Derivation of equation for star-planet system orbiting com

[moonkey]

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 ?

Homework Equations

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)<--r_star-->com<------r_planet------>(planet)

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

MV-mv=0
Mr_star-mr_planet=0

MV-mv=Mr_star-mr_planet
MV-Mr_star=mv-mr_planet

Period:
2pi*r_star=PV
2pi*r_planet=Pv

r_star=PV/2pi
r_planet=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...

Please help

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

 

[Delphi51]

Put the m1*v1 = m2*v2 together with v2 = 2πR2/T2 to get
m1*v1 = 2πm2*R2/T2
That is a formula relating the m and v of the star to the mass and period of the planet.

Note that the period of the star is the same as the period of the planet's orbit - the pair must remain opposite each other in their orbits so they must be perfectly synchronized.