WEBVTT 00:00.780 --> 00:04.760 Prof: Welcome back to more of Astronomy 160. 00:04.760 --> 00:07.410 The syllabus, if you haven't got it yet, 00:07.413 --> 00:09.253 is on the classes server. 00:09.250 --> 00:12.610 We have extra copies but I have to say I don't know where they 00:12.605 --> 00:15.215 are, and in any case, everything that you need to 00:15.220 --> 00:18.170 know is on the classes server and there's actually more than 00:18.170 --> 00:20.420 is on the printed syllabus there anywhere. 00:20.420 --> 00:24.200 Let me say a couple things; we're going to stick to the 00:24.197 --> 00:28.207 rule that there should be no science majors taking this 00:28.211 --> 00:28.881 class. 00:28.880 --> 00:32.110 And from the looks of the enrollment, there's going to be 00:32.107 --> 00:34.237 no problem for juniors and seniors. 00:34.240 --> 00:37.050 I think we've got plenty of sections to accommodate everyone 00:37.045 --> 00:39.275 in here, provided you're not a science major. 00:39.280 --> 00:41.990 But the way it's going to work in terms of signing up for the 00:41.991 --> 00:43.841 class, I think, as I understand it, 00:43.841 --> 00:45.721 is as follows: You need a section, 00:45.719 --> 00:48.109 sections are mandatory for this course. 00:48.110 --> 00:51.250 So, you have to go through the online section sign up process. 00:51.250 --> 00:54.490 That'll start on Monday but it'll only start for freshman 00:54.488 --> 00:56.048 and sophomores on Monday. 00:56.050 --> 01:00.260 Juniors and seniors will be able to sign up for the course 01:00.258 --> 01:04.538 and sign for sections on Tuesday, and so please do that. 01:04.540 --> 01:07.810 Sections will start--sections are going to be on Mondays, 01:07.810 --> 01:11.080 so they obviously won't start next week they'll start the 01:11.080 --> 01:12.190 following week. 01:12.189 --> 01:15.739 I have no doubt looking around the room that we're going to be 01:15.740 --> 01:19.410 able to accommodate juniors and seniors if they want to take the 01:19.407 --> 01:19.987 class. 01:19.989 --> 01:23.149 All right, other things, yes, the first problem set is 01:23.149 --> 01:24.519 assigned as of today. 01:24.519 --> 01:27.969 When I get back to my office I'll stick that on the classes 01:27.968 --> 01:30.468 server as well, and you can pick it up from 01:30.465 --> 01:31.055 there. 01:31.060 --> 01:34.690 It's due one week from today. 01:34.690 --> 01:38.230 The way I like to do this, I like to have problem sets due 01:38.228 --> 01:42.018 at the start of class because it bugs me to look out and watch 01:42.016 --> 01:44.806 people copying each other's problem sets. 01:44.810 --> 01:47.770 So, you have to hand it in at the beginning of class otherwise 01:47.768 --> 01:48.348 it's late. 01:48.349 --> 01:50.979 On the classes server, there's a whole little thing 01:50.977 --> 01:53.497 about lateness policy which you ought to read. 01:53.500 --> 01:58.370 And also something--let me make a couple general remarks about 01:58.371 --> 01:59.571 problem sets. 01:59.569 --> 02:04.169 The purpose of problem sets is different from the purpose of a 02:04.168 --> 02:05.448 take-home test. 02:05.450 --> 02:10.170 This is not something where the major purpose is to evaluate how 02:10.171 --> 02:11.371 much you know. 02:11.370 --> 02:15.000 The point of problem sets is to get you to actually do those 02:14.997 --> 02:18.497 particular problems because that's how one learns stuff in 02:18.502 --> 02:20.472 this kind of subject matter. 02:20.470 --> 02:22.200 It's kind of like doing the reading; 02:22.199 --> 02:24.269 if you don't do it, you're just going to have 02:24.269 --> 02:24.739 trouble. 02:24.740 --> 02:28.260 And so, the reason it's a substantial fraction of your 02:28.264 --> 02:31.194 grade is just to make sure that you do it. 02:31.189 --> 02:36.269 So, this has implications for working in groups. 02:36.270 --> 02:38.990 Working in groups is strongly encouraged. 02:38.990 --> 02:40.550 Please do that. 02:40.550 --> 02:43.090 It's a very good thing, make friends with the other 02:43.090 --> 02:44.970 people in the class, work together, 02:44.970 --> 02:47.640 talk to each other, start early, 02:47.640 --> 02:51.430 and make use of each other's intelligence. 02:51.430 --> 02:54.720 However, at the same time, we want to make sure that you 02:54.722 --> 02:55.982 actually do things. 02:55.979 --> 02:59.039 So, we want the work you hand in to be your own. 02:59.040 --> 03:00.770 This sounds contradictory. 03:00.770 --> 03:02.300 On the one hand, I'm tell you to work in groups; 03:02.300 --> 03:03.490 on the other hand, I'm telling you, 03:03.485 --> 03:04.735 you have to hand in your own work. 03:04.740 --> 03:05.780 What does that mean? 03:05.780 --> 03:08.730 So, as is written down on the classes server, 03:08.725 --> 03:12.735 the way we try and work this is when you are actually writing 03:12.741 --> 03:15.621 down the thing you're going to hand in, 03:15.620 --> 03:17.270 do that alone. 03:17.270 --> 03:19.380 So, if you work in a group, figure out what you're doing, 03:19.383 --> 03:22.223 get everything all set up, then split up and write down 03:22.217 --> 03:26.027 the answer, because the process of actually writing the thing 03:26.026 --> 03:29.766 down yourself will make sure that you've actually understood 03:29.772 --> 03:31.362 what you were doing. 03:31.360 --> 03:32.890 We'll check on this. 03:32.889 --> 03:35.199 We'll look for identical wording or identical 03:35.197 --> 03:38.127 multiplication mistakes and so forth, and so we'll keep a 03:38.134 --> 03:39.764 little bit of an eye on it. 03:39.759 --> 03:42.089 I do encourage you to work in groups. 03:42.090 --> 03:44.850 There's other ways of getting help if you need it. 03:44.849 --> 03:46.969 The primary source of help are the sections, 03:46.968 --> 03:48.988 but of course, those won't be meeting this 03:48.988 --> 03:51.548 week so not before the first problem set is due. 03:51.550 --> 03:55.160 Office hours; I'll be in Starbucks Monday 03:55.160 --> 03:58.920 from 9:30 to 11:00; please come and say hello, 03:58.919 --> 04:02.529 and ask any questions you might have. 04:02.530 --> 04:05.880 The teaching fellows are going to have office hours on 04:05.878 --> 04:08.228 Wednesdays; conveniently located the day 04:08.230 --> 04:10.800 before the problem sets are due you'll notice. 04:10.800 --> 04:15.550 And let me strongly recommend, don't send us email with 04:15.551 --> 04:17.401 questions about it. 04:17.399 --> 04:20.869 Send things into the forum on the classes server. 04:20.870 --> 04:24.370 That way--and look first on the forum to see if somebody has 04:24.366 --> 04:25.666 sent your questions. 04:25.670 --> 04:27.610 This works really well. 04:27.610 --> 04:29.510 We'll keep an eye on that all the time. 04:29.509 --> 04:32.709 We'll be answering it--not all the time, not probably after 04:32.706 --> 04:34.686 around 8:00 at night on Wednesday. 04:34.690 --> 04:37.310 If you have a question at 2:00 in the morning the day before 04:37.314 --> 04:39.364 something is due you're kind of on your own. 04:39.360 --> 04:40.350 Yes? 04:40.350 --> 04:42.900 Student: Is that Monday 9:30 to 11:00 in the morning or 04:42.895 --> 04:43.475 the evening? 04:43.480 --> 04:45.120 Prof: Excellent question, thank you so much. 04:45.120 --> 04:49.560 Morning, I'm afraid, sorry. 04:49.560 --> 04:53.030 Yeah, I'm not as young as I used to be. 04:53.030 --> 04:56.150 04:56.149 --> 04:59.529 Let me say, if you do want to meet at some ghastly hour of the 04:59.526 --> 05:01.626 evening or something, send me email. 05:01.629 --> 05:04.669 We can find a time, same with the teaching 05:04.674 --> 05:05.644 assistants. 05:05.639 --> 05:08.159 Send stuff into the classes server at all times, 05:08.156 --> 05:11.206 we just don't guarantee to look at it after 8:00 the night 05:11.209 --> 05:12.279 before it's due. 05:12.279 --> 05:15.659 Start the problem sets early, that way if you have a question 05:15.662 --> 05:16.792 you can answer it. 05:16.790 --> 05:19.770 Not only that, if you even just look at the 05:19.772 --> 05:23.962 problem set, your brain will work on it in your subconscious 05:23.963 --> 05:28.443 and you won't freak out quite so much when you get past midnight 05:28.437 --> 05:30.707 the night before it's due. 05:30.709 --> 05:34.469 Let's see, in general, I'm going to try and make it so 05:34.471 --> 05:38.871 that everything you need to know to do a particular problem set 05:38.871 --> 05:43.201 has been covered in class by the day the problem set is handed 05:43.200 --> 05:43.910 out. 05:43.910 --> 05:46.090 That would be today for this one. 05:46.089 --> 05:48.719 In the very first week this week I'm not quite going to make 05:48.722 --> 05:51.632 it and so there's one problem, it's noted on the problem set, 05:51.633 --> 05:54.703 there's one problem that you'll need stuff that I'll talk about 05:54.697 --> 05:55.387 on Tuesday. 