How would you find a new planet at the outskirts of our solar system that is too dim to be seen with the unaided eye and is so far away that it moves very slowly among the stars? This was the problem confronting astronomers during the nineteenth century as they tried to pin down a full inventory of our solar system.

If we could look down on the solar system from somewhere out in space, interpreting planetary motions would be much simpler. But the fact is, we must observe the positions of all the other planets from our own moving planet. Scientists of the Renaissance did not know the details of Earth’s motions any better than the motions of the other planets. Their problem was that they had to deduce the nature of all planetary motion using only their earthbound observations of the other planets’ positions in the sky. To solve this complex problem more fully, better observations and better models of the planetary system were needed.

The efforts of two other scientists dramatically advanced our understanding of the motions of the planets. These two astronomers were the observer Tycho Brahe and the mathematician Johannes Kepler. Together, they placed the speculations of Copernicus on a sound mathematical basis and paved the way for the work of Isaac Newton in the next century.

4.1.1 Tycho Brahe’s Observatory

Three years after the publication of Copernicus’ De Revolutionibus, Tycho Brahe was born to a family of Danish nobility. He developed an early interest in astronomy and, as a young man, made significant astronomical observations. Among these was a careful study of what we now know was an exploding star that flared up to great brilliance in the night sky. His growing reputation gained him the patronage of the Danish King Frederick II, and at the age of 30, Brahe was able to establish a fine astronomical observatory on the North Sea island of Hven (Figure 4.1). Brahe was the last and greatest of the pre-telescopic observers in Europe.

At Hven, Brahe made a continuous record of the positions of the Sun, Moon, and planets for almost 20 years. His extensive and precise observations enabled him to note that the positions of the planets varied from those given in published tables, which were based on the work of Ptolemy. These data were extremely valuable, but Brahe didn’t have the ability to analyze them and develop a better model than what Ptolemy had published. He was further inhibited because he was an extravagant and cantankerous fellow, and he accumulated enemies among government officials. When his patron, Frederick II, died in 1597, Brahe lost his political base and decided to leave Denmark. He took up residence in Prague, where he became court astronomer to Emperor Rudolf of Bohemia. There, in the year before his death, Brahe found a most able young mathematician, Johannes Kepler, to assist him in analyzing his extensive planetary data.

4.1.2 Johannes Kepler

Johannes Kepler was born into a poor family in the German province of Württemberg and lived much of his life amid the turmoil of the Thirty Years’ War (Figure 4.1). He attended university at Tubingen and studied for a theological career. There, he learned the principles of the Copernican system and became converted to the heliocentric hypothesis. Eventually, Kepler went to Prague to serve as an assistant to Brahe, who set him to work trying to find a satisfactory theory of planetary motion—one that was compatible with the long series of observations made at Hven. Brahe was reluctant to provide Kepler with much material at any one time for fear that Kepler would discover the secrets of the universal motion by himself, thereby robbing Brahe of some of the glory. Only after Brahe’s death in 1601 did Kepler get full possession of the priceless records. Their study occupied most of Kepler’s time for more than 20 years.

Through his analysis of the motions of the planets, Kepler developed a series of principles, now known as Kepler’s three laws, which described the behavior of planets based on their paths through space. The first two laws of planetary motion were published in 1609 in The New Astronomy. Their discovery was a profound step in the development of modern science.

4.1.3 Kepler’s First and Second Law

The path of an object through space is called its orbit. Kepler initially assumed that the orbits of planets were circles, but doing so did not allow him to find orbits that were consistent with Brahe’s observations. Working with the data for Mars, he eventually discovered that the orbit of that planet had the shape of a somewhat flattened circle, or ellipse. Next to the circle, the ellipse is the simplest kind of closed curve, belonging to a family of curves known as conic sections (Figure 4.2).

You might recall from math classes that in a circle, the center is a special point. The distance from the center to anywhere on the circle is exactly the same. In an ellipse, the sum of the distance from two special points inside the ellipse to any point on the ellipse is always the same. These two points inside the ellipse are called its foci (singular: focus), a word invented for this purpose by Kepler.

This property suggests a simple way to draw an ellipse (Figure 4.3). We wrap the ends of a loop of string around two tacks pushed through a sheet of paper into a drawing board, so that the string is slack. If we push a pencil against the string, making the string taut, and then slide the pencil against the string all around the tacks, the curve that results is an ellipse. At any point where the pencil may be, the sum of the distances from the pencil to the two tacks is a constant length—the length of the string. The tacks are at the two foci of the ellipse.

