Today we are going to talk about Doppler Effect.
It is best known for sound. That probably reminds you of
your high school days. If you move towards a whistle,
you hear the pitch of the whistle higher than the whistle,
and if you move away from the whistle then you hear a lower
pitch than the pitch of the whistle.
Or, if the whistle moves through you, you hear a higher
pitch. And, when the whistle moves
away from you, you hear a lower pitch.
You may think now, whether you move with a certain
speed towards the whistle or whether the whistle moves with
the same speed to you that you will hear the same increase in
pitch. But that is not true.
There is actually a big difference.
The situation is not symmetric. If you move towards the whistle
with the speed of sound then the frequency that you hear is twice
the frequency of the whistle. But, if the whistle moves
towards you with the speed of sound, you will hear nothing
because all the sound stays with the whistle until the whistle
passes you. And then you hear a huge boom
of sound, the frequency goes to infinity.
You break through the sound barrier, so to speak,
if you survive it. So there is a large asymmetry
between them. The speed of sound is about 340
meters per second at room temperature.
It is about 770 miles per hour. That is the reason why
commercial airplanes do not fly any faster relative to air than
770 miles per hour. Because they are not designed
to break through the sound barrier.
That is the limiting factor for commercial airlines.
Let this be the transmitter of sound and this the receiver of
sound. That could be you.
And let us assume that you move with a certain velocity.
I call this the plus direction. And you move with the velocity
vr. And let the transmitter move
with the velocity vt. And the transmitter produces a
sound f, but what you receive is a frequency f prime.
So, f is the frequency. If you go in velocity in this
direction, that is a positive, and a velocity in this
direction then becomes negative. Then f prime is f times the
speed of sound minus vr divided by the speed of sound minus vt.
And you should be able to derive that on your own.
And it is very asymmetric. And you can see that
immediately. Suppose I make vr the same as
vs, so the upstairs becomes zero, f prime goes to zero.
That is obvious. If you walk away from the sound
source with the speed of sound, the sound never reaches you.
You walk as fast as the sound comes to you so you hear
nothing. It is immediately obvious that
f prime then has to become zero. But, if we now make the sound
source go away from you with the speed of sound,
again, the distance between the two of you grows,
now v of the transmitter I make minus the speed of sound,
notice what happens. The frequency that I receive is
simply one-half the f. So, f prime is now one-half f.
You see an enormous difference in the two.
If you walk away from the source with the speed of sound
and you hear nothing. And, if the sound moves away
from you with the speed of sound, all you hear is half the
frequency of the sound source. I have a 4000 hertz tuning fork
there. And so f is 4000 hertz.
And I am going to move my hand towards you as fast as I can,
which is one meter per second. In other words,
I am the transmitter, so vt is approximately plus one
meter per second. And then you should hear an f
prime which is about 0.3% higher than f, so it is about 4012
hertz. And then I will move my hand
away from you with about one meter per second,
so I have to put the minus sign in there.
And then I will find that the frequency that you will hear is
about 3988 hertz. And the nice thing about this
demonstration is that the difference between the motions
is 24 hertz, and you can very easily hear that.
Here is this tuning fork 4000 hertz.
I will do it a few times. And I will treat you,
too, because I gave them preferential treatment.
Do you hear it? Higher pitch when I come to you
and clearly a lower pitch when I go away.
A difference of about 24 hertz. Suppose now that I rotate a
sound source in a circle. I rotate it around like so,
angular velocity omega, radius r, this is where you
are, and let the circumferential speed be v0.
It goes around with constant speed, so v0 is omega R.
And the sound source has frequency f.
It is clear that if it moves away from you then f prime will
be smaller than f. Here f prime will be f.
And here f prime will be larger than f.
And so, as you listen to this sound, you will hear the sound
change in a sinusoidal way. Because what matters is not the
speed of the source but the radial component in your
direction. And the radial component here
is v0 towards you, here away from you,
but the radial component here is zero.
If you arbitrarily call this time equal zero here then the
radial velocity, the component in your direction
is then v0 times the sine of omega t.
And so, if you not only listen, but if you actually record,
as a function of time, the frequency that you hear
then you hear something like this.
And this then is the mean value f which is transmitted by the
sound source. And now you can ask yourself
the questions what do you learn from this?
And it is amazing what you learn from this.
You can close your eyes and just record this sound signal as
a function of time. The first thing that you learn
from it is the period t, which is 2pi divided by omega.
You know the period of rotation, you know omega.
But you also record what f prime maximum is.
And you also record f prime minimum.
And so, with the Doppler shift equation that we have here,
you can immediately calculate what v0 is.
You know also v0 but, since v is omega r,
you also know R. Just imagine by recording that
sound signal as a function of time, you can determine the
period of rotation, this omega, the speed in
circular orbit and the radius even.
I want to demonstrate this to you just quantitatively.
I have here a whistle. I don't even know what
frequencies it produces. It maybe is a mixture of many
frequencies. That is not so much the point.
I want you to hear that if I twirl it around now it no longer
the matter of high/low, high/low, high/low,
but it now is a gradual change to its high pitch,
low pitch. And then you hear this in
between. Can you hear it?
It is clearly high at times and it is low at other times,
but it is now a cosinusoidal change in time.
Keep in mind, which is going to be important
in what follows today, that from this you can
determine the periods, the speed and the radius.
