Your test is this coming Wednesday. Be here
on time, bring a pencil and know your student
number. The format is like the sample test,
50 multiple-choice questions on material that
we covered here.
I won't be asking you anything that we haven't
discussed in the class so far, including today's
lecture. We talked last time about neutron
stars. They are formed when the collapsed
core doesn't have too much mass.
The mass is less than about three solar masses,
then the collapsed core will form a neutron
star, which is supported by the pressure created
by the genesee neutrons. The density of neutron
stars is enormous -- it's about 10-11 kg per
1^3 cm -- it's equal to the density of nucleus.
There's a tremendous gravitational forces
trying to compress the star and what is keeping
it alive is, so to speak, is the genesee pressure
of the neutrons.
But if the mass is too big so that the Chandrashekhar
limits for the general object composed of
neutrons is exceeded, the whole thing will
collapse.
The gravity eventually wins. Throughout the
life of the star, the crushing force of gravity
was always fought by either the pressure created
via fusion reactions in the core or the genesee
pressure in the case of white works in neutron
in stars.
But if the mass if the mass of the end object
is too big, the gravity will win in the end.
The mass of the collapsed core is greater
than about three solar masses. The pressure
exerted by the genesee neutrons cannot support
this collapse core and eventually it collapses
into single points singularity.
The resulting object is called "Black Hole."
To remind you of the concept that we talked
about when we discussed Newtonian theory of
gravity, the concept of escape velocity, how
did we know that it would be the escape from
a body of mass and radius r?
You see, any mass will exert the force the
gravitational attraction on the bodies around
it. So, if I take this car key and toss it
up with certain speed, it would climb up,
it would slow down as it goes up because it's
being pulled down by gravity.
It would reach a certain height that depends
on the initial speed and then it will fall
down. If I toss it up with a greater initial
speed it would climb a higher distance but
eventually it will stop and come down.
The escape velocity, by definition, is the
velocity with which I have to project this
so it never returns. It's speed stops when
it is at infinite distance from us.
That's what escape velocity means. For instance,
escape velocity from the Earth is about 11
kilometers per second. If I want to launch
something so it never comes back I have to
give it initial speed of at least 11 kilometers
per second. It will take off and never come
back.
If I have this object of mass and radius r,
it turns out that the escape speed squared
is given as a product between gravitational
constant, the mass of the body, over its radius.
This particular result is obtained with Newtonian
theory of gravity.
Any first year physics student can get this
result from the conservation of energy, by
saying that the kinetic energy of a moving
body that is associated with its speed but
its gravitational potential energy is constant,
it cannot change.
It turns out that if you apply general theory
of relativity, which is the correct theory
of the gravity, that the result is essentially
the same.
There is a different numerical prefactor here,
but what we are interested in is how this
escape velocity depends on the mass of the
object and its radius. If I have a given mass,
and somehow I compress it into smaller and
smaller volume, I keep on decreasing the radius.
The escape velocity will start increasing
and could become greater than the speed of
light. Then nothing, including light, can
escape the gravitational confine of this body.
Then nothing, including light, can escape
the gravity of the body.
If there was a source of light at a distance
from this body which is in the region where
the escape velocity's bigger than the speed
of light, then that light could not leave.
We could not receive that light. Hence, by
definition, it would be black, right?
A black body is something that does not produce
photons of any wavelengths. That's the reason
why it's called the black hole. There is a
distance called Schwarzschild radius. It's
the distance from a body where the escape
velocity is greater than the speed of light.
OK. If this is the singularity, the point
into which all of the mass of the collapsed
core has fallen, so to speak, then there is
a sphere around it with a radius equal to
this Schwarzschild radius, so that no information,
nothing, can leave this volume of radius RS.
The boundary around it is called event horizon.
We can't see any events, nothing, beyond the
event horizon. Nothing can leave this volume
of radius RS. It turns out one can show that,
the value of the Schwarzschild radius is related
to the mass, M, as T times the mass.
Where the units are such that then RS is expressed
in kilometers and the mass in solar masses.
If I have a black hole or an object of the
mass, say, 10 solar masses, then the corresponding
Schwarzschild radius would be 30 kilometers.