05:55.389 --> 05:59.539 There are going to be a bunch of help sheets for various 05:59.540 --> 06:04.370 topics that are going to also be linked to the classes server. 06:04.370 --> 06:07.480 By all means, take a look at the problem set 06:07.478 --> 06:11.018 and start thinking about it as soon as you can. 06:11.020 --> 06:12.220 Okay. 06:12.220 --> 06:16.020 06:16.019 --> 06:21.129 Let me remind you what we had started to talk about. 06:21.129 --> 06:26.269 The class is organized into three fairly specific topics. 06:26.269 --> 06:28.959 The first of which is extrasolar planets. 06:28.959 --> 06:31.229 Planets around stars other than the Sun. 06:31.230 --> 06:33.970 Exoplanets, so-called. 06:33.970 --> 06:35.710 That's our topic. 06:35.709 --> 06:38.769 One of the things that I pointed out last time is that, 06:38.768 --> 06:41.598 surprisingly enough, very--until ten years ago none 06:41.600 --> 06:42.960 of these were known. 06:42.959 --> 06:46.209 And it's only a very recent development that there's any 06:46.208 --> 06:48.688 actual evidence that these things exist. 06:48.690 --> 06:52.240 So, one question you might ask is why are these things so hard 06:52.238 --> 06:52.818 to find? 06:52.819 --> 06:55.359 The science fiction folks seem to have no trouble; 06:55.360 --> 06:58.100 they just sort of go around in their spaceships and find these 06:58.101 --> 06:59.361 things all over the place. 06:59.360 --> 07:04.260 And to consider that question let me show you a picture. 07:04.260 --> 07:07.800 Here's a picture of a star. 07:07.800 --> 07:10.320 This is the star Sirius. 07:10.320 --> 07:11.370 It's the brightest. 07:11.370 --> 07:13.200 It's a blowup obviously of a photograph plus a little 07:13.197 --> 07:14.777 Photoshopped [computer program to edit photos] 07:14.778 --> 07:15.128 arrow. 07:15.129 --> 07:17.459 That's not a celestial object [laughter]. 07:17.460 --> 07:20.870 07:20.870 --> 07:24.210 So, this is a blowup of a photograph of the star Sirius. 07:24.209 --> 07:27.959 Sirius is the brightest star in the sky to the--easily visible 07:27.959 --> 07:29.249 with the naked eye. 07:29.250 --> 07:31.890 In fact, as I say, the brightest star. 07:31.890 --> 07:33.430 It's one of the closest stars. 07:33.430 --> 07:36.930 It's a little bit brighter than the Sun intrinsically, 07:36.930 --> 07:39.440 but it's ten light years away or so. 07:39.440 --> 07:42.130 And let me comment on a couple features of this picture. 07:42.129 --> 07:44.959 First of all, you can see the star covers a 07:44.961 --> 07:48.131 fair amount of area on this blown up picture, 07:48.129 --> 07:51.999 but that has nothing to do with the actual size or shape of the 07:51.996 --> 07:52.866 star itself. 07:52.870 --> 07:56.760 It's not like it's a big round ball with spikes coming off it. 07:56.760 --> 07:58.490 That's not what's happening. 07:58.490 --> 08:02.690 In fact, if you had perfect optics and perfect vision, 08:02.691 --> 08:07.211 this star would be a point basically so small you couldn't 08:07.209 --> 08:10.379 resolve it right in the middle here. 08:10.379 --> 08:13.979 That's the physical extent of the star at this scale. 08:13.980 --> 08:15.490 It's something like this [pointing to photo], 08:15.490 --> 08:17.000 and you wouldn't be able to see it at all. 08:17.000 --> 08:19.340 The reason it's extended is for two reasons; 08:19.339 --> 08:21.909 first of all, the atmosphere distorts--this 08:21.906 --> 08:25.446 was a ground-based picture, the atmosphere distorts how you 08:25.451 --> 08:27.341 see the star, and in particular, 08:27.337 --> 08:30.017 it's like looking up from the bottom of a swimming pool. 08:30.019 --> 08:33.059 There's distortion in the atmosphere and it makes the 08:33.062 --> 08:34.702 stars seem to jump around. 08:34.700 --> 08:37.400 And so if you take a long photographic exposure of 08:37.401 --> 08:40.271 something the star has been jumping around during the 08:40.269 --> 08:42.529 exposure and it smears out the image. 08:42.529 --> 08:46.349 Plus also, the optics of the telescope aren't perfect, 08:46.348 --> 08:50.308 and these spiky things here are due to the optics of the 08:50.311 --> 08:51.321 telescope. 08:51.320 --> 08:54.860 This sort of--there are all these spikes and that has to do 08:54.860 --> 08:58.400 with how one of the mirrors of the telescope is held up. 08:58.399 --> 09:02.019 So, the combination of optics and atmosphere sort of spreads 09:02.016 --> 09:04.036 out the light by a whole bunch. 09:04.039 --> 09:08.949 This is always the case even when you're doing observations 09:08.949 --> 09:10.049 from space. 09:10.049 --> 09:14.999 Now, the little arrow here is pointing at a little bump off in 09:15.001 --> 09:19.951 the side, but you might not have thought anything more of than 09:19.953 --> 09:22.473 you did about these spikes. 09:22.470 --> 09:24.530 But in fact, that's not a property of 09:24.525 --> 09:26.525 optics; there's actually something 09:26.529 --> 09:26.929 there. 09:26.929 --> 09:30.169 That's an object in orbit around Sirius. 09:30.169 --> 09:33.529 And you might think that this is the way you would go about 09:33.530 --> 09:35.790 finding planets orbiting other stars. 09:35.790 --> 09:37.000 It would look kind of like this. 09:37.000 --> 09:39.170 Here's a great big star; there's a planet. 09:39.169 --> 09:42.269 And if you watch this thing over a course of many years, 09:42.270 --> 09:45.540 you actually see the position of this point move around. 09:45.540 --> 09:46.710 It's because it's in orbit. 09:46.710 --> 09:49.660 It's in about a forty-year orbit. 09:49.660 --> 09:51.720 Here's the thing. 09:51.720 --> 09:53.460 This isn't a planet. 09:53.460 --> 09:55.100 It's another star. 09:55.100 --> 09:56.710 This is another star somewhat fainter than this [pointing to 09:56.711 --> 09:57.341 the photo of Sirius]. 09:57.340 --> 10:00.950 A planet would be 10,000 times fainter than this thing, 10:00.952 --> 10:04.902 but probably in about the same relative position to this. 10:04.899 --> 10:09.259 Now, picture all this light, all this mess from this star, 10:09.264 --> 10:14.094 and try to find something right here that's 10,000 times fainter 10:14.088 --> 10:15.158 than this. 10:15.160 --> 10:16.830 That's the problem. 10:16.830 --> 10:21.760 The problem is that planets are relatively faint things. 10:21.759 --> 10:24.189 That, in itself isn't catastrophic. 10:24.190 --> 10:27.010 We have big telescopes; we can see pretty faint things. 10:27.009 --> 10:30.559 It's just they are faint things that are really near to things 10:30.561 --> 10:32.691 that are bright, and so basically, 10:32.691 --> 10:36.061 the problem with finding planets around other stars is 10:36.062 --> 10:40.262 the same as trying to find stars in the sky during the daytime. 10:40.259 --> 10:42.599 Why can't you see the stars during the daytime? 10:42.600 --> 10:45.960 Because the background of the sky is much, much brighter than 10:45.955 --> 10:46.845 the stars are. 10:46.850 --> 10:50.850 Similarly, if you had a little planet here you wouldn't be able 10:50.849 --> 10:54.529 to distinguish it from the atmospheric distortions and the 10:54.525 --> 10:58.265 optical distortions caused by the telescope that takes this 10:58.266 --> 10:59.166 picture. 10:59.169 --> 11:02.679 So, the problem is you've got stuff that's faint near things 11:02.679 --> 11:05.549 that are bright, and the bright things have all 11:05.547 --> 11:09.017 kinds of distortions on them anyway, and so you wouldn't be 11:09.023 --> 11:11.903 able to figure out what actually is a planet. 11:11.899 --> 11:16.399 So, just to summarize this, the problem is not that the 11:16.403 --> 11:20.993 planets are too faint, although they are quite faint. 