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3.7 Kepler's Laws

Johannes Kepler published his discoveries about the orbits of planets in two books, in 1609 and 1619. His findings can be distilled into three famous "laws," or rules, which have come to be called Kepler's laws of planetary motion. Kepler's laws are powerful because they are so general. They are simply an extension of Newton's laws of motion when two bodies are gravitationally bound to each other (although they predate Newton!). Kepler's laws apply to any orbital motion, whether of a planet around the Sun, the Moon around the Earth, or a star around the center of a galaxy. >br/> 
Kepler's first law is simple: all planets' orbits are ellipses, with the Sun at one focus. While simple, this law actually caught many people off guard. The philosopher Aristotle had declared the circle the perfect shape, and Plato had come up with many other ideal shapes. The ellipse was on none of their lists! Once this shape was sorted out, it was suddenly possible to understand planetary positions accurately.

The second and third laws were a result of Kepler's attempts to find patterns in the orbits of the planets. Kepler's second and third laws are mathematical relationships between the distance of a planet from the Sun and the rate of its motion around the Sun. They are both natural consequences of applying Newton's law of gravity and the law of conservation of angular momentum to an object moving on an elliptical path, but Kepler was remarkably able to derive them without either of these concepts! 

Kepler's second law of planetary motion states that a planet speeds up when it is near the Sun and slows down when it is far away. In other words, if you imagine a line connecting a planet to the Sun, the line sweeps out equal areas in equal intervals of time. This is a simple consequence of the conservation of angular momentum. The angular momentum of a planet-Sun system is conserved at all times. So when the planet is closer to the Sun, it must move faster to conserve angular momentum. (Remember, angular momentum is L = r × m v)


Kepler's First Law. Click here for original source URL.

Kepler's third law states that the square of the orbital period is proportional to the cube of the semi major axis:

P2 ∝ a3

In this equation, P is the orbital period, and a is the semi major axis. For our Solar System, we can set the constant of proportionality to one if we use units of years for the period and astronomical units for the semi major axis (1 A.U. = the mean Earth-Sun distance = Earth's semi major axis). In this case:

(PYears)2 = (aAU)3

Mars's average distance from the Sun is about 50% larger than the Earth's, or 1.52 A.U. Therefore the period squared is equal to (1.52)3 = 3.51, and the period is √(3.52) = 1.87 years. The Martian year is almost twice as long as an Earth year. Kepler's third law describes a steady increase in orbital period with distance from the Sun. Notice that in these calculations we only use three significant figures. This is because there are departures from Kepler's laws at a level below 0.1%.

Let's do a few more applications of this important relation. Say we can use geometry to show that Jupiter is 5.2 times farther from the Sun than the distance of the Earth from the Sun. What is Jupiter's orbital period? Following the previous example, the period squared is equal to (5.2)3 = 140.6, and the period is √(140.6) = 11.9 years. The Jovian year is nearly twelve times as long as the Earth year. Now let's imagine that with careful observation we were able to show that Mercury took 89 days to orbit the Sun. Converting to years, we have that the semi major axis cubed is equal to (89 / 365)2 = 0.0595, so the semi major axis is (0.0595)1/3 = 0.39 AU. Mercury lies at just over a third the Earth-Sun distance.

Kepler's work greatly advanced Copernicus' idea that all planets move in orbits around the Sun, rather than around the Earth. His simple relationships to describe the planets' orbits made it clear that the Sun governed planetary motions, not the Earth. Kepler's idea of ellipses swept away all the complications of the geocentric cosmology, in particular Ptolemy's system and its epicycles and deferents. More important, this simpler system was able to more accurately predict the motions of the planets. It still wasn't perfect, but it was an observationally based step in the right direction. Newton's laws were required to allow corrections to be made for the tugs of the planets on one another, and Einstein's relativity was needed to explain the details of Mercury's high speed orbit near the Sun. The systematic work of these three men shows that science is a process, with successive work building and improving on past discoveries.