The speed of sound is 340 meters per second--
it depends a little bit on the temperature--
about 770 miles per hour.
When I speak to you, my sound reaches you
with that speed.
I produce a certain frequency here,
a certain number of oscillations per second.
They reach you,
your eardrum starts to oscillate with the
same frequency
and you hear that tone.
I have here a tuning fork
which oscillates 440 times per second.
(tuning fork produces medium-pitched tone)
Your eardrum oscillates 440 times per second--
you hear this tone.
Here I have 256 oscillations per second.
(metal rod emits lower tone)
Your eardrum is now shaking,
going back and forth 256 times per second.
If you stay where you are and you don't move
and I move these tuning forks,
you will hear a different frequency
and that's what we call Doppler effect.
If my sound source approaches you,
you will hear a frequency f prime which is
larger
than the frequency of the tuning fork.
If it moves away from you, which I will call
receding,
then f prime equals lower... lower frequency.
For instance, I move to you a sound source--
I call that a transmitter--
with a speed of about one meters per second.
Transmitter is the sound transmitter.
Then if it approaches you here, you will hear
f prime,
which is 1.003 times f.
This three here is the one part out of 340
that you get an increase in frequency.
If I move it away from you, then f prime would
be
0.997 times the frequency of the source itself.
You stay where you are.
I have here a tuning fork which generates
4,000 hertz,
a very high frequency.
If I move it to you with the speed of one
meter per second,
which I can do, then you get an increase in
pitch of 0.3%.
That makes it 4,012 hertz.
And when I move it away from you there is
a decrease of 0.3%.
And you can clearly hear that difference.
I will first make you listen to the 4,000
hertz
without my moving.
(tuning fork produces very high pitched tone)
Can you hear it? Very high frequency.
Is it painful, really?
High frequency.
Most of you are young enough
you should be able to hear 4,000 hertz.
Okay, now I am going to move it to you one
meter per second
and away from you.
(high tone goes up and down slightly in pitch)
Did you hear it? Once more.
(tone goes up and down quickly again)
(class laughs)
When it comes to you, it's clear that the
frequency goes up,
and when it moves away from you, the frequency
is down.
Now imagine that I'm going to rotate the sound
source
around in a circle.
Now the sound that you receive, the frequency
that you receive
will change in a sinusoidal fashion.
If this is that circle, and this is the radius
of that circle,
and if you are here in the plane of the circle,
then when the source comes straight to you
with the velocity v-- let's say it's a uniform
circular motion--
f prime will be larger than f
and it will, in this case, reach a maximum.
When it is at 90 degrees relative to you--
I don't have to give it a vector notation--
f prime equals f.
When it moves away from you, f prime is smaller
than f,
you hear a minimum.
And when it is here again--
when the angle between the velocity and your
direction
is again 90 degrees-- then f prime equals
f again.
And so this phenomenon is called the Doppler
effect.
So if I twirl it around,
you will hear a sinusoidal fluctuation in
f prime.
Suppose I plot, as a function of time, f prime
the way you will receive it-- you sit still,
but I'm going to move the sound source around
like this.
Then you will have a curve that looks something
like this:
some sinusoidal-cosinusoidal fluctuation of
f prime.
This will be the value f
produced by the sound source itself.
This will be f prime maximum and this will
be f prime minimum.
If you could record this, there is an amazing
number of things
that you can deduce from this curve.
First of all, you can take...
You can measure f prime max divided by F,
because you see this curve, so you know what
f is here,
you see what f prime max is,
and that should allow you to retrieve immediately
v velocity of the transmitter.
If that number were 1.003, then you know
that the speed in the orbit was one meter
per second.
So this ratio immediately gives you the transmitter
velocity.
This time separation gives you immediately
the period of rotation,
but since two pi R-- if R is the radius
divided by the velocity of the transmitter--
since that is the... oh, I can reverse it,
it doesn't matter.
Two pi R divided by the time to go around
is the velocity of the transmitter.
Since you know the velocity of the transmitter
from this ratio
since you know the period, which is this,
you now also find the radius R.
So from that curve-- and keep that with you,
because it's going to be important in what
follows--
we can derive three things:
the radius, the period of rotation
and the speed of the object as I twirl it
around.
