1 00:00:00,012 --> 00:00:04,636 So, so far I've been telling you stories about what people tell us the galaxy is 2 00:00:04,636 --> 00:00:09,319 made of, I've told you how much the mass of the disc is, how much the mass of the 3 00:00:09,319 --> 00:00:14,273 halo is and most of these, other than the gas discovered by Chandra where estimates 4 00:00:14,273 --> 00:00:18,624 based on basically star counts. We count the stars and therefore there's 5 00:00:18,624 --> 00:00:23,498 some uncertainty, because rememeber. The number of stars in the Milky Way is 6 00:00:23,498 --> 00:00:28,108 uncertain to almost a factor of two, depending on what the prevalence of 7 00:00:28,108 --> 00:00:31,912 brown, brown dwarfs is. But, brown dwarfs don't comprise a 8 00:00:31,912 --> 00:00:37,524 significant fraction of the mass of the stars since they may be numerous but 9 00:00:37,524 --> 00:00:41,779 they're not very massive. So we have these estimates, but it would 10 00:00:41,779 --> 00:00:47,097 be good to make an independent Comparison of a direct measurement of the mass the 11 00:00:47,097 --> 00:00:51,437 galaxy, How do you weight a galaxy? Well, by now we know how to weigh anything. 12 00:00:51,437 --> 00:00:55,472 You just need something to orbit it. If only we had something orbiting the 13 00:00:55,472 --> 00:00:59,322 Milky Way, oh yea, we do. We orbit the Milky Way, so all we need is 14 00:00:59,322 --> 00:01:03,212 the orbital parameters of the sun and you think that's an easy problem. 15 00:01:03,212 --> 00:01:05,792 If fact it's a lot trickier then you think. 16 00:01:05,792 --> 00:01:09,927 the distance of the sun from the center of the galaxy that I quoted as a 8 17 00:01:09,927 --> 00:01:14,552 kiloparsecs is one of the least precisely low numbers in astronomy, in fact it's 18 00:01:14,552 --> 00:01:19,652 somewhere between 7 1/2 and 8 1/2 kiloparsecs, so there's a 12% uncertainty 19 00:01:19,652 --> 00:01:24,852 in or 6% uncertainty in this distance. It is a difficult measurement to make, 20 00:01:24,852 --> 00:01:29,502 likewise the speed with which the sun moves as it orbits the galaxy. 21 00:01:29,502 --> 00:01:33,796 You'd think that's an easy thing to measure, but it's not easy because 22 00:01:33,796 --> 00:01:38,820 remember all around us are stars that are moving in their own peculiar motion as 23 00:01:38,820 --> 00:01:43,275 an, along with that, with some average motion that is essentially ours. 24 00:01:43,275 --> 00:01:48,442 And so, getting correct value for the speed with which the sun moves around 25 00:01:48,442 --> 00:01:53,112 the, milky way is also not trivial. I'm going to use the established values. 26 00:01:53,112 --> 00:01:58,148 There are some uncertainties, but they're not going to affect what we're going to 27 00:01:58,148 --> 00:02:03,455 do significantly, and if you compute 2 pi R divided by, v, you find that the sun 28 00:02:03,455 --> 00:02:08,257 orbits the Milky Way once every 230 million years, which means that in the 5 29 00:02:08,257 --> 00:02:14,226 billion, year history of the sun, we've gone around the Milky Way, about 20 or 25 30 00:02:14,226 --> 00:02:16,902 times. So the sun has, sampled, all kinds of 31 00:02:16,902 --> 00:02:19,842 places, and space, in the course of its history. 