### Video Transcript

In this video, we’re going to learn about Kepler’s laws of planetary motion. We’ll see what these laws are, we’ll learn some new terminology to help us understand them, and we’ll see how these laws help describe the motion of planets.

To get started, imagine that one night you take your telescope outdoors and begin looking up into the sky. You notice a bright shining object. But when you consult your map of stars that you should be able to see, you’re not able to identify this bright light. Excited that you may be looking at something new, you begin to chart the position of this bright light as you come back night after night to observe its position. After some time, you get to the point where you wish you could predict where this bright light will appear in the sky on any given day. To be able to do that, it’ll be helpful to know something about Kepler’s laws of planetary motion.

Before we discuss these laws though, let’s talk about the mathematical shapes known as “ellipses.” The way an ellipse works is you pick two points: these are called the focal points or the foci. Once you have these two points selected, here’s the rule for creating an ellipse from them. First, we draw a line from the first focus point and call that line 𝑑 one. Then, we draw a line from the second focus point to meet the end of that first line. We call this 𝑑 two. The rule to make the rest of the shape of the ellipse is that 𝑑 one plus 𝑑 two — the line segments from each one of our two foci — must be constant when they add together. This has to be true for every point along the path of the ellipse.

Once an ellipse is formed this way by joining together the distances from its two foci, then it has some interesting geometry. If our two focal points 𝑓 one and 𝑓 two were located on top of one another at the origin of this coordinate system we’ve drawn, then our ellipse would form a perfect circle. But when that’s not the case — that is when 𝑓 one and 𝑓 two are not in the same location — then one of the two dimensions of that the ellipse makes will be longer than the other. Along that longer dimension, if we take half of what we can call the width of the ellipse, that’s called “the semimajor axis.” In a similar way, half of what we can call the height of the ellipse is called “the semiminor axis.”

Now, we may stop to wonder, what does this have to do with planetary motion? It turns out that if we put a large mass of body such as the Sun at one of the focal points of an ellipse, then as Kepler found the planets in our solar system follow an elliptical path around that focal point centred on the sun. Kepler came up with three laws of planetary motion and this it turns out is one of them. It’s sometimes called “the law of orbits.” This law says that all planets in our solar system move in elliptical orbits with the Sun at one focus.

Here’s a little bit of interesting terminology when it comes to elliptical motion around the Sun. If the planet we’re considering moves along in its orbit until it’s as close as it ever gets to the Sun, that planet is then said to be at its perihelion, where you may recognize helion as referring to the Sun. On the other hand though, if the planet in orbit was at its farthest point away from the Sun distancewise, then the planet is at what’s called its aphelion. Considering both perihelion and aphelion, it can be helpful to develop a mental way of remembering which one is the close one and which one is the farther one. Personally, when I hear “peri,” I think of a periscope — the very short optical device that is put up above water from a submarine. The shortness of the periscope helps me remember that the perihelion is the shorter of the two distances. If that mental reminder helps, then keep it and if it doesn’t, feel free to think of a different one.

One last word about these terms, we see that helion refers to the fact that it’s the Sun that’s at the focal point. But of course, we could encounter an orbital situation, where it’s not the Sun that’s at the centre, but something else, for example, the Earth. If the Earth was the focal point around which our objects are rotating, then our terms changed to reflect that. The shorter distance is now called “the perigee” and the longer distance is called “the apogee.” So that’s the first law of Kepler’s three laws of planetary motion that planets in our solar system move in elliptical orbits and the Sun is at one focus of that ellipse.

The second of Kepler’s three laws called the law of areas has to do with the area inside the ellipse that a planet sweeps out as it moves around its elliptical orbit. This law says that the line that connects a planet to the Sun sweeps out equal areas in equal times. So if we draw a line from our planet at one point to the Sun and then we let that planet move ahead in its orbit to a second spot where we keep our line between these two bodies, then the area inside the ellipse that this planet sweeps out over sometime — we can call that area 𝐴 sub one — is equal to the area that the planet would sweep out if we let it evolve the same amount of time as it continues on its elliptical orbit.

In other words, if we let our planet move along its orbit for some particular time — we can call it 𝑡 — and we noticed that over that time an area of 𝐴 one was swept out, then if we let the planet move for another increment of the time 𝑡 and so the area that swept out was 𝐴 two, this second law says that 𝐴 one is equal to 𝐴 two. And the second law claims that this relationship holds regardless of where in its orbit our planet is — whether it’s close or far to the focus.

This leads us to the third and final law of Kepler’s laws of planetary motion called the law of periods. An object’s period you’ll recall is the time it takes to move around one complete revolution. The period of a planet orbiting around the Sun will be the time it takes to move all the way round its ellipse once and return to its original location. The period for the Earth, for example, is about 365 days.

