[Music] [Applause] [Music] thank you thank you according to Newton's Laws all objects in a gravitational field Trace out conic sections the precise shape of an orbit depends on the interplay between energy and eccentricity [Music] all right we're all setting out the plunge in and see why trajectories in space follow the conic sections but before we do there's something curious I'd like to tell you about the conic sections have not only mathematical properties but also grammatical properties that is for each of the conic sections there's a construction in the English language for ellipse there is ellipsis for parabola parable and for hyperbola hyperbole Ellipsis means leaving something out of a sentence that can be understood for example when you say physics is more fun than chemistry that's ellipses because what you should say is physics is more fun than chemistry is the parable is a story that's intended to teach us something as in the story of Isaac and the Apple and hyperbole is an extravagant exaggeration as when you say this is the greatest Physics course in the world it's really true that these words are related to the conic sections you can find that out from any good dictionary but nobody seems to know why that's a mystery but today I'd like to go into a much deeper mystery I remember once when I was old enough to know about the conic sections but not yet old enough to know about Isaac Newton I went to a show at the planetarium and there was an exhibit there which said that the bodies that fly around in space planets and meteors and asteroids follow trajectories which are either ellipses or parabolas or hyperbolas and I remember being absolutely stunned by that why should those bodies fly around in courses which are curves having these special mathematical properties well it is a breathtaking fact because it brings us face to face with that same mystery that has awed everyone from Galileo right up to Albert Einstein nature obeys mathematics today our job is to do the mathematics that nature obeyed in designing the solar system from a scientific Viewpoint the strategy for finding the ultimate secrets of the universe is for the most part still unknown but scientists know that strategy has something to do with logic trial and error with taking risks and going to the Limit if viewed in a certain light not only the strategy but the universe itself may turn out to be some sort of cosmic game a game that has been played for the highest stakes and for the longest time sometimes playing conditions have been less than ideal it's been War poverty and poor health and there's been the threat of torture but no matter the odds or the obstacles there have been great players scientists such as Kepler in Galileo who in order to satisfy themselves about the rules have had to play the game a game of serious consequences and profound implications but like Star Wars like a hand of Five Card Stud or a stick of eight ball a game nonetheless after all human knowledge like any number of games evolves with increasing skill and the rewards of science pay off in the currency of mathematics of course not all games require genius and certainly not all games require the genius of Galileo Galilei even though he too was creative inspired and very competitive for example in a field along these lines when he was on a roll Galileo had no competition perhaps that was because at least on this level it was the only game in town [Music] Galileo rolled the ball hours on end but it wasn't all Play and No Work in fact with just such a game he was actually able to slow down falling motion and locked in competition with nature itself Galileo made some remarkable discoveries for example the distance Fallen is proportional to the square of the time though all bodies even his own fall with the same constant acceleration Galileo Rose to make many other remarkable discoveries in Europe toward the end of the 16th century another astronomer played with another collection of toys his name was Tico Bry and his playthings were sextants and quadrants of his own design and construction like Galileo with the inclined plane the game Tico played would reinterpret the rules that govern the universe in fact the strategy itself would be Rewritten because in 1600 Johannes Kepler sat down to play at this game and especially in Tico braai Kepler met his match but after almost two years a little luck finally came Kepler's way that look in the form of Tico's Heavenly observations kept Kepler at the table almost three decades the game was never easy to begin with and in some ways the longer he played the more complicated it got [Music] but by the end Johannes Kepler had won his final round of cosmic mathematics the victory became known as Kepler's three laws of planetary motion the law of ellipses the law of equal areas and the law of Harmony [Music] Isaac Newton would polish these laws to perfection in his mind Kepler was a giant and he realized those three laws were correct Newton imagined Galileo is a giant as well and saw that his law of falling bodies was likewise correct but in science as well as history timing is everything and it took Newton's era and Newton's genius to explain why they're correct he explained the laws of both Galileo and Kepler by deducing them from his own laws the laws of gravity and motion because of those laws the equation of the Earth's orbit is that of a conic section and in particular an ellipse this is one of the underlying