Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 346 / 4

For either component $j\in \{h,s\}$ and $0<\gamma <3$, direct integration of the power law density givesLet $B_j=4\pi G{A}_j/(3-\gamma )$. Since $d{\Phi }_j/dr=G{M}_j/r^2=B_j{r}^{1-\gamma }$,The additive constants depend on the chosen reference and cannot generally be set by requiring $\Phi \to 0$ at infinity for an untruncated power law.

For a satellite on a circular orbit, the linearized effective radial force gives the tidal radiusHere ${\Omega }^2=G{M}_h(<r_o)/r_o^3=B_h{r}_o^{-\gamma }$ and ${\Phi }_h^{\prime \prime }=(1-\gamma ){\Omega }^2$, soSubstitution of the two mass profiles yieldsand equivalently

Now set $\gamma =1$. ThenThe host circular speed is $v=B\sqrt{r_o}$ with $B=\sqrt{2\pi G{A}_h}$. The magnitude of Chandrasekhar dynamical friction becomesa constant. The specific angular momentum is $j=r_{o}v=B{r}_o^{3/2}$. Since the drag torque gives $dj/dt=-r_o{a}_{\mathrm{df}}$,and thereforeIntegration giveswhich is finite.

If stripping is switched off, hold the satellite mass at its initial value $M_0=\mu r_{o0}^2$. The frictional acceleration is then $a_{\mathrm{df}}=a_0{r}_{o0}^2/r_o^2$, where $a_0$ is the constant acceleration in the stripped calculation. The same torque equation givesThusWithout tidal stripping, the satellite retains its mass while the background density increases inward, so dynamical friction strengthens rapidly. Stripping instead gives $M_s\propto r_o^2$ and removes the very mass that creates the gravitational wake.

For a singular isothermal host, ${\rho }_h=A_h{r}^{-2}$, $M_h\propto r$, and $v=B$ is constant. The tidal formula with a satellite ${\rho }_s=A_s{r}^{-1}$ givesConsequently $a_{\mathrm{df}}\propto M_s{\rho }_h/v^2\propto r_o^2$. Since now $j=B{r}_o$, the torque equation yieldsIts solution satisfies $r_o^{-2}=r_{o0}^{-2}+2C_{2}t$, so $r_o$ approaches zero only as $t^{-1/2}$ and never arrives in finite time. This is dynamical-friction stalling by tidal stripping: the inward tidal field strips the satellite so aggressively that its wake and drag disappear. Observationally, disrupted satellites should deposit stars in streams and the stellar halo, surviving low-mass remnants can remain at finite radii for very long times, and merger times inferred from a constant satellite mass can be severe underestimates.

Solved by gpt-5.6-sol high.