User:Reddwarf2956/orbital period


 * $$T = \tau \sqrt{a^3/\mu}$$

$$\mu = GM $$

$$ M = 2 \tau a^3 \rho / 3 $$.

Substitute

$$ T = \tau \sqrt{3a^3/(G 2 \tau a^3 \rho)} $$.

If $$a^3 = a^3$$ then

$$ T = \tau \sqrt{3/(G 2 \tau \rho)} $$.

$$ \tau = \sqrt{\tau^2} $$ therefore,


 * $$T = \sqrt{ \frac {3\tau}{2 G \rho} }$$

Formula for the Schwarzschild radius
The Schwarzschild radius is proportional to the mass with a proportionality constant involving the gravitational constant and the speed of light:


 * $$r_\mathrm{s} = \frac{2Gm}{c^2},$$

where:
 * $$r_s\!$$ is the Schwarzschild radius;
 * $$G\!$$ is the gravitational constant;
 * $$m\!$$ is the mass of the object;
 * $$c\!$$ is the speed of light in vacuum.

The proportionality constant, 2G/c2, is approximately $1.48 m/kg$, or $2.95 km/solar mass$.

An object of any density can be large enough to fall within its own Schwarzschild radius,
 * $$V_s \propto \rho^{-3/2},$$

where:
 * $$V_s\! = \frac{ 2 \tau}{3} r_\mathrm{s}^3$$ is the volume of the object;


 * $$\rho\! = \frac{ m }{ V_s\! }$$ is its density.

$$\rho\! = \frac{ 3 m }{2 \tau r_\mathrm{s}^3\! }$$