# Is it possible to measure the density of a supermassive black hole?

##### 1 Answer

Yes, the density of a black hole can be calculated from its mass.

#### Explanation:

The mass of a supermassive black hole can be estimated from the period and semi-major axis distance of a star orbiting it.

In our galaxy there is a star called S2 which is orbiting the central supermassive black hole with a period of 15.2 years and a semi-major distance of about 970AU.

At its closest point it is 120AU from the central black hole.

These values have been obtained from observations.

So, given the period of the star

Kepler's third law relates

$$M=\frac{a^3}{T^2}$$

This gives the mass of the central supermassive black hole as

This simple calculation does not take into account relativistic effects and the mass has been calculated as

This makes the supermassive black hole have a mass of

The Schwarzschild radius

It is defined in terms of the gravitational constant

$$r_s=\frac{2GM}{c^2}$$

This makes radius of the black hole

The density

$$\rho = \frac{M}{4\pi r_s^3}$$

This makes the density

So, the density of a black hole can easily be determined from its mass.