Kepler's Three Laws: Why Planets Speed Up Near the Sun

Johannes Kepler's three laws say that every planet moves on an ellipse with the Sun at one focus, that a line from the Sun to the planet sweeps out equal areas in equal times, and that the square of a planet's orbital period grows in step with the cube of its average distance. In practice that means Earth moves about 3.4 percent faster in early January than in early July, and that a planet four times farther out takes eight times longer to circle the Sun. Newton later showed that all three follow from gravity, which is why astronomers still use them to weigh stars and to find planets around other suns.

From Tycho's notebooks to three laws

Before Kepler, nearly everyone who computed planetary positions assumed that heavenly bodies move in circles at constant speed. Copernicus had moved the Sun to the center, but his model still stacked circles on circles to match what the sky actually did. The trouble was that the match was never quite good enough, and nobody had data precise enough to say exactly how bad it was.

Tycho Brahe supplied that data. King Frederick II gave him the island of Hven in 1576, and he spent about two decades building and using naked-eye instruments there. His logged observations were accurate to roughly one arcminute (one sixtieth of a degree) in the best cases, according to the Tycho Brahe article on Wikipedia, which also notes that his published star catalogue carried a somewhat larger median error of about 1.5 arcminutes. That was an order of magnitude better than anything before him. Tycho died on October 24, 1601, and his Mars records passed into the hands of his assistant, Johannes Kepler, who had been working with him for roughly a year.

Mars was the hard case. Its orbit is more eccentric than that of any other planet except Mercury, so the circle-based models that worked tolerably for Venus or Jupiter failed visibly for Mars. Kepler spent years fitting models to Tycho's numbers. His circle-based "vicarious hypothesis," with the Sun off-center and the planet's speed made uniform about a second point, matched eight Mars oppositions (the moments when it sits opposite the Sun in the sky) to within about 2 arcminutes. But when he tried to make that model physically sensible by bisecting the eccentricity, the error rose to 8 arcminutes, roughly a quarter of the apparent width of the full Moon. Tycho's accuracy was good to about 2 arcminutes, so the discrepancy could not be blamed on sloppy observing.

Kepler took the gap seriously. In the Astronomia Nova (1609) he wrote that these eight minutes "alone will lead us along a path to the reform of the whole of Astronomy," as quoted in the Astronomia nova entry on Wikipedia. That book contains the first two of his laws, and its subject is almost entirely the orbit of Mars. The third law came about a decade later in Harmonices Mundi (1619).

Why this was a break with 2,000 years of astronomy

The old assumption was not stubbornness. Circles are the simplest closed curves, and constant speed looked like the only motion fit for heavenly bodies. Kepler's real innovation was to let the data overrule an aesthetic commitment. He gave up uniform circular motion because it disagreed with the numbers by an amount that could be measured, and then he searched for rules that did agree.

Here is how the three laws line up, at a glance.

Law	Statement	What it describes	Published
First (law of ellipses)	Each planet moves on an ellipse with the Sun at one focus	Shape of the orbit	1609
Second (law of equal areas)	The Sun-planet line sweeps equal areas in equal times	Speed along the orbit	1609 (modern form in 1621)
Third (harmonic law)	Period squared is proportional to semi-major axis cubed	How different planets relate	1619

The first two laws describe one planet; the third ties all the planets together.

The publication years come from the Wikipedia article on Kepler's laws of planetary motion, which notes that the clean modern statement of the equal-areas law appeared in Kepler's Epitome of Copernican Astronomy in 1621, while the Astronomia Nova contains the same idea in other forms.

Law one: orbits are ellipses, and the Sun is off-center

An ellipse is a stretched circle. It has two special points inside it called foci (singular: focus), and the defining property is that the distances from any point on the curve to the two foci always add up to the same total. If you pin the ends of a loop of string to two nails and trace a pencil around the taut string, you draw an ellipse, and the nails are the foci. The closer the nails are to each other, the closer the figure is to a circle.