Now, electromagnetic radiation also shows Doppler shift.
That means if you move towards a source of electromagnetic
radiation, you will record a higher frequency than was
transmitted. And, if the source of radiation
comes to you, you will also record a higher
frequency. And, if you move away from each
other, you will record a lower frequency.
And the Doppler shift equation for electromagnetic radiation is
not so easy to derive. You need special relativity for
that, but I will give you the result.
I want you to appreciate that there is no such thing as a
velocity of the receiver and a separate velocity of the
transmitter now. Because, in special relativity,
the only thing that matters is the relative velocity between
the two. It is an illegal question even
to ask who is moving towards whom and who is moving away from
whom. There is only one velocity in
special relativity. Let this be the transmitter of
electromagnetic radiation and this be the receiver.
And this is the relative velocity v and this angle is
theta. I am not trying to tell you
that it is this object that is moving in this direction.
It could be this one that is moving in this direction.
This is the relative velocity between the two and the line of
sight. The component of the velocity,
which we call the radial component, which plays an
important role there, too, is v cosine theta.
And, with this picture in mind, if this one radiates a
wavelength lambda with frequency f and this one receives a
wavelength lambda prime with frequency f prime then lambda
prime is c divided by f prime and lambda is c divided by f.
And I will give you the results of the Doppler shift equation in
terms of lambda, and then you can always use
that relationship to do it in terms of f.
And the complete relationship is that lambda prime,
the one that you will receive, is lambda times one minus beta
cosine theta divided by the square root of one minus beta
squared. And beta is v over c,
c being the speed of light. You see here that if theta is
smaller than 90 degrees but larger than zero that means you
are approaching each other. This stands for approaching.
That means that lambda prime is then smaller than lambda,
so you record smaller wavelengths when it is coming to
you. And we have a name for that,
which is blueshift. And the reason why that is
called blueshift is obvious. Because blue light has a
shorter wavelength than red light.
Even though this Doppler Effect may not even be in the visible
part of the spectrum, we still call that blueshift.
When relative velocity is approaching you go towards
shorter wavelengths, and we call that blueshift.
Now, if theta is between 180 degrees and 90 degrees that
means you are receding from each other, then lambda prime is
larger than lambda. And, of course,
you guessed it. That is called redshift.
And astronomers talk all the time about redshift and
blueshift.
If beta square is much, much smaller than one,
you can forget about the downstairs, if you like that.
If you take beta, say, 10% of the speed of light,
which is substantial, which is 30,000 kilometers per
second, so beta is 0.1, then beta squared is
approximately 0.01. And so downstairs one minus one
hundred you could, through reasonable
approximation, ignore that.
But that depends, of course, on the accuracy that
you want. In that case,
you would get lambda prime is approximately lambda times one
minus beta times cosine theta.
And if now you want to know it in terms of frequency then f
prime is roughly f times one plus beta cosine theta.
And I will leave you with proving that this is indeed then
a good approximation. Doppler shifts are being used
in sports to measure the speed of tennis balls and baseballs
when they are being pitched. Radar is sent to the object,
and it reflects off the object. And so you can measure this way
the speed. Weather radar works the same
way. You can measure the speed with
which weather approaches you or goes away from you.
The police can check the speed of your car.
Sends radar to you, it bounces off your car,
and then it can sense your speed.
And, in my problem set seven, I think I have a problem based
on that. If you are approaching the
police radially, if theta is zero then,
of course, you would simply get one minus beta here and one plus
beta. Theta can be zero,
of course. It doesn't have to be,
but it can be. You can imagine how important
the role was that this played in astronomy because this allows
you now to calculate the velocity of stars relative to
our solar system, relative to us,
because almost all stars in their spectrum show absorption
lines. And, if you know the absorption
line frequency or the absorption line wavelength as measured in
our laboratory, then you can measure the
velocity of that star. And I will give a particular
example so that you see this at work.
There is a star, and I picked only one star,
which is called Delta Leporis. The reason why I give you the
name is not to remember it but to know it is a real star.
And the star, in its absorption spectrum,
shows a very clear dark line which is the result of calcium
in the atmosphere. Perhaps you remember that I did
a demonstration here of sodium absorption that I had here
creating an atmosphere of sodium vapor which created an
absorption line in the spectrum. Well, in the same sense,
you will see calcium absorption line in the spectrum of this
star. It is actually called the
calcium K line. It has nothing to do with
potassium. In the laboratory,
we know wavelengths very well. It is 3933.664,
think, angstrom. I like to work in angstroms.
Some of you prefer nanometers. One angstrom is 10 to the minus
10 meters, just because we always worked,
when I was a student, in angstroms.
But, of course, if you prefer nanometers,
that is fine. And, when you observe the
wavelength in the spectrum, you will see that it is lambda
plus 1.298 angstroms. The accuracies are stunning.
You can determine wavelengths. And so you clearly see that
lambda prime is larger than lambda.
That means there is redshift. And when there is redshift that
means we are receding from each other.
And we can calculate the radial velocity.
All you know is the radial velocity in this case.
And so with that information you get that beta cosine theta,
if you use that relationship, and the Doppler shift equation
you get minus 3.3 times 10 to the minus 4.
And so you multiply that by c and then you get that v cosine
theta is then very close to minus 99 kilometers per second.