I have here what we call a wind organ.
When I twirl this around, it produces a particular
tone.
We will talk later about 801
why it produces a particular tone.
Sometimes you hear two tones.
I'll try to make you hear only one.
And as I swirl it around, the sound is coming...
the sound source, the transmitter is coming
to you.
This, when it goes like this, it's 90-degree
angle
so you should not hear any Doppler shift.
When it is here, it's moved away from you
and so you hear a sinusoidal change in f prime.
Try to hear that.
(wind organ producing tone that changes pitch)
Can you hear, when it's coming to you, that
it's higher-pitched
than when it's going away from you?
Can you hear that?
Just say no if you don't hear it.
Not very clear.
(wind organ again producing varying tone)
For me, it's impossible to hear
because I'm standing right under it, of course.
Well, I tried.
I now want to change to electromagnetic waves.
Electromagnetic waves travel with the speed
of light,
which is 300,000 kilometers per second.
And if you want to treat that correctly,
you would have to use special relativity.
In the case of sound, I stressed repeatedly
that you in the audience should not move
but that the sound source is moving.
In the case of electromagnetic radiation
when you deal with the speed of light,
you don't have to ask that question.
It is a meaningless question in special relativity.
To ask whether you are moving relative to
me
or whether I am moving relative to you, it
doesn't matter.
All that matters in special relativity
is the relative motion,
so you can always think of yourself as standing
still
and make the source of electromagnetic radiation
move to you, or away from you, relative to
you.
Electromagnetic radiation is optical light,
infrared,
ultraviolet, radio, x-rays, gamma rays.
All of that is electromagnetic radiation.
If the velocity of the source of electromagnetic
radiation--
the transmitter--
if that is way, way smaller than the speed
of light,
then it is very easy to predict
the change in frequency due to Doppler shift.
Let this be the transmitter which produces
frequency f,
and here is the receiver which receives the
frequency f prime.
And let the velocity
of the source of electromagnetic radiation
be v--
I could put transmitter here, but we can drop
that index--
and let this angle be theta.
Then this is the component in your direction--
we call that the radial component--
which is v cosine theta.
So I delete the tr.
This is just the velocity of the source
relative to you at that angle.
If now we want to know what f prime is,
then f prime equals f times one plus v over
c
times the cosine of theta.
What matters is only the radial component
of the velocity.
This is the radial component.
If theta is 90 degrees, just like we had with
sound,
then f prime equals f.
So 90 degrees, the cosine of theta is zero,
f prime equals f.
If theta is smaller than 90 degrees,
then it's coming towards you,
then f prime equals larger than f.
If theta equals larger than 90 degrees,
it's going away from you,
then f prime equals smaller than f.
You would get a similar equation for sound
by replacing this c by the speed of sound.
But I want to stress
that this only holds for electromagnetic radiation
if v over c is much, much smaller than one.
Now, when we deal with sound,
there is something mechanically oscillating.
Something is vibrating.
With electromagnetic radiation, charges are
vibrating.
Electrons are vibrating,
and they are vibrating with a certain frequency,
and that means
there is a certain period of one oscillation.
And that period of one oscillation is, of
course,
one over the frequency.
I can ask myself now the question,
how far does electromagnetic radiation,
how far does light travel in the time of one
period capital T?
Well, it goes with the speed of light,
so in T seconds, it moves a distance cT.
And that distance we call
the wavelength of electromagnetic radiation,
lambda equals cT, for which you can also write
c divided by F.
So this is the wavelength of the electromagnetic
radiation--
the speed of light, 300,000 kilometers per
second--
the period of one oscillation, say, of the
electrons,
and this is the frequency, which you can give
in hertz.
I could give you a specific example.
I, for instance, can take a period T
of two times ten to the minus 15 seconds.
That would give me a wavelength
of about six times ten to the minus seven
meters--
six times ten to the minus seven meters--
and that you would experience as red light.
If I make the period shorter--
say, 1.3 times ten to the minus 15 seconds--
I get a shorter wavelength.
I get four times ten to the minus seven meters,
and you would experience that as blue light.
