1 00:00:00,000 --> 00:00:25,000 2 00:00:25,000 --> 00:00:28,671 Today we are going to talk about Doppler Effect. 3 00:00:28,671 --> 00:00:32,968 It is best known for sound. That probably reminds you of 4 00:00:32,968 --> 00:00:36,814 your high school days. If you move towards a whistle, 5 00:00:36,814 --> 00:00:40,262 you hear the pitch of the whistle higher than the whistle, 6 00:00:40,262 --> 00:00:43,770 and if you move away from the whistle then you hear a lower 7 00:00:43,770 --> 00:00:45,887 pitch than the pitch of the whistle. 8 00:00:45,887 --> 00:00:49,153 Or, if the whistle moves through you, you hear a higher 9 00:00:49,153 --> 00:00:51,149 pitch. And, when the whistle moves 10 00:00:51,149 --> 00:00:53,387 away from you, you hear a lower pitch. 11 00:00:53,387 --> 00:00:56,350 You may think now, whether you move with a certain 12 00:00:56,350 --> 00:00:59,858 speed towards the whistle or whether the whistle moves with 13 00:00:59,858 --> 00:01:03,487 the same speed to you that you will hear the same increase in 14 00:01:03,487 --> 00:01:06,448 pitch. But that is not true. 15 00:01:06,448 --> 00:01:08,793 There is actually a big difference. 16 00:01:08,793 --> 00:01:13,068 The situation is not symmetric. If you move towards the whistle 17 00:01:13,068 --> 00:01:17,482 with the speed of sound then the frequency that you hear is twice 18 00:01:17,482 --> 00:01:21,206 the frequency of the whistle. But, if the whistle moves 19 00:01:21,206 --> 00:01:25,137 towards you with the speed of sound, you will hear nothing 20 00:01:25,137 --> 00:01:29,344 because all the sound stays with the whistle until the whistle 21 00:01:29,344 --> 00:01:33,355 passes you. And then you hear a huge boom 22 00:01:33,355 --> 00:01:36,604 of sound, the frequency goes to infinity. 23 00:01:36,604 --> 00:01:40,502 You break through the sound barrier, so to speak, 24 00:01:40,502 --> 00:01:44,319 if you survive it. So there is a large asymmetry 25 00:01:44,319 --> 00:01:47,893 between them. The speed of sound is about 340 26 00:01:47,893 --> 00:01:50,898 meters per second at room temperature. 27 00:01:50,898 --> 00:01:55,203 It is about 770 miles per hour. That is the reason why 28 00:01:55,203 --> 00:02:00,238 commercial airplanes do not fly any faster relative to air than 29 00:02:00,238 --> 00:02:05,372 770 miles per hour. Because they are not designed 30 00:02:05,372 --> 00:02:08,154 to break through the sound barrier. 31 00:02:08,154 --> 00:02:12,327 That is the limiting factor for commercial airlines. 32 00:02:12,327 --> 00:02:17,236 Let this be the transmitter of sound and this the receiver of 33 00:02:17,236 --> 00:02:19,199 sound. That could be you. 34 00:02:19,199 --> 00:02:23,699 And let us assume that you move with a certain velocity. 35 00:02:23,699 --> 00:02:28,690 I call this the plus direction. And you move with the velocity 36 00:02:28,690 --> 00:02:33,043 vr. And let the transmitter move 37 00:02:33,043 --> 00:02:38,586 with the velocity vt. And the transmitter produces a 38 00:02:38,586 --> 00:02:44,239 sound f, but what you receive is a frequency f prime. 39 00:02:44,239 --> 00:02:49,891 So, f is the frequency. If you go in velocity in this 40 00:02:49,891 --> 00:02:55,543 direction, that is a positive, and a velocity in this 41 00:02:55,543 --> 00:03:01,956 direction then becomes negative. Then f prime is f times the 42 00:03:01,956 --> 00:03:10,000 speed of sound minus vr divided by the speed of sound minus vt. 43 00:03:10,000 --> 00:03:13,830 And you should be able to derive that on your own. 44 00:03:13,830 --> 00:03:17,426 And it is very asymmetric. And you can see that 45 00:03:17,426 --> 00:03:20,631 immediately. Suppose I make vr the same as 46 00:03:20,631 --> 00:03:24,853 vs, so the upstairs becomes zero, f prime goes to zero. 47 00:03:24,853 --> 00:03:28,527 That is obvious. If you walk away from the sound 48 00:03:28,527 --> 00:03:34,000 source with the speed of sound, the sound never reaches you. 49 00:03:34,000 --> 00:03:38,429 You walk as fast as the sound comes to you so you hear 50 00:03:38,429 --> 00:03:41,605 nothing. It is immediately obvious that 51 00:03:41,605 --> 00:03:46,704 f prime then has to become zero. But, if we now make the sound 52 00:03:46,704 --> 00:03:50,632 source go away from you with the speed of sound, 53 00:03:50,632 --> 00:03:54,644 again, the distance between the two of you grows, 54 00:03:54,644 --> 00:03:59,325 now v of the transmitter I make minus the speed of sound, 55 00:03:59,325 --> 00:04:04,378 notice what happens. The frequency that I receive is 56 00:04:04,378 --> 00:04:08,367 simply one-half the f. So, f prime is now one-half f. 57 00:04:08,367 --> 00:04:11,512 You see an enormous difference in the two. 58 00:04:11,512 --> 00:04:15,731 If you walk away from the source with the speed of sound 59 00:04:15,731 --> 00:04:19,490 and you hear nothing. And, if the sound moves away 60 00:04:19,490 --> 00:04:23,863 from you with the speed of sound, all you hear is half the 61 00:04:23,863 --> 00:04:28,542 frequency of the sound source. I have a 4000 hertz tuning fork 62 00:04:28,542 --> 00:04:32,292 there. And so f is 4000 hertz. 63 00:04:32,292 --> 00:04:38,172 And I am going to move my hand towards you as fast as I can, 64 00:04:38,172 --> 00:04:42,657 which is one meter per second. In other words, 65 00:04:42,657 --> 00:04:47,840 I am the transmitter, so vt is approximately plus one 66 00:04:47,840 --> 00:04:52,425 meter per second. And then you should hear an f 67 00:04:52,425 --> 00:04:58,305 prime which is about 0.3% higher than f, so it is about 4012 68 00:04:58,305 --> 00:05:02,276 hertz. And then I will move my hand 69 00:05:02,276 --> 00:05:05,934 away from you with about one meter per second, 70 00:05:05,934 --> 00:05:09,186 so I have to put the minus sign in there. 71 00:05:09,186 --> 00:05:14,065 And then I will find that the frequency that you will hear is 72 00:05:14,065 --> 00:05:17,804 about 3988 hertz. And the nice thing about this 73 00:05:17,804 --> 00:05:22,276 demonstration is that the difference between the motions 74 00:05:22,276 --> 00:05:26,016 is 24 hertz, and you can very easily hear that. 75 00:05:26,016 --> 00:05:30,000 Here is this tuning fork 4000 hertz. 76 00:05:30,000 --> 00:05:35,000 77 00:05:35,000 --> 00:05:41,330 I will do it a few times. And I will treat you, 78 00:05:41,330 --> 00:05:47,798 too, because I gave them preferential treatment. 79 00:05:47,798 --> 00:05:54,128 Do you hear it? Higher pitch when I come to you 80 00:05:54,128 --> 00:05:59,633 and clearly a lower pitch when I go away. 81 00:05:59,633 --> 00:06:08,921 A difference of about 24 hertz. Suppose now that I rotate a 82 00:06:08,921 --> 00:06:16,474 sound source in a circle. I rotate it around like so, 83 00:06:16,474 --> 00:06:23,737 angular velocity omega, radius r, this is where you 84 00:06:23,737 --> 00:06:31,000 are, and let the circumferential speed be v0. 85 00:06:31,000 --> 00:06:36,296 It goes around with constant speed, so v0 is omega R. 86 00:06:36,296 --> 00:06:39,962 And the sound source has frequency f. 87 00:06:39,962 --> 00:06:45,972 It is clear that if it moves away from you then f prime will 88 00:06:45,972 --> 00:06:50,148 be smaller than f. Here f prime will be f. 89 00:06:50,148 --> 00:06:54,018 And here f prime will be larger than f. 90 00:06:54,018 --> 00:07:00,027 And so, as you listen to this sound, you will hear the sound 91 00:07:00,027 --> 00:07:06,842 change in a sinusoidal way. Because what matters is not the 92 00:07:06,842 --> 00:07:11,518 speed of the source but the radial component in your 93 00:07:11,518 --> 00:07:15,094 direction. And the radial component here 94 00:07:15,094 --> 00:07:18,487 is v0 towards you, here away from you, 95 00:07:18,487 --> 00:07:21,879 but the radial component here is zero. 96 00:07:21,879 --> 00:07:27,106 If you arbitrarily call this time equal zero here then the 97 00:07:27,106 --> 00:07:31,415 radial velocity, the component in your direction 98 00:07:31,415 --> 00:07:36,000 is then v0 times the sine of omega t. 99 00:07:36,000 --> 00:07:42,298 And so, if you not only listen, but if you actually record, 100 00:07:42,298 --> 00:07:47,619 as a function of time, the frequency that you hear 101 00:07:47,619 --> 00:07:51,203 then you hear something like this. 102 00:07:51,203 --> 00:07:57,719 And this then is the mean value f which is transmitted by the 103 00:07:57,719 --> 00:08:02,228 sound source. And now you can ask yourself 104 00:08:02,228 --> 00:08:05,490 the questions what do you learn from this? 105 00:08:05,490 --> 00:08:08,832 And it is amazing what you learn from this. 106 00:08:08,832 --> 00:08:13,527 You can close your eyes and just record this sound signal as 107 00:08:13,527 --> 00:08:17,427 a function of time. The first thing that you learn 108 00:08:17,427 --> 00:08:21,724 from it is the period t, which is 2pi divided by omega. 109 00:08:21,724 --> 00:08:25,464 You know the period of rotation, you know omega. 110 00:08:25,464 --> 00:08:30,000 But you also record what f prime maximum is. 111 00:08:30,000 --> 00:08:33,346 And you also record f prime minimum. 112 00:08:33,346 --> 00:08:38,797 And so, with the Doppler shift equation that we have here, 113 00:08:38,797 --> 00:08:42,622 you can immediately calculate what v0 is. 114 00:08:42,622 --> 00:08:46,448 You know also v0 but, since v is omega r, 115 00:08:46,448 --> 00:08:50,846 you also know R. Just imagine by recording that 116 00:08:50,846 --> 00:08:56,202 sound signal as a function of time, you can determine the 117 00:08:56,202 --> 00:09:00,314 period of rotation, this omega, the speed in 118 00:09:00,314 --> 00:09:05,000 circular orbit and the radius even. 119 00:09:05,000 --> 00:09:09,054 I want to demonstrate this to you just quantitatively. 120 00:09:09,054 --> 00:09:12,420 I have here a whistle. I don't even know what 121 00:09:12,420 --> 00:09:16,475 frequencies it produces. It maybe is a mixture of many 122 00:09:16,475 --> 00:09:19,688 frequencies. That is not so much the point. 123 00:09:19,688 --> 00:09:24,278 I want you to hear that if I twirl it around now it no longer 124 00:09:24,278 --> 00:09:27,491 the matter of high/low, high/low, high/low, 125 00:09:27,491 --> 00:09:31,163 but it now is a gradual change to its high pitch, 126 00:09:31,163 --> 00:09:35,891 low pitch. And then you hear this in 127 00:09:35,891 --> 00:09:38,666 between. Can you hear it? 128 00:09:38,666 --> 00:09:45,142 It is clearly high at times and it is low at other times, 129 00:09:45,142 --> 00:09:50,115 but it is now a cosinusoidal change in time. 130 00:09:50,115 --> 00:09:55,088 Keep in mind, which is going to be important 131 00:09:55,088 --> 00:10:00,176 in what follows today, that from this you can 132 00:10:00,176 --> 00:10:07,000 determine the periods, the speed and the radius. 133 00:10:07,000 --> 00:10:12,092 Now, electromagnetic radiation also shows Doppler shift. 