The star S2 orbits the compact object at the Galactic centre with orbital period and…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · Gravitation and central-force motion

Relativity and Gravitation, Ch Escape velocity, the Schwarzschild radius and the idea of a black hole

Problem

The star S2 orbits the compact object at the Galactic centre with orbital period

T = 16.05\ \mathrm{yr}

and semi-major axis

a_{\rm orb} = 1030\ \mathrm{au}

. Model the orbit as a Kepler ellipse about a single central mass

M

that is much larger than the star's own mass, and ignore all other matter. Data:

1\ \mathrm{yr} = 3.156 \times 10^{7}\ \mathrm{s}

1\ \mathrm{au} = 1.496 \times 10^{11}\ \mathrm{m}

G = 6.674 \times 10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}

c = 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}

M_\odot = 1.989 \times 10^{30}\ \mathrm{kg}

. ((a)) Estimate

M

in kg. ((b)) Express

M

in solar masses. ((c)) Compute the Schwarzschild radius of a black hole of this mass, and express it in au.