# Harvard Astronomy 201b

## Isolation Mass

In Uncategorized on April 27, 2011 at 3:45 pm

If we assume that all planets and planetesimals move on roughly circular orbits, then there is a finite supply of planetesimals for a growing planet to accrete.  The mass at which the growing planet has consumed all of its nearby planetesimal supply is called the isolation mass $M_\text{iso}$.

The “feeding zone” $\delta a_\text{max}$ of a planet accreting planetesimals is of order the Hill radius.  Thus we can write $\delta a_\text{max} = C r_H$, where $C$ is a constant of order unity, and $r_H$ is the Hill radius.  Furthermore, we note that the mass within this feeding zone can be calculated from the surface density of the disk $Sigma_p$: $(2 \pi a) (2 \delta a_\text{max}) \Sigma_p \propto M_\text{planet}^{1/3}$

Finally we can set the isolation mass to be the mass at which the planet mass is equal to the mass of the planetesimals in its feeding zone: $M_\text{iso} = 4 \pi a C (\frac{M_\text{iso}}{3 M_*})^{1/3} a \Sigma_p$.  We can solve this equation to find that $M_\text{iso} = \frac{8}{\sqrt{3}} \pi^{3/2} C^{3/2} M_*^{-1/2} \Sigma_p^{3/2} a^3$.*

*See discussion in Armitage 2007