Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 346 / 2 / Solution

The two halo centres orbit their centre of mass with separation $R$. Their relative coordinate obeys $\ddot{\mathbf{R}}=-G(M_1+M_2)\mathbf{R}/R^3$, hence circular motion requiresReflection symmetry about the orbital plane makes the vertical force point back toward $z=0$, while the centrifugal force has no vertical component. An equilibrium away from that plane is therefore impossible.

In the uniformly rotating frame, an equilibrium is a stationary point of the gravitational plus centrifugal effective potentialSet $R=1$, divide by $G(M_1+M_2)/R$, and write $\alpha =M_2/(M_1+M_2)$. The primary and secondary lie at $x=-\alpha$ and $x=1-\alpha$, so on their line

For the two roots near the secondary, put $x=1-\alpha \mp d$ in $F^{\prime }(x)=0$. Dominant balance gives $3d\simeq \alpha /d^2$, so $d=(\alpha /3{)}^{1/3}$. Expanding the root beyond the primary directly in powers of $\alpha$ givesto the requested orders. The other equilibria are the two Triangular Lagrange points

The distance from $H_2$ to either nearby collinear point is the Hill radius. Since $\alpha \simeq M_2/M_1$,Inside this tidal radius, the subhalo's gravity dominates the host's differential gravitational field; outside it, material can escape through the neighborhoods of $L_1$ and $L_2$. For an extended spherical host, $M_1$ is replaced by enclosed mass and the coefficient becomes $3-d\log M_1/d\log R$, giving the Jacobi tidal radius. An extended subhalo requires the bound mass inside $r_t$ to be found self-consistently. On an eccentric orbit there is no time-independent rotating potential or exact tidal boundary; stripping is strongest near pericentre and the instantaneous radius varies around the orbit.

Because the dark component is more extended, tidal stripping first sends dark matter through both $L_1$ and $L_2$, producing leading and trailing dark-matter tidal tails. The compact stellar component is stripped more deeply and also produces a leading and a trailing stellar tail. The two constituents therefore give four tails distinguished by composition, with the dark tails broader and more extended.