Before the discovery of gravity, before the invention of the telescope1, back when it was pretty well accepted that the Earth was the center of the universe, a German astronomer named Johannes Kepler figured out how planetary orbits work. This theory was organized into three laws, the first two published in 1609 and the third in 1619, and these laws of planetary motion, named Kepler’s Laws, are still used in science four centuries later! Quite a legacy.
Imperfect Circles: Kepler’s First Law
The first of Kepler’s laws of planetary motion states that planetary orbits are ellipses with the Sun at one focus. This sounds pretty innocuous, aside from the fact that you may not have thought about what the foci of an ellipse are since middle school geometry, but it was a radical theory at the time. (The foci of an ellipse, by the way, are the two interior locations where the sum of the distances from any point on the ellipse are constant; a circle is a special case of an ellipse where the two foci are at the same location, the center.)
For one thing, circles were considered perfect and simple (ellipses are a little weird, after all). Given that planets orbit in the heavens, only the perfection of circles would do. This concept dates at least back to Plato, whose proposed cosmological model was a combination of perfect, circular motions. Kepler himself also believed orbits would be circular, until he didn’t. We have Mars to thank for that.
Kepler was primarily a theoretical astronomer. His theory was built upon detailed observations taken by the Danish astronomer Tycho Brahe. These were unprecedented in their volume and accuracy, all the more remarkable when considering these were naked-eye, pre-telescopic observations. Tycho tasked Kepler, his assistant, with figuring out the orbit of Mars. And Tycho’s observations were so precise that, for Kepler, there was no ignoring the fact that a circular orbit just did not match the observations. Mars, in fact, has the second-least-circular orbit of all the planets—lucky thing, or it may have taken even longer to discover this law!
The other radical notion in Kepler’s first law is that the Sun is at one of the foci of the ellipse. Heliocentrism, though proposed over sixty years earlier by Nicolaus Copernicus, was far from established at the time. Tycho himself was not a heliocentrist, and heliocentrism would be banned by the Roman Inquisition in 1616. Yet Kepler was both able to realize and willing to publish his theory that the Sun was the focus of the planets’ orbits.
So don’t overlook the ingenuity and importance of Kepler’s First Law!
Sweeping Areas: Kepler’s Second Law
It’s impossible (for me, at least) to discuss Kepler’s Second Law without an obligatory xkcd reference.
This is hilarious, and if you don’t get the joke, you’ve come to the right place!
The second law of planetary motion states that planets sweep out equal areas over equal intervals of time. Now, a planet doesn’t actually have a broom, so what does it mean for it to “sweep out” an area? Imagine a line connecting the planet and the Sun (which you’ll recall is at one of the foci of the orbit). Over time, as the planet moves in its orbit, that imaginary line would also move, and the area that the line passes over is what is considered the area being swept. The second law tells us that if you measured that area over, for example, 100 hours during one part of the orbit, if you measured that area over 100 hours in a different part of the orbit it would be the same.
Now for a circle, this isn’t surprising. The line connecting the planet and the Sun would always have the same length, and you would expect it to sweep out a steady area. But since planetary orbits aren’t circles, this takes on much more significance. This imaginary sweeping line changes size over the course of the orbit, as the planet gets closer and farther from the Sun. If the planet was moving at the same speed, the area swept out would be less when the planet is close and more when the planet is far. But if the areas are equal, that means the planet’s speed must be changing throughout its orbit! Closer to the Sun, the planet moves faster. Farther from the Sun, the planet moves slower.
You may be wondering what I mean by “speed”, because an object moving along an arc has two kinds of speed: angular and linear. Linear speed is what we might more often encounter as speed, measured in distance per time (like mph or km/s). It’s just how much distance the planet is covering over time. Angular speed refers to how the angle changes over time and is measured in an angle per time (like deg/hr or rad/s)2. This angle is measured from the Sun, and because for angular speed it’s a difference in angle, the reference used to measure the angle doesn’t matter. But typically, the angle is measured relative to the planet’s perihelion, the place where it is closest to the Sun, and is called the true anomaly (f).
So which speed is the one that isn’t constant over time? My first inclination3 would be the angular speed. After all, we already know the distance is changing over time and that linear speed and angular speed are related by distance4, so it could be possible that the angular speed changes in a way opposite the changing distance to maintain the same linear speed over time. But if that were the case, the areas swept out would not be equal, so it is both speeds that must change over time!
Nowadays, we can [relatively] easily prove the areas must be equal using some calculus and Newton’s law of gravity, and it turns out to be a version of the conservation of angular momentum. But remember, Kepler didn’t have Newton’s law of gravity. Kepler didn’t even have calculus! He just noticed this from the observational data, and this observation in turn helped lead Newton to formulate his law of gravity.
So because of Kepler’s second law, we know that planets move fastest near perihelion and slowest near aphelion. You can see this for yourself—look at how the location of the sun at a set time changes between days over the course of a year (creating what is called an analemma). The change is biggest in January (Earth’s perihelion) and smallest in July (Earth’s aphelion), leading to asymmetrical lobes of the resulting figure 8 shape.5
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And you can now properly appreciate xkcd 21.
