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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 320

The enclosed masses of the $r^{-1}$ host and truncated satellite are
$$
M_h(r_o)=2\pi A_hr_o^2,
\qquad
M_s=2\pi A_sr_t^2.
$$
For a circular orbit, the <Jacobi tidal radius> is
$$
r_t=r_o\left[
\frac{M_s}{(3-d\log M_h/d\log r_o)M_h(r_o)}
\right]^{1/3}.
$$
Since $d\log M_h/d\log r_o=2$, substitution gives
$$
r_t^3=r_o^3\frac{A_sr_t^2}{A_hr_o^2},
$$
and hence
$$
\boxed{r_t=\frac{A_s}{A_h}r_o}.
$$
The assumption $A_s\ll A_h$ makes $r_t/r_o\ll1$, precisely the scale separation required by the local tidal approximation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 3 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 3 / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 3 / a / Solution

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