Professor Charles
Bailyn: Welcome to part
three of the course.
This is going to be about
cosmology.
One of the most amazing things
that's happened over the past
half century or so is that
cosmology,
which is the study of the
Universe as a whole,
has become a scientific
subject,
and something that one can say
something about in scientific
terms, rather than merely
philosophical terms.
In recent years,
in the past ten years or so,
there has been kind of a
revolution in cosmology,
which has come about because of
the discovery that the vast
majority of the Universe is made
out of stuff that we have no
idea what it is.
So, the discovery is,
we have no idea what the
Universe is made of.
And that actually,
you know, it doesn't sound so
good.
It sounds like this is a big
failure of science,
somehow.
But scientists are clever,
so we describe it differently.
What we claim is that what has
happened is that we have
discovered dark energy.
Dark energy, what is that?
We don't have a clue,
but it's most of the Universe.
And we've discovered that it
exists.
So, that's kind of where we are
right now.
What I want to do over the next
four or five weeks is talk about
the discovery of dark
energy--how that was done.
And now that we know that it's
there, but we don't what it is,
what people intend to do about
it.
And this is,
I should say,
only a relatively small
fraction of modern cosmology.
If you want to know about many
of the other interesting things
that are going in cosmology,
you have to take a whole
course, or several courses,
on this topic.
Such courses exist.
I recommend Astro 170 or 220 if
you find yourself interested in
this kind of thing.
But nevertheless,
this one particular
discovery--and,
in fact, what I'm going to be
focusing on is not just dark
energy,
but one particular way that
dark energy--the first way.
There are now other indications
that exist.
One particular way in which
dark energy has been discovered.
And so, it's a fairly narrow
focus that I'll be taking here,
but that will enable us to get
into some depth about how this
discovery was made.
So, let me let you in on the
secret before we even begin.
Dark energy is something that
you can't see.
That's sort of its name.
We've done this twice already,
right?
We've discovered planets around
other stars--that you couldn't
see the planets.
How do you do that?
You look at the star--the
motion of the star.
We've discovered the existence
of black holes which,
of course, you can't see.
How do you do that?
You look at the motion of some
star that is influenced by the
existence of the black hole.
So, what are we going to do
about dark energy?
Well, this isn't hard to
extrapolate.
What we're going to do is we're
going to look at the motion of
things that we can see,
and infer the presence of dark
energy.
And of course,
how do we look at the motion?
We look at Doppler shifts.
That will be another recurring
theme.
And so, the plan of action here
is actually not so different
from something we've done a
couple of times before,
although the context and the
implications,
and the particular details,
needless to say,
are.
Okay.
So, let's go back in time
almost ninety years,
back to 1920.
So, supposing one were to give
a course in Frontiers and
Controversies in Astrophysics in
the year 1920.
What would one have been
talking about?
Frontiers and Controversies.
The big issue,
at least in terms of cosmology,
was the question of the
so-called spiral nebulae.
These had been discovered over
the previous few decades after
the invention of photography,
allowed pictures of the sky to
be created.
And as people took pictures of
the sky, what they discovered
was, scattered all over the sky
are these little spiral clouds,
the so-called spiral nebulae.
What are these things,
they wondered.
And, in particular,
where are these things?
And there were two basic
hypotheses.
One was that these are,
sort of, clouds--shining clouds
of gas.
These were known to exist in
other kinds of shapes.
So, these are clouds of some
kind of glowing gas,
and that they are part of our
so-called galaxy.
Galaxy comes from the Greek
word for milk.
It's basically Greek for Milky
Way.
The Milky Way is this band of
stars across the sky.
It looks like just a white
streak.
You can't see it in the city
with city lights.
You know, if you've never seen
this, go somewhere dark.
Wait until the Moon goes down.
Get your eyes adapted and then
look for the Milky Way.
It's really very spectacular.
If you really want to do this
right, do it in the southern
hemisphere, because the center
of our galaxy is down there.
It's quite spectacular.
Anyway, it had been known since
Galileo's time that what the
Milky Way really was,
was a band of stars.
And so--that there's this huge
bunch of stars out there,
so faint, so packed together,
we see it just as a continuous
band of light.
And so, all these stars--that's
our galaxy.
