Consider the dense core of a gmc which has a mass of 103 m and a radius of 0.1 pc.

(a) assume that the core starts to collapse and that the gravitational energy released during the collapse is radiated away, i. e. that it escapes from the cloud so that the temperature remains roughly constant for this phase of the collapse. calculate the free-fall timescale for this phase.

(b) next, as the cloud collapses further and increases in density, assume that the gravitational energy can no longer escape and instead goes into the dissociation of molecular hydrogen (releasing 4.5 ev per h2 molecule) followed by the ionisation of atomic hydrogen (releasing 13.6 ev per h1 atom), thus increasing the temperature. calculate the final temperature on the assumption that all of the core is initially h2 which is first completely converted to h1 which in turn is completely ionised.

(c) during this collapse the core is expected to fragment into smaller clumps and we can assume that the collapse of a clump ends when the gas becomes optically thick and is in thermal equilibrium, at which point it can be assumed to be a blackbody. by equating the luminosity generated by the release of gravitational energy during free-fall to the blackbody luminosity at the end of the phase, calculate the minimum mass allowed for a clump (which will be the minimum mass of a star that can form by this process).

you can take the typical radius of the clump at this point to be ∼ 100 r (solar masses).