32 00:02:19,842 --> 00:02:23,542 And now, we're back to Newton. We know exactly how to do this. 33 00:02:23,542 --> 00:02:27,911 We have an object, that is orbiting. We know its period, we know its, the 34 00:02:27,911 --> 00:02:31,739 radius of its orbit. The sun's orbit is roughly circular And 35 00:02:31,739 --> 00:02:36,901 so we can make a calculation, the easiest way we know how to do that is to do our 36 00:02:36,901 --> 00:02:40,851 favorite scaling. Compared to the Earth's orbit around the 37 00:02:40,851 --> 00:02:46,726 sun the mass that the sun is orbiting is related to the solar mass by a The ratio 38 00:02:46,726 --> 00:02:52,527 of the periods to the -2, radius, ratio of the radii ^ 3, plug in all the numbers 39 00:02:52,527 --> 00:02:57,011 and you find 88 billion solar masses is what the sun is orbiting. 40 00:02:57,011 --> 00:03:02,375 Now, that is a little bit higher than the numbers I quoted for the galaxy, but 41 00:03:02,375 --> 00:03:06,677 remember That, and i am going to make a slight fudge here and i will explain if 42 00:03:06,677 --> 00:03:10,779 the galaxy were a symmetric object then the sun would be orbiting precisely that 43 00:03:10,779 --> 00:03:14,321 fraction of a galaxy that is within its own orbit the the galaxy being 44 00:03:14,321 --> 00:03:18,435 spherically symmetric everything outside the everything outside the suns orbit 45 00:03:18,435 --> 00:03:22,420 would have no impact on our orbit. Everything inside it could, It could 46 00:03:22,420 --> 00:03:26,482 equivalently be placed at the center. We talked about that when we did 47 00:03:26,482 --> 00:03:29,587 Newtonian gravity. Now the galaxy is not spherically 48 00:03:29,587 --> 00:03:34,759 symmetric it's distinctly a disc, this adds a a small change to the calculation. 49 00:03:34,759 --> 00:03:39,294 It is still true that everything outside the sun's orbit has zero impact on our 50 00:03:39,294 --> 00:03:43,602 orbital acceleration. however there's a small geometric factor. 51 00:03:43,602 --> 00:03:50,129 that the factor or order one that is associated to the fact that the mass 52 00:03:50,129 --> 00:03:55,392 inside our orbit is mostly located in a disc, were that the case, and, but our 53 00:03:55,392 --> 00:03:59,694 estimate is still good. And we found is what the sun is orbiting 54 00:03:59,694 --> 00:04:05,292 is slightly more than the total mass that we had for the entire Milky Way galaxy. 55 00:04:05,292 --> 00:04:11,202 Galaxy, that's interesting, typically, so, so we want to figure out what it is 56 00:04:11,202 --> 00:04:17,087 we've missed, are there a lot more brown dwarfs or whatever that we thought? To 57 00:04:17,087 --> 00:04:23,283 study this, the way this investigation is done, is typically instead of using this 58 00:04:23,283 --> 00:04:28,915 nice scaling relation, which I . confess is my favorite way to do it. 59 00:04:28,915 --> 00:04:34,766 The way we usually do it is we write that an object that is orbiting at a radius r 60 00:04:34,766 --> 00:04:40,072 with speed v has a c, centripetal acceleration which we computed in the 61 00:04:40,072 --> 00:04:46,357 second week of v^2 over R and that is given by the gravitational, acceleration 62 00:04:46,357 --> 00:04:52,905 about whatever it's orbiting, so, we would put R over here, where M(R) is the 63 00:04:52,905 --> 00:04:59,812 mass inside, the circle of radius R, which this object orbits and I should say 64 00:04:59,812 --> 00:05:03,447 R^2. And now that I've written the correct 65 00:05:03,447 --> 00:05:08,722 formula, I can cancel this R and we see that v^2 is GM(R)/R. 66 00:05:08,722 --> 00:05:12,952 This is exactly one half the escape velocity, if you remember that 67 00:05:12,952 --> 00:05:16,351 calculation. And so the way this is usually written, 68 00:05:16,351 --> 00:05:21,224 is that v squared gives you, if you measure the speed with which something is 69 00:05:21,224 --> 00:05:26,239 orbiting and the radius from the center of which it is orbiting, you can figure 70 00:05:26,239 --> 00:05:31,885 out the mass enclosed within its orbit. Okay the study of galactic rotation 71 00:05:31,885 --> 00:05:35,113 curves is something extremely interesting. 