11:20.990 --> 11:27.150 The problem is that the planets are too close to the star. 11:27.150 --> 11:30.320 11:30.320 --> 11:34.900 That prompts--we've got to quantify this statement I think 11:34.900 --> 11:39.160 and so the question arises, "Well how close is it?" 11:39.160 --> 11:43.230 11:43.230 --> 11:45.980 That leads into a discussion, which we already started last 11:45.981 --> 11:47.311 time, of planetary orbits. 11:47.310 --> 11:50.490 11:50.490 --> 11:54.570 If you want to answer that question in any quantitative 11:54.571 --> 11:55.631 sort of way. 11:55.629 --> 12:00.019 You will recall that in the last class I wrote down an 12:00.016 --> 12:04.316 important equation of--regarding planetary orbits, 12:04.320 --> 12:08.890 that looks like this: a^(3) = P^(2)M, 12:08.888 --> 12:13.458 where this [M] is the mass of the objects in 12:13.457 --> 12:18.297 orbit around each other in units of solar masses. 12:18.299 --> 12:24.199 This [P^(2)] is the orbital period in units 12:24.203 --> 12:25.753 of years. 12:25.750 --> 12:30.010 And this [a^(3)] is--orbits you will recall are 12:30.012 --> 12:35.292 ellipses, and this is the so-called semi-major axis of the 12:35.294 --> 12:39.654 ellipse in units called Astronomical Units, 12:39.649 --> 12:45.579 which is the distance from the Earth to the Sun. 12:45.580 --> 12:48.040 Or the semi-major axis of the Earth-Sun orbit, 12:48.036 --> 12:48.906 more properly. 12:48.910 --> 12:51.970 12:51.970 --> 12:54.130 So, that's in a particular set of units. 12:54.130 --> 12:57.270 12:57.269 --> 12:59.479 Before going on, let me make a couple general 12:59.482 --> 13:00.792 remarks about equations. 13:00.789 --> 13:03.549 First thing to say about equations is there's this idea 13:03.546 --> 13:04.926 that equations, mathematics, 13:04.925 --> 13:07.575 is some terribly different language, not at all. 13:07.580 --> 13:11.160 This is an English sentence. 13:11.159 --> 13:16.159 The sentence reads A cubed is equal to P squared times 13:16.157 --> 13:17.737 M, period. 13:17.740 --> 13:20.290 If you wrote this down in a textbook you'd actually see they 13:20.294 --> 13:21.424 put the period in there. 13:21.419 --> 13:23.919 It's got a verb, is equal to, 13:23.923 --> 13:27.593 it's got a subject, it's got an object. 13:27.590 --> 13:31.380 All this is is a different way of writing stuff down. 13:31.379 --> 13:33.109 So, it's not a different language; 13:33.110 --> 13:35.400 it's just a different writing system, that's all. 13:35.399 --> 13:37.569 So, don't freak out, that's the first thing. 13:37.570 --> 13:40.760 It's just ordinary English. 13:40.759 --> 13:44.659 The second thing to say is that this is a physics equation and 13:44.658 --> 13:47.788 not a math equation, and that's actually a subtler 13:47.790 --> 13:48.430 point. 13:48.429 --> 13:51.509 You will recall back in eighth-grade algebra when they 13:51.507 --> 13:55.047 started writing down X is equal to whatever X was equal to. 13:55.049 --> 13:57.579 And they said, you know, the amazingly cool 13:57.583 --> 14:00.603 thing about algebra is that X could be anything. 14:00.600 --> 14:03.330 That's why it's so powerful, and so you can solve whole 14:03.329 --> 14:06.059 classes of problems all at once just by letting X equal 14:06.058 --> 14:07.168 something unknown. 14:07.170 --> 14:11.500 That's math; physics--it isn't true. 14:11.500 --> 14:14.520 It is not the case that this equation is true regardless of 14:14.524 --> 14:16.984 what a is, or regardless of what P 14:16.975 --> 14:17.335 is. 14:17.340 --> 14:21.470 This is only true for very specific meanings of those 14:21.467 --> 14:22.497 quantities. 14:22.500 --> 14:27.590 And the amazing thing about the Universe is it turns out that 14:27.585 --> 14:32.665 math equations apply in very specific physical situations. 14:32.670 --> 14:34.280 This didn't have to be true. 14:34.279 --> 14:37.819 Either that you could express physical things in mathematical 14:37.816 --> 14:41.226 equations, and that math and physics would have anything to 14:41.234 --> 14:42.594 do with each other. 14:42.590 --> 14:46.260 And it is different to have a physics equation than to have a 14:46.256 --> 14:49.246 math equation because it has to be so specific. 14:49.250 --> 14:52.370 If you substitute something that isn't the semi-major axis 14:52.365 --> 14:54.815 in for a, then that equation isn't true 14:54.824 --> 14:55.484 anymore. 14:55.480 --> 14:59.350 It seems kind of obvious, I know, but worth keeping in 14:59.349 --> 14:59.859 mind. 14:59.860 --> 15:03.950 The other thing is that physics equations tend to have units 15:03.954 --> 15:05.554 associated with them. 15:05.549 --> 15:07.179 Not always, but most of the time. 15:07.179 --> 15:09.659 And so that's why it's so important to say, 15:09.655 --> 15:13.065 well, this is true not only if M stands for mass but 15:13.073 --> 15:16.553 also if mass is expressed in a particular set of units, 15:16.550 --> 15:18.750 in this case solar masses. 15:18.750 --> 15:22.830 You'll recall from last time that this is actually a specific 15:22.832 --> 15:26.512 version of a slightly more general equation which looks 15:26.507 --> 15:30.587 like this: a^(3) = P^(2)GM/4π^(2), 15:30.590 --> 15:37.170 where G is a constant that depends on what units you're 15:37.165 --> 15:38.135 using. 15:38.139 --> 15:41.599 And last time I derived this equation from that. 15:41.600 --> 15:49.710 But another way of saying it is if you choose your units to be 15:49.706 --> 15:55.816 solar masses, years, and astronomical units; 15:55.820 --> 16:02.340 then G is equal to 4π^(2) and so it cancels out. 16:02.340 --> 16:06.560 That's another way of thinking about why this specific version 16:06.562 --> 16:08.502 of the equation works out. 16:08.500 --> 16:10.260 All right. 16:10.259 --> 16:12.849 So, once you've got your physics equation, 16:12.849 --> 16:15.249 what are you supposed to do with it? 16:15.250 --> 16:18.180 Well, the first thing you should do is take a careful look 16:18.177 --> 16:21.257 at it and figure out what kinds of problems you can solve. 16:21.259 --> 16:23.539 This, again, is fairly straight-forward, 16:23.536 --> 16:24.466 obvious stuff. 16:24.470 --> 16:27.730 If you've got a^(3) = P^(2)M, 16:27.732 --> 16:31.692 there are obviously three different problems you can 16:31.693 --> 16:32.473 solve. 16:32.470 --> 16:35.560 If you know a and M you can compute 16:35.560 --> 16:36.720 P, right? 16:36.720 --> 16:39.450 If you know a and P you can compute 16:39.453 --> 16:40.083 M. 16:40.080 --> 16:42.210 And if you know P and M you can compute 16:42.209 --> 16:42.659 a. 16:42.659 --> 16:44.699 So, you know two of these things, you find out the third. 16:44.700 --> 16:49.240 This problem--an example of this problem we did last time. 16:49.240 --> 16:52.380 We had--we solved for the orbital period of Jupiter by 16:52.378 --> 16:55.868 knowing how far away from the Sun it is and what the mass of 16:55.871 --> 16:56.761 the Sun is. 16:56.760 --> 16:59.090 So, like Jupiter last time. 16:59.090 --> 17:05.900 So, let's do another one of these examples just quickly 17:05.897 --> 17:06.777 here. 17:06.780 --> 17:08.430 What is the mass of the Sun? 17:08.430 --> 17:11.660 17:11.660 --> 17:14.240 So, that's obviously this version of the problem [if you 17:14.244 --> 17:16.834 know a and M you can compute P]. 17:16.829 --> 17:19.919 And so, what we're going to do is we're going to take a 17:19.923 --> 17:21.853 and P from the Earth's orbit. 17:21.849 --> 17:25.979 Use Earth's orbit to determine the mass of the Sun. 17:25.980 --> 17:28.050 And this is, in fact, how one goes about 17:28.047 --> 17:29.847 determining the mass of the Sun. 17:29.849 --> 17:33.609 So, let's use this in the units we have, a = 1 17:33.610 --> 17:36.070 Astronomical Unit for the Earth. 17:36.069 --> 17:39.149 P is equal to 1 year for the Sun. 17:39.150 --> 17:42.730 1^(3) = 1^(2) x M. 17:42.730 --> 17:47.950 So, M = 1 in units of solar masses. 17:47.950 --> 17:50.460 That's not especially surprising, right? 17:50.460 --> 17:53.730 17:53.730 --> 17:57.460 Not particularly illuminating, right? 17:57.460 --> 18:02.520 That the mass of the Sun comes out to be one in units of solar 18:02.522 --> 18:03.272 masses. 18:03.269 --> 18:06.669 If we use a different set of units this is a more interesting 18:06.667 --> 18:07.457 calculation. 