---
title: Hill Regime Accretion in Astrophysical Disks
url: https://www.emergentmind.com/topics/hill-regime-accretion
type: topic
---

# Hill Regime Accretion in Astrophysical Disks

Hill regime accretion is the class of accretion processes in shearing astrophysical disks for which the relevant gravitational sphere of influence is the Hill sphere, $R_{\rm H}=a\left(M/(3M_\ast)\right)^{1/3}$, rather than a purely Bondi-type sphere set only by pressure support. In this regime, encounter kinematics are set primarily by local Keplerian shear, the characteristic relative speed is the Hill velocity $v_{\rm H}=\Omega R_{\rm H}$, and mass delivery is controlled by the structure and transport physics inside or across the Hill sphere. The phrase is used in several related literatures: pebble capture by planetary cores, gas accretion by superthermal protoplanets, trans-Hill planetesimal growth, collision-dominated inner-disk accretion, ring-moon accretion in Saturn’s rings, and Hill-limited gas retention by compact objects embedded in AGN disks [1205.3030] [1709.03530] [2011.02869] [1907.03746].

## 1. Common dynamical framework

The basic scales are the Hill radius,
$$
R_{\rm H}=a\left(\frac{M}{3M_\ast}\right)^{1/3},
$$
the Hill velocity,
$$
v_{\rm H}=\Omega R_{\rm H},
$$
and the Bondi radius,
$$
R_{\rm B}=\frac{GM}{c_s^2}
$$
or, in pebble-accretion notation with a headwind $\Delta v$,
$$
R_{\rm B}=\frac{GM}{\Delta v^2}.
$$
The Hill sphere is the region in which the accretor’s gravity dominates over stellar tides in the rotating frame; the Bondi radius is the region in which gravity dominates over pressure or headwind-driven drift. Hill-regime accretion begins when the gravitational sphere of influence is effectively truncated by stellar tides, so that capture and supply are regulated by the Hill geometry and by shear-driven encounters rather than by purely embedded Bondi flow [1709.03530] [1205.3030].

In gas-accretion studies, the thermal limit is the condition $r_{\rm H}\approx H$. One formulation writes
$$
M_{\rm th}\simeq 3h^3M_\ast,
$$
with $h\equiv H/a$, while another introduces
$$
q_{\rm th}\equiv \frac{(M_{\rm p}/M_\ast)}{(H_{\rm p}/R_{\rm p})^3}.
$$
The 3D global simulations of runaway gas inflow distinguish three ranges: $q_{\rm th}\lesssim0.3$ for Bondi-embedded accretion, $0.3\lesssim q_{\rm th}\lesssim10$ for 3D Hill accretion, and $q_{\rm th}\gtrsim10$ for a strongly superthermal regime in which the Hill sphere protrudes vertically out of the disk and inflow becomes effectively 2D [2305.01684]. In pebble accretion, the analogous transition is set by the “transition mass”
$$
M_t=\frac{\Delta v_K^3}{G\Omega},
$$
or, in the Lambrechts–Johansen notation,
$$
M_t=\sqrt{\frac{1}{3}}\,\frac{\Delta v^3}{G\Omega_{\rm K}},
$$
defined by $R_{\rm B}=R_{\rm H}$ or $v_{\rm H}=\Delta v$ [1709.03530] [1205.3030].

A second organizing distinction is geometric. Pebble accretion is “2D” when the pebble layer thickness is smaller than the capture radius or Hill radius, and “3D” when it is thicker. Gas accretion onto giant planets is likewise 3D when the Hill sphere remains embedded in the disk thickness, but becomes vertically limited when $R_{\rm H}\gg H$. These dimensional transitions alter the mass scaling even when the Hill sphere remains the governing gravitational scale [1709.03530] [2305.01684].

## 2. Pebble accretion onto planetary cores

In the pebble-accretion literature, the Hill regime is the high-efficiency mode entered once $M_c>M_t$. For marginally coupled pebbles, $\tau_s\sim0.1{-}1$, particles entering the Hill sphere are captured with near-unit efficiency in laminar flow because gas drag dissipates energy during the $\sim\Omega^{-1}$ encounter. The canonical 2D Hill rate is
$$
\dot{M}_{\rm H}=2\,r_{\rm H}\,\Sigma_{\rm p}\,v_{\rm H}\propto M_c^{2/3},
$$
while the 3D rate is reduced by the factor $r_{\rm H}/H_{\rm p}$ when $r_{\rm H}<H_{\rm p}$. For tightly coupled particles the effective Hill cross section shrinks, with
$$
r_{\rm eff}\lesssim \tau_f^{1/3}r_{\rm H},
$$
so that $\dot{M}_{\rm H,eff}\propto \tau_f^{2/3}$ [1205.3030].