72 00:05:35,113 --> 00:05:40,822 What is it that we expect and what is it that we find? So, here's what we expect, 73 00:05:40,822 --> 00:05:47,122 in the red here I've drawn 2 red graph, curves the first is the Measured stellar 74 00:05:47,122 --> 00:05:51,817 density, and you see that it's an exponential density in the disc, the 75 00:05:51,817 --> 00:05:56,507 stellar population is densest in the center of the disc, and then smoothly 76 00:05:56,507 --> 00:06:01,797 falls off, you see that by the time you get to the sun, it has fallen to about 77 00:06:01,797 --> 00:06:06,397 1/6th of its density at the center, and it continues to fall off rapidly. 78 00:06:06,397 --> 00:06:11,948 this, if you compute how much mass is with With this, density, you find this 79 00:06:11,948 --> 00:06:15,036 red curve. We're going to make an approximation 80 00:06:15,036 --> 00:06:19,794 because it's, it allows me to do a calculation, the approximation I'm 81 00:06:19,794 --> 00:06:24,804 going to do is I'm going to replace, this crazy disc by a uniform disc with a 82 00:06:24,804 --> 00:06:29,500 uniform mass distribution. So I'm going to imagine that I have a 83 00:06:29,500 --> 00:06:35,419 disc of some thickness, say, T. This is a, again assumed uniform, we can 84 00:06:35,419 --> 00:06:39,124 make it some kiloparsec or something like that. 85 00:06:39,124 --> 00:06:43,552 And, some density, row of, and kilo per meter cubed or. 86 00:06:43,552 --> 00:06:48,904 Billion, 10 billion solar masses per kiloparsec cubed, would be the more 87 00:06:48,904 --> 00:06:53,864 standard units to be using here. And so I have a density, and I have a 88 00:06:53,864 --> 00:06:57,879 thickness, and I'm going to give my disc some radius R. 89 00:06:57,879 --> 00:07:03,356 And the way I've fixed all of these parameters, is I fixed T and row, so that 90 00:07:03,356 --> 00:07:08,392 the stellar density, out where the sum is at 8 kiloparsecs agrees. 91 00:07:08,392 --> 00:07:14,361 With the actual stellar density, and then I fix the radius so that the total mass 92 00:07:14,361 --> 00:07:20,514 of this uniform blue disc, is the same as the total mass of the stellar disc. 93 00:07:20,514 --> 00:07:26,045 So, what is M of R, in this case? Well, in this case, M(R) is pretty easy to 94 00:07:26,045 --> 00:07:28,882 write. M(R) is row times the volume. 95 00:07:28,882 --> 00:07:35,821 Of, all of the stars m of by burning the mass of all of the stars enclosed in a 96 00:07:35,821 --> 00:07:42,436 circle of radius r, those stars enclosed in a circle of radius r, form a cylinder. 97 00:07:42,436 --> 00:07:47,683 I'm great at drawing cylinders. A cylinder of radius r and height t. 98 00:07:47,683 --> 00:07:49,132 So. its volume. 99 00:07:49,132 --> 00:07:56,147 Is pi R^2 times t, and I will sometimes fall into using, lower case r, for the 100 00:07:56,147 --> 00:08:01,356 radius, because that's, the way it's done in the field, 101 00:08:01,356 --> 00:08:07,635 So, this is M or R, our calculation said, that V of R, should be if this is what 102 00:08:07,635 --> 00:08:15,587 you are orbiting, V of R, Should be, let me make a correction to this in a minute. 