Table of Contents

Orbital Mechanics

Student Activity

A Monster at the Centre of the Galaxy: Answer Key

Orbits to the Rescue

This is an historic image. It shows a luminous star designated S2 which has been tracked by astronomer Andrea Ghez and a team at UCLA since the early 1990s.

The orbital period of S2 is approximately 15 years and it has a major axis of about 11 light days (the distance light can travel in 11 days).

M (in units of the Sun's mass) = a³/p²

where a is the semi-major axis in AUs and p is the orbital period in years.

Assignment

1. Using the observed orbital information calculate the mass of the object at the centre of the galaxy

   • in solar mass units.

     ANS.: approximately 3.9 x 10⁶ solar mass units
     Hint: Convert 5.5 light days into minutes and then into AUs to find the value of a, the semi-major axis of the stars orbit.

   • in kilograms.

     ANS.: 7.8 x 10³⁶ kg

2. The classical size of a black hole ( that is, the radius ( r ) of its event horizon), is given as

   r = 2MG/c²

   What is the radius of this black hole's event horizon,

   • metres?

     ANS.: 1.1 x 10¹⁰m

   • AUs?

     ANS.: 7.3 x 10^-2 AU

     This is an incredibly large mass of about 4 million solar masses in a volume of space less than 1/10 of an AU in radius!

G = 6.67 x 10^-11 Nm²/kg²
1AU = 8.3 light minutes
1AU = 1.5 x 10¹¹m
c = 3 x 10⁸ m/s
1 solar mass = 1.99 x 10³⁰ kg