And the question was whether
these spiral nebulae might be
little clouds of something or
other scattered around our
galaxy.
The alternative,
less popular at the time,
but perhaps more spectacular,
is that the spiral nebulae are
galaxies themselves--are whole
galaxies of stars,
different from our own galaxy.
These were sometimes referred
to as island universes.
So, the Milky Way is our own
galaxy, and each one of these
tiny little spiral things is a
galaxy itself,
located much,
much further away,
obviously, than any of the
stars that we can see.
And you can see that this is a
question of some importance,
in particular,
to how big you think the
Universe is.
If all these spiral nebulae are
part of our own galaxy,
then maybe what the Universe
is, is one galaxy.
And at that time,
it was already starting to be
known how big the galaxy was,
and so forth.
But if each one of them is its
own galaxy, then obviously,
the entire Universe must be
much,
much bigger,
because it contains thousands,
perhaps millions,
of individual galaxies,
each of them more or less like
our own.
So, we know what the answer's
going to turn out to be.
This is correct [points to
"island universes" on slide
notes].
But they didn't know that in
1920.
And, in fact,
they staged what was called the
Great Debate in 1920,
between a very famous
astronomer named Harlow Shapley,
who was at Harvard.
And he maintained that the
spiral nebulae must be part of
our own galaxy.
And so, he had many good
reasons to think this.
There was a lot of evidence
that the spiral nebulae couldn't
be that far away.
They must be really quite
nearby.
And so, Shapely had,
in the true Harvard manner,
all of the right arguments,
and was completely wrong.
And his opponent,
a man named Curtis,
educated at Yale,
but at that point working
somewhere else,
turned out to be entirely
right, but totally lost the
debate because he just didn't
have that much evidence backing
him up.
So, the intuition of the Yale
man comes through.
So, this is a famous incident
in the history of science.
It garnered enormous attention
at the time.
Remember, this is a year after
the eclipse expedition has
verified Einstein's Theory of
General Relativity,
so people are pretty excited
about modern science at the
time.
And this was reported widely in
the press.
So, this is another one of our,
sort of, science fables,
here, "The Great Debate."
And one version of the moral of
this story might be,
you can have all the right
arguments and still be
wrong--have many good arguments
and still be wrong,
as Mr.
Shapley turned out to be.
Interestingly,
at the time,
in 1920, they thought,
you know, it's going to take us
a long time to really get to the
truth of this,
and turned out not to be the
case.
Within just a few years after
the debate had taken place,
this problem was totally
solved.
And the man who solved it was
perhaps the greatest
observational astronomer of the
twentieth century--a man named
Edwin Hubble,
after whom a telescope was
subsequently named.
And what helped solve this was
simply better equipment.
And, in fact,
starting kind of a trend that
has continued for a century now,
what happened was-- So,
Hubble was out in California,
where there are nice,
clear skies,
and persuaded a rich man to
give a lot of money to build a
really big telescope.
This was a 100" telescope.
That's the diameter of the--the
diameter of the mirror.
This is actually still in use,
although the site is no good
because it's way too near to the
lights of San Francisco.
But he took this brand new
telescope.
He looked at the spiral nebulae.
And he did the same thing to
the spiral nebulae that Galileo
had done to the Milky Way.
He resolved it into individual
stars.
And he could see the individual
stars in these spiral nebulae,
and discovered in the nearest
of them that they were made up
of stars.
And then, he noticed that the
stars that were making them up
were incredibly faint.
And that even the brightest
stars in these things were
incredibly faint.
And so, he inferred the
distance to these things by
assuming that,
you know,
they're kind of ordinary stars,
like any other kind of stars,
except they're way,
way fainter,
and therefore they must be much
further away.
And this was compelling
evidence that these spiral
nebulae really were island
universes--galaxies like our
own.
And the key thing to do--that
is, to have a telescope powerful
enough to resolve some of the
nearby examples into individual
stars.
So, there are many galaxies.
This was known shortly after
the Great Debate took place.
And that made the Universe very
much larger than people had
previously suspected,
because there are all these
galaxies around.
So, Hubble was the great expert
at observing galaxies.
