Kepler's Laws of Planetary Motion

by the 16th century the laws of planetary motion were being challenged by new thinking largely led by Nicholas Copernicus and those who followed him so basically Nicholas Copernicus offered an alternative to ptolemies geocentric universe namely that the Sun was at the center of the cosmos and that the planets in turn orbited the Sun so the resolution to the geocentric versus heliocentric debate would largely be undertaken by Tycho Brahe hey this was the world's leading astronomer he was a nobleman from Denmark and his ideas basically merged some of the mathematical benefits of Copernicus system with the philosophical benefits of the Ptolemaic system namely the moon orbits the earth and the Sun carries the planets in its orbit around the earth so it was a kind of a hybrid universe a geocentric model with a heliocentric component to it no Tycho was very well connected the King of Denmark allowed him to use property among the kingdom not the least of which was a small island called ven on this island was an old fortress and Ptolemy converted this fortress into a palace he named uranium org so this was a palace where he could conduct his astronomical observations using the finest equipment available and in those days the finest equipment available largely meant things like sextant sand geodesics and other geometrical line of sight astronomical tools this was before the age of the telescope that would be assured in by Galileo nevertheless Tycho was able to capture a tremendous amount of highly detailed data and now he wanted to test his idea to see if his universe was in fact the correct model the problem though is that for all of Tycho's capabilities as an astronomer he really did not have the mathematical chops to analyze the data later in life he would meet yoha Kepler and invite him to come work for him a Kepler was an astronomer and mathematician and he had his own ideas of how the universe might be ordered like Copernicus he believed that the Sun would be at the center of the cosmos however he thought that he could determine the arrangement and orbital periods and locations of the planets by nesting the so-called platonic solids these are the solids that all have equal sides now he didn't seriously believe that there were geometric objects in space but rather the arrangement of these geometric objects would reveal the locations of the planets orbits but lacking the data he had no way to test this until Tycho Brahe he came a-calling so Kepler went to work for Tycho but the problem was that he could not get all of Tycho's data you see Tycho was a little bit of an insecure guy and he didn't want to just give this young mathematician all of his data so Kepler worked with what little scraps of data Tycho would pass along and this frustrated Kepler and he was about ready to quit and give up and go back home when luckily for Kepler Tycho died and now Kepler had the opportunity to finally test his idea to see if it was correct it wasn't in fact he could not get the planets to fit the perfectly circular orbits that his model required the data just would not agree with his beliefs and so he discarded his beliefs and he came out with the laws of planetary motion and it's these so-called Keplerian laws that were going to discuss now Kepler's first law is this rather than a circle a planet orbits the Sun in an ellipse with the Sun at one focus and nothing at the second focus now let's just describe ellipses for a moment and to do that let's talk briefly about circles so if we took a nail and we hammer it into the wall and we take our pencil and a piece of string and then just do what comes naturally well of course you get a circle so a circle as a center and in most of our dealings with circles were familiar with the diameter of a circle but in mathematics we only concern ourselves with one half of the debt of the diameter we call this the radius an ellipse on the other hand is made by hammering two nails into the wall taking your pencil and the piece of string and then just doing what comes naturally to generate an ellipse so the nail holes you might say are each called a focus or fossa for plural we can then go across the widest part of the ellipse giving us something called the major axis but just as we learned with radius versus a diameter we only concern ourselves with half of the major axis and therefore we call this the semi major axis and since we're going to be using semi major axis later on we're going to substitute the lowercase letter A to refer to semi major axis now there's another property of an ellipse called the s intricity the a centricity is a measure of an ellipses shape so for example this particular ellipse it has an S on tricity of 0.35 the value of the SN tricity is always going to be at least 0 and less than 1 and there's no unit associated with this it's not measured in feet or kilometers or astronomical units rather it's just a unitless number so as long as it's less than 1 all the way down to 0 you have an ellipse that means that if we bring the two foci together to form a center the essen tricity Falls to 0 or we could spread the two foci apart and in this case we can see a fairly flattened football shaped ellipse with an S intricity of points own to calculate the eccentricity it's really quite simple what you can do is take half of the major axis that is the semi-major axis and then you can go from the geometric centre to either of the two foci we call that lowercase letter C and then just take their ratio to give the SN tricity so to recap Kepler's first law states that planets orbit the Sun in an ellipse with the Sun at one focus and nothing at the second now when it comes to planets orbiting the Sun there are two locations we should point out first is the closest position to the Sun that's called perihelion and the most distant position from the Sun is called aphelion so you'll be hearing me refer to perihelion and aphelion a lot and that's just ways of describing the closest and farthest positions from the Sun so Kepler's second law states that if we allowed a planet to come past the Sun let's say in this case near perihelion over a 10-day period and if we were to look at that same planet towards aphelion over a same 10-day period so in other words the time intervals are going to be equal to each other now if we do what comes naturally and play connect the dots between the perihelion pass we can get some kind of an area and we can do the same thing with the aphelion pass and if we measured the area of these two triangles it turns out that both of these areas are equal therefore Kepler's second law states that a planet sweeps out equal areas during equal intervals as it orbits the Sun and there's another implication here in order for the perihelion interval to equal the aphelion interval the rate at which the plant is moving during perihelion must be faster and the rate which the planet moves during App Hylian must be slower so a planet's speeds up as it approaches perihelion and slows down as it approaches aphelion and repeats the cycle over Kepler's third law showed us a relationship between the orbital period of the planet when measured in years and its semi-major axis when measured in astronomical units and it turns out that the relationship is pretty simple the square of the planets orbital period is proportional to the cube of its semi-major axis in other words the orbital period squared is equal to its semi-major axes cubed to show you an example of this let's take a look at well the very simplest we're just going to use our own planet Earth so we know it's orbital period is exactly one year its semi-major axis is exactly one astronomical unit okay so one squared is equal to one and its semi-major axis cubed is also equal to one so P squared equals a cubed that is to say they both equal one not very exciting but at least you can see the logic here now let's use a more distant planet in this example we'll go ahead and use Jupiter it has an orbital period of just under twelve years and it has a semi-major axis of 5.2 astronomical units so if we take its orbital period eleven point nine years and square it we get a hundred forty-one and if we take its semi-major axis five point two a you--and qubit we also get one hundred and forty-one and every planet in our solar system you can perform this exact same exercise for and you'll find that it's period squared is always going to be equal to its semi major axis cubed so those are Kepler's three laws of planetary motion but one thing that's important to remember is that Kepler's laws are purely descriptive in other words Kepler had no way of knowing why this was the case he didn't understand things about gravity or whatever forces would be involved in keeping a planet in its orbit to observe these laws they were simply descriptive of the data that Tycho had collected in order to understand the laws of planetary motion at a deeper level and understand that the forces of gravity involved it would literally take a genius to discover those and we're gonna learn about that genius next

Channel: Launch Pad Astronomy

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Keywords: Astronomy, Kepler's Laws, Kepler

Id: lcDMv_bs4LY

Length: 10min 41sec (641 seconds)

Published: Mon Feb 19 2018