1 00:00:00,980 --> 00:00:05,000 Hi, welcome back. We ended last week with some success and 2 00:00:05,000 --> 00:00:09,691 a promise of more challenges. We realize that no matter how much we've 3 00:00:09,691 --> 00:00:14,248 added, we still have to make modifications to our little universe. 4 00:00:14,248 --> 00:00:19,608 We discovered the planets, we have to add them, we're going to see that they move, 5 00:00:19,608 --> 00:00:22,557 too. we're still pretty much on pace with 6 00:00:22,557 --> 00:00:27,784 Aristotle, but we're going to take that little thread, these extra moving parts, 7 00:00:27,784 --> 00:00:32,541 and we're going to pull on it. And by the time we're done unraveling it 8 00:00:32,541 --> 00:00:36,406 or following the way scientists unraveled it, we would have 9 00:00:36,406 --> 00:00:40,309 gone places that Aristotle could not even have begun to imagine. 10 00:00:40,309 --> 00:00:44,516 And we're going to spend much of this week actually away from astronomy, 11 00:00:44,516 --> 00:00:49,476 talking about fundamental physics. it'll be tough, but I promise you it will 12 00:00:49,476 --> 00:00:54,593 pay off. Because what we learn this week when we bring up the understanding we 13 00:00:54,593 --> 00:00:59,447 take away from it is going to help us understand everything we talk about 14 00:00:59,447 --> 00:01:03,514 throughout the class. And hopefully, way beyond this particular 15 00:01:03,514 --> 00:01:08,041 astronomy class. And, of course, the father of all of this is Sir Isaac 16 00:01:08,041 --> 00:01:11,124 Newton. So, let's start with the beginning. We'll 17 00:01:11,124 --> 00:01:16,438 start with the problem we found and Aristotle would have been perfectly aware 18 00:01:16,438 --> 00:01:22,300 that there are five planets, planets derived from the word for wandering. 19 00:01:22,300 --> 00:01:27,414 Because these were very bright stars, the five were Mercury, Venus, Mars, Jupiter 20 00:01:27,414 --> 00:01:31,127 and Saturn. And they, as we saw in our image follow 21 00:01:31,127 --> 00:01:36,101 paths very near to the ecliptic. We're beginning to sense a pattern here 22 00:01:36,101 --> 00:01:41,636 and they move along the ecliptic, so that means we need planetary spheres in 23 00:01:41,636 --> 00:01:46,050 addition to the lunar and solar and celestial spheres that we have. 24 00:01:46,050 --> 00:01:49,144 So far so good. They move around the ecliptic at 25 00:01:49,144 --> 00:01:52,626 different rates. Each planet moves at a different rate. 26 00:01:52,626 --> 00:01:57,654 So certainly, you can't put them all on a planetary sphere, you need a venution 27 00:01:57,654 --> 00:02:00,362 sphere separate from the Mercurean sphere. 28 00:02:00,362 --> 00:02:04,640 so, we're up to seven moving spheres and a celestial sphere. 29 00:02:04,640 --> 00:02:09,582 Their motion is not completely regular. Remember, even the sun doesn't completely 30 00:02:09,582 --> 00:02:14,153 move uniformly around the ecliptic. But, the deviations for the planets are 31 00:02:14,153 --> 00:02:17,366 far more serious. And, in fact, will end up telling us 32 00:02:17,366 --> 00:02:20,887 something deep. So, let's look closely at some of these 33 00:02:20,887 --> 00:02:25,829 deviations from regular motion that we see with planets but do not see with the 34 00:02:25,829 --> 00:02:31,143 sun and the moon. And the simulation, we're looking at a 35 00:02:31,143 --> 00:02:37,215 patch of sky near the constellation Gemini and Leo just to the east of Orion 36 00:02:37,215 --> 00:02:42,300 that we were looking at before. And, we see the ecliptic and we're going 37 00:02:42,300 --> 00:02:45,867 to focus on the behavior of the planet Mars. 38 00:02:45,867 --> 00:02:51,103 And this simulation starts little over a year ago, in October 2011. 