And so, the next thing he
decided he would look at
is--okay, now we know these
things are galaxies,
let's check out their-- you
know, he took Doppler shift
measurements.
He's trying to figure out the
motions of galaxies.
And so, he discovered a very
strange thing.
Namely, that they are all going
away from us.
Every galaxy he measured was
redshifted.
And so, by the usual Doppler
shift interpretation of these
things, they must all be moving
away from us.
And Hubble went further,
and he created one of the most
famous diagrams in all of
astronomy,
in which he plots
redshift--which is to say
Z, which is to say radial
velocity divided by the speed of
light,
at least in the Newtonian
approximation,
which he was working in,
versus distance.
So, he figured out how far away
these galaxies were.
He measured their redshift.
And what he discovered is that
these things were very tightly
correlated.
So, if you plot each galaxy,
measure a distance and a
redshift for each galaxy,
for a large number of galaxies,
what you find is that they line
up like this.
This is the so-called Hubble
Diagram, and it's basically the
thing we're going to be talking
about for the next five weeks or
so--is Hubble Diagrams,
and what you can infer from
them.
And the Hubble Diagram can be,
sort of, summed up in an
equation.
If this is a straight line,
then it must be true that the
velocity of a given galaxy is
equal to a constant,
which was given the letter
H, for the Hubble
Constant, times the distance.
This is Hubble's Constant.
And that just basically says in
algebra what this says
pictorially, that these things
line up in a straight line.
The Hubble Constant is an
extremely important number.
You measure it by creating
these Hubble Diagrams and just
measuring the slope of the line.
And the primary scientific
purpose of the Hubble Space
Telescope, the largest and most
expensive science project ever
created,
was to get an accurate
measurement of the Hubble
Constant.
So, all these beautiful
pictures you see are byproducts.
The purpose of the thing was to
measure the Hubble Constant
accurately.
There had been,
for many years,
a dispute over what the correct
value was.
This was resolved by the Space
Telescope and other things
within the past decade.
And this was a big success.
We now know what the Hubble
Constant is.
It's measured in somewhat weird
units.
It's seventy or so kilometers
per second, per megaparsec
[Mpc].
All right.
Now, let's pause there for just
a moment.
You can see why they use this
peculiar unit.
Because you want to--they're
measuring velocity in kilometers
per second, and they're
measuring distance in
megaparsecs.
Mega, of course,
doesn't just mean big.
It's a technical term.
It means a million.
It means 10^(6) parsecs.
A parsec, you will remember,
is about three light years,
or 3 x 10^(16) meters,
or 3 x 10^(13) kilometers.
And so, for each megaparsec
that a given galaxy is away--and
notice, now, we've changed our
units from parsecs,
which we used for stars,
to megaparsecs.
That's basically the change in
the scale of the Universe that
came about when people realized
that the spiral nebulae were
galaxies.
So, now you have to talk about
megaparsecs.
In fact, now we also talk,
from time to time,
about gigaparsecs,
which are billions of parsecs.
For each megaparsec that a
galaxy is away from us,
it moves 70 kilometers per
second more away from us.
So, something that's 10
megaparsecs away ought to be
moving 700 kilometers per
second.
Something that's 100
megaparsecs away would be moving
7,000 kilometers a second away
from us.
And that's just an expression
of what this relationships is.
Hubble, by the way,
got this quite wrong.
He correctly lined them all up,
but his value for the Hubble
Constant turned out to be about
500,
in these units,
which was quite wrong,
and we'll talk about that--why
that was, later on.
But, by now,
with the Hubble Space
Telescope, we totally know this,
and that's an important result.
Now, this diagram and this
relationship are basically the
key to what we're going to be
talking about.
So, in fact,
the fact that there's this
linear relationship tells you
several things.
It tells you that the Universe
is expanding.
And because of that,
it tells you that
the--indirectly,
there's other evidence that has
to be brought to bear,
as well, that the Big Bang
exists.
So, this is the basis of Big
Bang cosmology.
I don't want to talk about that
right now.
We'll talk about the Big Bang
on Tuesday.
We'll have a whole big
question-and-answer session,
so, bring all your questions
about the Big Bang and
cosmology.
We'll do that all up on Tuesday.
What I want to talk about now
is much more mundane.