The widest diameter of the ellipse is called its major axis. Half this distance—that is, the distance from the center of the ellipse to one end—is the semimajor axis, which is usually used to specify the size of the ellipse. For example, the semimajor axis of the orbit of Mars, which is also the planet’s average distance from the Sun, is 228 million kilometers.

The shape (roundness) of an ellipse depends on how close together the two foci are, compared with the major axis. The ratio of the distance between the foci to the length of the major axis is called the eccentricity of the ellipse.

If the foci (or tacks) are moved to the same location, then the distance between the foci would be zero. This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius.

Next, we can make ellipses of various elongations (or extended lengths) by varying the spacing of the tacks (as long as they are not farther apart than the length of the string). The greater the eccentricity, the more elongated is the ellipse, up to a maximum eccentricity of [latex]1.0[/latex], when the ellipse becomes “flat,” the other extreme from a circle.

The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity. Using Brahe’s data, Kepler found that Mars has an elliptical orbit, with the Sun at one focus (the other focus is empty). The eccentricity of the orbit of Mars is only about [latex]0.1[/latex]; its orbit, drawn to scale, would be practically indistinguishable from a circle, but the difference turned out to be critical for understanding planetary motions.

Kepler generalized this result in his first law and said that the orbits of all the planets are ellipses. Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an acceptable cosmos. The universe could be a bit more complex than the Greek philosophers had wanted it to be.

Kepler’s second law deals with the speed with which each planet moves along its ellipse, also known as its orbital speed. Working with Brahe’s observations of Mars, Kepler discovered that the planet speeds up as it comes closer to the Sun and slows down as it pulls away from the Sun. He expressed the precise form of this relationship by imagining that the Sun and Mars are connected by a straight, elastic line. When Mars is closer to the Sun (positions 1 and 2 in Figure 4.4), the elastic line is not stretched as much, and the planet moves rapidly. Farther from the Sun, as in positions 3 and 4, the line is stretched a lot, and the planet does not move so fast. As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves (the colored regions in our figure). Kepler found that in equal intervals of time (t), the areas swept out in space by this imaginary line are always equal; that is, the area of the region B from 1 to 2 is the same as that of region A from 3 to 4.

If a planet moves in a circular orbit, the elastic line is always stretched the same amount and the planet moves at a constant speed around its orbit. But, as Kepler discovered, in most orbits that speed of a planet orbiting its star (or moon orbiting its planet) tends to vary because the orbit is elliptical.

4.1.4 Kepler’s Third Law

Kepler’s first two laws of planetary motion describe the shape of a planet’s orbit and allow us to calculate the speed of its motion at any point in the orbit. Kepler was pleased to have discovered such fundamental rules, but they did not satisfy his quest to fully understand planetary motions. He wanted to know why the orbits of the planets were spaced as they are and to find a mathematical pattern in their movements—a “harmony of the spheres” as he called it. For many years he worked to discover mathematical relationships governing planetary spacing and the time each planet took to go around the Sun.

In 1619, Kepler discovered a basic relationship to relate the planets’ orbits to their relative distances from the Sun. We define a planet’s orbital period, (P), as the time it takes a planet to travel once around the Sun. Also, recall that a planet’s semimajor axis, a, is equal to its average distance from the Sun. The relationship, now known as Kepler’s third law, says that a planet’s orbital period squared is proportional to the semimajor axis of its orbit cubed, or

For instance, suppose you time how long Mars takes to go around the Sun (in Earth years). Kepler’s third law can then be used to calculate Mars’ average distance from the Sun. Mars’ orbital period (1.88 Earth years) squared, or [latex]P^2[/latex], is [latex]1.88^2 = 3.53[/latex], and according to the equation for Kepler’s third law, this equals the cube of its semimajor axis, or [latex]a^3[/latex]. So what number must be cubed to give 3.53? The answer is [latex]1.52\ (\text{since}\ 1.52\ ×\ 1.52\ ×\ 1.52=3.53)[/latex]. Thus, Mars’ semimajor axis in astronomical units must be 1.52 AU. In other words, to go around the Sun in a little less than two years, Mars must be about 50% (half again) as far from the Sun as Earth is.

Kepler’s three laws of planetary motion can be summarized as follows:

• Kepler’s first law: Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse.
• Kepler’s second law: The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time.
• Kepler’s third law: The square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit.

For an audiovisual summary of Kepler’s three laws of planetary motion watch the below video from NASA.

