Terence Tao: The Cosmic Distance Ladder, UCLA

ladies and gentlemen its vice president of the American Mathematical Society I'd like to welcome you here in turn burg hall and in the overflow locations to the einstein public lecture in mathematics launched in 2005 the 100th anniversary of einstein's annus mirabilis in which Einstein published his papers on the photoelectric effect Brownian motion and special relativity here's the list of the Einstein lectures Sir Michael Atia benoît mandelbrot Sir Roger Penrose Michael Waterman and today professor Terry Tao from here at UCLA in 2006 for her sertao won the Fields Medal generally considered the highest award in mathematics his single most famous result joined with Ben green is a proof that you can find billions of evenly spaced prime numbers indeed as many as you like Taos website on Google Buzz is in my opinion the best site on the internet with fascinating observations for a wide audience I go there every day now I go MacArthur Fellow fellow of the Royal Society for an associate of the United States National Academy of Sciences and member of the American Academy of Arts and Sciences speaking today on the cosmic distance ladder we will shortly have Tarrytown after his talk there will be time for questions and then you're all invited to a reception but first our host and co-sponsor UCLA Vice Chancellor James Economou well good evening I'm very pleased to represent Chancellor block at the Einstein lecture this evening and from I'm on the School of Medicine faculty and it was a privilege on July 1 to succeed professor Roberto Pichet as a vice chancellor for research I'd like to thank the American Mathematical Society for selecting professor Terence Tao as tonight's distinguished Einstein lecture in addition to the chancellor's office I'd also like to acknowledge the support of the division physical sciences the office of the Dean and the mathematics department that for hosting tonight's event you may be aware that the Department of Mathematics is one of six departments that comprise the division of physical sciences the others include physics and astronomy Earth and Space Sciences atmospheric and oceanic sciences statistics and chemistry and biochemistry within within this division there are also a number of preeminent Institute's the Institute for geophysics and planetary systems the instant for pure and applied mathematics and the Institute of the environment the distinguished faculty from this division include four recipients of the Nobel Prize six National medals of science and 20 members of the National Academy of Sciences professor at AU became the first mathematics professor in UCLA history to be awarded the Fields Medal often described as the Nobel Prize in mathematics in the seventy years of this prize has been awarded by the International mathematical Union only 48 researchers ever received it he's also received numerous other national and international honors outstanding graduate students from around the world travel to usally to study with the tairy at only 24 he was appointed to full professor and to this day he remains the youngest person ever appointed to that rank at this university in 2007 he became the first incumbent of the James and Carol Collins chair after a generous endowment by mr. and mrs. Collins so I joined Frank Morgan and the department in recognizing a Terry for as many accolades and it's privileged to be able to introduce him he'll speak this in this evening about cosmic the cosmic distance ladder thank okay thank you very much thank you all for coming here I'm very honored to be giving the Einstein lecture series because the very distinguished list of previous speakers but of course an assign himself is is such a legend he's such a great scientist he did so many great things but you know even someone as great as Einstein didn't work in a vacuum here the work that he did was built on the work of many many people before him and you know he was part of a larger scientific story in fact part of many many scientific stories and what I want to talk about today is just one of them the cosmic distance ladder and this is one my favorite stories actually a story about math and physics and astronomy in history it's a very long story it over 2,000 years old and it's still going the story has not ended you would have learned about parts of this story at school or in textbooks or on the internet but it's rare that you see the entire story at once and this is what I want to convey that the science is not just an isolated set of facts but it's really part of a very large narrative and this is a great example of one okay so the cosmic distance ladder is a foundation of a strong tree and what is the stroma tree astrometry is a major subfield of astronomy it is the study of positions and movements of celestial bodies planets moons Suns galaxies you want to know where they are where they're going and the typical questions you ask Innes trauma tree are things like this how far is it from the earth to the moon or from the earth to the Sun or from the Sun to Mars or from the Sun to Alpha Centauri or from the Sun to Alderaan Lord Rama dromeda galaxy questions like this now nowadays it's it's quite easy to answer these questions you just look it up Mallika pedia but how does Wikipedia know you know it comes from these the astronomers how do these phone numbers no I mean these are not easy distances to measure they're far too vast to measure them directly you know if you want to to measure the distance of a galaxy over there you can't just take a ruler or something I mean it is just immense distances that you can't measure directly but what astronomers have done is that they have found many many ingenious ways to measure distances indirectly so for example of the distance d1 to that galaxy over there we might not work out directly or the distance d2 to like galaxies over there but we might be able to work out the ratio between the first distance in the second distance and we have many ways of measuring ratios of distances by comparison by comparing one distance with another one and these methods they use a lot of technology in science but they often they also just often just use some basic mathematics also some sometimes some very advanced mathematics in this particular case one way to measure ratios between distances is to use the Hubble law which I'll talk about later the hub of all connects distances to galaxies with with their velocity and you can measure velocity by the redshift and from that and a bit of mathematics you can work out the ratio between these distances I'll explain that a bit later so we have always in direct methods and what they do is that they control large distances distances to very far objects in terms of slightly less far distances distances of slightly smaller objects and these distances in turn are controlled by slightly smaller distances and you can go keep going down and down and down until finally you can read your distance that you can actually measure directly and so you have this hierarchy of distances happies hierarchy ratios and you put them all together and you get the cosmic distance ladder and this is the way that we measure distances throughout the universe so what I'm going to do in this talk is climb this ladder from a historical perspective starting at the very base of the ladder which is distances on the earth and then moving up to the moon and then the Sun and the planets and so forth and we'll see how each one was developed and how each one builds on the previous one this is the story on hotel so the first rung is the earth okay so nowadays we know a lot about the earth we know the earth is round well approximately round it's about 660 300 kilometers in radius a little bit wider at the equator than other poles but we know pretty much exactly how how big it is and we have extremely accurate maps of it so then nowadays this is very easy because we have all these satellites and we have all these maps and and we have surveyed every single square inch above the planet really so we have all this but just imagine for now that we didn't have all this technology suppose we do not have satellites spacecraft or telescopes suppose we didn't have the ability to cross the ocean we didn't have airplanes we didn't even telescopes or sextants no technology at all is it still possible with no technology to calculate the radius of the earth these numbers I just I just gave you okay even more basic question can you even tell you this round and the amazing thing is that the answer is yes all you need is just some geometry and you can already work out that the earth is round and you and what is radius is and this was done