Transonic spherical flow in a power-law potential

A smooth transonic branch in a power-law gravitational potential has $u_{\ast }^2=c_{\ast }^2=A\beta /(2r_{\ast }^{\beta })$ at its sonic point. Substitution into the Bernoulli function gives $\mathcal{B}=c_{\ast }^2[1/(\gamma -1)+1/2-2/\beta ]$. The local sonic-point slope discriminant distinguishes a genuine crossing from a degenerate everywhere-sonic scaling solution.

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.

The dimensionless accretion factor $(1-q\delta {)}^{q-1/\delta }$ tends to $e^q$ as $\delta \to 0$. For $q>0$, it tends to $1$ as $\delta \uparrow 1/q$, although the sonic point moves to the origin. Writing the latter logarithm as $-qh\ln h/(1-h)$ makes the finite limit transparent. These limits connect transonic spherical accretion rate in a power-law potential to the isothermal equation of state and the limiting critical adiabatic exponent.

For $0<\beta <4$, a nondegenerate transonic spherical flow in a power-law potential connected to infinity requires $\gamma <(4+\beta )/(4-\beta )$. For $\beta \ge 4$, every finite $\gamma >1$ satisfies the local crossing condition; there is no finite upper bound. The uniformly valid inequality is $2\beta -(4-\beta )(\gamma -1)>0$.