Namely, how do you measure
these points?
How do you do this?
How do you create this diagram?
This diagram has these fabulous
implications,
but in order to understand
what's going on,
you'd better know how you get
the diagram in the first place.
Now, part of it's clear.
The x-axis we understand
fairly well.
Measuring radial
velocity--that's easy.
That's just the Doppler shift.
So, that comes directly out of
a particular kind of measurement
that you can readily do.
The y-axis is the
biggest problem in all of
astronomy;
namely, how do you measure the
distance to something?
And, if you think about it,
that's what this Great Debate
was all about.
Some people thought that the
distances to these spiral
nebulae were a few hundred,
or maybe a few thousand
parsecs.
Other people thought they were
millions of parsecs away,
and it wasn't clear which was
right.
And if you think about looking
up into the sky,
looking at stars in the sky at
night, you can see where there's
a problem.
You look up there.
You see a bunch of stars,
and suddenly one of them shoots
across the sky.
It's a shooting star.
Except for the motion across
the sky, they don't look that
different, and yet,
some of them--the fixed stars,
are hundreds of light years
away.
The planets,
which look much the same to the
naked eye are,
you know, a million times
closer.
And that shooting star is in
the top of the atmosphere.
And it's very hard to tell what
the distance is when you just
kind of look up there.
And so, distance is hard.
And that's basically the
problem I want to talk about for
the rest of today's class.
How do you measure distances?
Yes?
Student: You said that
redshift was easy,
but how do we know what the
original wavelength is?
Professor Charles
Bailyn: Oh,
how do you know what the
original wavelength was?
You make the assumption that
atomic physics and chemistry are
the same in the distant galaxies
as they are here,
and therefore,
hydrogen should emit lines at
the same frequency as they do in
the atmosphere.
Now, one of the things you can
do if you don't like what the
cosmological implications of
things is,
you can say,
well, rather than interpreting
this as cosmology,
let's just say that in distant
portions of the Universe,
physics is different.
If you go down that road,
you can get any answer you
want.
And this has been seriously
discussed from time to time.
All right.
So, how do you measure distance?
That's the key.
Well, there's one,
and only one,
kind of, direct way you could
imagine doing this.
Here's the Sun.
Here's the Earth going around
the Sun.
Here's a nearby star.
And here are a bunch of other,
much further away stars.
They're all scattered around
the sky.
And you observe this nearby
star during the course of the
year.
As the year goes on,
you observe it repeatedly.
You observe it from here,
for example.
And when you observe it from
here, it's in this direction,
and therefore,
it appears to be in that
position relative to the other
stars.
Whereas, when you observe it
here, it's in a little bit of a
different position with respect
to the other stars.
It looks like it's here.
And so, if you observe this
star repeatedly over the course
of the year, it appears to move
back and forth against the
background of the other stars.
This is a triangle.
We know what to do about
triangles.
You can measure this angle
because that's just the angular
separation of the two apparent
positions of this star in the
sky.
We know this distance.
That's 1 Astronomical Unit,
because this is the Earth going
around the Sun.
And then, what we want to know
is the distance to this star,
D.
Let's call that
D_1.
And we know the equation for
this already.
We've done this before.
This is α =
D_2 /
D_1,
where D_2,
in this case,
is exactly 1 Astronomical Unit.
And you may recall that if you
measure this in arc seconds,
and you measure this in
parsecs,
and you measure this in AU,
you get a consistent set of
answers.
And so, the way it works out is
1 / α, in arc seconds,
equals the distance in parsecs.
And the reason that works is
because D_2,
by this method,
is always equal to 1
Astronomical Unit.
If you try this on Jupiter,
you have to account for the
fact that Jupiter's further
away.
This is called the parallax
method.
It's common in surveying.
You know, you look at the same
thing from two different places.
You can figure out distances by
basically trigonometry,
here.
This is called the parallax
method.
And a parsec--this is the
definition of a parsec--is one
parallax second.
It's a contraction.
Because you measure an arc
second and the distance is one
parsec.
And that is the definition of a
parsec.
That's why we use parsec as a
distance measurement,
because it comes naturally out
of this parallax method.
And so, that's a fairly direct
geometric method.
This is a surveying technique.
