Introduction

Johannes Kepler was a German astronomer who actually lived before Newton formulated his Law of Universal Gravitation, figuring out his three laws of planetary motion from astronomical data. There's a bit of interesting history behind Kepler's story, but since this page isn't a science history site I won't be covering it. The general gist of how Kepler came to his discovery was that he noticed that a circle didn't exactly fit the astronomical figures, while an elliptical shape fit it much better. This leads us to Kepler's First Law:

Kepler's First Law

This was revolutionary at the time, as people thought circles were the perfect shape and the cosmos must be perfect. However, most planetary orbits are very close to circular, which is why most of the time they are approximated as circles for simplicity. But now it's commonly accepted fact, so we won't be disputing this law today. Instead, we should think about how it affects our conceptions of planetary motion.

Obviously, since the orbit is no longer a circle, the orbital radius is no longer constant. This means that the speed at any given point in the orbit will no longer remain constant, instead changing. In the diagram, it looks like the sun is at the center of the ellipse. This, however, is not correct; in fact, it should be located at one of the foci of the ellipse. (I put it at the center more for aesthetic feel and to prove that what most people immediately think isn't true.) Here's an updated and correct diagram: The orbit is no longer a circle, so the radius is no longer constant. Intuitively, if this distance is changing the oribiting speed is also changing. Additionally, the star shouldn't actually be at the center of the ellipse, but rather slightly off to one side. If you've learned about ellipses, this is called the focus of the ellipse, but if you haven't, just know that it's slightly off to one side. It would be good to read up on ellipses though, since most references to these orbits use that terminology.

There are some terms associated with the elliptical orbit. The perihelion is the point where the planet is closest to the star, while the aphelion is the point where it is furthest. Sometimes, the terms perigee and apogee are used instead. They have the same meaning.

Eccentricity

I (Eric) want to talk about a new quantity that describes ellipses, the eccentricity. The eccentricity of an ellipse is the ratio of its focal length $c$ to its semimajor axis length $a$. In mathematical terms, this means:

$$e = \dfrac{c}{a}$$
A circle, which has a focal length of zero, therefore has zero eccentricity. An ellipse will always have an eccentricity less than one but greater than zero, with a higher eccentricity indicating a more flattened shape. When eccentricity is equal to exactly one, we have a parabola. When it exceeds one, we have a hyperbola. The latter two are also valid paths for celestial objects under the influence of gravity, though they are both open paths that end with the object going off to infinity.

This concept is more of a math concept, but I think it links together the possible orbits quite nicely. By connecting the four conic sections with a single quantity $e$, it is shown how all the possible planetary orbits are indeed connected. I wanted to go on and talk about the polar form for conic sections which can be tweaked to include all four, but that is bordering too far into the scary realm of pure math.

While it is possible to use this law for calculations, it's not easy to do so without an understanding of energy. We will come back to gravitation after we learn about energy and use it to derive many more interesting results. However, for now we're stuck with a conceptual understanding of this topic.

Kepler's Second Law

This law is a bit unorthodox, to say the least. Imaginary line? Areas? It all seems so abstract, but I promsise it is anything but. It's certainly a bit verbose and can be hard to understand without a visual guide. Here is a basic visual to explain what I'm talking about:

In the visual, the red and blue areas are (approximately, I made this by hand don't judge me) equal. The time it takes the planet to travel through those distances is also equal. This is what the law means by "equal areas in equal times". What this also tells us is that the planet must be moving faster as it nears the star and slower when it is futher away, because the distance travelled in the red area is much greater than for the blue area. This also makes sense, as you would expect something to get faster as it "fell" closer towards a massive object.

In calculus terms, this law can be expressed in terms of a derivative. The wording "equal areas in equal times" can be translated to mean that the rate of change of the area with respect to time is constant. In mathematical terms, this can be written as:

$$\dfrac{dA}{dt} = \textrm{const.}$$
When deriving this law later on, it is sufficient to prove this condition. However, the derivation of this law requires a number of concepts like energy and angular momentum which we have not covered, so it won't be coming anytime soon.

This law is honestly not particularly useful, as applications of its findings are few and far between. The only real application of this law is to realize that the speed of orbit is fastest when at the perihelion and slowest at the aphelion, but this is rather intuitive and can be shown with other methods as well. The same cannot be said for Kepler's Third Law, which is actually a mathematical relation.

Kepler's Third Law

This final law is heavily mathematical and can be used to great effect in finding orbital periods. When you know the length of the semimajor axis of orbit, it is easy to find the orbital period, either by direct computation or by using a method of ratios with another planet whose radius and orbital period are known. This law leaves little to be desired, really. It also reinforces the idea that there is only one possible orbital velocity (and therefore one orbital period) for each radius.

While Kepler's Third Law does technically exist as a mathematical equation, at a conceptual level it is sufficient to know it as a proportionality. This makes the relation a lot less messy and much less daunting. In its most basic terms, Kepler's Third Law states:

$$T^2 \propto a^3$$
Here, $a$ is the length of the semimajor axis of orbit. In the cases where the orbit is circular, we can simply replace $a$ with the orbital radius $r$. The mass of the star is also a factor that isn't seen here, but that doesn't matter most of the time since we deal with the solar system or planets around the same star most of the time. However, it's still important to remember. (I think it's not in the proportionality because we didn't know about other star systems at the time of Kepler.)

This simple proportionality actually does allow us to directly compute the exact orbital periods of planets by setting up a ratio. I will actually challenge you to do so with this practice problem.



Kepler's Third Law can be written as a proportionality, but it also has an equation form. The proportionality form is as follows:

$$T^2 \propto a^3$$
However, the exact form takes into account the mass of the star around which the planet is orbiting. This makes sense, as the only two factors that orbital period depends on are the orbital radius and the mass of whatever's keeping the object in orbit. We'll actually do the relatively simple derivation for Kepler's Third Law for a circular orbit, but the generalized form that applies to elliptical orbits is much harder to derive.

From a previous section, we know that the orbital velocity of an object circling a star of mass $M_s$ at a radius of $R$ is equal to:

$$v = \sqrt{G\dfrac{M_s}{R}}$$
The orbital period is defined as the total distance travelled over the speed of travel, so we can write:

$$ T = \dfrac{2\pi R}{v}$$
Substituting in $v$ gives us:

$$ T = \dfrac{2 \pi R^{3/2}} {\sqrt{GM_s}}$$
This form is more useful for finding the actual orbital period, but admittedly it doesn't look exactly like the proportionality. To restore the resemblance, we can simply square both sides:

$$T^2 = \dfrac{4 \pi^2}{GM_s} R^3$$
And in the generalized form, $R$ is replaced with the semimajor axis length $a$:

$$T^2 = \dfrac{4 \pi^2}{GM_s} a^3$$
With the mathematical results established, let's try a simple practice problem.



Kepler's Third Law might seem challenging at first, but it really isn't all too bad once you can understand it. After all, at its core it just says that planets that orbit further away will take longer to complete their orbits, while planets that orbit closer to a star will take less time to complete a full orbit. This is not exactly groundbreaking news to us, so if you're struggling with this law just remember that at its core, it's a statement of the obvious, albeit a mathematical one.

Conclusion

Next, we're going to completely pivot and talk about a new fundamental concept, that of energy. This will enable us to solve many problems with ease, as long as we're careful about setting them up.