Orbital Period⚠ unverified

Understanding the Parameters

• Only $a$ matters — not eccentricity — the striking content of the law. Two satellites sharing a semi-major axis orbit in lockstep even if one traces a near-circle and the other a long cigar. On the ellipse the body races through perigee and dawdles at apogee, but the total time is identical. This is why the semi-major axis, not the shape, is the fundamental "size" of an orbit.

• The $3/2$ power law — $T \propto a^{3/2}$: doubling the orbit's size multiplies the period by $2^{3/2} \approx 2.83$. LEO is $\sim90$ min; a GPS satellite ($a \approx 26{,}560\ \text{km}$) takes $\sim12$ h; geostationary ($a \approx 42{,}164\ \text{km}$) exactly one sidereal day (23 h 56 min); the Moon 27.3 days.

• Semi-major axis $a$ — for an ellipse, $a = (r_{\text{perigee}} + r_{\text{apogee}})/2$, the average of the closest and farthest distances (from the primary's centre). For a circle it is just the radius.

• The primary's $\mu$ sets the clock — the same $a$ gives very different periods around different bodies. An orbit of radius $a$ around the Sun ($\mu \approx 1.327\times10^{20}$) is far faster than around Earth. Kepler's original third law ($T^2 \propto a^3$ with a shared constant) holds only for objects orbiting the same primary; Newton's form makes the constant $4\pi^2/\mu$ explicit and body-specific.

• Geostationary orbit — the practical headline: solve $T = 86{,}164\ \text{s}$ (one sidereal day) for $a$ to get $a \approx 42{,}164\ \text{km}$, the unique altitude where a satellite hovers over a fixed longitude. The whole geostationary communications belt is a solution of this equation.

Derivation (Approaching a Proof)

Circular case first (transparent). For a circular orbit the body travels the circumference $2\pi r$ at the constant circular speed $v = \sqrt{\mu/r}$. Period is distance over speed:

$$T = \frac{2\pi r}{v} = \frac{2\pi r}{\sqrt{\mu/r}} = 2\pi r \sqrt{\frac{r}{\mu}} = 2\pi\sqrt{\frac{r^3}{\mu}}.$$

With $r = a$ for a circle, this is already the result.

General ellipse (via Kepler's second law). For any orbit, Kepler's second law says the radius vector sweeps area at a constant rate $dA/dt = h/2$, where $h$ is the specific angular momentum. Integrating over one full period sweeps the entire area of the ellipse, $A = \pi a b$ (semi-axes $a$ and $b$):

$$T = \frac{A}{dA/dt} = \frac{\pi a b}{h/2} = \frac{2\pi a b}{h}.$$

For an ellipse the semi-minor axis is $b = a\sqrt{1-e^2}$, and the angular momentum is $h = \sqrt{\mu\,a(1-e^2)}$ (a standard orbit relation). Substitute both:

$$T = \frac{2\pi a\cdot a\sqrt{1-e^2}}{\sqrt{\mu\,a(1-e^2)}} = \frac{2\pi a^2\sqrt{1-e^2}}{\sqrt{\mu\,a}\,\sqrt{1-e^2}} = \frac{2\pi a^2}{\sqrt{\mu\,a}} = 2\pi\sqrt{\frac{a^3}{\mu}}. \qquad\blacksquare$$

The eccentricity cancels completely — the origin of the "$a$ only" rule. Squaring gives the familiar $T^2 = \dfrac{4\pi^2}{\mu}\,a^3$, Kepler's third law with Newton's constant.

Dimensional check. $$2\pi\sqrt{\frac{a^3}{\mu}} = \sqrt{\frac{\text{m}^3}{\text{m}^3/\text{s}^2}} = \sqrt{\text{s}^2} = \text{s}. \checkmark$$ ($2\pi$ is dimensionless.)

History and Development

• Kepler's harmony (1619). Johannes Kepler published the third law — "the square of the period is proportional to the cube of the semi-major axis" — in Harmonices Mundi, after a decade searching Tycho Brahe's data for a numerical relationship between the planets' periods and distances. It was, to him, the literal "music of the spheres."

• Newton's explanation (1687). Newton derived $T^2 = 4\pi^2 a^3/\mu$ from gravitation, revealing that Kepler's proportionality constant is $4\pi^2/\mu$ — the same for all bodies orbiting one primary, and encoding that primary's mass. Weighing the Sun and planets became a matter of measuring periods and distances.

• Mass from orbits. Because $\mu = GM = 4\pi^2 a^3/T^2$, timing a moon or satellite weighs the body it orbits. This is how the masses of the Sun, planets, and countless exoplanet host stars are determined — and, with the later measurement of $G$, how Earth itself was "weighed."

• Geostationary orbit (1928–1945). Setting $T$ to one sidereal day and solving for $a$ defines the geostationary radius — envisioned by Herman Potočnik and popularised by Arthur C. Clarke as the home of communications relays, now one of the most valuable real-estate bands in the Solar System.

Related Concepts: Circular Orbit Velocity, Orbital Velocity, Vis Viva Energy, Escape Velocity, Time Since Perigee, Gravitational Force, Angular Momentum

Notes: Kepler's third law (Newtonian form). Depends only on $a$, not eccentricity — same-$a$ orbits share a period. $T\propto a^{3/2}$ (LEO ~90 min, GPS ~12 h, GEO 1 sidereal day, Moon 27.3 d). $a$ = semi-major axis = mean of perigee/apogee radii. $T^2=4\pi^2a^3/\mu$ ⇒ orbits weigh the primary ($\mu=4\pi^2a^3/T^2$). Defaults ⇒ $T\approx92.4$ min.