And so we are moving away from each other radially,
I have no information in this direction radially at about 100
kilometers per second. It is illegal to ask who is
moving, they or we. It is only a matter of relative
motion which is all that matters in special relativity.
Now, Doppler shift measurement of stars and galaxies have had a
spectacular impact on the way that we think about the world,
about ourselves and about the universe.
And I will discuss now only two examples which,
in my view, are the most impressive of all.
The consequences are very far reaching in both cases.
The first thing I want to discuss with you is the
discovery of black hole binaries; how that came about as
a result of Doppler shift measurements.
There are many stars that are binary systems.
That means there are two stars going about the common center of
mass. Our sun is not.
Our sun is a single star. If you look at a star which is
a binary system, you may only see one star in
the sky because you couldn't spatially resolve them.
But you will know that it is a double star because you see the
absorption lines of the spectrum move.
Suppose you see only one of the two stars.
Suppose one is invisible because it is too faint,
but the other is very bright. You will see the absorption
lines do this. They move back and forth.
And you can tell what the orbital period of the star is.
If both stars produce enough light, if one star comes to you
that gives you blueshift, the other star must go away
from you because they go around each other like that.
Now you see one set of lines that go like this and the other
set of lines from the other star goes like this.
And so you now see the motion of the absorption lines of both
stars. In a situation like that,
that you see the spectral lines moving back and forth,
you know it is a binary system. And so we know of many stars
whereby we only really see a spectrum of one because the
companion is too faint. We still know that it is a
binary system because of the motion of the absorption lines
in the spectrum. What you can derive now from
that is the period of the orbit, and you can find the radii of
the orbits if you see both pairs of lines moving.
And, therefore, you can find the orbital speeds
in exactly the same way that we did with sound,
period, speed and radius. Now I want to work this out in
quite a bit more detail which then ultimately,
as you will see, led to the discovery of black
holes in binary systems. Let's make a picture of a
simple binary system whereby the orbits are circular.
That is a little special, but it will get the point
across. This is the orbit of one star
and this is the orbit of the other star.
It is a circular orbit. And you are here.
This is the center of mass. And let this star have a mass
m2. And let its speed in orbit be
v2. And let this separation from
the center of mass be r2. The other star must be exactly
here, m1, and that has a speed v1.
And let its distance to the center of mass be r1.
Circular orbits, the center of mass means that
m1 r1 is m2r2. If you don't remember,
go back to your 8.01 notes. And you are observing here,
on earth, the optical spectra. You see the absorption lines in
Doppler shifts. You see them moving like this.
And so you will find from star number one you get the periods
of the orbit, you get its velocity in orbit
and you get r1. Just like we got from the
sound. There is no difference.
No you have star number two. You get the period in orbit.
You can check now that you get the right answer.
You have v2 and you have r2. All of that comes out of the
Doppler shift measurements. But there is more.
You know Newton's Law of Gravity which leads to Kepler's
Third Law Which is that T2 equals 4pi
squared times r1 plus r2 to the power of three divided by m1
plus m2 times G, which is your gravitational
constant. Now look.
You know r1 and you know r2 from your Doppler shift
measurements, so you know this.
You know the period from your Doppler shift measurements so
you know this. But what you don't know is what
is m1 plus m2, but you know that m1r1 is m2r2.
So we have two equations with two unknowns and you find m1 and
m2. Think about this for a minute.
Out of the Doppler shifts of this binary system,
you get their orbital radii, you get their velocities,
you get their period in orbit, But you even get the masses of
the individual objects. Now, when you observe from
earth, you are probably not in the plane of the orbit.
Notice what I did very cleverly.
I put you in the plane of the orbit.
I said you are here, which is the plane of the
blackboard. In reality, you will probably
not be in that plane. You will not see the orbital
period [add on, like this it also adds on?],
but it will be tilted a little. And, if it is tilted,
then the radial velocity, which is the only one you will
measure, will be lower than v2. In fact, you can easily see
that suppose the orbit was like this, so you were on earth and
they go around each other just like this, then there is no
radial velocity at all. The inclination of the orbit is
very important, but I will not further address
this today as it does not affect the basic principle behind
Doppler shift. But you can imagine,
of course, that this is a key issue for astronomers to get a
handle on the inclination. In our galaxy there are a few
hundred, I will lower this and get it back up later again,
very special binaries whereby one star is more or less like
the sun. Pretty common.
Pretty boring. But there is another one very
small in size; a neutron star or a black hole,
as you will see very shortly, that is very close.
We call it a close binary, not because they are close to
us but the two objects are close together.
This one could be a neutron star or, as you will see very
shortly, a black hole. And they go around a common
center of mass. Say in the blackboard is the
plane and they go around like this.
And there is a point here between the two where the
gravitational pull in one direction toward this star is
the same as the gravitational pull in this direction.
It has a name. We call it the inner LaGrange
point. There is also such a point
somewhere between the earth and the moon.
It is very close to the moon, but there is such a point.
And so, if the inner Lagrange point lays here so that the
force in this direction on the test mass is the same as the
force in that direction, if it lies under the surface of
that star then the matter that is on this side here wants to
float towards the neutron star. It is energetically more
favorable because the force in this direction is larger.
Now, since they go around each other, it cannot fall radially.
And so what will happen is that this matter from this star is
going to spiral in through a disk and finally find its way
onto the neutron star. It shouldn't surprise you,
therefore, that this star is called the donor,
it provides the mass, and this small object here
could be a neutron star is called the accretor and this
disk is called the accretion disk.