103 00:08:15,587 --> 00:08:19,578 V of R should be the root of G M of R over R. 104 00:08:19,578 --> 00:08:27,812 Now, what is thing going to give me? Well M(R) is row times the area of a disc of 105 00:08:27,812 --> 00:08:34,527 radius R times T as long as R is less. Then R0, which is the radius of my disc, 106 00:08:34,527 --> 00:08:40,687 as long as your orbiting inside what I call the uniform disc, then the farther 107 00:08:40,687 --> 00:08:46,472 you are, the more you are orbiting. Of course if R is bigger than R0, then 108 00:08:46,472 --> 00:08:53,787 the mass inside your orbit is simply N or row times Pi R0^2 t, you're not orbiting 109 00:08:53,787 --> 00:08:56,860 any more. By orbiting farther outside in the 110 00:08:56,860 --> 00:09:00,134 vacuum, if you're orbiting at distances out here. 111 00:09:00,134 --> 00:09:04,813 And so, what is V of R I can do this calculation, and if R is less than R 0, 112 00:09:04,813 --> 00:09:09,918 there's a whole bunch of constants that I'm not going to worry about, but M goes 113 00:09:09,918 --> 00:09:14,811 like R^2. Right? And, and dividing by R, so V is 114 00:09:14,811 --> 00:09:20,409 going to like,well okay. The root of row pi t G times R. 115 00:09:20,409 --> 00:09:24,664 It' going to grow like the square root of R. 116 00:09:24,664 --> 00:09:31,212 So the farther you are from the middle, the faster your orbiting. 117 00:09:31,212 --> 00:09:36,452 by the square root of R, the reason, remember, in the solar system the farther 118 00:09:36,452 --> 00:09:41,081 you are from the sun, the slower you are, but that's because we all orbit the same 119 00:09:41,081 --> 00:09:43,676 mass. We're all orbiting the sun, here the 120 00:09:43,676 --> 00:09:48,289 farther you're orbiting, the more mass you're orbiting, therefore, you're 121 00:09:48,289 --> 00:09:52,851 orbital speed will increase. But, once you hit R0, then of course, the 122 00:09:52,851 --> 00:09:56,785 result is, M no longer depends on R will call of this M. 123 00:09:56,785 --> 00:10:01,662 And so, now it's just root of GM/R. The only R dependence is in the 124 00:10:01,662 --> 00:10:05,583 denominator. So we expect something that grows like 125 00:10:05,583 --> 00:10:11,266 the square root of R, and then decreases like 1 over the square root of R to be 126 00:10:11,266 --> 00:10:15,572 the plot of V as a function of R, which is what I'm about. 127 00:10:15,572 --> 00:10:19,233 To plot, but I wanted you to show, to show you that this is not some, 128 00:10:19,233 --> 00:10:21,818 misticism, we can actually do the calculation. 129 00:10:21,818 --> 00:10:25,631 And so, here is the prediction, Indeed, the blue line does what we 130 00:10:25,631 --> 00:10:28,089 expected, it grows like the square root of R. 131 00:10:28,089 --> 00:10:32,331 This is the plot of a square root, and then once you, are outside the disc, 132 00:10:32,331 --> 00:10:36,999 you've reached the maximum speed, you're orbiting as much as you're ever In orbit, 133 00:10:36,999 --> 00:10:41,888 farther out you're orbiting the same thing but at a larger radius, so your 134 00:10:41,888 --> 00:10:46,209 orbital velocity decreases. Notice that the mass curve the m of r 135 00:10:46,209 --> 00:10:51,048 curve or the blue curve is not a bad approximation considering the simple 136 00:10:51,048 --> 00:10:55,813 assumptions I made, to the red curve they're different inside the sun's 137 00:10:55,813 --> 00:11:00,582 radius, they certainly agree almost exactly past the sun's radius. 138 00:11:00,582 --> 00:11:06,097 Because most of the mass is within that. So, the red curve differs the prediction 139 00:11:06,097 --> 00:11:11,061 for v of r differs closer in, than the sum but past the sun's radius, they 140 00:11:11,061 --> 00:11:14,917 pretty much agree. Okay, this is not yet a real galaxy. 