In astronomy, in optical astronomy
we cannot measure the period or the frequency
of optical light.
All we can measure is the wavelength.
And so if I want to use this equation,
then I have to replace f prime by c divided
by lambda prime
and f I have to replace by c divided by lambda.
And when I do that, I get the following result.
I get lambda prime equals lambda
times one minus v over c cosine theta.
If this is a plus, this is a minus.
Check that for yourself.
You have to use the Taylor expansion,
namely that v over c is much, much smaller
than one.
So you can see now if the object comes to
you,
in other words, if f prime is larger than
f,
if the frequency is higher, then the wavelength
will be smaller.
And so let me write that down.
When the cosine of theta... so the object
is coming to you--
when the cosine of theta is larger than zero,
the object is approaching you,
then the wavelength lambda prime will be less
than lambda.
And that has a name-- we call that blue shift.
And the reason why we call that blue shift
is
that if the wavelengths become shorter,
it moves towards the blue end of the spectrum,
because blue has a lower wavelength than red.
If cosine theta is negative,
then the object is receding from you,
then lambda prime is larger than lambda,
and we call that red shift.
These are the terms that astronomers use all
the time.
When you make a spectrum of a star--
you can do that using prisms or by other means--
and you look at the light intensity
as a function of wavelength--
so here is the light intensity as a function
of wavelength--
then you may expect to see some kind of a
continuum.
But, in fact, what you do see is...
Superimposed on a continuum
you see sometimes very sharp absorption lines--
black, missing light, called absorption lines
And these absorption lines correspond
to elements in the atmosphere of the star.
In fact, if you see the absorption lines,
you can tell
what kind of elements are present in the star.
Some are very characteristic absorption lines
for hydrogen, some for calcium, some for silicon,
some for magnesium and so on.
It's actually interesting
that when you look at the spectrum of the
sun--
when people did that first,
when they had the means of doing that,
they found absorption lines in the spectrum
of the sun
which could not be identified.
They had never been seen here on Earth, these
lines,
and so they called these lines after the sun.
The sun is Helios, and so they called it helium.
So helium was first discovered on the sun
before it was later discovered on Earth
by looking at the absorption lines of the
solar spectrum.
If a star moves to you, then all the lines--
every single line-- will be blue-shifted.
And if the star moves away from you,
all the lines will be red-shifted.
If you take an example:
With lambda prime divided by lambda
and you pick any one of those lines--
it doesn't matter which you pick
because they will all do exactly the same.
If this were, for instance, 1.00333--
I just pick a very nice number;
that means lambda prime, as you see, is larger
than lambda;
the wavelengths get longer, so we have a red
shift--
and you substitute that in that equation,
then you'll find that the velocity
at which that star is moving relative to you--
that gives you immediately the answer there--
equals minus 0.00333 times the speed of light,
c,
and that is minus 100 kilometers per second.
And the minus, then, reminds you that
the object is receding from you.
So that gives you a red shift.
I just wrote down
that the velocity v is minus 100 kilometers
per second.
It's, of course, v cosine theta
that is minus 100 kilometers per second.
It's the radial velocity-- that's all you
can measure.
You have no information on theta.
So it is this component, v cosine theta--
which we call the radial velocity--
that is minus 100 kilometers per second.
Half of all the stars in the sky are binaries,
and so when you look at the spectra,
you will see them go around each other.
And so you, in principle, can measure the
red shifts
and the blue shifts as they go around each
other.
You see Doppler effect.
If they come to you, you see blue shift.
If they go away from you, you see red shift.
So, in principle,
you can determine for each one of those stars
the velocity in orbit, the radius of their
orbit
and, of course, the period of the binary system.
So it's an extremely powerful tool in astronomy
if you have a binary system, when the stars
exactly do this,
to determine
all these quantities that you would like to
know.
I first would like to show you now some slides.
The first slide... oh, I have to lower the
screen, by the way.
That would help, wouldn't it?
(chuckles)
The first slide is a spectrum made in the
laboratory
of hydrogen, helium and calcium and sodium.
It shows you emission lines, no absorption
lines.
These lines are produced by lamps,
and the frequencies are very well known.
Here you see the famous sodium yellow lines.
