1 00:00:01,160 --> 00:00:06,048 [INAUDIBLE] One last thing to catch up on is, we talked about the beautiful 2 00:00:06,048 --> 00:00:10,450 planetary nebula the sun will make. But then, what about that core? 3 00:00:10,450 --> 00:00:15,963 I want to look a little more closely at that white dwarf which is, in some sense, 4 00:00:15,963 --> 00:00:21,186 the end result of what the sun will be. And white dwarves have an interesting 5 00:00:21,186 --> 00:00:25,699 discovery history, shortly after measuring the first successful parallax 6 00:00:25,699 --> 00:00:30,934 measurement Bessel followed another star which he thought would be likely Sirius 7 00:00:30,934 --> 00:00:35,747 which we know is very nearby and as he measured its proper motion he noticed 8 00:00:35,747 --> 00:00:40,199 that Sirio Sirius wobbles and he immediately understood what was going on 9 00:00:40,199 --> 00:00:43,810 he knew about Newtonian physics, Sirius has a binary partner. 10 00:00:43,810 --> 00:00:51,188 the search for what was called the pup was on people wanted to find it it turned 11 00:00:51,188 --> 00:00:55,780 out to be a very challenging prospect but in 1846 12 00:00:55,780 --> 00:01:01,441 indeed the star was found and the reason it had been so difficult to detect was 13 00:01:01,441 --> 00:01:06,489 because it was so much dimmer than serious, serious has a luminosity of 23 14 00:01:06,489 --> 00:01:09,900 solar luminosities and serious b its partner only. 15 00:01:09,900 --> 00:01:16,164 .03 solar luminosities Clark was very fortunate the year that he was making the 16 00:01:16,164 --> 00:01:20,090 measurement turned out to be a year of ap astron. 17 00:01:20,090 --> 00:01:24,873 Which is akillian for a binary system. It's the time when, in their elliptical 18 00:01:24,873 --> 00:01:29,357 orbits, Sirius A and B were farthest apart, their period is a little over 50 19 00:01:29,357 --> 00:01:32,586 years, and this will play into the history in a minute. 20 00:01:32,586 --> 00:01:36,173 Measuring their orbits, we can figure out their masses. 21 00:01:36,173 --> 00:01:40,478 We're good at that by now, after the experience of the previous week, and 22 00:01:40,478 --> 00:01:45,022 Sirius, as we mentioned, has a mass a little over two solar masses, and, its 23 00:01:45,022 --> 00:01:48,730 partner has about a mass of one solar mass, so it's a dim star. 24 00:01:48,730 --> 00:01:53,896 With one solar mass, it's certainly not a main sequence star, we know that, because 25 00:01:53,896 --> 00:01:58,488 a main sequence star with one solar mass would have about 30 times that 26 00:01:58,488 --> 00:01:59,511 luminosity. Okay. 27 00:01:59,511 --> 00:02:03,303 Well the game was on to find the spectrum of these objects. 28 00:02:03,303 --> 00:02:08,123 You needed to measure them again. You needed to wait for a fast gen to come 29 00:02:08,123 --> 00:02:09,794 around. And it did in 1915. 30 00:02:09,794 --> 00:02:15,000 And an American astronomer named Adams managed to measure the temperature of the 31 00:02:15,000 --> 00:02:17,828 two stars. And his results shocked the world, 32 00:02:17,828 --> 00:02:20,978 because the temperature of Sirius had been known. 33 00:02:20,978 --> 00:02:23,677 It's the 9900 degrees Kelvin that we knew. 34 00:02:23,677 --> 00:02:28,819 But the temperature of the partner turned out to be not a lot less as you would 35 00:02:28,819 --> 00:02:32,290 expect from the low luminosity, but in fact a lot more. 36 00:02:32,290 --> 00:02:38,110 Hookie so now if you know the temperature and you know the luminosity. 37 00:02:38,110 --> 00:02:42,580 Bada bing bada beam, sigma t to the fourth, we can compute the radius of 38 00:02:42,580 --> 00:02:47,303 Sirius B, and the radius of Sirius B turns out to be less than a percent of 39 00:02:47,303 --> 00:02:51,837 the solar radius, more impressively, less than Earth's radius by slightly. 40 00:02:51,837 --> 00:02:54,608 It's the radius is less than Earth's radius. 