Edwin Hubble discovered that Andromeda is not a part of the Milky Way, and expanded
our universe exponentially. He classified galaxy types, and neither
of those is what he's really known for. What he's really known for had to do with
measuring distances to galaxies. Again like he did to Andromeda, but to
galaxies at a larger distances. so we're starting to measure distances at
which finding cepheides becomes difficult.
With todays' technology we can measure cepheides out to, as I say Maybe 10
million parsecs or 20 million parsecs. But at the time, this was still
difficult. Astronomers needed ways to measure
distances to galaxies. Galaxies are easier to see than Cepheids,
they're more luminous. What you'd like is a period luminosity
relation for galaxies. And indeed, there are such, at least
approximate relations. For spiral galaxies the Tully-Fisher
relation relates the rotation velocity in a galaxy to it's luminosity.
and there's a Tully-Fisher relation for every kind of, every type of galaxy.
So S B galaxies have a different Tully-Fisher relation than SA.
Galaxies and so on. And roughly, the idea is that, larger
galaxies are more luminous and have larger rotational velocities.
And you see here the accuracy with which this is relation holds.
It's not a very accurate distance measurement, but it tells you that if you
can measure with Doppler shifts, the rotational velocity of a galaxy You can
classify its type from its shape. You can now figure out the luminosity of
the galaxy. You have galaxies as standard candles.
This are very brilliant standard candles indeed you can see them out, to very
large distances and you can measure distances because if you have luminosity
and you measure brightness, you can figure out the distance to something.
There's an analogous relationship for elliptical galaxies which allows the
distance to ellipticals, to be measured, today at larger distances yet, we can
measure with type 1a supernovae. But remember, that if you want to measure
the distance to a galaxy in which alas, there is not a type 1a supernovae that
has been observed. Then you have been reduced to variable
stars or these kind of methods and refinements thereof.
So, using all these methods Hubble goes around and he measures the distances to
many of the galaxies and, of course, try to understand how they work and measures
their spectra, and this spectra take a while to identify because galaxies are
moving at rather high velocities relative to each other and therefore some are
approaching us at high velocity. Some are receding at high velocity, and
you have a case of big red shifts or big blue shifts, and you need to figure out
which spectral lines are which. Astronomers, as I said, are very good at
spectrometry. And what Hubble finds when he starts
working through these spectra Is that is, it looks farther and farther out into the
universe, and galaxies that are more distant, less and less of them appear to
be having negative radio velocity. Moving towards the Milky Way, more and
more of them appear to receding and furthermore, the farther they get, the
faster they appear to be receding, and so, here's a nice plate with an example
of the The shifted spectra that Doppler is, that Hubble is measuring, so here are
four images of four galaxies they become smaller, appear smaller as they get more
distance in these negatives. And here are their spectra, compared
against some standard spectrum above and below.
And we're looking for some particular line of potassium here.
And what we see is that the farther the galaxy is, the farther to the red this
line has shifted. And the bottom image gives you a sense of
how good these guys were with spectra graphs because it's not totally obvious
that this little gap right here is a spectra line and furthermore that it is
the potassium line. And it turns out that it was.
And so, with these Hubble can relate that known wavelength of the sode, of the
potasium line to the wavelength that which he measures it.
He can use the relativistic version of the doppler shift formula or the Newton
approximation. Non-realistic expression.
Both of those are things we are familiar by now.
And ye can convert the observed redshift into an observed recessional velocity.
And the statement that the redshift is increasing with distant is the statement
that the farther the galaxy is, the faster he observes it moving away from
us. And so, here is Hubble's collection of
data from the paper. that he published, and what we see here
the horizontal axis is distance. The vertical axis is recessional
velocity, radial velocity. We see that here's a galaxy that's
actually approaching the Milky Way. That is fine, but that the farther You
get the more the galaxies are all receding, and Hubble very bravely draws a
straight line through this bit of data, and declares that he's found a relation
between distance and relational velocity. Notice the Hubble Law, this is what he's
famous for, the recessional velocity of the galaxy is given by a constant,
Hubble's Constant, times its distance from the Milky Way or the Sun, it's the
same thing. And Hubble's Constant is therefore,
conversion of distance to velocity, so 1 way to express its units, because that's
the way we measure it, is a kilometer per second recession.
