The supermassive black hole at the centre of the Galaxy

What just happened?

We track stars (like S2) moving on tight, elliptical orbits around an invisible point — Sgr A* at the centre of the Milky Way. Knowing the orbit size $a$ and the orbital period $T$, we determine the mass of whatever holds them.

Kepler's third law (mass of the central object):

In convenient units ($a$ in AU, $T$ in years, $M$ in solar masses) this simplifies to $M = a^3/T^2$. For S2 ($a \approx 1000\,\text{AU}$, $T \approx 16$ yr) this gives about 3.9 million $M_\odot$.

Why must it be a black hole?

Millions of solar masses fit into a volume smaller than the star's orbit — no star cluster or gas cloud reaches such a density without collapsing. The only object that explains this is a supermassive black hole. The measurements of stellar orbits around Sgr A* were awarded the 2020 Nobel Prize.

Why this matters for your exam:

Kepler's third law $T^2 \propto a^3$ lets you "weigh" an invisible body from the motion of its satellites alone — the key to problems about orbits and the masses of the Sun, planets and stars. The trick: write down $a$ and $T$, substitute into $M = 4\pi^2 a^3/(GT^2)$.

Remember: the motion of the stars reveals the mass of whatever holds them. Millions of $M_\odot$ in a tiny volume ⇒ a supermassive black hole.