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← 8. Introduction to Black Holes

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Ukazujem Revíziu 1 vytvorenú 08/17/2012 od Amara Bot.

  1. Professor Charles
    Bailyn: Okay,
  2. welcome to the second part of
    Astro 160.
  3. This is going to be about black
    holes and relativity.
  4. And just to give you kind of a
    preview, the whole point of
  5. black holes is that,
    of course, they emit no light,
  6. so you can't see them directly.
    And so, the question arises,
  7. "How do you know that they're
    there?"
  8. And the reason you can
    demonstrate that black holes
  9. exist is because they're in
    orbit around other things and
  10. you can see the motion of the
    other things that interact
  11. gravitationally with the black
    hole.
  12. This concept should be
    familiar, to a certain extent,
  13. because it's exactly the same
    thing we've been doing for
  14. discovering exoplanets.
    You don't see the exoplanet
  15. directly.
    What happens is that there's
  16. something else that you can see
    that's affected by the presence
  17. of the exoplanet.
    So, exactly the same thing
  18. happens with black holes.
    And so, we're going to use the
  19. same equations,
    the same concepts,
  20. to explore this very different
    context.
  21. So, black holes can't be seen
    directly.
  22. And so, instead of detecting
    them directly,
  23. you use this combination of
    orbital dynamics and things like
  24. the Doppler shift to infer their
    presence,
  25. and more than just inferring
    their presence,
  26. to infer their properties.
    Now, the context is more
  27. complicated.
    And, in particular,
  28. we're no longer going to be
    using Newton's laws – Newton's
  29. Law of Gravity,
    Newton's Laws of
  30. Motion--because there is a more
    comprehensive theory that
  31. replaced Newton,
    which is necessary to
  32. understand these things.
    That more complex theory is
  33. Einstein's Theory of Relativity.
  34. So, we're going to be using
    some relativity rather than
  35. Newtonian physics.
    This gets weird very fast, okay?
  36. And so, I'm not going to start
    there.
  37. I'm going to start with a kind
    of Newtonian explanation for
  38. what black holes are,
    we'll do that this time,
  39. and then the weirdness will
    start on Thursday.
  40. So, the first concept and the
    easiest way, I think,
  41. to understand black holes is
    the concept of the escape
  42. velocity.
    This is a piece of high school
  43. physics.
    Some of you may have
  44. encountered it before.
    And it just means how fast you
  45. have to go to escape from the
    gravitational field of a given
  46. object.
    If you go outside and you shoot
  47. up a rocket ship or something
    like that, how fast do you have
  48. to shoot it up so that it
    doesn't fall back to the Earth?
  49. And so, you can define an
    escape velocity for the Earth,
  50. or for any other object for
    that matter,
  51. which is just how fast you have
    to go to escape its
  52. gravitational field.
    There is an equation associated
  53. with this.
    It looks like this,
  54. V_escape,
    that's the escape velocity,
  55. 2GM / R all to
    the 1/2 power.
  56. And this is the speed required
    to escape the gravitational
  57. field of an object;
    supposing that the object has
  58. mass equal to M and
    radius equal to R.
  59. Oh, one other assumption,
    I'm assuming here that you're
  60. standing on the--that you start
    from standing on the surface of
  61. the object.
    If you are on the surface.
  62. Okay.
    This equation should look
  63. vaguely, but not a hundred
    percent familiar to you,
  64. because you derived something
    that looked a lot like it on the
  65. second problem set,
    where you worked out the
  66. relationship between the
    semi-major axis of an orbit and
  67. the speed and object had to go
    in to be in that orbit,
  68. and what--the way that
    calculation worked out it was
  69. the velocity equals GM
    over the semi-major axis,
  70. a.
    So that 2 wasn't there in that
  71. derivation, but otherwise,
    the form of this is actually
  72. quite similar to that.
    And so, let me explain why that
  73. is.
    Here's some object.
  74. It's got a radius of R
    and a mass of M.
  75. And imagine that you've got
    something that's in orbit around
  76. this object, but it's in orbit
    right above the surface.
  77. It's just skimming the surface
    of the thing.
  78. That would be impossible in the
    case of Earth,
  79. because the friction from the
    atmosphere would slow you down,
  80. but in a planet without an
    atmosphere, this would be
  81. possible.
    Imagine you're just sort of
  82. skimming over the tops of the
    mountains, or whatever.
  83. So here you are in orbit,
    this is something in a nice
  84. circular orbit right above the
    surface of this object,
  85. and it's going around.