18:07.460 --> 18:12.460 So, let's use a more standard set of units, 18:12.459 --> 18:18.049 this so called mks units, this is meters, 18:18.053 --> 18:23.413 kilograms for mass, and seconds for time. 18:23.410 --> 18:28.340 All right, so now, turns out an Astronomical Unit 18:28.336 --> 18:32.846 is expressed in meters is 150 billion meters, 18:32.852 --> 18:34.702 approximately. 18:34.700 --> 18:38.530 It looks something like this [150,000,000,000]. 18:38.529 --> 18:41.129 And what's a year expressed in seconds? 18:41.130 --> 18:47.120 Well, let's see a year is 365.24 days, a day is 24 hours, 18:47.118 --> 18:52.358 an hour is 60 minutes, and a minute is 60 seconds, 18:52.358 --> 18:56.848 so that's a year expressed in seconds. 18:56.849 --> 19:02.039 And at this point we better pause and get our large numbers 19:02.036 --> 19:03.016 straight. 19:03.019 --> 19:06.879 You are all probably familiar I imagine with scientific 19:06.883 --> 19:07.673 notation. 19:07.670 --> 19:10.320 This becomes very important, otherwise the numbers, 19:10.315 --> 19:13.115 as you can clearly see, get completely out of hand. 19:13.119 --> 19:18.709 And because it's so important I'm going to tire you by writing 19:18.708 --> 19:22.738 down some facts about scientific notation. 19:22.740 --> 19:26.370 There's a help sheet on this if you have any trouble but recall 19:26.373 --> 19:27.373 how this works. 19:27.369 --> 19:31.939 You're supposed to express numbers as in the following 19:31.936 --> 19:34.436 form: N times 10^(m), this [N] 19:34.435 --> 19:39.255 is a decimal with one digit out in front of the dot. 19:39.259 --> 19:41.759 This [m] is supposed to be an integer, 19:41.759 --> 19:44.799 a whole number either positive or negative. 19:44.799 --> 19:50.339 And the reason that this is easy to deal with is because of 19:50.342 --> 19:55.122 one of the great limitations of the human brain. 19:55.119 --> 19:58.189 We only deal well with really small numbers, 19:58.188 --> 20:00.328 basically one-digit numbers. 20:00.329 --> 20:03.649 If I take a few coins here and I sort of throw them up you 20:03.651 --> 20:06.391 immediately know that there are four of them, 20:06.390 --> 20:09.480 you don't have to count one, two, three, four to satisfy 20:09.479 --> 20:12.739 yourself that there are four coins on this table because we 20:12.738 --> 20:15.778 have a concept of fourness, all of us in our minds, 20:15.782 --> 20:19.012 and it's clear that this pattern satisfies that concept. 20:19.009 --> 20:22.869 If I threw two coins or six coins, or seven coins you'll 20:22.867 --> 20:24.197 probably be okay. 20:24.200 --> 20:26.450 If I threw thirteen, you'd start to have real 20:26.448 --> 20:26.958 trouble. 20:26.960 --> 20:30.480 If there were 271 of them, there's no way you could guess 20:30.484 --> 20:33.384 just by looking at them how many there were. 20:33.380 --> 20:35.030 You'd have to count them one by one. 20:35.029 --> 20:38.519 So, the human brain really only deals well with one-digit 20:38.521 --> 20:42.261 numbers and you can tell this because of the names we give to 20:42.261 --> 20:43.011 numbers. 20:43.009 --> 20:46.869 When you start getting big numbers we call them millions or 20:46.865 --> 20:50.185 billions, or trillions and they all sound alike, 20:50.190 --> 20:52.440 that's because we can't tell the difference between a million 20:52.441 --> 20:54.861 and a trillion, it's just big. 20:54.859 --> 20:58.879 So, anytime you get up into big numbers we start to completely 20:58.879 --> 21:01.509 lose our grip on what these numbers mean, 21:01.514 --> 21:04.484 and you have to have elaborate metaphors; 21:04.480 --> 21:07.830 a trillion is a million millions. 21:07.829 --> 21:10.659 And if you fill up the Moon with ping-pong balls there 21:10.661 --> 21:11.731 are--I don't know. 21:11.730 --> 21:14.860 You have to do that kind of mental exercise to get a grip on 21:14.864 --> 21:15.134 it. 21:15.130 --> 21:19.710 The nice thing about scientific notation is that it turns big 21:19.711 --> 21:22.461 numbers into small numbers, right? 21:22.460 --> 21:25.950 N is a number between one and nine, m is an integer, 21:25.951 --> 21:30.061 it's one of those numbers four, five, six that we can get our 21:30.058 --> 21:31.768 minds wrapped around. 21:31.769 --> 21:39.159 Recall what this means, right, 10^(m) is 1 with m zeros 21:39.155 --> 21:40.655 after it. 21:40.660 --> 21:43.090 So, that's the point of scientific notation, 21:43.086 --> 21:46.356 it's to take big numbers and allow us to deal with them. 21:46.359 --> 21:49.049 I think this is a really valuable thing and I kind of 21:49.047 --> 21:52.297 wish that budget discussions in the political context would take 21:52.304 --> 21:55.414 place in scientific notation because people are always making 21:55.405 --> 21:58.605 mischief from the fact that all big numbers sound alike. 21:58.609 --> 22:01.489 You have congress critters saying things like, 22:01.487 --> 22:05.327 "oh terrible disaster," there's a million dollars of waste in 22:05.325 --> 22:06.855 the Pentagon budget. 22:06.859 --> 22:09.589 The Pentagon budget is $500 billion dollars. 22:09.589 --> 22:12.959 One million dollars of waste--that's me dropping a 22:12.958 --> 22:15.638 nickel behind the couch on my budget. 22:15.640 --> 22:19.910 So that's--nobody would accuse me of being wastefulness if I 22:19.910 --> 22:23.170 didn't move the couch and get that nickel out, 22:23.166 --> 22:24.176 all right. 22:24.180 --> 22:28.880 It's because a million sounds big, and so does 500 billion. 22:28.880 --> 22:32.980 We have no real sense of how big these things are relative to 22:32.981 --> 22:33.871 each other. 22:33.869 --> 22:37.799 But we do have a sense of the difference between 6 and 11, 22:37.795 --> 22:41.715 which is the difference in scientific notation between one 22:41.720 --> 22:44.820 million dollars and $100 billion dollars, 22:44.820 --> 22:46.810 so that's how this works. 22:46.810 --> 22:49.210 So, arithmetic rules. 22:49.210 --> 22:52.080 Again, there's a help sheet on this if you're rusty. 22:52.079 --> 22:56.949 If you have N times 10^(m) and you multiply it by another 22:56.948 --> 23:00.748 scientific notation number, A times 10^(B), 23:00.746 --> 23:04.996 that is equal to A times N (times 10 to the n+m). 23:05.000 --> 23:09.960 And the other nice thing about scientific notation is it turns 23:09.963 --> 23:14.363 multiplication into addition because the exponents add: 23:14.357 --> 23:17.447 A times N to--times 10^(m) plus B. 23:17.450 --> 23:24.680 Similarly, if you take N times 10^(m) and raise it to the K, 23:24.683 --> 23:29.713 that's N^(k) times 10 to the m times K, 23:29.710 --> 23:33.140 so it turns exponentiation into multiplication. 23:33.140 --> 23:36.410 So, the nice thing about dealing with exponents is it 23:36.410 --> 23:38.990 makes it one kind of arithmetic easier. 23:38.990 --> 23:42.760 So, a specific example of this might be N times 10^(m), 23:42.756 --> 23:46.866 the square root of that which is raising something to the ½ 23:46.872 --> 23:47.572 power. 23:47.569 --> 23:54.119 This is equal to N^(1/2) times 10^(m/2). 23:54.119 --> 23:57.509 That works if m is an even number but if m is 23:57.512 --> 23:59.912 an odd number you get a crazy exponent. 23:59.910 --> 24:02.440 This really has to be an integer over here. 24:02.440 --> 24:03.830 So, what do you do then? 24:03.829 --> 24:12.259 You say N times 10^(m) to the ½ is equal to 10 times N times 24:12.264 --> 24:15.924 N^(m-1) to the ½ power. 24:15.920 --> 24:20.820 So, I've just moved a 10 from here over to here, 24:20.818 --> 24:27.278 and then that is equal to 10 N to the ½ times 10^((m-1)/2). 24:27.279 --> 24:30.779 And that's how you recover when you take a square root or 24:30.783 --> 24:32.163 something like that. 24:32.160 --> 24:35.370 That's how you recover an integer on this side. 24:35.369 --> 24:37.519 We'll have lots of practice in this kind of thing. 24:37.520 --> 24:42.060 So, going back to the problem. 24:42.059 --> 24:47.429 1 AU is equal to 1.5 times 10^(11) meters; 24:47.430 --> 24:53.160 1 year is equal to 2.4 times 10^(1). 24:53.160 --> 24:59.700 That's 24 times 6 times 10^(1) times, 6 times 10^(1), 24:59.701 --> 25:04.231 times 3.6524 times 10^(2) seconds. 