Kepler's first law says the Sun sits on one of those nails. The other focus is empty space. Nothing is there.

Semi-major axis and eccentricity

Two numbers describe any ellipse well enough for orbital work. The semi-major axis, written a, is half the longest diameter, and it serves as the planet's "average distance" from the Sun. The eccentricity, written e, measures how stretched the ellipse is: 0 is a perfect circle, and values approaching 1 are long, thin ovals.

The nearest and farthest points of the orbit follow directly. Perihelion (closest approach to the Sun) is a(1 − e), and aphelion (farthest point) is a(1 + e). You can check this against NASA's Planetary Fact Sheet: Mars has a of about 227.9 million km and e of 0.094, and the formula gives a perihelion of about 206 million km. NASA lists 206.7 million km, the small gap coming from rounding the eccentricity.

How non-circular are real orbits?

Less than the textbook diagrams suggest. The diagrams exaggerate eccentricity so that you can see the Sun is off-center. NASA's figures show how modest the real values are.

Planet	Eccentricity	Perihelion (million km)	Aphelion (million km)
Mercury	0.206	46.0	69.8
Venus	0.007	107.5	108.9
Earth	0.017	147.1	152.1
Mars	0.094	206.7	249.3
Jupiter	0.049	740.6	816.4
Neptune	0.010	4,471.1	4,558.9

Planetary orbits are close to circular, except Mercury and Mars. Source: NASA Planetary Fact Sheet.

Take Mars. With e = 0.094, the short axis of its orbit is only about 0.4 percent shorter than the long axis, a difference you could not see on a printed page. What shows up on a page is the Sun's offset: it sits about 0.14 AU (about 21 million km) from the geometric center of the Martian orbit. Tycho's instruments could detect that offset, and that is the whole reason the Mars data forced the issue.

An analogy helps here. Imagine a running track that is almost a perfect circle, and a coach who stands not at the center but a few steps to one side. Runners would never notice the track is not round, but they would notice that they pass the coach at different distances on different parts of the lap. That small asymmetry is what Kepler found in Mars.

Law two: equal areas in equal times

The second law governs how fast a planet moves along its ellipse. Draw a line from the Sun to the planet. As the planet travels, that line sweeps across the orbital plane like the hand of a clock. The law says the hand always covers the same area in the same amount of time, whether the planet is near the Sun or far from it.

To sweep the same area when the line is short, the planet has to cover more arc. When the line is long, a smaller arc does the same job. So the planet moves fast near perihelion and slowly near aphelion.

The numbers confirm it

NASA gives Earth's maximum orbital velocity as 30.29 km/s and its minimum as 29.29 km/s, according to the Earth fact sheet. The ratio is 1.034, a difference of 3.4 percent, and it matches the ratio of Earth's aphelion distance (152.100 million km) to its perihelion distance (147.095 million km), which is also 1.034.

Mercury makes the pattern much easier to see. Its fact sheet lists a maximum orbital velocity of 58.97 km/s and a minimum of 38.86 km/s. The ratio is 1.52, and the ratio of aphelion to perihelion distance (69.818 to 46.000 million km) is also 1.52. At those two special points, the planet moves exactly in proportion to how much closer it is. Mercury at perihelion is a full 20 km/s faster than Mercury at aphelion.

Why equal areas? A figure skater explains

Kepler had no explanation for the second law beyond the observation that it worked, and he suspected the Sun somehow pushed the planets along. The real reason came with Newton, and it is one of the most general results in physics.

The area swept per second is proportional to the planet's angular momentum, a quantity that measures how much rotational motion a body has around a center. Angular momentum is conserved when no outside twist acts on the system. A figure skater spinning with arms out and then pulling them in speeds up, because the smaller radius has to be compensated by a higher rotation rate. A planet falling toward the Sun does the same thing: a shorter radius means a higher speed.