141 00:11:14,917 --> 00:11:20,047 To have a real galaxy, you have to take into account that there's also a bulge in 142 00:11:20,047 --> 00:11:23,337 the middle. So how do we add a bulge in the middle 143 00:11:23,337 --> 00:11:29,237 well, just to be very, sinful, I have added in this picture, this is the red 144 00:11:29,237 --> 00:11:34,762 graph we had before, I've added a bulge which is a uniform from a genius fear of 145 00:11:34,762 --> 00:11:37,937 radius 3 who's mass is adjusted 3 kiloparsecs. 146 00:11:37,937 --> 00:11:43,712 Who's mass is adjusted to be exactly the mass, I've predicted for the the galactic 147 00:11:43,712 --> 00:11:47,522 bulge and so up to radius 3 there's this extra increase. 148 00:11:47,522 --> 00:11:52,546 Because you're not orbiting more and more in the bulge and then past the outside of 149 00:11:52,546 --> 00:11:57,824 the bulge, the total mass orbited gets a constant contribution for the mass and so 150 00:11:57,824 --> 00:12:02,390 the total mass you're orbiting is now this black graph that I've drawn over 151 00:12:02,390 --> 00:12:05,020 here. And from this I can again reproduce a 152 00:12:05,020 --> 00:12:08,632 graph of v of R. I've broken it up into contributions. 153 00:12:08,632 --> 00:12:12,360 this is, this would be your orbit of velocity. 154 00:12:12,360 --> 00:12:18,039 If you only orbited the bulge if you want the fact that this increases in a 155 00:12:18,039 --> 00:12:22,315 straight line. it has to do with the fact that the mass 156 00:12:22,315 --> 00:12:27,645 you're orbiting inside of radius R. As long as you're inside the bulge the 157 00:12:27,645 --> 00:12:31,419 mass you're orbiting is M(R) for R inside the bulge. 158 00:12:33,581 --> 00:12:40,451 M($) is row of the bulge if we're just talking about the bulge * the volume of a 159 00:12:40,451 --> 00:12:43,565 sphere of radius R. Which is R^3. 160 00:12:43,565 --> 00:12:47,972 And so M of R / R which is what goes into G into V. 161 00:12:47,972 --> 00:12:53,707 Goes like lots of constants * R^2 and v^2 is going like R ^ 2 means v goes like R 162 00:12:53,707 --> 00:13:00,607 in a homogeneous ball of homogeneous density, your radial velocity increases 163 00:13:00,607 --> 00:13:04,544 linearly. Okay, and, not your radial, orbital 164 00:13:04,544 --> 00:13:10,212 velocity increases linearly, the outside has to orbit bit faster. 165 00:13:10,212 --> 00:13:15,192 This is great, because this also means, that, V of R proportional to R, is what 166 00:13:15,192 --> 00:13:19,157 would happen if you just took that ball and rotated it rigidly. 167 00:13:19,157 --> 00:13:24,597 So, a homogeneous ball rotating, orbiting itself under the force of gravity rotates 168 00:13:24,597 --> 00:13:27,957 like a rigid ball, at least in the plane of the orbit. 169 00:13:27,957 --> 00:13:32,984 In the plain of the equator, all a straight line drawn from the origin to 170 00:13:32,984 --> 00:13:37,608 the, equator will continue to stay a straight line, everybody orbits the 171 00:13:37,608 --> 00:13:40,432 period of everybody on the equator is the same. 172 00:13:40,432 --> 00:13:45,477 That's an interesting observation. It turns out that this does describe in, the 173 00:13:45,477 --> 00:13:50,271 real galaxy. In real galaxies, the behavior at very small radii. 174 00:13:50,271 --> 00:13:54,223 And, then we have the red curve, which is the same curve we had before. 