Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 57 / 1 / b / Solution

At a finite smooth sonic point $r_{\ast }$, the equation from (a) requiresSet $\delta =\gamma -1$ and $q=2/\beta -1/2$. Substitution into the Bernoulli function givesAn inflow from a warm reservoir at infinity has $\mathcal{B}=c_{\infty }^2/\delta >0$; an outflow reaching infinity has $\mathcal{B}=u_{\infty }^2/2+c_{\infty }^2/\delta \ge 0$. Therefore a nondegenerate transonic spherical flow in a power-law potential requiresFor the usual range $0<\beta <4$, the critical adiabatic exponent for spherical power-law flow is consequentlyThe printed assumption $\beta >0$ also permits $\beta \ge 4$. In that range the boxed inequality holds for every finite $\gamma >1$: there is no finite upper bound. Equivalently, take $f(\beta )=+\infty$ for $\beta \ge 4$. Extending the rational expression beyond $\beta =4$ would give an incorrect restriction.

The strict inequality also follows from local sonic-point slope discriminant analysis, including the potentially cold zero-energy endpoint. Put $X=(r_{\ast }/u_{\ast })u_{\ast }^{\prime }$. Since mass conservation implies $(c^2{)}^{\prime }/c^2=-\delta (u^{\prime }/u+2/r)$, differentiating the sonic point equation at $r_{\ast }$ yieldsIts discriminant is $8[2\beta -(4-\beta )\delta ]$. Moreover, the logarithmic Mach number derivative at the sonic point isA genuine crossing has two distinct branches and a nonzero derivative. At equality the crossing degenerates. In particular, for $\gamma =f(\beta )$ and $\mathcal{B}=0$, mass conservation and the polytropic equation of state make the Mach number constant along the scale-invariant solution; a solution that is sonic there is sonic everywhere, rather than crossing an isolated sonic point.