The bigger the mass, the larger is the radius
of the event horizon. Now, what observational
evidence do we have that such objects, black
holes, exist?
The possibility that these objects exist was
from the very beginning present in Einstein's
general theory of relativity that he put forward
in about 1916, but the question is, are these,
if you will, figments of imagination, something
that is theoretically possible but does not
exist?
The answer is no, these things do exist in
the universe. Let me explain how we can detect
them, in spite of them not emitting any light,
being black, right?
What you have is a binary system, consisting
of a normal star and a black hole, and the
two are bound together by the force of mutual
gravitational attraction. As those of you
who took Astronomy 1P01 will remember, when
we have two bodies gravitationally bound to
each other.
They orbit around a point between them, so-called
center of mass that is closer to the more
massive object. Both of them are revolving
around this common center of mass, so then
you see a star with invisible companion.
And you know that it has the companion because
you can see that the star is moving around
this fixed point, the center of mass. Moreover,
what happens is that the stellar wind particles
get accelerated by the strong gravity of the
black hole.
And as they are being accelerated to great
speeds, in the process they emit electromagnetic
radiation, in particular X-rays. The in-falling,
the material from the star that is falling
into a black hole, will produce X-rays.
Then, in order to detect them, one looks for
a binary system of a normal star plus its
invisible companion, which has a mass of at
least three solar masses. You can work out
the mass by observing the orbit of the normal
star and applying.
Essentially, Newton's laws of motion and gravitation,
in particular as formulated by the third Keplar's
law, as formulated by Newton, to work out
that that companion that you can't see has
a mass that is at least three solar masses.
It's also a very strong source 
of X-rays.
You look for binary systems where one mate
is not visible, but you know it must be there
from the way the visible companion is moving.
It has to have a mass of at least solar masses.
Moreover, it has to be a fairly strong source
of X-rays, produced as I've outlined here.
The first observation was done in Canada,
by an astronomer named Tom Bolton, in 1971,
using a 1.9-meter reflector telescope in now-defunct
David Dunlap Observatory in Richmond Hill.
Years ago, we used to take students who were
taking astronomy -- of course, the numbers
were less than they are now. We used to take
them for a field trip to that observatory.
We would get a couple of buses and go for
a day, but it's no longer possible because
the observatory has been closed.
Now, what he saw, the normal star was star
Cygnus, and the corresponding black hole was
named Cygnus X-1 -- as a source of X-rays,
X, Cygnus X-1. The mass of the star was found
to be 27 solar masses, and the mass of the
black hole 15 solar masses. That's a nice,
big value.
See, if you get something that is close to
the threshold of three solar masses, you're
not quite sure. Maybe you have a binary system
of a star and a neutron star, OK, because
neutron stars are very dense. They have a
strong gravity.
They can also pull in the solar wind particles,
which could then produce the X-rays. If you
get the mass of invisible companion near the
threshold, you don't know.
But when you get 15, which is way above the
Chandrasekhar limit for objects supported
by degenerate neutrons, they're fairly confident
that that must be a black hole. You cannot
have a neutron star with a mass of 15 solar
masses. It's got to be a black hole.
Now, as I said, in order to describe black
holes, we need to use Einstein's general theory
of relativity. It is fairly complex. Advanced
knowledge of math is required, and of course,
we cannot discuss it in any significant detail,
but I'll try to convey to you the gist of
it, what the theory is saying.
To describe vicinity of black holes, but also
very dense neutron stars, also very dense
white dwarfs, and stars, like our sun, one
needs to use Einstein's general theory of
relativity.
He has two theories of relativity, one that
he put forward in 1905, so-called special
theory of relativity, that has to do with
objects moving at a significant fraction of
the speed of light.
Turns out that our classical notion based
on everyday experience of space and time as
something totally separate are not quite correct.
If there was a spaceship moving at, say, 10
percent of the speed of light, and we could
observe a clock inside that spaceship.
And before the spaceship was launched the
clock was synchronized with the one that was
left behind on the Earth, it turns out that
we would see that clock moving together with
the spaceship at the speed of, say, 10 percent
of the speed of light, would run slower than
our clock.
It would fall behind. Also, the lengths shrink
in the direction of motion, so on and so forth.