    And it's got some velocity,
  86. which I'm going to call the
    circular velocity.
  87. And as we calculated in the
    previous section of the course,
  88. that's GM / a,
    which is, in this case,
  89. GM / R, because
    you're skimming the surface,
  90. to the 1/2 power.
    So that's how fast you have to
  91. go to stay in orbit.
    If you go slower than
  92. that--let's take a different
    color, here.
  93. If you go slower than that,
    you crash into the surface
  94. right away, because you
    don't--you're not going fast
  95. enough to stay in orbit.
    So, that's what happens in the
  96. case--if you have a velocity
    less than the circular velocity.
  97. What happens if you're going
    faster than the circular
  98. velocity?
    Well, you're going so fast that
  99. you move further away from this
    object.
  100. You don't stay skimming the
    surface.
  101. So, here you are,
    going a little bit faster.
  102. And now you're not in a
    circular orbit and your orbit
  103. ends up looking something like
    this--nice elliptical orbit.
  104. And this is a velocity greater
    than the circular velocity,
  105. but still less than the escape
    velocity because you haven't
  106. escaped the gravitational field,
    because you're still in an
  107. orbit.
    It's an elliptical orbit now,
  108. but you're still in an orbit,
    and it'll come back to the same
  109. place.
    Now, if you're going at
  110. somewhat faster than that at the
    escape velocity,
  111. then you never come back.
    You continue all the way out to
  112. infinity.
    Your orbit continues to change
  113. direction a little bit due to
    the gravitational force of this
  114. thing.
    You continue to slow down,
  115. but in the end you never come
    back.
  116. So, it's a non-repeating orbit.
    And, if you're going even
  117. faster than the escape velocity,
    then you get to infinity even
  118. faster.
    And, more importantly,
  119. when you get there,
    you're still moving pretty
  120. fast.
    This, you gradually slow down.
  121. So, all of these different
    kinds of orbits--the one that
  122. crashes into the planet,
    the one that skims the planet,
  123. the one that's an elliptical
    orbit, the one that escapes
  124. altogether--those are all within
    a factor of the square root of 2
  125. of each other.
    Because, remember,
  126. the escape velocity here is
    equal to (2GM /
  127. R)^(1/2).
    So, just by increasing your
  128. orbital speed by a factor of the
    square root of 2,
  129. you go from being in a nice
    circular orbit to escaping the
  130. gravitational field of the
    object altogether.
  131. So, you can calculate escape
    velocities of things.
  132. Let's do that once.
    The escape velocity of
  133. Earth--that would be--the escape
    is (2GM /
  134. R)^(1/2).
    2G = 7 x 10^(-11)
  135. M is, for the Earth,
    is 6 x 10^(24) R for the
  136. Earth is something like 7 x
    10^(6).
  137. This all has to be taken to the
    ½ power.
  138. The 7s cancel.
    What do we have here?
  139. Let's see, 12 x 10^(13).
    24 - 11 over 10^(6) 1/2--let's
  140. see, that's 1.2.
    14 - 6 = 8, times 10^(8),
  141. to the 1/2.
    That's something like 10^(4)
  142. meters per second,
    because I've used the value of
  143. G that's appropriate for
    meters per second.
  144. So, that's about 10 kilometers
    a second.
  145. So, that's the escape velocity
    of the Earth.
  146. If you go outside,
    you throw a football up into
  147. the air at 10 kilometers per
    second, it's not going to come
  148. back down.
    Try it--good exercise.
  149. Okay, you could calculate the
    escape velocity of any
  150. particular object.
    You could calculate the escape
  151. velocity of the human being--of
    a human.
  152. A human has a mass of 100
    kilograms, a radius of about 1
  153. meter;
    this is kind of a big human,
  154. but we'll go with it.
    Let's see,
  155. V_escape is
    equal to 2GM over--a
  156. half--there's a very famous book
    about physics with the title,
  157. "Assume A Spherical Cow," and
    this--so, this is the kind of
  158. thing that physicists like to
    do.
  159. It's an idealized situation.
    We have this perfectly
  160. spherical human being and--okay
    so let's do the calculation,
  161. 2 x 7 x 10^(-11) x 100,
    that's 10^(2),
  162. over 1, to the square root.
    That's 14 x 10^(-9).
  163. Call it (1.4 x 10^(-8))^(1/2).
    That's something like 1.2,
  164. let's call it 1.
    1 x 10^(-4) meters per second.
  165. Or--let's see,
    that's 1/10 of a millimeter per
  166. second.