Let us take a closer look at this neutron star.
I am going to make a blowup here.
This is the neutron star which has mass m and it has radius R.
And I take a little bit of matter, a little test mass from
a very large distance and I let it fall onto the neutron star.
29:02 This is the speed with which it
hits the neutron star. And so the little m cancels.
It does not make any difference whether this is a large mass or
a small mass. It will always reach the
neutron star with the same speed.
And you will see that that speed is then the square root of
2MG divided by R. You have seen this in 8.01.
You may also have seen it in terms of what we call escape
velocity. If you ask yourself the
question if you were on the surface of the neutron star what
is the speed that you need to make it out to infinity?
In other words, to break away from the
gravitational pull of the neutron star.
That is exactly that same speed, of course.
It has to do with the conservation of energy.
You can also think of this as escape velocity.
Now comes the amazing thing. If you take a neutron star,
the mass of a neutron star is very roughly one and a half
times the mass of the sun. This is the symbol for sun.
And the radius for a neutron star is about ten kilometers.
If you substitute those numbers in there you will get a
phenomenal speed. You get 200,000 kilometers per
second which is 70% the speed of light.
Whenever any matter falls onto a neutron star from a large
distance, it hits the neutron star with 70% of the speed of
light. This is kinetic energy.
It hits the surface and is all converted to heat.
And so, if enough kilograms per second fall onto the neutron
star, the surface of the neutron star gets very hot.
And we know there are cases where it can get as hot as about
10 million degrees. And at 10 million degrees this
neutron star will radiate almost all its energy in the x-rays,
very little in the optical, almost all in the x-rays.
And so it becomes a strong x-ray source.
To make you appreciate the energy release when something
hits a neutron star, if you take a marshmallow and
you throw a marshmallow from a large distance on a neutron
star, when it hits the surface the energy that is released,
that means the explosion that is caused by the impact is
comparable to the energy that was released when the atomic
bomb was thrown on Hiroshima at the end of the Second World War.
That enormous amount of energy that causes this incredible
explosion is comparable to the energy that is released when a
marshmallow hits the surface of a neutron star.
Now, when you look through an optical telescope from the
ground and you look at this system, you will only see the
light from the donor because there is very little light
coming from the neutron star because there is an enormously
small surface. Most of the energy comes out in
x-rays anyhow, so you only see the light from
the donor. Astronomers have learned over
the years, this has nothing to do with Doppler shift,
that when you look at the spectrum of a star they,
in general, can tell how massive that star is.
In other words, a star which is ten times more
massive than our sun has a very different spectrum than the
spectrum from the sun. And so they look at the
spectrum and say this is probably five times more massive
than the sun. This is probably 20 times more
massive than the sun. Astronomers can tell by simply
looking at the spectrum of the donor what the mass of the donor
is. Let's call this star number
one. They look at this star in
optical light from the ground, and they say we know roughly
m1. Now they see the Doppler shift
of this one. They see an absorption line in
the spectrum. It goes nicely back and forth.
And they say, ah-ha, we now know the orbital
period and we know the velocity v1 and we know the radius r1.
We know all that. But, of course,
astronomers also know, I hope, where is Kepler's Law?
I didn't erase anything. Did I hide it?
Yeah. They also know Kepler's Law.
Now look at Kepler's Law. You know r1,
you know m1, you know t, but what you don't
know is r2, which is what you would like to know,
that is the radius of the orbit of the equator.
You don't know this one and you don't know that one.
But you do know that m1 times r1 is m2 times r2.
You have two equations with two unknowns, and you can solve for
m2 and you can solve for r2. And so now you will find the
mass of the accretor. Immensely powerful.
This has been a game that has been going on for many years
whereby x-rays astronomers, this is what my specialty has
been for the past I cannot even remember, since 1966,
for the past decades, have made a living out of
trying to determine the mass of these accretors.
And, in many cases, do you find that the mass of
the accretor is very close to 1.4 solar mass.
We are sure that in that case we are dealing with a neutron
star. It is not an accident that we
find 1.4 times the mass of the sum.
Chandrasekhar demonstrated, quantum mechanical calculation,
in 1930 that if you take a white dwarf and you dump matter
onto the white dwarf. 35:36
Out of that theory it follows that if you keep dumping mass
onto a neutron star, the moment that you cross the
three solar mass line the neutron star can no longer
support itself and collapses into a black hole.
And now comes the question what is a black hole?
A black hole is an object from which light cannot escape.
I have to erase something, and I think I am going to erase
the left part of the blackboard. It is an object from which
light cannot escape. I have to be more specific than
that. A black hole is an object for
which the escape velocity is larger than the speed of light.
We know what the escape velocity is because this --
Remember that is why I paid attention to it and mentioned it
is also the escape velocity? Here is a black hole.
There is a radius around that black hole that has a wonderful
name. It is called the event horizon.
And all I have to do is make this c and I can calculate what
that radius is. That means if you are then
inside that radius then the escape velocity would be larger
than c so you cannot get out. And so the radius the radius of
the event horizon, I wouldn't want to call it the
radius of the black hole, but the radius of the event
horizon is then 2MG divided by c2.
If you are closer to the black hole, the escape velocity would
be larger than c which is not possible.