over 2,000 years ago by the ancient Greeks so for example the first person to convincingly argue that the earth was round was Aristotle over 23 centuries ago and so he gave an indirect argument all the arguments I will give are in direct indirect argument the earth is round and like all indirect arguments you don't just look at the object that you are trying to study I mean look at the earth you can't really tell that's round or flat but you have to use an indirect observation using a second object in this case by looking at the moon so how did he do it he used lunar eclipses you will see eclipses over and over again in this talk incredibly useful thing for astronomers so the new about lunar eclipses the Greeks knew that lunar eclipse has only occurred when the moon was opposite the Sun for the moon was in the opposite constellation was o diac that the sun was and so they knew because of this that the reason for lunar eclipses were because that the moon was actually falling into the earth shadow okay this is what causes a lunar eclipse but then if you look at a lunar eclipse you can see the the shadow of the earth falling on the on the moon and the shadow is always a circular arc okay no matter what position what constellation the moon is in no matter what part of the eclipse you're in the shadow of the earth as seen on the moon is always a circular arc so that means that every single shadow of the earth must be a circle and there's only one shape that does that it's a sphere okay so this is them this is a correct argument it explains why the it explains that the earth is round so you know for example it can't just be a flat disc okay if it was a flat disc then the shadows be like ellipses and that's not what you see by looking at the moon okay so this is one of the simplest proofs of the earth is round our sort of also knew so I said they didn't have have any technology it's not quite true they could travel a little bit for example from Greece they were able to travel to Egypt and even from Egypt it was enough of a distance that you could start seeing some constellations from the southern hemisphere that you could not see in Greece that the constellations are slightly different and I started realized that the reason for this is because even the relatively short distance from Egypt to Greece was enough to see some curvature of the earth and that changed the constellations that you could see and so that told him that the earth was not enormous ly huge even just from the trip to Egypt to Greece you could see some of the curvature of the earth so it's radius was finite but unfortunately this argument it's also correct but it didn't really give an accurate measurement of exactly what the radius of the earth was so that had to come a little later about 100 years later another Greek Eratosthenes did compute the radius of the earth and he competed it to be 40 thousand stadia last week unit in modern units are 1600 kilometers or 4,200 miles that's a very good estimate that's accurate with an 8% you know the astronomers nowadays who kill for 80% accuracy but you know and he had very little he had almost no technology to do this and he got an answer with an 8% which is really quite amazing and it's an indirect argument as before so and now it was it was it was done by looking at the Sun well not directly at the Sun by stupid but okay but indirectly other Sun Oh see okay so how did it work so you may have heard the story before without tell it again so Eratosthenes knew he wrote a book about oh maybe a scroll about a well in Syene Syene is a town in Egypt a village in Egypt which had a well and this war had the funny property that on the summer solstice June 21st the longest day of the year you could look into this world and you could see the Sun reflected directly overhead in the water below even though the world was very deep okay and most wells don't do that but the world in Syene did okay and the reason for that ultimately we now know is because Syene is just has the fortune to lie exactly almost on the Tropic of Cancer which is one of only two places in the world where this happens we have the subtropical Capricorn on the southern hemisphere so okay Eratosthenes read this but he didn't live in Syene he lived up network xandria which was also an Egypt but a bit to the north and so he decided being a good scientist to try this experiment he waited until June 21st and looked into a well and he did not see the Sun reflected in the in the world below okay and so the Sun was not at an angle so most people at this point was like well I'll get that book was rubbish and move on but ah sneeze kept going said okay what must be happening he knew about I was Aristotle we knew the earth was curved and so the reason why this the Sun was not reflecting down the world where he where he lived was because of the curvature of the earth and he happened to agree with him a measuring stick called in Oman can like a little supportable sundial and so he measured exactly the angle the deviation to which to which would be the Sun was was falling and it was not quite vertical it was about seven degrees in our units off the vertical okay and he also knew how far was too sighing so there were merchants going up another denial they had to know how many days it took to get from A to B they knew how many stadia they would cover in a day so Eratosthenes could just ask these merchants and and some legends say he actually paid paid some graduate students like just walk but it's a dispersed except that that's apocryphal but believable but anyway he knew that the distance from Alexandria to Syene was five thousand stadia and just those two pieces of data and just a little bit mathematics that's already enough to work out the radius of the earth just having no liberal trigonometry and you have this triangle with one side of 5000 stadia and that's enough to to work out the radius of the earth and that's how he got its forty thousand stadia Tonopah okay one thing just to mention I drew these lines from the Sun as parallel lines so I'm assuming the Sun is really really far away now that has to be justified we will see a justification of that a little bit later okay so let us that's the first rung the earth we will now go on to the next one which is the moon so the basic questions here are what shape is the moon there's not a silly question explain why how large is it and how far away is it these are the basic questions okay so the ancient Greeks also had good answers to these questions to again with no technology just geometry really and some observation so first of all the shape of the moon so the moon looks round but it could be round and flat no it could be like painted over the sky for we know okay we know now it's a sphere why is it a sphere well you can I was total realized there was a sphere just from the the familiar phases of the moon crescent moon Half Moon gibbous moon these phases are caused by the sudden lighting at one side of the of the of the moon and the other side being being dark and he looked at the Terminator the difference between the edge between the light and dark parts of the Moon and this was always an elliptical arc okay or maybe a straight line in the case of a Hoffman but it was always an ellipse and again the only shape with that property is is a sphere you have a sphere then one hemisphere is lit up and from our perspective the the great circle or great semicircle the pedestal Terminator comes across as an ellipse if the earth was if the moon was flat then in fact there'll be no flight phases at all they'll just the moon would just be bright or dim but there would be no no Terminator the Terminator comes because of the spherical nature of the moon not the not theif that nature and around the same time Aristarchus another greek was the first to compute the distance to the moon in terms of the Earth's radius so an indirect measurement couldn't measure the distance the moon directly but you could measure in terms of the radius of the earth and he said that it was about 60 Earth radii and this was again an amazingly accurate answer in fact it's basically could spot on the mood actually is has a slightly elliptical orbit it varies between 57 and 63 of the radii but on the average our stalkers was exactly correct he also computed the radius of the moon he said that the moon was about one-third the radius of the earth which is pretty close point three three it's actually 0.27 but still not not bad not bad accuracy for for green technology and of course we know the radius of the earth we computed it in the first rung so you just put that we just combined that would be information information I just told you and that gives