The mass of the element considered is: r dr. Then the force of attraction between M(r) e.g. the mass enclosed inside the sphere of radius, r and r dr is the same as that between a mass M(r) at the center and r dr at r. By Newton’s law this attractive force is given by:
F = G M(r) r dr/ r2
Since the attraction due to the material outside r is zero, we should have for equilibrium:
- dP = G M(r) r dr/ r2
dP/dr = - G M(r) r / 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.
Consider now the mass of the shell between outer layer A and deeper stellar layer C. This is approximately, 4p r2 r dr, provided that dr is small. The mass of the layer is the difference between M(r + dr) and M(r) which for a thin shell is:
M(r + dr) - M(r) = (dM/ dr) dr
Equating the two expressions for the mass of the spherical shell we obtain:
dM/dr = 4p r2 r
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.
A third equation of stellar structure may be derived using by using the equation for dM/dr in combination with the fact that a star’s luminosity is produced through the consumption of its own mass. This may be expressed mathematically as:
dL/dM = e
where e denotes the rate of energy generation. For the proton-proton cycle (for stars like the Sun- and designed for cgs units!):
e= 2.5 x 106 (r X2).· (106 /T)2/3 exp[-33.8(106 /T)1/3]
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. )