So Andrew's given us an introduction to the theoretical properties of black holes.
I'm going to give you an introduction to what he calls the practical properties, or what I call just essentially proofs of existence.
So here is a picture of our nearest neighbour, nearest big neighbour at the Andromeda Galaxy.
You see, it's companion M32 and then this boring galaxy, NGC 205 and know the many, many stars around us.
And in this picture, there are roughly 10,002 black holes.
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.
There is a slightly smaller one at the centre of M32.
There's probably not one at the centre of this dwarf elliptical NGC two or five.
And then in the surrounding environment, in the stars, among the stars surrounding each of these galaxies, there are roughly ten to the four.
Ten to the five. Who cares? Stellar mass black holes. Right.
So in astrophysics, there are at least two different types of black hole.
And the type just refers to the origin. So stellar mass black holes.
They have masses of around about ten times the mass of the sun.
Andre pointed out that the Schwarzschild radius of the sun is three kilometres, so therefore the horizon scale in these is about 30 kilometres.
Right. These occur every you have got stars,
particularly where you get massive stars because these are massive stars that have
gone supernova and then believe behind this stellar mass black hole remnant.
In contrast, the bombastic arena and supermassive black holes there are much bigger.
They're billions to billions times the mass of the sun.
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.
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.
Okay. Another thing that these have in common, apart from their fundamental physical properties,
is that whenever you try to study them with the best telescopes,
you find that the angular size of the black hole's horizon is roughly the same in the nearest cases in both situations.
So it's roughly ten to the minus 5 seconds of arc, right?
Or one arcseconds is basically. There's 3600 arcseconds in a degree.
Right. So to give you a fitting for that number, the best optical and infrared telescopes, they can resolve point 1 seconds of arc.
So factor ten to the four bigger than that.
So for the point of view of for most of the time in astrophysics these black holes, they just looked like Newtonian points masses.
Right. We can hope to measure the mass, but we can't hope to measure the speed, at least not directly.
Right. So the name of the game in astrophysics is to find a compact object,
then argue that this compact mass has to be a black hole just through lack of imagination.
Right. So here's an outline of my talk. Very simple.
I'll tell you about why we believe stellar mass black holes exist.
Then I'll tell you about why supermassive black holes probably almost certainly exist.
And I'll give you some consequences of these supermassive black holes. Okay.
So stellar mass black holes remind you we know where they come from. They're the remnants of dead stars.
Dead. Massive stars. Okay. So when you think about stars.
Well, not too. A question to ask is what stops a star from collapsing into a black hole?
And the answer is pressure. Support. Right.
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.
Then if special stars like white dwarfs a neutron stars.
And in those cases, the pressure that prevents the stars from collapsing.
And the first kiss is electron degeneracy pressure. And the second kiss is neutron degeneracy pressure.
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.
Right. And that's most naturally done, most easily done in terms of energy instead instead of in terms of mass.
So just run through that. So just to remind you of the Polish exclusion principle that says that fermions are antisocial.
They refuse to share states with one another. Identical fermions are antisocial and refused to share the same physical state.
So this means that if you take a gas of these objects and you reduce the temperature toward zero,
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.
There would be a range of minimum momentum.
States that have to be populated, right a corresponds to a certain character, certain energy called the Fermi Energy.
Right. So even when you cool these things down to zero temperature, they still have some internal energy.
And it's a nice exercise in second year undergraduate physics to show that if you have any of these identical
fermions per unit volume on their moving ultra relativistic li so other you squeeze their density enough
for the was run very very quickly you can show that this Fermi energy skills as the cube root of the number
density right then just with the standard factors to make it have the right dimensions here it's part time.
See? Okay, so the Fermi Energy's important.
Then here is Lando's argument for why there's a maximum mass and such to generate objects.
He did this in 1932. It was actually before the discovery of the neutron.
So just to show what these Russians can do things right.
So here is an example of a simple idealised model of a neutron star.
