Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 314 / 3 / a

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- Solution a

Solution

Steady spherical mass conservation gives

4 π r^{2} ρ u = \dot{M}

. For the polytropic equation of state,

c_{s}^{2} = d p / d ρ

, so

\frac{ρ ^{'}}{ρ} = - \frac{u ^{'}}{u} - \frac{2}{r}, \frac{p ^{'}}{ρ} = c_{s}^{2} \frac{ρ ^{'}}{ρ} .

Substitution into the radial Euler equations for an inviscid fluid gives the Parker wind equation

(u - \frac{c _{s}^{2}}{u}) \frac{d u}{d r} = \frac{2 c _{s}^{2}}{r} - \frac{G M _{*}}{r ^{2}} .

At a sonic point, the coefficient of

d u / d r

vanishes. A smooth transonic branch can pass it only if the right-hand side vanishes simultaneously. Hence

u_{s} = c_{s, s}, r_{s} = \frac{G M _{*}}{2 c _{s, s}^{2}} .

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