# Binary Star Mass

## Homework Statement

How do I show that for a binary star system, if one star has mass ##M_s##, speed ##V_s##, period ##P##, the mass of the other star is given by: ##M_P^3 \approx \frac{V_s^3}{2\pi G} PM_s^2##?

## The Attempt at a Solution

$$\frac{GM_pM_s}{(a_p+a_s)^2} = \frac{M_s v_s^2}{a_s}$$
Substituting in ##PV_s=2\pi a_s##:
$$M_p = \frac{2\pi(a_p+a_s)^2V_s}{PG}$$
Using kepler's second law: ## P^2 = \frac{(a_p+a_s)^3(2\pi)^2}{G(M_p+M_s)} ##:
$$M_p^3 = \frac{V_s^3}{2\pi G} P (M_p+M_s)^2$$

Seems correct to me. The mass of the planet is more often than not negligible compared to the mass of a star, so $M_{p} << M_{S}$
Seems correct to me. The mass of the planet is more often than not negligible compared to the mass of a star, so $M_{p} << M_{S}$