= Spherical polytropic flow with adiabatic exponent three halves
For steady spherical <isentropic flow> with $p=K\rho^{3/2}$ around a point mass, put $j=r^2\rho v>0$ and $\mathcal E=v^2/2+2c^2-GM/r>0$. Eliminating the <mass density> gives $c^2=[j(3K/2)^2]^{2/5}r^{-4/5}\mathcal M^{-2/5}$. With $r_s=GM/(4\mathcal E)$, $x=(r/r_s)^{1/5}$, $y=\mathcal M^{2/5}$ and $\lambda=[j(3K/2)^2]^{2/5}/(2\mathcal E r_s^{4/5})$, the <Bernoulli equation> reduces to $\lambda F(y)=F(x)$, where $F(t)=t^4+4/t$. Since $F$ has its unique minimum at $t=1$, a regular <sonic point> requires $\lambda=1$ and $r=r_s$. One <transonic branch> is $y=x$, which has $v=\sqrt{2\mathcal E}$, $\rho\propto r^{-2}$ and $p\propto r^{-3}$. The decreasing branch has inner <free fall> and an outer nearly static reservoir, giving <Bondi accretion> for inward <fluid flow>. A nonradiative stationary <shock wave> keeps $j,\mathcal E,r_s$ fixed but raises $K$, so $\lambda$ increases by the factor $(K_2/K_1)^{4/5}$. The inward transonic solution is the spherical $n=2$, $\gamma=3/2$ specialization of <transonic accretion in a power-law tube>.
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