Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 / 3 / iv / Solution

Using the supplied approximate conversion factors, $D=50\times 10^3\times 3\times 10^{16}=1.5\times 10^{21}\mathrm{m}$, ${\sigma }_d=10^4\mathrm{m}{\mathrm{s}}^{-1}$, and ${\sigma }_h=3\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$. The circular-orbit Jacobi tidal radius isThe dwarf mass isTo one significant figure,The rough equal-density convention instead gives $r_t=1.67\mathrm{kpc}$ and $m_d=7.14\times 10^7{M}_{\odot }$, or $2\mathrm{kpc}$ and $7\times 10^7{M}_{\odot }$ to one significant figure. The unrounded results are retained for subsequent ratios. For the circular prescription $r_t/D\simeq 0.024$, so the local expansion is well controlled.