Astronomers using the Hubble Space Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be at a distance of 60 light-years from the core. Deduce the mass of the core, and compare it to the mass of our Sun.

The distance of the gas clouds is\(d = 5.7 \times {10^{17}}\;{\rm{m}}\).

The speed of the gas clouds is\(v = 780\;\;{\rm{km/s}}\;{\rm{ = }}\;{\rm{780}} \times \;{\rm{1}}{{\rm{0}}^3}\;{\rm{m/s}}\).

The mass of the sun is\({m_s}\; = 2 \times {10^{10}}\;{\rm{kg}}\).

The gravitational attraction of the core provides centripetal acceleration.

The expression for the mass of the core is:

\({m_\varepsilon } = \frac{{{v^2}d}}{G}\)\( = \frac{{{{\left( {780 \times {{10}^3}\;{\rm{m/s}}} \right)}^2} \times 5.7 \times {{10}^{17}}\;{\rm{m}}}}{{6.67 \times {{10}^{ - 11}}\;{\rm{N}}{{\rm{m}}^{\rm{2}}}{\rm{/k}}{{\rm{g}}^{\rm{2}}}}}\)\( = \;5.2 \times {10^3}\;{\rm{kg}}\)

So, the mass of the core is \(5.2 \times {10^3}\;{\rm{kg}}\).

When comparing with the mass of the sun:

\(\begin{aligned}\frac{{{m_{\rm{C}}}}}{{{m_y}}} &= \frac{{5.7 \times {{10}^{10}}\;{\rm{kg}}}}{{2 \times {{10}^{10}}\;{\rm{kg}}}}\\\frac{{{m_C}}}{{{m_{\rm{y}}}}} &= 2.6\end{aligned}\)

So, the mass of the core is\(2.6\) times more than the sun.