So here is the red part of the spectrum
and there is the blue part of the spectrum.
So we know these frequencies,
we know these wavelengths very well.
And here you see the spectrum of the sun
with all these absorption lines that I mentioned
to you.
It's plastered with absorption lines,
and each of them can be identified.
These are due to calcium, iron, hydrogen and
so on.
Here is the blue part of the spectrum,
here is the green part, the green part,
and here is the red part of the spectrum.
And here you see the basic idea behind a binary
system.
Suppose you have a binary system
that only one star is visible and the other
one is invisible
and the one star shows you three clear absorption
lines.
Then as the star moves around the center of
mass,
you see that all these lines drift in unison.
And out of this information
you get the radius, the velocity and the period,
assuming that you are on Earth
in the plane of the orbit of the stars.
If you have a binary system whereby both stars
are visible
so you get the spectrum of both stars,
then you see the Doppler shift of both stars
in the spectrum.
Here we have a simple case
that we only have two absorption lines, not
to confuse the issue,
and so in one... in the case of one star,
the shift will be towards the left of the
two lines,
but the other star, the shift will be to the
right,
because if you have a binary system
when one star comes to you,
the other star goes away from you, and vice
versa.
So now you are very lucky, now you have an
ideal situation
that you can find for both stars the radius
of the orbit,
the velocity in orbit and the period for each
star,
which, of course, is the same for both.
And here you see real data.
You see here, first of all,
the emission lines which are measured in the
laboratory
that I just showed you.
They are always done simultaneously
with the measurements.
You always must be sure
that you have a good calibration of your wavelength.
And this spectrum a, the top spectrum, is
of a star,
a binary system, that has a period of 20.5
days.
And you see here single lines, if you have
good eyes.
That means at this very moment
both stars move relative to you at angles
of 90 degrees,
so you don't see any Doppler shift.
But now look here.
Later in time, you see that this line has
split in two lines
and this one has also split in two lines.
Clearly, one component is coming to you
and the other component is moving away from
you.
And so you get all this useful information
in astronomy
by making the Doppler shift measurements of
binary systems.
I want to pursue the idea of binary stars.
They give us
not only the information that we want regarding
the orbits,
but there is even more that we can get out
of it
which is even more exciting.
So I will remind you what a binary system
looks like.
Remember exam... the second exam.
I'm sure you will never forget that second
exam
and maybe never forgive me for that.
Binary system: star one, radius r1, mass m1,
velocity v1,
and star two-- going about their common center
of mass--
mass m2, radius r2 and velocity v2.
m1 r1 equals m2 r2.
That's the way the center of mass is defined.
Imagine that you as an observer
are somewhere in the plane of this orbit,
and you are here.
And you are observing the system going around.
Kepler's Third Law, which you derived on your
exam
as well as on an assignment: the period squared
equals
four pi squared times r1 plus r2 to the power
three.
divided by G times m1 plus m2.
Let me check that to make sure I have that
right.
Yes, that is correct.
Imagine now you can make
the Doppler shift measurements of both stars.
You make the Doppler shift measurement of
star number one,
so you measure lambda one prime as a function
of time.
Out of that pops immediately the period of
rotation.
Out of that pops the velocity, as we discussed.
Out of that pops the radius, r1.
And now you measure
the Doppler shift of star two as a function
of time.
Out of that pops the period
which, of course, better be the same.
Out of that pops its velocity in orbit
and out of it pops its radius.
All these things come out of the Doppler shift
measurements,
but if you know r1 and you know r2,
then you also know r1 plus r2,
so you know this part in Kepler's Third Law.
Since you also know the periods, you can find
what m1 plus m2 is.
So now you get an extra bonus.
You know now what the sum of the mass of the
two stars is
in the binary system, but you also know that
m1 r1 is m2 r2.
So now you have two equations.
You know what m1 plus m2 is
and you know this equation, and you can solve
for m1 and m2,
which is an amazing thing when you come to
think of it.
So we finally end up
with the mass of star one and the mass of
star two.
All this comes out of Doppler shift measurements:
the velocities, the radii, the periods
and even the masses of these objects.