To give you a little bit of feeling for what this means,
if you take the earth and you take the mass of the earth,
which you can look up in your book, and you substitute that in
there, you will find that the radius of the earth,
to become a black hole, is only one centimeter.
So you substitute in here the mass of the earth.
C2 you know. G you can look up.
You find that the radius is about one centimeter.
If you could squeeze the earth with its present radius of 6,400
kilometers, if you could squeeze it one centimeter then you would
have created a black hole. But, of course,
that may be problematic to do that.
If you take the sum which, by definition,
is one solar mass, and so you put in here the mass
of the sun, which you can look up in your book,
then you will find that the radius of the event horizon is
three kilometers. If you take a five solar mass
black hole then you will find that the radius of the event
horizon is 15 kilometers. Notice it scales linearly with
m. That is what you see here.
M is linearly upstairs. If you go from one solar mass
to five solar mass, the radius of the event horizon
scales linearly. In 1971, astronomers,
Tom Bolton and Murdin and Webster came with a daring
statement that one of these systems was a black hole binary.
They looked at the spectrum of the donor and concluded that it
probably was a 20/30 mass solar star.
And then they made the Doppler shift measurements of the
absorption line in the spectrum of the donor.
40:28 And a lot of people emotionally
could not handle it. And there were lots of
publications for years to come arguing that it was all
nonsense; that black holes did not exist, the data was
misinterpreted, they came with all kinds of
other weird explanations, but the black hole still
stands. And most sane astronomers,
that does not mean that all astronomers are sane,
but most sane astronomers do believe that Cygnus X-1 is a
black hole. And we now know of about two
dozen of these systems whereby the accretor is a black hole.
And I decided to make Cygnus X-1 part of your problem set so
you have some time to wrestle with it.
I want you to see a picture of Cygnus X-1.
It is not very dramatic, but I want you to see it for
emotional reasons. You will just see a star.
You cannot see the black hole. You just see a star and that is
all. But at least you can tell your
parents that you have seen a star that is in orbit with a
black hole. If we can get the first slide,
the only slide I think I have, then we should be able to show
you a star. It is a negative.
In astronomy we almost always work with negative.
The stars themselves are black and the sky is white.
And it is this star here that is the donor.
It is quite bright. You cannot see it with a naked
eye. For those of you who are
familiar with magnitudes of stars, it is a [nine?] magnitude
star. And it is in orbit with the
black hole. It itself has a mass of about
20 to 30 solar masses, and the black hole has about a
mass of about 50 solar masses. And all of this is the result
of Doppler shift measurements profound far reaching
consequences. I think this is a wonderful
moment to have a break. And I know that most of you
came here not to listen to my lecture but to take the
mini-quiz. And so let's do that now then.
If you can help handing this out.
For those of you who come here only for the mini-quiz,
I have some advice. You may have noticed that the
question that I asked on the mini-quiz on Tuesday almost
always goes back to the previous lecture on Thursday.
Think about that. OK.
If you can help me handing this out here.
The biggest impact that Doppler shift measurements of
electromagnetic radiation had on the perception of ourselves and
of the universe came in the 1920s.
An industry had already been developed by astronomers to
measure the radial velocity of stars.
I gave you one. Delta Leporis was a typical
one. We were receding 100 kilometers
per second. Velocities were typically 100,
200 kilometers per second. In some cases receding.
In some cases we were approaching each other.
44:30 And so that gives you then a
handle on the radial velocity of that galaxy.
And Edwin Hubble and Humason found, surprisingly,
a correlation between the distance of galaxies and their
velocities. To measure the velocity is
easy. That is a Doppler shift
measurement. That is relatively trivial.
All you have to measure is lambda prime and compare it with
lambda. Distance measure is very,
very difficult in astronomy, very problematic.
And it is a lecture all by itself.
And so, in the days of Hubble, his distance estimates were
very different from what they are today.
His distance estimates to the nearby galaxies were seven times
smaller than what we think today, but that does not matter.
He did find the linear relationship between velocity
and distance. And that was a spectacular
discovery. And this is known now as
Hubble's Law. The radially receding velocity
of galaxies is some constant, which was later called after
him Hubble's Constant, times the distance d to the
galaxies. V is the radial velocity.
It is receding. We don't give it a minus sign
anymore. That v is always receding.
The present most reliably value for H is approximately 70
kilometers per second per megaparsec.
I give it to you in this form because that is the way you will
always see it the literature, but I own an explanation,
of course. The megaparsec is simply a
distance in astronomy. It is the distance that light
travels in 3.26 million years, so it is also 3.26 million
light years. If you know what the speed of
light is, 300,000 kilometers per second, then you will find that
it is about 3.1 times 10 to the 19 kilometers.
Notice that the dimension of Hubble is one over seconds,
is one over time, because kilometers is distance
and megaparsec is distance, so it is one divided by a
second. Hubble himself found a value
which is 500 kilometers per second per megaparsec.
The reason was that he had a different estimate for the
distance. He thought that the distance to
the galaxies that were in his sample were seven times smaller
than what we think today. Suppose you trust Hubble's Law
at face value, as it is, and you and I look at
the particular galaxy in the sky, and we measure that lambda
prime is 1.02 times lambda. In other words,
it is redshift and the radial velocity is there for 2% of the
speed of light. You can measure that beta
cosine theta is then minus 0.02. I carry that minus sign because
I gave you that Doppler shift measurement in the form that I
want you to use. We don't even think of that
minus sign anymore because it is always receding.