41 00:02:54,608 --> 00:02:59,370 This in an object with the size of Earth. And the mass of the sun this is 42 00:02:59,370 --> 00:03:04,888 extraordinarily dense the surface gravity on serious b you just compute plug the 43 00:03:04,888 --> 00:03:10,202 mass you know divided by the radius squared scaled to earth and you find that 44 00:03:10,202 --> 00:03:15,516 the surface gravity on serious b is only 450,000 times the surface gravity on 45 00:03:15,516 --> 00:03:21,359 earth you do not really want to go there. astronomers in the early twentieth 46 00:03:21,359 --> 00:03:24,375 century scoffed they figured there's got to be a mistake. 47 00:03:24,375 --> 00:03:28,926 There's no way that such an object could exist but it was there so it was there to 48 00:03:28,926 --> 00:03:31,678 be studied. It's spectrum showed very broad hydrogen 49 00:03:31,678 --> 00:03:34,377 absorption lines. Well if you have an atmosphere of 50 00:03:34,377 --> 00:03:38,134 hydrogen in this kind of gravitational field it will certainly be very 51 00:03:38,134 --> 00:03:42,103 compressed and you expect very broad absorption lines but otherwise it was 52 00:03:42,103 --> 00:03:47,020 just a featureless black body continuum. you could make estimates of, given the 53 00:03:47,020 --> 00:03:51,603 mass and the radius of the central density and pressure and the central 54 00:03:51,603 --> 00:03:55,741 pressure is millions of times more than in the center of the sun. 55 00:03:55,741 --> 00:03:58,549 The central temperature is. 70,000,000 Kelvin. 56 00:03:58,549 --> 00:04:02,424 Well that tells me something. If, we know that there are hydrogen 57 00:04:02,424 --> 00:04:05,936 absorption lines. We know the universe is made of hydrogen. 58 00:04:05,936 --> 00:04:10,477 But there better not be that much hydrogen in this object because, if there 59 00:04:10,477 --> 00:04:15,200 was, it would fuse and, and then this luminosity at these temperatures would be 60 00:04:15,200 --> 00:04:19,498 way, way higher than this three one hundredths of a solar luminosity. 61 00:04:19,498 --> 00:04:22,526 So, this is an hydrogen free object. Something rare. 62 00:04:22,526 --> 00:04:25,254 It was poorly understood. But now we know. 63 00:04:25,254 --> 00:04:31,631 The way to make a hydrogen-free object is first condense the, heavy byproducts of 64 00:04:31,631 --> 00:04:34,744 fu, helium fusion into the core of a star. 65 00:04:34,744 --> 00:04:40,589 You get basically an object that is almost exclusively carbon and, carbon and 66 00:04:40,589 --> 00:04:44,309 oxygen. Blow away the hydrogen, envelope, and 67 00:04:44,309 --> 00:04:47,650 what you're left with will be a white dwarf. 68 00:04:47,650 --> 00:04:50,888 So now we know. White dwarves are the degenerate cores of 69 00:04:50,888 --> 00:04:54,923 stars whose mass is not too high. Turns out that you can have a mass as 70 00:04:54,923 --> 00:04:58,616 high as eight solar masses. We'll talk about what happens to more 71 00:04:58,616 --> 00:05:02,367 massive stars in a few clips. There composition is indeed hydrogen 72 00:05:02,367 --> 00:05:05,208 free, they're almost exclusively carbon and oxygen. 73 00:05:05,208 --> 00:05:09,697 Their masses, after we measured many of them, lie in this narrow range between.4 74 00:05:09,697 --> 00:05:14,542 and.7 solar masses for the most part. Given the much larger range of stellar 75 00:05:14,542 --> 00:05:20,215 initial masses this gives you a sense of the mass loss that a star undergoes in 76 00:05:20,215 --> 00:05:25,360 the course of its last thermal pulsing days when it leaves an exposed mass. 77 00:05:25,360 --> 00:05:28,780 And. And here comes this guy trying to guy 78 00:05:28,780 --> 00:05:32,455 Chandrasekhar again with another interesting discussion. 79 00:05:32,455 --> 00:05:35,743 And this time, I want us to follow though with this. 80 00:05:35,743 --> 00:05:39,290 So, let's understand what Chandrasekhar tells us about 81 00:05:39,290 --> 00:05:42,571 About the weird behavior of the generate matter. 82 00:05:42,571 --> 00:05:47,288 So what Chandrasekhar says is this. So consider an object here, which is 83 00:05:47,288 --> 00:05:53,577 completely degenerate and not luminous so we can image that it's isothermal, but 84 00:05:53,577 --> 00:05:58,363 temperature will play no roll. And, we'll give it a radius R and a mass 85 00:05:58,363 --> 00:06:01,710 M. So, what do we know about this object? 86 00:06:01,710 --> 00:06:07,611 Well, we can figure out its density. We could imagine, for the sake of 87 00:06:07,611 --> 00:06:13,426 argument, that the density is uniform, although it's not going to be. 88 00:06:13,426 --> 00:06:18,719 And so, if the density were uniform, then the density would be m4*pi))/(3*r^3). 