134 00:10:12,092 --> 00:10:17,370 That means if you move towards a source of electromagnetic 135 00:10:17,370 --> 00:10:22,277 radiation, you will record a higher frequency than was 136 00:10:22,277 --> 00:10:26,259 transmitted. And, if the source of radiation 137 00:10:26,259 --> 00:10:30,148 comes to you, you will also record a higher 138 00:10:30,148 --> 00:10:34,008 frequency. And, if you move away from each 139 00:10:34,008 --> 00:10:36,600 other, you will record a lower frequency. 140 00:10:36,600 --> 00:10:40,617 And the Doppler shift equation for electromagnetic radiation is 141 00:10:40,617 --> 00:10:44,051 not so easy to derive. You need special relativity for 142 00:10:44,051 --> 00:10:46,384 that, but I will give you the result. 143 00:10:46,384 --> 00:10:50,012 I want you to appreciate that there is no such thing as a 144 00:10:50,012 --> 00:10:53,511 velocity of the receiver and a separate velocity of the 145 00:10:53,511 --> 00:10:56,557 transmitter now. Because, in special relativity, 146 00:10:56,557 --> 00:11:00,380 the only thing that matters is the relative velocity between 147 00:11:00,380 --> 00:11:03,875 the two. It is an illegal question even 148 00:11:03,875 --> 00:11:07,625 to ask who is moving towards whom and who is moving away from 149 00:11:07,625 --> 00:11:09,750 whom. There is only one velocity in 150 00:11:09,750 --> 00:11:12,812 special relativity. Let this be the transmitter of 151 00:11:12,812 --> 00:11:15,937 electromagnetic radiation and this be the receiver. 152 00:11:15,937 --> 00:11:19,187 And this is the relative velocity v and this angle is 153 00:11:19,187 --> 00:11:21,250 theta. I am not trying to tell you 154 00:11:21,250 --> 00:11:24,687 that it is this object that is moving in this direction. 155 00:11:24,687 --> 00:11:29,000 It could be this one that is moving in this direction. 156 00:11:29,000 --> 00:11:33,645 It could be this one that is moving in this direction. 157 00:11:33,645 --> 00:11:38,904 This is the relative velocity between the two and the line of 158 00:11:38,904 --> 00:11:42,059 sight. The component of the velocity, 159 00:11:42,059 --> 00:11:46,354 which we call the radial component, which plays an 160 00:11:46,354 --> 00:11:51,000 important role there, too, is v cosine theta. 161 00:11:51,000 --> 00:11:56,106 And, with this picture in mind, if this one radiates a 162 00:11:56,106 --> 00:12:01,598 wavelength lambda with frequency f and this one receives a 163 00:12:01,598 --> 00:12:07,091 wavelength lambda prime with frequency f prime then lambda 164 00:12:07,091 --> 00:12:12,679 prime is c divided by f prime and lambda is c divided by f. 165 00:12:12,679 --> 00:12:18,750 And I will give you the results of the Doppler shift equation in 166 00:12:18,750 --> 00:12:22,893 terms of lambda, and then you can always use 167 00:12:22,893 --> 00:12:28,000 that relationship to do it in terms of f. 168 00:12:28,000 --> 00:12:33,747 And the complete relationship is that lambda prime, 169 00:12:33,747 --> 00:12:40,643 the one that you will receive, is lambda times one minus beta 170 00:12:40,643 --> 00:12:47,080 cosine theta divided by the square root of one minus beta 171 00:12:47,080 --> 00:12:50,413 squared. And beta is v over c, 172 00:12:50,413 --> 00:12:56,850 c being the speed of light. You see here that if theta is 173 00:12:56,850 --> 00:13:03,517 smaller than 90 degrees but larger than zero that means you 174 00:13:03,517 --> 00:13:10,454 are approaching each other. This stands for approaching. 175 00:13:10,454 --> 00:13:15,362 That means that lambda prime is then smaller than lambda, 176 00:13:15,362 --> 00:13:20,007 so you record smaller wavelengths when it is coming to 177 00:13:20,007 --> 00:13:22,812 you. And we have a name for that, 178 00:13:22,812 --> 00:13:26,756 which is blueshift. And the reason why that is 179 00:13:26,756 --> 00:13:32,144 called blueshift is obvious. Because blue light has a 180 00:13:32,144 --> 00:13:35,094 shorter wavelength than red light. 181 00:13:35,094 --> 00:13:40,545 Even though this Doppler Effect may not even be in the visible 182 00:13:40,545 --> 00:13:45,014 part of the spectrum, we still call that blueshift. 183 00:13:45,014 --> 00:13:49,572 When relative velocity is approaching you go towards 184 00:13:49,572 --> 00:13:53,772 shorter wavelengths, and we call that blueshift. 185 00:13:53,772 --> 00:13:58,688 Now, if theta is between 180 degrees and 90 degrees that 186 00:13:58,688 --> 00:14:03,961 means you are receding from each other, then lambda prime is 187 00:14:03,961 --> 00:14:08,625 larger than lambda. And, of course, 188 00:14:08,625 --> 00:14:12,850 you guessed it. That is called redshift. 189 00:14:12,850 --> 00:14:18,375 And astronomers talk all the time about redshift and 190 00:14:18,375 --> 00:14:20,000 blueshift. 191 00:14:20,000 --> 00:14:25,000 192 00:14:25,000 --> 00:14:29,291 If beta square is much, much smaller than one, 193 00:14:29,291 --> 00:14:34,346 you can forget about the downstairs, if you like that. 194 00:14:34,346 --> 00:14:38,923 If you take beta, say, 10% of the speed of light, 195 00:14:38,923 --> 00:14:43,787 which is substantial, which is 30,000 kilometers per 196 00:14:43,787 --> 00:14:47,888 second, so beta is 0.1, then beta squared is 197 00:14:47,888 --> 00:14:52,656 approximately 0.01. And so downstairs one minus one 198 00:14:52,656 --> 00:14:56,089 hundred you could, through reasonable 199 00:14:56,089 --> 00:15:00,000 approximation, ignore that. 200 00:15:00,000 --> 00:15:04,363 But that depends, of course, on the accuracy that 201 00:15:04,363 --> 00:15:06,363 you want. In that case, 202 00:15:06,363 --> 00:15:11,727 you would get lambda prime is approximately lambda times one 203 00:15:11,727 --> 00:15:15,000 minus beta times cosine theta. 204 00:15:15,000 --> 00:15:20,000 205 00:15:20,000 --> 00:15:25,696 And if now you want to know it in terms of frequency then f 206 00:15:25,696 --> 00:15:31,000 prime is roughly f times one plus beta cosine theta. 207 00:15:31,000 --> 00:15:34,655 And I will leave you with proving that this is indeed then 208 00:15:34,655 --> 00:15:37,862 a good approximation. Doppler shifts are being used 209 00:15:37,862 --> 00:15:41,646 in sports to measure the speed of tennis balls and baseballs 210 00:15:41,646 --> 00:15:45,237 when they are being pitched. Radar is sent to the object, 211 00:15:45,237 --> 00:15:49,213 and it reflects off the object. And so you can measure this way 212 00:15:49,213 --> 00:15:51,650 the speed. Weather radar works the same 213 00:15:51,650 --> 00:15:53,831 way. You can measure the speed with 214 00:15:53,831 --> 00:15:58,000 which weather approaches you or goes away from you. 215 00:15:58,000 --> 00:16:01,376 The police can check the speed of your car. 216 00:16:01,376 --> 00:16:04,834 Sends radar to you, it bounces off your car, 217 00:16:04,834 --> 00:16:07,407 and then it can sense your speed. 218 00:16:07,407 --> 00:16:12,150 And, in my problem set seven, I think I have a problem based 219 00:16:12,150 --> 00:16:14,884 on that. If you are approaching the 220 00:16:14,884 --> 00:16:17,939 police radially, if theta is zero then, 221 00:16:17,939 --> 00:16:23,005 of course, you would simply get one minus beta here and one plus 222 00:16:23,005 --> 00:16:24,854 beta. Theta can be zero, 223 00:16:24,854 --> 00:16:27,427 of course. It doesn't have to be, 224 00:16:27,427 --> 00:16:32,080 but it can be. You can imagine how important 225 00:16:32,080 --> 00:16:36,455 the role was that this played in astronomy because this allows 226 00:16:36,455 --> 00:16:40,257 you now to calculate the velocity of stars relative to 227 00:16:40,257 --> 00:16:42,552 our solar system, relative to us, 228 00:16:42,552 --> 00:16:46,641 because almost all stars in their spectrum show absorption 229 00:16:46,641 --> 00:16:49,295 lines. And, if you know the absorption 230 00:16:49,295 --> 00:16:53,742 line frequency or the absorption line wavelength as measured in 231 00:16:53,742 --> 00:16:56,540 our laboratory, then you can measure the 232 00:16:56,540 --> 00:17:00,126 velocity of that star. And I will give a particular 233 00:17:00,126 --> 00:17:04,000 example so that you see this at work. 234 00:17:04,000 --> 00:17:07,064 There is a star, and I picked only one star, 235 00:17:07,064 --> 00:17:11,269 which is called Delta Leporis. The reason why I give you the 236 00:17:11,269 --> 00:17:15,261 name is not to remember it but to know it is a real star. 237 00:17:15,261 --> 00:17:18,112 And the star, in its absorption spectrum, 238 00:17:18,112 --> 00:17:22,246 shows a very clear dark line which is the result of calcium 239 00:17:22,246 --> 00:17:25,738 in the atmosphere. Perhaps you remember that I did 240 00:17:25,738 --> 00:17:29,730 a demonstration here of sodium absorption that I had here 241 00:17:29,730 --> 00:17:33,578 creating an atmosphere of sodium vapor which created an 242 00:17:33,578 --> 00:17:39,561 absorption line in the spectrum. Well, in the same sense, 243 00:17:39,561 --> 00:17:45,857 you will see calcium absorption line in the spectrum of this 244 00:17:45,857 --> 00:17:49,059 star. It is actually called the 245 00:17:49,059 --> 00:17:53,328 calcium K line. It has nothing to do with 246 00:17:53,328 --> 00:17:56,316 potassium. In the laboratory, 247 00:17:56,316 --> 00:18:01,118 we know wavelengths very well. It is 3933.664, 248 00:18:01,118 --> 00:18:05,939 think, angstrom. I like to work in angstroms. 249 00:18:05,939 --> 00:18:10,163 Some of you prefer nanometers. One angstrom is 10 to the minus 250 00:18:10,163 --> 00:18:12,933 10 meters, just because we always worked, 251 00:18:12,933 --> 00:18:15,288 when I was a student, in angstroms. 252 00:18:15,288 --> 00:18:18,058 But, of course, if you prefer nanometers, 253 00:18:18,058 --> 00:18:20,689 that is fine. And, when you observe the 254 00:18:20,689 --> 00:18:24,637 wavelength in the spectrum, you will see that it is lambda 255 00:18:24,637 --> 00:18:29,000 plus 1.298 angstroms. The accuracies are stunning. 256 00:18:29,000 --> 00:18:33,035 You can determine wavelengths. And so you clearly see that 257 00:18:33,035 --> 00:18:35,442 lambda prime is larger than lambda. 258 00:18:35,442 --> 00:18:39,690 That means there is redshift. And when there is redshift that 259 00:18:39,690 --> 00:18:42,309 means we are receding from each other. 260 00:18:42,309 --> 00:18:45,141 And we can calculate the radial velocity. 261 00:18:45,141 --> 00:18:48,539 All you know is the radial velocity in this case. 262 00:18:48,539 --> 00:18:52,716 And so with that information you get that beta cosine theta, 263 00:18:52,716 --> 00:18:56,893 if you use that relationship, and the Doppler shift equation 264 00:18:56,893 --> 00:19:01,000 you get minus 3.3 times 10 to the minus 4. 265 00:19:01,000 --> 00:19:05,503 And so you multiply that by c and then you get that v cosine 266 00:19:05,503 --> 00:19:09,931 theta is then very close to minus 99 kilometers per second. 