39 00:02:51,103 --> 00:02:57,155 And its set up so that time advances by sidereal day so that the stars repeat 40 00:02:57,155 --> 00:03:00,205 themselves. And we that Mars, like all the objects 41 00:03:00,205 --> 00:03:04,811 that we've been encountered so far, is moving slowly to the east along the 42 00:03:04,811 --> 00:03:09,231 ecliptic. We see the moon here moving much faster to the east along the 43 00:03:09,231 --> 00:03:12,156 ecliptic. And as time rolls by, the moon will 44 00:03:12,156 --> 00:03:15,953 quickly overtake Mars. But then, as the moon overtakes Mars, in 45 00:03:15,953 --> 00:03:18,630 December last year, something weird happens. 46 00:03:18,630 --> 00:03:22,141 Mars, in fact, stopped it's eastward progress. 47 00:03:22,141 --> 00:03:28,126 And then, for about three months, Mars was actually moving west along the 48 00:03:28,126 --> 00:03:31,956 ecliptic. And then, it recanted and continued its 49 00:03:31,956 --> 00:03:38,340 normal eastward motion. This retrograde motion of planets happens 50 00:03:38,340 --> 00:03:43,418 to most planets is a slightly more involved engineering challenge than what 51 00:03:43,418 --> 00:03:45,190 we've met so far. Challenging? 52 00:03:45,190 --> 00:03:48,393 For sure. It's not easy to construct a model that 53 00:03:48,393 --> 00:03:53,231 leads to the apparent motions of the planets in the sky as we see them. 54 00:03:53,231 --> 00:03:58,410 But, very brilliant people were working on this and they were very competent 55 00:03:58,410 --> 00:04:02,022 geometers. And they came up with the solution this 56 00:04:02,022 --> 00:04:07,270 is first the germ of the idea first proposed in an 150 BC by Aristocles. 57 00:04:07,270 --> 00:04:14,218 And, the basic idea is that to understand how Mars moves, you imagine that there is 58 00:04:14,218 --> 00:04:17,776 a circle around Earth called the deferent. 59 00:04:17,776 --> 00:04:20,689 And essentially, Mars moves along this 60 00:04:20,689 --> 00:04:23,759 deferent, but it's not on the deferent itself. 61 00:04:23,759 --> 00:04:27,375 On the deferent rolls an epicycle which turns, itself. 62 00:04:27,375 --> 00:04:33,364 and Mars is at a point, a fixed point, on this epicycle. So that as the animation 63 00:04:33,364 --> 00:04:37,240 runs, you see that the motion of Mars rather 64 00:04:37,240 --> 00:04:43,041 resembles what we saw in the picture, in the simulation. Mars is sometimes moving 65 00:04:43,041 --> 00:04:46,637 faster, sometimes slower and sometimes even retrograde. 66 00:04:46,637 --> 00:04:51,498 And so, along these principles by combining a deferent with an epicycle, 67 00:04:51,498 --> 00:04:55,559 one could describe the motions of the planets as we see them. 68 00:04:55,559 --> 00:05:00,820 Explaining how it is that sometimes a planet moves faster and sometimes slower 69 00:05:00,820 --> 00:05:06,346 is one thing what we were after or what they were after was an actually precise 70 00:05:06,346 --> 00:05:11,660 matching of the predictions to the data. And this leads to a rather elaborate 71 00:05:11,660 --> 00:05:17,225 model that Ptolemy comes out with 300 years after Hipparchus floated the idea. 72 00:05:17,225 --> 00:05:22,327 For example, in order to understand why the sun does not move at a fixed rate 73 00:05:22,327 --> 00:05:27,495 around the celestial sphere the solar sphere is a little bit off center. 74 00:05:27,495 --> 00:05:32,730 Its center is not exactly on the Earth and there are various such elaborations. 75 00:05:32,730 --> 00:05:36,101 The end result is that the model is extremely successful. 76 00:05:36,101 --> 00:05:40,045 It makes good predictions. Far into the future about, exactly where 77 00:05:40,045 --> 00:05:44,616 in the sky, about where on the celestial sphere, you'll find which planets. 78 00:05:44,616 --> 00:05:49,428 And because of these slight deviations of getting the precise agreement, you now 79 00:05:49,428 --> 00:05:52,556 have an actual ordering of the sizes of the spheres. 80 00:05:52,556 --> 00:05:57,247 the moon is on the inner most sphere of Mercury, Venus, the Sun, Mars, Jupiter, 81 00:05:57,247 --> 00:06:01,096 Saturn, and then outside the celestial sphere of the fixed stars. 82 00:06:01,096 --> 00:06:05,728 And an example of how this process works out would be the planet Venus. 