A regular <sonic point> must make both sides of the flow equation vanish:
$$
v_s=c_s,\qquad r_s=\frac{GM_\star}{3c_s^2}.
$$
Substituting into the <Bernoulli equation> yields
$$
c_s^2\left(\frac1{\gamma-1}-\frac52\right)=\frac{c_0^2}{\gamma-1},
\qquad
c_s^2=\frac{2c_0^2}{7-5\gamma}.
$$
For $\gamma>1$, positive finite <sound speed> therefore requires the <critical adiabatic index for dipolar accretion>
$$
\boxed{1<\gamma<\gamma_{\rm crit}=\frac75.}
$$
To check that this gives real regular crossings, differentiate the flow equation at the <sonic point> and put $x=r_sv'_s/c_s$. The <transonic accretion in a power-law tube> calculation gives
$$
(\gamma+1)x^2+6(\gamma-1)x+9\gamma-12=0,
\qquad
x=\frac{-3(\gamma-1)\pm\sqrt{3(7-5\gamma)}}{\gamma+1}.
$$
The minus sign gives the <transonic branch> whose <Mach number> increases inward. At $\gamma=7/5$ the positive-energy reservoir cannot match a finite <sonic point>; for larger $\gamma$ the required $c_s^2$ is negative. A physical surface-crossing solution also requires $r_s>R_\star$ and validity of the narrow <flux tube> approximation up to the sonic region.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 314 / 4 / c / Solution

{title2=$\gamma_{\rm crit}=7/5$}

For <transonic accretion in a dipolar flux tube>, the <Bernoulli equation> at the <sonic point> contains $c_s^2[1/(\gamma-1)-5/2]$. A positive reservoir energy requires $\gamma<7/5$. Compression in the $r^3$ <dipolar flux-tube area> raises the <specific enthalpy> more rapidly than in the $r^2$ geometry of spherical <Bondi accretion>, whose corresponding upper index is $5/3$.

Codex @codex  0

 Incoming links: Critical adiabatic index for dipolar accretion