According to the Wikipedia article on Kepler's laws, the second law is exact for any force that points straight toward the center, whatever the force's dependence on distance. That is a notable property: the first and third laws need gravity's specific inverse-square form, but the second does not. It would hold for a planet bound to the Sun by a spring.

What it means for the calendar

Because Earth moves a little faster when nearer the Sun, the half of the year that includes perihelion is slightly shorter than the other half. The effect is a few days, and it is one reason the seasons are not exactly equal in length. The tilt of Earth's axis, not its distance from the Sun, is what causes the seasons, as the article on why we have seasons explains.

Law three: the harmonic law that ties planets together

The first two laws look at one planet at a time. The third connects them. It says that if you square a planet's orbital period T and divide by the cube of its semi-major axis a, you get the same number for every planet orbiting the Sun.

If you measure T in years and a in astronomical units (AU, the average Earth-Sun distance), the constant is exactly 1, because Earth has T = 1 year and a = 1 AU. The law becomes T² = a³.

A worked example

Take Jupiter, which orbits at 5.2 AU. Cube the distance: 5.2 × 5.2 × 5.2 is about 140.6. The square root of 140.6 is about 11.86. So Jupiter's period should be about 11.86 years. NASA's fact sheet gives 4,331 days, which is 11.86 years. The prediction takes about ten seconds with a calculator.

Now try Neptune at 30.07 AU. The cube is about 27,190, and its square root is about 165. Neptune's measured period is 59,800 days, or about 164 years. Kepler died in 1630 and never knew Neptune existed, yet the planet obeys his rule to within the rounding of the inputs.

Here is the comparison between Kepler's own 1618 data and modern values, from the Wikipedia article on Kepler's laws. The last column shows a³/T² in units of 10⁻⁶ AU³ per day², which the law says should be identical for every planet.

Planet	Kepler's 1618 distance (AU)	Kepler's 1618 period (days)	Kepler's ratio	Modern ratio
Mercury	0.389	87.77	7.64	7.496
Venus	0.724	224.70	7.52	7.496
Earth	1.000	365.25	7.50	7.496
Mars	1.524	686.95	7.50	7.495
Jupiter	5.20	4,332.62	7.49	7.504
Saturn	9.510	10,759.2	7.43	7.498

Kepler's seventeenth-century numbers hold within about 2 percent across a range of distances spanning a factor of 25.

Kepler wrote that he "first believed I was dreaming" when he saw the pattern, a line the Wikipedia article quotes from his own account. The ratio he found is the closest thing in early astronomy to a law of nature that you can verify with a hand calculator.

Think of it as a gear ratio

An analogy: the third law works like the gear ratio of a bicycle with fixed proportions. If you know how many teeth sit on one gear, the rule fixes the speed of the other. For planets, if you know how far from the Sun a world sits, the rule fixes how long its year must be. Outer planets do not just have farther to travel; they also move more slowly, so their years stretch out faster than their distances. Mercury's year is 88 days. Neptune's is about 164 Earth years. The distance ratio is only 78, but the period ratio is nearly 700.

A test on the most eccentric case

Pluto is a harder test, because its orbit is far more stretched than any planet's. NASA's fact sheet lists a perihelion of 4,436.8 million km, an aphelion of 7,375.9 million km, and an eccentricity of 0.244. The semi-major axis is the average of the two extremes, about 5,906 million km, or 39.5 AU. Raising 39.5 to the power of 1.5 gives about 248, so the law predicts a period of about 248 years. NASA's period for Pluto is 90,560 days, which is 248 years. The law does not care how eccentric the orbit is, because the semi-major axis already averages over the stretch.

The same numbers show something else. Pluto's perihelion distance of 4,436.8 million km is smaller than Neptune's aphelion distance of 4,558.9 million km, so for part of its orbit Pluto is closer to the Sun than Neptune's farthest point. The example makes the point that an ellipse can be very different from a circle and still obey the same rule.

