Introducing Some Advanced Astrophysics (3)

The fact stable stars like the Sun exist shows that a pressure-gravity balance must obtain within them - layer by layer. To see how this can be quantified, consider an element of  a defined layer of width dr and lying between layers at distances r and (r + dr) from the center. Let P be the pressure at r and let the increment in P be dP. The difference in pressure dP represents a force, -dP,  acting on the mass element considered in the direction of increasing r.

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

Or:

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. )