I carry the minus sign. And so v cosine theta then is
this number times the speed of light, and that becomes then
minus 6,000 kilometers per second receding.
The radial velocity receding speed is 6,000 kilometers per
second. This is about 2% of the speed
of light. If you trust Hubble's Law,
you can now use Hubble's Law and calculate what d is.
49:34 It means that the light that
you were processing on the ground was emitted by that
galaxy 280 million years ago. In astronomy,
we look back in time. I would like you to see now the
spectra of three galaxies which at vastly different distance
from us. Let me first turn on the slide
projector. Otherwise, I may end up in the
darkness and not know what I am doing.
That is right. I will show you the spectra of
three galaxies. This is a galaxy which is the
closest of the three. That should be rather obvious
because it is also the largest one.
And the spectrum of the galaxy is what you see in the middle
here. These are measurements of known
lines with known wavelengths in the laboratory.
This is just a calibration here and this is a calibration here.
This is the spectrum of that galaxy.
And what you see here are two dark lines.
Those are absorption lines. The average of the stars in
that galaxy. They have shifted relative to
the wavelength in our laboratory.
By the way, one of those lines is the same line that I used for
Delta Leporis, which is the calcium K
absorption line. The other one is the calcium H
absorption line. H has nothing to do with
hydrogen. And it had been shifted over
the distance that you see indicated by this small arrow.
And, if you measure now the velocity, you find 1,150
kilometers per second. And, if you use today's value
of Hubble's Law, this object is 52 million light
years away from us. Now you go to this object,
which is further away. According to Hubble's Law,
it has a higher speed, the receding speed.
And you see the spectrum here. This is the spectrum.
And here you see those two absorption lines.
They have shifted all the way. And out of that then follows
the radial velocity which in this case is 7% of the speed of
light, 22,000 kilometers per second, which pushed this object
at one billion light years away from us.
And, when I go to this one here in the corner,
it is hard to see which one it is, the absorption lines have
shifted even further. You see the absorption lines
here. And that object is going 20% of
the speed of light receding. Then it would be roughly at a
distance of 2.8 billion light years.
The most modern work on Hubble's Law was done by a group
under the leadership of Wendy Freedman, California.
Wendy used data from the Hubble Space Telescope,
and she measured the Doppler shifts and the distance to a few
hundred objects. Distance is always the can of
worms. And she came up with presently
the most reliable value for Hubble's constant.
Let me get the next slide, if I succeed.
This is Wendy Freedman's work. Forget what you see here.
Just look at what you see here. The farthest object that she
was able to get a reliable distance for is 400 megaparsecs.
That is a stunning distance, by the way.
That is about 1.3 billion light years.
And she was able to measure radial velocities up to about
10% of the speed of light, velocities of 30,000 kilometers
per second. And so she can draw this line,
which is the linear relationship between velocity
and distance. And out of that line comes
Hubble's Law. And she concludes that it is
very close to 72 kilometers per second per megaparsec,
plus or minus a few. I used 70 just to round it off.
What is interesting is that Hubble, in his entire data,
the velocities were all less than 1,100 kilometers per
second. Put that in scale here.
This already is 2,000 kilometers per second.
Hubble has only data in this teeny weenie little portion
here. Wendy doesn't even have any
data. She doesn't even use them.
And the reason why she doesn't use them, she says it is too
close. It is not really representative
for what we call the Hubble Flow, so she stays away from the
local stuff. But, nevertheless,
based on that very local stuff, which is not even put in here,
Hubble came with his linear relationship which was a
monumental statement in the history of mankind because it
puts the universe at a very different level than what people
felt before. The universe is expanding.
Let's now ask ourselves the question what does this all
mean? The first thing that may come
to mind, since all objects are moving away from us,
you may think that you are very special.
We all like to think that we are very special.
You may even think that you are at the center of the universe
because everything moves away from you.
And persuasive is to think that there was a time in the past
where there was a huge explosion, which we refer to as
the Big Bang, when all of this happened and
there were galaxies with large speeds in the explosion.
They are now the ones the farthest away from us so they
have the highest speed. And some had a lower speed.
That is a picture that presents itself.
It is a little bit na�ve, though, and partially wrong.
However, the idea of the Big Bang Theory, that there was such
an explosion a long time ago, that still holds.
And that is considered the birth of our universe.
Now comes the question when did this Big Bang occur?
Well, we could make a simple assumption, perhaps not quite
accurate, that the velocity with which all these galaxies are
moving never changed in time. We can do that as a start.
No matter which galaxy you take, the velocity now is the
same that it was in the past. Then we have 8.01 which has the
distance that that galaxy traveled is the velocity with
which it traveled, the radial velocity times the
time that it traveled, which is then the age of the
universe, how many seconds the universe lasted.
This is simply 8.01. But we also know Hubble's Law.
In other words, we can replace this v here Hd
by this one, so we get d equals H times d times t universe.