25:04.230 --> 25:09.420 All right, so let's do that one, two, three, 25:09.421 --> 25:16.791 four, five--2.4 times 6 times, 6 times 3.6 times 10^(5) to the 25:16.785 --> 25:20.215 5; 2.5 times 6 is around 15,15 25:20.223 --> 25:25.653 times 6 is around 90,90 is a little less than 100,3.6 a 25:25.649 --> 25:27.859 little more than 6. 25:27.859 --> 25:32.909 Those two things multiplied together is around 300. 25:32.910 --> 25:38.560 And 300 times 10^(5) is equal to 3 times 10^(7). 25:38.559 --> 25:42.989 So, there are around 3 times 10^(7) seconds in 1 year. 25:42.990 --> 25:46.870 That's a good number to remember. 25:46.869 --> 25:53.079 In mks units, G is this magical constant, 25:53.075 --> 25:57.955 is something like 7 times 10^(-11). 25:57.960 --> 25:59.370 That's an approximation. 25:59.369 --> 26:02.229 It's actually 6.6 something, something, something times 26:02.233 --> 26:03.933 10^(-11), but we'll call it 7. 26:03.930 --> 26:08.350 So, now we're in a position to calculate what the solar mass is 26:08.350 --> 26:09.420 in kilograms. 26:09.420 --> 26:15.510 So remember a^(3) equal to GMP squared over 26:15.512 --> 26:16.602 4π^(2). 26:16.599 --> 26:21.099 So, let's see, 1.5 times 10^(11); 26:21.099 --> 26:24.609 that's an astronomical unit cubed. 26:24.609 --> 26:26.869 P squared, 3 times 10^(7); 26:26.870 --> 26:31.090 that's 1 year squared; 7 times 10^(-11) that's 26:31.087 --> 26:34.637 G over M over 4π^(2). 26:34.640 --> 26:37.560 I'm going to do the arithmetic really fast. 26:37.559 --> 26:40.869 I would like you to learn how to do this yourself rather than 26:40.874 --> 26:43.474 relying on calculators so this is an example. 26:43.470 --> 26:46.180 Let me blaze through this at warp speed; 26:46.180 --> 26:48.480 then we'll stop and consider what's happened. 26:48.480 --> 26:49.300 All right. 26:49.299 --> 26:56.869 On the right-hand side, 3^(2) is--that's 3^(2) times 26:56.866 --> 27:03.836 10^(7) times 2 is 14, times 7 times 10^(-11). 27:03.839 --> 27:09.459 M over 4 times π, times π. 27:09.460 --> 27:11.900 3^(2) is 10. 27:11.900 --> 27:15.230 10 times 10^(14) is 10^(15). 27:15.230 --> 27:20.140 And so on the top we've got 10^(15) times 7, 27:20.139 --> 27:22.079 times 10^(-11). 27:22.079 --> 27:25.709 15 minus 11 is 4 so that's 7 times 10^(4) on top. 27:25.710 --> 27:28.020 π times π is 10 that's 40. 27:28.019 --> 27:32.269 4 times 10 to the 1 on the bottom, 7 divided by 4 is 2, 27:32.270 --> 27:35.340 and so that's 2 times 10^(3) times m. 27:35.339 --> 27:41.129 Good, left-hand side, all right 1.5 times, 27:41.131 --> 27:47.351 1.5 times, 1.5, that's 1.5^(3) times 10^(33), 27:47.346 --> 27:48.896 why 33? 27:48.900 --> 27:50.400 That's 11 times 3. 27:50.400 --> 27:54.400 1.5 times 1.5 that's a little bit more than 2. 27:54.400 --> 27:56.620 If you multiply something that's a little bit more than 2 27:56.619 --> 27:58.839 by something that's a little bit less than 2 that's 4, 27:58.840 --> 28:02.570 and so this is 4 times 10^(33). 28:02.570 --> 28:04.830 Now we want m, right? 28:04.829 --> 28:07.279 So we've got to divide both sides by 2 times 10^(3); 28:07.279 --> 28:14.219 m is equal to 4 halves times 10^(33) over 10^(3). 28:14.220 --> 28:16.570 That's 10^(-33) minus 3. 28:16.569 --> 28:22.159 That's 2 times 10^(30) kilograms, got to put the units 28:22.162 --> 28:22.692 in. 28:22.690 --> 28:25.150 Okay, how we doing? 28:25.150 --> 28:27.550 Questions, comments, abuse. 28:27.550 --> 28:28.240 Yes sir? 28:28.240 --> 28:29.430 Student: How accurate is that number? 28:29.430 --> 28:30.790 Prof: How accurate is that number? 28:30.789 --> 28:34.339 It's accurate to about one digit, which is why I only wrote 28:34.335 --> 28:36.715 down one digit of accuracy out front. 28:36.720 --> 28:39.320 This is a pro--excellent question thank you very much. 28:39.319 --> 28:44.089 This is appropriate because remember I'm using 7 times 10 28:44.093 --> 28:48.273 minus 11 for G, where it's actually 6.6. 28:48.269 --> 28:51.729 So, I'm already about 10% off from there. 28:51.730 --> 28:55.630 I did my little calculation to come up with one year equals 3 28:55.633 --> 28:57.133 times 10^(7) seconds. 28:57.130 --> 29:00.140 That's about accurate to one digit or so. 29:00.140 --> 29:03.220 And so the whole thing is done to one digit accuracy. 29:03.220 --> 29:05.910 If you're dealing with one digit accuracy, 29:05.913 --> 29:08.413 it is true that 7 divided by 4 is 2. 29:08.410 --> 29:11.320 It really is true, because if that wasn't true, 29:11.321 --> 29:14.801 then you would have to have more digits on your unit for 29:14.801 --> 29:17.081 G or something like that. 29:17.079 --> 29:21.009 In particular, let me make an official rule 29:21.007 --> 29:22.687 for this course. 29:22.690 --> 29:27.170 Three equals π, equals the square root of 10, 29:27.174 --> 29:28.374 all right. 29:28.369 --> 29:31.979 That will solve an enormous amount of arithmetic problems 29:31.981 --> 29:35.271 and it will not get you into any serious trouble. 29:35.269 --> 29:39.049 So, we don't have to worry about the .14159 and however 29:39.049 --> 29:42.059 many more digits you all memorized it to. 29:42.059 --> 29:45.669 And when you multiply it together you get ten. 29:45.670 --> 29:46.230 Yes? 29:46.230 --> 29:47.800 Student: Are you expecting this kind of 29:47.799 --> 29:48.879 calculation for problem sets? 29:48.880 --> 29:50.610 Prof: Yes. 29:50.609 --> 29:52.169 The question was, "Am I expecting this kind of 29:52.170 --> 29:53.280 calculation for problem sets?" 29:53.280 --> 29:54.360 The answer is "yes." 29:54.359 --> 29:56.459 Here's the rule about calculators. 29:56.460 --> 30:00.050 Let me put it this way: You can only use calculators if 30:00.051 --> 30:02.381 I can't tell that you've done it. 30:02.380 --> 30:06.520 So, that means you can check your work to make sure you've it 30:06.520 --> 30:07.970 right or something. 30:07.970 --> 30:11.830 But if you start coming up with numbers like 7.1516397, 30:11.833 --> 30:16.273 that's eight digits of accuracy and I'm pretty sure you haven't 30:16.268 --> 30:18.198 worked it out yourself. 30:18.200 --> 30:24.090 So important, no calculators on the tests, 30:24.085 --> 30:25.085 okay? 30:25.089 --> 30:28.429 So, get some practice doing this kind of thing. 30:28.430 --> 30:32.300 And this will--this I promise you will be useful to you in 30:32.295 --> 30:36.225 everyday life because this is how you catch the politicians 30:36.229 --> 30:39.009 doing screwy things with big numbers. 30:39.009 --> 30:42.649 You do it in your head in scientific notation and you 30:42.645 --> 30:46.345 figure out whether the answer is meaningful or not. 30:46.349 --> 30:49.299 This whole business of significant digits, 30:49.300 --> 30:52.990 I think, is badly distorted; by the way, it's taught in high 30:52.986 --> 30:53.376 school. 30:53.380 --> 30:55.710 In high school you, and also I should say in 30:55.711 --> 30:58.531 laboratory courses sometimes at the college level, 30:58.529 --> 31:02.219 you often get situations where people say--give you a whole 31:02.223 --> 31:06.173 sheet of rules on how to figure out how many significant digits 31:06.172 --> 31:07.002 you have. 31:07.000 --> 31:08.200 This is nonsense. 31:08.200 --> 31:10.730 All you have to do is behave like a human being. 31:10.730 --> 31:13.360 We say to each other, I'll meet you in the dining 31:13.363 --> 31:14.573 hall in ten minutes. 31:14.569 --> 31:17.509 That doesn't mean--that means something different from I'll 31:17.514 --> 31:20.514 meet you in the dining hall in eleven minutes and twenty-six 31:20.510 --> 31:21.120 seconds. 31:21.119 --> 31:24.359 Even if the person happens to show up in the dining hall in 31:24.359 --> 31:27.039 exactly eleven minutes and twenty-six seconds. 31:27.039 --> 31:29.109 Ten minutes means I'll meet you there in ten minutes, 31:29.107 --> 31:30.297 we all know what that means. 31:30.299 --> 31:33.239 I'll meet you there in eleven minutes and twenty-six seconds 31:33.241 --> 31:35.881 means you're a character in a bad spy novel who's just 31:35.883 --> 31:37.233 synchronized his watch. 31:37.230 --> 31:41.130 So, this shows up in science fiction too. 31:41.130 --> 31:43.190 I don't know how many of you are Star Trek fans, 31:43.191 --> 31:44.421 I certainly am [laughter]. 31:44.420 --> 31:47.430 And in all the different Star Trek movies [inaudible 31:47.431 --> 31:48.731 comment]--thank you. 31:48.730 --> 31:50.890 In all the different--a friend [referring to person who made 31:50.887 --> 31:51.287 comment]. 