83 00:06:05,728 --> 00:06:10,419 Venus is a planet that as we observe it in the sky, is never very far from the 84 00:06:10,419 --> 00:06:12,906 Sun. It's sometimes a little but the west of 85 00:06:12,906 --> 00:06:15,825 the Sun, and then it's a little bit ahead of the Sun. 86 00:06:15,825 --> 00:06:20,318 And so, we see it rising just before Sun rise, some times it's to the east of the 87 00:06:20,318 --> 00:06:23,294 Sun then we see it in the west in the evening sky. 88 00:06:23,294 --> 00:06:28,680 So then, it's the evening star or the morning star the way that's arranged in a 89 00:06:28,680 --> 00:06:33,655 Ptolemaic model is that the deferent for Venus moves at about the same rate, as 90 00:06:33,655 --> 00:06:36,721 the Sun does. In other words, the deferent orbits the 91 00:06:36,721 --> 00:06:39,961 Earth once a year. And then, the epicycle rotation around 92 00:06:39,961 --> 00:06:44,300 that, which is not too large, accounts for the periodic change in elongation. 93 00:06:44,300 --> 00:06:48,771 In other words, the distance in right ascension between the Sun and the planet, 94 00:06:48,771 --> 00:06:53,179 and Ptolemy builds in tilts, and the, the model as I said, is somewhat elaborate. 95 00:06:53,179 --> 00:06:56,739 But at the end of the day, it's a successful mathematical model. 96 00:06:56,739 --> 00:07:01,316 And the reason I'm bringing this up is because it's often presented as the wrong 97 00:07:01,316 --> 00:07:03,519 model. It is not the way we think today. 98 00:07:03,519 --> 00:07:08,617 But, it is certainly wonderfully successful mathematical model of how 99 00:07:08,617 --> 00:07:11,729 things work. what was its competition? 100 00:07:11,729 --> 00:07:16,630 Well, from early on, there was an, an idea first attributed 101 00:07:16,630 --> 00:07:21,373 definitively to our star clusters as early as 270 BC which is that the 102 00:07:21,373 --> 00:07:26,564 perceived motion of the planets along the celestial sphere follows from the fact 103 00:07:26,564 --> 00:07:31,885 that both the planets and the Earth orbit the Sun. And what we see is basically the 104 00:07:31,885 --> 00:07:35,666 result of the relative motion of the planets and the earth. 105 00:07:35,666 --> 00:07:40,922 in other words, the solar system is now looking kind of like a race track along 106 00:07:40,922 --> 00:07:44,896 which the planets race. One of the principles of this model is 107 00:07:44,896 --> 00:07:49,876 that the planets closer to the sun move around the Sun, orbit the Sun, in shorter 108 00:07:49,876 --> 00:07:53,099 periods, faster. And so, the inside lane always wins. 109 00:07:53,099 --> 00:07:57,356 And as a result, when one planet is passing the other, you get the same 110 00:07:57,356 --> 00:08:02,343 effect when you're looking at the planet that you get when you're looking out the 111 00:08:02,343 --> 00:08:06,904 window of your car and it looks like relative to the distant mountains, the 112 00:08:06,904 --> 00:08:11,769 trees near the road are moving backwards. Trees are not going anywhere, it's just 113 00:08:11,769 --> 00:08:15,600 that your line of sight is changing. We'll see that in a second. 114 00:08:15,600 --> 00:08:19,310 So, this has a nice, simple explanation for retro-grade motion. 115 00:08:19,310 --> 00:08:23,245 Fine. the model that Aristocles proposed was 116 00:08:23,245 --> 00:08:25,334 primitive. It needed elaboration, 117 00:08:25,334 --> 00:08:29,706 and nobody paid it enough attention to bring it up to the level of 118 00:08:29,706 --> 00:08:35,448 sophistication that it needed until Copernicus in the 16th century brings out 119 00:08:35,448 --> 00:08:40,538 Heliocentric model that is as detailed and as predictive and as successful as 120 00:08:40,538 --> 00:08:44,584 the Ptolemaic model. And the planets move along circles and 121 00:08:44,584 --> 00:08:48,630 they move at uniform speed. And the way that this explains the 122 00:08:48,630 --> 00:08:53,263 non-uniform motion along the sky is because of this aspect of relative 123 00:08:53,263 --> 00:08:56,100 motion. We can do a nice calculation that might 124 00:08:56,100 --> 00:08:58,712 help us get a handle on this racetrack story. 