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

Question

Claire Anctil · Accepted Answer

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

A supermassive black hole's mass can be inferred from its orbiting star's semi-major axis distance and period. A star in our galaxy by the name of S2 orbits the central supermassive black hole at a semi-major distance of roughly 970 astronomical units (AU) and a period of 15.2 years. These values, which come from observations, show that it is 120 astronomical units away from the central black hole at its closest point. 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 imes 10^6# solar masses. This simple calculation does not take into account relativistic effects and the mass has been calculated as #4.1 * 10^6# solar masses. A solar mass is #1.989 imes 10^{30}Kg#. This makes the supermassive black hole have a mass of #8.15 imes 10^{36}Kg#. 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 imes 10^{10}m=0.085AU#. The density # ho# of the supermassive black hole can be calculated. $$ ho = \frac{M}{4\pi r_s^3}$$ This makes the density #3 imes 10^5kg /m^3#. This is much less dense than a neutron star which has a density of about #4 imes 10^{17}kg /m^3#. Thus, one can easily infer a black hole's density from its mass.

Owen Atkinson · Answer

undefined Indeed, it is impossible to obtain a direct measurement of a supermassive black hole's density.