= Solution
For a spherical <planet> of <mass density> $\rho_p$, $R_p=(3M_p/4\pi\rho_p)^{1/3}$. The point-comet surface <escape velocity> and circular <Kepler orbit> speed therefore give
$$
\chi^2=\frac{2GM_p/R_p}{GM_\star/a_p}
=\frac{2M_pa_p}{M_\star}\left(\frac{4\pi\rho_p}{3M_p}\right)^{1/3}.
$$
Raise this equation to the power $3/2$ and solve for $M_p$:
$$
\boxed{M_p=\sqrt{\frac{3}{32\pi}}\,
\chi^3 M_\star^{3/2}a_p^{-3/2}\rho_p^{-1/2}}.
$$
Thus the constant of proportionality is $\boxed{\sqrt{3/(32\pi)}}$, and the <gravitational constant> cancels. Dimensional checking gives <mass> on the right: $M_\star^{3/2}a_p^{-3/2}\rho_p^{-1/2}$ has that dimension.
This uses the <planet>'s own <escape velocity>. If instead $\chi$ is defined at finite-comet contact, an additional factor $(1+R/R_p)^{3/2}$ appears on the right, and $R_p$ itself depends on $M_p$; the quoted pure-planet scaling is then only the $R/R_p\to0$ limit.
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