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

cap m sub Jeans equals one divided by cap sigma times left parenthesis two times k sub cap b times cap t divided by cap g times m macron right parenthesis squared comma

where T and m macron are the temperature and mean molecular mass of the gas. Noting that the sound speed is given by c sub s squared equals k sub cap b times cap t solidus m macron (Equation 3), so the Jeans mass may be written as

cap m sub Jeans equals four times c sub s super four divided by cap g squared times cap sigma full stop

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

equation sequence part 1 cap m sub Jeans equals part 2 four times c sub s super four divided by cap g squared times pi times cap g divided by c sub s times omega sub cap k equals part 3 four times pi times c sub s cubed divided by cap g times omega sub cap k full stop

Then, recognising that cap h equals c sub s solidus omega sub cap k (Equation 6) and omega sub cap k squared equals cap g times cap m sub asterisk operator solidus r cubed (Equation 2), we have

cap m sub Jeans equals four times pi divided by cap g times omega sub cap k multiplication left parenthesis cap h times omega sub cap k right parenthesis cubed

cap m sub Jeans equals four times pi divided by cap g multiplication cap h cubed multiplication cap g times cap m sub asterisk operator divided by r cubed full stop

Therefore, this gives the final result:

cap m sub Jeans equals four times pi times cap m sub asterisk operator times left parenthesis cap h divided by r right parenthesis cubed full stop

Equation label:(Equation 25)

Estimate the Jeans mass for a typical disc with H/r = 0.05, centred on a Sun-like star.
Inserting values into Equation 25:

(This is about 1.6 Jupiter masses.)

OpenLearn - The formation of exoplanets
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