MRI turbulence does not erase the Hill-regime capture mechanism. In shearing-box simulations, accretion for $\tau_s\sim0.1{-}1$ remains intrinsically efficient even in strong ideal-MHD turbulence once the pebble layer thickness is normalized out. The paper defines
$$
\dot{M}_{\rm 2D}=3\Sigma_{\rm p}\Omega r_{\rm H}^2
$$
and a modified 2D/3D normalization
$$
\dot{M}_{\rm mod}=\int_{-r_{\rm H}}^{+r_{\rm H}}3\rho_p(z)\Omega(r_{\rm H}^2-z^2)\,dz,
$$
and finds comparable $k_{\rm mod}=\dot{M}_{\rm sim}/\dot{M}_{\rm mod}$ in hydro, ambipolar-diffusion, and ideal-MHD runs. Turbulence broadens the feeding zone and lowers the capture probability within the Hill sphere, but these effects largely cancel; the remaining change in the absolute rate is primarily geometric, through the turbulence-set pebble layer thickness
$$
H_p\approx \sqrt{\alpha_z/\tau_s}\,H
$$
[1709.03530].

Analytical generalizations retain the same Hill control but replace monodisperse capture by size-integrated rates. Lyra et al. derive an exact monodisperse bridging formula between 3D and 2D,
$$
\dot{M}=\pi R_{\rm acc}^2\rho_{d0}\delta v\,e^{-\xi}\left[I_0(\xi)+I_1(\xi)\right],\qquad
\xi\equiv\left(\frac{R_{\rm acc}}{2H_d}\right)^2,
$$
with the 2D Hill limit
$$
\dot{M}_{\rm mono,2D,Hill}=2\times10^{2/3}\Omega R_{\rm H}^2{\rm St}^{2/3}\Sigma_d.
$$
For a polydisperse MRN distribution with $k=3.5$, $q=0$, and ${\rm St}={\rm St}_{\max}$, the exact Hill-regime correction is
$$
\left(\frac{\dot{M}_{\rm poly}}{\dot{M}_{\rm mono}}\right)_{\rm 2D,Hill}=\frac{3}{7},
$$
so the Hill rate is reduced by an exact factor $3/7$ relative to the monodisperse case [2301.03825].

Radiative thermodynamics does not necessarily alter Hill-regime pebble capture. For a $0.95\,M_\oplus$ embryo at $1\,{\rm AU}$, $R_{\rm B}<R_{\rm H}$ under the disk conditions studied, so the embryo resides in the Hill regime. Radiative transfer lowers temperatures in the outer Hill sphere by $\sim100\,{\rm K}$ and raises inner densities by up to a factor of two, while convective velocity amplitudes increase by roughly an order of magnitude near the embryo; nevertheless the measured pebble accretion rates remain essentially identical to the purely convective runs and preserve the linear scaling $\dot{M}_p\propto s$ over $10\,\mu{\rm m}$ to $1\,{\rm cm}$ [1810.07048].

## 3. Gas accretion by giant planets in the Hill regime

For gas giants, Hill-regime accretion is not simply geometric capture from the full Hill cross section. After partial gap opening, residual gas diffuses into the co-orbital region, settles onto horseshoe streamlines, and enters the Hill sphere near the Lagrange channels. Inside $r_{\rm H}$, the incoming horseshoe flow encounters the outer circumplanetary disk (CPD), where gas is on nearly circular, shear-dominated orbits around the planet. Above the thermal mass, stellar tidal and Coriolis forces deflect the converging streamlines, and the interface between horseshoe flow and outer CPD develops discontinuities in vortensity and Bernoulli constant. In the inviscid limit these quantities are conserved along streamlines, so the interface acts as a “tidal barrier” that impedes free mixing and makes accretion diffusion-limited rather than purely supply-limited [2011.02869].

