= Transonic accretion in a power-law 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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