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So Andrew's given us an introduction to the theoretical properties of black holes.

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I'm going to give you an introduction to what he calls the practical properties, or what I call just essentially proofs of existence.

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So here is a picture of our nearest neighbour, nearest big neighbour at the Andromeda Galaxy.

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You see, it's companion M32 and then this boring galaxy, NGC 205 and know the many, many stars around us.

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And in this picture, there are roughly 10,002 black holes.

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Right? Right. So the two black holes that we know about, definitely there's a big black hole at the centre of the Andromeda Galaxy.

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There is a slightly smaller one at the centre of M32.

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There's probably not one at the centre of this dwarf elliptical NGC two or five.

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And then in the surrounding environment, in the stars, among the stars surrounding each of these galaxies, there are roughly ten to the four.

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Ten to the five. Who cares? Stellar mass black holes. Right.

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So in astrophysics, there are at least two different types of black hole.

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And the type just refers to the origin. So stellar mass black holes.

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They have masses of around about ten times the mass of the sun.

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Andre pointed out that the Schwarzschild radius of the sun is three kilometres, so therefore the horizon scale in these is about 30 kilometres.

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Right. These occur every you have got stars,

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particularly where you get massive stars because these are massive stars that have

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gone supernova and then believe behind this stellar mass black hole remnant.

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In contrast, the bombastic arena and supermassive black holes there are much bigger.

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They're billions to billions times the mass of the sun.

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When you work out the horizon scale that corresponds to solar system scales, they seem to locate it at the centres of almost every biggish galaxy.

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And the only problem about them is we have no idea how they form. We know they're there, but we're not sure how they got there.

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Okay. Another thing that these have in common, apart from their fundamental physical properties,

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is that whenever you try to study them with the best telescopes,

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you find that the angular size of the black hole's horizon is roughly the same in the nearest cases in both situations.

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So it's roughly ten to the minus 5 seconds of arc, right?

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Or one arcseconds is basically. There's 3600 arcseconds in a degree.

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Right. So to give you a fitting for that number, the best optical and infrared telescopes, they can resolve point 1 seconds of arc.

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So factor ten to the four bigger than that.

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So for the point of view of for most of the time in astrophysics these black holes, they just looked like Newtonian points masses.

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Right. We can hope to measure the mass, but we can't hope to measure the speed, at least not directly.

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Right. So the name of the game in astrophysics is to find a compact object,

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then argue that this compact mass has to be a black hole just through lack of imagination.

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Right. So here's an outline of my talk. Very simple.

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I'll tell you about why we believe stellar mass black holes exist.

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Then I'll tell you about why supermassive black holes probably almost certainly exist.

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And I'll give you some consequences of these supermassive black holes. Okay.

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So stellar mass black holes remind you we know where they come from. They're the remnants of dead stars.

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Dead. Massive stars. Okay. So when you think about stars.

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Well, not too. A question to ask is what stops a star from collapsing into a black hole?

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And the answer is pressure. Support. Right.

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In a normal stars, the pressure is provided by just the thermal pressure that run the motion due to the atoms that make up the star.

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Then if special stars like white dwarfs a neutron stars.

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And in those cases, the pressure that prevents the stars from collapsing.

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And the first kiss is electron degeneracy pressure. And the second kiss is neutron degeneracy pressure.

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Right. And I want to argue that there's a I going to reproduce the arguments that say that there's a maximum mass for neutron stars.

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Right. And that's most naturally done, most easily done in terms of energy instead instead of in terms of mass.

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So just run through that. So just to remind you of the Polish exclusion principle that says that fermions are antisocial.

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They refuse to share states with one another. Identical fermions are antisocial and refused to share the same physical state.

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So this means that if you take a gas of these objects and you reduce the temperature toward zero,

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then the kinetic energy of the system doesn't go towards your internal energy doesn't go towards zero, but at any given fixed point in space.

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There would be a range of minimum momentum.

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States that have to be populated, right a corresponds to a certain character, certain energy called the Fermi Energy.

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Right. So even when you cool these things down to zero temperature, they still have some internal energy.

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And it's a nice exercise in second year undergraduate physics to show that if you have any of these identical

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fermions per unit volume on their moving ultra relativistic li so other you squeeze their density enough

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for the was run very very quickly you can show that this Fermi energy skills as the cube root of the number

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density right then just with the standard factors to make it have the right dimensions here it's part time.

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See? Okay, so the Fermi Energy's important.

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Then here is Lando's argument for why there's a maximum mass and such to generate objects.

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He did this in 1932. It was actually before the discovery of the neutron.

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So just to show what these Russians can do things right.

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So here is an example of a simple idealised model of a neutron star.

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We've could and neutrons each of mass m on their own bundled together in some radius are there is the Fermi Energy and Lando's assumption

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was the very reasonable one is that you assume that the equilibrium state of this system is the state of minimum total energy.

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Okay. Kind of hard to argue with that.

