# How do astronomers measure the mass of a supermassive black hole?

Jan 9, 2016

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

#### Explanation:

In our galaxy there is a star called S2 which is orbiting the central black hole with a period of 15.2 years and a distance of about 970 AU. These values have been obtained from observations.

So, given the period of the star $T$ in seconds and the semi-major distance from the black hole $a$ in metres 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 gravitational constant $G$, the mass of the central body (in this case the black hole) $M$ and the mass of the star $m$ using the following equation:
${T}^{2} / {a}^{3} = \frac{4 {\pi}^{2}}{G \left(M + m\right)}$

Now the mass of the star is unknown but is is so small compared to the black hole's mass that it can be ignored. So now we have the mass of the black home defined by the equation:
$M = \frac{4 {\pi}^{2} {a}^{3}}{G {T}^{2}}$

The supermassive black hole at the centre of our Milky Way galaxy has been estimated at 4.1 million times the mass of the Sun. A staggering $8 \cdot {10}^{36} k g$