175 00:13:54,223 --> 00:13:56,500 Adding them together we get this black curve. 176 00:13:56,500 --> 00:14:00,622 The sharp corner here is an artifact, of the sharp corner here, of my, approximate 177 00:14:00,622 --> 00:14:04,528 model of the bulge, but I wasn't about to insert bars, it gets to complicated. 178 00:14:04,528 --> 00:14:08,301 So, I wanted to see that we could make a reasonable prediction as to what the 179 00:14:08,301 --> 00:14:12,633 velocity curve should look like. We already know that we're falling short 180 00:14:12,633 --> 00:14:17,321 because I inserted the measured masses. So this gives the sun a velocity of about 181 00:14:17,321 --> 00:14:22,660 180 meter, kilometers per second, whereas the measured value is 230 kilometers per 182 00:14:22,660 --> 00:14:25,456 second. But we knew we had a short falling in the 183 00:14:25,456 --> 00:14:28,227 mass there. The big question is going to be what 184 00:14:28,227 --> 00:14:33,832 happens out here? Of outside the solar radius and, we have measurements, here 185 00:14:33,832 --> 00:14:38,317 are the measurements. So remember, 8 kilo parsecs is where the 186 00:14:38,317 --> 00:14:42,157 sun is, so this is, the solar radius, the sun is here. 187 00:14:42,157 --> 00:14:48,207 Are measured at about, 225 kilometers per second, I told you that measurement is 188 00:14:48,207 --> 00:14:52,397 subject to corrections. And here is a function of R, is the 189 00:14:52,397 --> 00:14:59,797 observed rotation, speed and, what it is that's being measured are carbon monoxide 190 00:14:59,797 --> 00:15:05,237 clouds and H molecular H. hydrogen clouds and 21 centimeter 191 00:15:05,237 --> 00:15:11,137 emission from atomic hydrogen, Various things have been used to make this 192 00:15:11,137 --> 00:15:15,618 measurement and what we see is indeed the steep. 193 00:15:15,618 --> 00:15:22,275 linear with R, when R = 0, then they decline, when you leave the dense region, 194 00:15:22,275 --> 00:15:27,976 this structure is associated with the galactic bulge, and modelling it more 195 00:15:27,976 --> 00:15:33,030 precisely, we get out to the sun and we are little underestimating. 196 00:15:33,030 --> 00:15:38,942 And then what goes on here is a disaster, the BFR graph starts, Decreasing, as we 197 00:15:38,942 --> 00:15:41,585 predicted. Remember, it was supposed to start 198 00:15:41,585 --> 00:15:46,052 declining, because you're outside, essentially, the entire mass of the disc, 199 00:15:46,052 --> 00:15:48,757 and it's not declining. It's staying constant. 200 00:15:48,757 --> 00:15:53,001 So not only are we missing some mass inside the sun's orbit, but the farther 201 00:15:53,001 --> 00:15:56,552 and farther out we get, the more and more mass we're missing. 202 00:15:56,552 --> 00:16:00,677 Because what we're finding is that M(R)/R is a constant. 203 00:16:00,677 --> 00:16:04,802 The mass enclosed within radius R is growing linearly with R. 204 00:16:04,802 --> 00:16:10,052 And where it stops growing linearly with R? This graph is not going to show us. 205 00:16:10,052 --> 00:16:15,552 it becomes tricky to measure it here, we'll have more luck actually measuring 206 00:16:15,552 --> 00:16:19,353 this with other galaxies. So the resolution of this problem will 207 00:16:19,353 --> 00:16:23,170 wait until we see whether it's some crazy quirk of the Milky Way and of us being 208 00:16:23,170 --> 00:16:26,987 stuck in a disc and not measuring very well, or actually a more general thing. 209 00:16:26,987 --> 00:16:29,919 But I wanted to bring it up, and also we have to do a calculate.