Planets and Satellites: Kepler's Laws — Question 6

Question 6

A binary period measures total mass and a relative semimajor axis

Two isolated stars have masses in ratio $M_1/M_2=3$. The semimajor axis of their relative separation is $a=10^{10}\,\mathrm m$ and the period is $T=2\times10^6\,\mathrm s$. Use $T^2=4\pi^2a^3/[G(M_1+M_2)]$ and $G=6.67\times10^{-11}$ SI.

1. Find their total mass and individual masses.

2. Find the two semimajor axes about the common mass center.

Question 6 - Solution

A binary period measures total mass and a relative semimajor axis

(a) Derivation and calculation

The relative orbit measures total mass: $$M_{\mathrm{tot}}=\frac{4\pi^2a^3}{GT^2} =\frac{4\pi^2(10^{10})^3}{(6.67\times10^{-11})(2\times10^6)^2} =1.47970\times10^{29}\ \mathrm{kg}.$$ With mass ratio $3:1$, $$\boxed{M_1=\tfrac34M_{\mathrm{tot}}=1.10978\times10^{29}\ \mathrm{kg}},$$ $$\boxed{M_2=\tfrac14M_{\mathrm{tot}}=3.69925\times10^{28}\ \mathrm{kg}}.$$

(b) Result and interpretation

The center-of-mass conditions are $$M_1a_1=M_2a_2,\qquad a_1+a_2=a.$$ Therefore $$\boxed{a_1=\frac{M_2}{M_{\mathrm{tot}}}a=2.5\times10^9\ \mathrm m},$$ $$\boxed{a_2=\frac{M_1}{M_{\mathrm{tot}}}a=7.5\times10^9\ \mathrm m}.$$ The period formula uses the relative axis $a=a_1+a_2$.

Physical check

The more massive star has the smaller orbit, and the two semimajor axes sum to the stated relative separation axis.