This is how--and this is a
straightforward way of getting
the distance.
And the problem with this is
that it only works on things
that are really nearby.
Because we can measure,
you know, maybe,
a hundredth of an arc second
change in position,
but no better than that.
And so, you can only get
measurements of distances in
this way out to a few hundred
parsecs.
Works to a kind of maximum,
given our current
instrumentation of a few hundred
parsecs.
But the center of our galaxy is
8,000 parsecs away.
These other galaxies are
megaparsecs away.
We can't be measuring 1
one-millionth of an arc second,
at least not at the moment,
in terms of parallax.
So, this only works for the
very nearest stars.
And that's why there was all
this confusion about the spiral
nebulae because,
you know, if they were 500
parsecs away,
you'd never be able to tell.
So, most other distance
measurements--methods of
distance--are some form of
what's called the Standard
Candle Method.
So, here's how the Standard
Candle Method works.
It's a three-step process.
Part one, you look at something.
You know how bright something
is--how bright something is.
This is basically a version of
what Hubble did when he figured
out that the spiral nebulae must
be very,
very distant,
and must, therefore,
be island universes.
You know how bright something
is.
In the case of Hubble,
he's looking at the brightest
stars in the galaxy.
He figures they're about as
bright as the brightest stars in
our own galaxy,
so, he knows how bright they
are.
Part two, you measure how
bright the object looks.
You take a picture of it,
or you count photons,
or whatever it is that you do,
and you figure out how bright
it looks.
And obviously,
for something of a given
brightness, the further away it
is, the fainter it looks.
And we know exactly how that
works.
It's a distance-squared thing.
If it's twice as far away,
it's a fourth as bright.
If it's three times further
away, it's 1/9th as bright.
It goes as one over the
distance, squared.
This is a well-known fact.
You can try this out with light
bulbs at home.
And so, if you know how bright
it is, and you measure how
bright it appears to be,
then you can compute the
distance.
Okay.
And this is why it's called a
standard candle--oh,
so, the way this is--how do you
know this?
That's the big question.
How do you know how bright the
thing is in the first place?
And the answer is,
usually, that you're looking at
something which is an example of
a class of objects,
like stars, or bright stars,
whose brightness is known.
And that's why you use the
term, standard candle.
Because here's a bunch of
things.
They're all the same brightness.
They all have the same standard
candle power.
Some of them look fainter than
others, but ones that look
fainter obviously have to be
further away.
And if you know the true
brightness of this class of
objects, you can figure out how
far away any example of it is.
Hence, standard candles.
And the problem with this is
this phrase here,
as you can pretty clearly see.
If you get it wrong--if you
make the wrong assumption about
how bright these objects are,
you're going to screw this up
completely, and that's what
Hubble did.
Hubble was looking at a
particular kind of bright star,
which he thought he knew how
bright it was,
and he was wrong.
And so, he got the wrong Hubble
Constant.
Now, because he used the same
kind of star in all his
galaxies, he got it the same
amount wrong for all these
different galaxies.
So, they still lined up.
They just lined up along the
wrong track.
So, it was still true that
something that he thought was
twice as far away as something
else was in fact twice as far
away as something else.
He just got both of those
numbers wrong by the same
factor.
Okay.
This brings us to the awkward
question of, "How do you measure
brightness?"
And now, we have to talk about
one of the great impediments to
learning astronomy--namely,
the magnitude scale.
Astronomers count brightness
upside down and logarithmically.
And I am now obliged,
by my membership in the
astronomical community,
to inflict this upon you.
So, the magnitude scale--this
is how we measure brightness.
It's upside down and
logarithmic.
And the key numerical
relationship that works looks
like this.
If you subtract the magnitude
of one object from another,
that equals minus five halves
times the log of the ratio of
the brightnesses [-5/2 log
(b_1/
b_2)].
Don't panic.
The magic word,
here, is "help sheet," which
will be posted later this
afternoon.
So, this is a key equation.
So, let me just define the
terms.
This is the magnitude of two
different objects and this is
the brightness of the two
objects.
Whereby "brightness," I mean
something sensible,
like how many photons per
second do you get from them,
or some other kind of true
measure of how bright they are.
Now, this is a somewhat awkward
equation, because it's a
relative equation.