Now, if you as an observer on Earth
are not exactly in the plane of the orbit,
then the situation is a little bit more complicated,
and I will not discuss that here today
because, in principle,
it doesn't affect the idea behind Doppler
shift.
But for astronomers, it is very important.
It's really a nuisance,
but I will not discuss that in any detail.
I want to discuss a fascinating application
that we have in x-ray astronomy.
Namely, we have x-ray binaries.
What is an x-ray binary?
Well, it is a binary system.
This is a star not unlike our sun.
It has as certain mass, has a certain radius,
and it is in orbit, let's say, with a neutron
star,
even though it could be a black hole.
But for now, let's just assume it is a neutron
star.
And if these two masses are the same,
which I only use for the sake of simplicity--
in practice, they could be very different--
then there is a point between these two, right
in the middle,
whereby the gravitational pull in one direction
is the same
as the gravitational pull in the other direction.
And we call that the inner Lagrangian point.
In other words, if you were there,
the neutron star would pull at you
with exactly the same force as the other star.
So you wouldn't know where to go.
If this inner Legrangian point lies
below the surface of this star, that means
if the stars are a little closer than I have
drawn them here,
then the matter of this star will fall
towards the neutron star,
because the pull in this direction is, then,
larger
than the pull in this direction.
Now, of course, this system is a binary system.
They go around in the plane of the blackboard,
say,
and so this matter cannot fall radially in
but it will fall in and spiral in
and forms what we call an accretion disk
around the neutron star.
This is called the accretor and this is called
the donor.
There is mass transfer from the donor to the
neutron star.
Oops, I just noticed I misspelled the word
"accretion."
There is an "r" in "accretion."
And as that occurs,
there is a tremendous amount of energy that
is released.
I want to blow up the neutron star.
Very simple 801 considerations, now.
What comes is extremely pedestrian.
This is the mass of the neutron star
and this is the radius of the neutron star.
And I take a little bit of matter m,
and I drop it from a large distance onto the
neutron star.
At what speed
will that little piece of matter reach the
neutron star?
You should almost be able to close your eyes
and give me that answer right now.
The kinetic energy when it reaches the neutron
star
equals one-half m v squared.
That is the speed
at which it will crash onto the neutron star,
and that must be
mM neutron star G divided by the radius of
the neutron star.
You always lose your m, and so you find
that the speed at which it reaches the neutron
star
is the square root of two M neutron star
times G divided by R neutron star.
You should remember this equation.
This was the equation that we had for escape
velocity.
If you were here, and you go back to infinity,
you reach exactly that speed, so if you fall
in from infinity
that is exactly the speed
at which you reach the neutron star.
It should obviously be the same number.
And you don't really have to be infinitely
far away;
you just have to be much further away
than the radius of the neutron star.
When this matter crashes onto the neutron
star,
the kinetic energy that is released is one-half
mv squared.
It is converted to heat, and to give you some
feeling
for the incredible power of a neutron star,
if you make this little m as little as ten
grams--
think of it as a pretty full-sized marshmallow--
and you throw a marshmallow
from a large distance onto a neutron star,
the energy that is released is comparable
to the atomic bomb that was used on Hiroshima.
A ten-gram object thrown onto a neutron star--
the reason being that this velocity
becomes enormously high.
If you put in for the neutron star
a mass of about three times ten to the 30th
kilograms,
and you take for the radius of the neutron
star
about ten kilometers,
you will find that that velocity becomes
about two times ten to the eighth meters per
second,
which is about 70% of the speed of light.
And because of this enormous speed--
one-half mv squared is horrendously high--
it is a conversion
of gravitational potential energy to kinetic
energy
and then ultimately to heat.
Now, nature is transferring mass at an extraordinarily
high rate
in many of these binary systems.
There are several hundreds in our own galaxy.
The mass transfer rate, which I call dm/dt--
so that is the transfer rate
from the donor onto the neutron star--
that transfer rate
is roughly ten to the 14th kilograms per second.
It is ahorrendous mass transfer rate.
You can calculate--
by multiplying it with one-half v squared--
how many joules per second are released
in the form of kinetic energy.
That means in the form of heat.
