Outline

The Virial Theorem and Kelvin-Helmholtz Time Scale

(1/2) d ²I /dt ² = 2K + Ω [ = 0 in hydrostatic equilibrium].
I = total moment of inertia = ∫ r ²dm
K = total kinetic energy
Ω = total gravitational potential energy.

Derived from ideal gas of individual particles:
assume a collisionless perfect gas, then for individual particle i of mass m_i, F = ma is

m_id ²r_i /dt ² = − Gm_i∑ _j m_j (r_i − r_j) / |r_i − r_j| ³ = ∑ _j F_ij .

Start with the time derivative:
d ∑ _i m_i v_i ^. r_i /dt = ∑ _i (d m_i v_i /dt) ^. r_i + ∑ _i m_i v_i ^. d r_i /dt .

The left-hand side is
(1/2) d ²I /dt ².
The second term on the right-hand side is 2K.
The first term on the right-hand side is
∑ _i F_i ^. r_i = ∑ _pairs F_ij ^. r_i + F_ji ^. r_j = ∑ _pairs F_ij ^. (r_i − r_j)
which from above is
− ∑ _pairs Gm_im_j / |r_i − r_j| = Ω.

2.2. Pressure

The pressure of an isotropic fluid is
P = (1/3) ∫ n(p) pv d ³p
where n(p) is the number distribution of particles having momentum p per unit geometrical volume per unit momentum volume.

Homework: show that 2K = 3 ∫ P dV = 3 ∫ (P /ρ) dm

For ideal gasses, integrated internal energy U = K, and integrated total energy W = U + Ω = Ω/2 .

For gasses described by a γ-law, we write
P = (γ −1)ρE,
where E is the local specific internal energy per gram of material.
Ideal gas: γ = 5/3.
Radiation: γ = 4/3.
Relativistic Fermi degeneracy: γ = 4/3.
Then 2K = 3(γ −1) ∫ E dm = 3(γ −1)U, and the virial theorem becomes

(1/2) d ²I /dt ² = 3(γ −1)U + Ω .

Homework: write U and Ω in terms of M, R, and f for our two-layer star. Assume all the material is at the same temperature. Pretty easy, since everything can be derived from K. Now, find the Kelvin-Helmholtz time scale t_KH = R / (dR /dt) by assuming L = − dW /dt = constant. It should, of course, be t_KH = constant × GM ²/LR.

Homework answers