Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 70 / 3 / a

Solution

Let

v = - u_{r} > 0

c = v_{s}

and

c_{0} = v_{s 0}

. Steady smooth adiabatic flow has

p = K ρ^{γ}

with the entropy parameter fixed by infinity, and

\dot{M} = 4 π r^{2} ρ v, \frac{v ^{2}}{2} + \frac{c ^{2}}{γ - 1} - \frac{GM}{r} = \frac{c _{0}^{2}}{γ - 1}, c^{2} = γ K ρ^{γ - 1} .

The second expression follows by integrating the radial momentum equation with

d p / ρ = d (c^{2} / (γ - 1))

and imposing the outer rest condition. Differentiate continuity to give

ρ^{'} / ρ = - 2/ r - v^{'} / v

, and substitute into momentum. This yields the Bondi accretion wind equation

(v - \frac{c ^{2}}{v}) v^{'} = \frac{2 c ^{2}}{r} - \frac{GM}{r ^{2}} .

At a smooth positive-radius sonic point,

v = c = c_{s}

and both sides' vanishing factors require

GM / r_{s} = 2 c_{s}^{2}

. Evaluate the Bernoulli relation there:

\frac{5 - 3 γ}{2 ( γ - 1 )} c_{s}^{2} = \frac{c _{0}^{2}}{γ - 1}, c_{s}^{2} = \frac{2 c _{0}^{2}}{5 - 3 γ}, r_{s} = \frac{5 - 3 γ}{4} \frac{GM}{c _{0}^{2}} .

Both the squared sound speed and sonic radius must be positive and finite. Hence

1 < γ < 5/3.

For

γ = 5/3

no finite-radius regular sonic crossing is possible; the critical solution discussed next approaches sonic speed only at the central limit.

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