Planets and Satellites: Kepler's Laws — Question 10

Question 10

Repeated alignments measure a relative angular period

Two satellites follow coplanar circular prograde orbits about the same central mass, with periods $10$ and $15$ days. They start on the same outward radial line.

1. Find the time to their next alignment on the same outward radial line.

2. Find their orbital-radius ratio and explain why the alignment time is not their period difference.

Question 10 - Solution

Repeated alignments measure a relative angular period

(a) Derivation and calculation

For prograde motion, relative phase grows at $$\Delta\omega=\frac{2\pi}{10}-\frac{2\pi}{15} =\frac{2\pi}{30}\ \mathrm{rad/day}.$$ A same-direction alignment requires a full relative revolution: $$\boxed{T_{\mathrm{syn}}=\frac{2\pi}{\Delta\omega} =\frac1{1/10-1/15}=30\ \mathrm{days}}.$$

(b) Result and interpretation

Kepler scaling gives $$\boxed{\frac{r_{15}}{r_{10}}=\left(\frac{15}{10}\right)^{2/3} =1.31037}.$$ Subtracting periods gives $5$ days, which is not an alignment condition. The phase difference after that time is only $$\Delta\theta=(2\pi/30)(5)=\pi/3.$$

Physical check

After 30 days the inner satellite completes three revolutions and the outer two, returning to the same outward radial line.