It doesn't tell you what the
magnitude of either one of these
things is.
What it tells you is the
magnitude of one compared to
another.
So, in order to figure out the
magnitude of something,
you have to know the magnitude
of something else.
And so, you need one other
piece of information to have
this be useful,
which is that they have defined
a particular star to have
magnitude 0.
The star, Vega,
is defined to have magnitude 0.
So, if you start with Vega,
you can figure out the
magnitude of any other given
object.
Actually, this is now causing
trouble, because it turns out
that in the far infrared,
Vega is variable.
And that's kind of unfortunate
for the basis of the whole
magnitude system to turn out to
be variable.
But, they've coped with that in
various ways.
All right.
Let me pause and remind you of
some things about logarithms,
which you may have forgotten.
What's the definition here?
The logarithm of 10^(x)
= x.
That's the definition of a
logarithm.
So, for example,
the logarithm of 3 x 10^(2) is
equal to the logarithm of
10^(1/2) that's 3 times 10^(2),
is equal to the logarithm of
10^(2.5).
Because when you multiply
10^(x) by 10^(y)
you get 10^(x) + y,
which is equal to 2.5.
That's an example.
More examples on the help sheet.
Just, in general,
log (10^(x) x
10^(y)) = x +
y.
Because when you add those
together, that's how it works.
log (10^(x) /
10^(y)) = x -
y.
Again, because of the way
logarithms multiply together.
And the logarithm of--let's
see, 10^(x),
raised to the mth power,
is equal to mx
[log([10^(x)]^(m)) =
mx].
This, by the way,
is why logarithms are so
incredibly useful.
You should always do all your
arithmetic in logarithms.
You should just automatically
convert everything in your head
into logarithms.
In fact, in the days before
calculators, this is how people
used to do arithmetic.
This is how slide rules work,
I should say.
They mark the thing off
logarithmically and then you
move them back and forth.
And people, you know--if you
memorize, like,
ten logarithms of a few
convenient numbers,
you can do all sorts of
calculations in your head
because multiplication turns
into addition,
which is much,
much easier to do.
And so, if you can convert
things into logarithms in your
head, then all you have to do is
add numbers.
That's easy.
Similarly, taking something to
the mth power,
for example,
taking the square root of a
number,
or taking the cube of a number,
or something like that,
reduces down to multiplication.
It's hard to take a square root
of a number in your head,
but it's easy to divide a
number by two,
which is the equivalent of
taking a square root.
So, if you're thinking in
logarithms, all you got to do is
divide it by two,
and you can amaze your friends
by doing square roots in your
head--if your friends are the
kinds of people who are amazed
by that kind of thing.
Mine are.
And so, I recommend this to you.
If you have to prove that
you've learned something in
college, spend a half hour
memorizing ten different
logarithms and then just blow
people's minds by taking square
roots on bets.
Okay.
Or you can do problem sets in
Astronomy 160.
For example,
Sirius--the star Sirius,
which is the brightest star in
the sky, is about three times
brighter than Vega.
So, what's its magnitude?
What's its magnitude?
Well, let's write down the
equation.
Magnitude of Sirius minus the
magnitude of Vega is equal to
-5/2 log of the brightness of
Sirius, divided by the
brightness of Vega.
Now, any time you see three
times brighter or twenty times
fainter, or anything of that
kind, what you're really talking
about is a ratio.
If Sirius is three times
brighter than Vega,
that means the brightness of
Sirius divided by the brightness
of Vega is three.
That's what that statement
means.
And so, we can just plug it
right in.
-5/2 log (3).
Because Sirius is three times
brighter than Vega,
so that ratio is equal to 3.
What's the log of 3?
One half. Thank you very much.
This is equal to - 5/2 log
(10^(1/2)).
You will remember that the
square root of 10 is 3.
So, 3 is also equal to π,
but for logarithms,
the important thing is that 3
is equal to the square root of
10.
And so, this is (-5/2) x (1/2).
-5/4.
Now, the magnitude of Vega,
we know.
That's 0.
So, the magnitude of Sirius is
equal to -5/4.
This is what I mean by the fact
that the scale is upside down.
The brighter the star is,
the lower--or,
if it goes through 0,
the more negative the number
becomes.