Time/Distance Relation: Third Law
Kepler’s third law stands a bit apart from the first two; it was published a decade later6, and it makes a quantitative statement rather than the somewhat more qualitative statements of the first two laws. The mathematical relation expressed in this law is still very useful to this day, even in its original, approximated form!
The third law of planetary motion is easiest expressed as an equation, but the general idea is that the further a planet is from the Sun, the longer its orbit is, with a very specific relationship between its orbital distance and its orbital period: the square of the periods of the planets is proportional to the cube of their distance. The “proportional to” means that there is some constant value that relates the period squared and the distance cubed—the same constant for all of the planets!
There was one very important invention that happened between the publication of Kepler’s first two laws and of the third law: logarithms.7 The Scottish mathematician John Napier published the method of logarithms in 1614. Logarithms make working with exponents a lot easier. In particular, using a logarithmic scale for a mathematical plot reduces any exponential relationship to a straight line (with a slope equal to that exponent). A relationship with an exponent of 3/2, like Kepler’s third law, can be difficult to ascertain in a linear plot. Transform into a log-log plot (meaning both axes are changed to logarithmic scale) and the relationship just jumps right out!
Kepler had data regarding only six planets to make this observation, the discovery of Neptune and Uranus being some centuries away, and he found that common constant to be about 7.5 millionths. Consider Earth, for example. At the time, the exact orbital distance of the Earth was unknown8, and so that distance was taken to be the base unit of orbital distances and called an astronomical unit. With an orbital distance of 1 AU and an orbital period of 365.25 days, 1 AU cubed is 1 AU3 and 365.25 days squared is 133,407.5625 days2, and 1/133,407.5625 is 7.496×10-6. It is left as an exercise to the reader to try this for another planet, perhaps Jupiter, which has an orbital distance of 5.204 AU and an orbital period of 4,332.59 days.
Now Kepler did not have any insight into why this constant had that value, just that such a constant value existed. With the advent of Newtonian gravity, the constant is explained.9 For the simplest case of a circular orbit, using Newton’s equation for gravitational force to describe the centripetal velocity of a planet, it can be rearranged to provide the numerical form of Kepler’s third law. We can see that the constant is GM/(4π2).
A more rigorous derivation would reveal the constant is not actually constant but rather G(M+m)/(4π2). Because m is the mass of the planet, this term cannot be constant for all the planets. But! Because the mass of a planet is so small compared to the mass of the Sun, the planet’s mass term is essentially negligible. Thus the constant approximation, GM/(4π2), works for any small body orbiting the Sun. The third law can be generalized to orbits around any primary body, but with a different constant, because the M is just the mass of the central body. Indeed, in 1621 Kepler observed that the Galilean moons of Jupiter, discovered in 1610, followed his 3/2 law.10
In my own research, I have frequently applied this relationship to exoplanet systems to convert between distance and period11, and I can appreciate why Kepler considered this part of the harmony of the world, a relationship so clear and beautiful that at first he believed he was dreaming.
And thus we reach the end of our exploration of these three laws of planetary motion. Still remarkable over four hundred years after Kepler’s discovery!
1. Debatably, given that the first patent application for a telescope was in 1608, and Kepler’s planetary motion theory was first published in 1609, so the two were likely developed in a very similar time frame. But certainly his theory was created without the benefit of telescopes, and in fact Kepler himself proposed telescope improvements in 1611 that went on to be widely used, which is entirely outside of the scope of this article but a fun fact nonetheless. [back]
2. While knowing the unit of an angle is important for clarity, angular units are dimensionless. Often in science, angular speeds are represented simply as per time, e.g., s-1. In this case, the implied unit is a radian, which is really just a ratio of distances. We don’t tend to think in radians much in our day-to-day life, though, so it’s often easier to conceptualize in degrees. A full circle is 2π (about 6.3) radians or 360°. [back]
3. Pun intended. [back]
4. This is true, v=r⍵, but only for purely circular motion where v and r are perfectly perpendicular. The first law strikes again! [back]
5. The details of how eccentricity and obliquity (axial tilt) interact to form the resulting analemma shape are fascinating, but beyond the scope of this article, so I guess I’m adding to my to-do list! [back]
6. At the time, Kepler’s Laws were not grouped together into three statements and were not even called laws. That nomenclature dates to Voltaire over a century later, and the grouping of the laws into three was Robert Small in 1804, almost two centuries after Kepler’s publication! The article by Curtis Wilson in the Newsletter of the Historical Astronomy Division of the AAS that details this is a quick and interesting read, I definitely recommend it! [back]
7. How weird is it to think about logarithms being invented? It’s crazy to imagine doing science without even knowing it is possible to make a log-log plot. [back]
8. A fascinating history in and of its own right! [back]
9. Well, the can is at least kicked a little further down the road, because there is still an empirical constant involved, the universal gravitational constant G, which equals 6.6743×10-11 m3/(kg s2). [back]
10. Because this law refers to a proportionality relationship, any units can be used to observe the relationship, such as using AU for the planetary orbits. For the Galilean moons, their orbital distances can be measured in Jupiter radii without needing to know anything about how far that is in a different unit. [back]
11. And had to double check every time to make sure I had the 2 and 3 in the right spot in the exponent! [back]