4.1.5 Orbital Motion and Mass

Kepler’s laws describe the orbits of the objects whose motions are described by Newton’s laws of motion and the law of gravity. Knowing that gravity is the force that attracts planets toward the Sun, however, allowed Newton to rethink Kepler’s third law. Recall that Kepler had found a relationship between the orbital period of a planet’s revolution and its distance from the Sun. But Newton’s formulation introduces the additional factor of the masses of the Sun (M₁) and the planet (M₂), both expressed in units of the Sun’s mass. Newton’s universal law of gravitation can be used to show mathematically that this relationship is actually

[latex]a^3=(M_1\ +\ M_2)\ ×\ P^2[/latex]

where a is the semimajor axis and P is the orbital period.

How did Kepler miss this factor? In units of the Sun’s mass, the mass of the Sun is 1, and in units of the Sun’s mass, the mass of a typical planet is a negligibly small factor. This means that the sum of the Sun’s mass and a planet’s mass, [latex](M_1+M_2)[/latex], is very, very close to 1. This makes Newton’s formula appear almost the same as Kepler’s; the tiny mass of the planets compared to the Sun is the reason that Kepler did not realize that both masses had to be included in the calculation. There are many situations in astronomy, however, in which we do need to include the two mass terms—for example, when two stars or two galaxies orbit each other.

Including the mass term allows us to use this formula in a new way. If we can measure the motions (distances and orbital periods) of objects acting under their mutual gravity, then the formula will permit us to deduce their masses. For example, we can calculate the mass of the Sun by using the distances and orbital periods of the planets, or the mass of Jupiter by noting the motions of its moons.

Indeed, Newton’s reformulation of Kepler’s third law is one of the most powerful concepts in astronomy. Our ability to deduce the masses of objects from their motions is key to understanding the nature and evolution of many astronomical bodies.

4.1.6 How We Use Kepler’s Laws Today

Kepler didn’t know about gravity, which is responsible for holding the planets in their orbits around the Sun, when he came up with his three laws. But Kepler’s laws were instrumental in Isaac Newton’s development of his theory of universal gravitation, which explained the unknown force behind Kepler’s third law. Kepler and his theories were crucial in the understanding of solar system dynamics and as a springboard to newer theories that more accurately approximate planetary orbits. However, his third law only applies to objects in our own solar system.

Newton’s version of Kepler’s third law allows us to calculate the masses of any two objects in space if we know the distance between them and how long they take to orbit each other (their orbital period). What Newton realized was that the orbits of objects in space depend on their masses, which led him to discover gravity.

Newton’s generalized version of Kepler’s third law is the basis of most measurements we can make of the masses of distant objects in space today. These applications include determining the masses of moons orbiting the planets, stars that orbit each other, the masses of black holes (using nearby stars affected by their gravity), the masses of exoplanets (planets orbiting stars other than our Sun), and the existence of mysterious dark matter in our galaxy and others.

In planning trajectories (or flight plans) for spacecraft, and in making measurements of the masses of the moons and planets, modern scientists often go a step beyond Newton. They account for factors related to Albert Einstein’s theory of relativity, which is necessary to achieve the precision required by modern science measurements and spaceflight.

However, Newton’s laws are still accurate enough for many applications, and Kepler’s laws remain an excellent guide for understanding how the planets move in our solar system.

Johannes Kepler died Nov. 15, 1630, at age 58. NASA’s Kepler space telescope was named for him. The spacecraft launched March 6, 2009, and spent nine years searching for Earth-like planets orbiting other stars in our region of the Milky Way. The Kepler space telescope left a legacy of more than 2,600 planet discoveries from outside our solar system, many of which could be promising places for life.

4.1.7 The Discovery of Neptune Using Mathematics

Astronomers during the 1800s noticed that the planet Uranus deviated from the orbit predicted by Kepler’s and Newton’s laws. Two astronomers, Le Verrier and John Couch Adams, went on a mathematic hunt to figure out why, and they ended up discovering Neptune! Watch the below video to learn how this happened. 

Text Attributions

The text of this chapter is adapted from:

• Section 3.1 of OpenStax’s Astronomy 2e (2022) by Andrew Fraknoi, David Morrison, and Sidney Wolff. Licensed under CC BY 4.0. Access full book for free at this link.
• The article Orbits and Kepler’s Laws (2024) by the NASA Science Editorial Team posted to the NASA website. Licensed under public domain.

Media Attributions

• “Solar System Dynamics: Orbits and Kepler’s Laws.” YouTube, uploaded by NASA Solar System, 9 Nov 2010, https://www.youtube.com/watch?v=wjOOrr2uPuU.
• “The crazy way we found Neptune in the 1800s.” YouTube, uploaded by SolarSystemSnacks, 18 Jul 2024, https://www.youtube.com/watch?v=xO_zng-jeww.