That was revolutionary. It changed our understanding
of space and time, because before that, space
was considered as an empty stage on which
events take place, OK, but it's much more
subtle than that.
Now, in his general theory of relativity,
he extended his ideas from special theory
of relativity in order to describe the force
of gravity. One problem that he had with Newton's
theory of gravity was that in Newtonian theory.
The force of gravity is an instantaneous force
between two objects at a distance. He could
not stomach that, because one of the hallmarks
of the special theory of relativity is that
no interaction, no signal, can propagate faster
than the speed of light.
There is nothing like infinite speed of propagation.
So he was trying to reconcile the phenomenon
of gathering with these ideas, and that's
how working for, at least ten years, he was
able to formulate the general theory of relativity.
And basic idea is that mass or energy bends
space and other objects move in that bent
space along the paths of shortest distance,
so-called geodesics. So let me illustrate
this as shown here in this drawing. First
of all it is very hard for us, and it has
to do with the way our brain evolved.
You see we are not moving, and things around
us are not moving, at speeds close to the
speed of light. So we don't observe this mixing
of space and time, and therefore our brains
have evolved that we react very quickly to
the events happening in three-dimensional
space.
We have really hard time imagining spatial
dimensions greater than three. Because we
needed to process the information quickly
in three-dimensional space - if there's a
tiger jumping out of the bush to eat us we
had to react very quickly. Either fight or
flee; the basic instinct.
So we can mathematically write down the equation
for say, four dimensional sphere but we really
cannot imagine it. On the other hand, we can
understand things happening in dimensions
lower than three. For instance, imagine here
that the entire universe is in this two-dimensional
sheet.
I have three objects of different masses,
the smallest, the one medium size, and one
really massive one. And all three of them
bend the space and the larger the mass, the
bigger the bending. So if I had another object
that was nearby it would fall into this curved
space.
And that's how Einstein's general relativity
treats the gravitational attraction. This
big body bends the space-time around it and
then another body, that I drew here, moves
in that bent space-time along the path of
shortest distance which is not going to be
a straight line, but a curved path.
This is the gist of his general theory of
relativity. And it turns out that our Sun
bends the starlight because it's massive enough.
It has mass of ten times...ten-to-the-thirty
kilograms or about 300,000 times the mass
of the Earth and its mass will also curve
the space-time around it.
So does the Earth but the Earth's mass is
not big, the curving of space and time caused
by the Earth's mass is not that great, it's
hard to detect. However Sun is much more massive,
and in principle.
And that was one of the initial predictions
of the general relativity, was that it should
bend the starlight as the starlight from distant
stars passes near the Sun.
Those rays will move in the curved space-time
near the Sun, and as a result it would look
to us that the light is coming from a different
direction where they are. And let me illustrate
that.
So schematically this is illustrated on this
drawing here. So here is the Sun. Its mass
will bend the space around it. Here is the
Earth. This is the actual position of the
star.
And as this starlight passes near the Sun
it will have to move in the curved space and
eventually it would come to us in a direction
seemingly going this way. So this here is
the apparent position and this bending, this
angle here, Einstein predicted it had to be
1.75 seconds of arc.
And then indeed it was confirmed by Arthur
Eddington, same Eddington responsible for
mass luminosity relation for the main sequence
star, and same Eddington who didn't believe
Chandrasekhar, the young starring graduate
student about his proposal about the limiting
mass.
Of course this can be observed only during
the total solar eclipse when the light coming
from the Sun is blocked by the Moon. Otherwise
we are blinded by the sunlight and we cannot
see the stars that are behind the Sun.
So to see this, the Moon has to be between
the Earth and the Sun so we have here a total
solar eclipse. And Eddington went on two expeditions
to the places in the world at that time late
twenties to see this effect. And indeed he
was able to confirm.
You take the photograph of the night sky when
the Sun is not in the way, and then during
the eclipse you take a photograph of the same
part of the sky. And then you compare the
two photographs, two photographic plates and
you see that the given star has shifted.
It apparently moved from here, where you saw
it when the Sun was not in the way, to here.
That amount of shift.
The angle of the displacement of 1.75 seconds
of arc was exactly what general theory of
relativity predicted. So it's not just something
that exists on paper but these things could
be observed.
