Bondi accretion

Bondi accretion is steady spherical accretion from a uniform gas reservoir onto a compact gravitating object. In its adiabatic form, compression raises internal energy without providing the irreversible shear heating of a radiatively efficient disk.

The thermal sphere-of-influence scale $r_B=GM/c_s^2$ compares gravitational potential per unit mass with the ambient gas sound-speed squared. A convention based on escape speed instead introduces a factor of two. The scale is not a capture cross-section or an accretion rate without a flow and dissipation model.

With planet-to-star mass ratio $q$ and disk aspect ratio $h$, $r_B/a=q/h^2$ and $r_H/a=(q/3{)}^{1/3}$. Equating the two gives $q=h^3/\sqrt{3}$. The thermally limited radius grows faster with planet mass than the tidally limited radius; an envelope outer scale is often the smaller of the two.

For isentropic spherical accretion with this adiabatic index, define $\lambda =\dot{M}{c}_{\infty }^3/(4\pi G^2{M}^2{\rho }_{\infty })$ and $x=r{c}_{\infty }^2/(GM)$. The steady-flow equation is $\sqrt{\lambda }[{\mathcal{M}}^{3/2}/2+3{\mathcal{M}}^{-1/2}/2]=1+3x/2$. The bracket has minimum two at Mach one. Reaching arbitrarily small positive radii therefore requires $\lambda \le 1/4$. For smaller rates the outer-rest branch remains subsonic; at the maximal rate it approaches Mach one only as $r\to 0$, not at a finite sonic point. Supersonic outer branches fail the specified resting finite-density condition, and larger rates terminate at a positive minimum radius. An additional inner boundary prescription is needed to select a unique rate among the globally defined subsonic solutions.

Relativistic spherical accretion is steady radial gas inflow in Schwarzschild spacetime, with conserved rest-mass flux and relativistic Bernoulli function. A regular solution passes a finite sonic point outside the event horizon. For an isentropic gas with $\gamma =5/3$ and cold asymptotic sound speed $c_{\infty }\ll c$, the critical radius is approximately $3GM/(4c{c}_{\infty })$ and the leading mass accretion rate is $\pi G^2{M}^2{\rho }_{\infty }/c_{\infty }^3$. Relativistic corrections resolve the zero-radius critical point of the Newtonian $\gamma =5/3$ model without eliminating accretion.

Bondi--Hoyle--Lyttleton accretion is gravitational capture by a compact object moving through a gas. In the pressureless supersonic limit, ballistic streamlines focus into a downstream wake.

In a fixed central Newtonian gravitational potential, pressureless free fall has speed $u=a{R}^{-1/2}$, and randomly moving stars have typical gas-relative speed $V=b{R}^{-1/2}$. A steady, undepleted spherical mass flux $A$ gives $\rho =A{R}^{-3/2}/(4\pi a)$. The Bondi--Hoyle--Lyttleton accretion rate therefore becomes $\dot{M}=\lambda M^2$ with $\lambda =G^{2}A/(a{b}^3)$, independent of radius. Integrating $d(1/M)/dt=-\lambda$ gives the displayed solution before $\lambda M_{0}t=1$. Its finite-time divergence signals failure of the fixed potential, undepleted gas supply and negligible feedback assumptions; it is not a physically unlimited mass supply.

If stars are neither created nor lost and all follow quadratic accretion in a central Keplerian potential, conservation of number gives $F(M,t)dM=F_0(M_0)d{M}_0$. Inverting the trajectory gives $M_0=M/(1+\lambda tM)$ and its Jacobian determinant is $(1+\lambda tM{)}^{-2}$. This proves the displayed mass-density-in-mass-space transformation. In a broad high-mass range with $\lambda tM\gg 1$, whose inverse images remain in a narrow interval where $F_0$ varies little, the power law is $F\propto M^{-2}$. A strictly monochromatic initial population remains monochromatic. Finite initial support imposes a finite upper cutoff before the earliest formal blow-up, so the power law is an intermediate asymptotic subject to the accretion model's limits.

In a cold pressureless stream of asymptotic speed $v$, conservation of specific angular momentum and the eccentricity vector makes a streamline at impact parameter $d$ cross the downstream axis at $b=d^2{v}^2/(2GM)$. Its axial velocity there remains $v$, while opposing transverse velocities collide in a shock.

If transverse kinetic energy is dissipated at the downstream axis, the remaining specific orbital energy is $v^2/2-GM/b$. Binding requires $b<2GM/v^2$, corresponding to incident impact parameter $d<2GM/v^2$. Integrating the captured mass flux gives the Bondi--Hoyle--Lyttleton accretion rate.