    That's less than 1 meter per
  167. hour.
    And that's why we don't go into
  168. orbit around each other.
    Okay?
  169. Because you're always moving
    way, way, way faster than the
  170. escape velocity of the people
    you're interacting with.
  171. The escape velocity of the
    Earth is 10 kilometers a second.
  172. So, you're not moving that fast.
    So, you're bound to the Earth.
  173. But when you're hanging around
    with your friends,
  174. you're probably going faster
    than 1 meter per hour,
  175. and therefore you're moving
    faster than the escape velocity.
  176. You don't feel a great effect
    of the gravitational force of
  177. other people.
    This is kind of
  178. anti-Valentine's Day
    calculation, right?
  179. Because it proves that human
    beings are not attracted to each
  180. other;
    at least not by the force of
  181. gravity.
    And so, you can do this
  182. calculation for any given
    object.
  183. Fine, so what's a black hole?
    Black hole has,
  184. actually, a very simple
    definition.
  185. A black hole is simply
    something in which the escape
  186. velocity is greater than or
    equal to the speed of light.
  187. c is the expression for
    the speed of light.
  188. This is 3 x 10^(8) meters per
    second.
  189. And, if you've got something
    where the escape velocity is
  190. greater than the speed of light,
    that's what a black hole is.
  191. And it makes a certain amount
    of sense.
  192. If the escape velocity's
    greater than the speed of light,
  193. then light won't escape this
    object, so you won't be able to
  194. see it, right?
    Because, the way you see
  195. something is you see the photons
    that come off it.
  196. And, there's nothing
    particularly extraordinary about
  197. this, as far as it goes.
    In fact, the middle of--this
  198. was already talked about and
    worked out in some detail in the
  199. middle of the eighteenth century
    by an otherwise obscure English
  200. clergyman named,
    I think, John Michell,
  201. and he did the following
    calculation.
  202. He said, okay now,
    if this is going to be true,
  203. how big is such an object?
    How dense does it have to be?
  204. And, you can work this out.
    Supposing the escape velocity
  205. is equal to the speed of
    light--so c = (2GM
  206. / R) ^(1/2).
    You can square both sides and
  207. regroup, and you have R =
    2GM / c^(2).
  208. This is now--wasn't called this
    in the eighteenth century,
  209. but it's now called the
    Schwarzschild radius.
  210. Schwarzschild was a
    contemporary of Einstein.
  211. And, this is the size something
    has to be in order for its
  212. escape velocity to be equal to
    the speed of light.
  213. And a black hole--another
    definition of a black hole is
  214. something in which the radius of
    the object is less than the
  215. Schwarzschild radius--because if
    the radius is less,
  216. then the escape velocity will
    be even greater.
  217. So as I say,
    this was worked out in
  218. obscurity in the middle of the
    eighteenth century,
  219. and nobody thought anything
    much about it.
  220. Michell sort of pondered for a
    little bit what such an object
  221. would look like,
    and he decided it would look
  222. dark.
    And true enough, but who cares.
  223. And nobody thought anything
    further about it for 150 years
  224. until Einstein came along at the
    start of the twentieth century,
  225. and came up with the Theory of
    Relativity.
  226. And one of the pieces,
    important pieces of relativity
  227. is that the speed of light,
    c is a very,
  228. very special velocity,
    and has rather bizarre and
  229. extremely profound properties.
    And it was only at that point
  230. that the concept of the black
    hole was recognized as something
  231. that was in any way out of the
    ordinary.
  232. And so, for 150 years this idea
    kind of lay dormant until it was
  233. resurrected by a profound change
    in the thinking about the
  234. underlying physics.
    So, this is one of these kinds
  235. of historical fables that we've
    hit a couple of times.
  236. This is the fable of Michell's
    discovery of black holes.
  237. And the moral of this little
    fable might be that the
  238. importance of a result changes
    depending on the context;
  239. result changes,
    sometimes dramatically,
  240. with context.
    So, something that looks quite
  241. unimportant for a long time can
    all of a sudden become a very
  242. big deal.
    And so, it happened in this
  243. case.
    Okay, so, one question you can
  244. ask is that's all fine,
    how big is the Schwarzschild
  245. radius for some bunch of
    objects?
  246. Let's try the Sun;
    here, let me start a new piece
  247. of paper.
  248. So, how big is the
    Schwarzschild radius of the Sun?
  249. Schwarzschild radius,
    again, 2GM /
  250. c^(2) = 2 x 7 x 10^(-11).
