Thayer Watkins
Silicon Valley
& Tornado Alley

Models of Stars

The purpose of this analysis is to derive the governing equations for three simple models of a star:

Model I

In the absence of rotation the body of the star is spherically symmetric so the only significant spatial variable is the distance r from the center. The gravitational attraction at r is balanced by the pressure gradient force; i.e.,

GM(r)/r2 = -(1/ρ)∂p/∂r

where G is the gravitational constant, M is the mass contained within a radius r, ρ is the mass density and p is pressure.

The mass contained within a radius r, M(r), is given by:

M(r) = ∫0r4πρ(s)s2ds

It is assumed that the star material obeys the ideal gas law; i.e.,

p = ρRT

where R is the gas constant for the star material and T is the temperature in degrees Kelvin. For a steady state the temperature is uniform throughout the star. This means that

∂p/∂r = RT∂ρ/∂r

The governing equation for the internal structure of the star is:

M(r) = ∫0r4πρ(s)s2ds = (r2/G)(RT/ρ)(∂ρ/∂r)
= (RTr2/G)(∂(ln(ρ))/∂r)

The usual procedure for dealing with integral-differential equations is to differentiate in order to eliminate the integration operation. But first a change of variable is in order. Let ln(ρ)=W so ρ=exp(W) and thus the governing equation is:

0rexp(W)4πs2ds = (RT/G)∂W/∂r

Now differentiation with respect to r gives:

exp(W)(4πr2) = (RT/G)∂2W/∂r2

Upon rearrangement this becomes:

e-W2W/∂r2 = (4πG/RT)r2

Model II

In this model there is the creation of heat within the star matter. There is a balance between forces due to gravitation and the pressure gradient and the ideal gas law is assumed to apply as in Model I but the temperature is not uniform. Steady state conditions require that the heat created within the star be transferred to the surface and this requires a radial temperature gradient.

The heat energy passing through a surface is proportion to the area times the temperature gradient. The proportionality factor is negative because heat is transferred in the direction that temperature is declining. The net flow outflow from an infinitesimal volume is therefore proportional to the divergence of the temperature gradient. But for steady state conditions this net outflow must be equal to the heat generated within the infinitesimal volume. Thus

cρ - D∇2T = 0

where c is the rate of production of heat per unit mass and D is the coefficient of heat conduction. As in Model I, ρ and T represent the mass density and temperature of the star material. This equation is of the form of a Poisson equation. The Laplacian of T, ∇2T, for spherical coordinates when there is spherical symmetry is:

2T = (1/r2)∂(r2∂T/∂r)/∂r).

The full version of Model II is:

GM(r)/r2 = (1/ρ)∂p/∂r
M(r) = ∫0r4πρ(s)s2ds
p = ρRT
(1/r2)∂(r2∂T/∂r)/∂r) = (c/D)ρ

This last equation can be put into the form

∂(r2∂T/∂r)/∂r) = (c/4πD)ρ4πr2

which upon integration with respect to the radius variable gives

r2(∂T/∂r) = (c/4πD)∫0rρ4πs2ds
= (c/4πD)M(r)

This means that

M(r)/r2 = (c/4πD)(∂T/∂r)
M(r)/r2 = (1/G)(1/ρ)(∂p/∂r)
(Gc/(4πD))∂T/∂r = (1/ρ)(∂p/∂r)

Since from the ideal gas equation

∂p/∂r = RT∂ρ/∂r + Rρ∂T/∂r
and hence
(1/ρ)(∂p/∂r) = RT(1/ρ)(∂ρ/∂r) + R∂T/∂r
it follows that
γ∂T/∂r = RT(1/ρ)(∂ρ/∂r) + R(∂T/∂r)
and thus
(γ-1)(1/T)∂T/∂r = (1/ρ)∂ρ/∂r

where γ = Gc/(4πDR).

This last equation above implies that

ρ/ρ0 = (T/T0)γ-1

where ρ0 and T0 represents a standardized density and temperature.

Because from the ideal gas equation

(p/p0) = (ρ/ρ0)(T/T0)
it follows that
(p/p0) = (T/T0)γ

The temperature profile is determined thus from the Poisson equation

2T = CTγ

Model III

When radiation is produced within the star radiation pressure as well as gas pressure support the gravitational attraction of the star matter. The radiation does not simply radiate away. The star material is opaque and a quantum of radiation is absorbed and re-radiated randomly so the radiation must make its way to the surface by diffusion. A crucial parameter in this diffusion process is the average path length.

(To be continued.)

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