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

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> gives
$$
c_s^2=\frac{2c_0^2}{7-5\gamma},
\qquad
r_s=\frac{GM(7-5\gamma)}{6c_0^2},
\qquad
\rho_s=\rho_0\left(\frac2{7-5\gamma}\right)^{1/(\gamma-1)}.
$$
Here $c_0$ 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> is
$$
\dot M=A_\star\left(\frac{r_s}{R_\star}\right)^3\rho_s c_s.
$$
For two equal polar caps of surface angular radius $\theta_\star\ll1$, $A_\star\simeq2\pi R_\star^2\theta_\star^2$. 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 $\theta_\star^2r_s/R_\star\ll1$ for the sonic region to remain a narrow <flux tube>.

Transonic accretion in a dipolar flux tube

Consider steady <isentropic flow> toward a <Newtonian gravitational potential> $-GM/r$ through a tube of cross-sectional area $A(r)=Cr^n$. Write $v>0$ for the inward speed and use a <polytropic equation of state> with <specific-heat ratio> $\gamma>1$. <Mass conservation> and the <Euler equations for an inviscid fluid> give
$$
\rho v A=\dot M,
\qquad
\left(v-\frac{c_s^2}{v}\right)v'=\frac{nc_s^2}{r}-\frac{GM}{r^2},
$$
where $c_s$ is the <adiabatic sound speed>. A regular <sonic point> therefore has $v_s=c_s$ and $r_s=GM/(nc_s^2)$. The <Bernoulli equation>, matched to a nearly stationary reservoir with <sound speed> $c_0$, gives
$$
\frac{v^2}{2}+\frac{c^2}{\gamma-1}-\frac{GM}{r}=\frac{c_0^2}{\gamma-1},
\qquad
c_s^2=\frac{2c_0^2}{(2n+1)-(2n-1)\gamma}.
$$
For $n>1/2$, a finite positive <sonic point> requires $1<\gamma<(2n+1)/(2n-1)$. Differentiating the flow equation at that point, with $x=r_sv'_s/c_s$, gives
$$
(\gamma+1)x^2+2n(\gamma-1)x+n^2(\gamma-1)-n=0.
$$
Its <discriminant> is $4n[(2n+1)-(2n-1)\gamma]$, so the same bound permits real regular slopes. The branch on which the <Mach number> rises inward selects the negative sign in
$$
x=\frac{-n(\gamma-1)\pm\sqrt{n[(2n+1)-(2n-1)\gamma]}}{\gamma+1}.
$$
For a spherical tube $n=2$, this recovers the $5/3$ threshold of <Bondi accretion>; for a <dipolar flux-tube area>, $n=3$ gives $7/5$. The tube approximation must remain valid between the reservoir matching region and the accretor.

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 Incoming links: Transonic accretion in a power-law tube