    Mass of the Sun,
  251. now, 2 x 10^(30).
    Divided by c^(2).
  252. c is 3 x 10^(8) and
    we'll square it.
  253. Okay, so let's see,
    2 x 2 = 4, times 7 is 30.
  254. 30 x 10^(19) 30 - 11 = 19
    Divided by--3 x 3 = 10.
  255. (10^(8))^(2) = 10^(16).
    So, that's 3 x 10^(3),
  256. because 19 - 16 = 3--oh,
    in meters, because we're still
  257. in MKS units,
    because that what we're using.
  258. That's the units of G
    that we've chosen.
  259. 3 x 10^(3);
    that's 3 kilometers.
  260. That's pretty small for the Sun.
    And so, you might imagine that
  261. such objects are rare,
    or perhaps even non-existent,
  262. because it would have to be
    incredibly small,
  263. and therefore,
    incredibly dense,
  264. to have a strong enough
    gravitational field to prevent
  265. light from escaping.
    And so, you might have thought
  266. that this is an entirely
    theoretical concept,
  267. and that no matter how
    interesting it is,
  268. there's no point in really
    studying it further,
  269. because you're unlikely to ever
    encounter one of these things in
  270. real life.
    But one of the strange things
  271. that was--has been known for
    quite some time is that black
  272. holes really should exist,
    and that they are predicted to
  273. exist.
    This has been known for quite a
  274. while--known for at 70 years,
    that black holes should exist.
  275. And the reason they should is
    that they are one of the
  276. possible end points of stellar
    evolution--the evolution of
  277. stars.
  278. And so, now I want to summarize
    how stars evolve.
  279. I should say,
    this is the subject of whole
  280. courses.
    You can take Astro 350 and they
  281. will talk about this for the
    entire semester.
  282. You can take Astro 110 and then
    they'll talk about it for a
  283. month.
    But we're going to do it in
  284. about twenty minutes,
    save you time and effort,
  285. here.
    So, a star's life is determined
  286. by the competition between two
    forces;
  287. so, star's lifetime and
    evolution is determined by two
  288. forces.
    One is gravity,
  289. which has the tendency to hold
    the star together.
  290. And now, the thing about
    gravity is, it ought to work.
  291. But stars have no solid surface.
    So, if you can imagine gravity
  292. pulling on an atom on the
    surface of the star,
  293. why shouldn't that atom just
    fall all the way down to the
  294. bottom of the star?
    Because there's no solid
  295. surface to prevent it from doing
    so.
  296. And the answer is that gravity
    isn't the only force operating,
  297. there's also pressure.
    So, there's gravity,
  298. which pulls stuff in,
    and pressure,
  299. which has the tendency to push
    out.
  300. And these things balance in
    most stars.
  301. In the Sun, for example,
    these two forces are in balance
  302. at all points in the Sun,
    and this balance goes by the
  303. technical name hydrostatic
    equilibrium.
  304. Hydro, because it's a fluid,
    it's not a solid surface.
  305. Static--nothing's moving,
    and equilibrium is just
  306. balance.
    And, to be a little more
  307. precise, the way this
    works--here's the surface of
  308. some star.
    Here's some point within the
  309. star, and there are two kinds of
    forces acting on this point.
  310. There's gravity,
    which is pulling the thing
  311. toward the center of the star,
    and then there's pressure
  312. forces in two different ways.
    The outer regions of the star
  313. exert a pressure inward.
    So, there's an inward pressure.
  314. And the inner regions of the
    star exert a pressure outward.
  315. And the outward pressure has to
    be greater than the inward
  316. pressure by exactly the right
    amount to counteract gravity.
  317. So, it basically looks like
    P_out -
  318. P_in = gravity.
    And this holds true at all
  319. points.
    Now, in order--it's the other
  320. way around right?
    Thank you.
  321. P_in -
    P_out--yeah,
  322. right.
    P_in minus
  323. P_out has to
    equal gravity.
  324. And that requires that the
    pressure on the inside has to be
  325. bigger than the pressure on the
    outside, because you don't have
  326. negative gravity;
    at least, not until the third
  327. part of this course.
    But, at the moment,
  328. you don't have any negative
    gravity, and so the pressure on
  329. the inside has to be greater
    than the pressure on the
  330. outside.
    Okay, so what is pressure?
  331. Cast your mind back to high
    school chemistry.
  332. Remember high school chemistry?
    It doesn't matter if you don't.
  333. I'll tell you everything you
    need to know.
  334. There's something called an
    ideal gas, and there's something
  335. that the ideal gas does,
    which is to exert gas pressure.
