# Derivation of equation for star-planet system orbiting com

## Homework Statement

(b) For a mass m1 in circular orbit with velocity v1, and a mass m2 orbiting with velocity
v2 (about a common center of mass), the following holds: m1/m2 = v2/v1. Use this
relation to derive an expression relating the mass and velocity of a star with the period
and mass of an orbiting planet.

(c) The planet in (a) orbits with a period of 1 month. If the velocity of the star as a
result of this is 100 m s−1, calculate the density of the planet. Is this an Earth-type
planet, or a gaseous giant ?

## The Attempt at a Solution

I can't figure out part (b) and I need it to do part (c) and I think part (d) which I haven't shown here.

Here's what I've tried so far

(star)<--rstar-->com<------rplanet------>(planet)

let mass and velocity of star be M and V
let mass and velocity of planet be m and v

MV-mv=0
Mrstar-mrplanet=0

MV-mv=Mrstar-mrplanet
MV-Mrstar=mv-mrplanet

Period:
2pi*rstar=PV
2pi*rplanet=Pv

rstar=PV/2pi
rplanet=Pv/2pi

MV-MPV/2pi=mv-mPv/2pi

As you can probably see, that brings me back around to square 1.
I just can't seem to work it out in such a way that I won't need at least 1 of the distances and as you can see in part (c) I'm only given two values to calculate the density so...

I'll thank, anyone who helps me, in advance because if I leave it too long I won't want to bump this thread, so thank you in advance