And so you will find that the age of the universe is one over
H. And remember I mentioned that H
has the dimension one divided by seconds, so this has the
dimension of seconds. And if you use the value for H
of 70 kilometers per second per megaparsec then you will find
that the age of the universe, you can check that for
yourself, is 4.3 times 10 to the 17 seconds, so tu comes about
4.3 times 10 to the 17 seconds, which is about 14 billion
years. You would have to conclude then
that the Big Bang explosion took place about 14 billion years
ago. Now you can easily imagine that
due to gravitational attraction that the speeds in the past were
probably higher. You would think that this
expansion probably decelerated. And so, on that basis,
you could conclude that it is probably less than 14 billion
years. We have very good reasons to
believe that the oldest stars in our own galaxy are 10 to the 10
years old. That means 10 billion years
old. So the Big Bang must have been
somewhere in 10 to the 14 and 10 to the 10.
And most astronomers now would say it is probably something
like 12 billion years ago that the Big Bang took place.
And the issue of the deceleration is,
of course, at the heart of research is cosmology.
It is now believed that the very early universe there was an
acceleration of the expansion. It is called inflation.
It was followed by a deceleration,
which is what you would expect due to gravity.
And it is now also believed, very mysteriously,
that there is again, at a later time,
again acceleration not very well understood.
We have MIT experts here. Professor Alan Guth is a world
authority on this. Ed Bertschinger and Scott
Burles do also research in this. Allan Guth is the godfather of
inflation and our universe. I will get back to that.
The question now that I would like to raise is how far can we
look back in time? All I can tell you is that the
largest redshift that has been measured to date,
lambda prime divided by lambda, is 7.5.
That is enormous. And so now comes the question
what is the speed and what is the distance?
And I cannot answer that. The reason is that the equation
that I have given you, the Doppler shift equation was
only derived using special relativity.
It did not take general relativity into account.
And general relativity introduces also redshift which
has nothing to do with the Doppler redshift.
And so I will leave it up to my colleagues, Professor Guth and
Scott Burles and Ed Bertschinger,
to tell you what perhaps the speed is of this object and what
the distance is. And they may not even give you
one answer. The will probably give you more
than one answer, depending upon certain
assumptions. General relativity changes the
picture quite dramatically. There is no doubt,
however, that they will all agree this object is very far
away, probably more than 10 billion light years.
So you are looking back in time more than 10 billion years.
And so you are almost getting to the point that you are
beginning to see the actual birth of the universe,
the Big Bang itself. What is also interesting is
that remember I mentioned the calcium absorption line,
the K line was roughly at 3,933 angstroms.
If you multiply that wavelength, this is lambda,
by 7.5 it ends up at 29,500 angstroms which is outside the
optical spectrum. Impossible to observe from the
ground. And so how can astronomers ever
find redshifts that are as large as that ratio 7.5?
The way they do that is look at a line in the spectrum that
cannot be observed from the ground at all,
it is a line emitted by hydrogen called the Lyman alpha
line. It is really of hydrogen.
And the wavelength is 1,216 angstroms which is in the UV
that you cannot observe from the earth.
The atmosphere absorbs the UV. But, if you multiply this by
7.5, then you get it all the way down to about 9,100 angstroms
which you can observe from the ground.
And so it is remarkable, when you think of it,
that lines which are invisible due to Doppler shift are shifted
all the way to the extreme red part to the infrared part of the
spectrum where they can just barely be seen.
Now comes the question are we at the center of the universe?
Well, not likely. Suppose you have a raisin
bread. You have made the dough,
you put the raisins in there and you put it in the oven.
This raisin bread is going to expand.
Each raisin will see all other raisins move away from it
radially. Think about it.
Because, if the dough expands uniformly in all directions,
everything moves away from you radially.
And one raisin will see the nearest raisin going with very
low velocity, but when it is farther out in
the cake it goes with a higher speed.
That raisin can actually be the discoverer of the expansion of
its own universe. And it can measure the
distances to the other raisins and the velocities,
and it will come up with Hubble's Law.
Not only will one raisin come up with Hubble's Law,
but any other raisin that you pick will come up with the same
Hubble's Law. And it will also think,
perhaps, that is it at the center of that cake,
which is, of course, not the case.
Anything that expands uniformly in all directions will fit the
idea of Hubble's Law. And also will make you think
erroneously that you may be at the center of all that.
There is a very nice and even better analogy that I can draw,
so I will abandon the raisin cake now and will discuss with
you the universe of flatlanders. Flatlanders are people who do
not have three dimensions but only two dimensions,
and so they can only move in a plane.
They do not know, but we do in this flatland
world, that they really live on the surface of a balloon.
But they do not know that. And so the lights in their
flatland world can only travel along the surface of the
balloon. But they do not know that they
live on the surface of the balloon.
And now this balloon can expand.
And the galaxies, which are on the surface of
that balloon, here they are.
Here are the galaxies. There is a spiral galaxy.
You see that? Nice galaxies.
When you blow up this balloon, each galaxy will see other
galaxies move radially away from each other because light travels
along the surface. And, the farther the galaxy is
away, the higher is the velocity.
And so let us expand this balloon and see what this world
-- I first have to connect it to
the source of the expansion.
These poor flatlanders have no idea what is happening to them.
They are living on the surface of this balloon.
They cannot see the third dimension so they have no way of
knowing. And this balloon grows and
grows and grows. And so the distance between the
galaxies gets larger and larger and larger.
And each one of those galaxies may have astronomers,
and each one of them will discover Hubble's Law.
And they will find the same constant for Hubble's Law when
they make the measurement at the same moment in time.
