\documentclass[compose]{exam-n}
\begin{document}
\begin{question}{20} \author{John Brown and Declan Diver}
\part An earth satellite in a highly eccentric orbit of (constant)
perigee distance $q$ undergoes a targential velocity impulse
$-\Delta V$ at each perigee passage. By considering the mean rate
of change of velocity at perigee, show that the mean rate of
change of the semi-major axis $a$ ($\gg q$) satisfies
\begin{equation*}
\frac{1}{a^2} \Diffl at =
\left(\frac{8}{GMq}\right)^{1/2}\frac{\Delta V}{T},
\end{equation*}
where $M$ is the Earth's mass and $T$ the orbital period.
\partmarks{3}
\begin{questiondata}
You may assume $\displaystyle v^2(r)=GM\left(\frac{2}{r}-\frac{1}{a}\right)$.
\end{questiondata}
Using $T=2\pi(a^3/GM)^{1/2}$ show that with $a_0=a(0)$, (where
$a(t)$ is the semimajor axis at time $t$)
\begin{equation*}
\frac{a(t)}{a_0}=\left[1-\frac{t\Delta V}{2^{1/2}\pi
a_0(1-e_0)^{1/2}}\right]^2
\partmarks{2}
\end{equation*}
and
\begin{equation*}
\frac{T(t)}{T_0}=\left[1-\frac{t\Delta V}{2^{1/2}\pi
a_0(1-e_0)^{1/2}}\right]^3
\partmarks{1}
\end{equation*}
and the eccentricity satisfies (with $e_0=e(0)$)
\begin{equation*}
e(t)=1-\frac{1-e_0}{\left[1-\frac{t\Delta V}{2^{1/2}\pi
a_0(1-e_0)^{1/2}}\right]^2}.
\partmarks{2}
\end{equation*}
Show that, once the orbit is circular, its radius decays
exponentially with time on timescale $m_0/2\dot{m}$ where $m_0$
is the satellite mass and $\dot{m}$ the mass of atmosphere
`stopped' by it per second. \partmarks{2}
\part What is meant by (a) the sphere of influence of a star, and
(b) the passage distance?
\partmarks{2}
Consider a system of $N$ identical stars, each of mass $m$.
\part Given that the change $\delta u$ in the speed of one such star
due to the cumulative effect over time $t$ of many gravitational
encounters with other stars in the system can be approximated by
\begin{equation*}
(\delta u)^2 \propto [\nu tm^2\log(p_{\rm max}/p_{\rm
min})]/\bar{u},
\end{equation*}
where $\bar{u}$ is the rms mutual speed, $\nu$ is the stellar
number density, and $p_{\rm max, min}$ are the maximum, minimum
passage distances for the system, show that this leads to a natural
time $T$ for the system, where
\begin{equation*}
T\propto\frac{\bar{u}u^2}{m^2\nu\log N}.
\partmarks{5}
\end{equation*}
\begin{questiondata}
You may assume that the sphere of influence radius of a star is
approximated by $(m/M)^{2/5}R$ where $R$ and $M$ are the radius
and mass of the whole system respectively.
\end{questiondata}
\part Deduce that $T$ is the disintegration timescale for the system,
by showing that a star with initial speed $u_0$ in a stable circular
orbit reaches escape speed after time $T$.
\partmarks{3}
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\end{question}
\end{document}