To talk about the law of periods, we’ll introduce some shorthand notation for the two axes we’ve identified on this ellipse. We’re gonna refer to the semimajor axis as lowercase 𝑎 and the semiminor axis as lowercase 𝑏. Referring to a planet’s period by the symbol capital 𝑇, as a mathematical relationship, the third law of planetary motion says that the square of a planet’s period around its orbit is proportional to the cube of that orbit’s semimajor axis. In other words, if we took the semimajor axis 𝑎 and we cubed it, then that would be equal to a planet’s period 𝑇 squared multiplied by some constant factor.

The way we’ve written this third law currently is a proportionality. But we can also write it as an inequality. We can say that a planet’s period squared is equal to four 𝜋 squared over the gravitational constant 𝐺 multiplied by the sum of the mass of the sun and the planet orbiting the Sun all multiplied by the semimajor axis of that planet’s orbit cubed. Oftentimes we’re able to simplify this expression further because the mass of any planet orbiting the Sun is typically much, much less than the mass of the Sun, meaning that we can approximate their sum as simply the mass of the Sun by itself. This law of periods shows us the mathematical relationship between an orbiting planet’s period and its semimajor axis 𝑎.

Now, let’s get some practice with Kepler’s laws of planetary motion through an example.

A satellite orbits Jupiter with an average orbital radius of 963700 kilometers and an orbital period of 3.269 days. Find the mass of Jupiter, considering a day as 864.0 times 10 to the third seconds.

We’ll label this mass of Jupiter that we want to solve for capital 𝑀. And to start on our solution, we’ll recall Kepler’s third law of planetary motion. For a much smaller body orbiting a much larger one, Kepler’s third law tells us that the period of the smaller orbiting body squared is equal to four 𝜋 squared over 𝐺 the gravitational constant times the mass of the planet it’s orbiting all multiplied by the semimajor axis of the orbiting planet’s ellipse cubed.

If we draw a sketch of this orbiting satellite, we have it in an elliptical orbit moving around a focus point of the planet Jupiter. We’re told the average orbital radius of this satellite as it moves throughout its elliptical path and we’re also told how long it takes for the satellite to move all the way around its orbit once — that is its period. We can call 𝑟 sub avg that average orbital radius and capital 𝑇 the period of the satellite’s orbit given as the product of 3.269 days times the number of seconds in a day.

When we apply Kepler’s third law to our scenario, where capital 𝑀 the mass of Jupiter is what we want to solve for, we see that we know the period capital 𝑇 and big 𝐺 the universal gravitational constant is a known constant. We will let big 𝐺 be exactly 6.67 times 10 to the negative 11th cubic meters per kilograms second squared. This means that when it comes to our relationship to solve for the mass of Jupiter 𝑀, the only unknown is 𝑎 the semimajor axis of our elliptical orbit. But there’s a special relationship for elliptical path which says that the average distance from the path to the centre of the path at what we’ve drawn as an origin is equal to the semimajor axis of that orbit.

In other words, 𝑎 the semimajor axis is equal to 𝑟 sub avg. So we now know everything in this expression except the mass value we want to solve for. We rearrange the expression so that 𝑀 is by itself on one side. When we plug in these values, we’re careful to convert our expression for 𝑎 from units of kilometers to units of meters. Entering this expression on our calculator, to four significant figures, we find that 𝑀 is equal to 6.641 times 10 to the 25th kilograms. That’s the mass of Jupiter based on these numbers.

Let’s summarize now what we’ve learned about Kepler’s laws of planetary motion. In this segment, we learned some planetary ellipse terms. We learned how do identify the semimajor axis of an ellipse — sometimes identified with the letter 𝑎 — as well as the semiminor axis — sometimes identified with 𝑏. We also learned that when a planet is on an elliptical orbit with the Sun at one of its focus points, when that planet is closest to the Sun it’s said to be at its perihelion and when it’s furthest away from the Sun, it’s said to be at its aphelion. And we noted that the suffix helion specifically refers to a Sun-centred orbit.

We also learned about Kepler’s three Laws of planetary motion. The first of which is called the law of orbits and this says that all planets move in elliptical orbits with the Sun at one focus. This law of course is based on observations of our own solar system. The second law called the law of areas says that a line connecting a planet with the Sun sweeps out equal areas in equal amounts of time passing. And finally, we learned the law of periods. This law states that if the period of a planet 𝑇 is squared, the net value is proportional to the cube of the orbit’s semimajor axis 𝑎. and we also saw this law can be expanded on into an equation that 𝑇 squared is equal to four 𝜋 squared all divided by the universal gravitational constant multiplied by the mass of the planet being orbited — often the Sun — all times the cube of the semimajor axis of the planet’s orbit. These three laws of planetary motion are strongly geometric laws and are often easiest to understand when we have a drawing of a planetary ellipse in front of us.