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So when you write down the star's total energy as a function of the number of neutrons and the radius over which they're spread,

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that's just going to be the kinetic energy, which is end times, the Fermi energy up to factors of order one.

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Plus the gravitational potential energy, which again of two factors of 4 to 1.

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It's just going to be GMM squared divided by the radius. Okay.

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You plug in the expressions for this.

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You get this thing here, you get a numerator that depends only on the total number of neutrons on the mass of the neutron and dependent radius.

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And then the denominator just radius. Right. So then if you look at this numerator, you see that if this numerator is less than zero,

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then if you happened to squeeze your star, if you reduce this radius, then the energy will go down.

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Right. So then come back to this principle that says that this system is unstable.

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And so then by looking at the numerator you range and you get an expression for the maximum mass that a neutron star can have.

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Right. It's given by this. There's a numerical factor in front of this.

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But when you work out what this thing here is, it's about 1.8 times the mass of the sun.

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Right. Of course, this calculation was very, very simplified.

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And you could do more complete models. That includes the equation of state of the neutrons.

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You know, here assume that they're not interacting.

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So you're going to create the effect of general relativity, which also becomes important in working out the gravitational potential energy.

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And this gives you numbers that are run about 3.1 solar masses.

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I think actually the most recent one is by 2.5. Right.

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Very dependent on the unknown equation of state and QCT. Right.

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Okay. So we know that there's this maximum mass that a neutron star can have.

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So that's if we find a dark object. It has a mass bigger than this.

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It's probably not a neutron star. Must be something else. It must be a black hole.

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Right. So in order to find these objects to measure their masses, the best place to go is to look at X-ray binary stars.

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Here's an artist's impression of one here. Imagine you've got a circular binary here.

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You've got a dark object of mass and blob on a star of mass and star there separated it by a radius, are there in a circular orbit.

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We don't see the compact object correctly. We only see the star.

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Right. And then it's a nice exercise in first year mechanics or maybe even high school mechanics to write down the equations of motion for this.

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So you immediately do the period of the system in terms of this radius and the two masses,

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because we we see only this star, we have to count for the reflex motion of that one.

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And then you can do that very easily.

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Then just the velocity we observe is given by this, where the circular velocity, the system is given by this normal central circular motion.

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The thing from O-level or GCSE physics probably. Right.

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And then so but the thing, the reason I am showing this, if you take this, you cube it and you multiply it by that, the radius cancels out, right?

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Really, our business are very hard to measure in astrophysics. And so you have this quantity here.

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The period times the velocity cubed divided by two pages is something that you can measure directly.

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And when you plug in what that corresponds to in the model,

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it corresponds to something which is going to give you a lower bound on the mass of the compact object.

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But it's very hard to argue with that. It's just Newtonian physics. Okay, so then you go off and you measure some of these systems.

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Here's an example. V for four cygni the V means as variable as a new V does all kinds of exciting things.

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But for the purposes of this talk, they're not interesting. Right.

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And here is the heliocentric radial velocity as a function of time in the system of the companion star.

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The companion star is very boring. It's just a bit less than the mass of the sun.

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It's totally boring. It's distorted in ancient ways because of the companion.

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But so you measure of lost to here a peak velocity of 220 kilometres a second.

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The period is just over six days.

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You plug this into this minimum mass expression, this F of M and you get the compact object must be at least seven times the mass of the sun.

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So a company, a neutron star, presumably a black hole. Right.

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Then you could do a bit more detailed modelling of you models, light curve.

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You can model the properties of the companion star and that gives you more detail.

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It gives you better constraints on inclination and on the mass of the companion, and therefore better constraints on the mass of the central object.

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This is what you get then for a sample of many such objects. This shows the mass is divided into two categories here.

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This is roughly 3.1.

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This is the theoretical limit as of a few years ago for neutron stars.

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And you see, you get a population here of things that are creating neutron stars.

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Interestingly, their masses look very, very similar despite this upper bound.

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That's probably tell us something about the equation of state of this book.

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But you also see a separate population. Of things that have masses that are much larger than that.

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Right. And so these are mass black holes. Right.

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I think this is hard to argue with unless you want to believe in something even more exotic and black holes.

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Right. Okay. So that's stellar remnants.

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What about supermassive black holes? Right. In order to go through those.

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Let me be long, long suspected to be a descendant of galaxies.

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But the observational evidence for these really began in the 1960s when quasars or quasi stellar objects were discovered.

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So just to remind you,

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they were known for a long time because these point like sources and they had seemingly a weird distribution of spectral lines until 1963,

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I think Martin Schmidt pointed out. Yeah, these were just the normal spectral lines we knew and love from laboratory astrophysics.

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It's just been red shifted so much that you don't recognise them unless you're looking for them.

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They're not what you expect them to be. Right. And so these sort of high redshifts, they're very distant.

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Right. And whenever you work with the parapets of these, they outshine galaxies.

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So 10 to 12 times, the luminosity of the sun is a characteristic one, and some of them are variable on timescales of about a day.

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Right. So you translate that into a distance.