The corresponding semi-analytic rate is written as
$$
\dot{M}_{\rm p}\simeq A(M_{\rm p},h,\alpha)\,\Sigma_{\rm min},
$$
with partial-gap depletion
$$
\frac{\Sigma_{\rm min}}{\Sigma_{\rm p}}=\frac{1}{1+0.034\,K},\qquad
K\equiv \frac{q^2}{h^5\alpha},
$$
and a Hill-regime inflow coefficient
$$
A_{\rm Hill}\simeq 2\pi r_{\rm H}H\Omega \exp\!\left[-\beta\left(\frac{r_{\rm H}}{H}\right)^2-\frac{1}{2}\right].
$$
The exponential term is the tidal barrier. It becomes severe when $r_{\rm H}>H$, so low-viscosity superthermal planets experience an exponential throttling,
$$
\dot{M}_{\rm p}\propto \exp\!\left[-\beta\left(\frac{q}{3h^3}\right)^{2/3}\right].
$$
This is the principal departure from the classical supply-limited estimate $\dot{M}_{\rm Hill}\sim \Sigma_{\rm min}\Omega r_{\rm H}^2$, which overestimates accretion above the thermal limit because it ignores CPD coupling and the barrier [2011.02869].

The mass dependence is therefore sensitive to viscosity and thickness. In weakly viscous, modest-thickness disks with $\alpha\sim10^{-3}$ and $h\sim0.03{-}0.05$, growth stalls at $M_{\rm p}\sim1{-}2\,M_{\rm J}$ because $\tau_{\rm grow}\gtrsim\tau_{\rm dep}$; in strongly viscous and/or thick disks with $\alpha\sim10^{-2}$ and $h\sim0.05$, planets can reach several $M_{\rm J}$. Two-dimensional calculations show eccentric horseshoe instabilities and accretion surges for very massive planets, whereas eccentric streamlines remain stable in 3D over $\gtrsim2000$ orbits, keeping the barrier effective [2011.02869].

A complementary 3D global calibration measures maximum one-way inflow into a sphere around the planet. It finds
$$
\max\dot{M}_{\rm p}=C_{\rm Hill}\rho_{\rm g}\Omega_{\rm p}r_{\rm H}^3
$$
for $0.3\lesssim q_{\rm th}\lesssim10$, with $C_{\rm Hill}\approx4$ without sink cells and $\approx2$ with sink cells, and
$$
\max\dot{M}_{\rm p}=C_{\rm sup}\rho_{\rm g}\Omega_{\rm p}r_{\rm H}^2H_{\rm p}
$$
for $q_{\rm th}\gtrsim10$, with $C_{\rm sup}\approx9$. These are strict upper limits to the true accretion because they quantify inflow into the Hill sphere rather than permanent accretion through the CPD and envelope [2305.01684].

## 4. Circumplanetary delivery, envelope structure, and reduced-order prescriptions

Local 3D hydrodynamic simulations of gas delivery to circumplanetary disks separate a Bondi-dominated low-mass regime from a Hill-dominated high-mass regime at roughly $M_p\simeq0.2\,M_{\rm Jup}$ at $5.2\,{\rm au}$. In the high-mass regime the CPD radius scales as
$$
R_{\rm CPD}\simeq0.2R_{\rm H},
$$
gas accretes vertically from above and below the midplane, and the radially averaged accretion-band width obeys
$$
\bar{\hat{w}}=0.12\hat{r}_{\rm H}^3
$$
for $\hat r_{\rm H}\gtrsim0.8$. A robust result is that
$$
\dot{M}_{\rm acc,CPD}/\dot{M}_{\rm acc,Hill}\simeq0.4
$$
over $0.05{-}1\,M_{\rm Jup}$. When combined with the Kanagawa gap formula,
$$
\frac{\Sigma_{\min}}{\Sigma_0}=\frac{1}{1+0.04K},
$$
this yields a semi-analytical 3D CPD accretion law with the same $M_p$ dependence as the earlier 2D case but a normalization lower by $\simeq40\%$ [2207.03664].