Minus 1 is a brighter star than
0.
0 is a brighter star than 2;
2 is a brighter star than 5,
and so forth.
So, low numbers are bright.
The magnitude of the Sun,
obviously extremely bright when
we look at it,
turns out to be -26.5.
And the magnitude of the
faintest star that can be seen
with the Hubble Telescope is
about +30, which is incredibly
faint.
And that, by the way,
is why, again,
why logarithms are such good
news.
Because the entire range of
things we can see in the sky
goes from -26 to +30.
Those are numbers you can get
your mind around.
In fact, the difference in
brightness between those things
is a number that I can't even
pronounce.
And so, much easier to deal
with magnitudes.
All right.
Now, that has to do with how
bright things are,
how bright things appear to be,
or how bright they are.
But remember what we're trying
to do.
We're trying to compare how
bright something is,
intrinsically,
to how bright it looks.
And this gets you into the
question of--there are actually
two kinds of magnitude.
The intrinsic brightness of an
object is something called
absolute magnitude,
whereas the observed brightness
is referred to as apparent
magnitude.
And now, the astronomers screw
you up again,
because we have different
symbols for the apparent
magnitude and the absolute
magnitude.
The absolute magnitude is given
a symbol capital M,
and the apparent magnitude is
given the symbol small m.
And the problem is that you
can't actually tell the
difference between those two in
my handwriting,
and probably yours as well.
So, this leads to untold
confusion, but that's the way
life is.
And the relationship between
these two has to relate to
distance.
Okay.
Before we get there,
the definition of absolute
magnitude.
You'll notice that in the
previous comparison of Sirius to
Vega, I was talking about
apparent magnitude.
How bright it looks in the sky.
Absolute magnitude is defined
as the apparent magnitude if the
object in question were exactly
10 parsecs away.
So, the Sun has an apparent
magnitude, as I said before,
-26.5, but it's actually not
that bright of a star.
If you took the Sun out to a
distance of 10 parsecs,
its magnitude would be 4.7,
which is among the fainter
stars you could see in the sky.
And so, the absolute magnitude
of the Sun is 4.7,
even though its apparent
magnitude is - 26.5.
So, it turns,
for example,
that the star Sirius is about 3
parsecs away.
And so--oh, I haven't told you
an important thing yet.
We'll come back to this in a
second, though.
The problem set problem
obviously is going to be what is
the absolute magnitude of
Sirius, but I have to write down
the formula first.
Little m,
that's the apparent magnitude,
minus big M,
that's the absolute magnitude,
is equal to 5 times the
logarithm of the distance over
10 parsecs.
Now, notice what happens.
If the distance is equal to 10
parsecs, then you've got the log
of 1.
What's the log of 1?
Zero.
Thank you very much.
Because this is the log of
10^(0).
log 10^(0) = 0.
And so, if the log of this
thing is 0, then this right-hand
side is 0.
So, at a distance of 10
parsecs, the apparent magnitude
is equal to the absolute
magnitude,
because m - M =
0, which is exactly what the
definition was.
So, that works.
So, example.
Sirius.
So, what is the absolute
magnitude of Sirius?
All right.
Here we go.
The apparent magnitude,
we figured out just a minute
ago, is - 5/4,
minus the absolute magnitude,
which is what we're trying to
figure out.
5 log (3 /10),
which is 1/3,
which is equal to 5
log(10^(-1/2)).
3 = 10^(1/2);
1/3 = 10^(-1/2), which is - 5/2.
That 5 comes from here.
The -1/2 comes from here.
So, M = 5/2,
let's see, minus 5/4,
is equal to 5/4.
And that is the absolute
magnitude.
I'm sorry.
Oh you want me to do--okay,
sure.
So, we okay to here?
Student: [Inaudible]
Professor Charles
Bailyn: So oh,
oh, okay, fine.
Log of--this is actually an
important point.
Log(1/3), that's what we're
going to try and do,
is equal to log (1/10^(1/2)).
1/10^(n) =
10^(-n).
That's the key thing.
And so, this is the log of 10
to the minus one half
[log(10^(-1/2))]
That, then, means that 5 log
(1/3) = 5 x (-1/2) = - 5/2.