you the radius of the moon and the distance to the moon okay so how did i stockist do it so it isn't again an indirect argument to find the distance to the moon we will use the Sun and once again we will use oopss lunar eclipses first so lunar eclipses as I told you already are caused by the the moon passing through the earth shadow now how big is the earth shadow well the earth is two or three di diameter and that's basically how wide the Earth's shadow is as well assuming the Sun is far away but we'll get to Latin in a bit okay but the yeah the other shadow is about two Earth radii wide and lunar eclipses well the Greeks already observed many many lunar eclipses they knew that the longest that a lunar eclipse ever lasted was three hours no eclipses that lasted more than three hours this was the maximum so that means that it took three hours for the moon to to cross two or three D I on the other hand it takes stringent off one month one lunar month twenty-eight days for the moon to go around the earth you can see their moon phases so it takes it takes 28 days to traverse an entire circumference of the moon's orbit and it takes three hours to traverse to Earth radii and that's a simple high school math word problem you can that's enough information to work out the distance to the moon in terms of the radius the radius of the 60th radii so that's the distance to the moon what about the size of the moon so you can look though many ways to do it but one way is just to wait under the moon sets and the moon takes about two minutes to set you can time it and so from our point of view the apparent motion of the moon covers two moon radii in two minutes on the other hand the moon goes around makes a full rotation at least apparently to us once a day or not quite once a day but pretty close ones once a day takes 24 hours to go to complete and entire orbit of the moon from a person from our perspective and it takes two minutes to do two moon radii and again this is enough information to compute the radius of the moon in terms of the distance to the moon which we've just computed in terms of the earth radius and you put that together you get that the moon radius is one third of the earth radius by the way just to emphasize how little technology they had they didn't so not only do they not have much technology they do not even have a value of pi at this point pi the first person to get an accurate idea of Pi was Archimedes this was about a couple decades after Aristarchus so here to use triangles instead some some other trigonometry so just to emphasize it they had basically zero technology at this point but they could still get good answers okay so this that's the earth and the moon the next stop is the Sun and so the type of questions we care about here are how large is the Sun and how far away is the Sun and it's still quite amazing the Greeks could still answer these questions although this time the accuracy began to get a little bit degraded because here they were really bumping up against the limits of the technology so again it's an indirect argument to work out the distance to the Sun you need to look at the moon so this was done by Aristarchus so he ready I was talking as we already saw computed the radius of the moon and the distance to the moon and the radius of the moon will turn out to be actually one 180th of the distance to the moon distance the moon is 60 radii radius of the moon's 1/3 earth radius so the moon radius is 180th of the distance to the moon on the other hand are doing the solar eclipse now we have this lucky coincidence that the moon and covers the Sun was almost a perfect fit it's an amazing coincidence okay but but basically the angle the width of the moon and the angle width of the Sun are exactly the same almost and so we can just use similar triangles what you learn in high school and therefore the radius of the Sun must also be 180th of the Radian of the distance to the Sun just by similar triangles okay so that tells you that if you know one of these numbers like if you know the distance to the Sun that tells you the radius of the Sun but you still need to work out the distance to the Sun but he did that too and so once again you have to use an indirect argument you again use the moon so we saw that moon has phases so for instance them the moon can be a new moon and this happens when the moon is between the Earth and the Sun then we see a new moon and conversely if the moon is opposite the Sun we see a full moon unless it's exactly opposite then you get in the clips but it is almost opposite you get a full moon and then you get part moon's where you see half of the moon so you might think that half moons occur exactly halfway between the full moons and new moons but they don't quite if you do the trigonometry half moons actually occur when the moon makes when the earth when the Sun make a right angle I went viewed from the moon feels drawn over there but that when when they get a right angle then exactly half of the moon is visible from the earth okay so half moons not occur exactly halfway between numerous and full moons they occur just a little bit closer to a numeron than they do to a full moon there is there's some angle which is a little bit less than a right angle you can't really see it here but it isn't there's a the angle here is slightly less than a right angle and if he knew that angle there you can use trigonometry and you can start working out the distance to the Sun so Aristarchus tried to compute this he thought that half moons occurred about 12 hours before the midpoint of a new moon and a rumor that there were 12 hours closer to a new moon than the halfway mark and of course they take a whole month to go around the earth and he just do some trigonometry then he was able to conclude that the distance to the Sun was about 20 times the distance to the moon now the math was correct but his observation was wrong he thought that the Hoffman came 12 hours earlier than the midpoint but he was using ancient technology first of all he had no telescopes it was hard to time exactly when the half moon occurred secondly he had no good clocks the best clocks include in Greek times were sundial was and they didn't work well at night so so so it was difficult so his answer was actually off like a fair bit the truth the true time discrepancy was not 12 hours as he had thought back in only a half an hour and so the distance to the Sun is not twenty times the distance the moon exactly 390 times the distance to the moon so the math was correct but because of the limitations of his technology the the answers he got were inaccurate but nevertheless the basic method was correct and even with the inaccurate data the inaccurate conclusion that he had Aristarchus computations led him to for the first time to a very important conclusion in science that he put all the data together you get that the Sun is much larger than the earth so in fact his computations said it was seven times larger so you think this is obvious nowadays but it is not obvious it was not obvious back then you know you look at the Sun dream at this big okay this is this big okay it is not obvious that the Sun is much bigger than the earth now our staff misses computations were wrong he thought they were the sun was seven times because the earth is actually a hundred nine times is because the earth but the basic fact is still this that the Sun is much bigger than the earth and because of that he was the first to realize that the raining period of the time about the solar system which was a the Sun went around the earth the geocentric model was really absurd because the Sun is much bigger than the earth and so he was the first to propose the heliocentric model that in fact instead the earth went around the Sun now you may think that's not what I learned in school I thought that that Copernicus did that seven seventeen hundred years later and he did but if you read the first page the dedication of Copernicus book he will quick you will say that the heliocentric theory was first proposed by Aristarchus now there's a reason why I was there's there's a ironic reason actually why at the time I was taught his theory was not accepted by the other ancient Greeks and I won't tell you why but not right now okay tease you a little bit okay so but anyway in principle at least aerosurfaces method gives you the distance from the earth to the Sun now this is a very important unit is called the astronomical unit and it is used for the next three or four rungs of the ladder lots and lots of other measurements were made in terms of the astronomical unit so it is incredibly important as I said I was talkin twas not very accurate but