89 00:06:18,719 --> 00:06:22,445 divided by pi over 3 r cubed. Now, that's not true, because the density 90 00:06:22,445 --> 00:06:26,603 is not uniform. But, I want to compute some kind of 91 00:06:26,603 --> 00:06:32,235 characteristic density. So I'm going to say that it's going to be 92 00:06:32,235 --> 00:06:38,040 some number, which I shall call A, say, times m divided by r cubed. 93 00:06:38,040 --> 00:06:43,611 A will incorporate 4s and pis. And the relative geometry of the 94 00:06:43,611 --> 00:06:50,066 situation in two-fifths and whatever else, some number of order one, times 95 00:06:50,066 --> 00:06:54,134 m/r^3. What this tells me is that if this object 96 00:06:54,134 --> 00:07:00,147 is held up by electron degeneracy pressure, then that pressure, Pe, is 97 00:07:00,147 --> 00:07:04,454 given by k. -Fer electrons times rho to the 5/3, and 98 00:07:04,454 --> 00:07:10,951 I plug this expression into this, and I get a new constant - eight to the 5/3 99 00:07:10,951 --> 00:07:14,627 times, times k, which I might as well call c. 100 00:07:14,627 --> 00:07:18,440 And then I get. M over R cubed. 101 00:07:18,440 --> 00:07:23,105 Raised to the five thirds power. So the electron degeneracy pressure is 102 00:07:23,105 --> 00:07:28,427 dependent on mass and radius in this form, that's n to the five thirds divided 103 00:07:28,427 --> 00:07:31,646 by r to the fifth if you write it out explicitly. 104 00:07:31,646 --> 00:07:35,654 That's worth writing out cleanly. Let me write it out cleanly. 105 00:07:35,654 --> 00:07:40,319 So, for some number c the electron degeneracy pressure is given by this. 106 00:07:40,319 --> 00:07:44,524 Now, what is that statement? The statement is that this object is 107 00:07:44,524 --> 00:07:47,350 supposed to hold itself up against gravity. 108 00:07:47,350 --> 00:07:53,044 that means when you draw my object again since I erased it at the edge of the 109 00:07:53,044 --> 00:07:56,560 object pressure is zero its touching space and so. 110 00:07:56,560 --> 00:08:02,081 I can ask how much pressure do you have to have in the center in order for the, 111 00:08:02,081 --> 00:08:06,054 pre, the thing for that to be enough to hold that object up. 112 00:08:06,054 --> 00:08:09,420 In other words, what is the gravitational pressure? 113 00:08:09,420 --> 00:08:14,605 And just dimensional analysis tells us about how to figure out the effective 114 00:08:14,605 --> 00:08:19,655 contribution to pressure by gravity. Because, well of course it'll depend on 115 00:08:19,655 --> 00:08:23,576 G. Right and we can make an acceleration out 116 00:08:23,576 --> 00:08:29,971 of G and N and R which are the only parameters we have by taking G M over R 117 00:08:29,971 --> 00:08:32,735 square. And we can turn that into a force by 118 00:08:32,735 --> 00:08:35,970 multiplying by the only other thing we have which is N. 119 00:08:35,970 --> 00:08:40,240 So, gm squared over r squared is a characteristic force of gravity. 120 00:08:40,240 --> 00:08:45,546 We want the pressure, so that needs to be the divided by area so I multiply by the 121 00:08:45,546 --> 00:08:50,140 only relevant thing that could be an area and this is completely wrong. 122 00:08:50,140 --> 00:08:55,754 this is not the pressure due to gravity. But if put some number here which I will 123 00:08:55,754 --> 00:09:01,403 call c prime then it is and exactly what depends where exactly your measuring the 124 00:09:01,403 --> 00:09:07,121 gravitational pressure due to gravity and what exactly the distribution of mass is 125 00:09:07,121 --> 00:09:12,150 it so on but some version c prime times g m squared over r to the fourth. 126 00:09:12,150 --> 00:09:15,673 Is always a good estimate for the gravitational pressure. 127 00:09:15,673 --> 00:09:20,186 Or, if you want, the difference in pressure from here to here, which is the 128 00:09:20,186 --> 00:09:24,761 amount basically the, by which the pressure in the center has to be higher 129 00:09:24,761 --> 00:09:27,914 than the pressure at the edge to hold the thing up. 130 00:09:27,914 --> 00:09:32,488 So let's take note of this gravitational pressure calculation that we did. 131 00:09:32,488 --> 00:09:35,456 I just copied it and rated, wrote it out cleanly. 