125 00:08:58,712 --> 00:09:02,485 So, imagine that we have planets and they're all orbiting the Sun. 126 00:09:02,485 --> 00:09:07,499 take some planet and let P, for period, be the time it takes that planet to 127 00:09:07,499 --> 00:09:12,221 complete one circuit around the Sun because its one complete circuit relative 128 00:09:12,221 --> 00:09:15,671 to the stars, we'll call this the sidereal period. Earth itself is a 129 00:09:15,671 --> 00:09:19,969 planet, it has its own period. we'll distinguish that with the name E 130 00:09:19,969 --> 00:09:23,662 for earth's period. That is exactly the sidereal year that we 131 00:09:23,662 --> 00:09:28,444 discussed, the time it takes Earth to orbit around the Sun once with respect to 132 00:09:28,444 --> 00:09:31,336 the stars. As with the moon, we'll define a synodic 133 00:09:31,336 --> 00:09:35,800 period which we will call S, which is the time between consecutive occurrences of 134 00:09:35,800 --> 00:09:39,883 some particular alignment between the Earth, the planet in question, and the 135 00:09:39,883 --> 00:09:43,912 Sun, just as with the Lunar phases. And we are going to try to relate these 136 00:09:43,912 --> 00:09:48,430 mathematically, something we sorted it for the moon but will be instructed to do 137 00:09:48,430 --> 00:09:52,513 it here using this race track model. This is another one of those wonderful 138 00:09:52,513 --> 00:09:55,290 animations from the University of Nebraska-Lincoln. 139 00:09:55,290 --> 00:09:59,510 It's a very elaborate animation, you can do lots of things with it. 140 00:09:59,510 --> 00:10:03,833 I'm going to do something very simple. I want to understand the relation between 141 00:10:03,833 --> 00:10:07,060 synodic and sidereal periods. So, what do we have here? 142 00:10:07,060 --> 00:10:09,770 Well, we have two planets. We have a blue planet, 143 00:10:09,770 --> 00:10:13,948 that's this one. And, I'm going to assume that it has a 144 00:10:13,948 --> 00:10:18,436 siderial period P1. That means, P1 is the time it takes the 145 00:10:18,436 --> 00:10:23,233 blue planet to a 360 degree circumlocution of the Sun. 146 00:10:23,233 --> 00:10:29,501 I have another planet, I will call it the green planet, or number two because I do 147 00:10:29,501 --> 00:10:34,638 not have a working gray marker. And this one takes a period P2 to 148 00:10:34,638 --> 00:10:41,654 circumnavigate the Sun to complete a full sidereal orbit of 360 degrees around the 149 00:10:41,654 --> 00:10:46,980 sun in right ascension. And because of our assumptions, P1 being 150 00:10:46,980 --> 00:10:51,967 the inner planet takes a shorter time to go around the sun. 151 00:10:51,967 --> 00:10:54,250 So, P1 is less than P2. Okay. 152 00:10:54,250 --> 00:10:59,102 Suppose that I know these or they are what they are, what I want to know is the 153 00:10:59,102 --> 00:11:02,560 following thing, I'm going to understand the synodic period. 154 00:11:02,560 --> 00:11:06,382 So, what is the synodic period? That's the time between repeated 155 00:11:06,382 --> 00:11:08,929 alignments of the two planets and the Sun. 156 00:11:08,929 --> 00:11:12,144 So, for example, I am starting my animation right here, 157 00:11:12,144 --> 00:11:16,997 h, very nicely and the situation where the three bodies are aligned, they are in 158 00:11:16,997 --> 00:11:19,909 a line. The synodic period will be the time that 159 00:11:19,909 --> 00:11:22,700 needs to pass before this pattern is repeated. 160 00:11:22,700 --> 00:11:27,955 And to see how long that takes, all we need to do is set the runners off and let 161 00:11:27,955 --> 00:11:31,058 them go. So, I'm going to press the button and off 162 00:11:31,058 --> 00:11:35,364 they're running. When you stop and see what's going on, we were write, 163 00:11:35,364 --> 00:11:40,050 the blue planet is much faster than the green planet and is outrunning it. 164 00:11:40,050 --> 00:11:44,990 And as I let them run indeed within a short time, the blue planet is back home. 165 00:11:44,990 --> 00:11:48,468 What does that mean? Well, this means the time that we just 166 00:11:48,468 --> 00:11:53,086 watched was precisely this period P1. The blue planet completed a 360 degrees 167 00:11:53,086 --> 00:11:56,865 sidereal rotation. Of course, the green one is nowhere near home. 168 00:11:56,865 --> 00:12:01,783 To keep track of what has gone on, let me give these guys trails, imagine they were 169 00:12:01,783 --> 00:12:05,921 trailing smoke or something. This is what the blue planet has done so 170 00:12:05,921 --> 00:12:09,160 far, and this is what the green planet has done so far. 171 00:12:09,160 --> 00:12:13,230 but we're not back, as usual, we're not back to alignment. 