In the ballistic supersonic limit, streamline collisions bind gas with impact parameter $b<2GM/v_{\infty }^2$, giving

Isothermal Bondi accretion has $p=c_s^2\rho$ with constant sound speed. Its regular transonic solution crosses the Bondi sonic point and approaches free fall near the accretor.

For steady Isothermal Bondi accretion from gas at rest at infinity, define $\lambda =\dot{M}{c}_s^3/(4\pi G^2{M}^2{\rho }_{\infty })$, $x=r/r_s$ and $y=u/c_s>0$, where $r_s=GM/(2c_s^2)$. The Bernoulli function and mass conservation give $y^2/2-\ln y=2\ln x+2/x-\ln (4\lambda )$. The left side has minimum $1/2$ at $y=1$, and the radius-dependent part on the right has minimum two at $x=1$. Existence over all positive radii therefore requires the displayed bound. Equality permits the smooth accelerating-inward transonic branch and gives the usual isothermal Bondi accretion rate. Smaller positive rates have an everywhere-subsonic reservoir-connected branch, and zero rate gives a hydrostatic atmosphere. An absorbing inner boundary inside the sonic radius selects the critical inflow; steady isothermal equations and the outer density alone do not specify that inner condition.

The regular spherical isothermal inflow from mass density ${\rho }_{\infty }$ at rest at infinity has sonic mass density $e^{3/2}{\rho }_{\infty }$. Combining it with the Bondi sonic point and mass conservation gives $\dot{M}=\pi e^{3/2}{G}^2{M}^2{\rho }_{\infty }/c_s^3$.

For inward speed $u>0$, steady spherical mass conservation and radial momentum balance give

With $x=r/r_s$ and $\mathcal{M}=|u_r|/c_s$, the radial isothermal equation integrates to ${\mathcal{M}}^2/2-\ln \mathcal{M}=2/x+2\ln x+C$. A regular transonic branch has $C=-3/2$, and the reservoir-fed accretion branch crosses $(x,\mathcal{M})=(1,1)$ with slope $-1$.

The isothermal Bondi sonic point is $r_s=GM/(2c_s^2)$, where the regular transonic solution has $u=c_s$.

At the critical point, both the coefficient of $du/dr$ and the remaining right-hand side of the Isothermal Bondi equation vanish, allowing a finite derivative through the sonic transition.

The mass accretion rate is the mass entering an accretor per unit time. In a steady spherical flow, $\dot{M}=4\pi r^2\rho u$ is independent of radius.

A steady full-sphere radial inflow has mass flux $\dot{M}=4\pi r^2\rho |v_r|$ by conservation of mass. With Newtonian free-fall speed from rest at infinity, $|v_r|=\sqrt{2GM/r}$, this becomes the displayed expression. The assumed velocity then requires $\rho \propto r^{-3/2}$ along the steady flow. A local density and speed do not determine the total mass accretion rate of a general nonspherical inflow, and this estimate does not impose the sonic boundary condition of Bondi accretion.

Consider steady spherical accretion of a gas obeying a polytropic equation of state in isentropic flow, with fixed background potential $\Phi =-A{r}^{-1/2}$, where $A=16\pi G{\Psi }_0$, with reservoir mass density ${\rho }_0$ and sound speed $c_0$ at infinity. At a regular sonic point, radial momentum balance requires $c_c^2=A/(4\sqrt{r_c})$. The Bernoulli function isSetting $u=c_s=c_c$ at the sonic point givesA finite positive critical sound speed therefore requires $1<\gamma <9/7$. Mass conservation gives $\dot{M}=4\pi r_c^2{\rho }_c{c}_c$, while the constant-entropy gas law gives ${\rho }_c/{\rho }_0=(c_c^2/c_0^2{)}^{1/(\gamma -1)}$. Since $r_c\propto A^2{c}_0^{-4}$ at fixed $\gamma$, these equations yield $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$. The steep sound-speed dependence comes from both the selected capture scale and the critical flow speed. The background potential is prescribed; gas self-gravity and finite outer boundaries require a different model.

For a background gravitating density ${\rho }_{\mathrm{dm}}={\Psi }_0{r}^{-5/2}$, the potential is $\Phi =-16\pi G{\Psi }_0{r}^{-1/2}$. A steady polytropic inflow has a regular sonic point only for $1<\gamma <9/7$, and its accretion rate scales as $\dot{M}\propto {\rho }_0{G}^4{\Psi }_0^4{c}_0^{-7}$.

The Newtonian free-fall speed from rest at infinity toward a point mass is $v_{\mathrm{ff}}=\sqrt{2GM/r}$.

For an adiabatic gas with $\gamma =5/3$, asymptotic density ${\rho }_0$, and sound speed $v_{s0}$ accreting onto a Newtonian point mass $M$, the largest globally subsonic steady rate is