mathematical patterns of the universe planets move in ellipses determined by the angular momentum the mass of the planet the mass of the sun and the eccentricity of the orbit but Newton's solution doesn't say that the orbit of every Heavenly Body must be elliptic it can be a parabola or a hyperbola or even a perfectly platonic Circle is this really the shape of things in the heavens and if so what determines the precise shape of a body's orbit all conic sections do appear in the heavens and the key to the precise shape is to be found in the energy of the body picture what happens to that energy as a planet Zips around its orbit when the planet falls close to the Sun its potential energy is low but the planet speeds up so the kinetic energy becomes High at the outer reach of the ellipse with its low kinetic energy the planet sort of loafs along at the same time because of the greater distance from the Sun its potential energy is high in other words the potential energy and kinetic energy of the planet keep trading back and forth like the potential and kinetic energy of a swinging pendulum one goes up the other goes down [Music] however because the planet moves in a vacuum with no drag the total energy does not change and this total energy is the key to the shape of the orbit [Music] the quest to find the shapes of orbits was undertaken long before Kepler's attempt Ptolemy sighted the stars with a small quadrant and such instruments have enhanced astronomical observations for centuries with the help of urania the Muse of astronomy Ptolemy saw to the limits of his vision and Imagination alike later Tico Bry called his Island Observatory uranabork named after the same Muse at uranaborg Tico built stargazing gadgets that were Beyond Compare before the invention of the telescope Tico's Brilliance in seeing afar was only out distanced by his vanity fancying himself a scientist King in one sweeping gesture Tico pointed out his kingdom which took in nothing less than all of Heaven itself with instruments almost as strong as his sense of self-worth Tico was Head and Shoulders above all the other astronomers on Earth his two minutes of Arc which is equivalent to about 1 15 of the diameter of the Moon as seen from the Earth was five times more accurate than any measurement in history only after 1609 the year Galileo looked through his telescope did astronomical Vision improve to any degree it's improved considerably by means of radio telescopes accuracy has increased to one thousandth of one second of Arc in human perspective that's about the width of a hair seen at a distance of 10 kilometers featuring both power and precision modern telescopes are able to seek countless objects that before now could be seen only in the astronomers dreams for instance Pluto though invisible to the naked eye stands out clearly in these photographs [Music] trying to explain irregularities in the orbits of Uranus and Neptune an astronomer named Percival Lowell predicted Pluto's dreamlike existence before its visual reality came true later Clyde Tombaugh searched hundreds of photographs for one tiny little blip moving amidst the fixed Stars and when he found it he found Pluto and when he found Pluto he found a rather eccentric ellipse it actually crosses inside the orbit of Neptune Halley's Comet discovered by Newton's friend Edmund Holly has an elliptical orbit that's even more eccentric these orbits are in marked contrast to those of Venus and the Earth which go around the Sun in very nearly circular fashion but all of them elliptic or almost round are Paths of constant energy this fact is nothing less than a mathematical miracle Behold a planet moving in an ellipse sweeping out equal areas in equal times it has kinetic energy and potential energy at any point in the orbit the total energy is the sum of kinetic energy and potential energy [Music] first calculate the potential energy [Music] [Music] the potential energy follows a cosine curve [Music] now calculate the kinetic energy [Music] foreign [Music] foreign the kinetic energy follows another cosine curve [Music] the total energy kinetic plus potential is constant that's because the cosine terms cancel in the addition and that's it the connection between energy and eccentricity it's true for a hyperbola a parabola or an ellipse of course Johannes Kepler's discovery that the planetary orbits are elliptical wasn't made on the basis of telescopic observations before the existence of the most Elementary telescope Kepler made his discoveries based on totally human observations and those eyes and observations alike had belonged to Tico Brian by the time he met Kepler Tico had recorded 777 Stars considering all the stars in heaven that may not seem like many but as Johannes Kepler knew it was the sum total of all the naked I could see [Music] when Kepler asked to take a look at the data Tico told him to take a walk [Music] Tico considered the serious-minded Kepler as somewhat of a play thing [Music] [Music] the game which was designed as cleverly as his instruments was Tico's version of Follow the Leader Kepler couldn't begin to play without the observations but eventually as fate would have it the tables turned [Music] on his Deathbed Tico Brian would have a change of heart as his dying wish he begged Kepler to take his data to use it well and to conclude the most serious game of their lives almost 30 years later Kepler finally managed to get Tico's data published but not before he had