Per megapersec distance, and Hubble has an estimate for the value.
he this estimate has been refined over the years, we have a lot more data.
but this consant is of such importance that long before its exact value was
known Astronomers needed to use it in their expressions and so they developed a
method of sort or parameterizing their ignorance.
The Hubble constant is characteristically writen and as 100 times h kilometer per
second per megaparsec. In other words if someone were to measure
the Hubble constant and its value were to be discovered to be 100 kilometers per
second per megaparsec, we'd say h is 1. And we'd go around into all the formulas,
and all the books and papers that have various powers of H in them, and set H to
1 and get an answer, but at the moment, until we're absolutely certain of the
value of the Hubble constant, people use this little h to parameterize.
The dependence on that measurement because so much depends on this, as we
shall see, but the value of h is not too far from one, We now have far better
measurements that hubble had, we have measured much more statistics, the most
recent measurements suggest a value of H on the order of point seventy one, and as
you see we know it quite accurately, We have here a plot of recent measurements
in the vein of Hubble of recessional velocities, a function of distance out to
1.6 billion light years away, so 500 megaparsecs.
We see that Hubble's relation, linear relationship Holds up very, very well
what's exciting about this plot is that little rectangle at the lower left, that
represents the data that Hubble actually had.
So he was predicting the straight line velocity on the base, the straight line
plot, and it's slope on the basis of a very small amount of data and was
gloriously vindicated. Now typically in astronomy we write the
Doppler shift as lamda is 1 + z * lamda 0.
And we call z the red shift, so z is a positive number.
And assuming something is red shifted and z = 0 is a red shift of 0, in other words
lamda = lamda 0. And so clearly z is related to velocity.
The expression is written down here, lets figure it out.
So, lamda / by lamda 0 is 1 + z. And lamda / by lamda 0, remember we had
lamda / by lamda 0 is the square root of 1 + v / c.
This is the completely Relativistic formula, and this is supposed to be 1+z.
So, the first thing I do, of course, is I, I want to figure out how z determines
v, and so I want to solve this for v over c, and the first thing I do, of course,
is I square both sides. And I write that, 1+z^2=1+vC/1-vC.
Then I multiply both sides by 1-v/C, and I find 1-v/C.
*(1+z)²=1+V/C and then, I move all the V/C over here, so I have V/C * (1+(1+z) ^
2) and then everything without V/C goes over here and that (1+z)²-1.
So, I solve for V/C and I get that V/C = (1+z)²-1 /(1 + z)² + 1 and again if z is
small, if the red-shift is small, you can use Newton's approximation to show that
this is approximately z. Which is indeed when this is
approximately 1+v over c, lambda 0, then z is approximately v over c, but this is
the exact relativistic expression. So, astronomers will rarely tell you,
unless they're talking to, you know, the Hoi polloi like us, that an object is at
a distance of 2.8 billion parsecs, they will tell you that it has a redshift of
.3. The reason is that we'll see next week
that distances are ill defined but certainly redshifts are measured,
observable quantities and The larger H redshift and so we basically can speak
about redshift as a surrogate for distance and indeed the Hubble Law tells
us, remember we solve for v over c as a function of z here, but the Hubble Law
tells us that v over c is the same as H0 D over c.
So if you know z you can figure out D as well as v, so.
Knowing the red shift tells you not only the speed obviously with which the galaxy
is receding, but also its distance. So we have to digest this.
In whatever direction we look, galaxies are receding from us.
At the rate of recession does not depend on the direction of the galaxy.
Galaxies at the same distant at all directions are receding with a equal
velocity, but the farther they are the faster they are receding.
Now, depending on your taste, you could either conclude that were not popular
galaxy, or more technically that we're in the center of some cosmic expansion,
everything is moving away from us. It turns out that this is not exactly
correct. Your intuition is in this case actually
misleading you. Imagine this, collection of galaxies here
they are all going to be moving away from each other.
