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Kepler's Three Laws of Planetary Motion, Explained with Numbers

Three simple rules describe how every planet moves around the Sun. Step through each law, see how to use it with actual numbers, and learn how the laws still help us design satellite orbits.

  1. First law β€” the law of ellipses

    Every planet orbits the Sun not in a circle but in an ellipse, with the Sun located not at the center of the ellipse but at one of its two foci. An ellipse is described by its semi-major axis (half of its longest diameter) and its eccentricity, a number from 0 (a circle) to just under 1 (a very long, thin ellipse). Kepler's first discovery was that planetary orbits, believed for thousands of years to be perfect circles, are actually slightly flattened ellipses. For most planets the flattening is so slight that a drawing of the orbit would look round.

  2. Second law β€” the law of equal areas

    An imaginary line segment connecting the Sun and a planet sweeps out equal areas in equal amounts of time. As a result, a planet moves faster near perihelion, when it is close to the Sun, and slower near aphelion, when it is far from the Sun. Orbital speed is therefore not constant. Today this law is understood as a consequence of the conservation of angular momentum: because the force of gravity points straight toward the Sun, it cannot change the angular momentum of the planet around the Sun.

  3. Third law β€” the law of harmonies

    The square of a planet's orbital period is proportional to the cube of its orbit's semi-major axis, written T squared proportional to a cubed. If you measure the period T in years and the semi-major axis a in astronomical units (AU, the average Earth-Sun distance), the constant becomes 1 and the law reads T squared equals a cubed. Once you know a planet's average distance you can calculate its period, and the other way around. This relation played a decisive role in establishing the relative scale of the whole solar system, since periods can be timed from Earth far more easily than distances can be measured.

  4. Worked example: how long is a year on Mars?

    Mars orbits at an average distance of about 1.52 AU from the Sun. Cube that distance to get about 3.51. The square root of 3.51 is about 1.87, so the law predicts a Martian year of roughly 1.87 Earth years. The accepted figure is about 1.88 years, close to the calculated value. The small difference comes from rounding the distance. You can run the same check on other planets: Venus at 0.72 AU gives about 0.61 years, or roughly 224 days, and Jupiter at 5.2 AU gives about 11.9 years.

  5. Perihelion and aphelion

    In an elliptical orbit, the point closest to the Sun is called perihelion, and the point farthest away is called aphelion. For Earth the perihelion distance is about 147.1 million km and the aphelion distance is about 152.1 million km. The orbit's eccentricity is about 0.017, so it is very close to a circle. Earth passes through perihelion in early January each year and aphelion in early July. This also shows that the seasons are not caused by Earth's distance from the Sun: the Northern Hemisphere has winter when Earth is nearest the Sun, because the seasons come from the tilt of Earth's axis.

  6. Which orbits are most stretched?

    Eccentricity differs widely. Venus (about 0.007) and Earth (about 0.017) have almost circular orbits. Mars is more eccentric at about 0.09, and Mercury is the most eccentric of the eight planets at about 0.21. Halley's Comet has an eccentricity of about 0.97, so it swings from near the Sun out beyond Neptune. With a semi-major axis of about 18 AU, the third law gives a period of about 76 years, which matches observations of its return.

  7. The eight-arcminute problem

    Kepler first tried to fit Mars's positions with circular orbits, as astronomers had done for centuries. His best circular model disagreed with Tycho Brahe's observations by about eight arcminutes, a very small angle but larger than Tycho's measurement error. Kepler decided that the observations were right and the circle was wrong. This decision to trust precise data over a favored idea is often cited as a turning point in the history of science.

  8. The connection to Newton's law of universal gravitation

    Kepler found these laws empirically but could not explain why they held. In 1687 Newton published his Principia, in which the law of universal gravitation, with its inverse-square dependence on distance, allowed him to derive all three of Kepler's laws mathematically. This showed that the same gravity that makes an apple fall also governs the planets.

  9. What Newton added to the third law

    Newton showed that for a small body orbiting a much larger one, T squared equals 4 pi squared times a cubed divided by G times M, where G is the gravitational constant and M is the mass of the central body. Kepler's constant therefore depends on the mass of the object being orbited. That is why the formula can be used to weigh planets, stars and even galaxies: measure the period and size of an orbit, and you can calculate the central mass.

  10. Still applied today in designing artificial satellite orbits

    Kepler's laws apply not only to the planets but also to calculating the orbits of artificial satellites and spacecraft orbiting Earth. Principles like the law of harmonies are central to calculating the altitude of a geostationary satellite, which stays above one spot on the equator at about 35,786 km, or the transfer orbit a probe uses to travel to another planet. The International Space Station, at roughly 400 km, completes an orbit in about 90 minutes, while the Moon, at about 384,000 km, takes just under 28 days.

  11. Where the laws stop being exact

    Kepler's laws describe an idealized system of two bodies. In the real solar system, planets also pull on one another, which makes orbits slowly drift. For Mercury, a small leftover shift in the orbit could not be explained by Newtonian gravity and was later explained by Einstein's general relativity. For very precise work, such as spacecraft navigation, engineers add these extra effects to Kepler's simple picture.

  12. How Kepler discovered these laws

    The German astronomer Johannes Kepler (1571 to 1630) inherited the precise naked-eye observations of Mars that his predecessor Tycho Brahe had collected over a lifetime, and published the first and second laws in his 1609 book Astronomia Nova (New Astronomy). The third law followed in Harmonices Mundi (The Harmony of the World) in 1619. Kepler found these mathematical laws purely from accumulated observational data and without a telescope, an achievement that gave later scientists a model for combining careful measurement with mathematics.

Why do planets trace elliptical rather than circular orbits?

For a long time people believed celestial bodies moved in perfect circles because the circle was seen as the most perfect shape. Kepler's analysis of precise data in the early 1600s showed otherwise. In Newton's physics, an ellipse is simply one of the shapes allowed when a body moves under an inverse-square force; a circle is the special case of zero eccentricity. A planet follows a perfect circle only if its speed is exactly right at every moment, which almost never happens, so in practice orbits are slightly elliptical.

It's easier to understand alongside Newton's laws of motion

If Kepler's laws explain how planets move, Newton's laws explain why they move that way. If you are curious about the relationship between force and acceleration, take a look at our guide to understanding Newton's three laws of motion as well.

Frequently Asked Questions

Are all planetary orbits noticeably flattened ellipses?

No. Planets with a small orbital eccentricity, like Earth or Venus, trace an ellipse very close to a circle, while Mercury traces a noticeably flattened one. Some comets have extremely large eccentricities and follow very elongated paths.

How is the law-of-harmonies formula actually used?

If you express the orbital period (T) in years and the semi-major axis (a) in AU, the constant becomes 1, so T squared equals a cubed. For example, Mars is about 1.52 AU from the Sun, so 1.52 cubed is about 3.51, and the square root gives a period of about 1.87 years, which closely matches the observed value.

Do Kepler's laws work for moons and exoplanets?

Yes. They apply to any body orbiting a much larger one under gravity, including moons around planets and planets around other stars. For the third law the constant depends on the mass of the central body, so the T squared equals a cubed form in years and AU is specific to the Sun. Astronomers use the generalized form to estimate the masses of stars from the orbits of their planets.

Why does Earth have seasons if it is only slightly closer to the Sun in January?

Because the seasons depend on the tilt of the Earth's axis, not the distance. The Northern Hemisphere is tilted away from the Sun in December and January, giving shorter days and lower sun, even though Earth is at perihelion. The roughly 3 percent distance difference has a much smaller effect.