31:51.289 --> 31:54.359 In all the different Star Trek movies there's always a second 31:54.359 --> 31:56.609 in command who isn't a human being, right? 31:56.609 --> 32:00.719 A Vulcan or an android or some damn thing or another. 32:00.720 --> 32:03.470 And to emphasize the non-humanness of these 32:03.465 --> 32:06.995 characters, what they do is they make them use too many 32:06.996 --> 32:08.496 significant digits. 32:08.500 --> 32:12.100 And so that makes them inhuman and so the captain will say, 32:12.099 --> 32:14.829 "When are we landing on omicron M?" 32:14.829 --> 32:18.209 The second in command will say, "Well, we should assume 32:18.211 --> 32:21.531 standard orbit in 2.6395 minutes," emphasizing somehow 32:21.530 --> 32:23.910 superior brain power or something. 32:23.910 --> 32:28.560 But it's nonsense because it takes the guy ten seconds to say 32:28.564 --> 32:32.064 that sentence, so what is this time calculated 32:32.055 --> 32:34.145 to a 100th of a second? 32:34.150 --> 32:36.730 Does it start from when he begins the sentence? 32:36.730 --> 32:38.780 From when he ends the sentence? 32:38.779 --> 32:41.679 What's the other end of that time interval? 32:41.680 --> 32:47.190 Can you say you assume standard orbit to the 100th of a second? 32:47.190 --> 32:48.230 What does that even mean? 32:48.230 --> 32:50.570 When you start beaming down? 32:50.570 --> 32:52.230 When you end beaming down? 32:52.230 --> 32:55.800 Also, keep in mind it takes more than a 100th of second for 32:55.795 --> 32:59.355 the sound to travel from his lips to the captain's ears, 32:59.359 --> 33:01.939 so the whole thing is just nonsense. 33:01.940 --> 33:04.930 And so, you don't need any special rules, 33:04.931 --> 33:08.731 just behave like a human being; don't behave like an android. 33:08.730 --> 33:13.000 So, no androids. 33:13.000 --> 33:15.320 And that's the only rule I'm going to give you [laughter]. 33:15.319 --> 33:19.419 These two are the only rules I'm going to give you about 33:19.421 --> 33:22.631 significant digits, just do the right thing, 33:22.628 --> 33:23.298 okay. 33:23.299 --> 33:27.999 All right, okay so this is all lovely but we haven't actually 33:28.004 --> 33:32.634 started to answer the question we started the class with. 33:32.630 --> 33:35.380 Let me remind you where we began this disquisition on 33:35.380 --> 33:35.910 numbers. 33:35.910 --> 33:41.650 The question was supposed to be how far away from stars are 33:41.650 --> 33:42.640 planets? 33:42.640 --> 33:49.630 So, we haven't answered the question, when the question was, 33:49.632 --> 33:53.782 "How close are planets to stars?" 33:53.779 --> 33:56.329 This notation will confuse the people who didn't make it to 33:56.333 --> 33:58.013 class today; that's probably just as well 33:58.010 --> 33:58.200 too. 33:58.200 --> 34:01.380 All right, how close to planets--are planets to stars? 34:01.380 --> 34:06.050 Now, we have a problem because closeness--we usually think of 34:06.048 --> 34:09.158 this as being measured with distances. 34:09.159 --> 34:13.129 And distances are a serious problem in astronomy. 34:13.130 --> 34:15.930 Supposing you--Distance, by the way, is more or less the 34:15.933 --> 34:17.363 same thing as size, right? 34:17.360 --> 34:20.200 Size is the distance from one side of something to another 34:20.202 --> 34:21.202 side of something. 34:21.199 --> 34:24.079 So, size and distance are the same. 34:24.079 --> 34:26.939 So, supposing you ask the question, "How big is the Moon?" 34:26.940 --> 34:28.840 What is the size of the Moon? 34:28.840 --> 34:31.390 What is the distance from one side of the Moon to the other? 34:31.389 --> 34:34.319 You go out one night and you look up at the Moon and you do 34:34.319 --> 34:37.399 this [holding fingers apart], one on each side of the Moon. 34:37.400 --> 34:38.640 You say, well, it's about an inch and a half. 34:38.639 --> 34:41.579 Now, obviously that's meaningless, because if you had 34:41.581 --> 34:44.071 done it here, it would have been about a half 34:44.070 --> 34:44.580 inch. 34:44.579 --> 34:48.169 And we all know that isn't--that somehow is not the 34:48.172 --> 34:49.252 right answer. 34:49.250 --> 34:52.560 If you look at the screen over here, for example, 34:52.558 --> 34:56.008 and you hold out your hands about eight inches from 34:56.005 --> 34:59.135 your--close one eye, hold out--you can try this, 34:59.135 --> 35:01.975 hold out your hands about here, you'll find that that 35:01.976 --> 35:04.976 screen--the lit part of the screen is about four or five 35:04.981 --> 35:06.021 inches across. 35:06.019 --> 35:09.469 But if you go closer to your eye, it's only about an inch 35:09.466 --> 35:10.016 across. 35:10.019 --> 35:13.069 If you're at arm's length it's maybe six, eight inches where 35:13.074 --> 35:14.114 I'm standing here. 35:14.110 --> 35:16.040 And if you try it at the back of the class, 35:16.041 --> 35:18.481 you'll get a different answer from what I just got. 35:18.480 --> 35:21.720 So, distance is a problematical thing. 35:21.719 --> 35:24.549 And so--But on the other hand, obviously, the size of the 35:24.547 --> 35:27.727 screen hasn't changed depending on where my arms were or whether 35:27.727 --> 35:31.057 I'm standing at the back of the room or the front of the room. 35:31.059 --> 35:33.239 So, something has to be measurable there, 35:33.235 --> 35:34.535 but it isn't distance. 35:34.540 --> 35:37.190 What it is, is angle. 35:37.190 --> 35:40.200 What you can measure in astronomy are angles. 35:40.199 --> 35:42.539 So, here's what you really want to know. 35:42.540 --> 35:45.110 Here's a star; here's a planet going around 35:45.110 --> 35:48.510 the star, and here is you [draws diagram on overhead]. 35:48.510 --> 35:50.960 This is the international symbol for an observer; 35:50.960 --> 35:53.690 it's supposed to be a little stylized eyeball. 35:53.690 --> 35:57.950 So, this observer is looking at this star-planet system. 35:57.949 --> 36:02.399 And looks at the star and looks at the planet. 36:02.400 --> 36:06.430 And what you can measure is the angle, which I'm going to call 36:06.427 --> 36:07.547 α here. 36:07.550 --> 36:10.920 And if you simulate the angle by putting your hands up close 36:10.922 --> 36:13.382 to your eye, you'll get a small distance; 36:13.380 --> 36:16.220 if you do it further away you'll get a bigger distance. 36:16.219 --> 36:18.709 But what you are measuring is the angle. 36:18.710 --> 36:22.910 What you would like is a way of being able to convert from that 36:22.913 --> 36:24.543 angle into a distance. 36:24.539 --> 36:28.609 And now, the ugly specter of trigonometry rears its head. 36:28.610 --> 36:31.160 Because that's what trigonometry is. 36:31.159 --> 36:34.199 It's a way of turning angles into distances and vice versa. 36:34.199 --> 36:36.899 And you may remember constructions that look vaguely 36:36.895 --> 36:38.665 like this; let's put some labels on it. 36:38.670 --> 36:41.620 Here's distance 1, here's distance 2, 36:41.620 --> 36:43.260 here's distance 3. 36:43.260 --> 36:47.380 And this is the definition of sine, the sine of α is 36:47.381 --> 36:50.951 equal to the opposite over the hypotenuse, so that's 36:50.945 --> 36:53.525 D_2 over D_3. 36:53.530 --> 36:55.700 And now--so that's how you do it. 36:55.699 --> 36:59.029 Now, I promised you on Tuesday that the sine was going to 36:59.033 --> 37:00.643 cancel out, and so it is. 37:00.640 --> 37:01.730 Here's how. 37:01.730 --> 37:06.330 If you use small angles it turns out that the sine of 37:06.325 --> 37:10.975 α is equal to α, if α is expressed in 37:10.977 --> 37:14.387 radians, which is a particular kind of measure of angles. 37:14.389 --> 37:17.009 And so for small angles, first of all, 37:17.011 --> 37:21.331 the hypotenuse and the longer side are the same length more or 37:21.332 --> 37:21.972 less. 37:21.969 --> 37:26.429 So, you've got a situation like this where D_2 over 37:26.427 --> 37:29.247 D_1 is equal to α. 37:29.250 --> 37:32.580 That is--let's circle that one in red; 37:32.580 --> 37:33.520 it's going to be important. 37:33.519 --> 37:38.849 That's called the small angle formula. 37:38.849 --> 37:41.599 And once again, you have to be careful of the 37:41.599 --> 37:42.099 units. 