Yes?
Student: [Inaudible]
Professor Charles
Bailyn: There are no
units--yeah, magnitudes are just
numbers.
Yes?
Student: [Inaudible]
Professor Charles
Bailyn: Ah,
well, what I did--sorry,
I skipped a step.
To start from that.
-5/4 - M = -5/2.
So, I multiply both sides by
negative 1, then I subtract 5/4
from both sides.
You got to keep the minus signs
straight.
The easiest way to make
mistakes in this is to get the
thing upside down and lose track
of where your minus signs are,
which is really easy to do,
because the whole scale is
backwards.
So, you know,
-5/4 is bright,
whereas 5/4 is faint.
And so, you're--one is
constantly getting these things
upside down.
Be careful.
Look at your answers to see it
makes sense.
If some incredibly faint galaxy
turns out to have a magnitude of
-50, that's almost certainly
wrong, because that's much
brighter than the Sun.
All right.
So, now.
Now, we can actually do the
Standard Candle Method.
Here's the key problem.
If you observe a star like
Sirius and you--I don't know,
you take its spectrum,
or some alien comes down and
tells you this star is just like
Sirius, or however you work this
out.
If you observe a star like
Sirius and measure its apparent
magnitude to be 8.75,
how far away is it?
And now, we're back to where we
were twenty minutes ago,
before I started all this
nonsense.
Namely, we're trying to measure
the distance of something,
which was the whole purpose,
as you may recall.
So, how far away is it?
Let's see.
m - M = 5 log
(D /10 parsecs).
The apparent magnitude is 8.75.
The absolute magnitude,
we just figured out,
is 5/4--1.25--is equal to 5 log
(D / 10 pc).
And you'll notice I've chosen
my numbers carefully,
because this is going to work
out well.
This is 7.5,
and I'm going to divide both
sides by 5;
7.5 / 5 = log (D / 10
pc).
Or, 1.5 = log (D / 10
pc).
Now what do I do?
Yeah, exactly.
You have to take 10 to the
power of both sides.
Whenever you're stuck with log
of something and you don't want
the log of the something,
you want the actual something
itself, what you got to do is
10^(1.5) = 10^(log
(D/10pc)).
Ten to the log of anything is
equal to itself.
So, this is D / 10
parsecs.
And thus, what's 10^(1.5)?
Student: Thirty.
Professor Charles
Bailyn: Thirty,
yes.
10^(1.5) = 10^(1) x 10^(1/2) =
3 x 10^(1) = 30.
So, 30 = D / 10 parsecs.
And D is equal to 300
parsecs.
And for the person who asked
about units, which was a very
good question,
this is where the units come
back.
It's because there's this 10
parsecs embedded inside the
equation.
And so, the unit of length
comes back here.
And so, that's how you measure
distance.
You know how bright something
is, probably by having looked at
some other, more nearby example.
You measure how bright it looks.
You compare those two things,
and out pops the distance.
So, this is done,
usually, in the form of what's
called the distance ladder,
which we will talk about much
more in section.
So, you'll get many more
opportunities to do this.
Nearby stars,
you measure the distance in
parallax--from the parallax
method.
Then you find examples of
similar stars.
You measure the apparent
magnitude.
You assume the absolute
magnitude to be the same as the
absolute magnitude for things
you already know the distance
to.
You assume the absolute
magnitude.
And you compute the distance.
This, then, gets you new--this
tells you that you learned from
this, the absolute magnitudes of
brighter things.
Not just stars,
but whole galaxies,
supernovae, all sorts of
things--brighter things,
which you can then measure
further away.
And just to conclude,
you can see pretty clearly that
this method is fraught with
potential problems.
Because every time you go
through this--you know,
that's a swear word in science.
You're making an assumption,
and that assumption can lead
you astray.
And so, the whole history of
cosmology, since 1925,
is in that word.
Assuming the absolute magnitude
of things for which you're
trying to measure the distance,
so you can put them on the
Hubble Diagram,
so that you can deduce the rate
of expansion of the Universe,
and thus its age and ultimate
fate.
So, it all rests on this little
point embedded inside the
Standard Candle Method,
which will be discussed at
great length.
Okay, that's it for today.
