Physics · Astrophysics · A2 Level

Planetary Motion

The circular orbit treated earlier in this topic is a useful approximation, but real planets don't move in perfect circles. Working out exactly how they do move — using nothing but careful observation and geometry — was one of the great achievements of pre-Newtonian astronomy, and it still frames how we describe orbital motion today.

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Kepler's first law: orbits are ellipses

Johannes Kepler's first law states that planets orbit the Sun on elliptical paths, with the Sun at one focus of the ellipse — not at the centre. An ellipse is a stretched-out oval with two focus points; the sum of the distances from any point on the ellipse to the two foci is constant. A circle is simply the special case of an ellipse where both foci coincide at the centre.

This matters because it explains why a planet's distance from the Sun changes over the course of one orbit. The point of closest approach to the Sun is called perihelion, and the point of greatest distance is called aphelion. For most planets in the Solar System, however, the orbital ellipse is only very slightly stretched — close enough to circular that treating the orbit as a genuine circle, as in the earlier Circular Orbits lesson, is an excellent and very widely used approximation for most calculations. The elliptical nature of orbits becomes important mainly for objects with markedly stretched orbits, such as many comets.

Kepler's second law: equal areas in equal times

Kepler's second law states that a line joining a planet to the Sun sweeps out equal areas in equal intervals of time. Picture the imaginary line connecting the planet to the Sun as it moves along its elliptical path: over any fixed time interval — say, one month — the area swept out by that line is the same, no matter which part of the orbit the planet is currently in.

The consequence of this is that a planet's orbital speed is not constant. Near perihelion, where the planet is close to the Sun, the connecting line is short, so it must sweep quickly through a wide angle to cover the same area in the same time — meaning the planet moves faster when closer to the Sun. Near aphelion, the connecting line is long, so it only needs to sweep a narrow angle to cover the same area — meaning the planet moves slower when further away.

This isn't an arbitrary rule — it follows directly from the conservation of angular momentum. Since the gravitational force from the Sun acts along the line joining the Sun and the planet, it exerts no turning moment about the Sun, so the planet's angular momentum about the Sun stays constant throughout the orbit.
Angular momentum depends on both the planet's distance from the Sun and its speed perpendicular to that line, so as distance decreases, speed must increase to keep angular momentum constant — and vice versa.
This is the same underlying physics that makes a spinning ice skater speed up when pulling their arms in: reducing the radius of the motion increases the speed needed to conserve angular momentum.

The key insight

Kepler's second law is really a statement about angular momentum conservation, not a separate law of nature bolted onto gravity. Because gravity is a central force — always directed along the line joining the two bodies — it can never exert a turning moment about that central point, so angular momentum about the Sun is conserved throughout the orbit. Equal areas in equal times, and the resulting speeding-up near perihelion and slowing-down near aphelion, both follow automatically from that one conservation principle.

Kepler's third law: connecting back to circular orbits

Kepler's third law states that the square of a planet's orbital period is proportional to the cube of its orbital radius: $T^2 \propto r^3$. This is exactly the relationship already derived from first principles for a circular orbit in the previous lesson, where combining $v = \sqrt{GM/r}$ with $v = 2\pi r/T$ gave $T = 2\pi\sqrt{r^3/GM}$, which rearranges to

$$T^2 = \dfrac{4\pi^2}{GM}\,r^3$$

For a genuinely elliptical orbit, the same proportionality holds, but $r$ must be replaced by the semi-major axis of the ellipse — half the length of the ellipse's longest diameter, which plays the same role that the fixed radius plays for a circle. The full derivation for an ellipse requires more advanced mathematics than is needed here, but the result is the natural generalisation of the circular-orbit case already derived: same law, same $4\pi^2/GM$ constant of proportionality, just with the semi-major axis standing in for the orbital radius.

Worked example: comparing two orbits

A key practical use of Kepler's third law is comparing two orbits around the same central body without needing to know $G$ or $M$ explicitly, since both cancel in a ratio.

Worked exampleFinding an unknown orbital period

Take the ratio form of Kepler's third law for two bodies orbiting the same central mass:

$$\left(\dfrac{T_1}{T_2}\right)^2 = \left(\dfrac{r_1}{r_2}\right)^3$$

Both $T^2 = \frac{4\pi^2}{GM}r^3$ relations share the same constant $4\pi^2/GM$, so it cancels when a ratio is taken.

Suppose planet 1 has orbital radius $r_1 = 1.50\times10^{11}\,\text{m}$ and period $T_1 = 1.00$ year (Earth-like), and planet 2 orbits the same star at $r_2 = 2.28\times10^{11}\,\text{m}$. Substitute into the ratio:

$$\left(\dfrac{1.00}{T_2}\right)^2 = \left(\dfrac{1.50\times10^{11}}{2.28\times10^{11}}\right)^3$$

Both radii are measured from the same central star, so the ratio form applies directly.

Evaluate the right-hand side and solve for $T_2$:

$$\left(\dfrac{1.00}{T_2}\right)^2 = (0.658)^3 \approx 0.285 \quad\Rightarrow\quad T_2 \approx \sqrt{\dfrac{1.00}{0.285}} \approx 1.87\ \text{years}$$

This matches Mars's real orbital period of roughly 1.88 years, at roughly its real orbital radius from the Sun — a useful check on the method.

Checks out

Key takeaways

Kepler's first law: planets orbit the Sun on ellipses with the Sun at one focus, not at the centre — though most planetary orbits are close enough to circular that the circular-orbit approximation from the previous lesson works well.
Kepler's second law: a line from the planet to the Sun sweeps equal areas in equal times, so planets move faster near perihelion (closest approach) and slower near aphelion (furthest point).
Kepler's second law is a direct consequence of conservation of angular momentum, since gravity is a central force and exerts no turning moment about the Sun.
Kepler's third law, $T^2 \propto r^3$, is the same relationship already derived for circular orbits in the previous lesson, $T^2 = \frac{4\pi^2}{GM}r^3$.
For elliptical orbits, $r$ is replaced by the semi-major axis of the ellipse, but the same proportionality and constant apply.
The ratio form $\left(T_1/T_2\right)^2 = \left(r_1/r_2\right)^3$ lets two orbits around the same central body be compared without needing $G$ or $M$ explicitly.

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