  336. Your high school chemistry
    teacher probably wrote down
  337. something that looks like this:
    PV = nRT. And
  338. here's the thing about this.
    V is volume in this case;
  339. P is the pressure;
    n is the number of
  340. particles per volume.
    And so, the key thing here is
  341. that n divided by
    V is equal to the
  342. density, basically,
    times a constant.
  343. Because, remember,
    density is mass per volume.
  344. If you take the number of
    particles and you multiply by
  345. the mass of each particle,
    that'll give you the total
  346. amount of mass in a given
    region.
  347. You divide by the volume,
    that equals the density.
  348. And so, this also is a
    constant, this R thing.
  349. And so, what you get is
    P is equal to a constant,
  350. times the density,
    times the temperature.
  351. So, this is how physicists
    think of the ideal gas law,
  352. because we prefer to work in
    terms of the density.
  353. Okay, so here's the pressure.
    And the pressure on the inside
  354. had better be bigger than the
    pressure on the outside,
  355. or this isn't going to balance,
    which means,
  356. either the density or the
    temperature,
  357. or both, had better be larger
    in the middle of the star than
  358. it is in the outer parts of the
    star.
  359. So, inside of the star,
    the T and/or the ρ has
  360. to be bigger than it is on the
    outside.
  361. Now, it turns out that if you
    just keep the temperature
  362. constant all the way through the
    star, you never achieve this
  363. balance.
    So, if only the density varies,
  364. then inner regions do have
    higher pressure but the increase
  365. in density also increases the
    force of gravity,
  366. because gravity is dependent on
    how much mass there is.
  367. And, if you increase the
    density, you also increase the
  368. amount of mass.
    So, you have higher pressure,
  369. but you also have higher
    gravity.
  370. And it turns out that you can
    prove, mathematically speaking,
  371. that no balance is possible,
    because it always ends up being
  372. the case, for gas pressure,
    that the amount you have to
  373. increase the density by,
    if you're only increasing the
  374. density, will also increase the
    gravity, and you'll never get a
  375. balance.
    So, the consequence of that is
  376. that the inner parts of a star
    must be hotter than the outer
  377. parts.
    Otherwise, the star wouldn't
  378. exist;
    it would collapse.
  379. And this is true.
    The inside of the Sun turns out
  380. to be something like 10^(7)
    degrees.
  381. The surface of the Sun turns
    out to be something like 6 x
  382. 10^(3).
    So yes, indeed,
  383. the inside very much hotter
    than the Sun,
  384. that is very much hotter than
    the outside,
  385. in the Sun, and that's what
    keeps the Sun in balance.
  386. And there's a problem with this.
    And the problem is that there
  387. is something called
    thermodynamics.
  388. And one of the laws of
    thermodynamics is that heat
  389. tends to flow from places where
    it's hot to places where it's
  390. cool.
    This is evident in everyday
  391. life.
    If you take a little piece of,
  392. I don't know,
    molten lead or something,
  393. and you drop it in a bucket of
    water,
  394. the heat from the lead will
    spread into the water.
  395. The water will increase in
    temperature very slightly.
  396. The heat will come out of that
    piece of lead.
  397. The lead will solidify,
    and everything will come into a
  398. kind of temperature balance.
    Similarly, if you put a
  399. snowball in some hot place,
    it'll melt.
  400. Why?
    Because the heat from around it
  401. will go into the snowball.
    The temperatures will try and
  402. equalize each other and
    they'll--it'll come out even.
  403. So, this law of thermodynamics
    is why the snowball has no
  404. chance in hell.
    And so, this happens in stars
  405. too, right?
    So, the heat in the center of
  406. the star flows out.
    And when it gets to the
  407. surface, it's radiated.
    At the surface,
  408. it radiates,
    and that's the energy that we
  409. see coming from the star.
    It's this heat that was in the
  410. center.
    It's gotten to the surface.
  411. It's now radiating away out
    into the cold depths of space,
  412. and that's what we see.
    But, that means that the
  413. temperature in the center of the
    star, which is holding the star
  414. up, decreases,
    and then the star wouldn't hold
  415. itself up.
    So you require--in order for
  416. the star to hold itself out--an
    energy source at the center of
  417. the star.
    And this does two things.
  418. It replaces all that lost heat,
    and it preserves the
  419. equilibrium of the star so it
    doesn't collapse.
  420. Okay, so this is all pretty
    abstract.
  421. And it was known that this had
    to be true before they figured
  422. out what the energy source was.
