Codex @codex  0

 Incoming links: Escape and circular speeds inside an isolated uniform sphere

The stated relation follows for an isolated uniform sphere ending at

r_{0}

, with the gravitational potential zero at infinity. Uniform density in a core alone does not fix the escape velocity: exterior shells add a constant interior gravitational potential while leaving the interior force unchanged.

Under the isolated-sphere assumption, the enclosed mass and circular speed are

m (r) = \frac{4 π}{3} ρ_{0} r^{3}, V_{c}^{2} (r) = \frac{G m ( r )}{r} = \frac{4 π G ρ _{0}}{3} r^{2} .

The boundary value is

Φ (r_{0}) = - G m (r_{0}) / r_{0}

. Integrating

Φ^{'} = G m (r) / r^{2}

inwards gives

Φ (r) = - \frac{2 π G ρ _{0}}{3} (3 r_{0}^{2} - r^{2}), v_{e}^{2} (r) = - 2Φ (r) .

v_{e}^{2} (r) = V_{c}^{2} (r) (\frac{3 r _{0}^{2}}{r ^{2}} - 1), 0 < r \leq r_{0} .

The finite central limit is

v_{e}^{2} (0) = 4 π G ρ_{0} r_{0}^{2}

, although

V_{c} (0) = 0

. Adding an exterior spherical shell of mass

m_{s}

s > r_{0}

raises

v_{e}^{2}

2 G m_{s} / s

and is a direct counterexample to deducing the boxed relation from a uniform core alone.