3.4 The Jeans mass for fragmentation

The typical mass of a planet formed through fragmentation can be estimated starting from the definition of its Jeans mass. The Jeans criterion states that a gas cloud will collapse if the cloud’s kinetic energy is less than the magnitude of its gravitational energy; the minimum mass of a gas cloud for which the Jeans criterion is met is known as the Jeans mass. For the geometry of a protoplanetary disc, the Jeans mass in terms of the surface density is

where T and are the temperature and mean molecular mass of the gas. Noting that the sound speed is given by (Equation 3), so the Jeans mass may be written as

Using Q = 1 to express the surface density Σ for a disc that is just becoming unstable (Equation 22) this becomes:

Then, recognising that (Equation 6) and (Equation 2), we have

Therefore, this gives the final result: