 Hello, so in the last capsule, we began this application of Fourier analysis to a problem in celestial mechanics. The last capsule, I gave you the background or the stage setup for the problem. And we came to the point that the problem amounts to two things, getting first integrals, a quadrature and the inversion. In the first integrals, we discussed last time and I also mentioned that using the Kepler's laws of planetary motion, we are going to get the equation which we have to invert the last phase of the problem. And now let us turn to this part and so let me introduce some terminology that you see in the slides. So you see the word perihelion, remember that in planetary motion, when the planet revolves around the sun, the sun is at the focus, the sun is not at the center, the orbit is not a circle, the orbit is an ellipse. So you think of an ellipse and you think of the sun at the focus, I am going to use the notation s for the focus because that is where the sun is going to be sitting and at some particular instant, the planet is going to come closest to the sun and there will be another time where the planet is going to be farthest away from the sun. So the points on the orbit at which the planet is closest to the sun and farthest from the sun, they are called perihelion and aphelion. Now you can think of the moon revolving around the earth, again it is an elliptical orbit and the position of the moon, when the moon is closest to the earth, it is at the perigee position and when it is farthest, it is apogee. So for the earth and moon, it is perigee and apogee because it is a geological phenomenon, the earth geo and for the sun, it is helios, so perihelion and aphelion. If you are studying the motion of the satellites of Jupiter, you will call it perigee and apogee. So this particular point, the perihelion will be denoted by the letter p and we are going to start our time, that we are going to start measuring time from the time exactly at the instant where the planet passes the perihelion. So when the planet passes the perihelion, we are going to reset time to be 0 and the sun is at the focus s and all angles will be measured from the focus. So we got the focus s and we got the perihelion. The radius vector sp and that is going to be your initial vector and xt, capital xt is the position of the planet at time t and so from the sun to the planet sx, radius vector sx. The angle between sx and sp will be denoted by theta t and in astronomy, this theta t is called the true anomaly, it is called the true anomaly, there will be a related angle which will be called the eccentric anomaly. If you have been looking at classical textbooks in coordinate geometry, you will know about the eccentric angle or the eccentric anomaly in the study of the ellipse and the auxiliary circle. We will come to that in a moment. The Kepler problem says find the function, find the true anomaly theta t as explicitly as possible. Can you find an explicit formula for the true anomaly theta t? So it is this theta t as a function of t that we are interested in ok. So now we have to look at this picture. In this picture, it depicts the perihelion p and the sun is of the focus s, the planet is at the position x, the point xt is the position of the planet. Now remember that in coordinate geometry, you consider an ellipse and the auxiliary circle. What are the ellipse? x squared upon a squared plus y squared upon b squared equal to 1. And the auxiliary circle x squared by a squared plus y squared by a squared equal to 1 or x squared plus y squared equal to a squared. So the ellipse and the auxiliary circle, they touch at the two extremities a comma 0 and minus a comma 0. So in coordinate geometry, you will see that the corresponding points. So take a point on the ellipse, say a cos theta comma b sin theta. The point x is a cos theta comma b sin theta. Look at the corresponding point y, a cos theta comma a sin theta. These two points are called auxiliary points. These two points are called auxiliary point and the auxiliary point is denoted by y of t. The point auxiliary to x of t is y of t. So now you have this auxiliary point and the center of the ellipse and the circle is being denoted by the letter capital C, C for center. Now from the center, draw the radius joining the auxiliary point y. And what is the point y? The point y is a point on the auxiliary circle corresponding to the point x. And the angle P C y, the angle P C y in this picture is depicted by e. This is called the eccentric anomaly or the eccentric angle. So we will have to come back to this picture in the next few minutes. So now let us take x and y to be corresponding points on the ellipse and the auxiliary circle. Then P is the perihelion, the perihelion, the focus and the point x. That is a sector of the ellipse. Area PSX refers to the area of the sector of the ellipse and area PSY refers to the corresponding sector of the auxiliary circle. Then equation 8.11, area PSX upon area PSY is equal to B upon A which is the same as the area of the ellipse upon the area of the circle. The second equation is very clear because the area of the ellipse is pi AB and the area of the circle is pi A squared. So the area of the ellipse upon the area of the circle is exactly B by A but this is exactly the same as the area of the sector PSX upon area of the sector PSY. Exercise Pro equation 8.11 using integral calculus. Now we are going to use the Kepler's second law of planetary motion. Now if you look at equation 8.11 which are easy, area of the ellipse is easy, area of the circle is easy and area PCY is very easy because area of PCY is a sector of a circle and we can compute the area of the sector of a circle. What is difficult? Area PSX, area of the sector of the ellipse is a difficult business. But what is Kepler's second law of motion? Kepler's second law of motion says that the radius vector joining the sun and the planet. Where is the sun? S. Where is the planet? X. So this SX, this radius vector SX sweeps out equal areas in equal intervals of time. This is a Kepler's second law that must be kept in mind. Capital T is the complete period of the revolution and little t is a time that has elapsed after the planet has passed the perihelion. So the planet was at the perihelion P, so SP that is the radius vector joining the sun and the planet at time t equal to 0 and SX is a radius vector joining the sun and the planet at time t. So the radius vector joining the sun and the planet has swept out an area PSX in time t. What happens when time at time capital T, the planet has made a complete circuit. So the radius vector PS has swept out the full ellipse. So the corresponding area of the, the area is going to be the area of the ellipse. So Kepler's second law of planetary motion says that area PSX upon the area of the ellipse is little t by capital T. So equation 8.12 that you see in the slide. So now equation 8.12 immediately gives you area of the sector of the ellipse PSX. Now we come to area PSY. Area PSY is area PCY minus area triangle SCY. Let us look at the picture and you will immediately see that area PSY plus area triangle SCY equal to area PCY. The area PCY is very easy to compute because it is the area of the sector of a circle. What is the area of the sector of a circle? One half radius squared times the central angle. The central angle is eccentric angle. Remember the picture? Capital E. What are the area of the triangle? SCY half base into height. The height is going to be given by a sin E. So we get area PSY is one half a squared capital E minus a squared epsilon sin E. Where does the epsilon come in? One is what? Epsilon is eccentricity of the ellipse. Remember what is the base? Where is the focus? From coordinate geometry, we must recall that the focus S is A epsilon comma 0 for an ellipse. So you got the base of the triangle and the base of the triangle is A epsilon and the height of the triangle is A sin E. So A squared epsilon sin E with a one half. Remember the area triangle SCY. So area sector PSY is basically one half a squared E minus a squared epsilon sin E. So now we know from this equation 8.12 area PSX upon area PSY is area of the ellipse upon the area of the circle. Area of PSY upon area of the circle is capital E minus epsilon sin capital E upon 2 by the A squared cancels out and we get capital E minus epsilon sin capital E equals 2 pi t by capital T. Equation 8.15 that is the Kepler equation that is a famous Kepler equation. So we have derived the Kepler equation using some very basic things like 8.11, 8.12 and that is a calculus exercise 8.11 is integral calculus, 8.12 is Kepler's second law of planetary motion and then a little bit of trigonometry and coordinate geometry and we get the final result 8.15 the Kepler equation has been determined. Now what is our problem we are going to find E as a function of T. What is in equation 8.15? T is given as a function of E we have to invert 8.15 and we go to get the eccentric anomaly as a function of time. Once we get the E as a function of time there is a relation between the true anomaly and the eccentric anomaly that I am going to tell you later. So if you understand the eccentric anomaly as a function of time the true anomaly can be determined as a function of time and vice versa. And this right hand side of 8.15 2 pi little t by capital T also has an interesting name it is called the mean anomaly in astronomy. So let us now focus on equation 8.15 and now that we have derived 8.15 couple of interesting exercises. Now we want to invert 8.15 the left hand side of 8.15 is an interesting function of E look at this function E minus epsilon sin E that is an increasing function from R to R and it maps bijectively on to R. It is a strictly increasing function remember epsilon is the eccentricity of the ellipse and eccentricity of ellipse is strictly between 0 and 1 the orbit is not circular the orbit is elliptic. The inverse function E t is also a strictly increasing function because that is elementary calculus. Now the inverse function is going to be infinitely differentiable I would like you to tell me why this is so. The next objective is E of t is an odd function what is E of t I am going to invert 8.15 t has been given as a function of E I want to invert it and I want to get E as a function of t the inverse function is also an odd function. Now this oddity of this inverse function should be derived using calculus and geometry in two different ways you should think about what happens when time is negative where was the planet before it crossed the perihelion and where is the planet after it crossed the perihelion try to use second law and try to derive the fact that E t is an odd function from geometrical or physical considerations and also using mathematical analysis. Of course at the point p or at the perihelion what is the true anomaly 0 what is the eccentric anomaly 0 what are the corresponding points p and p the ellipse and the circle they touch at the extremities. So that the two corresponding points they merge or they coalesce at the extremities so E of 0 is 0. What about E of capital T by 2 at the other extremity when you are looking at half the period capital T by 2 the planet has made half the orbit so it is from the perihelion it has gone to the aphelion so the angles are pi for the true and eccentric anomalies simultaneously. The problem of inverting the Kepler equation has been studied by many eminent mathematician such as Joseph Louis Lagrange in the memoirs the Berling Academy 1768-69. See for example second volume page 22 of his Mechanic Analytique which was published in 1815 it is exactly in this connection that Lagrange discovered the famous Lagrange inversion formula that bears his name and the Lagrange inversion formula today is used in many areas of mathematics which are quite unrelated to celestial mechanics such as combinatorics for instance. The Lagrange inversion formula is used very frequently and the Lagrange inversion formula arose from his investigations of the inversion of the Kepler equation. Now let us go a little further we want to now look into the periodicity after all we want to apply Fourier analysis and we want to understand the periodicity is this function E periodic let us look at what happens to E of t plus capital T what are the Kepler equation give you E of t plus capital T minus epsilon E of capital T plus t will be 2 pi by capital T t plus capital T. So, let us write this as 2 pi t by capital T plus 2 pi or let us write this 2 pi t by capital T as E t minus epsilon sin E t so this is the equation that we get let us write E t as E t plus 2 pi and then so we get this equation. So, what we see is that E of t plus t is not periodic, but it is going to become periodic after a subtract a 2 pi t by capital T. The injectivity of the function because this function lambda going to lambda minus epsilon sin lambda is injective from this equation we conclude that E of t plus capital T is E of t plus 2 pi equation 8.16 it is this equation 8.16 that is quite critical. And now we see from this equation directly a little algebra will convince you that E t minus 2 pi t by capital T is a periodic function this periodic function I am denoting by psi and what is the period the period is capital T. Again I would like you to derive 8.16 from physical consideration namely when time changes over one epoch that is over one period the planet returns back to the original position on the orbit, but the angle has changed the eccentric angle has changed by an additive 2 pi that is all that 2 8.16 is saying either you can derive it analytically or you can derive it physical considerations and geometrical consideration. But whatever method you may derive 8.16 the bottom line is this function psi t is a capital T periodic function. Now I do not want to work with periods capital T I want to get periods of period 2 pi I want 2 pi periodicity so I simply rescale the variables and psi of tt by 2 pi is a 2 pi periodic function because a 2 pi periodic odd function remember. So therefore it will have a Fourier series which only signs there will be no cosines present there and remember that capital E t was a smooth function infinitely differentiable and so psi t is also a smooth function and a smooth 2 pi periodic function certainly I can use the point wise convergence theorem from chapter 1. What is the formula for B n the usual formula 2 upon pi integral 0 to pi because it is an odd function I picked up a 2 upon pi and the integration is from 0 to pi psi of tt by 2 pi sin nt dt and what I do is that I just put in the definition of psi what is the definition of psi psi of s equal to E s minus 2 pi s upon capital T instead of s I have tt by 2 pi and I get this. And write the sin function as a derivative of a cosine with a 1 upon n integrate by parts in other words and throw the derivative from this factor to this factor I picked up an n in the denominator I got a minus sign because the derivative of cosine is a minus sign and when I integrate by parts the minus sign goes away I get B n equal to 2 upon n pi integral 0 to pi d dt of E tt by 2 pi minus t cos nt dt what happens to the boundary terms when little t is 0 you get E 0 and you get 0 but E of 0 is 0 remember when t is pi the pi cancels out E of capital T by 2 is pi and little t is also pi so the other boundary term also cancels out so no boundary terms to worry about and of course the derivative should be taken inside and I am going to get a a solo integral integral cos nt dt from 0 to pi that is sin nt upon n that vanishes at both ends this isolated minus t can be ignored compute the derivative and I am going to get E prime of s ds and I am going to get I mean I differentiate I am going to get a t upon 2 pi and I lump that along with that little t here and so this integral is exactly 2 upon n pi integral 0 to capital T by 2 E prime of s cos 2 pi ns upon t ds this is the elementary now we do the following we use the Kepler equation again we use the Kepler equation again 2 pi s upon t 2 pi s upon capital T what is 2 pi s upon capital T I write it as E of s minus epsilon sin E of s and this whole business is sitting inside the cosine inside the cosine and so now what do we have here we got this formula for b n and now I am going to make a change of variables I am going to put E of s equal to lambda so that E prime s ds is going to be d lambda and the variables of integration will become 0 to pi and I am going to get the coefficient b ns 2 upon n pi integral 0 to pi cosine n lambda minus n epsilon sin lambda d lambda this is our good old friend 1 upon pi 0 to pi cos n phi minus n epsilon sin phi d phi is the Bessel's function the integral representation of Bessel's function that we derived in chapter 1 and so the Fourier coefficient has been written in terms of the Bessel's function of integer order and so the Fourier series now reads psi of t t by 2 pi which is E of t t by 2 pi minus t is summation n from 1 to infinity to j n n epsilon upon n sin n t so the eccentric angle Et can be written as a series 8.17 2 pi by t plus summation n from 1 to infinity to j n n epsilon upon n sin 2 pi n t by capital T this kind of series Fourier series where the coefficients depend on the Bessel's functions are called captain series. So, we have obtained a Fourier expansion for the eccentric anomaly but we wanted the true anomaly and the relation between the true and eccentric anomaly is given in the next slide and I am going to leave this as a elementary exercise in trigonometry. So, for a short and quick historical survey see C.A. Ronan's article science is history and development among the world's culture in this book see page 336 to 337 the book of D.C. Knight Johanna Skepler and planetary motion that contains a poignant account of the life and times of the great astronomer and mathematician. For more on the Kepler problem and the mathematical principles that underlies celestial mechanics highly recommended book J.M. Danby celestial mechanics and dynamical astronomy Clure academic 1991 this is the second edition it contains computer experiments as well. And if you are really ambitious you should read Boca Latte and Pocahco's theory of orbits volume 1 and volume 2 Springer Verlag 2004 I give you lots of references now one can try to look at this captain series and try to expand these Bessel's function as a power series and rearrange it the whole thing as a power series in epsilon. But this power series in epsilon will have limited radius of convergence 0.667 the orbits of most comets exceed this number the Halley's comet has epsilon equal to 0.96 it's almost a parabolic orbit very close to 1. It seems that the investigation is to why the series in epsilon fails to converge beyond this threshold value seems to have led Cauchy to develop the theory of functions of one complex variable. There's an imaginary singularity when you invert this equation the ET is very nice it smooths on the real axis but there is a singularity in the complex domain which is responsible from preventing the power series from having a larger radius of convergence. And so you see that classical astronomy has given rise to certain developments in the theory of functions of one complex variables and these are things which are amazing developments in classical parts of mathematics. I think with this I would like to close this short chapter on astronomy as an application of techniques of Fourier analysis. Thank you very much.