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Vertically integrating the continuity equation and imposing a steady axisymmetric flow givesThus the inward-positive mass accretion rate isThe zero-torque condition at $R_{\mathrm{ISCO}}$ gives the standard Shakura--Sunyaev thin disk relationCombining this relation with mass conservation, or substituting it into the stated drift equation, yieldsFar outside the inner edge this reduces to $u_R\simeq -3\nu /(2R)$.

Combining the two expressions for the sonic sound speed givesFor a polytropic gas, $c_s^2\propto {\rho }^{\gamma -1}$, and henceThe transonic accretion in a steep dark-matter cusp has mass accretion rateThus $\dot{M}\propto c_0^{-7}$, whereas classical Bondi accretion onto a point mass has ${\dot{M}}_{\mathrm{B}}\propto c_0^{-3}$. The steeper dependence comes from the cusp potential, whose sonic radius scales as $c_0^{-4}$ instead of $c_0^{-2}$.

At the Bondi sonic point, $u_s=c_s$ and $GM/r_s=2c_s^2$. Evaluating the Bernoulli function there givesand henceThe conserved mass accretion rate is thereforeThus

For $p=K{\rho }^{1+1/m}$, the barotropic enthalpy function is $m{c}_s^2$. In $\Phi =-\lambda /R^2-\mu /R$, a regular sonic point obeys $c_{s,s}^2=\lambda /R_s^2+\mu /(2R_s)$, and its Bernoulli integral yields the displayed quadratic. For $K,\lambda ,\mu >0$ and the usual positive polytropic index, its unique positive root exists precisely when $m>1/2$. The mass accretion rate is $4\pi R_s^2{\rho }_{\infty }{c}_{s,s}(c_{s,s}^2/c_{\infty }^2{)}^m$. If indices with $0<1+1/m<1$ are admitted, $m<-1$ also permits a positive sonic root; negative compressibility indices do not.

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.

The thermal timescale compares the internal energy stored in a disc annulus with its local heating or cooling. In a gas-pressure-supported alpha disk, $t_{\mathrm{th}}\sim (\alpha {\Omega }_K{)}^{-1}$, whereas the viscous timescale is longer by approximately $(R/H{)}^2$. A local thermal perturbation is therefore studied at fixed surface density of a disk, rather than fixed steady-state mass accretion rate.

The dipolar flux-tube area has $A(r)=A_{\star }(r/R_{\star }{)}^3$. Applying transonic accretion in a power-law tube with $n=3$ and a polytropic equation of state givesHere $c_0$ and ${\rho }_0$ are the reservoir sound speed and mass density, and $c_s$ is the sound speed at the sonic point. The mass accretion rate isFor two equal polar caps of surface angular radius ${\theta }_{\star }\ll 1$, $A_{\star }\simeq 2\pi R_{\star }^2{\theta }_{\star }^2$. One loaded cap gives half the total rate. The finite-radius transonic branch exists for $1<\gamma <7/5$, must cross outside the star ($r_s>R_{\star }$), and requires ${\theta }_{\star }^2{r}_s/R_{\star }\ll 1$ for the sonic region to remain a narrow flux tube.

With reservoir density ${\rho }_{\infty }$ and adiabatic sound speed $c_{\infty }$, put $q=2/\beta -1/2$ and $h=1-q(\gamma -1)>0$. The selected mass accretion rate is $4\pi {\rho }_{\infty }(A\beta /2{)}^{2/\beta }{c}_{\infty }^{1-4/\beta }{h}^{q-1/(\gamma -1)}$. It follows by evaluating the conserved spherical flux at the sonic point.