The center of expansion could be the center of the screen, or if I take this
and add to it a motion of everybody to the right, the center of expansion could
be way over to our left. Each galaxy, as it observes each of the
others will see that each of of the other galaxies is growing farther and farther
away from it. So all that is going on is that as these
galaxies grow farther and farther away, all of their distances are increasing and
the fact that in this case. It looks like the expansion is about the
point of the center of the screen is completely irrelevant.
They are all moving away and the farther they are, the faster they are moving.
There is no centrality involved, this is just the way it is.
So we need not invoke some particular central location for ourselves.
Everything is moving away from everything else, all distances are growing, every
observer as he looks around him or she looks around her will see all other
observers receding and the mathematical expression of this is that if you
measure. Any distance in the universe at time t*0,
by t*0 I mean now and then you measure it either later or earlier.
If you measure it later, it will have grown by a factor which depends on time
in this fashion. What this tells you is that over a time,
t Distances grow, by an amount that is H*D, oh, yeah.
The speed of recession is H*D. Now this is valid both into the future
and of course this didn't just start happening now, so it's valid into the
past. If T is less T0, if you're looking into
the past, then this is a negative number, distance in the past were smaller.
Than they are now. Fine.
If everything is receding, that means everything used to be closer together.
I should point out immediately that what we're seeing is the galaxies, all
distances between all galaxies appear to be growing.
This does not mean that the distance between say, the sun and the Earth is
growing, first of all, or that my belly's growing That's not cosmic expansion.
This is a description that is valid for objects too far away to be
gravitationally bound. The Earth and the Sun and the Milky way
are all moving as one object. The Milky is not in particular.
Expanding this is the motion of distant galaxies, that are not gravitationally
bound to each other. It's to that case the Hubble applies.
The other point, Point is that of course, there are still peculiar motions,
galaxies and are still orbiting various things and that they do on top of this
global expansion. So everything is expanding but that
doesn't necessarily the only motion that galaxies do so for example one could ask
everything is expanding how do galaxies collide that is, That is a case where the
parculiar, are larger than the Hubble expansion for the relatively small
distance between them and that usually happens in cases where their
gravitationally strongly interacting or bound.
Okay. So, Hubble gives us this picture, where
the entire universe is growing, every galaxy is receding from every other
galaxy. What's it growing into? What is beyond
the place where these galaxies are now? More of the universe, the galaxies are
receding into the universe. We'll talk about the, sort of
mathematical implications of this expansion a lot next week.
But at the moment there need not be an edge to the universe in order for all
distances to grow. All we have is a bunch of galaxies that
are receding from each other, and beyond them are probably more galaxies that are
receding even faster from where we sit, and no matter which galaxy you sit on,
everything recedes at a rate proportional to its distance, because all distances
grow by the same factor. This is the symmetric picture.
Okay, so Hubble teaches us that all distances are increasing.
Right from there, you have a big discovery.
Look, this can't have been going on forever.
Because, if in the future, distances are going to bigger, as I said in the past,
they are going to be smaller. And, farther in the past they were
smaller yet, And farther in the past they were smaller yet.
At, oh, and at some point in the past you can solve the equation that says D(t) =
0. The distance is zero.
Which distance? It doesn't really matter. If I make this factor zero, all distances
are zero. All of the galaxies that we see and the
ones that are too far for us to see.
What is that time well solve the equation, one plus H times t minus t0 is
zero, move the one over to there you find that H t minus Hto equal.
To -1, and so what does that tell me? That tells me dividing by H, I call that
H0 ^ -1 and cross out the H's over here and I find that t = t0 - H0 ^ -1.
Wait, so H0 ^ -1 is a time? I thought H was something in a kilometer per second
per megaparsec. Well, once you remember that a megaparsec
and a kilometer are both distance, you realize that you can convert one to the
other. And indeed, kilometer per second per
megaparsec is one over seconds times some small number or megaparsecs per kilometer
per second The units of Ho inverse are indeed the units of time.
So what is this time? This is the so called Hubble Time.
It's very important. This is the time where if you imagine if
this equation governs the universe, this is the time in the past of go, the time
in the past in which everything was on top of everything else.
So let's compute it, the 100 gets converted to .01.
