Question

Show explicitly that for very massive stars (opacity dominated by Thompson scattering, pressure from radiation) that luminosity is proportional to mass.

Answer 1

We know that, for very massive stars for which the opacity is dominated by Thompson scattering, The main source of radiation is the Luminosity .

The star emits spherically symmetric radiation.

Now Let us Calculate the force that the photon exerts on surrounding gas when some of it is absorbed.

The Energy Flux at a Distance r is given by=

Now the Momentum of the photon is given by

So the momentum flux is given by

Let us assume the source is surrounded by the gas of opacity and density

then the fraction of radiation absorbed by traveling a distance is,

Now is defined as the column density of the gas.

so the can be labeled as the fraction of radiation absorbed by the unit column density of the gas.

So the outward force exerted by the radiation of the gas is,

  

Inward force due to the gravity is,

In the equllibrium, we have so that we have,

SO that Luminosity is directly proportional to the mass of the star .