Solution

By the gravitational shell theorem, the inward acceleration at radius

r

G m (r) / r^{2}

, where

m (r) = 4 π \int_{0}^{r} ρ (s) s^{2} d s

. Force balance for a thin fluid shell gives

- \nabla P = ρ \nablaΦ

, and hence

P^{'} (r) = - G m (r) ρ (r) / r^{2} .

Assembling shells from infinity gives

E_{grav} = - \int_{0}^{R} G m (r) d m (r) / r = - 4 π G \int_{0}^{R} m ρ r d r

E_{grav} = 4 π \int_{0}^{R} r^{3} P^{'} (r) d r = 4 π R^{3} P (R) - 12 π \int_{0}^{R} r^{2} P (r) d r .

For a star with negligible external pressure,

P (R) = 0

, this is $E_{grav} = - 3 ⟨ P ⟩ V$ . Nonzero external pressure would add the displayed surface term.

For a nonrelativistic ideal gas, the virial relation is

E_{grav} = - 2 E_{kin}

R^{- 2}

, faster than the magnitude

R^{- 1}

of gravity, providing restoring support; its adiabatic index

5/3

exceeds

4/3

. For an ultrarelativistic gas,

E_{grav} = - E_{kin}

and both terms scale as

R^{- 1}

4/3

in this idealized Newtonian model. Corrections such as general relativity can then destabilize it.

Integrating the kinetic energy

p^{2} / (2 m_{p})

over filled momentum states gives

E_{kin} = \frac{4 π g _{s} V}{h ^{3}} \int_{0}^{p_{F}} \frac{p ^{4}}{2 m _{p}} d p = \frac{4 π g _{s} V h ^{2}}{10 m _{p}} (p_{F} / h)^{5} .

Since

p_{F} = h [3 n / (4 π g_{s})]^{1/3}

, the pressure is

P = \frac{h ^{2}}{5 m _{p}} (\frac{3}{4 π g _{s}})^{2/3} n^{5/3} \sim \frac{h ^{2}}{m _{p}} n^{5/3} .

Finally

∣ E_{grav} ∣ \sim G M^{2} / R

, whereas

P V \sim (h^{2} / m_{p}) [M / (m_{p} R^{3})]^{5/3} R^{3}

. Equating these by the virial theorem gives

R \sim \frac{h ^{2}}{G m _{p}^{8/3}} M^{- 1/3} .

This is an order-of-magnitude estimate for nonrelativistic degeneracy support, not a uniform-density exact stellar solution.

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