most of the disc’s accretion flow—as it can in a viscously diffusing disc—carving out a consumption-limited gap that divides a gas-rich outer disc with surface density $\Sigma_+$ from a gas-poor inner one with surface density $\Sigma_-$. Transitional discs have just such an outer/inner structure (e.g., @emj_2014; @dvh_2017), suggesting that they represent viscous discs whose inner regions are cleared by accreting planets (with dust filtration at the outer gap edge, and grain growth in the inner disc, enhancing the surface density contrast in dust over gas; @drz_2012; @znd_2012).

In a viscous disc, a single accreting planet suffices to deplete the entire disc interior to its orbit. The 2D single-planet simulations of @znh_2011 bear this out; they find an outer vs. inner disc contrast of $\Sigma_+/\Sigma_- \sim 10$ for a $1 \, M_{\rm J}$ planet that accretes at the Hill rate from a disc of $h \simeq 0.05$ (their fig. 1, model P1A1). This numerical result agrees with our analytic theory, which predicts according to equations (\[eqn:mdot-0\]) and (\[eq:A\_B\_visc\_Hill\]) that $$\begin{aligned}
\frac{\Sigma_+}{\Sigma_-} \simeq \frac{A_{\rm Hill}}{3\pi B} \simeq 7 \left( \frac{M_{\rm p}}{M_{\rm J}} \right)^{-4/3} \left( \frac{h}{0.05} \right)^3\end{aligned}$$ for a consumption-dominated and deep gap with $A_{\rm Hill}/(3\pi) > B > \nu$, where $\nu$ is the disc viscosity. In steady state, $\Sigma_+/\Sigma_- = \dot{M}_+/\dot{M}_-$, the ratio of outer-to-inner disc accretion rates. A value of $\dot{M}_+/\dot{M}_- \sim 10$, as we have found for the above parameters, accords with the observation that the median accretion rate for stars hosting transitional discs is lower than that of stars hosting non-transitional discs by a factor of $\sim$10 (@nsm_2007; @kwm_2013). However, the corresponding factor-of-10 reduction in $\Sigma$ seems too small to match observed gas depletions in transitional disc cavities. In the disc studied in CO by @dvh_2017, the gas surface density declines by $\sim$$10^3$ from $r = 70$ au to 15 au. As recognized by Zhu et al. ([-@znh_2011]; see also @o_2016), it is a challenge to simultaneously explain how disc inner cavities can be strongly depleted in density while their central stars continue to accrete at near-normal rates. This challenge seems more easily met in the repulsion limit, where deep gaps are carved by planets which alter the disc accretion flow only modestly—assuming radial accretion velocities within the gap are large enough to maintain mass transport rates across it.

The repulsion limit is attained in viscous discs by planets having $M > M_{\rm repulsion,visc} \simeq 5.3 \, M_{\rm J}\, [r/(10\,{\rm au})]^{9/16}$, or in inviscid discs by planets having $M > M_{\rm repulsion,inv} \simeq 0.02 \,M_{\rm J}
[r/(10 \,{\rm au})]^{489/392}$. In both cases, multiple planets with adjoining gaps would be required to evacuate transition disc cavities spanning decades in radius—more planets in a viscous scenario where each gap has a radial width closer to $H$, and fewer in an inviscid scenario where each gap is of order $r_{\rm p} > H$ wide (@gs_2018; note that widths are not captured by our single-grid-point treatment of gaps). The inviscid picture requires only super-Earth masses and appeals more, insofar as observations seem to have already ruled out transitional discs containing families of super-Jupiters as required in the viscous scenario. Inviscid discs can still accrete, either by virtue of magnetized winds (@b_2016; @wg_2017), or by the repulsive torques of their embedded planets (@gr_2001; @sg_2004; @fc_2017).

Acknowledgements {#acknowledgements .unnumbered}
================

We thank Xuening Bai, Jeffrey Fung, Willy Kley, Eve Lee, James Owen, and Hidekazu Tanaka for helpful exchanges. An anonymous referee provided an encouraging report. EC acknowledges NASA grants 80NSSC19K0506 and NNX15AD95G/NEXSS. SG is supported by the Heising-Simons Foundation through a 51 Pegasi b Fellowship. MMR and RMC acknowledge support from NSF CAREER grant number AST-1555385.

Data availability {#data-availability .unnumbered}
=================

The code underlying this article will be shared on reasonable request to the corresponding author.

Analytic Steady-State Solution for $\Sigma$ and $\dot{M}$ For Viscous Disc with Planet  {#sec:analy_ss}
=======================================================================================

In this appendix we provide an analytic expression for the surface density profile of a disc with an embedded planet. Our derivation here is more careful than our order-of-magnitude sketch in section \[sec:oom\], and similar to that presented in Lubow & D’Angelo ([-@ld_2006], their section 2.4), with a couple of differences: we reduce the surface density at the planet’s location by a factor $1 + B/\nu$ to account for repulsive Lindblad torques (see section \[sec:oom\]), and we express our solution in terms of the surface density at infinity as opposed to the surface density at the planet’s location.

