Kepler's Laws of Planetary Motion | Law of Ellipses | Law of Areas | Law of Periods

Hello everyone. Today we're starting on astronomical laws. Here we'll be discussing some of the laws describing planetary motions that were come up with by a fellow named Kepler, who of course was one of the astronomers during the 17th century. He was one of the first to use mathematics to describe the motions of the planets. So Johannes Kepler, there's a portrait of him over here, was a student of Tycho Brahe, we've learned about him before in our section on the geocentric models, who was put in charge of Brahe's tables in 1601 after Brahe's death. And so given this huge store of astronomical data he decided to do something with it. Kepler strongly believed that the universe was heliocentric. He was a Copernican so he strongly believes in Copernicus's heliocentric model of the universe. Kepler was able to use Brahe's observations, particularly of the planet Mars, to make observations and laws about the orbits of the planets. So the laws aren't something that he made up, it's something that he observed and decided to describe with mathematical equations. So Kepler's three laws that we remember him for are the law of ellipses, the law of areas, and finally the law of periods. So they are empirical laws which means that they are based on observations which were made by Brahe and Kepler. They're not based on a sort of mathematical model that Kepler already had. He sort of created his own mathematical model based only on the observations that he made without having any sort of way to explain them. So for thousands of years the orbits of the planet were thought to be circular or even spherical. So Kepler proposed due to his studies on all the careful observations made by himself and Brahe that each planet moves in an ellipse with the sun at one focus. And so geometrically an ellipse, as we know, is a set of points for which the sum of the distance between two focuses or two foci is a constant. So that means that at any point on this ellipse here, if we drew a line to A and to B, then the distances from A to B would always add to be the same number. So A and B are called the foci of the ellipse. Of course if A and B are in exactly the same spot, that means that any point on the ellipse will be the same distance from the center and that would make it a circle. You'll probably learn a little more about this in geometry later on. So the second law is the law of areas. Kepler noticed that the planets moved faster when they were closer to the sun. So we can see in this picture here the planet that travels slowest at this end when it's the furthest distance from the sun and the planet travels fastest here at this end when it's closest to the sun. So why is it called the law of areas? Well he worked out that each planet sort of sweeps out the same area in the same times. So that means that it takes a planet, for example, the same time to go from A to B as it does to go from C to D because of course it's moving faster when it's close to the sun. And in this diagram the areas ASB, that is this area here, and DSC, that is this area here, are equal and they're covered by the planet in the same time. So that's why it's called the law of areas because in an equal amount of time a planet sweeps out equal areas no matter where it is in its orbit. So finally we have the law of periods. Kepler realized the period of each planet's orbit depends on the radius of the orbit. So the square of the period is proportional to the cube of the radius. It's a little complicated. So we have the radius cubed, it's in R3, and we have the period squared. Period is represented by a capital T because it's a time. And so if the square of the period is proportional to the cube of the radius then it will look something like this. KR3 equals T2. Of course we can rearrange this by dividing both sides by say R3 to say that this equation on the right here must be a constant. And so for different planets that have different periods and different radii you can compare them to each other and make predictions on them based on this relationship because this will be the same for every planet around the sun. And so Kepler included that. He said that for every planet going around the sun this is the same number. Now he used a mathematical equation to describe this relation. That equation is of course the period squared over the radius cubed equals K. He used a capital R for a radius because planets are very big things, they deserve big letters. So all planets revolving around the sun have the same value of K. So if you wanted you could say this was the period of planet 1 and the radius of planet 1 and it equaled the period of planet 2 squared over the radius of planet 2 cubed. So it's possible to compare different planets using this as well. So that's especially useful if you don't have the value for K. However if T is in years and the radius is in astronomical units or AU, that is multiples of the Earth's distance from the sun, then K simply equals 1 which makes it a very easy rule to follow. So I think we have a table here of the various different radii and periods of different planets. Here we go. So we have the five different planets that were discovered at the time and we have their period in years and their average radius in astronomical units which is exactly what we need to get a K of 1. So in each case if we take the period and we square it, so it's the T squared, and we take the radius and we cube it, so it's the R cubed, we arrive at this number that's on the rightmost column. So let's take a look at these numbers for each of the different planets. 1.06, 1.01, 1.01, 0.99 and 1.12. So this is in fact the actual data that Kepler used for his observations and you can see that pretty much every number here is almost exactly 1. And so this seems to say that this number T squared over R cubed is a constant for every planet in our solar system that was discovered at the time. So in 1687, so later on in that century, Isaac Newton put forward the idea of universal gravitation which I'm sure you've heard of before. He proposed that gravity was an invisible force acting over long distances and this sort of went against the current scientific understanding at the time that thought that everything had to be a contact force. So this was one of the first non-contact forces. He showed mathematically that his proposition supported Kepler's laws. He said that if the law of universal gravitation held true then it should explain all the things that Kepler is observing. So Newton could use mathematical equations like this one to describe the gravitational force. Here we have the two masses of the different objects. We have a gravitational constant over here. We have the radius between them squared. And so this in fact shows us the force on any object in a gravitational field. We can see that the heavier the object is, the greater the force it'll experience. This is consistent with our earlier equation of the weight force of an object equals mg. Now at the surface of the Earth, R is going to be a constant and the mass of the Earth is a constant. Capital G is already a constant. So this whole thing just becomes the lowercase g in this equation. So it matches up with what we already know. So these equations could be used to make accurate predictions about the universe when combined