I'll show you later on there are much better ways to measure it nowadays then I've been in now stock is this time okay so that's the Sun the next one is the planets okay we now you know Mercury Venus Jupiter Saturn na Pluto okay so what are the planets do so in ancient times you know they had astrology just like they do now and astrologers other some things they did figure out that the planets did lie if they kept moving around times the word planet means wandering star but they only moved through the zodiac is this this ring of 12 constellations around around the earth you know counselor Libra and so forth and so this already tells you that the all the planets lie on a plane the plane is called the ecliptic so did whatever the motion of the planets are is a two-dimensional problem not a three-dimensional one but they didn't know what the orbit was in fact it looked very funny you look at one planet say Mars and sometimes Mars go west and there summers ago East it just jumps back and forth and it's very weird pencil it to begin where there was not easy to answer basic questions like how far away is a Mars what orbit does Mars how does Mars move and how long does it take to go around one orbit so the Greeks tried to answer these questions too and tomley for example was the first to make a really good effort but here remember they had they had discounted Aristarchus model and they were working with it with the geocentric model and as a consequence unfortunately despite his best efforts tomley got basically absolute rubbish as his answers is it I'm not even siding in here because they're so inaccurate okay because he was working with the wrong model so the first person to get good measurements of distances to planets was Copernicus okay now this is what he's known for this is why is in the textbooks and so Copernicus he started out by looking at the records of the ancient Babylonians actually so even before the Greeks the Babylonians watched the Stars and the planets they knew about the planets they made records the Babylonians pass them on to the Greeks the Greeks pass among the Arabs they I was possible to the Europeans so Copernicus knew had leads records and he knew for example that Mars at least from the point of view of the earth it went in a funny orbit but it would but every 780 days it would come back to the same constellation that I started with that it had a period was called the apparent period or smaller period of 780 days now working with a heliocentric model he knew that this was not the actual period of Mars because not only was Mars going on the Sun but that was going on the Sun as well it was only it was not so the the actual angle of angular velocity of Mars was not one for 780 days only the difference between the angular velocity of Mars and the angle of us leap earth was one of 170 on the other hand he knew that the earth took one year strangely enough to go around the Sun my solar year 365 days and then again high school math problem you just add together or subtract these these angular velocities and you soon find out that Mars therefore the true angular velocity of Mars is 687 days it takes 687 days for marked Mars to complete one full orbit around the Sun and you can do the same for all the other planets now once you have this he also knew at various times he knew at various times what constellation the Sun was in what constellation Mars was in that gave him some angle measurements if you have some angle measurements and then you just keep waiting a few days to move the planet around a little bit you can do some trigonometry and he was good enough in math that he could he could do the math and take some more measurements and he could actually work out assuming all the orbits were circular he could have work out the radius of all these orbits in terms of the radius of the earth so for instance he computed that the radius of the Mars orbit is about 1.5 astronomical units mars is 50% further away from the Sun than the earth and these are really good answers they are acted by two decimal places on 1% accuracy okay but not perfectly accurate a little later on couple centuries later Tycho Brahe he for some reason actually decided to make some really really detailed measurements exactly quite obsessed with with the planets and he made he managed to convince something a prince in Denmark to give him a an island and some peasants to Bournemouth observatory and he sat there for 10 years measuring the locations of all the planets and then but they didn't quite fit the Copernican model if you actually try to plug in component models of the circular numbers circular orbits the data that Tycho Brahe he got the declination z-- of all the planets here for example are the definitions of Mars didn't quite fit the Copernican model and this was picked up by kepler a little bit later kepler actually had his own theory of the solar system he thought that it was brought up using actually tectonic solids of all things and he needed to he wanted to publish this he wanted some data to back this up and so he yeah he stole Tycho Brahe huge data there's a story behind that but then he found that the data didn't fit he didn't fit his theory didn't fit the Copernican theory and finally he was forced to concede that what was going on was that the orbits of Earth and Mars were not actually perfect circles which is what Copernicus had assumed that there was something else so his job now was to take the data he had and work out what was the orbit of Mars more than orbit of the earth okay but how are we going to work out this data you know if you learn some math one of the things you learn when you solve equations that you need at least as many unknowns at least as many equations unknowns you wanted one piece of data and there are two unknowns here the orbit of Earth and the orbit of Mars I kids even worse than this because Tycho's data only gives you the direction like any given time it tells you where Mars the direction of where Mars is the declination but it does not tell you the distance okay if you know polar coordinates you know that you need two numbers to determine in a location in the plane her declination and distance there was no way to measure distance only declination so he a key when he had half a piece of data and hit her so for two unknowns so so it so it looks like this is insufficient information to solve the problem but nevertheless Kepler did it he finally an ingenious way to solve the problem so he had two great ideas the first idea was that if you wanted to compute Libra Mars - precisely you must first work out the orbit of the earth okay because if you don't know what the earth is doing and you're sitting on the earth watching all the other planets you there's there's no chance to work out what the other planets are doing either so how do you work out the orbit of the earth this was this next genius idea he would use Mars so how did this work so to explain how this works let me take a simpler case so of course Earth and Mars both move but suppose for simplicity that Mars did not move that Mars was fixed it was nailed to space somehow okay just set over there in the corner okay and only the earth moves okay and but we don't know how the earth is doing something but no matter where you are no matter when you are the earth is somewhere and because you can see my eyes and you can see the Sun and you can see what direction what constellation the Sun is in and you see what constellation Mars is in and so you have these two directions and on the other hand you have this fixed axis the the axis between Earth and Mars so the southern Mars is fixed okay I'm assuming masters don't move so we have this fixed side and two angles and if you know some high school geometry the angle side angle theorem that if you know a side and two angles of a triangle that tells you the triangle that tells you where you are this is a very ancient technique is called triangulation into navigators if they wanted to know where on the ocean they were they could see two landmarks that they recognized and they can measure their directions that's enough to tell them where they are this is triangulation you may think it's so archaic we don't use it nowadays but I bet many of you use it to come here GPS works by triangulation or tetra angulation use for satellites but it's the same principle okay now so if Mars was fixed then you can work out the orbit of the earth but Mars is not fixed Mars moves as well and so you are not triangulating the circular fixed axis you're trying to triangulate in respect to an axis that keeps moving in a way that you don't understand the orbit of Mars is unknown so it appears that triangulation does