This lab works on a phone. On a bigger screen you can see the orbit, the graph and the readouts at the same time.

Kepler's Laws

In 1609 Johannes Kepler, working from Tycho Brahe's twenty years of naked-eye measurements of Mars, found three rules that every planet obeys. They were the first laws of nature written as mathematics, eighty years before Newton explained why they are true. Stretch an orbit and watch a planet keep all three.

The Sun sits at a focus Equal areas in equal times T² = a³ Why the orbits look like circles

Pro

Shape the orbit

Stretch it, size it, and watch the planet speed up near the Sun while it sweeps equal areas

Earth e = 0.0167 Year: 1.00 yr Now: 29.8 km/s Orbit to scale · the Sun and planet are not

Pick a body

Eccentricity e 0.017

Semi-major axis a 1.00 AU

Readout

The ellipse

Semi-major axis a1.00 AU · 149.6 million km Eccentricity e0.0167 Semi-minor axis b– Sun's offset from the centre– Closest (perihelion) q– Farthest (aphelion) Q– Long vs short width–

The motion

Year T = √a³– Speed at perihelion– Speed at aphelion– Fast ÷ slow = Q ÷ q– Right now– Area swept per twelfth–

What you're learning

First law. An orbit is an ellipse with the Sun at one focus, not at the centre. The other focus is empty. How stretched the ellipse is comes from its eccentricity e = (Q − q) ÷ (Q + q), where q and Q are the closest and farthest distances; a circle has e = 0. Going the other way, q = a(1 − e) and Q = a(1 + e).

Second law. A line from the Sun to the planet sweeps equal areas in equal times. Near the Sun the line is short, so the planet must cover more ground to sweep the same area: it moves faster. The twelve dots are the planet's position every twelfth of its year; watch them bunch up at aphelion.

Third law. The square of the year is proportional to the cube of the semi-major axis. In Earth years and astronomical units the constant is exactly 1: T² = a³.

Why the orbits look like circles

Earth's e is 0.0167. Its orbit's long width is only 0.014 % more than its short width: drawn to scale it is a circle to the eye. What is off-centre is the Sun, by 1.7 % of the radius, so Earth is 5 million km closer to the Sun in early January than in early July. Six of the eight planets are under e = 0.10; only Mercury (0.21) is visibly oval. Textbook drawings exaggerate, which is why so many people wrongly think distance from the Sun causes the seasons.

Press Stretch it to see the shape the worksheet pack draws, e = 0.35, or pick Halley's Comet, e = 0.967, which swings from inside Venus's orbit to beyond Neptune's.

Test the third law

Every planet on one line: a year that grows with the orbit as a to the three halves

Jupiter T² ÷ a³ = 0.998 Both axes logarithmic · the line is T = a^3/2

Highlight a planet

Your own object 45.0 AU

Readout

Your object

Semi-major axis45.0 AU Year T = √a³– The constant, in SI–

Check it on the planets

Planet	a (AU)	T (yr)	T²	a³	T² ÷ a³

Reading the graph

On ordinary axes the planets would be crammed into a corner with Neptune off the page. On logarithmic axes every factor of 10 takes the same space, and a power law becomes a straight line whose slope is the power. The eight planets sit on a line of slope 3/2, which is T = a^3/2, or T² = a³.

The law does not care about mass, shape or anything else about the planet: only the size of the orbit. Halley's Comet, on a wildly stretched orbit, sits on the same line at a = 17.8 AU, T = 75 years. Newton later showed the line's height depends on the Sun's mass: T² = 4π²a³ ÷ GM, which is how the Sun was weighed.

Good to know

The tiny wobbles in the last column (0.988 to 1.003) are rounding in the three-figure a and T, not a failure of the law. The same three laws hold for moons round a planet, for the Moon round Earth, and for planets round other stars, which is how the Exoplanets lesson turns a planet's year into the size of its orbit.

Practice

Answer a quick Kepler question. Score counts answers right on the first try.

Challenge

Press New challenge.

Score: 0 Streak: 0 Attempted: 0

Choose the answer