267 00:19:09,931 --> 00:19:13,748 And so we are moving away from each other radially, 268 00:19:13,748 --> 00:19:18,328 I have no information in this direction radially at about 100 269 00:19:18,328 --> 00:19:22,068 kilometers per second. It is illegal to ask who is 270 00:19:22,068 --> 00:19:25,885 moving, they or we. It is only a matter of relative 271 00:19:25,885 --> 00:19:31,000 motion which is all that matters in special relativity. 272 00:19:31,000 --> 00:19:36,799 Now, Doppler shift measurement of stars and galaxies have had a 273 00:19:36,799 --> 00:19:42,319 spectacular impact on the way that we think about the world, 274 00:19:42,319 --> 00:19:45,874 about ourselves and about the universe. 275 00:19:45,874 --> 00:19:50,177 And I will discuss now only two examples which, 276 00:19:50,177 --> 00:19:54,106 in my view, are the most impressive of all. 277 00:19:54,106 --> 00:20:00,000 The consequences are very far reaching in both cases. 278 00:20:00,000 --> 00:20:03,096 The first thing I want to discuss with you is the 279 00:20:03,096 --> 00:20:06,645 discovery of black hole binaries; how that came about as 280 00:20:06,645 --> 00:20:09,096 a result of Doppler shift measurements. 281 00:20:09,096 --> 00:20:11,935 There are many stars that are binary systems. 282 00:20:11,935 --> 00:20:15,935 That means there are two stars going about the common center of 283 00:20:15,935 --> 00:20:17,225 mass. Our sun is not. 284 00:20:17,225 --> 00:20:20,774 Our sun is a single star. If you look at a star which is 285 00:20:20,774 --> 00:20:23,612 a binary system, you may only see one star in 286 00:20:23,612 --> 00:20:28,000 the sky because you couldn't spatially resolve them. 287 00:20:28,000 --> 00:20:32,348 But you will know that it is a double star because you see the 288 00:20:32,348 --> 00:20:34,986 absorption lines of the spectrum move. 289 00:20:34,986 --> 00:20:37,909 Suppose you see only one of the two stars. 290 00:20:37,909 --> 00:20:41,331 Suppose one is invisible because it is too faint, 291 00:20:41,331 --> 00:20:45,323 but the other is very bright. You will see the absorption 292 00:20:45,323 --> 00:20:48,104 lines do this. They move back and forth. 293 00:20:48,104 --> 00:20:52,025 And you can tell what the orbital period of the star is. 294 00:20:52,025 --> 00:20:56,231 If both stars produce enough light, if one star comes to you 295 00:20:56,231 --> 00:20:59,938 that gives you blueshift, the other star must go away 296 00:20:59,938 --> 00:21:05,000 from you because they go around each other like that. 297 00:21:05,000 --> 00:21:10,474 Now you see one set of lines that go like this and the other 298 00:21:10,474 --> 00:21:14,835 set of lines from the other star goes like this. 299 00:21:14,835 --> 00:21:20,402 And so you now see the motion of the absorption lines of both 300 00:21:20,402 --> 00:21:23,278 stars. In a situation like that, 301 00:21:23,278 --> 00:21:28,195 that you see the spectral lines moving back and forth, 302 00:21:28,195 --> 00:21:33,937 you know it is a binary system. And so we know of many stars 303 00:21:33,937 --> 00:21:37,742 whereby we only really see a spectrum of one because the 304 00:21:37,742 --> 00:21:41,132 companion is too faint. We still know that it is a 305 00:21:41,132 --> 00:21:45,144 binary system because of the motion of the absorption lines 306 00:21:45,144 --> 00:21:48,188 in the spectrum. What you can derive now from 307 00:21:48,188 --> 00:21:52,408 that is the period of the orbit, and you can find the radii of 308 00:21:52,408 --> 00:21:55,729 the orbits if you see both pairs of lines moving. 309 00:21:55,729 --> 00:21:58,911 And, therefore, you can find the orbital speeds 310 00:21:58,911 --> 00:22:02,094 in exactly the same way that we did with sound, 311 00:22:02,094 --> 00:22:08,010 period, speed and radius. Now I want to work this out in 312 00:22:08,010 --> 00:22:12,525 quite a bit more detail which then ultimately, 313 00:22:12,525 --> 00:22:17,041 as you will see, led to the discovery of black 314 00:22:17,041 --> 00:22:21,958 holes in binary systems. Let's make a picture of a 315 00:22:21,958 --> 00:22:27,176 simple binary system whereby the orbits are circular. 316 00:22:27,176 --> 00:22:32,193 That is a little special, but it will get the point 317 00:22:32,193 --> 00:22:37,625 across. This is the orbit of one star 318 00:22:37,625 --> 00:22:42,500 and this is the orbit of the other star. 319 00:22:42,500 --> 00:22:48,000 It is a circular orbit. And you are here. 320 00:22:48,000 --> 00:22:53,000 321 00:22:53,000 --> 00:22:59,426 This is the center of mass. And let this star have a mass 322 00:22:59,426 --> 00:23:04,264 m2. And let its speed in orbit be 323 00:23:04,264 --> 00:23:08,823 v2. And let this separation from 324 00:23:08,823 --> 00:23:16,911 the center of mass be r2. The other star must be exactly 325 00:23:16,911 --> 00:23:21,764 here, m1, and that has a speed v1. 326 00:23:21,764 --> 00:23:30,000 And let its distance to the center of mass be r1. 327 00:23:30,000 --> 00:23:34,233 Circular orbits, the center of mass means that 328 00:23:34,233 --> 00:23:37,620 m1 r1 is m2r2. If you don't remember, 329 00:23:37,620 --> 00:23:42,701 go back to your 8.01 notes. And you are observing here, 330 00:23:42,701 --> 00:23:48,440 on earth, the optical spectra. You see the absorption lines in 331 00:23:48,440 --> 00:23:52,674 Doppler shifts. You see them moving like this. 332 00:23:52,674 --> 00:23:58,319 And so you will find from star number one you get the periods 333 00:23:58,319 --> 00:24:02,271 of the orbit, you get its velocity in orbit 334 00:24:02,271 --> 00:24:06,940 and you get r1. Just like we got from the 335 00:24:06,940 --> 00:24:09,191 sound. There is no difference. 336 00:24:09,191 --> 00:24:13,537 No you have star number two. You get the period in orbit. 337 00:24:13,537 --> 00:24:17,185 You can check now that you get the right answer. 338 00:24:17,185 --> 00:24:21,531 You have v2 and you have r2. All of that comes out of the 339 00:24:21,531 --> 00:24:25,023 Doppler shift measurements. But there is more. 340 00:24:25,023 --> 00:24:29,292 You know Newton's Law of Gravity which leads to Kepler's 341 00:24:29,292 --> 00:24:34,303 Third Law Which is that T2 equals 4pi 342 00:24:34,303 --> 00:24:41,523 squared times r1 plus r2 to the power of three divided by m1 343 00:24:41,523 --> 00:24:46,784 plus m2 times G, which is your gravitational 344 00:24:46,784 --> 00:24:48,987 constant. Now look. 345 00:24:48,987 --> 00:24:55,105 You know r1 and you know r2 from your Doppler shift 346 00:24:55,105 --> 00:25:00,000 measurements, so you know this. 347 00:25:00,000 --> 00:25:04,340 You know the period from your Doppler shift measurements so 348 00:25:04,340 --> 00:25:07,707 you know this. But what you don't know is what 349 00:25:07,707 --> 00:25:11,074 is m1 plus m2, but you know that m1r1 is m2r2. 350 00:25:11,074 --> 00:25:15,639 So we have two equations with two unknowns and you find m1 and 351 00:25:15,639 --> 00:25:18,108 m2. Think about this for a minute. 352 00:25:18,108 --> 00:25:21,625 Out of the Doppler shifts of this binary system, 353 00:25:21,625 --> 00:25:25,591 you get their orbital radii, you get their velocities, 354 00:25:25,591 --> 00:25:30,081 you get their period in orbit, But you even get the masses of 355 00:25:30,081 --> 00:25:34,766 the individual objects. Now, when you observe from 356 00:25:34,766 --> 00:25:38,366 earth, you are probably not in the plane of the orbit. 357 00:25:38,366 --> 00:25:40,472 Notice what I did very cleverly. 358 00:25:40,472 --> 00:25:42,850 I put you in the plane of the orbit. 359 00:25:42,850 --> 00:25:45,907 I said you are here, which is the plane of the 360 00:25:45,907 --> 00:25:48,625 blackboard. In reality, you will probably 361 00:25:48,625 --> 00:25:51,953 not be in that plane. You will not see the orbital 362 00:25:51,953 --> 00:25:54,875 period [add on, like this it also adds on?], 363 00:25:54,875 --> 00:25:59,420 but it will be tilted a little. And, if it is tilted, 364 00:25:59,420 --> 00:26:03,140 then the radial velocity, which is the only one you will 365 00:26:03,140 --> 00:26:07,063 measure, will be lower than v2. In fact, you can easily see 366 00:26:07,063 --> 00:26:11,190 that suppose the orbit was like this, so you were on earth and 367 00:26:11,190 --> 00:26:15,045 they go around each other just like this, then there is no 368 00:26:15,045 --> 00:26:18,698 radial velocity at all. The inclination of the orbit is 369 00:26:18,698 --> 00:26:21,742 very important, but I will not further address 370 00:26:21,742 --> 00:26:25,665 this today as it does not affect the basic principle behind 371 00:26:25,665 --> 00:26:27,965 Doppler shift. But you can imagine, 372 00:26:27,965 --> 00:26:31,956 of course, that this is a key issue for astronomers to get a 373 00:26:31,956 --> 00:26:37,955 handle on the inclination. In our galaxy there are a few 374 00:26:37,955 --> 00:26:43,764 hundred, I will lower this and get it back up later again, 375 00:26:43,764 --> 00:26:49,675 very special binaries whereby one star is more or less like 376 00:26:49,675 --> 00:26:51,917 the sun. Pretty common. 377 00:26:51,917 --> 00:26:56,299 Pretty boring. But there is another one very 378 00:26:56,299 --> 00:27:00,885 small in size; a neutron star or a black hole, 379 00:27:00,885 --> 00:27:07,000 as you will see very shortly, that is very close. 380 00:27:07,000 --> 00:27:10,780 We call it a close binary, not because they are close to 381 00:27:10,780 --> 00:27:13,598 us but the two objects are close together. 382 00:27:13,598 --> 00:27:17,447 This one could be a neutron star or, as you will see very 383 00:27:17,447 --> 00:27:20,815 shortly, a black hole. And they go around a common 384 00:27:20,815 --> 00:27:23,771 center of mass. Say in the blackboard is the 385 00:27:23,771 --> 00:27:26,108 plane and they go around like this. 386 00:27:26,108 --> 00:27:29,545 And there is a point here between the two where the 387 00:27:29,545 --> 00:27:33,257 gravitational pull in one direction toward this star is 388 00:27:33,257 --> 00:27:38,000 the same as the gravitational pull in this direction. 389 00:27:38,000 --> 00:27:40,415 It has a name. We call it the inner LaGrange 390 00:27:40,415 --> 00:27:42,213 point. There is also such a point 391 00:27:42,213 --> 00:27:44,460 somewhere between the earth and the moon. 392 00:27:44,460 --> 00:27:47,550 It is very close to the moon, but there is such a point. 393 00:27:47,550 --> 00:27:50,696 And so, if the inner Lagrange point lays here so that the 394 00:27:50,696 --> 00:27:53,955 force in this direction on the test mass is the same as the 395 00:27:53,955 --> 00:27:57,044 force in that direction, if it lies under the surface of 396 00:27:57,044 --> 00:28:00,359 that star then the matter that is on this side here wants to 397 00:28:00,359 --> 00:28:04,673 float towards the neutron star. It is energetically more 398 00:28:04,673 --> 00:28:08,507 favorable because the force in this direction is larger. 