And I call that the power of that neutron
star,
and that, then, for this mass transfer rate,
that's about two times ten to the 30th joules
per second,
which is watts.
And that is about 5,000 times larger
than the power of our own sun.
But the temperature of this neutron star--
because of this enormous amount of energy
released,
the temperature would reach values
of about ten million degrees Kelvin,
and at that high temperature,
the neutron star would emit almost exclusively
x-rays.
You and I are very cold bodies, only 300 degrees
Kelvin.
We radiate electromagnetic radiation
in the infrared part of the spectrum.
We have warm bodies.
When you hold someone in your arms, you can
feel that.
If I would heat you up to 3,000 degrees Kelvin,
you would become red-hot.
And you actually...
we could turn the light off and I would see
you.
You're just emitting red light.
If I would heat you up to three million degrees,
you would start to begin to radiate in x-rays.
You may not like it, but that's a detail,
of course.
So I want you to appreciate the fact that
the...
the kind of radiation that you get depends
strongly
on the temperature, and at ten million degrees
you're dealing almost exclusively with x-rays.
So these binary systems are very potent sources
of x-rays.
The neutron stars rotate around, we discussed
that earlier--
conservation of angular momentum--
and they have strong magnetic fields.
The matter that falls onto the neutron star
already heats up during the infall
because there is gravitational potential energy
released,
and so the matter is so hot
that, in general, it's highly ionized.
And highly ionized material cannot reach
a magnetic neutron star
in all locations that it prefers to do so.
In 802, you will learn why that's the case.
However, the matter can reach
the neutron star at the magnetic poles.
And so what you're going to see now is
you're going to have a neutron star with magnetic
poles,
so the matter streams in onto the magnetic
poles,
which gives you two hot spots.
And if the axis of rotation doesn't coincide
with the line through the two hot spots,
if the neutron star rotates,
you're going to see x-ray pulsations.
When the hot spot is here, you will see x-rays,
and when the hot spot is here, you will not
see x-rays.
And so we observe from these systems x-ray
pulsations.
Now think of the following.
The x-ray pulsations are a clock.
It is the clock of the rotating neutron star.
If the neutron star in a binary system--
because all of these are in a binary system,
these x-ray binaries--
if it's coming to you, you see Doppler shift.
The ticks of the clock come a little closer
together.
If the neutron star moves away from you,
the ticks of the clocks are a little bit further
apart.
That is exactly what Doppler shift is all
about.
So by timing the pulses of the x-rays,
you can get a handle
on the Doppler shift of the neutron star.
That means you can get the speed of the neutron
star,
you can get the radius of the orbit,
and you can get the period, just like we discussed
before.
But now you take an optical...
The x-ray observations, by the way, have to
be made
from outside the Earth's atmosphere,
because x-rays are absorbed by the Earth's
atmosphere.
Now you take an optical telescope
and you look from the ground,
and now you see the optical spectrum of the
donor.
And what do you see in the donor?
You see these absorption lines.
And as the donor moves around the center of
mass,
these absorption lines move back and forth.
The Doppler shift of the donor.
So you know the velocity of the donor,
you know the radius of the orbit, and you
know the period.
So now we have a situation that I just described
earlier
that you have the Doppler shift of both objects,
and remember, I told you that you also get
the masses.
You get the mass of the donor and the mass
of the accretor.
Before I go ahead, let me show you some slides.
So we have to lower this again, if that's
possible-- yep.
I want to show you
an artist's conception of such a binary system.
So this is what it may look like.
You see the donor there
and you see the neutron star right here,
so small, of course, that it's invisible.
And this is the accretion disk.
Swirls in, the matter ends up on the neutron
star.
And this is another view that gives you an
idea of the donor
and then the swirl of matter,
and then it swirls in and ends up here on
the neutron star.
And here you see data that were obtained in
1971.
It's clear evidence for the existence
of these rotating neutron stars with these
x-ray hot spots.
You see here the observed x-ray intensity
as a function of time.
And the actual data are these very thin lines.
And this bold line was drawn over by the authors
to convince you that you see a signal which
is highly periodic.
The time from here to here is 1.24 seconds.
This object was called Hercules X-1.
So this is one of the magnetic poles
and this is the other magnetic pole.
