Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 314 / 2 / e

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- Solution e

Solution

Put

Y = y^{2}

. On the inner supersonic branch, the equation from part (d) is

Y - ln Y = \frac{4}{x} + 4 ln x - 3.

Its asymptotic expansion as

x \to 0^{+}

begins

Y = \frac{4}{x} + 3 ln x + ln 4 - 3 + O (x ln x) .

Taking the positive square root gives

y = \frac{2}{x} [1 + \frac{x}{8} (3 ln x + ln 4 - 3) + O (x^{2} ln^{2} x)] .

Since

x = r / r_{s}

and

y = u / c_{s}

,

u (r) = 2 c_{s} \frac{r _{s}}{r} [1 + \frac{r}{8 r _{s}} (3 ln \frac{r}{r _{s}} + ln 4 - 3) + O (\frac{r ^{2}}{r _{s}^{2}} ln^{2} \frac{r}{r _{s}})] .

The leading term is the free-fall speed

2 GM / r

; the logarithmic term is the first pressure correction to the inner Isothermal Bondi accretion flow.

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