= Solution
Using the supplied approximate conversion factors, $D=50\times10^3\times3\times10^{16}=1.5\times10^{21}\,\mathrm m$, $\sigma_d=10^4\,\mathrm{m\,s^{-1}}$, and $\sigma_h=3\times10^5\,\mathrm{m\,s^{-1}}$. The circular-orbit <Jacobi tidal radius> is
$$
r_t=\frac{1.5\times10^{21}}{30\sqrt2}\,\mathrm m=3.54\times10^{19}\,\mathrm m=1.18\,\mathrm{kpc}.
$$
The dwarf mass is
$$
m_d=\frac{2(10^4)^2(3.54\times10^{19})}{7\times10^{-11}}\,\mathrm{kg}=1.01\times10^{38}\,\mathrm{kg}=5.05\times10^7M_\odot.
$$
To one significant figure,
$$
\boxed{r_t\simeq1\,\mathrm{kpc},\qquad m_d\simeq5\times10^7M_\odot.}
$$
The rough equal-density convention instead gives $r_t=1.67\,\mathrm{kpc}$ and $m_d=7.14\times10^7M_\odot$, or \b[$2\,\mathrm{kpc}$ and $7\times10^7M_\odot$] to one significant figure. The unrounded results are retained for subsequent ratios. For the circular prescription $r_t/D\simeq0.024$, so the local expansion is well controlled.
Back to article page