37:42.099 --> 37:44.459 And once again, there's two sets of units in 37:44.456 --> 37:46.426 which you might possibly use this. 37:46.429 --> 37:50.449 So D_2 over D_1 is equal to α. 37:50.449 --> 37:56.059 If D_2 and D_1 are in the same 37:56.061 --> 38:00.071 units, then α has to be radians. 38:00.070 --> 38:03.230 A radian--there are 2π radians in a circle. 38:03.230 --> 38:08.390 38:08.389 --> 38:11.899 You'll recall there are 360 degrees in a circle that's how 38:11.900 --> 38:14.180 you convert from degrees to radian. 38:14.179 --> 38:15.819 But there's a better set of units. 38:15.820 --> 38:18.930 It turns out also that if you do this, use this equation 38:18.934 --> 38:22.184 D_2,/D_1, α, and D_2 is 38:22.183 --> 38:25.173 measured in Astronomical Units, remember those? 38:25.170 --> 38:26.750 That's the Earth-Sun distance. 38:26.750 --> 38:31.460 And D_1 is measured in parsecs, which is equal to 3 38:31.456 --> 38:35.026 times 10^(16) meters; that's about three light years, 38:35.030 --> 38:37.030 for those of you counting at home. 38:37.030 --> 38:42.760 And α is in arc seconds. 38:42.760 --> 38:44.180 Then it also works out. 38:44.179 --> 38:49.379 Let me tell--an arc second, let's see, 60 arc seconds is 38:49.384 --> 38:51.754 equal to an arc minute. 38:51.750 --> 38:53.190 Just so you know how these units work; 38:53.190 --> 38:57.600 60 arc minutes is equal to 1 degree. 38:57.600 --> 39:00.060 So, that's a really small angle. 39:00.059 --> 39:03.519 So, it turns out this set of units is particularly useful for 39:03.523 --> 39:05.663 these kinds of planet calculations, 39:05.659 --> 39:09.389 because distances from planets to stars tend to be a few 39:09.394 --> 39:10.824 astronomical units. 39:10.820 --> 39:13.030 Distances from us to the nearest star. 39:13.030 --> 39:15.010 The nearest star, aside from the Sun, 39:15.014 --> 39:17.114 turns out to be about 1 parsec away. 39:17.110 --> 39:20.330 So, distances to stars are some number of parsecs. 39:20.329 --> 39:23.969 And so, you'll get answers that are a few arc seconds, 39:23.967 --> 39:28.087 and this is yet another way of turning big numbers into small 39:28.086 --> 39:28.906 numbers. 39:28.909 --> 39:31.559 You've used the right set of units so that everything's 39:31.558 --> 39:32.538 approximately one. 39:32.539 --> 39:36.889 Another way of compensating for the weakness of the human brain. 39:36.889 --> 39:42.919 Okay, so now we can solve a problem. 39:42.920 --> 39:47.560 We can actually solve numerically the question of how 39:47.559 --> 39:50.859 far away from a star is our planet? 39:50.860 --> 39:52.890 So, here's an example. 39:52.889 --> 39:56.719 Here's a problem of the kinds that might show up on a problem 39:56.717 --> 39:57.097 set. 39:57.100 --> 40:00.800 40:00.800 --> 40:03.990 I told you that that object orbiting Sirius has about a 40:03.988 --> 40:05.168 forty-year period. 40:05.170 --> 40:11.030 So, let's take a planet with a forty-year period; 40:11.030 --> 40:14.250 that's kind of between Saturn and Uranus in our own Solar 40:14.252 --> 40:14.772 System. 40:14.770 --> 40:23.060 Around a star 3 parsecs away; pc means parsecs, 40:23.055 --> 40:27.495 whatever they may tell you in the humanities classes. 40:27.500 --> 40:34.530 Around a star 3 parsecs away, what is the angular separation? 40:34.530 --> 40:38.740 Angular separation, because that's what we can 40:38.741 --> 40:40.521 actually measure. 40:40.519 --> 40:43.429 We can't measure the distance but we can measure the angle. 40:43.429 --> 40:49.329 Okay, so this is a hard problem, why is it hard? 40:49.329 --> 40:51.379 It's hard for two different reasons. 40:51.380 --> 40:53.340 First of all, well, you guys know how to 40:53.335 --> 40:54.685 solve problems like this. 40:54.690 --> 40:55.730 You all went to high school. 40:55.730 --> 40:58.570 What you do is you say, okay what do I know? 40:58.570 --> 41:01.530 I have P = 40, I have D = 3, 41:01.527 --> 41:05.127 I want α, so I look for an equation with D, 41:05.133 --> 41:07.733 P, D and α in it. 41:07.730 --> 41:11.460 I shove these two things into--you substitute them in for 41:11.458 --> 41:14.718 P and D, and then I calculate α. 41:14.719 --> 41:18.199 And you don't have to actually know anything about anything in 41:18.204 --> 41:20.444 order to do that; you just have to have a good 41:20.443 --> 41:21.023 set of notes. 41:21.019 --> 41:23.959 And you go through all the different equations until you 41:23.959 --> 41:27.219 find the equation that contains all the variables you have, 41:27.219 --> 41:29.519 and the one that you're asked for. 41:29.519 --> 41:31.149 And then you can just plug it in, you don't have to know 41:31.149 --> 41:31.919 anything about anything. 41:31.920 --> 41:33.850 This won't work, right? 41:33.849 --> 41:37.279 Because if you go through the lecture notes thus far, 41:37.277 --> 41:41.167 you will not find an equation with P and D and 41:41.166 --> 41:44.656 α in it because it's two different equations that you 41:44.660 --> 41:46.440 have to put together. 41:46.440 --> 41:49.370 And that requires you to actually know what you're doing. 41:49.369 --> 41:51.179 And that of course is much harder. 41:51.180 --> 41:55.080 So, plug and chug won't work. 41:55.079 --> 42:02.849 Chug fails because there's no one equation. 42:02.850 --> 42:05.970 42:05.969 --> 42:08.469 This is a hard problem for another reason, 42:08.472 --> 42:11.832 which is that I haven't told you something important. 42:11.829 --> 42:14.739 If you try and go through and solve this you will discover 42:14.743 --> 42:17.353 that you don't have all the information you need, 42:17.349 --> 42:20.289 and in particular, you need to know the total mass 42:20.289 --> 42:22.569 of the system, the mass of the star. 42:22.570 --> 42:24.520 Most of the mass is in the mass of the star. 42:24.520 --> 42:28.610 So, there's missing information. 42:28.610 --> 42:30.300 That's one way to make a problem hard, 42:30.303 --> 42:32.963 the other way to make a--if I don't tell you everything you 42:32.957 --> 42:35.197 need to know, it's a somewhat harder problem. 42:35.199 --> 42:39.209 The other way to make a problem hard--this is a classic, 42:39.208 --> 42:42.268 be warned--I will do this at some point, 42:42.269 --> 42:45.419 is to give you one extra piece of information that you don't 42:45.416 --> 42:45.786 need. 42:45.789 --> 42:50.369 This screws students up totally because you go looking for the 42:50.373 --> 42:54.283 equation that uses the extra piece of information, 42:54.280 --> 42:56.900 and there is no such an equation and you don't need it, 42:56.896 --> 42:58.346 then you're cast into doubt. 42:58.349 --> 43:01.909 I must need this somehow or he wouldn't have given it to me. 43:01.910 --> 43:03.600 Nonsense. 43:03.599 --> 43:07.509 In real life it is not the case that all problems come to you 43:07.511 --> 43:11.231 with all the information you need and only the information 43:11.226 --> 43:12.396 that you need. 43:12.400 --> 43:14.510 That's not how things work in real life. 43:14.510 --> 43:18.260 And it shouldn't be how things work in preparation for real 43:18.260 --> 43:22.010 life, which is I guess what classes sort of ought to be. 43:22.010 --> 43:25.190 So, missing information, too much information, 43:25.187 --> 43:27.867 sometimes although not in this case; 43:27.869 --> 43:31.519 I could have told you the temperature of the star, 43:31.517 --> 43:32.557 for example. 43:32.560 --> 43:34.520 I'll do that. 43:34.519 --> 43:39.439 The temperature of Sirius is 10,000 degrees, 43:39.441 --> 43:41.961 surface temperature. 43:41.960 --> 43:45.640 So, now I've given you both; I've left out information and 43:45.635 --> 43:47.055 told you too much. 43:47.059 --> 43:51.749 All right, so what is the mental process that you need to 43:51.745 --> 43:56.345 go through in solving a problem which has these kinds of 43:56.347 --> 43:59.357 difficulties associated with it? 43:59.360 --> 44:00.130 Okay. 44:00.130 --> 44:08.700 So, how to think about such problems? 44:08.699 --> 44:12.099 The first thing to notice is what you would do in the 44:12.096 --> 44:13.726 ordinary plug and chug. 44:13.730 --> 44:15.180 What have you got, what do you need? 44:15.179 --> 44:17.499 You have a value for the orbital period, 44:17.497 --> 44:20.707 you have a value for the distance, and what you want is 44:20.707 --> 44:21.477 an angle. 