Overview of Kepler's Laws - Online Tutor, Practice Problems & Exam Prep

Johannes Kepler established three fundamental laws of planetary motion. Kepler's first law states that orbits are elliptical, with the sun at one focus. The second law indicates that equal areas are swept out in equal times, meaning planets move faster when closer to the sun. The third law relates the square of a planet's orbital period (T) to the cube of its average distance (r) from the sun, expressed as T^2 ∝ r^3. These laws are crucial for understanding celestial mechanics.

Overview of Kepler's Laws

Hey everyone. So, back before the days of Isaac Newton, there was a German astronomer by the name of Johannes Kepler, who was studying the motion of planets and different objects in our solar system. And basically, what he determined is that all orbits, whether they're stars or planets or different satellites like comets, obeyed three laws. And these are called Kepler's laws. Now, I'm going to go ahead and give you a brief overview of Kepler's laws in this video, and then later on, we'll cover each one of them in a little bit more detail. So, let's check this out here. Alright? The first one is called Kepler's first law. And, basically, what he said is that all objects are ellipses. So, all orbits are elliptical even the circular ones. It's kind of like how a square is always a rectangle. So, all orbits, even circular ones, are ellipses, and ellipses sort of have some special mathematical points. There are two points in an ellipse, they're called foci. And what Kepler's first law said is that the Sun is at one of those focus points, for example, right here. There's also another spot which is the center of the ellipse. But what's really important about the center and the focus points, or leave these two, is that there's nothing physical actually here at these points. So, if you went out to this point in space, it's not like you would actually see anything. This is sort of more of a mathematical thing here. Alright? So, there's no planet or physical object at the other focus. Now what Kepler was studying, he was studying the motion of planets and orbits, and he noticed that some of them were more circular than others. And he came up with a way to measure this, which is called the eccentricity, and it is given by the letter, little e. It's basically just a number between 0 and 1 and it's a measure of how elliptical or how weird this orbit is. Alright? Now, let me give you two examples. One of them is that most objects in our solar system orbit in circles like the Moon around the Earth. It's almost perfectly circular. Right? So if you have the Earth here and you have the Moon as it goes around like this, then what happens here is this is a lower number eccentricity. Lower numbers generally mean that the orbit is more circular. So, this would correspond to an e value of something that's very nearly 0. On the other hand, when you have higher numbers, that corresponds to something that is more elliptical. A good example of this is a lot of comets in our solar system have extremely elliptical orbits. They get very close to the Sun and then they go very, very far away and sometimes it takes tens and hundreds or even thousands of years. Alright? So, this would be a number that is sort of very close to 1. That would be a very elliptical or very weird orbit. Alright. So, let's move on. That's all for Kepler's first law. Kepler's second law says in orbits, equal areas of the orbit's arc are swept out in equal times. So, what does that mean? Well, what happens here is that in an orbit, as you go closer in towards the object that you're orbiting, you tend to go faster. So, you're going faster here. Right? So, imagine the Earth is going around in this orbit like this. And as it travels in its orbit, it's sweeping out this arc. And this arc here has an area. I'm going to call this a from a to b. Later on that the Earth goes farther away, what happens is it's traveling slower, and so the angle of the arc is a little bit lower. Right? It's thinner, but it's at a farther distance r. Well, basically, what Kepler's second law says is if the time that it takes to go from a to b is equal to the time it takes to go from c to d in this diagram, by the way, it could be weeks or months or years, it could be whatever number, then the areas of these arcs are actually going to be the same. So, then a from a to b is equal to a from c to d. Alright? So, in this case, you have a bigger angle, but it's a shorter distance. In this case, you have a smaller angle, but it's a bigger distance. But they actually turn out to represent the same exact area between these two arcs if the times are the same. Alright? Now, you won't have to do any mathematical calculations with this, but this is a really important concept that you probably should know if you run across on a homework or a quiz. Alright? So, last but not least, there's Kepler's third law, which is basically a relationship between the square of the period and the cube of the orbital distance. So, in other words, if I have the radius of the Earth here, it's going to be at some r, then it takes some time for the Earth to go around in its orbit. That's going to be t. Now, if you have another object like Mars, which is at a bigger orbital distance, so in other words, this r_m is greater than r of the Earth, then what happens is that the time that it takes to go around, which I'm going to call t_Mars, is going to be greater than t_Earth. Right? So, the farther you are away, the more time it takes for you to go around, but it's not quite a linear relationship. There's some powers involved. What he said was that t² is proportional to r³. Alright. So, there are some exponents involved. Again, we'll talk about this a little bit more in detail. Now, moreover, what he also found was that if two satellites orbit the same mass, for example, like all of the planets in our solar system orbit the Sun, then the ratio of r³ over t² is the same. So basically, what he found out is that if you do r³ over t² of the Earth, you're going to get a constant. You're going to get a number. And, by the way, this number here depends only on the mass. So, it only depends on n on big M, which in this case is the mass of the Sun. But if you take the r³ over t² of Mars, you'll actually get the same exact ratio. So basically, these ratios here only depend on the mass of the Sun and you can actually compare the two different objects that are orbiting the same object. Alright? So anyway, that's it for this one. Let me know if you have any questions.

Here’s what students ask on this topic:

What are Kepler's three laws of planetary motion?

Kepler's three laws of planetary motion are fundamental principles that describe the motion of planets around the sun. The first law, known as the Law of Ellipses, states that planets orbit the sun in elliptical paths with the sun at one focus. The second law, or the Law of Equal Areas, indicates that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time, meaning planets move faster when closer to the sun. The third law, the Law of Harmonies, relates the square of a planet's orbital period (T) to the cube of its average distance (r) from the sun, expressed as $T^2\propto r^3.$

How does Kepler's first law describe planetary orbits?

Kepler's first law, also known as the Law of Ellipses, describes planetary orbits as elliptical rather than circular. According to this law, each planet orbits the sun in an ellipse with the sun located at one of the two foci of the ellipse. This means that the distance between a planet and the sun varies throughout its orbit. The elliptical nature of orbits helps explain the varying speeds of planets as they move closer to or farther from the sun.

What is the significance of Kepler's second law?

Kepler's second law, or the Law of Equal Areas, is significant because it explains the variable speed of planets in their orbits. The law states that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time. This means that planets move faster when they are closer to the sun and slower when they are farther away. This principle helps us understand the dynamics of planetary motion and the conservation of angular momentum in celestial mechanics.

How does Kepler's third law relate the orbital period and distance of planets?

Kepler's third law, known as the Law of Harmonies, establishes a relationship between the orbital period (T) of a planet and its average distance (r) from the sun. The law states that the square of the orbital period is proportional to the cube of the average distance from the sun, expressed mathematically as $T^2\propto r^3.\; This\; means\; that\; planets\; farther\; from\; the\; sun\; have\; longer\; orbital\; periods.\; The\; law\; provides\; a\; way\; to\; compare\; the\; motion\; of\; different\; planets\; and\; is\; crucial\; for\; understanding\; the\; scale\; and\; structure\; of\; our\; solar\; system.$

What is the eccentricity of an orbit and how is it measured?

The eccentricity of an orbit is a measure of how much the orbit deviates from being circular. It is denoted by the letter e and ranges from 0 to 1. An eccentricity of 0 corresponds to a perfect circle, while an eccentricity close to 1 indicates a highly elongated ellipse. Eccentricity is calculated using the formula $e=\frac{\sqrt{a^2-b^2}}{a}$, where a is the semi-major axis and b is the semi-minor axis of the ellipse. Understanding eccentricity helps in analyzing the shape and characteristics of planetary orbits.