## FANDOM

155 Pages

Let us consider an accretion disc around a star with mass $M_s$, with surface mass density $\Sigma$. Suppose a planet starts forming at a distance $a$ from the star. Assuming the planet can easily accrete gas that passes through its Hill sphere, but not anything from outside. How large can the planet grow? The upper limit on the planet's mass $M$ is given by the condition that the Hill radius of the planet is equal to all the disc mass it can accrete, i.e. an annulus of radius $a$ and width of the hill radius. The Hill radius is given by

$r_h \approx a \left(\frac{M}{M_s}\right)^{1/3}$

so the upper limit is given by

$M \approx \Sigma a r_h \Rightarrow M \approx \frac{\left(\Sigma a^2 \right )^{3/2}}{M_s^{1/2}}$

This mass is also called the isolation mass.