not work but Kepler did not give up at this point he had one additional piece of information okay something he knew from Copernicus every 687 days Mars comes back to where it was before so if you don't take all the data but if you just take Bryce data in intervals of 687 days or in that interval Mars is fixed and for that interval you can triangulate and this is barely enough data in in brought his data that you can do this no it was it was a good reason why he's good it was good that he sat Lee for 10 years doing this so you take you take this interval of of six of 687 day data and this is enough to give you the orbit of the earth relative to one fixed point on Mars orbit and so that tells you Mars's orbit that tells you the Earth's orbit relative to one fixed point on Mars so once you have that once you have this orbit now that you can use as a fixed reference and then you take another interval of 697 day data and then you can figure out another point on Earth of Mars's orbit relative to the Earth's orbit already computed and you just keep doing this and this will give you back Mars's orbit okay so this was Kepler's idea Einstein of course our sponsor in some sense who wrote one sort of preface to a book on astronomy which book on Kepler actually and he mentioned the study of Kepler he called it and I do a pure genius and you know Einstein calls you what genius you really doing well and so as a couple worked out you know you worked out all the orbits of course you know nowadays we don't have to do all this because we have Kepler's laws right which over had given all the reasons for us so you didn't have them back then but because he had worked out painstakingly the orbits of Mars and Saturn and so forth he worked out all his orbit and he formulated the three laws of planetary motion Kepler's laws and this was very important in physics it led later honors a century or two later Newton to then deduce from Kepler's laws actually his own law the law of universal gravitation his famous inverse square law for gravity okay so Kepler's methods one thing they did was that they gave all the distances to all the other planets in terms of the astronomical unit the distance to the earth and very precisely okay I mean they're ellipses so they they they wobble back and forth but you can measure exactly how much they wobble so one thing this does for you is that it actually gives you a new way to measure the astronomical unit that if you can measure say that just seems to say Venus and because you then combine that couples measurements and that will give you back the astronomical unit and so the first really accurate measures of the astronomical unit were done using Kepler's methods combined with measurements of things like the distance to Venus and want me to do it do this is by parallax if if you are able to have on two different two different ends of the earth in the northern hemisphere in southern hemisphere if you at the same time you observe the same planet and you measure the angle that you see from from one location and the angle from other location and you take it it's so accurately you can look at the difference you get the angle we already know the radius of the earth such as like trigonometry once you have an angle and the radius of the earth that tells you the distance to say Venus and then using Kepler that will give you the distance say to the Sun okay so you could do this but it's tricky you need first of all you need to travel to the other side of the planet you need accurate clocks you need good telescopes in a good sexton's okay Copernicus I don't have this Kepler did not have this but the 18th century the Europeans had this and the first person to like you do all this and get a really good accurate measurement of the astronomical unit was James Cook all right well he did one of the two measurements you know so someone had to go to the thermogram is hemisphere and measure the other measurement and James could get this James could voyages famous especially in my home country of Australia but he was there really for scientific mission that was that was his the purpose of his voyage in part to do things like this okay discovering Australia was a key just a bonus okay nowadays actually we have much even more precise ways to compute things like Western astronomical unit because we have things like radar we just even the distance the sunniest bounce a radar wave awful off of the Sun waiting to come back multiply the speed of light which we also know how to compute and that gives you incredibly precise information on things like the astronomical unit so we have everything up to this point is now extremely precise and it's important that we have all these precise measurements because doing that we found that that capitalist theory Kepler's laws were not quite correct that if you look at the orbit of mercury in particular Kepler says that mercury must movement in ellipse and it doesn't quite hit Bob was a little bit the the ellipse wobbles every time it goes around and Newton's low gravity did not quite explain this and and it was an important observation it was one of the first major confirmations of Einstein again I'll sponsored his uh his few general relativity the precession of mercury was explained by his theory it was one of the first famous applications of his theory and we'll need general relativity later on in his ladder in fact we will always see that astronomy has helped the developer of physics and conversely physics has helped the development of astronomy the two go hand-in-hand ok so the next round is always the speed of light now I needed a picture to depict a speed of light this is the best picture I could come up with actually they did the the UCLA math department actually managed to get permission from Lucasfilm so that you can see this image today okay so thank you to the math department for that now this is not a distance so it doesn't strictly speaking belong on a distance ladder but what you see why I put it in here but it's important to know in order to get to later parts of the ladder okay so the speed of light is tricky to measure for a long time there's even an argument where the light was had traveled with finite speed or infinite speed it was not know Galileo back in the 17th century tried to do this he got a friend it said okay I'm gonna I'm going to get up on one hill with a lantern and you're going to get up on the other hill I'm going to unshut on my lantern and when you see that light you and shatter your Lantern now time how long it takes come back and it from this outside work at the speed of light so gal I reported that this this experiment did not work but the the method was correct it's the math fact is just the technology and nowadays we can actually make the experiment work with with nanosecond clocks and so forth we can actually do something like Galileo's experiment on a tabletop in the lab with with you know lasers and LCD displays and so forth and and we can actually do this but the first accurate measurement of the speed of light was done in the 17th century it I can use the same idea but two different hilltops were not far enough apart to make the augment work so they used two planets instead so in particular they used a Jupiter and a moon of Jupiter yield in fact so eeo is one of the four big moons of Jupiter and is the closest moon of Jupiter to the planet itself so close that it always incredibly quickly it I uh goes around Jupiter wants to be forty two and a half hours two days you know the moon our moon takes 28 days to go around it's this little it's slow I'll goes only two days to go and do but it was a key much bigger planet so IO scoots around Jupiter and you can see it in telescopes is this load dot algebra it goes left and right and it falls into Jupiter equal if it goes in and out of Jupiter shadow so it it so again there's a these eclipses again and you can see you can see IO disappear and reappear disappear reappear and so Roma just made many many measurements for for over a year of IO going in and out in and out just timing when when I enter in and out and it was very regular every 42 and a half hours I would go in and out in and out but not quite if he did this over a year over he discovered that half the time the orbit came ahead of schedule and half the time it became a behind schedule that when Jupiter was on the same side as the earth the orbit was a little bit ahead of schedule it came earlier than anticipated and when Jupiter was in the opposite side of the Sun the orbit came to his telescope a little bit behind schedule not by much you know but but the lag was about 20 minutes okay but by that point the clocks were good enough that you