Kepler's Laws Explained: How Planets Really Move

Johannes Kepler spent nearly two decades turning careful observations into three simple rules. He published the first two laws in 1609 and the third in 1619, and together they describe how planets move around the Sun without any knowledge of gravity itself. Those rules still start every orbit calculation, from planets to the satellites that carry phone signals today.

Where the laws came from

Kepler worked with observations made by Tycho Brahe, whose naked-eye measurements of Mars were far more precise than anything before them. Astronomers had assumed for nearly two thousand years that planets moved in perfect circles, possibly with smaller circles riding on top of larger ones. Brahe's Mars data did not fit that picture. Kepler tried circle after circle and found that an ellipse matched the observed positions far better than any combination of circles he could build.

First law: orbits are ellipses

Every planet travels along an ellipse with the Sun at one of the two foci. The second focus is empty and has no physical meaning. It only exists because the shape is defined by two points.

Semi-major axis and eccentricity

Two numbers describe any ellipse. The semi-major axis, written a, is half of the longest diameter. The eccentricity, written e, measures how stretched the shape is. A value of 0 is a perfect circle, and values near 1 are long, thin ellipses. The table below uses rounded values from standard references.

Body	Eccentricity e	Closest to Sun (million km)	Farthest from Sun (million km)
Earth	0.017	147	152
Mars	0.093	207	249
Mercury	0.206	46	70
Halley's Comet	0.967	88	5,250

Earth's orbit is almost round, which is why the seasons are not caused by distance. Halley's Comet has an extreme ellipse that carries it from inside the inner planets out past Neptune. The first law applies to comets, asteroids and spacecraft just as well as to planets.

Second law: equal areas in equal times

A line drawn from the Sun to a planet sweeps out equal areas in equal intervals of time. The consequence is that a planet moves fastest at perihelion, its closest approach, and slowest at aphelion, its farthest point.

Why the law holds

The Sun pulls a planet along the line joining the two bodies. That pull has no sideways component, so it exerts no torque about the Sun. Angular momentum stays constant, and the rate at which the radius sweeps area stays constant too. The law is really a statement of angular momentum conservation written in geometric form.

Earth shows the effect clearly. Its speed is about 30.3 km/s at perihelion and about 29.3 km/s at aphelion. The difference is only about 3 percent, but it is large enough to measure and to explain seasonal timing in the northern and southern hemispheres.

Third law: period and distance

The square of the orbital period is proportional to the cube of the semi-major axis. When the period is measured in years and the distance in astronomical units, where 1 AU is the average Earth-Sun distance, the proportionality constant becomes exactly 1. That gives the compact form T² = a³.

Planet	Semi-major axis a (AU)	Period T (years)
Mercury	0.387	0.241
Venus	0.723	0.615
Earth	1.000	1.000
Mars	1.524	1.881
Jupiter	5.203	11.86
Saturn	9.537	29.46

Kepler found this pattern by searching through the numbers, without a physical theory. The table shows why it works. Saturn is about nine and a half times farther from the Sun than Earth, and its year is about 29.5 Earth years long.

Newton's general form

Isaac Newton later showed that Kepler's third law follows from his law of gravity. For two bodies with masses M and m, the general form is T² = 4π²a³ / G(M + m), where G is the gravitational constant. Because the Sun is so much heavier than any planet, the planet's mass is negligible in most cases. The Pluto-Charon pair is an exception, because Charon is a large fraction of Pluto's mass, so the combined mass term matters.

Worked example: a geostationary satellite

A geostationary satellite stays over one point on the equator. To do that, its period must match Earth's rotation, which is about 23.93 hours. Using a semi-major axis of 42,164 km and the value GM = 398,600 km³/s² for Earth, the formula T = 2π √(a³ / GM) gives about 86,160 seconds, which is 23.93 hours.

The radius of 42,164 km measured from Earth's center corresponds to an altitude of about 35,786 km above the surface. This is the reason that television and weather satellites sit at that height and nowhere else.

Worked example: the ISS and GPS

The International Space Station flies at roughly 400 km altitude. Its semi-major axis is near 6,778 km, and the same formula gives a period of about 92 minutes, so the crew sees about 16 sunrises per day.

GPS satellites fly at about 20,200 km altitude. Their period works out to roughly 12 hours, so each satellite completes two orbits per sidereal day and its ground track repeats once per sidereal day. These numbers come directly from the third law, and they show why different orbit types suit different jobs.

Energy ties the laws together

The three laws describe the shape, the speed and the timing of an orbit. Energy links them through the vis-viva equation, v² = GM(2/r − 1/a). Here r is the current distance from the Sun and a is the semi-major axis. For a circular orbit, the speed is simply √(GM/r).

At 400 km altitude, the circular speed is about 7.7 km/s. That is the speed a spacecraft needs to stay in low Earth orbit without thrust. The same equation predicts the speed at any point on any ellipse once a and r are known.

Common misunderstandings

Many people think the seasons come from Earth being closer to the Sun in summer. The truth is that Earth reaches perihelion in early January, which is winter in the northern hemisphere. The seasons come from the 23.4 degree tilt of Earth's axis, which changes how directly sunlight strikes each hemisphere during the year.

Another common idea is that planets speed up on purpose. They do not. Their speed changes because the gravitational pull is strongest near the Sun, and the energy exchange between kinetic and potential forms follows directly from the conservation laws.

Key takeaways

Kepler's first law says orbits are ellipses with the Sun at one focus. The second law says a planet sweeps equal areas in equal times, which means it moves faster near the Sun. The third law says the square of the period grows with the cube of the orbit size.

Newton later explained all three laws with gravity and conservation of angular momentum and energy. The same mathematics now guides satellite placement, mission planning and the search for planets around other stars.