Three common mistakes about the laws

The first is picturing the Sun at the middle of the ellipse. It is at a focus, off to one side, and for most planets the offset is small enough that the center and the focus nearly coincide. The second is thinking the first law says orbits are "oval" in the loose sense of an egg. An ellipse is a precise curve with two foci and perfect symmetry, unlike an egg. The third is assuming that "equal areas" means equal distances. The planet covers very different distances in equal times, and the equal part refers to the area swept by the line to the Sun.

How Newton explained all three

Kepler's laws are descriptions. They tell you what planets do, not why. The explanation arrived in 1687, when Isaac Newton published the Principia, and its central claim is that one force explains all three laws: gravity, pulling with a strength that falls off as the inverse square of distance.

The story of how Newton got there involves Edmond Halley. According to the traditional account, which historians have debated in its details, Halley visited Newton in Cambridge in August 1684 and asked what orbit an inverse-square force would produce. Newton answered that it would be an ellipse but, asked to show his calculation, could not immediately find it. He later reconstructed it and sent a short treatise, De Motu, whose central result became Proposition 11 of Book I of the Principia: an elliptical orbit with the center of force at a focus requires an inverse-square force. The same propositions run in the other direction too: a body under an inverse-square pull toward a focus moves on a conic section.

The third law, with mass included

Newton's version of the third law has an extra ingredient, and it is what makes the law useful for weighing things. For two bodies orbiting each other,

T² = 4π² a³ / [G(M + m)]

where G is the gravitational constant, M is the mass of the central body, and m is the mass of the orbiting one. When m is tiny compared with M, as for a planet around the Sun, the formula reduces to Kepler's original form with the constant set by the Sun's mass alone. The Wikipedia article on Kepler's laws gives that constant as about 7.496 × 10⁻⁶ AU³/day², which matches the "modern ratio" column in the table above.

Newton's form changes the third law from an observation into a measuring tool. If you can measure the period and the distance of anything orbiting something else, you can solve for the central mass. Kepler's original statement hid this because the ratio looked like a pure number. It is in fact a measurement of the Sun's mass in disguise.

For more on the force itself, see the explainer on what gravity is and how it works, and the biography of Isaac Newton for the history of the Principia.

Weighing worlds with a period and a distance

Once the third law is in Newton's form, you can weigh any gravitating object from the motion of something orbiting it. The recipe is always the same: measure how long one orbit takes, measure how big it is, and solve for mass.

Weighing the Earth with the Moon

The Moon takes 27.3 days to orbit and sits about 384,400 km away, according to NASA's fact sheet values for its perigee and apogee (0.363 and 0.406 million km). Plugging a period of 2.36 million seconds and a distance of 3.844 × 10⁸ meters into Newton's formula gives a mass for Earth of about 6.0 × 10²⁴ kg. That is within about one percent of the accepted value of 5.97 × 10²⁴ kg, and the small gap comes from ignoring the Moon's own mass. No scale or balance is involved.

Weighing a black hole

The same logic works at the largest scales. At the center of the Milky Way, a star called S2 loops around an invisible object in about 16 years. The Wikipedia article on S2 gives its orbital period as 16.05 years and its semi-major axis as about 970 AU, with a very high eccentricity of 0.88. Dividing the cube of that distance by the square of the period, in solar units, gives roughly 3.5 million solar masses with these rounded numbers. More careful analyses that include the distance to the galactic center arrive at about 4.3 million solar masses, according to the same article.

This is the measurement behind the 2020 Nobel Prize in Physics, which was shared by Reinhard Genzel and Andrea Ghez for the discovery of a supermassive compact object at the center of our galaxy. The Nobel Foundation's material, as reported in coverage of the prize, put the mass at about four million Suns packed into a region no larger than the solar system. Nothing about the method is more exotic than what Kepler used on Mars. The astronomers watched stars orbit for years, measured their paths, and applied T² = a³ with Newton's correction.