Now, they may not call it Hubble's Law.
The name of that astronomer may be different,
of course, but that is a detail.
As their universe expands, and we are going to work with
that a little now, you will see what kind of
conclusions they draw.
There goes the universe.
Here is the balloon. And the balloon has a radius R
which they do not know but you and I do.
And this is today. Tomorrow the balloon is there,
or next month or next year. And so the radius of that
balloon next year is R plus dR. This radius is R plus dr.
And here is a galaxy which tomorrow is here.
And here is a galaxy which tomorrow is there.
And the distance between these galaxies I call S,
but tomorrow the distance is S plus dS.
In this universe, it is clear that S plus dS
divided by R plus dR, that is this triangle,
must be the same as S divided by R.
And so I can work that out. I get SR plus RdS is SR plus
SdR. Now I can look at the evolution
in time. I can give that radial
expansion certain value, so I divide by dt.
What is the meaning of S? That is the distance between
the galaxies as you measure them.
We call that d in our universe. What is the meaning of ds/dt?
That is the receding velocity with which they move away from
each other. Light travels along the surface
of the balloon. This ds/dt is that v.
What do we find? We find that v equals d times
one over R dR/dt. And this dR/dt is what I would
call the expansion rate of this balloon, expansion rate of the
universe. And so the flatlanders measure
v, they measure d. And at a particular moment in
times there is a value of dR/dt which they don't know,
there is a value of R which they don't know.
But, at a particular moment in time, this is a constant.
And so they call this Hubble's constant.
It has, indeed, the units of one over seconds,
because dR is in meters and R is in meters so it has,
indeed, the units of one divided by seconds.
They have no way of knowing what it means,
but we do. We can look in the third
dimension. We know that R in the past was
smaller. They don't know that,
but we do. Even if dR/dt never changed,
that the universe expanded with a constant rate,
we know that Hubble's constant in their past must have been
different. Because, if R is smaller in the
past, Hubble's constant must have been larger in the past.
That is the reason why we claim that Hubble's constant in our
universe is 70. We always put a little zero
there, so we always say H0 is in Wendy's case 72.
Even when we looked at the slide, she had that little zero
there. That means as we measure it
now. We make no claim about Hubble's
constant, how it was five billion years ago or how it will
be five billion years from now. We make no statement about
that. If the expansion rate of the
balloon decreases in time then there comes a time maybe that
the balloon will come to a halt. That means when dR/dt becomes
zero, Hubble's constant will then be zero.
All galaxies on the balloon will stand still.
And then gravity will take over and the balloon will start to
shrink. And Hubble's constant will then
become negative. Sign will change.
All redshifts will change to blueshifts.
Everything will approach each other.
And, ultimately, the whole universe may collapse
onto itself, which we generally refer to in the literature as
the Big Crunch as opposed to the Big Bang.
We used to talk about a closed universe, up to a few years ago.
A closed universe would be a universe that would collapse
onto itself, and so there would be a big crunch.
We compared that up to a few years ago with an open universe
which would never collapse onto itself, would always be
expanding. And we even introduced the idea
of what we call a flat universe. It is just in between the two.
That means the universe will be expanding forever and ever and
ever. And it would come to a halt,
but you would have to wait infinitely long.
It is like having a tennis ball throwing up from the earth.
It would just make it to infinity with zero speed.
That is basically that idea. But a lot has changed with the
discovery a few years ago with what we now refer to as dark
energy. And so we now believe that the
universe will expand forever and that there will not be a Big
Crunch. You should not carry the
analogy between the flatland balloon universe too far,
but it is very interesting. It gives you some interesting
insight, and it certainly gives you food for thought.
One thing it suggests is that our 3D universe perhaps is
curved in the fourth dimension, just like the 2D universe of
the flatlanders is curved in the third dimension.
A very fascinating thought. And if you want to learn more
about it, I would advise you very strongly to take a course
in cosmology. If you are so lucky that Alan
Guth is teaching it then don't miss that opportunity.
He gives wonderful courses on the early universe.
But also you may want to learn a little bit more about general
relativity which, of course, plays a very key
role in cosmology. Doppler shift measurements of
electromagnetic radiations from galaxies have played a key role
in the way that we think about ourselves and that we think
about the universe. It was in 1979 that Professor
Alan Guth, who was still at MIT, he was also then at MIT,
made an exciting prediction that the universe may have gone
through an expansion phase very early on after the Big Bang,
exponential growth in expansion, and that the universe
doubled in size in about 10 to the minus 34 seconds.
The universe doubled in its size, the early universe,
called the inflationary universe in 10 to the minus 34
seconds. How long this phase lasted is
not very well known, but it may have lasted 10 to
the minus 24 seconds. After about 10 to the minus 24
seconds our universe had increased in size by a factor 2
to the power 10 billion. Unimaginable.
Beyond comprehension. This idea of the inflationary
universe, which came first from Alan Guth, has been the driver
behind NASA's cosmology program. There is an observatory about
1.5 million kilometers from earth called the Wilkinson
Microwave Anisotropy Probe that was launched in June 2001.
And that is now confirming many of the predictions that follow
from Alan Guth's inflationary universe.
There is no doubt in my mind that Alan will get the Nobel
prize for this. It is not a matter of if,
it is only a matter of when you will see the day,
and I hope I will see that day, too.
Thank you. And see you next Tuesday.