= Kepler's third law
{c}
{title2=$T^2=4\pi^2a^3/[G(M+m)]$}
= Kepler third law
{c}
{synonym}
For an isolated <two-body problem> in <Newtonian gravity>, the relative <elliptic orbit>'s <orbital period> $T$ obeys
$$
T^2=\frac{4\pi^2a^3}{G(M+m)}=\frac{4\pi^2a^3}{\mu}.
$$
Here $a$ is the relative <semi-major axis>, and $M+m$ is the sum of the two <masses>. The separation $r$ varies during an <elliptic orbit>, so replacing $a$ by $r$ works only for a <circular orbit>, where $r=a$. Likewise, either body's individual <semi-major axis> about the <centre of mass> must first be converted to the relative one. When $m$ is negligible compared with $M$, the denominator can be approximated by $GM$.
For fixed <gravitational parameter>, the <orbital period> depends on the <semi-major axis> and is independent of <orbital eccentricity>. Our $a=1$, $\mu=1$ example gives $T=2\pi$ both for $e=0.6$ and for a <circular orbit> with $e=0$. The eccentric body's rapidly changing <speed> compensates for its different path. A compact way to see the cancellation uses <Kepler's second law>: divide the <ellipse>'s area $\pi ab$ by its <areal velocity> $h/2$. With $b=a\sqrt{1-e^2}$ and <specific angular momentum> $h=\sqrt{\mu a(1-e^2)}$, the result is $2\pi\sqrt{a^3/\mu}$.
Compare two separate <Kepler orbits> with the same <gravitational parameter>, $a_1=1$ and $a_2=4$:
$$
\frac{T_2}{T_1}=\left(\frac{a_2}{a_1}\right)^{3/2}=4^{3/2}=8.
$$
With $\mu=1$, their <orbital periods> are $2\pi$ and $16\pi$; measured in units of $T_1$, these are $1$ and $8$. Quadrupling the size therefore takes eight times as long to complete a circuit. This <power law> makes comparisons possible without first choosing physical units.
\Image[/kepler-period-scaling.png]
{title=Quadrupling the relative semi-major axis multiplies the orbital period by eight at fixed total mass}
The same relation can be inverted to weigh a system: $M+m=4\pi^2a^3/(GT^2)$. An observed relative <semi-major axis> and <orbital period> determine the total <mass>, while additional information is needed to separate the two <masses>. This is useful for <binary stars>, as discussed in https://science.nasa.gov/solar-system/orbits-and-keplers-laws/[NASA, Orbits and Kepler's Laws: measuring masses]. Geometry and a clock together become a weighing instrument.
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