Non-isothermal 3D simulations of a $1\,M_{\rm J}$ planet at $1\,{\rm au}$ add a thermodynamic boundary inside the Hill sphere. In that model, $R_{\rm B}\approx2.8R_{\rm H}$ and the flow is a 3D, vertically dominated inflow that becomes supersonic near the Hill sphere. The inflow halts at an ionization surface defined by $X=0.5$, with quasi-steady values $R_{\rm ion}\approx0.33{-}0.34\,R_{\rm H}$ without feedback and $R_{\rm ion}\approx0.42{-}0.43\,R_{\rm H}$ with radiative feedback. The mean accretion rate into the ionized envelope rises from $\langle\dot M\rangle\approx1.2\times10^{-12}\,M_\odot\,{\rm yr}^{-1}$ to $\approx7.8\times10^{-12}\,M_\odot\,{\rm yr}^{-1}$, a gain factor of $\approx5.4$, while only $\sim2.5\%$ of gas crossing $R_{\rm H}$ reaches $R_{\rm ion}$ without feedback and $\sim5\%$ with feedback. This redefines the effective accretion boundary: line emission and accretion luminosity should be anchored to $R_{\rm ion}$ rather than to $R_{\rm H}$ [2503.15256].

One-dimensional evolutionary models calibrated to 3D hydrodynamics implement disc-limited Hill accretion through an inverse growth time
$$
\frac{1}{\tau_{\rm H}}=\frac{1}{3}C_{\rm H}\Omega \frac{\Sigma r^2}{M_\ast}\left(\frac{r_p}{H}\right),
$$
with recommended parameters $R_f=3R_{\rm H}$, $C_{\rm B}=10.0$, and $C_{\rm H}=0.19$. The gas sink is applied cell-by-cell across the feeding zone,
$$
\dot M_{\rm gas}=\sqrt{2\pi}\,C_{\rm B,H}\!\int_{r_p-R_f}^{r_p+R_f}\!\rho_0(r)H(r)\,
{\rm erf}\!\left(\frac{\sqrt{R_f^2-(r-r_p)^2}}{\sqrt{2}H(r)}\right)
v_{\rm rel}\,dr,
$$
so mass is removed from the same radial location from which it is accreted. In the calibrated Hill regime this prescription reproduces 3D hydro growth and migration tracks at the $\lesssim10{-}20\%$ level across the parameter sets tested [2205.02858].

## 5. Trans-Hill growth, oligarchy, and collision-dominated inner disks

In coagulation theory, “trans-Hill” denotes the stage at which the bodies dominating viscous stirring satisfy $u\approx v_{\rm H}$, where $u$ is the random speed of the small bodies. The trans-Hill radius is
$$
R_{\rm th}\equiv \frac{u}{\sqrt{G\rho\alpha}},
$$
with $\alpha\equiv R/R_{\rm H}\ll1$. After onset, $u$ and the characteristic stirrer size evolve in lockstep,
$$
R_{\rm stir}(t)\sim \frac{\sigma\Omega}{\rho\alpha}\,t,
$$
and the size spectrum below $R_{\rm stir}$ becomes self-similar. In the collisionless trans-Hill substage the efficiency remains very low, with $\epsilon\approx10\alpha$ at the peak and differential slope $q=4$; in the collisional trans-Hill substage, entered at
$$
R_{\rm stir}^{\rm (col)}\approx \frac{s}{\alpha},
$$
the efficiency rises as $\Sigma/\sigma\sim \alpha^2(R_{\rm stir}/s)$ and the differential slope becomes $q=3$. Termination occurs either by mutual accretion of big bodies, at
$$
R_{\rm stir}^{\rm (b.b.\ accrete)}=s\,\alpha^{-3/2},
$$
or by oligarchy when the oligarchy parameter
$$
{\rm OP}\equiv \frac{\rho\alpha R^2}{\Sigma a}
$$
approaches unity [1303.3890].

The approach to Hill-regime pebble accretion is a distinct bottleneck. In a streaming-instability filament model, the transition mass is
$$
M_t=\frac{1}{\sqrt{3}}\eta^3M_\ast,
$$
and in the irradiated disk used there it scales as $M_t\propto r^{12/7}$. The paper finds that planetesimal accretion reaches the Hill-transition mass only inside $\approx20\,{\rm AU}$ around a solar-mass star within $10\,{\rm Myr}$, while beyond $5{-}10\,{\rm AU}$ there is little or no collisional growth. This implies that, in cold giant-planet formation zones, pebble accretion must act directly on the most massive planetesimals already present in the initial mass function, with little or no help from mutual planetesimal collisions [2208.01902].