could actually measure 20 minute difference over a period of a year so Huygens our enrollment figure this figure out what was going on the reason why there is lag was that when the earth was on the opposite side of the Sun offered to Jupiter the light from Jupiter and IO had to travel longer to get to the earth how much longer to astronomical units okay so it takes 20 minutes for light to travel to astronomical units and that's enough information to work out the speed of light and so they got to take out the first reasonable measurement of the speed of light as about 220,000 kilometers a second not completely accurate the actual truth is more like 300,000 but considering the technology they had this is an amazingly good estimate and is shortly afterwards with these observations further observations of the astronomical unit they got much better measurements of see pretty pretty pretty shortly so this was a very important number to get the speed of light for instance it led James Clerk Maxwell at shortly afterwards he was working out his laws of electromagnetism so he had unified electricity and magnetism into his four equations of electromagnetism that you learn in you learn here classes and and one of the consequences of his theory was that was that there were these electromagnetic waves but radiation and they traveled at a certain speed given by the permittivity and permeability of free space and so he computed the speed how fast does electromagnetic radiation travel and he got a number and he looked at a number that look familiar and so he looked up the speed of light the best estimate was speed of light at the time and it was almost the same number and so he was the first to realize that light is a form of electromagnetic radiation it is the visible portion the electromagnetic spectrum so this was a very important advance in science among other things it led to Einstein's theory of special relativity famously worked this out by will by asking what would happen if I ride on a beam of light it also led to the development of spectroscopy the ability to determine very precisely the color of various things like atoms or stars and this will okay and both of these we need to keep going up the ladder okay so we've done Sun Moon planets next stop is the nearby stars things like Proxima Centauri so okay we've already seen parallax if you if you make it this measurement of the same object at two different places on the earth you can use that to work out distance just like your two eyes can work out distance to objects but there's a limit to how much how to tell our our distance vision works and it's also a limit as to how well this parallax method works it does not if you if you do James Cook's method to try parallax to the difference in the Stars even the nearest star Proxima Centauri it's so far away it's a it's about 270,000 astronomical units away so it's two hundred thousand times further away than the Sun and there's just not enough parallax to make two different measurements on the earth and see this distance in fact the parallax is only about one ten thousandth of an arc second an arc segment is one sixtieth of an arc minute now committed is 1/60 of a degree a degree is 136 years of a before rotation is it tiny even a best telescopes cannot do this right now but you don't have to just do to two or three di worth of parallax if you just make a measurement and you just sit on the earth for six months the Sun will move you conveniently to astronomical units over this way and that gives you a lot more parallax and then so that is enough to start seeing some deviation in stuff but if you take a photograph of one portion of all twelve of them of the night sky and then you wait six months you take the same photograph most of the stars that are far away stay where they are but a few stars the nearby stars will shift and so there is enough parallax once you have some good telescopes you can see this and if you know the astronomical unit you can start computing distances to nearby stars and all stars that are recently nearby 100 light-years within a hundred light years all about 30 parsecs a parsec is exactly how much distance gives you a parallax of one arcsecond that is about the limit of what we can do with accurate telescopes to measure or distances of stars nearby and that's fortunate because it gives you a lot of stars about 10,000 stars how close enough that we can actually compute the distances and this is very important they have all this data because we need this data to climb the next rung this is a this is a very common thing in this business in order to keep going up we must keep collecting lots and lots of data I mean you've already seen some heroic efforts to collect data we'll see more okay now I'll get back to something I said before you remember I talked about I was darkest as his heliocentric theory oops not yet yeah so these parallax things were first on my Bessel in the 19th century think now okay yeah so when our stock is just supposed to have this entry model is the other ancient Greeks dismissed it because they said that this this theory cannot possibly work because if the earth moved around the Sun then we would see parallax that discover the stars in the night sky would be different in a different position in June as they would in January and we don't see that at least they didn't with it with their technology no telescopes and so the the reason that there's no way that they had essentially we could work unless the stars was a ridiculous distance away which of course copy because if they were that far away we wouldn't see them unless they were incredibly bright and big like as big as the Sun and you know that's that's ridiculous so it's a shame actually because I mean the Greeks did amazing things they had all this logic and mathematics and they did so many wonderful things but sometimes they still made mistakes which which is a real shame because they could have hit upon the truth much earlier but but they didn't okay because they didn't I mean it was inconceivable you know that the stars are about that far away but they are ok so let's new buy stars but nearby stars that only a very very small portion of the Milky Way our own galaxy so in this picture maybe just a little a little circle like that that's that's about 1 that's about how much 1 100 light years out there's a tiny tiny portion of the Milky Way parallax does not work for the rest of the galaxy but you can use all this data about nearby stars and extrapolate it to work out distances to two distant stars and the way you do this is a very clear ingenious method now that you have all these tools like spectroscopy you can look at a star new or file and you can work out its color alright size of blue some size I've read some stars in yellow in in the spectral sense and you can also take a photograph or use some electronic photo detector and you can measure how bright the star is some stars are brighter than others okay even the Greeks knew this but we can make precise measurements but that's only the apparent brightness that's how much the star how bright the star looks to us now a star could be bright either because it really is very very bright or because it's very close okay it could be a dim star very close and it looks bright like the Sun for example is not it's not one of the brightest stars out there but it's definitely much brighter than the other stars we see because it's much closer but if you have a new buy star we already know its distance okay you know how far away these stars are you know the apparent brightness and then you can just use the inverse square law and that tells you the absolute brightness you know how bright all the for all the nearby stars are out there you can use the distance measurements and the apparent brightness measurements and that tells you the exact so the brightness how bright these all these nearby stars actually are so yeah at least 10,000 stars you know how bright they are and you know the color and so to astronomers hertzsprung-russell plotted thousands of stars all the stars that they could find nearby we plotted their brightness and their color about a hundred years ago and they got this diagram which we now call the Vespa muscle diagram so they found that there was a relationship between color and coverage the absorb the x-axis and magnitude which was the the y-axis that red stars are less bright blue and white stars are more bright Sun is somewhere in the middle and so we have this this relationship between color and brightness and every time you have a relationship like this and some sort of curve this is this is great for astronomy you can use this curve to do things in fact