Watch a comet swing past the Sun: it whips around at its closest and crawls out at the far end. Four centuries ago Johannes Kepler found the exact rule — a planet sweeps out equal areas in equal times — which is why nearness means speed.
The line from the sun to the disc sweeps the same area each second — so when the disc is close it must move fast, and when it is far it crawls. The more stretched the orbit, the bigger that speed swing.
Kepler's second law — the law of equal areas — says a line drawn from the Sun to a planet sweeps out equal areas in equal times. Since a thin sweep is a triangle (½ × base × height), a short radius forces a long, fast arc and a long radius needs only a short, slow one. So the planet is fastest at its closest approach and slowest when farthest away.
It follows from angular momentum conservation. Gravity pulls straight toward the Sun, with no sideways twist, so the planet's angular momentum, m·r·v, cannot change. Holding r·v fixed means a small r demands a large v — fast when close. The swept-area rate works out to ½·r·v, which is therefore constant: equal areas in equal times.
In The Field, launch the disc onto a stretched orbit and watch it tear through perihelion (closest to the sun) and crawl at aphelion (farthest). The radius line from the sun sweeps the same area each second whether the disc is near or far — the speed difference is Kepler's law made visible.
Kepler's law is easiest to believe when you watch the planet itself speed up near the Sun and slow far away — while the little swept triangle stays the same size each step.
Distance and speed move in opposite directions — close means fast, far means slow — exactly so the swept area per second stays constant.
Halley races through the inner Solar System in a matter of weeks near the Sun, then takes decades crawling out beyond Neptune and back — a vivid case of fast-when-close, slow-when-far.
Earth's orbit is slightly stretched, so we move fastest in early January (closest to the Sun) and slowest in July. It is why the seasons are not exactly equal in length.
Mission planners use Kepler's laws to time launches and slingshots. Knowing exactly where and how fast a planet will be — years ahead — is what lets a probe arrive at the right place at the right moment.
Kepler turned Tycho Brahe's painstaking observations into three clean laws of planetary motion. This FAQ unpacks the second law — equal areas — and connects it to ellipses, angular momentum, comets, and the timing of real space missions.
Kepler's second law says that an imaginary line from the Sun to a planet sweeps out equal areas in equal amounts of time. Because that swept shape is essentially a thin triangle, a planet close to the Sun (short line) must travel a long, fast arc to cover the same area, while a far planet (long line) covers it with a short, slow arc. So planets speed up near the Sun and slow down far away.
| Law | What it says |
|---|---|
| First (ellipses) | Orbits are ellipses with the Sun at one focus |
| Second (equal areas) | The Sun-planet line sweeps equal areas in equal times |
| Third (T squared = a cubed) | Period squared is proportional to orbit size cubed |
Because the Sun-planet line must sweep the same area every second. Near the Sun the line is short, so to enclose that fixed area the planet has to swing through a long arc quickly. Far from the Sun the line is long, so the same area is covered by only a short, slow arc. The result, guaranteed by the equal-areas law, is maximum speed at the closest point and minimum speed at the farthest.
It is the conservation of angular momentum. Gravity always pulls a planet straight toward the Sun, which is a purely inward force with no sideways component, so it cannot change the planet's angular momentum, m·r·v. Keeping r·v constant means a small distance forces a large speed. The rate at which area is swept turns out to be exactly half of r·v, so a constant angular momentum is the same as equal areas in equal times.
Perihelion is the point on an orbit closest to the Sun, and aphelion is the point farthest from it. By Kepler's second law a planet moves fastest at perihelion and slowest at aphelion. Earth, for instance, reaches perihelion in early January and aphelion in early July, so it is actually moving a little faster in northern winter than in northern summer.
Kepler's third law says the square of a planet's orbital period equals the cube of its orbit size: T² ∝ a³. Measured in years and astronomical units it is simply T = a^1.5. It lets you predict a planet's year from its distance: Mars at 1.52 AU takes 1.88 years, Jupiter at 5.2 AU takes about 11.9 years. The same rule sizes the orbits of moons and exoplanets too.
Yes. Kepler's first law established that planetary orbits are ellipses, not the perfect circles assumed for centuries, with the Sun sitting at one focus rather than the centre. Most planets, including Earth, have orbits so close to circular that the ellipse is hard to notice, but comets follow dramatically stretched ellipses that make the equal-areas speed swing obvious.
Kepler inherited decades of remarkably precise naked-eye observations of the planets from the astronomer Tycho Brahe. Working especially with the troublesome orbit of Mars, he spent years testing shapes and speeds until the circles failed and the ellipse, with the equal-areas rule, finally fit the data. His three laws, published between 1609 and 1619, became the empirical foundation that Newton later explained with gravity.
Yes. Kepler's laws apply to anything orbiting under gravity — comets around the Sun, the Moon and satellites around Earth, even moons around other planets. A comet on a long, stretched orbit shows the equal-areas effect most vividly, sprinting through the inner Solar System and creeping along its distant arc. Artificial satellites in elliptical orbits speed up and slow down in exactly the same way.
Eccentricity measures how stretched an orbit is, from 0 for a perfect circle up toward 1 for a very elongated ellipse. The equal-areas law holds for every orbit, but a higher eccentricity makes its effect dramatic: the planet's distance, and therefore its speed, varies far more between perihelion and aphelion. A near-circular orbit shows almost no speed change, while a comet's high-eccentricity orbit shows an enormous one.
Newton showed that an inverse-square gravitational pull mathematically produces all three of Kepler's laws. The inward-only force conserves angular momentum, which gives the equal-areas second law; solving the motion yields elliptical orbits, the first law; and the period-distance relation falls out as the third. Kepler found the patterns from data; Newton revealed the single force law underneath them.
No. The Sun sits at one focus of the ellipse, which is off to one side, not at the geometric centre. That off-centre position is exactly why a planet's distance changes around its orbit, giving the perihelion and aphelion. Areas in Kepler's second law are always measured from that focus, the Sun, not from the centre of the ellipse.
Mission planners use Kepler's laws to know precisely where every planet will be, and how fast it will be moving, years into the future. That lets them design transfer orbits and gravity-assist slingshots that meet a target planet at just the right moment. The Voyager and Cassini missions, for example, were timed using these laws to thread past several worlds in a single journey.
For a small planet around a much larger star, no — the period depends on the orbit's size and the Sun's mass, not on the planet's own mass, which is why a pebble and a planet at the same distance would share a period. Kepler's third law in its simple form, T² ∝ a³, assumes the orbiting body is light compared with the Sun, which holds well across the Solar System.
Everywhere we navigate space. It sets the schedules of GPS and communication satellites, the launch windows for missions to Mars, and the predicted returns of comets. It guides the hunt for exoplanets, whose tiny tugs reveal orbit sizes through the third law. Four centuries on, Kepler's quiet rule about sweeping equal areas still underpins how we map and travel the Solar System.
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