So let's now find out how do black holes generate these amazing luminosities?
After all, black holes are black because nothing can escape them, not even
radiation. The answer is, the material around them
can still be shining and be observed from far away.
Now, we do know that black holes, or something like them, does exist in
galactic nuclei. We know that because of the kinematical
tracers, either stars or gas. We know that there is some depth to the
potential well. But then, right in the middle, there is an
extra mass that requires gas or stars to move faster.
And so, if it is not the black hole, then it's something really similar to it, a lot
of mass packed into a very small radius. So for all practical purposes it is like
just like black hole, regardless of the details of its physics, very compact, very
large masses near the centers of galaxies. We've seen evidence for it from variety of
reasons, like the motions of water masers in the Seyfert galaxy.
Now, there are about 20 of these that have been studied by now.
And are like point masses, that are like solar system, except they're round giant
blank hole. And the kinematics indeed indicates that
this black hole is some tens of millions of solar masses.
In the Milky Way itself, the motions of star in the intergalactic center indicates
there is an extra point mass there of the order of 3 or 4 million solar masses.
And even though it doesn't do much in terms of activity, we know it's there
because of its gravitational influence. So something like black holes, or indeed
black holes, exist in centers of the galaxies, but how do we get energy from
them? The key to those is dissipation of binding
energy. The fuel, gas comes from kiloparsec type
radiis because of galaxy mergers. The size of the black hole is given by so
called Schwarzschild radius. The formula is given here.
For solar mass, that's about three kilometers.
For a typical large black hole that might power a quasar, which has 100 million
solar masses. It would be 300 million kilometers.
Or about twice the distance of Earth from the sun.
So these really are regions about the size of a solar system.
That happens to be also about 10 microparsecs.
And therefore, the material crosses approximately eight or nine orders of
magnitude in radius to get to the center energy.
Well, that means it will be coming in on roughly radial orbits.
But however, it does get there. That net amount of energy lost will be a
substantial fraction of it's rest mass energy.
If it could get directly to Schwarzschild radius, then it could radiate away all of
it's m c squared. But it doesn't.
There is a smallest orbit which is stable. Remember that you can dissipate energy,
but not the angular momentum. So the material that falls in has to
settle into a small accretion disk. The inner edge of that is the smallest
stable orbit. Now this is very much a general relativity
concept. There is no such thing in Newtonian
gravity. A Newtonian gravity of two point masses,
you can get as close as you want. In general relativity, no, there is a
smallest stable orbit. And so that is how deep the material has
to get, and so therefore it will dissipate less than its full mc squared but a
fraction thereof. Typically, we think that on average, it's
about 10%. So it doesn't matter what you make black
hole from, the regular matter, dark matter.
It's just a mass that you know. But there are also a couple other things.
In principle, you can also have electric or charge or magnetic field going through
a black hole because of electromagnetic radiation.
So electromagnetic interaction and gravity are the only two infinite range
interactions. And black hole can be spinning, so three
numbers really characterize the black hole.
It's mass, it's angular momentum. And its electric charge.
Now, charges are well mixed in the universe, protons and electrons, so black
holes are not likely to have a significant net electric charge.
And that leaves mass and the angular momentum.
For a Schwarzchild black hole, classical black hole so to speak, the angular
momentum is zero. And its radius is given by Schwarzschild
formula. The smallest stable orbit around
stationary black hole is three times the Schwarzschild radius.
And this is how close material has to come in order to be eventually absorbed.
But how do we know there really is black holes?
Well it turns out there is another way to probe this, and this is from the X-ray
emission. The material that's so close to black hole
is very hot. It's plasma and millions of degrees of
kelvin. And therefore, x-rays are a good way to
probe it. There are emission lines in x-rays, which
we can use to probe kinematics of the gas just as we did with non-relativistic cases
earlier. And if you have a Newtonian case, there is
just a big mass and the stuff is going round even at relativistic speeds, if you
will. Then there'll be a two horn profile, one
for gas coming towards you, the other one going away from you.
But once we throw in relativity new things happen.
First there is so called a transverse doppler effect.
Moving clocks run slower. And then the, also the beaming, boosts of
radiation is coming, when material's coming towards you, and the emission is
the one going away from you. Those two would create asymetric profile,
one form would be larger than the other. Then there is also gravitational redshift,
that's the general altruistic effect. The photons would lose energy getting out
of potential role. So the whole thing would be moved toward
lower energies. And therefore you expect broad, asymmetric
shifted line to come up. And that's exactly what's observed.
Here is a profile of the x-ray emission line of iron seen in one of the active
nuclei. You can see it's very asymmetric and also
hapeens to be redshifted compared systemic velocity.
And so this is exactly the prediction of what.
Things should happen in case of point mass or something close enough to black hole
and the material orbiting around it. So that's probably the most direct
evidence that yes, really close in what we see are black holes or compact masses
compressed to such small sizes that for all practical purposes, they're black
holes. All right, now what in black hole is
spinning? We introduced the spin parameter, given by
this formula, little a. Which is ratio of angular momentum to mass
squared, a couple constants thrown in. This was studied by physicist named Kerr,
and those are called Kerr black holes, as opposed to Schwarzschild black holes which
are stationary. And maximum amount of angular momentum
that you can put in a black hole, spinning black hole, corresponds to the value of a
parameter of 1. For no rotation, it is zero.
It now turns out that the smallest radius of a smallest stable orbit around the
rotating black holes is closer in than it would be for stationery Schwarzschild
black hole. Therefore, you can extract more energy
from the spinning black hole and. And thats just from stuff falling in.
Now if you can somehow extract rotational kinetic energy from the black hole, then
you can get some more. There is a mechanism to do this.
And that is if there is a magnetic field threaded through black hole, and so the
mechanism called Blandford-Znajek effect that does this.
Alright, so this is how you can extract energy but how much.
And we have to now turn to a concept of Eddington limit which was really
introduced in stellar astrophysics and that's what limits the.
Largest mass stars and most luminous stars and it works as follows.
There is ionized gas, plasma, near the surface of a star.
In this case, in the region around the black hole and electrons can be
susceptible to radiaion pressure, absorb photons and so there is.
Essentially radiation driven wind. On the other hand, the gravitational field
pulls them back. So, for a given mass, there is certain
critical luminosity above which the stuff will be blown away by the radiation
pressure. And you can compute that using cross
section for absorption, so called Thompson cross-section for free electrons.
And it can put in now the, so they have outward force due to the photon wind
that's pushing the material that's directly proportional to luminosity.
And that has to be balanced exactlty with the gravitational force pulling back on
our protons and electrons together. It's a problem because that's where all
the mass is. They just latch to the electrons as
they're well mixed. So, the equation of these gives you the
limiting luminosity If luminosity is larger then that, radiaion pressure wins
and thing blows itself apart. If it's less then that, it can still be
sort of, of equilbrium. So the largest luminosity you can have
around any given mass is given by this formula, the Eddington luminosity.
And that turns out to be of the order of 10 to the 38 ergs.
Times the mass of the object in solar mass units.
So for something like 10 to the 8th solar mass black hole, Eddington luminosity
would be of the order of 10 to the 46 ergs.
Remember that solar luminosity is few times 10 to the 33 ergs.
So that will be on the order of 10 to the 13 solar luminosities and that's about as
much as the most luminous quasars that we do see.
Next, we will consider qualimated emissions from black holes in the form of
radio jets and other phenomena associated with them.