Kepler's Laws of Planetary Motion

Long before Newton wrote down his law of gravitation, a German astronomer named Johannes Kepler spent twenty years poring over the meticulous observations of the Danish astronomer Tycho Brahe, trying to make sense of how the planets actually move. The orbits he deduced were not the perfect circles of Greek astronomy, but slightly flattened ellipses. From these ellipses he distilled three empirical laws that even today are the basis for understanding planetary, lunar, and satellite orbits.

Newton later showed that Kepler's three laws are direct consequences of the inverse-square law of gravitation — a mathematical triumph that laid the foundation of modern physics. This lesson states the laws, explains their physical meaning, and derives Kepler's third law for the special case of circular orbits. It is OCR A-Level Physics A Module 5.4 (Gravitational fields).

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Kepler's First Law — the Law of Ellipses

The orbit of every planet is an ellipse, with the Sun at one focus.

An ellipse is a squashed circle. Technically, it is the set of points for which the sum of the distances to two fixed points — the foci — is constant. If the two foci coincide, the ellipse reduces to a circle.

For planetary orbits, the Sun sits at one focus. The other focus is empty. Most planetary orbits are only slightly elliptical — the Earth's orbit, for example, has an eccentricity of only 0.017, meaning it is almost indistinguishable from a circle. Mars is slightly more elliptical (eccentricity 0.093), which is why Tycho Brahe's observations of Mars were the key to Kepler's breakthrough.

Definitions Worth Knowing

Term	Meaning
Perihelion	The point in the orbit closest to the Sun
Aphelion	The point in the orbit farthest from the Sun
Semi-major axis a	Half the longest diameter of the ellipse; the average of the perihelion and aphelion distances
Semi-minor axis b	Half the shortest diameter of the ellipse
Eccentricity e	A measure of how squashed the ellipse is; e = 0 is a circle, e → 1 is an extreme ellipse

Exam Tip: For OCR A-Level questions, you can usually treat planetary and satellite orbits as perfect circles. The full ellipse geometry is not examined; what matters is understanding that "circular orbit" is the simplification of Kepler's first law.

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Kepler's Second Law — the Law of Equal Areas

A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

This is also known as the law of equal areas. It is a deeply geometric statement: if you draw a line from the Sun to a planet and trace it for, say, one week, the wedge-shaped area swept out will be the same no matter where in the orbit the planet happens to be during that week.

The practical consequence is that a planet moves faster near the Sun (at perihelion) and slower far from the Sun (at aphelion). At perihelion, a short arc has to sweep out the same area as a much longer arc at aphelion, so the planet must be moving along that short arc quickly.

Conservation of Angular Momentum in Disguise

Kepler's second law is, in modern language, a statement of conservation of angular momentum. Because gravity is a central force (always directed from the planet to the Sun), it exerts zero torque about the Sun, so the planet's angular momentum is conserved. The area swept out per unit time turns out to be L/(2m), a constant — and that is exactly Kepler's law.

For a circular orbit, the planet moves at constant speed — the areas swept out per second are automatically equal everywhere. Kepler's second law then reduces to "the orbital speed is constant", which is the assumption we will make for the rest of this lesson.

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Kepler's Third Law — the Law of Periods

The square of the orbital period is directly proportional to the cube of the semi-major axis.

In symbols, for any two bodies orbiting the same central mass,

T² ∝ r³

or, with the constant of proportionality written out,

T² = (4π² / GM) r³

where M is the mass of the central body, r is the semi-major axis (or, for a circular orbit, simply the radius), G is the universal gravitational constant, and T is the period.

Why T² ∝ r³? A Derivation for Circular Orbits

For a body in a circular orbit, gravity supplies the centripetal force:

GMm/r² = mv²/r

Cancelling m and rearranging:

v² = GM/r
v = √(GM/r)

The period is the circumference divided by the speed:

T = 2πr / v = 2πr / √(GM/r) = 2πr × √(r/GM) = 2π √(r³/GM)

Squaring both sides:

T² = 4π² r³ / (GM)

This is Kepler's third law, derived directly from Newton's law of gravitation and Newton's second law. It tells us that the ratio T²/r³ is the same for every object orbiting the same central mass.