One magnetic pole and the other magnetic pole.
So you see here unmistakably the rotation
of the neutron star
and the x-ray pulsations.
Here you see data from the same object,
but now the time scale is very different.
From here to here, this is two days.
And when you look at this data alone, forget
this for now,
notice that you see the source is active in
x-rays--
the 1.42-second oscillations
you cannot see, of course, anymore
because the time scale is different,
but notice here there are no x-rays at all:
1.7 days later, no x-rays at all.
1.7 days later, no x-rays at all.
And so what you're looking at here
are what we call x-ray eclipses.
When the neutron star moves behind the donor
star,
all the x-rays are absorbed by the donor star
and you get x-ray eclipses.
In other words, you get...
Independently from the Doppler shift
you also get the period of the orbit by the
x-ray eclipses.
And this really changed our whole concept
of astronomy,
the existence of these neutron star binaries.
And now comes a part, what are the masses
of these objects?
I already alluded you to the idea
of the possibility that there may be black
holes.
All the mass measurements that have been done
to date
of these neutron stars where you see the pulsations...
all of them are very close to 1.4 solar mass.
And there's a good reason for that-- that's
not an accident.
In 1930, the physicist Chandrasekhar predicted
that white dwarfs could not exist
if their mass is larger than 1.4 solar mass.
It was a quantum mechanical calculation
for which he received in 1983 the Nobel Prize.
Remember we discussed white dwarfs earlier.
A white dwarf has about a radius of 10,000
kilometers,
about the same as the Earth.
And imagine that you have a white dwarf,
and you add matter to the white dwarf
and you pass the 1.4-solar-mass mark.
Then the white dwarf will collapse
and becomes a neutron star.
And so when we measure the masses of neutron
stars,
it turns out, maybe somewhat by surprise,
that they're all very close to 1.4.
If you could add more matter to the neutron
star
by accreting more and more matter
and you reach the point that the neutron star
becomes
as massive as three times the mass of the
sun,
we believe that the neutron star can no longer
support itself
and becomes a black hole.
And so now comes the question, what is a black
hole?
A black hole is the most bizarre object that
you can imagine,
and it is something
that you want to stay away from, too.
A black hole has no size, unlike a neutron
star.
It has no size, but it does have mass, and
it has a lot of mass--
three times the mass of the sun, ten times
the mass of the sun,
a hundred times the mass of the sun.
So it has mass, but it has no size.
We identify around the black hole a sphere
with radius R
which we call the event horizon.
Imagine you are at the event horizon
and you want to get away from the black hole.
What kind of speed do you need?
You should be able to give me that answer
immediately.
The escape velocity must be
2 MG divided by the radius of that event horizon.
In other words, the radius of the event horizon
itself
equals 2 MG divided by c squared.
If you tell me what m is,
I will tell you what the radius of the event
horizon is.
I went a little fast here.
I skipped an important step.
v is the escape velocity from the event horizon,
which is at a distance capital R from the
mass M.
So we see that here.
Now, this escape velocity can never be larger
than the speed of light, so the maximum value
possible is c.
And if now you look at this part of this equation
and you take the radius on one side,
you'll get that the radius of the event horizon
equals 2 MG divided by c squared,
and that's how I found that equation.
Sorry that I went a little too fast.
If M is the mass of the Earth,
the radius of the event horizon is one centimeter.
If M is the mass of the sun,
the radius of the event horizon is three kilometers.
If M is three times the mass of the sun,
the radius of the event horizon would become
nine kilometers.
Because it scales linearly with the mass.
If you were inside the event horizon,
you could never escape the black hole
because you would need
a speed which is larger than the speed of
light.
Therefore, you can never escape from inside
the event horizon.
Nothing can get out of it,
not x-rays, no radio emission, no light, nothing.
Once you're inside the event horizon, you've
had it.
You cannot escape it.
And so the question now that comes up:
Can we see x-rays from a black hole?
Because if nothing can come out of a black
hole,
how can we see x-rays?
And the answer is yes, we can,
because as long as the matter that swirls
in
is outside the event horizon, it would still
be very hot.
Because gravitational potential energy
would already have been released, it would
be very hot
and it would emit x-rays.
