# Gravitational Radius

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### Gravitational Radius

The formula for the gravitational radius of an object is

\[

r_{g} \equiv \frac{GM}{c^{2}}

\]

where

$G$ is the gravitational constant,

$M$ is the black-hole mass, and

$c$ is the speed of light.

We can conveniently express this in terms of solar masses:

\[

r_{g} = 1.4822 \left(\frac{M}{M_{\odot}}\right) \ {\rm km}

\]

or simply,

\[

r_{g} \sim 1.5 \left(\frac{M}{M_{\odot}}\right) \ {\rm km}

\]

where $M_{\odot}$ is the mass of the sun. For much larger black-hole masses it is more convenient to write:

\[

r_{g} = 1.4822 \times 10^{13} \ M_{8} \ {\rm cm},

\]

where $M_{8}$ is the mass of the black hole in units of $10^{8}$ solar masses.

The following is also useful:

\[

r_{g} \sim \left(\frac{M}{10^{8}M_{\odot}}\right) \ {\rm AU} \ = \ M_{8} \ \ {\rm AU},

\]

where AU is the astronomical unit.

Note that the event horizon of a non-spinning black hole is at $2r_{g}$, which is also equal to the Schwarzschild radius.