At very short orbital periods, Hill-regime accretion enters a different limit because the physical body occupies a substantial fraction of its Hill sphere. For a rocky body,
$$
\alpha\equiv \frac{R_p}{R_{\rm H}}=\frac{1}{a}\left(\frac{9M_\ast}{4\pi\rho}\right)^{1/3},
$$
so $\alpha$ is mass-independent and scales as $a^{-1}$. Direct N-body simulations show that when $\alpha\approx0.1$ the system transitions to a collision-dominated Hill regime in which most close encounters end in mergers rather than scattering, accretion becomes nearly $100$ percent efficient, embryos grow beyond the classical isolation mass, and the familiar oligarch/planetesimal bimodality does not develop. For realistic rocky densities the inferred boundary is $P\simeq3{-}5$ days around a $0.08\,M_\odot$ star [2307.09712].

## 6. Ring moons, AGN disks, and observational consequences

In Saturn’s rings, Hill-regime accretion governs the growth of equatorial ridges on embedded moons when ring material approaches through the Hill sphere with shear-driven speeds $v_{\rm H}=nR_{\rm H}$. The relevant eccentricity scale is the Hill eccentricity,
$$
e_H=\frac{R_H}{a}=\left(\frac{m}{3M_S}\right)^{1/3},
$$
and synchronous rotation imposes a torque limit. The accretion torque is
$$
T_{\rm acc}\sim \frac{3}{2}\dot M\,n\,R_H^2,
$$
while the Saturnian tidal torque is
$$
T_{\rm tide}\simeq \frac{3}{2}\frac{GM_S^2}{a}\left(\frac{R}{a}\right)^5\frac{k_2}{Q}.
$$
Equating them gives a critical accretion rate
$$
\dot M_{\rm cr}=M\,n\,\left(\frac{M_S}{M}\right)^{5/3}\left(\frac{R}{a}\right)^5\left(\frac{k_2}{Q}\right)3^{2/3}.
$$
For Pan, Atlas, and Daphnis the paper quotes $\dot M_{\rm cr}\approx2\times10^{10}$, $3\times10^{10}$, and $2\times10^7\,{\rm kg\,yr^{-1}}$, with minimum ridge-growth times of $\approx2\times10^4$, $6\times10^4$, and $4\times10^4\,{\rm yr}$, and argues that the observed non-axisymmetric ridges imply actual durations $\gtrsim10^5\,{\rm yr}$ [2007.13227].

In AGN disks, Hill-regime accretion appears around stellar-mass black holes embedded in the disk of a supermassive black hole. The pre-merger gas reservoir is limited by the Hill sphere rather than by the Bondi sphere because shear truncates capture at
$$
R_H=a_{\rm BH}\left(\frac{m}{3M_{\rm SMBH}}\right)^{1/3}.
$$
After a gravitational-wave kick, ram-pressure stripping of this Hill-sphere gas produces a prompt electromagnetic transient if $R_H\ge H$. The onset time scales as
$$
t_{\rm UV}\sim \mathcal{O}(2\,{\rm days})
\left(\frac{a_{\rm BH}}{10^3r_g}\right)
\left(\frac{M_{\rm SMBH}}{10^8M_\odot}\right)
\left(\frac{v_{\rm kick}}{10^2\,{\rm km\,s^{-1}}}\right),
$$
the duration is $\sim R_H/v_{\rm kick}\approx \mathcal{O}(5t_{\rm UV})$, and the luminosity is
$$
L_{\rm UV}\sim \mathcal{O}(10^{42}\,{\rm erg\,s^{-1}})
\left(\frac{t_{\rm UV}}{2\,{\rm days}}\right)^{-1}
\left(\frac{M_{\rm Hill}}{1M_\odot}\right)
\left(\frac{v_{\rm kick}}{10^2\,{\rm km\,s^{-1}}}\right)^2.
$$
If $R_H<H$, the flare is delayed and weakened by photon diffusion through the disk atmosphere [1907.03746].

Across these applications, Hill-regime accretion has a common structural meaning: stellar tides and disk shear set the relevant sphere of influence, but the mass-growth law depends on the transport physics specific to the system. In pebble accretion, that dependence is set by stopping time and pebble-layer thickness; in giant-planet gas accretion, by gap depletion, CPD coupling, viscosity, and, in some models, a thermodynamic boundary inside the Hill sphere; in planetesimal growth, by the relation between $u$ and $v_H$; and in ring or AGN environments, by torque constraints or ram-pressure stripping. This suggests that “Hill regime accretion” is not a single rate law but a family of shear-dominated accretion problems linked by a common dynamical scale, $R_H$, and differentiated by the dissipative and supply processes that operate within it.

Source: https://www.emergentmind.com/topics/hill-regime-accretion