For related methods applied to whole galaxies, see the article on how astronomers weigh galaxies.

Kepler's laws and the hunt for exoplanets

The most direct modern use of the third law is finding planets around other stars. Most of the detection methods measure a planet's orbital period, and the period is the one number that almost every method delivers cleanly. If the host star's mass is known from its spectrum, Kepler's third law turns the period into a distance.

An example: 51 Pegasi b

On October 6, 1995, Michel Mayor and Didier Queloz announced a planet orbiting the Sun-like star 51 Pegasi, detected as a periodic wobble in the star's velocity. The Wikipedia article on 51 Pegasi b gives its period as 4.23 days and its semi-major axis as 0.0523 AU, about 7.8 million km.

You can check that against the third law. Convert 4.23 days to years: 0.01158. Square it to get 1.34 × 10⁻⁴. For a star about as massive as the Sun, the cube root gives a distance of roughly 0.05 AU. Mercury, the innermost planet in our own system, orbits at 0.387 AU, so 51 Pegasi b sits about seven times closer to its star. A "year" there lasts four days.

That result, an object with at least about half the mass of Jupiter orbiting far closer than Mercury, was unexpected, and it is one reason the search for exoplanets has produced so many surprises. NASA's Kepler space telescope, named for Johannes Kepler, searched for the small dips in starlight caused by planets passing in front of their stars. A dip that repeats every T days is an orbit, and the law supplies the planet's distance for free.

What the law gives and what it does not

The third law gives the orbital distance, not the planet's size or composition. To go from period to mass you need the star's mass and a velocity measurement. To go from period to temperature you need the star's brightness as well. The law is the foundation of the calculation, not the whole building.

Where Kepler's laws stop being exact

Kepler's laws describe an idealized situation: two bodies, alone in space, attracted by gravity that follows the inverse-square rule. The real Solar System breaks that ideal in several ways. Each exception has taught astronomers something.

Planets tug on each other

Every planet attracts every other planet. Jupiter, the most massive, pulls on all of them. As a result no planet's orbit is a perfect fixed ellipse. The ellipse itself slowly rotates, a drift called precession. For two bodies the Newtonian ellipse would stay fixed; with eight planets the long axes drift.

Mercury's case became famous. Newtonian calculations that included the pull of the other planets explained about 531 arcseconds per century of Mercury's perihelion drift, but observers measured about 574. The leftover, about 43 arcseconds per century in modern figures, resisted explanation for decades after Urbain Le Verrier flagged the anomaly in 1859. In 1915 Einstein showed that his theory of gravity predicts an extra advance of 43.03 arcseconds per century, which matched observation, as a review of tests of general relativity on arXiv notes. The explanation of that result is in the article on Einstein's relativity.

More than two bodies

When three or more bodies interact with comparable strength, there is no closed-form solution like Kepler's. Orbits of that kind can be computed numerically, but they are not ellipses, and they can be chaotic over long periods. Kepler's laws work well for the Solar System because the Sun carries almost all the mass, which makes every planet's orbit a lightly disturbed two-body problem.

Not just the Sun

The same laws describe moons around planets, satellites around Earth, and binary stars. A satellite in low Earth orbit obeys T² proportional to a³ with Earth's mass in place of the Sun's. The article on how satellites orbit Earth covers that engineering side, including why a satellite 35,786 km up takes one day to circle the planet.

What the laws taught about doing science

Kepler's real achievement may be methodological. He had a workable model, and he had data that disagreed with it by 8 arcminutes. He trusted the data. The result was not a more accurate version of the old circles but a different kind of astronomy: one in which the shapes and speeds of orbits are things to be measured, not assumed.

There is also an irony in the history. Kepler arrived at the laws partly through ideas that no modern scientist would accept, including a belief in musical harmonies of the planets. The third law appears in a book titled Harmonices Mundi ("Harmony of the World"). What survived was the arithmetic, which was correct, and not the mysticism, which was not. The laws made their way into Newton's physics because they were precisely stated and testable, and not because of the thinking that produced them.

