Worked example 12.5: Mass of star

Question: A planet is in circular orbit around a star. The period and radius of the orbit are $T= 4.3\times 10^7 {\rm s}$ and $r=2.34\times 10^{11} {\rm m}$, respectively. Calculate the mass of the star.

Answer: Let $\omega$ be the planet's orbital angular velocity. The planet accelerates towards the star with acceleration $\omega^2 r$. The acceleration due to the star's gravitational attraction is $G M_\ast/r^2$, where $M_\ast$ is the mass of the star. Equating these accelerations, we obtain

$$\omega^2 r = \frac{G M_\ast}{r^2}.$$

Now,

$$T = \frac{2 \pi}{\omega}.$$

Hence, combining the previous two expressions, we get

$$M_\ast = \frac{4 \pi^2 r^3}{G T^2}.$$

Thus, the mass of the star is

$$M_\ast = \frac{4\times\pi^2\times (2.34\times 10^{11})^3}{(6... ...11})\times (4.3\times 10^7)^2} = 4.01\times 10^{30} {\rm kg}.$$