H converted to H inverse, this is the inverse of a kilometer per second per
megaparsec. A megaparsec is how many kilometer's?
Well a megaparsec is this many Au. And then Au is this many kilometer's,
except, I also, while I was at it, Introduced the mega parsecs.
So, there's 10 ^ 11, this is how many, a mega, a mega parsec is and the seconds
come out. I plug in all the numbers, I get this
number. If I allow multiply through, I get 9.8h
inverse, plug in the most Recent value for H of .71 I get finally my result for
the Hubble time, 13.8 billion years. What is this? Well this is the maximum
amount of time the universe can exist because before that, at that time
everything was on top of everything else and I can't go farther back.
[UNKNOWN]. So we have a finite age for the universe,
given Hubble's expansion. And this is our rough and ready estimage
for how old the universe is. It turns out to be surprisingly good.
We will need to improve it. It will turn out that this precise
behavior is a Newton approximation to some More complicated time dependents of
distances, that'll be the topic of next week.
But first, we can get even more out this story.
Remember that the red shift, z, determines distance, at least for small
z, where the Newton Approximation holds, as, v/ c is 1 +, z/ c is z Remember that
V/C was supposed to be Z and V/C is therefore H0/C, D/C and so D is ZC/H0.
So if you know the red-shift you can figure out the number, the distance.
This, allows you to figure out something else though.
Someting called the look back time. If a galaxy is a distance d away, then,
what I am seeing is the light that left that galaxy a while ago, and the farther
the galaxy is, the longer the light has taken to get here.
The Andromed galaxy is 2 1/2 million light years away.
We see the Andromeda galaxy as it was 2 and a half million years ago.
The farther an object is the farther back in time we see it.
The look back time is simply the time that the light was emitted is t0 which is
our notation for the present minus D over c, this is how long it takes light at the
speed c to cover that distance. PLUGGING IN THE VALUE FOR D WE GET THAT
THE TIME WE SEE A GALAXY ACT IS T ZERO MINUS Z TIMES EIGHT ZERO INVERSE.
AGAIN Z IS THE FRACTION OF THE HUBBLE TIME IF YOU WANT A FRACTION OF THE AGE OF
THE UNIVERSE, THAT HAS PASSED SINCE THE LIGHT WAS EMITTED FROM THIS GALAXY.
[INAUDIBLE] us another exciting way to think about things.
let's think about it this way. So, remember that I'm going to compute
the wavelength at which light was emitted / the wavelength at which I'm going to
observe it. This is kind of backwards to the usual
calculation of the Redshift, the redshift is lambda observed divided by lambda
emitted, so this is actually 1+z, inverse.
it'll be clear why I'm doing it this way. Now I'm going to assume that I'm in the
small And that, reasonably small redshift neighborhood of non relativistic motion
so that this expression is correct, first of all and second of all that because I
can use Newton's approximation because z is much less than 1, then I can use
Newton's approximation to approximate 1 + z to the -1.
As one minus z. Ok, and then from here i have an
expression for minus z which says that this is equal to one minus to get minus z
i take h zero times minus z is plus H 0 times T emission minus T 0.
Oh, but I know what 1 plus H 0 times T minus T 0 is.
Remember that our expression for the Hubble change in distances, was D of T,
is D of T 0. Times 1 + H zero T - T zero.
Oh good, so this is exactly D at the time of emission, for any distance / that
distance at T zero. Okay.
So what do we have? That, the wavelength at the time of omission, / by the light
wave length that we observe, is the, which is 1 - Z approximately is the same
as distance at the time of emission, and distance now.
Which distance? Doesn't matter. They all scale by the same factor.
Write that out cleanly. And now you see, that it's telling me
something, What is this? This is a distance, the
wave length. This is telling me, that the wavelength
over the time between emission and observation or absorption has scaled the
same way any other distance would. So the red, Hubble's redshift formula is
basically telling me that the wave length of a light wave grows along with the
distances between galaxies. That's a very strange way to think about
it. We'll see next week that it's very
natural, this is thinking of the redshift, of the redshift not as a
Doppler shift but as a cosmological shift.
The wave lengths of light stretch just like any other distance, we'll come back
to this next week.