Using the same notation as in section \[sec:oom\], and neglecting for the moment the Lindblad torque, the equations of mass and angular momentum conservation with a mass sink at $r = r_\mathrm{p}$ read $$\begin{aligned}
\frac{1}{r}\frac{d\left( \mu u_r r / \nu\right)}{dr} & =-\frac{\dot{M}_\mathrm{p}}{2\pi r}\delta\left(r-r_\mathrm{p}\right) \label{eq:mass_cons_ss}\\
\frac{r^{2}\Omega \mu u_r}{\nu} & \label{eq:ang_cons_ss} =-\frac{d}{dr}\left(3\mu\Omega r^{2}\right)\end{aligned}$$ where $u_r$ is the radial velocity and $\mu \equiv \Sigma \nu$. Equation indicates that the mass flow rate $\dot{M}_+ = -2\pi \mu_+ u_r r/\nu$ is spatially constant in regions exterior to the planet’s orbit (the outer disc), and likewise for $\dot{M}_-$ in regions interior to the planet’s orbit (the inner disc): $$\begin{aligned}
 \label{eq:m_dot_cons}
    \dot{M}_- = \dot{M}_+ - A \frac{\mu_\mathrm{p}}{\nu_\mathrm{p}}\end{aligned}$$ where we have used $\dot{M}_{\rm p} = A \mu_{\rm p}/\nu_{\rm p}$ and $\nu_\mathrm{p} \equiv \nu \left( r_\mathrm{p} \right)$. Since $\dot{M}_-$ and $\dot{M}_+$ are constants, equation can be solved to yield $$\begin{aligned}
    3 \pi \mu_\pm(r) = \dot{M_\pm} + \frac{C_\pm}{\sqrt{r}}\end{aligned}$$ where $C_\pm$ are integration constants. For the inner disc we use the boundary condition $\mu_-(r_\star) = 0$, whence $$\begin{aligned}
 \label{eq:mu_sol}
3\pi\mu_{-}\left(r\right)=\dot{M}_{-}\left(1-\sqrt{\frac{r_\star}{r}}\right) \,.\end{aligned}$$ Following our treatment in the main text, we encode the planetary gap caused by Lindblad torques at a sub-grid level, i.e., we force the surface density at the planet’s location to be depleted relative to the surface density just interior to the planet according to $$\begin{aligned}
 \label{eq:inner_mu_p}
\mu_{\rm p} = \mu_-(r_{\rm p}) \left( 1 + B/\nu_{\rm p} \right)^{-1}\end{aligned}$$ where subscript $p$ denotes the planet’s location. For the outer disc, we fix the surface density at infinity, $\mu(\infty) = \mu_\infty$, so that $$\dot{M}_+ = 3 \pi \mu_\infty \,.$$ Then from equations (\[eq:m\_dot\_cons\]), (\[eq:mu\_sol\]), and (\[eq:inner\_mu\_p\]) we have $$\begin{aligned}
 \label{eq:ss_mup}
\frac{\mu_\mathrm{p}}{\mu_\infty}=\frac{1}{\frac{1}{1-\sqrt{r_\star/r_\mathrm{p}}} + \left( \frac{A}{3 \pi} + \frac{B}{1-\sqrt{r_\star/r_\mathrm{p}}}\right)/\nu_\mathrm{p}}\end{aligned}$$ which can be compared to equation (\[eqn:oom2\]). We may also solve for $$\begin{aligned}
    \frac{\dot{M}_\mathrm{p}}{\dot{M}_+} &= \frac{A/(3\pi \nu_\mathrm{p})}{\frac{1}{1-\sqrt{r_\star/r_\mathrm{p}}} + \left(\frac{A}{3 \pi} + \frac{B}{1-\sqrt{r_\star/r_\mathrm{p}}}\right)/\nu_\mathrm{p}}\\
    \frac{\dot{M}_-}{\dot{M}_+} &= \frac{1 + B/\nu_\mathrm{p}}{1 + \left[ \frac{A}{3 \pi} \left(1-\sqrt{\frac{r_\star}{r_\mathrm{p}}}\right) + B \right]/\nu_\mathrm{p}}\end{aligned}$$ which can be compared to equations (\[eq:m\_dot\_ratio\]) and (\[eqn:mdot-0\]). Finally, stitching the outer disc solution to the inner disc solution implies $\mu_+(r_{\rm p}) = \mu_-(r_{\rm p}) = \mu_{\rm p} (1+B/\nu_{\rm p})$ and $$\begin{aligned}
 \label{eq:plus}
    3 \pi \mu_+(r) = 3\pi \mu_\infty \left[1 - \sqrt{\frac{r_\mathrm{p}}{r}} \left(1-\frac{\mu_\mathrm{p}\left( 1 + B/\nu_\mathrm{p} \right)}{\mu_\infty} \right) \right] \,.\end{aligned}$$ The equations above mirror the results in section \[sec:oom\], with the addition of a factor of $3 \pi$ (see section \[sec:num\]) and the factor of $1-\sqrt{r_\star/r_\mathrm{p}}$ which accounts for the star’s ability to divert material from the planet.