399 00:28:08,507 --> 00:28:12,760 Now, since they go around each other, it cannot fall radially. 400 00:28:12,760 --> 00:28:16,944 And so what will happen is that this matter from this star is 401 00:28:16,944 --> 00:28:20,918 going to spiral in through a disk and finally find its way 402 00:28:20,918 --> 00:28:24,264 onto the neutron star. It shouldn't surprise you, 403 00:28:24,264 --> 00:28:27,402 therefore, that this star is called the donor, 404 00:28:27,402 --> 00:28:30,679 it provides the mass, and this small object here 405 00:28:30,679 --> 00:28:34,444 could be a neutron star is called the accretor and this 406 00:28:34,444 --> 00:28:38,000 disk is called the accretion disk. 407 00:28:38,000 --> 00:28:46,761 Let us take a closer look at this neutron star. 408 00:28:46,761 --> 00:28:52,857 I am going to make a blowup here. 409 00:28:52,857 --> 00:29:04,476 This is the neutron star which has mass m and it has radius R. 410 00:29:04,476 --> 00:29:15,333 And I take a little bit of matter, a little test mass from 411 00:29:15,333 --> 00:29:26,952 a very large distance and I let it fall onto the neutron star. 412 00:29:26,952 --> 00:29:32,089 29:02 This is the speed with which it 413 00:29:32,089 --> 00:29:35,460 hits the neutron star. And so the little m cancels. 414 00:29:35,460 --> 00:29:39,640 It does not make any difference whether this is a large mass or 415 00:29:39,640 --> 00:29:42,134 a small mass. It will always reach the 416 00:29:42,134 --> 00:29:44,292 neutron star with the same speed. 417 00:29:44,292 --> 00:29:48,202 And you will see that that speed is then the square root of 418 00:29:48,202 --> 00:29:51,168 2MG divided by R. You have seen this in 8.01. 419 00:29:51,168 --> 00:29:54,943 You may also have seen it in terms of what we call escape 420 00:29:54,943 --> 00:29:57,101 velocity. If you ask yourself the 421 00:29:57,101 --> 00:30:01,078 question if you were on the surface of the neutron star what 422 00:30:01,078 --> 00:30:06,000 is the speed that you need to make it out to infinity? 423 00:30:06,000 --> 00:30:08,402 In other words, to break away from the 424 00:30:08,402 --> 00:30:10,870 gravitational pull of the neutron star. 425 00:30:10,870 --> 00:30:13,597 That is exactly that same speed, of course. 426 00:30:13,597 --> 00:30:16,454 It has to do with the conservation of energy. 427 00:30:16,454 --> 00:30:19,376 You can also think of this as escape velocity. 428 00:30:19,376 --> 00:30:22,948 Now comes the amazing thing. If you take a neutron star, 429 00:30:22,948 --> 00:30:26,584 the mass of a neutron star is very roughly one and a half 430 00:30:26,584 --> 00:30:31,000 times the mass of the sun. This is the symbol for sun. 431 00:30:31,000 --> 00:30:35,101 And the radius for a neutron star is about ten kilometers. 432 00:30:35,101 --> 00:30:38,987 If you substitute those numbers in there you will get a 433 00:30:38,987 --> 00:30:42,369 phenomenal speed. You get 200,000 kilometers per 434 00:30:42,369 --> 00:30:45,104 second which is 70% the speed of light. 435 00:30:45,104 --> 00:30:49,205 Whenever any matter falls onto a neutron star from a large 436 00:30:49,205 --> 00:30:53,379 distance, it hits the neutron star with 70% of the speed of 437 00:30:53,379 --> 00:30:55,466 light. This is kinetic energy. 438 00:30:55,466 --> 00:31:00,000 It hits the surface and is all converted to heat. 439 00:31:00,000 --> 00:31:04,313 And so, if enough kilograms per second fall onto the neutron 440 00:31:04,313 --> 00:31:08,041 star, the surface of the neutron star gets very hot. 441 00:31:08,041 --> 00:31:12,355 And we know there are cases where it can get as hot as about 442 00:31:12,355 --> 00:31:15,937 10 million degrees. And at 10 million degrees this 443 00:31:15,937 --> 00:31:20,396 neutron star will radiate almost all its energy in the x-rays, 444 00:31:20,396 --> 00:31:24,198 very little in the optical, almost all in the x-rays. 445 00:31:24,198 --> 00:31:28,000 And so it becomes a strong x-ray source. 446 00:31:28,000 --> 00:31:31,918 To make you appreciate the energy release when something 447 00:31:31,918 --> 00:31:35,410 hits a neutron star, if you take a marshmallow and 448 00:31:35,410 --> 00:31:39,471 you throw a marshmallow from a large distance on a neutron 449 00:31:39,471 --> 00:31:43,604 star, when it hits the surface the energy that is released, 450 00:31:43,604 --> 00:31:47,523 that means the explosion that is caused by the impact is 451 00:31:47,523 --> 00:31:51,584 comparable to the energy that was released when the atomic 452 00:31:51,584 --> 00:31:57,000 bomb was thrown on Hiroshima at the end of the Second World War. 453 00:31:57,000 --> 00:32:00,852 That enormous amount of energy that causes this incredible 454 00:32:00,852 --> 00:32:04,908 explosion is comparable to the energy that is released when a 455 00:32:04,908 --> 00:32:08,017 marshmallow hits the surface of a neutron star. 456 00:32:08,017 --> 00:32:11,735 Now, when you look through an optical telescope from the 457 00:32:11,735 --> 00:32:15,520 ground and you look at this system, you will only see the 458 00:32:15,520 --> 00:32:19,170 light from the donor because there is very little light 459 00:32:19,170 --> 00:32:23,091 coming from the neutron star because there is an enormously 460 00:32:23,091 --> 00:32:26,133 small surface. Most of the energy comes out in 461 00:32:26,133 --> 00:32:29,107 x-rays anyhow, so you only see the light from 462 00:32:29,107 --> 00:32:33,066 the donor. Astronomers have learned over 463 00:32:33,066 --> 00:32:36,771 the years, this has nothing to do with Doppler shift, 464 00:32:36,771 --> 00:32:40,262 that when you look at the spectrum of a star they, 465 00:32:40,262 --> 00:32:43,469 in general, can tell how massive that star is. 466 00:32:43,469 --> 00:32:46,675 In other words, a star which is ten times more 467 00:32:46,675 --> 00:32:50,808 massive than our sun has a very different spectrum than the 468 00:32:50,808 --> 00:32:54,014 spectrum from the sun. And so they look at the 469 00:32:54,014 --> 00:32:58,004 spectrum and say this is probably five times more massive 470 00:32:58,004 --> 00:33:02,284 than the sun. This is probably 20 times more 471 00:33:02,284 --> 00:33:06,167 massive than the sun. Astronomers can tell by simply 472 00:33:06,167 --> 00:33:10,888 looking at the spectrum of the donor what the mass of the donor 473 00:33:10,888 --> 00:33:13,172 is. Let's call this star number 474 00:33:13,172 --> 00:33:15,380 one. They look at this star in 475 00:33:15,380 --> 00:33:19,796 optical light from the ground, and they say we know roughly 476 00:33:19,796 --> 00:33:22,309 m1. Now they see the Doppler shift 477 00:33:22,309 --> 00:33:25,507 of this one. They see an absorption line in 478 00:33:25,507 --> 00:33:30,000 the spectrum. It goes nicely back and forth. 479 00:33:30,000 --> 00:33:34,795 And they say, ah-ha, we now know the orbital 480 00:33:34,795 --> 00:33:41,486 period and we know the velocity v1 and we know the radius r1. 481 00:33:41,486 --> 00:33:45,055 We know all that. But, of course, 482 00:33:45,055 --> 00:33:50,855 astronomers also know, I hope, where is Kepler's Law? 483 00:33:50,855 --> 00:33:55,092 I didn't erase anything. Did I hide it? 484 00:33:55,092 --> 00:34:00,000 Yeah. They also know Kepler's Law. 485 00:34:00,000 --> 00:34:03,157 Now look at Kepler's Law. You know r1, 486 00:34:03,157 --> 00:34:06,741 you know m1, you know t, but what you don't 487 00:34:06,741 --> 00:34:10,837 know is r2, which is what you would like to know, 488 00:34:10,837 --> 00:34:14,762 that is the radius of the orbit of the equator. 489 00:34:14,762 --> 00:34:19,114 You don't know this one and you don't know that one. 490 00:34:19,114 --> 00:34:23,125 But you do know that m1 times r1 is m2 times r2. 491 00:34:23,125 --> 00:34:28,416 You have two equations with two unknowns, and you can solve for 492 00:34:28,416 --> 00:34:34,402 m2 and you can solve for r2. And so now you will find the 493 00:34:34,402 --> 00:34:37,834 mass of the accretor. Immensely powerful. 494 00:34:37,834 --> 00:34:42,724 This has been a game that has been going on for many years 495 00:34:42,724 --> 00:34:47,529 whereby x-rays astronomers, this is what my specialty has 496 00:34:47,529 --> 00:34:51,991 been for the past I cannot even remember, since 1966, 497 00:34:51,991 --> 00:34:55,937 for the past decades, have made a living out of 498 00:34:55,937 --> 00:35:01,000 trying to determine the mass of these accretors. 499 00:35:01,000 --> 00:35:12,435 And, in many cases, do you find that the mass of 500 00:35:12,435 --> 00:35:23,141 the accretor is very close to 1.4 solar mass. 501 00:35:23,141 --> 00:35:37,254 We are sure that in that case we are dealing with a neutron 502 00:35:37,254 --> 00:35:45,526 star. It is not an accident that we 503 00:35:45,526 --> 00:35:53,799 find 1.4 times the mass of the sum. 504 00:35:53,799 --> 00:36:07,911 Chandrasekhar demonstrated, quantum mechanical calculation, 505 00:36:07,911 --> 00:36:21,780 in 1930 that if you take a white dwarf and you dump matter 506 00:36:21,780 --> 00:36:32,000 onto the white dwarf. 35:36 507 00:36:32,000 --> 00:36:37,106 Out of that theory it follows that if you keep dumping mass 508 00:36:37,106 --> 00:36:41,421 onto a neutron star, the moment that you cross the 509 00:36:41,421 --> 00:36:45,911 three solar mass line the neutron star can no longer 510 00:36:45,911 --> 00:36:49,962 support itself and collapses into a black hole. 511 00:36:49,962 --> 00:36:54,100 And now comes the question what is a black hole? 512 00:36:54,100 --> 00:37:00,000 A black hole is an object from which light cannot escape. 513 00:37:00,000 --> 00:37:05,343 I have to erase something, and I think I am going to erase 514 00:37:05,343 --> 00:37:10,781 the left part of the blackboard. It is an object from which 515 00:37:10,781 --> 00:37:15,562 light cannot escape. I have to be more specific than 516 00:37:15,562 --> 00:37:18,750 that. A black hole is an object for 517 00:37:18,750 --> 00:37:24,281 which the escape velocity is larger than the speed of light. 518 00:37:24,281 --> 00:37:30,000 We know what the escape velocity is because this -- 519 00:37:30,000 --> 00:37:34,953 Remember that is why I paid attention to it and mentioned it 520 00:37:34,953 --> 00:37:39,066 is also the escape velocity? Here is a black hole. 521 00:37:39,066 --> 00:37:44,103 There is a radius around that black hole that has a wonderful 522 00:37:44,103 --> 00:37:47,125 name. It is called the event horizon. 