    It was known for people just
  423. sort of thinking in very general
    terms about how the Sun could
  424. exist--understood that there had
    to be some kind of large source
  425. of energy down in the middle of
    the star.
  426. And, notice it has to be in the
    middle of the star.
  427. It does no good for the energy
    to be created all the way
  428. through the star,
    because if it's created all the
  429. way through the star,
    then the temperature is
  430. distributed throughout the star
    and you don't get a situation
  431. where it's much hotter in the
    middle than it is on the
  432. outside.
    So, everybody knew for quite a
  433. while that there was energy
    being created in the center of
  434. the star.
    They just didn't understand how
  435. that was done.
    And then when people invented
  436. nuclear physics in the 1930s and
    '40s and '50s,
  437. it was understood that this
    comes from nuclear
  438. reactions--nuclear fusion,
    in particular.
  439. In the case of the Sun,
    it's the fusion of hydrogen,
  440. atoms together to make helium,
    that does this.
  441. And that releases energy in the
    same way that a nuclear bomb
  442. does.
  443. The problem with this is that
    eventually you run out of
  444. hydrogen, or whatever your
    nuclear fuel is,
  445. because you've got only a
    limited amount of it in any one
  446. star.
    So, eventually,
  447. the nuclear fuel runs out.
  448. And then the star has many
    adventures before it settles
  449. down.
    And for these,
  450. I will have to refer you either
    to a textbook or to some other
  451. course,
    because I'm not going to take
  452. you through the whole exciting
    life of a star once its nuclear
  453. fuel is exhausted.
    Suffice it to say that you know
  454. in advance what the outcome has
    to be, because there's no way it
  455. can hold itself up,
    in the long run,
  456. because it doesn't have an
    energy source down at the center
  457. of the star.
    So, the consequence of this has
  458. to be that the star collapses.
    Now, it doesn't necessarily
  459. collapse all the way down to
    being a black hole,
  460. because there are other kinds
    of pressure besides the pressure
  461. exerted by an ideal gas.
    So, at very high densities,
  462. you get other kinds of
    pressure.
  463. In particular,
    there's something called
  464. "electron degeneracy pressure."
  465. This is sometimes called Fermi
    pressure, after the guy who
  466. thought it up.
  467. Degeneracy is another one of
    those words that means something
  468. different to physicists than
    they do to ordinary,
  469. normal people.
    I remember the first time I
  470. taught Astro 110,
    right after I came here,
  471. about fifteen years ago.
    I started talking in the middle
  472. of a class about degenerate
    white dwarfs.
  473. And you could feel this sort of
    wave of something between
  474. confusion and anxiety permeate
    through the class,
  475. and I had no idea what was
    going on.
  476. Some teaching assistant had to
    pull me aside afterward and
  477. remind me that these words mean
    different things.
  478. So, I apologize if this sounds
    like a sort of adult version of
  479. a Grimm's fairy tale,
    you know, with degenerate white
  480. dwarfs wandering around and
    stuff, but such are the words we
  481. have to work with.
    Okay.
  482. So you have this electron
    degeneracy pressure--that's a
  483. different kind of pressure –
    and that can stabilize the star.
  484. So it stabilizes a
    star--stabilizes the star
  485. at--around the radius of the
    Earth.
  486. So, that's very high density
    material.
  487. We can calculate the density.
    You'll remember our equation
  488. for density: mass over volume.
    And so let's see,
  489. how does this work?
    This will be the mass of the
  490. Sun, let's say,
    divided by the volume of the
  491. Earth.
    That's 4/3πr^(3).
  492. Radius of the Earth is (7 x
    10^(6))^(3).
  493. So, let's see,
    (2 x 10^(30)) / 4.
  494. 7^(3)-- 7^(2) = 57,
    cubed is 350.
  495. Times 10^(18).
    That's 6 x 3--yeah,
  496. okay, we're doing all right.
    And let's say that that's 2,000
  497. x 10^(27) over,
    I don't know,
  498. 1,400 x 10^(18).
    Let's cancel those.
  499. 10^(9) kilograms per meters
    squared.
  500. That's about a million times
    denser than water.
  501. Water, you'll remember,
    is 10^(3) kilograms per meters
  502. squared.
    And water is defined to have a
  503. density of 1 gram for a cubic
    centimeter.
  504. So, if you were to pick up one
    gram of this white dwarf--one
  505. cubic centimeter of this white
    dwarf,
  506. it would be a million times
    more massive than a gram.