you can reverse this curve and now you can measure distances to other stars stars very far away much further than parallax can reach and the way you do that is that if you have a stylus really far away you can again use fixed spectroscopy find out the color of the star use photography and you can find the the apparent brightness of the star okay these things you can always do as long as the stars not too too far away and then we can use the husband muscle diagram gives a relationship between color and absolute brightness I want you have absolute brightness at apparent brightness use the inverse square law and that gives you distance okay so you just combine all these measurements together and you can get the distance to stars are quite far away and this works quite well yeah it's called main sequence fitting and now it's it works up two stars out even like three hundred thousand light years away so not 100 light years but three hundred thousand light years away much much further and in particular enough to cover the the Milky Way which is only about a hundred thousand light-years in diameter so using main sequence fitting you can work out basically distances all throughout the galaxy beyond that what happens is that they're too faint the galaxy stars are too faint to measure them properly also they may be too close together to other stars and so it's hard to work out the brightness of any given star because other signs were in the way okay so that's the Milky Way so but that's only one of billions of galaxies out there we got to do the other galaxies do so this was first done by Henry just wanna leave it in the 1930 20th century so she was observing a special type of star a very bright big star called a Cepheid because they're first found in the constellation cepheus these stars are very big and unstable did they wobble they become very bright and then dim and bright and dim they oscillate in a very periodic way like maybe every 14 days for example they might oscillate in brightness and so some of them were in our galaxy and some were in distant galaxies the ones in our galaxy she was able to work out the distance to and she could work out their apparent blindness so she could work out how bright each each one of these separates were if they were in our own galaxy and she could also work out a period and so she collected lots lots of data and she plotted it this is actually her original plot in 1912 she plotted the period and and and apparent brightness of many many say feeds become a two types type one and type two cepheid's and she got she got a curve the bigger the the bigger and brighter the star this fe the longer the period and so there was this curve the data does not fit very well here because his ninth row of data okay with with with 21st century day we have a much better curve right now but even back then she could plot this curve and much like the husband muscle diagram you can once you have this curve we can use it to extract you can extrapolate it beyond our own galaxy and to to work out distances in other galaxies and okay and and this works because selfies are so bright that you can see them even in galaxies are very very far away so for example the the most distant sefie that we've ever seen through the hubble space telescope it's about 108 million light-years away okay in contrast the Milky Way is only a hundred thousand light years away so one thousand times further away in the time Mogi way we can still see Stephanie's I can still measure distances out to a huge distance or it looks huge but still not so huge compared with the size universe okay so 100 million light-years sounds pretty big until you realize the universe is a key about 76 billion light uses why so we're still not at the top of the ladder okay but this is another significant round okay you now get a whole bunch of galaxies that you can measure and that gives you lots of data for the next rung of the ladder okay so lots of all the galaxies within about 100 million light years you can measure distance too but the biases have in measurements it's not just surface that we use that's one of the that's the oldest way of measuring distances galaxies we now have many other ways of doing it too you can watch supernovas there's certain properties of supernovas that you can use to measure distances but you have to be very lucky because supernovas don't happen very often and you have to wait and you get lucky but if a supernova happens in the galaxies then you can use it to work out how far away that galaxy is and that works really quite well supernovas are one of the brightest objects we can in the universe where they happen and so we can see them out to really far distances the the most distant supernovae we've ever seen is about 11 billion light years away that's getting close actually to the limit of the observable universe is unlikely to see much further than that so you can sometimes you can see a lot further and you can use these alternate ways to measure distances to calibrate and confirm the Cepheid measurements - okay so this gives you a huge chunk of universe but not the whole thing okay so finally the top around at least we think as the top row is the whole universe okay so the way we measure distances throughout the rest of the universe is well we do it thanks to this great observation of Hubble so Hubble did the same thing that husband and Russell and Leavitt did before him he collected a lot of data and he tried to get a curve so he was measuring all these galaxies he knew now distances to many many galaxies but he saw that stomach galaxies were redder than there should be and some of the spectral lines were shifted to the red to the from what theory predicted and so there was his redshift for for certain galaxies and he found that the further away the galaxy was the ready the river was and relativity - told him that the reason why you're red shifted is because these galaxies are moving further away from you walking more precisely because the the the space between the galaxies is expanding actually more precisely but so but he plotted for you know hundreds of galaxies how far away they were the distance versus how far they were moving away from us and he got a nice linear law which is known as Hubble's law that the recessional velocity of of anything of our of any galaxy is proportional to the distance from the galaxy okay so this was again very important for physics if the universe was expanding then you run time backwards I mean the universe was contracting and so at some point it must all have been a why were in one location and this led it was is one of the major confirmations of the big bang theory which is now now universally accepted and confirm not just by it by these observations but by many other observations but this curve this Hubble's law gives us yet another way to measure distances but this time it works all the crossing universe if you have any galaxy out there you can measure its redshift using saccade spectroscopy you know the speed of light this will tell you how far is receding from you and Hubble's law then it gives you the distance and so you can make any object you actually see you can work out as distance using Hubble's law not always very accurately but over time we have we have managed to refine Hubble's law at the point where this gets fairly accurate so this is great I told you before that one of the difficulties of astronomy is that you only measure declamations our directions but not distances but once you have distances then you can start making two-dimensional three-dimensional maps of the universe because you now have all the polar coordinates you need so you can now what take a loss of data collect data of all the galaxies out there there measure the direction and the redshift which gives you their distance and that gives you in polar coordinates location of all the galaxies out there and you can start making maps and so people started doing this so this is this is called a to defeat to degree field it's a two degree slice of the universe every dot here is a galaxy and what people found when they assemble these maps is that the galaxies were not randomly strewn across space they tend to cluster in these little strands and we've given them some names this big big string on the left is called the great wall a stone the great wall so you've heard the Great Wall of China this is a bigger war this is a group of galaxies much much bigger it is in fact almost the bigger structure known in the universe yeah so we've discovered these very very large-scale structures and all this data has told us a lot about the shape of the universe and the Big Bang and we now have a fairly convincing