132 00:09:35,456 --> 00:09:40,649 And now Chandrasekhar notes, set one equal to the other up for the case of an 133 00:09:40,649 --> 00:09:45,768 objects that's held up by a. Degenerate matter and what do we do we 134 00:09:45,768 --> 00:09:52,894 set this equal to that and I find that c primed times g times m squared over r to 135 00:09:52,894 --> 00:09:57,207 the fourth. Is equal to C times N to the five thirds 136 00:09:57,207 --> 00:10:02,119 divided by R cube to the five thirds is just R to the fifth. 137 00:10:02,119 --> 00:10:04,821 Okay so. I get some cancellations. 138 00:10:04,821 --> 00:10:09,652 I have all kinds of constants I don't know what to do with. 139 00:10:09,652 --> 00:10:13,910 But these R's go away. And four of those R's go away. 140 00:10:13,910 --> 00:10:17,513 And five thirds is less than two by a third. 141 00:10:17,513 --> 00:10:22,545 So I can get rid of this. And raise n to the one third. 142 00:10:22,545 --> 00:10:30,368 So, multiplying by r here I find that n to the one third times r and then 143 00:10:30,368 --> 00:10:38,809 dividing by this constant is some number c over c prime g and then it's easier to 144 00:10:38,809 --> 00:10:43,750 take the cube of all of this which is n r cubed. 145 00:10:43,750 --> 00:10:46,198 Is this whole constant, some number cubed. 146 00:10:46,198 --> 00:10:50,857 Notice that this constant depends a little bit on the geometry but it doesn't 147 00:10:50,857 --> 00:10:53,485 change too much. Well, that's very reasonable. 148 00:10:53,485 --> 00:10:58,263 The mass of an object times the cube of its radius, essentially times its volume 149 00:10:58,263 --> 00:11:01,076 is a constant. That is the weirdest thing ever. 150 00:11:01,076 --> 00:11:03,978 Let's write that out and think what that means. 151 00:11:03,978 --> 00:11:08,240 That means that if you have two white dwarfs and one is more massive. 152 00:11:08,240 --> 00:11:11,891 The one with the bigger end has a smaller radius. 153 00:11:11,891 --> 00:11:15,542 Degenerate matter works in a very, very weird way. 154 00:11:15,542 --> 00:11:21,428 The more massive a white dwarf is, the more it has to compress in order to hold 155 00:11:21,428 --> 00:11:25,750 itself up against gravity. And therefore the smaller it is. 156 00:11:25,750 --> 00:11:30,228 This already hints at something very bizarre as you make the white dwarf more 157 00:11:30,228 --> 00:11:32,979 and more massive and. Become smaller and smaller. 158 00:11:32,979 --> 00:11:37,571 The electrons are pushed to higher and higher energy levels, and eventually 159 00:11:37,571 --> 00:11:40,632 relativity comes into this, that is next weeks stuff. 160 00:11:40,632 --> 00:11:44,930 But when you make the correction for relativity, you find that eventually. 161 00:11:44,930 --> 00:11:48,026 If you make the mass too big the radius becomes zero. 162 00:11:48,026 --> 00:11:52,408 the white dwarf cannot exist with a mass bigger than what is called the 163 00:11:52,408 --> 00:11:56,030 Chandrasekhar limit. And the Chandrasekhar limit is about 1.44 164 00:11:56,030 --> 00:11:59,127 solar masses. You cannot have a white dwarf with mass 165 00:11:59,127 --> 00:12:03,509 bigger than that, A degenerate object held up by electron degeneracy cannot 166 00:12:03,509 --> 00:12:07,365 have a mass bigger than that. Our observations are consistent with 167 00:12:07,365 --> 00:12:10,170 this, here is the weird mass radius relationship. 168 00:12:10,170 --> 00:12:14,062 For White dwarfs for degenerate matter would 169 00:12:14,062 --> 00:12:16,909 be computed as the non-relativistic blue curve. 170 00:12:16,909 --> 00:12:20,482 You notice that the larger the mass, the smaller the radius. 171 00:12:20,482 --> 00:12:24,297 And that's pretty weird. the relativistic corrections become 172 00:12:24,297 --> 00:12:28,657 important at large masses and small radii and they show you that at the 173 00:12:28,657 --> 00:12:33,441 Chandrasekhar limit, that's the end, you cannot pack more than 1.44 solar masses 174 00:12:33,441 --> 00:12:37,075 into a chunk of degenerate matter, something bad will happen. 175 00:12:37,075 --> 00:12:39,013 What will happen, we'll see soon.