In Figure \[fig:sig\_analy\_sol\] we plot equations (\[eq:mu\_sol\]), (\[eq:inner\_mu\_p\]) and (\[eq:plus\]), adopting parameters as close as possible to those used in the top panel of Figure \[fig:sigma\_detailed\] so that we may compare the numerical result there to the analytic result here (see caption to Figure \[fig:sig\_analy\_sol\] for details).

![Analytic solution (black solid curve) for the surface density profile of a viscous disc perturbed by a planet, as given by equations (\[eq:mu\_sol\]), (\[eq:inner\_mu\_p\]) and (\[eq:plus\]), using parameters as close as possible to those used in the top panel of Figure \[fig:sigma\_detailed\], whose numerical result is overlaid here for comparison (orange dashed curve). For our analytic unperturbed “no planet” disc (blue dashed curve) we use a power law of slope -1 and normalization at 1 au equal to the corresponding “no planet” curve in Figure \[fig:sigma\_detailed\]. The $A$ and $B$ coefficients are taken from equations and for $M_\mathrm{p} = 0.3 \,M_\mathrm{J}$. The differences between the analytic and numeric curves mainly arise from the behaviour of the outermost disc near the turn-around “transition radius” (@lp_1974; @hcg_1998). This transition radius, which varies with time, does not appear in our steady-state solution.[]{data-label="fig:sig_analy_sol"}](ss_cons_sol_v4.pdf){width="\linewidth"}

Magnetized winds and disc accretion {#sec:winds}
===================================

We motivate here our simple, constant accretion velocity model for a wind-driven disc using the numerical simulations of Bai and collaborators. From continuity (equations 1, 6, and 9 of @b_2016), $$\begin{aligned}
 \label{continuity}
\frac{\partial \Sigma}{\partial t} &= +\frac{1}{2\pi r} \frac{\partial 
  \dot{M}_{\rm disc}} {\partial r} - \frac{1}{2\pi r} \frac{\partial 
  \dot{M}_{\rm wind}}{\partial r} \nonumber \\
&= + \frac{1}{2\pi r}
\frac{\partial}{\partial r} \left[ 2(\lambda-1) r \frac{\partial \dot{M}_{\rm wind}}{\partial r} \right] 
 - \frac{1}{2\pi r} \frac{\partial 
  \dot{M}_{\rm wind}}{\partial r} \end{aligned}$$ where $$\begin{aligned}
\dot{M}_{\rm wind} (r) = \int^r \frac{\partial \dot{M}_{\rm
  wind}}{\partial r} dr\end{aligned}$$ is the cumulative rate at which mass is carried to infinity by the wind (integrated over the disc within $r$). From equation (20) of @b_etal_2016, $$\label{eqn:rhov}
\frac{\partial \dot{M}_{\rm wind}}{\partial  r} = 2\pi r \rho_0 u_{\rm p0}$$ where $\rho_0$ and $u_{\rm p0}$ are the volumetric mass density and poloidal velocity of the wind where it is launched, near the disc surface. All quantities subscripted with 0 are evaluated at the wind base $(r_0,z_0)$.

The disc accretion rate $$\label{eqn:usual}
\dot{M}_{\rm disc} \equiv -2\pi \Sigma r u_r = 2(\lambda-1) r \frac{\partial \dot{M}_{\rm wind}}{\partial r}$$ for surface density $\Sigma$ and radial velocity $u_r$ is identical in definition to the variable $\dot{M}_{\rm disc}$ used throughout our paper. Unlike $\dot{M}_{\rm wind}$, $\dot{M}_{\rm disc}$ is not a cumulative quantity, but measures the mass crossing a circle of radius $r$ per unit time, and uses a sign convention such that $\dot{M}_{\rm disc} > 0$ for $u_r <0$.