44:21.480 --> 44:26.140 44:26.140 --> 44:28.400 Okay. 44:28.400 --> 44:34.600 Now, so this doesn't lead you to where you want to go. 44:34.599 --> 44:36.989 The next thing you've got to do is make an assumption. 44:36.990 --> 44:39.670 This is a star; all stars are more or less the 44:39.671 --> 44:40.051 same. 44:40.050 --> 44:43.950 So, let us imagine that this star has a mass more or less 44:43.950 --> 44:45.970 similar to that of the Sun. 44:45.969 --> 44:48.159 So, the M is approximately equal to a solar 44:48.157 --> 44:48.467 mass. 44:48.470 --> 44:49.950 How far wrong could you go? 44:49.949 --> 44:53.109 In fact, you go wrong by about a factor of two, 44:53.113 --> 44:57.173 but at a certain point you're going to take the cube root of 44:57.170 --> 44:59.990 two, and so it actually isn't so bad. 44:59.989 --> 45:02.309 So suppose M is equal to the mass of the Sun, 45:02.309 --> 45:03.719 most stars are approximately. 45:03.719 --> 45:07.349 Ah ha, now we're getting somewhere, because now we have 45:07.354 --> 45:08.974 M and P. 45:08.969 --> 45:13.039 And if you have put that equation--Kepler's Third 45:13.042 --> 45:17.202 Law--and interpreted it deeply into your brain, 45:17.199 --> 45:20.019 and it has become part of your heart and soul and very being, 45:20.024 --> 45:22.714 you will immediately realize that if you have M and 45:22.708 --> 45:25.048 P, of course you can calculate 45:25.049 --> 45:25.689 a. 45:25.690 --> 45:29.150 In fact, you probably realized this first before you make the 45:29.148 --> 45:31.768 assumption in step two, because that's why you're 45:31.768 --> 45:34.308 interested in the mass, because you realize that if you 45:34.313 --> 45:38.473 happened to have M, you would be able to determine 45:38.469 --> 45:39.399 a. 45:39.400 --> 45:42.220 Next thing you have to do is, is interpret what this a 45:42.224 --> 45:43.894 is; a is the semi-major axis 45:43.885 --> 45:45.905 of the orbit of the planet around the star. 45:45.909 --> 45:52.609 That's something very close to--very similar to--the 45:52.612 --> 45:59.842 distance between the star and the planet is a. 45:59.840 --> 46:04.070 So, a in this equation is the same as D_2 in 46:04.066 --> 46:06.316 the other equation, all right. 46:06.320 --> 46:09.770 So, a is D_2 and the distance you have up 46:09.774 --> 46:12.804 there is the distance from us to that whole system, 46:12.804 --> 46:14.324 that's D_1. 46:14.320 --> 46:24.180 And now, you can compute α, right? 46:24.180 --> 46:26.060 That's the thought process. 46:26.059 --> 46:30.769 And you may have to go--sort of worry a little bit around here 46:30.767 --> 46:34.697 before you have the "ah ha" moment of how to get the 46:34.703 --> 46:35.633 problem. 46:35.630 --> 46:39.980 This is the moment when you've solved the problem. 46:39.980 --> 46:45.060 The key to solving this is to have an instinctive reaction to 46:45.064 --> 46:48.034 knowing that you've got a period; 46:48.030 --> 46:51.140 namely, if I had a mass I could figure out the semi-major axis. 46:51.139 --> 46:53.929 Or if I had the semi-major axis, I could figure out the 46:53.928 --> 46:54.288 mass. 46:54.290 --> 46:55.940 And that has to click. 46:55.940 --> 47:00.210 And the way that clicks is to have this equation really--I was 47:00.214 --> 47:04.704 joking before--but really become instinctive in a certain way, 47:04.699 --> 47:07.189 and there won't be many equations in this class. 47:07.190 --> 47:12.260 But it will really help you to assimilate them in that kind of 47:12.264 --> 47:12.934 depth. 47:12.929 --> 47:15.789 All right, so now in the few minutes remaining, 47:15.791 --> 47:18.841 let me actually solve this problem numerically. 47:18.840 --> 47:22.430 47:22.429 --> 47:28.839 P is equal to 40 years, D is equal to 3 parsecs. 47:28.840 --> 47:32.370 Here's something not to do: 40 times 3 times 10^(7), 47:32.369 --> 47:35.759 because that'll give you the answer in seconds. 47:35.760 --> 47:38.580 No don't do that, because both of the equations 47:38.582 --> 47:42.202 would rather have things in Astronomical Units and years and 47:42.201 --> 47:43.491 things like that. 47:43.489 --> 47:46.959 So a^(3) equals P^(2)M, 47:46.959 --> 47:52.289 M is 1 by assumption, P is 40 so this is 47:52.291 --> 47:57.861 1,600, that's 40^(2) on this side, a^(3) that's 1.6 47:57.858 --> 47:59.908 times 10 to the 3. 47:59.909 --> 48:04.019 Now, we're going to take the cube root of both sides, 48:04.023 --> 48:08.773 that's the cube root of 1.6 times, the cube root of 1,000. 48:08.769 --> 48:12.369 3 times a third is equal to 1, conveniently enough. 48:12.369 --> 48:15.359 The cube root of 1.6, I don't know that's 1.2 or 48:15.357 --> 48:17.007 something but call it 1. 48:17.010 --> 48:22.200 So, that's 10, and so it's 10 Astronomical 48:22.197 --> 48:23.207 Units. 48:23.210 --> 48:25.470 Remember we're doing this in solar masses, 48:25.473 --> 48:27.023 years, Astronomical Units. 48:27.019 --> 48:29.869 10 Astronomical Units, that's how far away it is. 48:29.869 --> 48:33.369 Now, we can do this [D_2/D_1] 48:33.368 --> 48:37.738 where this is in Astronomical Units, and this down here is in 48:37.740 --> 48:40.510 parsecs, and 10 divided by 3 is 3. 48:40.510 --> 48:46.340 So, the answer is a is equal to 3 arc seconds. 48:46.340 --> 48:50.750 Now, this is an important result because the--let me go 48:50.751 --> 48:52.631 back to this picture. 48:52.630 --> 48:56.410 48:56.409 --> 49:00.209 The angular scale over which light is kind of thrown around 49:00.207 --> 49:03.477 by the atmosphere and by the optics of the kinds of 49:03.482 --> 49:07.282 telescopes we can build is some number of arc seconds. 49:07.280 --> 49:10.680 49:10.679 --> 49:14.009 Good ground-based observations, maybe half the light is inside 49:14.010 --> 49:16.270 1 arc second; the rest of the light is 49:16.269 --> 49:19.329 scattered all the way out to maybe 10 arc seconds. 49:19.329 --> 49:21.439 From space it's a little bit better. 49:21.440 --> 49:37.900 But, light from observations of stars is scattered over angular 49:37.902 --> 49:44.012 sizes of arc seconds. 49:44.010 --> 49:46.800 So, you are guaranteed that there's going to be a bunch of 49:46.799 --> 49:49.929 light from the star right on top of where you're looking for your 49:49.931 --> 49:50.471 planet. 49:50.469 --> 49:52.459 And in fact, it's going to get worse, 49:52.458 --> 49:56.048 because one of the things about this equation is D_1; 49:56.050 --> 49:57.410 the distance to the star. 49:57.410 --> 49:58.700 We did this for Sirius. 49:58.699 --> 50:00.789 Sirius is one of the closest stars; 50:00.790 --> 50:02.150 it's 3 parsecs away. 50:02.150 --> 50:07.360 Most--the center of the galaxy is 8,000 parsecs away. 50:07.360 --> 50:11.200 So, if you take a typical star in our galaxy this number isn't 50:11.202 --> 50:13.552 going to be 3; it's going to be 8,000. 50:13.550 --> 50:19.130 And this number isn't going to be 3, it's going to something 50:19.129 --> 50:22.249 like 1/1000th of an arc second. 50:22.250 --> 50:25.880 And the space telescope scatters light over tenths of 50:25.878 --> 50:30.208 arc seconds, so you have factors of 100 buried inside the light 50:30.205 --> 50:32.015 coming from that star. 50:32.019 --> 50:38.689 So, that's why you don't care about more than one digit. 50:38.690 --> 50:41.570 In fact, you probably don't even care about anything other 50:41.574 --> 50:44.514 than the exponents of this answer because why are you doing 50:44.509 --> 50:45.369 this problem? 50:45.369 --> 50:48.409 You're doing this problem to figure out whether it's 50:48.406 --> 50:51.916 plausible to think that you might be able to see a planet as 50:51.918 --> 50:54.578 a separate dot, separate from the star. 50:54.579 --> 50:57.769 And the fact that your answer comes out in arc seconds or 50:57.771 --> 51:01.591 fractions of an arc second tells you everything you need to know. 51:01.590 --> 51:05.250 You don't need anymore decimal places to realize that the idea 51:05.250 --> 51:08.430 of looking up and seeing a planet is going to fail. 51:08.429 --> 51:13.019 So, that's not the way that we're going to find planets. 51:13.019 --> 51:16.459 So, we have to come up with different scheme and that's what 51:16.457 --> 51:18.727 we're going to talk about on Tuesday. 51:18.730 --> 51:23.000 Thanks.