523 00:37:47,125 --> 00:37:52,079 And all I have to do is make this c and I can calculate what 524 00:37:52,079 --> 00:37:55,520 that radius is. That means if you are then 525 00:37:55,520 --> 00:38:00,390 inside that radius then the escape velocity would be larger 526 00:38:00,390 --> 00:38:07,179 than c so you cannot get out. And so the radius the radius of 527 00:38:07,179 --> 00:38:12,102 the event horizon, I wouldn't want to call it the 528 00:38:12,102 --> 00:38:17,435 radius of the black hole, but the radius of the event 529 00:38:17,435 --> 00:38:20,820 horizon is then 2MG divided by c2. 530 00:38:20,820 --> 00:38:27,076 If you are closer to the black hole, the escape velocity would 531 00:38:27,076 --> 00:38:32,000 be larger than c which is not possible. 532 00:38:32,000 --> 00:38:35,783 To give you a little bit of feeling for what this means, 533 00:38:35,783 --> 00:38:39,636 if you take the earth and you take the mass of the earth, 534 00:38:39,636 --> 00:38:43,832 which you can look up in your book, and you substitute that in 535 00:38:43,832 --> 00:38:47,203 there, you will find that the radius of the earth, 536 00:38:47,203 --> 00:38:50,368 to become a black hole, is only one centimeter. 537 00:38:50,368 --> 00:38:53,601 So you substitute in here the mass of the earth. 538 00:38:53,601 --> 00:38:55,665 C2 you know. G you can look up. 539 00:38:55,665 --> 00:39:00,000 You find that the radius is about one centimeter. 540 00:39:00,000 --> 00:39:04,366 If you could squeeze the earth with its present radius of 6,400 541 00:39:04,366 --> 00:39:08,873 kilometers, if you could squeeze it one centimeter then you would 542 00:39:08,873 --> 00:39:11,760 have created a black hole. But, of course, 543 00:39:11,760 --> 00:39:14,154 that may be problematic to do that. 544 00:39:14,154 --> 00:39:16,971 If you take the sum which, by definition, 545 00:39:16,971 --> 00:39:20,422 is one solar mass, and so you put in here the mass 546 00:39:20,422 --> 00:39:23,661 of the sun, which you can look up in your book, 547 00:39:23,661 --> 00:39:27,676 then you will find that the radius of the event horizon is 548 00:39:27,676 --> 00:39:32,530 three kilometers. If you take a five solar mass 549 00:39:32,530 --> 00:39:37,503 black hole then you will find that the radius of the event 550 00:39:37,503 --> 00:39:42,302 horizon is 15 kilometers. Notice it scales linearly with 551 00:39:42,302 --> 00:39:44,744 m. That is what you see here. 552 00:39:44,744 --> 00:39:49,281 M is linearly upstairs. If you go from one solar mass 553 00:39:49,281 --> 00:39:53,644 to five solar mass, the radius of the event horizon 554 00:39:53,644 --> 00:40:00,005 scales linearly. In 1971, astronomers, 555 00:40:00,005 --> 00:40:09,734 Tom Bolton and Murdin and Webster came with a daring 556 00:40:09,734 --> 00:40:20,988 statement that one of these systems was a black hole binary. 557 00:40:20,988 --> 00:40:32,624 They looked at the spectrum of the donor and concluded that it 558 00:40:32,624 --> 00:40:39,491 probably was a 20/30 mass solar star. 559 00:40:39,491 --> 00:40:49,982 And then they made the Doppler shift measurements of the 560 00:40:49,982 --> 00:40:58,375 absorption line in the spectrum of the donor. 561 00:40:58,375 --> 00:41:03,948 40:28 And a lot of people emotionally 562 00:41:03,948 --> 00:41:06,588 could not handle it. And there were lots of 563 00:41:06,588 --> 00:41:09,920 publications for years to come arguing that it was all 564 00:41:09,920 --> 00:41:13,251 nonsense; that black holes did not exist, the data was 565 00:41:13,251 --> 00:41:15,891 misinterpreted, they came with all kinds of 566 00:41:15,891 --> 00:41:18,971 other weird explanations, but the black hole still 567 00:41:18,971 --> 00:41:21,045 stands. And most sane astronomers, 568 00:41:21,045 --> 00:41:24,062 that does not mean that all astronomers are sane, 569 00:41:24,062 --> 00:41:27,582 but most sane astronomers do believe that Cygnus X-1 is a 570 00:41:27,582 --> 00:41:30,034 black hole. And we now know of about two 571 00:41:30,034 --> 00:41:35,000 dozen of these systems whereby the accretor is a black hole. 572 00:41:35,000 --> 00:41:39,173 And I decided to make Cygnus X-1 part of your problem set so 573 00:41:39,173 --> 00:41:41,790 you have some time to wrestle with it. 574 00:41:41,790 --> 00:41:44,690 I want you to see a picture of Cygnus X-1. 575 00:41:44,690 --> 00:41:48,368 It is not very dramatic, but I want you to see it for 576 00:41:48,368 --> 00:41:51,409 emotional reasons. You will just see a star. 577 00:41:51,409 --> 00:41:55,724 You cannot see the black hole. You just see a star and that is 578 00:41:55,724 --> 00:41:58,129 all. But at least you can tell your 579 00:41:58,129 --> 00:42:02,090 parents that you have seen a star that is in orbit with a 580 00:42:02,090 --> 00:42:06,698 black hole. If we can get the first slide, 581 00:42:06,698 --> 00:42:12,096 the only slide I think I have, then we should be able to show 582 00:42:12,096 --> 00:42:14,615 you a star. It is a negative. 583 00:42:14,615 --> 00:42:18,934 In astronomy we almost always work with negative. 584 00:42:18,934 --> 00:42:23,522 The stars themselves are black and the sky is white. 585 00:42:23,522 --> 00:42:27,301 And it is this star here that is the donor. 586 00:42:27,301 --> 00:42:31,744 It is quite bright. You cannot see it with a naked 587 00:42:31,744 --> 00:42:33,372 eye. For those of you who are 588 00:42:33,372 --> 00:42:36,802 familiar with magnitudes of stars, it is a [nine?] magnitude 589 00:42:36,802 --> 00:42:38,662 star. And it is in orbit with the 590 00:42:38,662 --> 00:42:40,988 black hole. It itself has a mass of about 591 00:42:40,988 --> 00:42:44,011 20 to 30 solar masses, and the black hole has about a 592 00:42:44,011 --> 00:42:47,441 mass of about 50 solar masses. And all of this is the result 593 00:42:47,441 --> 00:42:50,348 of Doppler shift measurements profound far reaching 594 00:42:50,348 --> 00:42:52,674 consequences. I think this is a wonderful 595 00:42:52,674 --> 00:42:56,975 moment to have a break. And I know that most of you 596 00:42:56,975 --> 00:43:00,780 came here not to listen to my lecture but to take the 597 00:43:00,780 --> 00:43:04,000 mini-quiz. And so let's do that now then. 598 00:43:04,000 --> 00:43:10,000 599 00:43:10,000 --> 00:43:14,000 If you can help handing this out. 600 00:43:14,000 --> 00:43:21,000 601 00:43:21,000 --> 00:43:25,179 For those of you who come here only for the mini-quiz, 602 00:43:25,179 --> 00:43:28,964 I have some advice. You may have noticed that the 603 00:43:28,964 --> 00:43:33,301 question that I asked on the mini-quiz on Tuesday almost 604 00:43:33,301 --> 00:43:37,401 always goes back to the previous lecture on Thursday. 605 00:43:37,401 --> 00:43:38,978 Think about that. OK. 606 00:43:38,978 --> 00:43:43,000 If you can help me handing this out here. 607 00:43:43,000 --> 00:43:47,822 The biggest impact that Doppler shift measurements of 608 00:43:47,822 --> 00:43:53,665 electromagnetic radiation had on the perception of ourselves and 609 00:43:53,665 --> 00:43:56,725 of the universe came in the 1920s. 610 00:43:56,725 --> 00:44:01,826 An industry had already been developed by astronomers to 611 00:44:01,826 --> 00:44:06,000 measure the radial velocity of stars. 612 00:44:06,000 --> 00:44:17,101 I gave you one. Delta Leporis was a typical 613 00:44:17,101 --> 00:44:26,352 one. We were receding 100 kilometers 614 00:44:26,352 --> 00:44:37,189 per second. Velocities were typically 100, 615 00:44:37,189 --> 00:44:50,140 200 kilometers per second. In some cases receding. 616 00:44:50,140 --> 00:45:01,770 In some cases we were approaching each other. 617 00:45:01,770 --> 00:45:08,105 44:30 And so that gives you then a 618 00:45:08,105 --> 00:45:11,415 handle on the radial velocity of that galaxy. 619 00:45:11,415 --> 00:45:15,025 And Edwin Hubble and Humason found, surprisingly, 620 00:45:15,025 --> 00:45:19,161 a correlation between the distance of galaxies and their 621 00:45:19,161 --> 00:45:21,944 velocities. To measure the velocity is 622 00:45:21,944 --> 00:45:24,050 easy. That is a Doppler shift 623 00:45:24,050 --> 00:45:26,983 measurement. That is relatively trivial. 624 00:45:26,983 --> 00:45:31,345 All you have to measure is lambda prime and compare it with 625 00:45:31,345 --> 00:45:34,969 lambda. Distance measure is very, 626 00:45:34,969 --> 00:45:38,515 very difficult in astronomy, very problematic. 627 00:45:38,515 --> 00:45:41,115 And it is a lecture all by itself. 628 00:45:41,115 --> 00:45:45,606 And so, in the days of Hubble, his distance estimates were 629 00:45:45,606 --> 00:45:48,678 very different from what they are today. 630 00:45:48,678 --> 00:45:53,484 His distance estimates to the nearby galaxies were seven times 631 00:45:53,484 --> 00:45:59,000 smaller than what we think today, but that does not matter. 632 00:45:59,000 --> 00:46:04,204 He did find the linear relationship between velocity 633 00:46:04,204 --> 00:46:08,183 and distance. And that was a spectacular 634 00:46:08,183 --> 00:46:11,653 discovery. And this is known now as 635 00:46:11,653 --> 00:46:16,040 Hubble's Law. The radially receding velocity 636 00:46:16,040 --> 00:46:21,857 of galaxies is some constant, which was later called after 637 00:46:21,857 --> 00:46:26,857 him Hubble's Constant, times the distance d to the 638 00:46:26,857 --> 00:46:31,327 galaxies. V is the radial velocity. 639 00:46:31,327 --> 00:46:35,423 It is receding. We don't give it a minus sign 640 00:46:35,423 --> 00:46:38,588 anymore. That v is always receding. 641 00:46:38,588 --> 00:46:43,801 The present most reliably value for H is approximately 70 642 00:46:43,801 --> 00:46:47,153 kilometers per second per megaparsec. 643 00:46:47,153 --> 00:46:52,831 I give it to you in this form because that is the way you will 644 00:46:52,831 --> 00:46:57,858 always see it the literature, but I own an explanation, 645 00:46:57,858 --> 00:47:03,116 of course. The megaparsec is simply a 646 00:47:03,116 --> 00:47:09,229 distance in astronomy. It is the distance that light 647 00:47:09,229 --> 00:47:15,941 travels in 3.26 million years, so it is also 3.26 million 648 00:47:15,941 --> 00:47:20,856 light years. If you know what the speed of 649 00:47:20,856 --> 00:47:28,407 light is, 300,000 kilometers per second, then you will find that 650 00:47:28,407 --> 00:47:35,000 it is about 3.1 times 10 to the 19 kilometers. 651 00:47:35,000 --> 00:47:40,344 Notice that the dimension of Hubble is one over seconds, 652 00:47:40,344 --> 00:47:44,910 is one over time, because kilometers is distance 653 00:47:44,910 --> 00:47:49,963 and megaparsec is distance, so it is one divided by a 654 00:47:49,963 --> 00:47:53,364 second. Hubble himself found a value 655 00:47:53,364 --> 00:47:59,000 which is 500 kilometers per second per megaparsec. 