  507. That's about a ton.
    So, a thimbleful of this stuff
  508. weighs about a ton,
    very dense.
  509. And the Sun will end its life
    as a white dwarf with this
  510. electron pressure balancing the
    gravity.
  511. These--such stars are called
    white dwarfs,
  512. and there are many of them
    known.
  513. White dwarfs--this is the end
    point of the Sun.
  514. And then, in the 1930s,
    what happened to make black
  515. holes inevitable,
    was that one of the great
  516. theoretical astrophysicists of
    the twentieth century,
  517. a man named Subramanyan
    Chandrasekhar,
  518. discovered that electron
    degeneracy pressure doesn't
  519. always do the job.
    So, in the 1930s,
  520. Chandrasekhar discovers--proves
    that if the mass of an object is
  521. greater than 1.4 times the mass
    of the Sun,
  522. electron--this kind of electron
    pressure is insufficient.
  523. And the star continues to
    collapse.
  524. Now, Chandra was a graduate
    student in England at the time
  525. he figured this out,
    and he presented this rather
  526. dramatic result at a big meeting
    of the Royal Astronomical
  527. Society in London.
    And then, every graduate
  528. student's worst nightmare took
    place.
  529. Chandra's thesis advisor was a
    very famous man named Arthur
  530. Eddington, one of the great
    physicists of the early
  531. twentieth century.
    And after Chandra had presented
  532. his results to all these
    assembled scientific
  533. dignitaries,
    Eddington got up and denounced
  534. his own student,
    and said, "This can't possibly
  535. be true."
    And Eddington made the famous
  536. remark, "There ought to be a law
    of nature to prevent stars from
  537. behaving in this foolish
    manner."
  538. And the consequence of that is
    that a lot of people didn't
  539. follow up on Chandra's idea.
    Chandra got miffed,
  540. as you might understand.
    He got on the boat to the
  541. United States.
    He spent the rest of his career
  542. at the University of Chicago.
    On the boat,
  543. he wrote a great textbook still
    in use on the structure of
  544. stars,
    in which he laid out in detail
  545. all of the arguments that the
    Chandrasekhar limit must
  546. actually exist.
    And fifty years later he got
  547. the Nobel Prize for it.
    But that--there was this kind
  548. of time lag there,
    and it is, I think,
  549. important for those of us,
    say, on the faculty to recall
  550. that if Eddington had listened
    to his student instead of to his
  551. intuition,
    the study of black holes would
  552. probably be forty years more
    advanced than it is.
  553. So, another fable for our times.
    Chandra was always very
  554. gracious about this,
    actually.
  555. He would praise Eddington to
    the skies as a wonderful
  556. advisor, and then with this
    little asterisk.
  557. So, fable: Chandrasekhar's
    limit is the title,
  558. and the moral here is,
    "believe your student,
  559. not your intuition."
    And actually,
  560. the story of Eddington was
    actually kind of interesting.
  561. As Eddington got older,
    he became more and more
  562. convinced that,
    you know, he could guess the
  563. right answer.
    And so, it wasn't just the
  564. Chandrasekhar limit,
    it was other things.
  565. He got a little weird toward
    the end of his life,
  566. and he started believing his
    intuition.
  567. Einstein did this too, right?
    Einstein famously discovers all
  568. this great stuff,
    and then the second half of his
  569. life is completely useless,
    scientifically,
  570. because he becomes convinced
    that his gut is telling him that
  571. quantum mechanics is wrong.
    This famous remark,
  572. "God does not play dice with
    the universe"--but it turns out,
  573. that isn't true.
    And so, there is this
  574. probabilistic nature of reality
    that quantum mechanics shows.
  575. And so, Einstein spent the
    second half of life trying to
  576. prove that his intuition was
    correct,
  577. that quantum mechanics couldn't
    really be true and thus,
  578. did no physics worth doing for
    about thirty or forty years.
  579. So, you have to watch out for
    this.
  580. If you're too smart and start
    believing yourself,
  581. you can get into trouble.
    All right.
  582. Now, having said all that,
    Eddington was partly right.
  583. There is, sort of,
    a law of nature that prevents
  584. stars from behaving in this
    foolish manner.
  585. So, to understand that,
    what happens when white
  586. dwarfs--when the white dwarf
    collapses?
  587. It's got to get rid of its
    electrons, because the problem
  588. is, the way this electron
    degeneracy pressure works--you
  589. can't squeeze electrons any
    closer together than they are in
  590. a white dwarf.
    So now, you have to get rid of
  591. electrons.
