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

Jun 4, 2018

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 $T$ in years and the semi-major distance from the black hole $a$ in AU we can calculate the mass of the black hole it is orbiting around.

Kepler's third law relates $T$ and $a$ in terms of the mass of the central body (in this case the black hole) $M$ where the mass is in solar masses.

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

This gives the mass of the central supermassive black hole as $3.95 \setminus \times {10}^{6}$ solar masses.
This simple calculation does not take into account relativistic effects and the mass has been calculated as $4.1 \cdot {10}^{6}$ solar masses. A solar mass is $1.989 \setminus \times {10}^{30} K g$.
This makes the supermassive black hole have a mass of $8.15 \setminus \times {10}^{36} K g$.

The Schwarzschild radius ${r}_{s}$ defines the radius of the event horizon of a black hole.
It is defined in terms of the gravitational constant $G$, the mass of the black hole $M$ and the speed of light $c$.

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

This makes radius of the black hole ${r}_{s} = 1.27 \setminus \times {10}^{10} m = 0.085 A U$.

The density $\setminus \rho$ of the supermassive black hole can be calculated.

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

This makes the density $3 \setminus \times {10}^{5} k g \frac{\setminus}{m} ^ 3$. This is much less dense than a neutron star which has a density of about $4 \setminus \times {10}^{17} k g \frac{\setminus}{m} ^ 3$.

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