contraction of a star

Question

When a star of mass M is devoid of a nuclear energy source it will keep the same luminosity L for quite a while by consuming its gravitational energy through contraction. Assume the star can remain in a state of hydrostatic equilibrium and its gravitational energy is given by:

Ω = −α(GM^2)/ R

where G is the gravitational constant, α is a constant and R is the radius of the star.

1. Prove that the rate of contraction of its radius R can be expressed by:

dR/dt =R=-(R0/τ)/`[1+ (t /τ)]^2

where R0 is the star’s initial radius when it starts to contract and:

1. τ = α(GM^2)/(2R0L)

Answer 1

Start from the gravitational potential energy of the star

Ω = − α GM² / R

As the star contracts, gravitational energy is released and powers the luminosity. Therefore the luminosity equals the rate at which gravitational energy decreases:

L = − dΩ/dt

Differentiate Ω with respect to time. Since M and G are constant,

Ω = − α GM² R⁻¹

dΩ/dt = − α GM² d(R⁻¹)/dt

But

d(R⁻¹)/dt = − (1/R²) dR/dt

so

dΩ/dt = α GM² (1/R²) dR/dt

Substitute into L = − dΩ/dt:

L = − α GM² (1/R²) dR/dt

Solve for dR/dt:

dR/dt = − (L R²)/(α GM²)

This gives a differential equation for the radius evolution.

Separate variables:

dR / R² = − (L/(α GM²)) dt

Integrate:

∫ R⁻² dR = − (L/(α GM²)) ∫ dt

−1/R = − (L/(α GM²)) t + C

Multiply by −1:

1/R = (L/(α GM²)) t + C'

Use the initial condition R(0) = R₀:

1/R₀ = C'

Thus

1/R = (L/(α GM²)) t + 1/R₀

Solve for R:

R = 1 / [ (L/(α GM²)) t + 1/R₀ ]

Multiply numerator and denominator by R₀:

R = R₀ / [1 + (L R₀ /(α GM²)) t]

Define the characteristic timescale

τ = α GM² / (2 R₀ L)

Rewriting the solution in terms of τ gives

R(t) = R₀ / (1 + t/τ)

Now differentiate this with respect to time:

dR/dt = − (R₀/τ) / (1 + t/τ)²

which is the required expression for the contraction rate.

Thus the radius decreases with time according to

dR/dt = − (R₀/τ) / (1 + t/τ)²

with

τ = α GM² / (2 R₀ L)

This timescale is essentially the Kelvin–Helmholtz contraction timescale describing how long a star can shine by releasing gravitational energy.