656 00:47:59,000 --> 00:48:04,215 The reason was that he had a different estimate for the 657 00:48:04,215 --> 00:48:08,079 distance. He thought that the distance to 658 00:48:08,079 --> 00:48:13,875 the galaxies that were in his sample were seven times smaller 659 00:48:13,875 --> 00:48:19,187 than what we think today. Suppose you trust Hubble's Law 660 00:48:19,187 --> 00:48:23,534 at face value, as it is, and you and I look at 661 00:48:23,534 --> 00:48:29,232 the particular galaxy in the sky, and we measure that lambda 662 00:48:29,232 --> 00:48:34,545 prime is 1.02 times lambda. In other words, 663 00:48:34,545 --> 00:48:40,729 it is redshift and the radial velocity is there for 2% of the 664 00:48:40,729 --> 00:48:44,851 speed of light. You can measure that beta 665 00:48:44,851 --> 00:48:51,343 cosine theta is then minus 0.02. I carry that minus sign because 666 00:48:51,343 --> 00:48:57,423 I gave you that Doppler shift measurement in the form that I 667 00:48:57,423 --> 00:49:03,893 want you to use. We don't even think of that 668 00:49:03,893 --> 00:49:10,813 minus sign anymore because it is always receding. 669 00:49:10,813 --> 00:49:18,311 I carry the minus sign. And so v cosine theta then is 670 00:49:18,311 --> 00:49:26,674 this number times the speed of light, and that becomes then 671 00:49:26,674 --> 00:49:32,730 minus 6,000 kilometers per second receding. 672 00:49:32,730 --> 00:49:40,948 The radial velocity receding speed is 6,000 kilometers per 673 00:49:40,948 --> 00:49:46,139 second. This is about 2% of the speed 674 00:49:46,139 --> 00:49:51,186 of light. If you trust Hubble's Law, 675 00:49:51,186 --> 00:49:58,683 you can now use Hubble's Law and calculate what d is. 676 00:49:58,683 --> 00:50:05,695 49:34 It means that the light that 677 00:50:05,695 --> 00:50:12,559 you were processing on the ground was emitted by that 678 00:50:12,559 --> 00:50:18,103 galaxy 280 million years ago. In astronomy, 679 00:50:18,103 --> 00:50:24,967 we look back in time. I would like you to see now the 680 00:50:24,967 --> 00:50:32,755 spectra of three galaxies which at vastly different distance 681 00:50:32,755 --> 00:50:38,092 from us. Let me first turn on the slide 682 00:50:38,092 --> 00:50:42,216 projector. Otherwise, I may end up in the 683 00:50:42,216 --> 00:50:46,030 darkness and not know what I am doing. 684 00:50:46,030 --> 00:50:50,567 That is right. I will show you the spectra of 685 00:50:50,567 --> 00:50:55,103 three galaxies. This is a galaxy which is the 686 00:50:55,103 --> 00:51:00,257 closest of the three. That should be rather obvious 687 00:51:00,257 --> 00:51:05,000 because it is also the largest one. 688 00:51:05,000 --> 00:51:09,652 And the spectrum of the galaxy is what you see in the middle 689 00:51:09,652 --> 00:51:12,492 here. These are measurements of known 690 00:51:12,492 --> 00:51:16,119 lines with known wavelengths in the laboratory. 691 00:51:16,119 --> 00:51:21,009 This is just a calibration here and this is a calibration here. 692 00:51:21,009 --> 00:51:23,769 This is the spectrum of that galaxy. 693 00:51:23,769 --> 00:51:26,924 And what you see here are two dark lines. 694 00:51:26,924 --> 00:51:31,928 Those are absorption lines. The average of the stars in 695 00:51:31,928 --> 00:51:34,857 that galaxy. They have shifted relative to 696 00:51:34,857 --> 00:51:37,142 the wavelength in our laboratory. 697 00:51:37,142 --> 00:51:41,571 By the way, one of those lines is the same line that I used for 698 00:51:41,571 --> 00:51:44,142 Delta Leporis, which is the calcium K 699 00:51:44,142 --> 00:51:47,428 absorption line. The other one is the calcium H 700 00:51:47,428 --> 00:51:50,285 absorption line. H has nothing to do with 701 00:51:50,285 --> 00:51:52,928 hydrogen. And it had been shifted over 702 00:51:52,928 --> 00:51:58,000 the distance that you see indicated by this small arrow. 703 00:51:58,000 --> 00:52:01,255 And, if you measure now the velocity, you find 1,150 704 00:52:01,255 --> 00:52:04,510 kilometers per second. And, if you use today's value 705 00:52:04,510 --> 00:52:07,510 of Hubble's Law, this object is 52 million light 706 00:52:07,510 --> 00:52:10,382 years away from us. Now you go to this object, 707 00:52:10,382 --> 00:52:13,446 which is further away. According to Hubble's Law, 708 00:52:13,446 --> 00:52:16,063 it has a higher speed, the receding speed. 709 00:52:16,063 --> 00:52:19,319 And you see the spectrum here. This is the spectrum. 710 00:52:19,319 --> 00:52:22,063 And here you see those two absorption lines. 711 00:52:22,063 --> 00:52:27,020 They have shifted all the way. And out of that then follows 712 00:52:27,020 --> 00:52:31,278 the radial velocity which in this case is 7% of the speed of 713 00:52:31,278 --> 00:52:35,608 light, 22,000 kilometers per second, which pushed this object 714 00:52:35,608 --> 00:52:38,422 at one billion light years away from us. 715 00:52:38,422 --> 00:52:41,670 And, when I go to this one here in the corner, 716 00:52:41,670 --> 00:52:45,927 it is hard to see which one it is, the absorption lines have 717 00:52:45,927 --> 00:52:49,463 shifted even further. You see the absorption lines 718 00:52:49,463 --> 00:52:52,061 here. And that object is going 20% of 719 00:52:52,061 --> 00:52:56,175 the speed of light receding. Then it would be roughly at a 720 00:52:56,175 --> 00:53:00,000 distance of 2.8 billion light years. 721 00:53:00,000 --> 00:53:05,842 The most modern work on Hubble's Law was done by a group 722 00:53:05,842 --> 00:53:11,153 under the leadership of Wendy Freedman, California. 723 00:53:11,153 --> 00:53:16,146 Wendy used data from the Hubble Space Telescope, 724 00:53:16,146 --> 00:53:22,520 and she measured the Doppler shifts and the distance to a few 725 00:53:22,520 --> 00:53:27,300 hundred objects. Distance is always the can of 726 00:53:27,300 --> 00:53:32,141 worms. And she came up with presently 727 00:53:32,141 --> 00:53:36,854 the most reliable value for Hubble's constant. 728 00:53:36,854 --> 00:53:40,939 Let me get the next slide, if I succeed. 729 00:53:40,939 --> 00:53:46,699 This is Wendy Freedman's work. Forget what you see here. 730 00:53:46,699 --> 00:53:52,878 Just look at what you see here. The farthest object that she 731 00:53:52,878 --> 00:54:00,000 was able to get a reliable distance for is 400 megaparsecs. 732 00:54:00,000 --> 00:54:02,692 That is a stunning distance, by the way. 733 00:54:02,692 --> 00:54:05,247 That is about 1.3 billion light years. 734 00:54:05,247 --> 00:54:09,113 And she was able to measure radial velocities up to about 735 00:54:09,113 --> 00:54:13,048 10% of the speed of light, velocities of 30,000 kilometers 736 00:54:13,048 --> 00:54:15,879 per second. And so she can draw this line, 737 00:54:15,879 --> 00:54:19,193 which is the linear relationship between velocity 738 00:54:19,193 --> 00:54:21,886 and distance. And out of that line comes 739 00:54:21,886 --> 00:54:24,717 Hubble's Law. And she concludes that it is 740 00:54:24,717 --> 00:54:28,376 very close to 72 kilometers per second per megaparsec, 741 00:54:28,376 --> 00:54:33,530 plus or minus a few. I used 70 just to round it off. 742 00:54:33,530 --> 00:54:37,938 What is interesting is that Hubble, in his entire data, 743 00:54:37,938 --> 00:54:42,265 the velocities were all less than 1,100 kilometers per 744 00:54:42,265 --> 00:54:44,714 second. Put that in scale here. 745 00:54:44,714 --> 00:54:48,224 This already is 2,000 kilometers per second. 746 00:54:48,224 --> 00:54:52,714 Hubble has only data in this teeny weenie little portion 747 00:54:52,714 --> 00:54:55,326 here. Wendy doesn't even have any 748 00:54:55,326 --> 00:54:59,000 data. She doesn't even use them. 749 00:54:59,000 --> 00:55:03,492 And the reason why she doesn't use them, she says it is too 750 00:55:03,492 --> 00:55:06,357 close. It is not really representative 751 00:55:06,357 --> 00:55:10,927 for what we call the Hubble Flow, so she stays away from the 752 00:55:10,927 --> 00:55:13,250 local stuff. But, nevertheless, 753 00:55:13,250 --> 00:55:17,974 based on that very local stuff, which is not even put in here, 754 00:55:17,974 --> 00:55:21,924 Hubble came with his linear relationship which was a 755 00:55:21,924 --> 00:55:26,261 monumental statement in the history of mankind because it 756 00:55:26,261 --> 00:55:30,831 puts the universe at a very different level than what people 757 00:55:30,831 --> 00:55:35,166 felt before. The universe is expanding. 758 00:55:35,166 --> 00:55:39,666 Let's now ask ourselves the question what does this all 759 00:55:39,666 --> 00:55:42,500 mean? The first thing that may come 760 00:55:42,500 --> 00:55:46,666 to mind, since all objects are moving away from us, 761 00:55:46,666 --> 00:55:49,916 you may think that you are very special. 762 00:55:49,916 --> 00:55:53,666 We all like to think that we are very special. 763 00:55:53,666 --> 00:55:58,666 You may even think that you are at the center of the universe 764 00:55:58,666 --> 00:56:03,000 because everything moves away from you. 765 00:56:03,000 --> 00:56:06,952 And persuasive is to think that there was a time in the past 766 00:56:06,952 --> 00:56:10,502 where there was a huge explosion, which we refer to as 767 00:56:10,502 --> 00:56:13,315 the Big Bang, when all of this happened and 768 00:56:13,315 --> 00:56:16,933 there were galaxies with large speeds in the explosion. 769 00:56:16,933 --> 00:56:20,550 They are now the ones the farthest away from us so they 770 00:56:20,550 --> 00:56:23,899 have the highest speed. And some had a lower speed. 771 00:56:23,899 --> 00:56:26,444 That is a picture that presents itself. 772 00:56:26,444 --> 00:56:31,000 It is a little bit naïve, though, and partially wrong. 773 00:56:31,000 --> 00:56:35,958 However, the idea of the Big Bang Theory, that there was such 774 00:56:35,958 --> 00:56:39,760 an explosion a long time ago, that still holds. 775 00:56:39,760 --> 00:56:43,727 And that is considered the birth of our universe. 776 00:56:43,727 --> 00:56:47,942 Now comes the question when did this Big Bang occur? 777 00:56:47,942 --> 00:56:52,652 Well, we could make a simple assumption, perhaps not quite 778 00:56:52,652 --> 00:56:57,611 accurate, that the velocity with which all these galaxies are 779 00:56:57,611 --> 00:57:03,032 moving never changed in time. We can do that as a start. 780 00:57:03,032 --> 00:57:07,331 No matter which galaxy you take, the velocity now is the 781 00:57:07,331 --> 00:57:12,021 same that it was in the past. Then we have 8.01 which has the 782 00:57:12,021 --> 00:57:16,242 distance that that galaxy traveled is the velocity with 783 00:57:16,242 --> 00:57:19,916 which it traveled, the radial velocity times the 784 00:57:19,916 --> 00:57:23,824 time that it traveled, which is then the age of the 785 00:57:23,824 --> 00:57:27,420 universe, how many seconds the universe lasted. 786 00:57:27,420 --> 00:57:32,875 This is simply 8.01. But we also know Hubble's Law. 787 00:57:32,875 --> 00:57:37,092 In other words, we can replace this v here Hd 788 00:57:37,092 --> 00:57:42,651 by this one, so we get d equals H times d times t universe. 