    And so what do you do?
  592. You combine the electrons and
    the protons and you turn them
  593. into neutrons plus neutrinos.
    These are neutrinos.
  594. They stream out.
    And so, you end up with
  595. something that's made entirely
    out of neutrons.
  596. So, the whole star turns into
    neutrons.
  597. A chemist would think of this
    as essentially turning the whole
  598. star into one atom,
    into one atomic nucleus.
  599. An atomic nucleus with no--an
    atom with no protons,
  600. no electrons,
    and 10^(57) neutrons.
  601. And you could imagine putting
    that somewhere on the Periodic
  602. Table.
    Astronomers call these things
  603. neutron stars,
    and they exist.
  604. They were discovered in the
    1960s.
  605. And a typical neutron star,
    a couple times the mass of the
  606. Sun has--mass equals 2 times the
    mass of the Sun.
  607. Radius of about 10 kilometers.
    And you can work out the
  608. density for that.
    Density is a billion times
  609. greater--I'll leave this as an
    exercise--greater than for white
  610. dwarfs.
    So, instead of a cubic
  611. centimeter of the stuff weighing
    a ton, it now weighs a billion
  612. tons.
    And you're having a tough time
  613. moving it around.
    But 10 kilometers – that's
  614. getting close to the
    Schwarzschild radius.
  615. Remember, we calculated the
    Schwarzschild radius of the Sun.
  616. It was about 3 kilometers.
    And, in fact,
  617. if you calculate the
    Schwarzschild radius of a star
  618. in terms of the Schwarzschild
    radius of the Sun--let's see,
  619. you get 2GM /
    c^(2),
  620. where M is the mass of
    the star,
  621. divided by 2G mass of
    the Sun, over c^(2).
  622. So the Gs and the
    cs all cancel,
  623. here.
    And you get
  624. M_star /
    M_sun.
  625. So, if the Chandrasekhar mass
    of the Sun is equal to 3
  626. kilometers, as we calculated,
    the Schwarzschild radius of a
  627. star with--whose mass happens to
    equal 3 times the mass of the
  628. Sun,
    is going to be 3 x three
  629. kilometers, or 10 kilometers.
    That's equal to the radius of a
  630. neutron star.
    So, a neutron star with mass
  631. greater than 3 times the mass of
    the Sun has a radius less than
  632. its Schwarzschild radius.
    And that's a black hole.
  633. Remember?
    And the key thing here is that
  634. there are lots of stars with
    mass more than 3 times the mass
  635. of the Sun.
    We don't see them as black
  636. holes because they're still in
    hydrostatic equilibrium.
  637. But eventually,
    they're going to run out of
  638. nuclear fuel,
    and they're going to collapse.
  639. Now, in fact,
    during the course of the star's
  640. life, one of the things I
    glossed over is stars tend to
  641. lose mass as they live.
    And so, they don't end up with
  642. the same mass they started with.
    But stars with initial masses,
  643. at the beginning of their
    lifetime, greater than--I don't
  644. know,
    something like thirty times the
  645. mass of the Sun,
    will end up with masses greater
  646. than three times to the mass of
    the Sun.
  647. And then, there's nothing to
    stop their collapse.
  648. What happens is,
    they turn into neutron stars,
  649. but they turn into neutron
    stars whose Schwarzschild
  650. radius--whose radii are smaller
    than their Schwarzschild radius,
  651. and that is a black hole.
    So, they collapse down into
  652. black holes.
    And so, you expect a large
  653. number of black holes to
    actually exist.
  654. This is what happens to massive
    stars at the end of their life.
  655. And so, we expect there to
    be--that there are many black
  656. holes.
  657. And so, the question we'll be
    exploring in the rest of this
  658. segment of the class is,
    how can you find these things?
  659. What are the properties of
    these things from a theoretical
  660. point of view?
    What does Einstein's Theory of
  661. Relativity suggest that these
    things are going to behave like?
  662. And then, the big question is,
    once you've found some,
  663. and you have a theory for what
    they behave like--then you can
  664. ask the question,
    does the actual behavior of
  665. these objects conform to the
    theoretical expectations?
  666. Another way of saying that is,
    was Einstein right?
  667. Is general relativity the
    correct theory to describe these
  668. very exotic objects?
    And so, that's what we'll be
  669. talking about in the rest of
    this section of the course.
  670. Now, let me turn back to the
    previous section of the course,
  671. which was--which culminated
    last time in this little test.
  672. And I think we're ready to hand
    these back.
  673. Is that true?