For readers who want a broader view of the solar system these laws describe, the planets of the solar system article has the physical properties of each world, and the Complete Space Guide links to the rest of the topic cluster.

The takeaway

Kepler's three laws compress the motion of every planet into three short statements: the path is an ellipse, the speed varies so that areas are swept equally, and the periods of different planets are locked to their distances. Newton showed that these are not three separate facts but three consequences of one inverse-square force, with the second law resting on the even more general conservation of angular momentum. That is why the laws still earn their keep. Any time two objects orbit each other and you can time the orbit and measure its size, you can weigh them, whether they are the Earth and the Moon, a star and a distant planet, or a star and the black hole at the heart of the Milky Way.

Frequently asked questions

Q: Do planets really speed up when they get closer to the Sun? A: Yes. NASA lists Earth's orbital speed as 30.29 km/s at its fastest and 29.29 km/s at its slowest, and Mercury's as 58.97 km/s at its fastest and 38.86 km/s at its slowest. At perihelion and aphelion the speed ratio equals the distance ratio, a direct consequence of conservation of angular momentum.

Q: Is Earth closest to the Sun in summer? A: No, for the Northern Hemisphere. Earth's distance varies only from 147.1 million km to 152.1 million km, a change of about 3 percent. Seasons come from the tilt of Earth's axis, which changes how directly sunlight strikes each hemisphere, and not from the distance.

Q: Why is Kepler's third law written with the semi-major axis and not the distance? A: Because the distance changes continuously along an ellipse. The semi-major axis, half the longest diameter, is a single number that stands in for the average size of the orbit. For a circle it equals the radius. Using it lets the law apply to eccentric orbits such as Mercury's or a comet's.

Q: Can Kepler's laws be used for anything other than the Solar System? A: Yes. With Newton's correction, which includes the masses of both bodies, they apply to moons, satellites, binary stars, exoplanets, and stars orbiting the central black hole of the Milky Way. In each case, a measured period and orbit size give the total mass of the system.

Q: Did Kepler know why the laws worked? A: No. He found them by fitting Tycho Brahe's observations and suspected that the Sun somehow drove the planets along. The explanation came in 1687, when Newton's Principia showed that an inverse-square force toward the Sun produces ellipses, equal-area sweeping, and the period-distance relation.

Q: Are Kepler's laws still exactly true? A: They are exact for two bodies attracted by an inverse-square force. In the real Solar System, planetary tugs cause small drifts, and general relativity adds a further correction, such as the 43 arcseconds per century of extra advance in Mercury's perihelion. For most practical purposes the laws remain extremely accurate.

Q: How does Kepler's third law help find exoplanets? A: Detection methods such as transits and velocity wobbles measure a planet's orbital period directly. If the star's mass is known, the third law converts that period into the planet's distance from the star. For 51 Pegasi b, a 4.23-day period corresponds to about 0.05 AU.

Sources

• NASA Planetary Fact Sheet: perihelion, aphelion, period, and eccentricity for each planet.
• NASA Mercury Fact Sheet: maximum and minimum orbital velocity, distances, and eccentricity.
• NASA Earth Fact Sheet: orbital velocities, distances, and eccentricity.
• Kepler's laws of planetary motion (Wikipedia): publication dates, Kepler's 1618 data, modern values, Newtonian form.
• Astronomia nova (Wikipedia): the vicarious hypothesis, the 8-arcminute discrepancy, and the structure of the book.
• Tycho Brahe (Wikipedia): Hven, observing accuracy, date of death.
• S2 (star) (Wikipedia): orbital elements and the mass of the Milky Way's central black hole.
• 51 Pegasi b (Wikipedia): discovery date, period, and orbital distance.
• Tests of General Relativity: A Review (arXiv:1705.04397): Mercury's perihelion advance of 43.03 arcseconds per century.