Math Problem Solver

Finding Mass in the Cosmos (Groups 2, 4, 6)

One of the neatest things in astronomy is being able to figure out the mass of a

distant object, without having to 'go there'. Astronomers do this by employing a very

simple technique. It depends only on measuring the separation and period of a pair

of bodies orbiting each other.

Imagine a massive body such as a star, and around it there is a small planet in orbit.

The force of gravity, Fg, of the star will be pulling the planet inwards, but there will

also be a centrifugal force, Fc, pushing the planet outwards.

This is because the planet is travelling at a particular speed, V, in its orbit. When the

force of gravity and the centrifugal force on the planet are exactly equal, the planet

will travel in a circular path around the star with the star exactly at the centre of the

orbit.

Problem 1: Research and find out equations that give the force of gravity, centrifugal

force and the velocity of the planet at any instant.

Problem 2: Use the three equations found in Problem 1, to derive the mass of the

primary body (M), given the period T and radius R of the companion's circular orbit.

Problem 3: Use the formula M =

4π

2R

3

GT2

, where G = 6.6726 x 10-11 N m2

/kg2 and M is

the mass of the primary body in kilograms, R is the orbit radius in metres and T is the

orbit period in seconds, to find the masses of the primary bodies in the table below.

Make a table showing the different values of R and T in each case and find the mass

of primary body. (Note: 1 light year = 9.5 trillion kilometres)

Primary Earth Jupiter Sun Milky Way Pluto

Companion Moon Callisto Earth Sun Charon