789 00:57:42,651 --> 00:57:48,402 And so you will find that the age of the universe is one over 790 00:57:48,402 --> 00:57:51,565 H. And remember I mentioned that H 791 00:57:51,565 --> 00:57:56,932 has the dimension one divided by seconds, so this has the 792 00:57:56,932 --> 00:58:03,186 dimension of seconds. And if you use the value for H 793 00:58:03,186 --> 00:58:09,560 of 70 kilometers per second per megaparsec then you will find 794 00:58:09,560 --> 00:58:14,978 that the age of the universe, you can check that for 795 00:58:14,978 --> 00:58:21,457 yourself, is 4.3 times 10 to the 17 seconds, so tu comes about 796 00:58:21,457 --> 00:58:27,406 4.3 times 10 to the 17 seconds, which is about 14 billion 797 00:58:27,406 --> 00:58:31,333 years. You would have to conclude then 798 00:58:31,333 --> 00:58:35,849 that the Big Bang explosion took place about 14 billion years 799 00:58:35,849 --> 00:58:38,483 ago. Now you can easily imagine that 800 00:58:38,483 --> 00:58:43,225 due to gravitational attraction that the speeds in the past were 801 00:58:43,225 --> 00:58:46,311 probably higher. You would think that this 802 00:58:46,311 --> 00:58:50,301 expansion probably decelerated. And so, on that basis, 803 00:58:50,301 --> 00:58:54,666 you could conclude that it is probably less than 14 billion 804 00:58:54,666 --> 00:58:57,225 years. We have very good reasons to 805 00:58:57,225 --> 00:59:01,967 believe that the oldest stars in our own galaxy are 10 to the 10 806 00:59:01,967 --> 00:59:05,942 years old. That means 10 billion years 807 00:59:05,942 --> 00:59:08,389 old. So the Big Bang must have been 808 00:59:08,389 --> 00:59:11,411 somewhere in 10 to the 14 and 10 to the 10. 809 00:59:11,411 --> 00:59:15,585 And most astronomers now would say it is probably something 810 00:59:15,585 --> 00:59:19,471 like 12 billion years ago that the Big Bang took place. 811 00:59:19,471 --> 00:59:22,062 And the issue of the deceleration is, 812 00:59:22,062 --> 00:59:25,516 of course, at the heart of research is cosmology. 813 00:59:25,516 --> 00:59:29,761 It is now believed that the very early universe there was an 814 00:59:29,761 --> 00:59:34,673 acceleration of the expansion. It is called inflation. 815 00:59:34,673 --> 00:59:37,075 It was followed by a deceleration, 816 00:59:37,075 --> 00:59:40,350 which is what you would expect due to gravity. 817 00:59:40,350 --> 00:59:43,698 And it is now also believed, very mysteriously, 818 00:59:43,698 --> 00:59:46,318 that there is again, at a later time, 819 00:59:46,318 --> 00:59:49,447 again acceleration not very well understood. 820 00:59:49,447 --> 00:59:53,450 We have MIT experts here. Professor Alan Guth is a world 821 00:59:53,450 --> 00:59:56,579 authority on this. Ed Bertschinger and Scott 822 00:59:56,579 --> 1:00:02,712 Burles do also research in this. Allan Guth is the godfather of 823 1:00:02,712 --> 1:00:07,324 inflation and our universe. I will get back to that. 824 1:00:07,324 --> 1:00:12,750 The question now that I would like to raise is how far can we 825 1:00:12,750 --> 1:00:17,090 look back in time? All I can tell you is that the 826 1:00:17,090 --> 1:00:21,340 largest redshift that has been measured to date, 827 1:00:21,340 --> 1:00:24,776 lambda prime divided by lambda, is 7.5. 828 1:00:24,776 --> 1:00:28,936 That is enormous. And so now comes the question 829 1:00:28,936 --> 1:00:34,000 what is the speed and what is the distance? 830 1:00:34,000 --> 1:00:38,016 And I cannot answer that. The reason is that the equation 831 1:00:38,016 --> 1:00:41,746 that I have given you, the Doppler shift equation was 832 1:00:41,746 --> 1:00:44,400 only derived using special relativity. 833 1:00:44,400 --> 1:00:47,772 It did not take general relativity into account. 834 1:00:47,772 --> 1:00:51,502 And general relativity introduces also redshift which 835 1:00:51,502 --> 1:00:54,586 has nothing to do with the Doppler redshift. 836 1:00:54,586 --> 1:00:58,962 And so I will leave it up to my colleagues, Professor Guth and 837 1:00:58,962 --> 1:01:01,257 Scott Burles and Ed Bertschinger, 838 1:01:01,257 --> 1:01:05,561 to tell you what perhaps the speed is of this object and what 839 1:01:05,561 --> 1:01:10,403 the distance is. And they may not even give you 840 1:01:10,403 --> 1:01:13,769 one answer. The will probably give you more 841 1:01:13,769 --> 1:01:16,814 than one answer, depending upon certain 842 1:01:16,814 --> 1:01:20,179 assumptions. General relativity changes the 843 1:01:20,179 --> 1:01:23,785 picture quite dramatically. There is no doubt, 844 1:01:23,785 --> 1:01:28,272 however, that they will all agree this object is very far 845 1:01:28,272 --> 1:01:33,000 away, probably more than 10 billion light years. 846 1:01:33,000 --> 1:01:38,929 So you are looking back in time more than 10 billion years. 847 1:01:38,929 --> 1:01:44,450 And so you are almost getting to the point that you are 848 1:01:44,450 --> 1:01:49,460 beginning to see the actual birth of the universe, 849 1:01:49,460 --> 1:01:54,265 the Big Bang itself. What is also interesting is 850 1:01:54,265 --> 1:01:59,683 that remember I mentioned the calcium absorption line, 851 1:01:59,683 --> 1:02:05,000 the K line was roughly at 3,933 angstroms. 852 1:02:05,000 --> 1:02:11,304 If you multiply that wavelength, this is lambda, 853 1:02:11,304 --> 1:02:18,951 by 7.5 it ends up at 29,500 angstroms which is outside the 854 1:02:18,951 --> 1:02:25,256 optical spectrum. Impossible to observe from the 855 1:02:25,256 --> 1:02:29,924 ground. And so how can astronomers ever 856 1:02:29,924 --> 1:02:34,641 find redshifts that are as large as that ratio 7.5? 857 1:02:34,641 --> 1:02:40,113 The way they do that is look at a line in the spectrum that 858 1:02:40,113 --> 1:02:43,981 cannot be observed from the ground at all, 859 1:02:43,981 --> 1:02:49,075 it is a line emitted by hydrogen called the Lyman alpha 860 1:02:49,075 --> 1:02:51,905 line. It is really of hydrogen. 861 1:02:51,905 --> 1:02:57,094 And the wavelength is 1,216 angstroms which is in the UV 862 1:02:57,094 --> 1:03:02,000 that you cannot observe from the earth. 863 1:03:02,000 --> 1:03:06,566 The atmosphere absorbs the UV. But, if you multiply this by 864 1:03:06,566 --> 1:03:11,370 7.5, then you get it all the way down to about 9,100 angstroms 865 1:03:11,370 --> 1:03:14,283 which you can observe from the ground. 866 1:03:14,283 --> 1:03:17,826 And so it is remarkable, when you think of it, 867 1:03:17,826 --> 1:03:22,708 that lines which are invisible due to Doppler shift are shifted 868 1:03:22,708 --> 1:03:27,590 all the way to the extreme red part to the infrared part of the 869 1:03:27,590 --> 1:03:32,000 spectrum where they can just barely be seen. 870 1:03:32,000 --> 1:03:37,000 871 1:03:37,000 --> 1:03:42,345 Now comes the question are we at the center of the universe? 872 1:03:42,345 --> 1:03:46,151 Well, not likely. Suppose you have a raisin 873 1:03:46,151 --> 1:03:48,869 bread. You have made the dough, 874 1:03:48,869 --> 1:03:53,852 you put the raisins in there and you put it in the oven. 875 1:03:53,852 --> 1:03:57,114 This raisin bread is going to expand. 876 1:03:57,114 --> 1:04:02,097 Each raisin will see all other raisins move away from it 877 1:04:02,097 --> 1:04:05,112 radially. Think about it. 878 1:04:05,112 --> 1:04:09,341 Because, if the dough expands uniformly in all directions, 879 1:04:09,341 --> 1:04:12,234 everything moves away from you radially. 880 1:04:12,234 --> 1:04:16,462 And one raisin will see the nearest raisin going with very 881 1:04:16,462 --> 1:04:19,578 low velocity, but when it is farther out in 882 1:04:19,578 --> 1:04:22,249 the cake it goes with a higher speed. 883 1:04:22,249 --> 1:04:26,774 That raisin can actually be the discoverer of the expansion of 884 1:04:26,774 --> 1:04:30,578 its own universe. And it can measure the 885 1:04:30,578 --> 1:04:34,094 distances to the other raisins and the velocities, 886 1:04:34,094 --> 1:04:36,750 and it will come up with Hubble's Law. 887 1:04:36,750 --> 1:04:40,337 Not only will one raisin come up with Hubble's Law, 888 1:04:40,337 --> 1:04:44,643 but any other raisin that you pick will come up with the same 889 1:04:44,643 --> 1:04:47,226 Hubble's Law. And it will also think, 890 1:04:47,226 --> 1:04:50,527 perhaps, that is it at the center of that cake, 891 1:04:50,527 --> 1:04:52,895 which is, of course, not the case. 892 1:04:52,895 --> 1:04:57,273 Anything that expands uniformly in all directions will fit the 893 1:04:57,273 --> 1:05:02,352 idea of Hubble's Law. And also will make you think 894 1:05:02,352 --> 1:05:06,806 erroneously that you may be at the center of all that. 895 1:05:06,806 --> 1:05:11,848 There is a very nice and even better analogy that I can draw, 896 1:05:11,848 --> 1:05:16,722 so I will abandon the raisin cake now and will discuss with 897 1:05:16,722 --> 1:05:21,848 you the universe of flatlanders. Flatlanders are people who do 898 1:05:21,848 --> 1:05:25,966 not have three dimensions but only two dimensions, 899 1:05:25,966 --> 1:05:30,000 and so they can only move in a plane. 900 1:05:30,000 --> 1:05:33,800 They do not know, but we do in this flatland 901 1:05:33,800 --> 1:05:38,750 world, that they really live on the surface of a balloon. 902 1:05:38,750 --> 1:05:43,345 But they do not know that. And so the lights in their 903 1:05:43,345 --> 1:05:48,118 flatland world can only travel along the surface of the 904 1:05:48,118 --> 1:05:51,477 balloon. But they do not know that they 905 1:05:51,477 --> 1:05:54,482 live on the surface of the balloon. 906 1:05:54,482 --> 1:05:57,222 And now this balloon can expand. 907 1:05:57,222 --> 1:06:01,111 And the galaxies, which are on the surface of 908 1:06:01,111 --> 1:06:05,000 that balloon, here they are. 909 1:06:05,000 --> 1:06:09,075 Here are the galaxies. There is a spiral galaxy. 910 1:06:09,075 --> 1:06:11,416 You see that? Nice galaxies. 911 1:06:11,416 --> 1:06:16,271 When you blow up this balloon, each galaxy will see other 912 1:06:16,271 --> 1:06:21,820 galaxies move radially away from each other because light travels 913 1:06:21,820 --> 1:06:25,982 along the surface. And, the farther the galaxy is 914 1:06:25,982 --> 1:06:28,757 away, the higher is the velocity. 915 1:06:28,757 --> 1:06:33,612 And so let us expand this balloon and see what this world 916 1:06:33,612 --> 1:06:40,000 -- I first have to connect it to 917 1:06:40,000 --> 1:06:45,000 the source of the expansion. 918 1:06:45,000 --> 1:06:55,000 919 1:06:55,000 --> 1:06:58,881 These poor flatlanders have no idea what is happening to them. 920 1:06:58,881 --> 1:07:02,000 They are living on the surface of this balloon. 921 1:07:02,000 --> 1:07:06,253 They cannot see the third dimension so they have no way of 922 1:07:06,253 --> 1:07:08,791 knowing. And this balloon grows and 923 1:07:08,791 --> 1:07:12,298 grows and grows. And so the distance between the 924 1:07:12,298 --> 1:07:15,432 galaxies gets larger and larger and larger. 