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
was the very reasonable one is that you assume that the equilibrium state of this system is the state of minimum total energy.
Okay. Kind of hard to argue with that.
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,
that's just going to be the kinetic energy, which is end times, the Fermi energy up to factors of order one.
Plus the gravitational potential energy, which again of two factors of 4 to 1.
It's just going to be GMM squared divided by the radius. Okay.
You plug in the expressions for this.
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.
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,
then if you happened to squeeze your star, if you reduce this radius, then the energy will go down.
Right. So then come back to this principle that says that this system is unstable.
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.
Right. It's given by this. There's a numerical factor in front of this.
But when you work out what this thing here is, it's about 1.8 times the mass of the sun.
Right. Of course, this calculation was very, very simplified.
And you could do more complete models. That includes the equation of state of the neutrons.
You know, here assume that they're not interacting.
So you're going to create the effect of general relativity, which also becomes important in working out the gravitational potential energy.
And this gives you numbers that are run about 3.1 solar masses.
I think actually the most recent one is by 2.5. Right.
Very dependent on the unknown equation of state and QCT. Right.
Okay. So we know that there's this maximum mass that a neutron star can have.
So that's if we find a dark object. It has a mass bigger than this.
It's probably not a neutron star. Must be something else. It must be a black hole.
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.
Here's an artist's impression of one here. Imagine you've got a circular binary here.
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.
We don't see the compact object correctly. We only see the star.
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.
So you immediately do the period of the system in terms of this radius and the two masses,
because we we see only this star, we have to count for the reflex motion of that one.
And then you can do that very easily.
Then just the velocity we observe is given by this, where the circular velocity, the system is given by this normal central circular motion.
The thing from O-level or GCSE physics probably. Right.
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?
Really, our business are very hard to measure in astrophysics. And so you have this quantity here.
The period times the velocity cubed divided by two pages is something that you can measure directly.
And when you plug in what that corresponds to in the model,
it corresponds to something which is going to give you a lower bound on the mass of the compact object.
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.
Here's an example. V for four cygni the V means as variable as a new V does all kinds of exciting things.
But for the purposes of this talk, they're not interesting. Right.
And here is the heliocentric radial velocity as a function of time in the system of the companion star.
The companion star is very boring. It's just a bit less than the mass of the sun.
It's totally boring. It's distorted in ancient ways because of the companion.
But so you measure of lost to here a peak velocity of 220 kilometres a second.
The period is just over six days.
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.
So a company, a neutron star, presumably a black hole. Right.
Then you could do a bit more detailed modelling of you models, light curve.
You can model the properties of the companion star and that gives you more detail.
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.
This is what you get then for a sample of many such objects. This shows the mass is divided into two categories here.
This is roughly 3.1.
This is the theoretical limit as of a few years ago for neutron stars.
And you see, you get a population here of things that are creating neutron stars.
Interestingly, their masses look very, very similar despite this upper bound.
That's probably tell us something about the equation of state of this book.
But you also see a separate population. Of things that have masses that are much larger than that.
Right. And so these are mass black holes. Right.
I think this is hard to argue with unless you want to believe in something even more exotic and black holes.
Right. Okay. So that's stellar remnants.
What about supermassive black holes? Right. In order to go through those.
Let me be long, long suspected to be a descendant of galaxies.
But the observational evidence for these really began in the 1960s when quasars or quasi stellar objects were discovered.
So just to remind you,
they were known for a long time because these point like sources and they had seemingly a weird distribution of spectral lines until 1963,
I think Martin Schmidt pointed out. Yeah, these were just the normal spectral lines we knew and love from laboratory astrophysics.
It's just been red shifted so much that you don't recognise them unless you're looking for them.
They're not what you expect them to be. Right. And so these sort of high redshifts, they're very distant.
Right. And whenever you work with the parapets of these, they outshine galaxies.
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.
Right. So you translate that into a distance.