So we can see x-rays outside the event horizon.
You will never see pulsations,
because a black hole has no surface, like
a neutron star.
So there's no such thing
as two hot spots which rotate around.
And so now comes the problem for astronomers:
How can you determine the mass of the accretor
if the accretor is not a pulsating neutron
star
but if the accretor is a black hole?
Well, you can only now measure the Doppler
shift of the donor,
because the donor, in general, is quite well
visible.
It's an optical star.
But you will not be able to measure
the Doppler shift of the black hole-- no pulsations.
If, however, an astronomer can make an estimate
of the mass of that donor,
then you will find the mass of the accretor.
In other words, instead of having
the Doppler shift measurements of both stars--
the neutron star and the donor star,
which gives you the mass of two stars--
now you have to settle for the Doppler shift
of only the donor
and the mass of the donor itself.
And if you have a reasonable idea of what
that mass will be,
then you can find the mass of the accretor.
And there is a very famous case
that was the first one discovered in the early
'70s,
which is called Cygnus X-1.
Cygnus X-1 is an x-ray binary
which has an orbital period of 5.6 days.
The Doppler shift measurements of the donor
were made,
and astronomers simply looking at the spectrum--
at the absorption lines
and the structure of the absorption lines
and the kind of absorption lines--
were able to say, "Yeah, the mass of the donor
is probably approximately 30 solar masses."
And with that information and with the Doppler
shift,
you can now arrive at the mass of the accretor,
and that is, in this case...
oh, by the way
there is an r missing in the word "accretion"
there--
that mass turns out to be about 15 solar masses.
Now, when this was found in the early '70s,
most people concluded this has to be a black
hole.
It is a very compact object.
Otherwise it wouldn't emit x-rays in the first
place.
And clearly, if the mass of that compact object
is way larger than three solar masses,
then there is no doubt in our minds that this
is a black hole.
Since that time,
many black hole x-ray binaries have been discovered.
So, if I summarize, the amazing thing is
from studying the Doppler shift
of binary systems like x-ray binaries,
you can derive the orbital parameters, orbital
radius,
orbital periods, the speed of the stars in
orbit,
but you can also find the masses.
And whenever you make a measurement of the
mass
when it is a neutron star
when you see the x-ray pulsations,
you almost always find
that it is very close to 1.4 times the mass
of the sun.
But in a few cases, you will find
that the mass is substantially larger.
Admittedly you have to do without the Doppler
shift, then,
of the accretor, but you have to use some
other information,
and then you can conclude in most cases
with pretty good confidence
that you're dealing with something like...
bizarre as a black hole,
which you can only define the event horizon...
And you can never escape a black hole
when you're inside the event horizon,
because that is when the escape velocity
would be larger than the speed of light.
So this is the escape velocity.
If you set that equal to c,
then you can solve for the radius of the event
horizon,
and out of it pops this equation.
I would like to show you now a slide of Cygnus
X-1,
which is the oldest known black hole x-ray
binary.
I have to lower 
the screen.
And there it comes.
This was really a bombshell when this was
discovered.
I still remember reading that first publication.
Two people discovered this independently,
by the way.
They came independently to the same conclusion.
Tom Bolton and it was Paul Merlin--
two independent groups.
All right, here is an optical picture-- it
is a negative,
so you see the stars dark and you see the
sky bright--
and right here is the star that is Cygnus
X-1.
It is the donor.
It is a very large star, a supergiant, huge
radius,
and it is believed to have
a mass of 30 times that of the sun.
You see here the close-up.
This is not the companion, believe me.
This is just an image of that star.
The position was... it was hard to get an
accurate position.
Various groups made a major contribution
to finding the position.
One of the rocket flights of MIT
found a position that is quite precise
and there was no doubt later...
When the orbital period was found of 5.6 days,
there was no doubt that this was the x-ray
source.
And so this is a system
whereby you can only see the donor in the
optical light.
You can measure the Doppler shift of the donor,
and by looking at the spectrum of this star
alone,
you come to the conclusion
that the mass must be about 30 solar masses,
and then you can argue
that the invisible x-ray source must be a
black hole.