Disc accretion by a wind hinges on the “magnetic lever arm” $$\lambda = (r_{\rm A}/r_0)^2 \label{eqn:lambda}$$ where $r_{\rm A}$ is the Alfvén radius for the wind streamline running through $r_0$. A lever arm $\lambda > 1$ enables $\dot{M}_{\rm disc} > 0$ by having the wind carry away more specific angular momentum than the Keplerian disc has at $r_0$. The fiducial wind model of Bai ([-@b_2016], their fig. 2) has $(\lambda-1)$ ranging from $\sim$30 at $r = 0.3$ AU to $\sim$2 at 30 AU; therefore the first term in (\[continuity\]) dominates the second term by a factor of order $2(\lambda-1) \sim 4$–60. Only the first term is modeled in our paper.

@b_2016 and the magnetized disc wind literature dating back to @bp_1982 parameterize the wind mass-loss rate in terms of the dimensionless mass loading parameter $$\mu = \frac{\omega r_0}{B_{\rm p0}} \times k = \frac{\omega
  r_0}{B_{\rm p0}} \times \frac{4\pi \rho u_{\rm p}}{B_{\rm p}}$$ where $k$ is the ratio of poloidal mass flux to poloidal field strength $B_{\rm p}$ ($k$ is constant along a magnetic field line), and $\omega$ is the angular velocity of a field line, approximately equal to the Keplerian frequency $\Omega_{\rm K}$ at $r_0$. Note that $\mu$ (not to be confused with $\mu$ in Appendix \[sec:analy\_ss\]) varies with $r$ from field line to field line. Evaluating $\mu$ at the wind base, we rewrite (\[eqn:rhov\]) as $$\frac{\partial \dot{M}_{\rm wind}}{\partial r} = \frac{\mu B_{\rm p0}^2}{2\omega}$$ (@b_etal_2016, equation 21). Now parameterize $B_{\rm p0}$ in terms of the midplane plasma beta: $$\beta_0 = \frac{8\pi}{B_{\rm p0}^2} \frac{\Sigma k_{\rm B} T}{\sqrt{2\pi} \overline{m} H }$$ where $k_{\rm B}$ is Boltzmann’s constant, $T$ is the disc temperature, $H = c_{\rm s}/\Omega_{\rm K}$ is the disc scale height, $c_{\rm s} = \sqrt{k_{\rm B}T/\overline{m}}$ is the gas sound speed, and $\overline{m}$ is the mean molecular weight. Then $$\label{eight}
\frac{\partial \dot{M}_{\rm wind}}{\partial  r} =  \frac{\sqrt{8\pi}k_{\rm B}}{\beta_0 \overline{m}} \frac{T}{H} \frac{\mu \Sigma}{\omega} = \frac{\sqrt{8\pi k_{\rm B}}}{\sqrt{\overline{m}}\beta_0} T^{1/2} \mu  \Sigma \,.$$ Combine (\[eight\]) and (\[eqn:usual\]) to find $$\label{eqn:end}
u_r = -\sqrt{\frac{8}{\pi}} \frac{\sqrt{k_{\rm B}}}{\sqrt{\overline{m}}\beta_0} T^{1/2} \mu (\lambda-1) \sim -\frac{\mu(\lambda-1)}{\beta_0} c_{\rm s} \,.$$ In the fiducial model of Bai ([-@b_2016], see their fig. 2), $\mu$ increases from $\sim$0.06 at $r = 0.3$ AU to $\sim$4 at 30 AU, and $(\lambda-1)$ decreases from $\sim$30 to $\sim$2 over the same range; therefore the product $\mu(\lambda-1)$ increases from $\sim$2 to $\sim$8, scaling roughly as $r^{0.3}$. Their model temperature scales as $T \propto r^{-1/2}$; therefore the combination $T^{1/2}\mu (\lambda-1)$ is nearly constant with $r$. Assuming it to be constant implies from (\[eqn:end\]) that $u_r$ is similarly constant (cf. @kdk_2020), if $\beta_0$ is constant: $$u_r \sim -4 \left( \frac{10^5}{\beta_0} \right) {\rm cm}/{\rm s} \,.$$ Taking $\beta_0$ to be a strict constant corresponds to a model intermediate between the conserved-flux model of @b_2016 (dashed line in the right panel of their fig. 5) and their flux-proportional-to-mass model (solid line). Using their initial fiducial $\beta_0 = 10^5$ implies the disc at $r = 30$ AU drains out in $r/|u_r| \simeq 3$ Myr.

\[lastpage\]

[^1]: E-mail: mmrosent@ucsc.edu

[^2]: The planet excites waves in the inner disc which carry negative angular momentum inward. This is equivalent to transmitting positive angular momentum outward.

[^3]: If the Shakura-Sunyaev $\alpha \lesssim 10^{-4}$, discs respond to planetary torques as if they were inviscid (@gc_2019a, their fig. 1).

[^4]: Our final planet masses are a factor of $\sim$3 lower than theirs, a consequence largely of their choice for $h$ which is 50% larger.