model of how the universe was created there's still some parameters we have to we have to hammer out but it's given us the confidence to talk to to work out a lot about the universe even beyond that which we can directly see so we can only see the most distant object we've ever seen actually directly seen it's about 13 billion light years away that those it is a gamma-ray burst as it turns out but using the information we have now on the shape of the shape of the universe we can extrapolate to to way beyond that and we now know for instance that the universe even the part that we can't currently see they totally accept the universe must be at least 78 billion light-years in diameter the way we do this is that we don't look at the universe as it is today because we can't have that distance but we look at the very distant past if I can get very near the Big Bang and there's a residue of the Big Bang which is called the Cosmic Microwave Background and you can use that and the models of space-time that we have to get to get this number okay now the ancient Greeks were able to do all this using high school trigonometry no technology we need a lot more mathematics and a lot more technology to get these numbers that for each one of the latter you need much more stuff much more technology much more physics much more math much more statistics and so forth so there's there's there's lots muscle of math that comes in master general relativity some statistics and some regressions and probability fluid mechanics on the optics lots and lots of stuff but and lots of computations lots of mad hours of the computer hours but we can do it we also need cutting-edge technology lots and lots of telescopes including of course the important space-based telescopes like Hubble and w map which unfortunate has recently decommissioned about last week actually it's finished his mission I mean it got us it got us this map which was basically the primary mission instead of decaying orbit but anyway yeah we we need the latest technology latest mathematics and we're so not done we we don't yet have a complete picture of the universe and so astronomy is not not by no means a dead subject people are still very busily fixing you know making everything on that I said currently more accurate and then extrapolating it to even further and that's where we are today so to close I wanted to show you four maps which I think summarize quite quite clearly where we are today so the map on the top left this is called Tom Lee's map it's a very famous map in Kanagawa it was made by Tom Lee the same Tom Lee who tried to work out distances the planets in the first century AD and so he lived in Egypt but so he didn't he didn't travel to all these other places because he couldn't but what he could do is that first of all he bought every map that he could get his hands on but he could talk to merchants who would come say from India by the Silk Road or from from Britain or from North Africa and he would ask them okay how long did it take me to youthful for you to get say from China to India from India to to Persia whatever and from these time measurements he try to work out distances and try to put this all together as best he could to get a map of the world and this this is this was what he got and for first in technology it's a really good map you compare it to the actual map of the world and if you know you know you try to do better for first originality than this map here for a thousand years this was the best map of the world in existence it was copied and copied over and over again but okay so down here is the two-degree field which we seen before this is our Tommy's map this is our pretty much still our best Mapple universe we are currently and actually works very much on the same principles Tom Lee you know we look at every galaxy we ask how far away it is and we get which we put together as best we can and that is that is our map and we get these three strands now on the right here what this should be what I should be putting here isn't the actual map of the universe but I couldn't find that online so what I have here instead is a simulation of our fake universe in the computer where it's been hard to see with the lights I don't know if you can dim them a little bit but but every dot hidden in this simulation is a single galaxy and you just evolve it was for several billion years using gravity to to to move move these around and you find that gravity organizes these are when you run the simulation over billions of years the gallic is organized into strands and the strands look very much like the strategy we actually see from from the distance ladder and all our data and observations and that's a very good sign that means that that's a great confirmation that that all the computations that I got up in spoil it because they have matching the assimilations of the models of the universe that we are using now maybe in the future in you know in a few more decades we can actually replace that computer simulation of an actual map of the universe I'll be a fantastic achievement but that we're not there yet but hope but maybe in in this century we will see that okay thank you very much well thank you very much for assert out if anyone would like to ask some questions you can walk to the front of the aisles here and line up at the microphones now we won't have time for comments but if but just short questions so if you want to come down to the microphones we're ready we'll see who will be first here good we have a little more light come right down here your boyfriend yeah go ahead how long did it take you to get the field badge the fewest medal okay well I got in 2006 so but I wasn't working directly for it I care was very surprised we got a phone call actually I just woke my bat and baseball fan I prove a theorem and sometimes people notice these maps the stars like that one in the lower left and that's based on where we see the stars right hi yes have there been any attempt to adjust those maps to where they actually are based upon the fact that we see them as they were it's a little hard because we don't know fully the velocities of each of these things that's it's a week um you can do a little bit of this so for example I said that the diameter of the universe is about 78 billion light years and the way this was done was that we looked way into the past this picture here is a picture of space and time space is the vertical axis time is the horizontal axis the W net probe was able to look way into the past and see the cosmic background radiation from the distant past and that holds the shape will universe a long time ago and they extrapolated that forward to the present day and that's where they got this business 78 billion from the extrapolation so we do do some of that but it's hot yeah thank you just as you mentioned in your lecture Lomer Romer discovered that it takes aisle 2 days to to circle around Jupiter yes ah well it's take 28 days for the moon to Savannah to swim around our earth but that's that's before the instance relativity actually two days two days for Jupiter is not actually two days in the earth so so is that correct that's true relativity does give some corrections to it but but very small corrections the the gravity field of Jupiter is like you quite weak from the standards of relativity yeah so I think it only affects things by a second or two it doesn't really affect I mean and there are many many other measurement errors made so I think the effect of relativity was like you've quite minor because nowadays when we do compute things like the speed of light we do definitely take volatility into account even our GPS units you know we could weaken these GPS units are accurate women meters and that's because they adjust for generality otherwise they'd be had to be like miles off but yeah we do know how to adjust for that in second what is your creative and most enjoyable pure period in our life in Flinders University or Princeton or in UCLA feel free if you don't choose the last one we won't tell the DNA I'm obliged to say that me CLA has been the most productive period of life thank you thank you but is also the truth right so I'm told that the reception is now ready for us outside so let's give one more

Channel: UCLA

Views: 123,427

Rating: 4.9042783 out of 5

Keywords: uclachannel, UCLA, professor, terence, tao, AMS, Einstein, cosmic, distance, ladder, fields, medal, science

Id: 7ne0GArfeMs

Length: 76min 16sec (4576 seconds)

Published: Tue Oct 26 2010

Reddit Comments

[u/jhanschoo]

This talk seems like the best thing to base a few high school math word problems around!

[u/solvorn]

That suit is awful. Is it even his I wonder?