used those data to discover the shape of Mars orbit the shapes of orbits depend on their energies for a given angular momentum the energy determines the eccentricity and the eccentricity determines the orbit the ellipse when the total energy is negative and the eccentricity is less than one the orbit is an ellipse positive kinetic energy is too small to overcome the negative potential energy therefore trapped in a closed orbit the body can never escape the solar system [Music] is exactly one the orbit is a parabola if a body started with zero kinetic energy and fell from Infinity it would whip Once Around the Sun and return to Infinity it's highly unlikely that positive kinetic energy and negative potential energy would exactly balance this way but it's possible foreign the hyperbola when the total energy is positive and the eccentricity is greater than one the orbit is a hyperbola if an object were shot toward the Sun from very far away its positive kinetic energy would overcome its negative potential energy some comets have hyperbolic orbits Circle if the energy has a special value the eccentricity is zero and the orbit is a perfect circle this is the lowest energy a planet can have for a given angular momentum the rings of Saturn for all their beauty are nothing more than a pile of evolving junk Cosmic scraps that have rubbed together for a billion years and as a result of all this brushing and bruising their orbits were chipped and counter by encounter into nice smooth circles that's known to be true just because of the relation between energy and eccentricity even though Computing a million interacting orbits is beyond the mathematical capability of the most powerful computers on Earth to the degree that modern astronomy is effective it owes its Effectiveness to the invention of the reflector telescope the mirror of this one has been ground and Polished to tolerances that are measured in thousands of a centimeter such accuracy wasn't always possible but the principle was nearly always the same to focus the light properly the reflecting mirror must be ground in the shape of a parabola the grinding requires skill patience and most important a compound of messy mud-like lens grinding polish first man to complete this task that somebody had to do was rewarded with an invitation to join the Royal Society was Isaac Newton and it was the reflecting telescope not his other discoveries that first made him famous a Fame that like the parabolic mirror endures and evolves today's giant telescope and this is no Small Wonder is an extension of Newton's principle but again human exploration into space the grandest game in the solar system didn't begin with Sir Isaac Newton there were after all those shoulders upon which he stood to play Galileo and his toys the stuff of which the law of falling bodies was made Tico and his observations pieces of a puzzle that changed the course of cosmic history Kepler who played with Tico's observations in the most clever and calculating Manner and who in the end rewrote the rules of course while the game didn't begin with Newton's genius neither did it end there modern scientists liked Newton before them take their turns at trying to unlock the cosmos still a fantastic and ongoing game in which now the best players see 15 billion years into the past and the answer the Winner's prize is a peek at the very origin of the universe so the question is who's up next as I told you before the first to know what Newton had done was his young friend Edmond Halley how he realized that according to Newton although the orbits of the planets happened to be very nearly circular it was also possible for a body to have a very eccentric elliptical orbit like this if they were such a body it would be cited by us and then it would go far far away for many years before it came back to be cited again so he consulted the historical record and he found sightings in 1531 and 1607 and again in 1682 that was so nearly identical that they have to be the same object that of course was Halley's Comet but the critical test was to predict when it would be seen again and he did unfortunately in order to find out whether he was right he would have had to live to be 102 and he didn't make it he only lived to be 86. nevertheless just as predicted on Christmas day of 1758 Ali's Comet was spotted once again as I'm sure you know its most recent round trip brings it by us in 1910 and again in 1986. so that's it we've explained the structure of the heavens that mystery from time out of mind has finally been solved many years ago there was something called the Pythagorean Brotherhood they were followers of the master Pythagoras of Samos it was a semi-mystical mathematical priesthood neophytes would have to go through a long and difficult apprenticeship to purify the mind and cleanse the soul before they could be entrusted with the awesome power of mathematical knowledge well today after a long and difficult apprenticeship you have been permitted to peek into the inner Temple of human knowledge I hope you'll use that power wisely but right now you can get out of here the eccentricity of an orbit depends on the energy of the moving body negative energy means an elliptical orbit when the total energy is zero the orbit is a parabola when the total energy is positive the orbit is hyperbolic foreign media for information about this and other Annenberg media programs call 1-800 learner and visit us at www.learner.org [Music]