An Introduction To The Basics Of Stellar Structure

 

                                    Detailed structure of the Sun

Stellar structure, the study of the physical structure of stars and how it influences their evolution, is among the most fundamental aspects of astrophysics. To get some physical idea of stellar structure the reader may refer to the diagram below showing the inner onion-like structure of a spherical gas system such as found in a star.

                                                        Basic Stellar structure 

Here, three layers, A, B and C are depicted in the interior of the ten solar mass star. The layer B was the one under consideration in the problem identified by the increment or differential radius, dr. Thus, since layer A is further removed from the core, it would be at LTE but at a cooler temperature, while C would be at LTE at a hotter temperature. (Thus a gradient of temperature, DT/ DR, exists from the core to the surface).

    From the gas equations of state we can easily see that greater (combined) gas pressures accompanies greater stellar depths. Note therefore from the diagram that the respective layer pressures are in the relation:

P_C > P_B > P_A

And in all cases the pressure characteristic of the layer is directed outwards, i.e. toward the surface.  Clearly then, some force must be acting in the opposite direction to resist the outward pressure and maintain the star in a state of balance. Otherwise the star would simply inflate, or contract. 

The fact stable stars like the Sun exist, shows that a pressure-gravity balance must obtain. To see how this can be quantified, consider the element of layer B shown of width dr and lying between r and (r + dr). 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 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. Consider now the mass of the shell between layer A and 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]

Suggested Problem:

a) Derive a third equation for stellar structure where dL/dr is the subject. (Hint: Make use of the energy generation form for dL/dM).

b) 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. )