925 1:07:15,432 --> 1:07:19,238 And each one of those galaxies may have astronomers, 926 1:07:19,238 --> 1:07:22,746 and each one of them will discover Hubble's Law. 927 1:07:22,746 --> 1:07:27,000 And they will find the same constant for Hubble's Law when 928 1:07:27,000 --> 1:07:32,000 they make the measurement at the same moment in time. 929 1:07:32,000 --> 1:07:34,814 Now, they may not call it Hubble's Law. 930 1:07:34,814 --> 1:07:38,074 The name of that astronomer may be different, 931 1:07:38,074 --> 1:07:40,370 of course, but that is a detail. 932 1:07:40,370 --> 1:07:44,444 As their universe expands, and we are going to work with 933 1:07:44,444 --> 1:07:47,629 that a little now, you will see what kind of 934 1:07:47,629 --> 1:07:50,000 conclusions they draw. 935 1:07:50,000 --> 1:07:55,000 936 1:07:55,000 --> 1:07:57,000 There goes the universe. 937 1:07:57,000 --> 1:08:12,000 938 1:08:12,000 --> 1:08:19,058 Here is the balloon. And the balloon has a radius R 939 1:08:19,058 --> 1:08:24,564 which they do not know but you and I do. 940 1:08:24,564 --> 1:08:31,341 And this is today. Tomorrow the balloon is there, 941 1:08:31,341 --> 1:08:39,735 or next month or next year. And so the radius of that 942 1:08:39,735 --> 1:08:48,103 balloon next year is R plus dR. This radius is R plus dr. 943 1:08:48,103 --> 1:08:54,528 And here is a galaxy which tomorrow is here. 944 1:08:54,528 --> 1:09:02,000 And here is a galaxy which tomorrow is there. 945 1:09:02,000 --> 1:09:08,251 And the distance between these galaxies I call S, 946 1:09:08,251 --> 1:09:13,199 but tomorrow the distance is S plus dS. 947 1:09:13,199 --> 1:09:18,800 In this universe, it is clear that S plus dS 948 1:09:18,800 --> 1:09:24,399 divided by R plus dR, that is this triangle, 949 1:09:24,399 --> 1:09:30,000 must be the same as S divided by R. 950 1:09:30,000 --> 1:09:39,763 And so I can work that out. I get SR plus RdS is SR plus 951 1:09:39,763 --> 1:09:45,976 SdR. Now I can look at the evolution 952 1:09:45,976 --> 1:09:51,301 in time. I can give that radial 953 1:09:51,301 --> 1:10:00,000 expansion certain value, so I divide by dt. 954 1:10:00,000 --> 1:10:04,787 What is the meaning of S? That is the distance between 955 1:10:04,787 --> 1:10:07,677 the galaxies as you measure them. 956 1:10:07,677 --> 1:10:13,096 We call that d in our universe. What is the meaning of ds/dt? 957 1:10:13,096 --> 1:10:18,425 That is the receding velocity with which they move away from 958 1:10:18,425 --> 1:10:22,219 each other. Light travels along the surface 959 1:10:22,219 --> 1:10:25,470 of the balloon. This ds/dt is that v. 960 1:10:25,470 --> 1:10:30,978 What do we find? We find that v equals d times 961 1:10:30,978 --> 1:10:35,806 one over R dR/dt. And this dR/dt is what I would 962 1:10:35,806 --> 1:10:42,072 call the expansion rate of this balloon, expansion rate of the 963 1:10:42,072 --> 1:10:46,078 universe. And so the flatlanders measure 964 1:10:46,078 --> 1:10:50,906 v, they measure d. And at a particular moment in 965 1:10:50,906 --> 1:10:56,350 times there is a value of dR/dt which they don't know, 966 1:10:56,350 --> 1:11:02,000 there is a value of R which they don't know. 967 1:11:02,000 --> 1:11:06,294 But, at a particular moment in time, this is a constant. 968 1:11:06,294 --> 1:11:09,339 And so they call this Hubble's constant. 969 1:11:09,339 --> 1:11:12,852 It has, indeed, the units of one over seconds, 970 1:11:12,852 --> 1:11:16,912 because dR is in meters and R is in meters so it has, 971 1:11:16,912 --> 1:11:20,270 indeed, the units of one divided by seconds. 972 1:11:20,270 --> 1:11:23,471 They have no way of knowing what it means, 973 1:11:23,471 --> 1:11:26,126 but we do. We can look in the third 974 1:11:26,126 --> 1:11:30,307 dimension. We know that R in the past was 975 1:11:30,307 --> 1:11:32,538 smaller. They don't know that, 976 1:11:32,538 --> 1:11:35,461 but we do. Even if dR/dt never changed, 977 1:11:35,461 --> 1:11:39,076 that the universe expanded with a constant rate, 978 1:11:39,076 --> 1:11:43,538 we know that Hubble's constant in their past must have been 979 1:11:43,538 --> 1:11:46,692 different. Because, if R is smaller in the 980 1:11:46,692 --> 1:11:51,076 past, Hubble's constant must have been larger in the past. 981 1:11:51,076 --> 1:11:55,692 That is the reason why we claim that Hubble's constant in our 982 1:11:55,692 --> 1:12:00,277 universe is 70. We always put a little zero 983 1:12:00,277 --> 1:12:04,325 there, so we always say H0 is in Wendy's case 72. 984 1:12:04,325 --> 1:12:09,132 Even when we looked at the slide, she had that little zero 985 1:12:09,132 --> 1:12:11,915 there. That means as we measure it 986 1:12:11,915 --> 1:12:14,867 now. We make no claim about Hubble's 987 1:12:14,867 --> 1:12:19,674 constant, how it was five billion years ago or how it will 988 1:12:19,674 --> 1:12:24,481 be five billion years from now. We make no statement about 989 1:12:24,481 --> 1:12:28,272 that. If the expansion rate of the 990 1:12:28,272 --> 1:12:33,059 balloon decreases in time then there comes a time maybe that 991 1:12:33,059 --> 1:12:38,009 the balloon will come to a halt. That means when dR/dt becomes 992 1:12:38,009 --> 1:12:41,336 zero, Hubble's constant will then be zero. 993 1:12:41,336 --> 1:12:44,906 All galaxies on the balloon will stand still. 994 1:12:44,906 --> 1:12:49,775 And then gravity will take over and the balloon will start to 995 1:12:49,775 --> 1:12:52,859 shrink. And Hubble's constant will then 996 1:12:52,859 --> 1:12:55,536 become negative. Sign will change. 997 1:12:55,536 --> 1:13:00,000 All redshifts will change to blueshifts. 998 1:13:00,000 --> 1:13:03,296 Everything will approach each other. 999 1:13:03,296 --> 1:13:07,722 And, ultimately, the whole universe may collapse 1000 1:13:07,722 --> 1:13:13,373 onto itself, which we generally refer to in the literature as 1001 1:13:13,373 --> 1:13:17,235 the Big Crunch as opposed to the Big Bang. 1002 1:13:17,235 --> 1:13:23,074 We used to talk about a closed universe, up to a few years ago. 1003 1:13:23,074 --> 1:13:28,349 A closed universe would be a universe that would collapse 1004 1:13:28,349 --> 1:13:34,000 onto itself, and so there would be a big crunch. 1005 1:13:34,000 --> 1:13:38,202 We compared that up to a few years ago with an open universe 1006 1:13:38,202 --> 1:13:42,049 which would never collapse onto itself, would always be 1007 1:13:42,049 --> 1:13:44,969 expanding. And we even introduced the idea 1008 1:13:44,969 --> 1:13:49,386 of what we call a flat universe. It is just in between the two. 1009 1:13:49,386 --> 1:13:53,731 That means the universe will be expanding forever and ever and 1010 1:13:53,731 --> 1:13:56,082 ever. And it would come to a halt, 1011 1:13:56,082 --> 1:14:00,000 but you would have to wait infinitely long. 1012 1:14:00,000 --> 1:14:04,790 It is like having a tennis ball throwing up from the earth. 1013 1:14:04,790 --> 1:14:08,837 It would just make it to infinity with zero speed. 1014 1:14:08,837 --> 1:14:13,628 That is basically that idea. But a lot has changed with the 1015 1:14:13,628 --> 1:14:18,418 discovery a few years ago with what we now refer to as dark 1016 1:14:18,418 --> 1:14:21,474 energy. And so we now believe that the 1017 1:14:21,474 --> 1:14:26,430 universe will expand forever and that there will not be a Big 1018 1:14:26,430 --> 1:14:29,759 Crunch. You should not carry the 1019 1:14:29,759 --> 1:14:33,645 analogy between the flatland balloon universe too far, 1020 1:14:33,645 --> 1:14:37,751 but it is very interesting. It gives you some interesting 1021 1:14:37,751 --> 1:14:41,564 insight, and it certainly gives you food for thought. 1022 1:14:41,564 --> 1:14:45,596 One thing it suggests is that our 3D universe perhaps is 1023 1:14:45,596 --> 1:14:49,922 curved in the fourth dimension, just like the 2D universe of 1024 1:14:49,922 --> 1:14:53,441 the flatlanders is curved in the third dimension. 1025 1:14:53,441 --> 1:14:57,547 A very fascinating thought. And if you want to learn more 1026 1:14:57,547 --> 1:15:01,800 about it, I would advise you very strongly to take a course 1027 1:15:01,800 --> 1:15:06,138 in cosmology. If you are so lucky that Alan 1028 1:15:06,138 --> 1:15:09,972 Guth is teaching it then don't miss that opportunity. 1029 1:15:09,972 --> 1:15:13,511 He gives wonderful courses on the early universe. 1030 1:15:13,511 --> 1:15:18,009 But also you may want to learn a little bit more about general 1031 1:15:18,009 --> 1:15:21,253 relativity which, of course, plays a very key 1032 1:15:21,253 --> 1:15:24,718 role in cosmology. Doppler shift measurements of 1033 1:15:24,718 --> 1:15:29,290 electromagnetic radiations from galaxies have played a key role 1034 1:15:29,290 --> 1:15:33,493 in the way that we think about ourselves and that we think 1035 1:15:33,493 --> 1:15:38,936 about the universe. It was in 1979 that Professor 1036 1:15:38,936 --> 1:15:44,607 Alan Guth, who was still at MIT, he was also then at MIT, 1037 1:15:44,607 --> 1:15:50,481 made an exciting prediction that the universe may have gone 1038 1:15:50,481 --> 1:15:56,455 through an expansion phase very early on after the Big Bang, 1039 1:15:56,455 --> 1:16:01,822 exponential growth in expansion, and that the universe 1040 1:16:01,822 --> 1:16:08,000 doubled in size in about 10 to the minus 34 seconds. 1041 1:16:08,000 --> 1:16:13,012 The universe doubled in its size, the early universe, 1042 1:16:13,012 --> 1:16:18,120 called the inflationary universe in 10 to the minus 34 1043 1:16:18,120 --> 1:16:21,686 seconds. How long this phase lasted is 1044 1:16:21,686 --> 1:16:26,313 not very well known, but it may have lasted 10 to 1045 1:16:26,313 --> 1:16:31,228 the minus 24 seconds. After about 10 to the minus 24 1046 1:16:31,228 --> 1:16:36,530 seconds our universe had increased in size by a factor 2 1047 1:16:36,530 --> 1:16:41,614 to the power 10 billion. Unimaginable. 1048 1:16:41,614 --> 1:16:47,826 Beyond comprehension. This idea of the inflationary 1049 1:16:47,826 --> 1:16:55,403 universe, which came first from Alan Guth, has been the driver 1050 1:16:55,403 --> 1:17:02,537 behind NASA's cosmology program. There is an observatory about 1051 1:17:02,537 --> 1:17:07,175 1.5 million kilometers from earth called the Wilkinson 1052 1:17:07,175 --> 1:17:12,162 Microwave Anisotropy Probe that was launched in June 2001. 1053 1:17:12,162 --> 1:17:17,500 And that is now confirming many of the predictions that follow 1054 1:17:17,500 --> 1:17:20,824 from Alan Guth's inflationary universe. 1055 1:17:20,824 --> 1:17:25,725 There is no doubt in my mind that Alan will get the Nobel 1056 1:17:25,725 --> 1:17:30,210 prize for this. It is not a matter of if, 1057 1:17:30,210 --> 1:17:34,454 it is only a matter of when you will see the day, 1058 1:17:34,454 --> 1:17:37,549 and I hope I will see that day, too. 1059 1:17:37,549 --> 1:17:38,433 Thank you. And see you next Tuesday.