172 00:12:13,230 --> 00:12:17,987 Because in the time that it took the blue planet to get back where he was, the 173 00:12:17,987 --> 00:12:22,135 green planet has run away. And as we're by now familiar to reproduce 174 00:12:22,135 --> 00:12:25,490 the synodic alignment, we need to keep going until, boom. 175 00:12:25,490 --> 00:12:32,061 There we go. Notice, we're back in alignment. So, 176 00:12:32,061 --> 00:12:37,794 this, the time that passed until this happened, from here to here is the 177 00:12:37,794 --> 00:12:40,701 synodic period. How do we compute it? 178 00:12:40,701 --> 00:12:47,160 Well, let's finish our star trails here. So, this is a very ugly trail, but this 179 00:12:47,160 --> 00:12:53,377 is how far the green planet has gone, and this is how far the blue planet has gone. 180 00:12:53,377 --> 00:12:57,764 A full circle. And then ooh, and then the same amount. 181 00:12:57,764 --> 00:13:01,175 Why? Because they started in the same place, 182 00:13:01,175 --> 00:13:06,807 they ended in the same place, but the blue planet has done an extra lap. 183 00:13:06,807 --> 00:13:12,122 This has taken a period S. Well, how much of a circle is this little 184 00:13:12,122 --> 00:13:14,820 arc? We've played that game before. 185 00:13:14,820 --> 00:13:18,072 This little arc, I will write it in green. 186 00:13:18,072 --> 00:13:23,466 The green arc is simply the fraction that S is out of the period P2. 187 00:13:23,466 --> 00:13:27,830 Because a time S has passed and it takes a longer time, 188 00:13:27,830 --> 00:13:32,720 P2 for the green planet to complete a full circumlocution of the sun. 189 00:13:32,720 --> 00:13:38,727 Now, because they started at the same place and ended at the same place, one 190 00:13:38,727 --> 00:13:45,274 could be confused and imagine that this is the same as the fraction of the circle 191 00:13:45,274 --> 00:13:50,819 that the blue planet has done because, after all, it is the same fraction. 192 00:13:50,819 --> 00:13:56,750 However, the blue planet has gone through this arc but also a complete circle. 193 00:13:56,750 --> 00:14:01,910 So, in fact, this number is much bigger than this number by how much? 194 00:14:01,910 --> 00:14:04,760 By a full circle. In other words, this, 195 00:14:04,760 --> 00:14:10,184 how much the green planet has done is how much the blue planet has done minus one. 196 00:14:10,184 --> 00:14:15,214 And that is the equation we wanted. The three variables, synodic period, time 197 00:14:15,214 --> 00:14:19,285 between alignments, sidereal periods for the two planets are 198 00:14:19,285 --> 00:14:23,492 related by this relation. And then, if you know any two of them, 199 00:14:23,492 --> 00:14:26,400 you can get the third. Hooray. 200 00:14:26,400 --> 00:14:30,666 The actual resolution of the question, which of these two models describe the 201 00:14:30,666 --> 00:14:35,154 universe? Was going to have to wait for better observations within the technology 202 00:14:35,154 --> 00:14:38,866 they would definitively settled this it the question. Along the way, 203 00:14:38,866 --> 00:14:43,076 understanding how the, the heliocentric model will turn out works out led to 204 00:14:43,076 --> 00:14:47,287 insights that are much deeper than anything that you've been discussing now 205 00:14:47,287 --> 00:14:51,553 and far more broad in their implications than just understanding the stars. 206 00:14:51,553 --> 00:14:55,930 And it is following that trier that is going to occupy us for the rest of this 207 00:14:55,930 --> 00:14:56,208 week.