Foundations — Kepler's third law — T² ∝ a³ — derivation

Before you can read the derivation on the parent page — Kepler's Third Law — you must own every squiggle it uses without apology. This page builds each one from nothing, in an order where every symbol leans only on the ones before it.

0 · How to read a symbol

A symbol is a shorthand for a picture. If you cannot draw the picture, you do not yet own the symbol. So every entry below has three parts: plain words → the picture → why the topic needs it. Read them in order; each one is a brick for the next.

1 · $r$ — distance from the centre

Look at figure 1. The Sun sits at the middle. The planet sits on the ring. The white arrow labelled $r$ is that gap, measured in metres.

Why the topic needs it. Everything in orbital mechanics gets weaker or slower with distance. Gravity fades with $r$, speed drops with $r$, the lap-time grows with $r$. If you have no name for "how far out," you cannot say any of that. $r$ is the first brick.

2 · $\pi$ and the circumference $2\pi r$ — the length of one lap

Look at figure 2. Unroll the circle into a straight strip: its length is exactly $2\pi r$. Double the radius, and the strip doubles too — the lap-length grows in step with $r$.

Why the topic needs it. One lap of the orbit is one circumference. When the parent note writes "the planet covers $2\pi r$ in one period," this is the picture. This is the "longer track" penalty for far-out planets: bigger $r$ means a literally longer road to travel.

3 · $v$ — orbital speed

If one lap is $2\pi r$ metres long and takes a time $T$ to finish, then

$v=\frac{2\pi r}{T}\text{(lap length divided by lap time)}.$

Why the topic needs it. Speed is the bridge between the size of the orbit and the time it takes. Gravity will fix the speed (next section), and speed will fix the period. Without $v$ we could never connect the shape of the orbit to a clock reading.

4 · $T$ — the period, and what "squared" means

Why $T^2$ and not just $T$? The parent law is stated as $T^2\propto a^3$, not $T\propto \text{something}$. That is because the honest relationship, $T\propto r^{3/2}$, contains an awkward square root ($r^{1/2}=\sqrt{r}$). Squaring both sides clears the root: $(r^{3/2}{)}^2=r^3$. So $T^2$ is chosen deliberately — it turns an ugly half-power into a clean whole-power cube. That is the only reason the "2" and the "3" show up together.

5 · $M$ and $m$ — the two masses (never mix them up)

Look at figure 3: big amber dot in the centre ($M$), small cyan dot on the ring ($m$). Mass just means "how much stuff," measured in kilograms.

Why the topic needs it. Gravity depends on both masses. But the punchline of Kepler's Third Law is that the little mass $m$ cancels out and vanishes from the answer — only the central $M$ survives. You cannot appreciate that magic trick unless you first keep the two masses strictly separate.

6 · $G$ — the strength of gravity

Why the topic needs it. Gravity's pull is written $\frac{GMm}{r^2}$ (see Newton's Law of Universal Gravitation). $G$ is the dial that converts "masses and distance" into an actual force in newtons. It carries through the whole derivation and lands in the final constant $\frac{4{\pi }^2}{GM}$.

7 · Force, centripetal pull, and $\frac{m{v}^2}{r}$

Look at figure 4. Without a pull the planet would shoot off along the dashed tangent line. Gravity (amber arrow) is the leash that bends it back onto the ring every instant.

Why the tool $\frac{m{v}^2}{r}$ and not another? We need a formula that says "how hard must I pull inward to hold this speed on this size circle?" That is exactly the centripetal force result from Centripetal Force & Circular Motion. Notice it grows with speed-squared (fast things resist turning) and shrinks with radius (gentle wide circles need less pull). The whole derivation is one sentence: set gravity's pull equal to this required pull.

8 · $\propto$ — the "proportional to" arrow

Why the topic needs it. Kepler discovered $T^2\propto a^3$ purely from telescope data, decades before anyone knew the multiplier was $4{\pi }^2/GM$. The $\propto$ sign lets us state the pattern first and fill in the exact constant later. When you compare two moons of the same planet, the constant cancels and only the $\propto$ pattern is left.

9 · $a$ — the semi-major axis (the ellipse's honest radius)

Since $r$ changes over an elliptical orbit (section 1), we need one steady number to plug into the law. That number is $a$, the average of the closest approach and the farthest point:

$a=\frac{r_{\min }+r_{\max }}{2}.$

Why the topic needs it. The parent law's final form is $T^2=\frac{4{\pi }^2}{GM}{a}^3$. Real orbits are ellipses, and it is $a$ — not any single distance — that controls the year. The circle derivation uses $r$; the honest ellipse version simply swaps $r\to a$. Everything you learn on the circle transfers.

The prerequisite map

Read it top-down: distance and masses feed the two forces; setting those forces equal, together with the period and the average size $a$, produces the law.

Equipment checklist

Cover the right side and test yourself. If any answer surprises you, re-read its section above before opening the parent derivation.