Kepler's Laws

Hello, and welcome. In this educational aid, we're going to talk about Kepler's three laws of planetary motion. For 400 years, these laws have been the foundation to understanding orbits. These laws not only apply to planets orbiting the Sun, but also satellites orbiting the Earth. Prior to Kepler, it was believed that all planets orbited the Sun in perfect circles. However, Mars didn't quite behave this way. It was through the observational data collected by Taha Brahat, a friend and mentor to Kepler, and additional observations by Kepler himself, which showed that Mars' orbit around the Sun is actually slightly elliptical. This insight resulted in the development of Kepler's three laws of planetary motion. His first two laws were published in 1609, and his third law in 1619. Kepler's first law states that all planets orbit the Sun in an ellipse, with the Sun at one of two foci. Let's make this more clear. Let's start by examining what is an ellipse. To make an ellipse, we need a foci. A foci is simply the plural form of focus, and they are the two defining characteristics of an ellipse. They help determine the ellipse's shape. With the foci placed on a straight line, we now add a string and a marker attached. Now, we can draw out the ellipse. Now, let's do this again, but with foci closer together. As you can see, the location of the foci will affect the shape of the ellipse. Using the same size string, and adjusting the distance between the foci, we change the shape of the ellipse. As we move the foci closer and closer together, the ellipse will start to look more and more like a circle. And when the foci are on top of each other, the ellipse becomes a perfect circle. Now, let's go back to Kepler's first law. All planets orbit the Sun in an ellipse, with the Sun at one of the two foci. This should now make a little bit more sense. Kepler's first law, that all orbits are ellipses, contradicted the belief held by most 16th century astronomers, including Brahe himself. This law was mathematically verified a half a century later by Newton in his model of gravity. Kepler's second law states that all orbits sweep out equal areas at equal times. With his second law, Kepler was observing that mechanical energy is conserved. Mechanical energy is the sum of both potential and kinetic energy. Potential energy deals with distance, and kinetic energy deals with speed or velocity. When the planet is closest to the Sun, our periapsis, kinetic energy of the orbit is at its maximum, and potential energy is at its minimum. At aphiapsis, when the planet is furthest from the Sun, the orbit's kinetic energy is minimized, and the potential energy is at its max. This conservation of energy is a key concept that is emphasized in other educational aims. Kepler's third law states that the period, the time it takes a planet to complete one orbit around the Sun, is proportional to the semimajor axis, or the planet's average distance from the Sun. More specifically, the planet's period squared is proportional to the semimajor axis cubed. To better explain the semimajor axis, let's go back to the example we used in Kepler's first law. As stated earlier, the semimajor axis is the planet's average distance from the Sun. But this can be more simply stated as just half the major axis, where the major axis is the distance from aphiapsis to periapsis. This can be seen mathematically as well. Don't worry, I'll keep this simple. Using the equation p equals 2 pi times the square root of a cubed over mu, where p is the period and a is the semimajor axis. Now, we will simplify this equation by removing all constants. This leaves us with p squared is proportional to a cubed. As a, the semimajor axis, increases, so does p, the period. And we can see this visually as well. Starting with Mercury, we can observe the size and the speed of Mercury's orbit. As we move further and further away from the Sun, with Venus and Earth, the size of the orbit increases, and so does the period. As we move even further out to the gas giants, the period increases even more. Newton, once more, 50 years later, proved this constant of proportionality by forming his gravitational constant. The Born-Newton gravity and other educational agents. The principles of these laws do not apply only to planetary orbits around the Sun, but all orbits, specifically, satellites orbiting the Earth. So each law can be restated. The first law, all orbits are ellipses, and a mass, which is being orbited, is at one focus, and the other focus is empty. An example of this is how a satellite orbits the Earth, and the Earth is at one focus of the ellipse. The second law, all orbits sweep out equal areas at equal times, and the third law, the square of the orbit's period is proportional to the cube of the semimajor axis. Well, that is it on Kepler-3 laws of planetary motion. I am Jeremy Brown with the National Security Space Institute, and I hope you enjoyed this educational aid.