with Kepler's laws. In fact in 1846, so 100 years later, Newton's equations were used to predict the position of Neptune based on the movements of the other planets which they assumed were caused by the gravitational pull of Neptune further out. This is one of the greatest triumphs of Newton and Kepler's laws of the universe. So Newton's law of universal gravitation provided an explanation of the empirical model that Kepler had set up. So this meant that the model could be improved based on the explanation. And this of course is what allowed us to discover Neptune. So finding mathematical rules to explain known empirical laws is very, very useful as we can see. If we have an explanation for the model that we've come up with then that lets us to expand and improve upon the model. As you can see our universe today, our model of the universe today has been greatly improved by mathematical understandings and equations originally produced by, for example, Newton. Well that's the end of the theory. So we've learned about Kepler's laws and how Newton contributed to them. So let's do some questions. Question 11. Which of the following is not one of Kepler's laws? Well I hope you can get this one right. So Kepler has three laws. So there are four options here. Three of them will be Kepler's laws and one of them will not be. Let's go through our options. A. A planet sweeps out equal areas of orbit in equal times. This is Kepler's law of areas, also known as Kepler's second law. So it's not A. B. The square of the time it takes for a planet to orbit the sun is proportional to the cube of the distance from it. Well that's a mouthful. So that's Kepler's third law and it's not quite right either. So that's the equation that we had, remember? So that's capital T squared over capital R cubed. That's Kepler's third law right there. C. A planet's orbit is elliptical with the sun at one focus. This is in fact Kepler's first law, the law of ellipses. So the sun is at one focus, the other focus is empty. And in fact this path can be completely described by Newton's universal law of gravitation. Speaking of which, D. Bodies with mass attract one another with gravity. While this does manage to explain all of Kepler's laws, it is in fact not one of Kepler's laws but it's a Newton's law of universal gravitation. So we see that D is the correct answer here. As it turns out, if you're good enough at maths, from Newton's law of universal gravitation, you can work out mathematically every single one of Kepler's laws. All right, on to the next question. Question 12. How did Kepler obtain the data he needed to derive his laws of planetary motion? Of course he had to get the data from somewhere. So we have a few options here. Was it A, sorry, was it C? He built the largest telescope in the world. Well, at the time, telescopes weren't actually all that popular yet. So he inherited the data in 1601 and made his laws a few years later. The third law he came up with sometime after that. So he did not build the largest telescope in the world in order to accomplish his observations. So C is wrong. Is it A, he was the first astronomer to use a telescope? Well, he didn't use a telescope at all to make these observations. So no, he did not use a telescope to derive his model. And A is incorrect. That was Galileo. D, he used mathematics to create a model and derive the laws from the model. Well, we've just mentioned in the last question, that's completely possible if we start with Newton's laws of universal gravitation. The problem is Newton based his universal laws on gravitation on the observations that Kepler had already made. So it was in fact the other way around. So Kepler's laws are empirical. They're based purely on observed data. He didn't create a mathematical model first and then base the laws on that. So D is incorrect. Finally, we have B. He used Tycho Brahe's enormous stores of data. And we see that this is the right answer. Kepler was in fact one of Brahe's favorite students, made a lot of observations with Brahe, and was put in charge of his entire store of astronomical tables when Brahe died. Question 13. Uranus is about 12 and a bit astronomical units. That's Earth orbital radii from the Sun. To the nearest year, how long does it take to complete an orbit? So we've got a math question. Now the relevant equation that we can use here is Kepler's third law, which looks something like this. The period squared over the average radius cubed equals one. But I should probably point out these aren't SI units. The radius has to be measured in astronomical units, and the period has to be measured in years, rather than the seconds and the meters that we're used to. If we used years and meters instead, we'd get... So if we use seconds and meters, we'd get a very different number to one, but the law would still hold true. It would still be a constant. We're just using astronomical units and years because it's more convenient. So we're actually looking for the period in this question. That's a t. So we can rearrange the equation to look something like this. Period squared equals radius cubed, assuming they're using the right units. So if we substitute in the 19.2 from here, we end up with the period squared is 19.2 cubed. That's simple enough so far. So that evaluates to about 7,078. It's actually a tiny bit less, but we're pretty close. We only need it to the nearest year. Once we have that, we can say that the period is simply the square root of 7,078. So sticking that into a calculator gives us an answer, and we find that it's about 84 years. So as you can see, a Uranian year is a lot longer than an Earth year. The further out you go, the longer the years get. Question 14. Outline Newton's contribution to Kepler's model of the universe. Well, Newton did a couple of things, didn't he? So Newton used Kepler's law to develop a mathematical model of the universe. So he was able to develop mathematics and explained all of the observations of Kepler. So the law of ellipses, the law of areas, and the law of periods, and was able to explain them based on a new fundamental force that had never been heard of before called gravity or gravitational attraction. So his mathematics were used to predict very accurately the position of Neptune in 1846. Without the work of Kepler and Newton, we would not be able to make observations on the universe with very much accuracy. Who knows how far we might be set back? So Newton was in fact critical, if you like, to Kepler's model of the universe. Question 15. Recall the significance of Newton's work in terms of the support it gave other models. Now, what exactly did Newton do in physics stuff? Well, he came up with equations and mathematics. So the application of mathematical formulae within a particular model, which is able to predict both the positions of planetary objects and explain their motion, that's of course the law of gravitational attraction, gives very strong support indeed to the model. And so with Newton's equations and mathematics, he was able to provide strong support for Kepler's model of the universe. Well, that concludes the questions. So we've learned about Kepler's three different laws. We've learned about Newton's contributions to Kepler's laws and his contribution with gravitational attraction in order to explain Kepler's laws and come up with an even better model of the universe.