F = G M(r) r dr/ r2
Note that P has been used to denote the total pressure. Thus P is the sum of the gas pressure and the radiation pressure.
The two equations, for dP/dr and dM/dr represent the basic equations of stellar structure, without which the innards of a star would be inaccessible to investigation.
Problems for the budding astrophysicist:
1) In a given layer for a 10 solar mass star, the composition is 90% hydrogen, 10% helium. Assume the layer is in local thermodynamic equilibrium. The temperature is taken to be 3.14 x 106 K. Take the density of the material as r = 20.2 kg m-3. Find the relative contributions of gas and radiation pressure in the layer and the thermal energy per kilogram of the material. (Assume the gas is totally ionized).
2) Derive a third equation for stellar structure where dL/dr is the subject. (Hint: Make use of the energy generation form for dL/dM and the chain rule for derivatives).
3) From one or more equations of stellar structure, obtain an estimate for the Sun’s central temperature and pressure. (Take the solar radius R = 7 x 108 m and the solar mass M = 2 x 1030 kg, and the density r = 1400 kgm-3. )