Planets and Satellites: Kepler's Laws — Question 5

Question 5

An orbital measurement determines the central gravitational parameter

A negligible satellite has semimajor axis $a=2a_0$ and period $T=T_0$. A reference system has central mass $M_0$, semimajor axis $a_0$ and period $T_0$, also with negligible satellite mass.

1. Infer the first central mass in units of $M_0$.

2. If measured $a$ is biased high by $1\%$ and $T$ is exact, find the exact fractional mass bias.

Question 5 - Solution

An orbital measurement determines the central gravitational parameter

(a) Derivation and calculation

Kepler’s law determines the gravitational parameter: $$\mu=GM=\frac{4\pi^2a^3}{T^2},\qquad \frac M{M_0}=\left(\frac a{a_0}\right)^3\left(\frac{T_0}T\right)^2.$$ With $a=2a_0$, $T=T_0$, $$\boxed{M=8M_0}.$$ Converting the directly inferred $GM$ to mass uses a value for $G$.

(b) Result and interpretation

If $a_{\mathrm{meas}}=1.01a_{\mathrm{true}}$ while $T$ is exact, $$\frac{M_{\mathrm{meas}}}{M_{\mathrm{true}}}=(1.01)^3 =1+3(.01)+3(.01)^2+(.01)^3.$$ $$\boxed{\frac{\Delta M}{M}=.030301=3.0301\%}.$$ The differential estimate $dM/M=3,da/a$ gives $3\%$ and omits the positive higher-order terms.

Physical check

The cubic dependence amplifies a positive semimajor-axis bias into a larger positive mass bias.