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Hi gang, and welcome back to Analyzing the Universe.

Today I want to talk to you about measurement of distances.

Easy, right? You just take a ruler,

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But what about if you can't even get to the other place?

Or if the distances are so vast, that ordinary rulers are impractical?

Such are the problems we have when we try to measure the distances to the stars.

We need some sort of stellar bootstrap in order to extrapolate

our earthly distance measurements into the realm of the cosmos.

We begin, as we must, with our home, the Earth.

And as usual, our story begins with the Greeks.

And, in particular,

Aristarchus of Samos around 250 BC.

What you see on your screen right now is Aristarchus's working diagram by which he

concluded that the Sun, on the left, was much bigger than the Earth in the middle.

And hence, was most likely to be at the

center of the Solar System rather than our planet.

Let's dissect this diagram carefully, and see how he did it.

Our dilemma is that all we can measure is the angular

size of an object in the sky, and sometimes not even that.

This means that the size of an object is ambiguous.

Let's consider this drawing. Here, we have the Earth.

And we imagine that we're looking out into space with a certain angular size.

You see that the Sun, or the Moon, or any object with the

same angular diameter can be either close and small,

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and still appear to be the same size in the sky.

So our angle here is the same, but the object, depending on

where it is, can be either large or small.

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Now, Aristarchus knew the angular sizes of the Sun and the Moon were the same.

Otherwise we could never have a solar eclipse where

the Moon almost exactly covers the surface of the Sun.

He also knew that the distance to the Sun

was much bigger than the distance to the Moon.

How did he know this? By realizing that the times from first

quarter to third quarter of the lunar cycle was almost

the same as from third quarter to first quarter.

Let's look at this carefully.

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We imagine that the Earth is here, and that the

Moon, is going around the Earth in a circular orbit.

Let's make that circle just a little bit better, huh?

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we have the Sun, here's the Earth,

and for the quarter Moons what has to happen is

the Moon has to make a 90 degree angle between the Earth and the

Sun. So, if we have the Moon over here,

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this will be a 90 degree angle and we will see the first quarter

of the Moon. Now, when the Moon is

down, oh about here, we'll have another right angle

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from third quarter to first quarter. Now,

note that as the distance to the Sun increases, the

differences between the two arcs of the circle, this

arc and this arc become smaller.

Let's see what happens. If we put the Sun

much, much further away, somewhere off to the,

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And the 90 degree angle will be

formed, something like this, in

a way that will start making this arc

almost exactly the same as the other one.

So, as the distance to the Sun increases, these arcs

become more equal. This is so ingenious, no?

Although Aristarchus's result was not particularly accurate, it was good enough

to realize that the Sun must be much farther away from the Earth than the Moon.

His value was 19 times further. Thus, the Sun must

be 19 times larger since the angular extent

was the same as the Moon in the sky.

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With the Sun much further away from the

Earth than the Moon, the angle of the Earth's

shadow is about the same as the angular size of the Sun and the Moon in the sky.

Let's look at that part carefully.

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Here is the shadow that must

be cast behind the Earth due to the fact that the Earth

has, more or less eclipsed the Sun, for any object

that is in this region over here.

So, this theta two is the

angular size of the Earth's shadow.

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And you can see that if the Sun is very far away, the angular

size of the Sun,

which is

theta 1, is almost

the same as this angle in here, the angular

size of the Earth's shadow. Notice that these objects

are not drawn to scale since the angles depicted are for clarity

of understanding, much greater than the one half degree that the

objects actually appear as in the sky. Now,

as the final step, let's look at the Earth's shadow

in detail during a lunar eclipse. Here

we have the Earth, and here we have the Earth's shadow.

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is over here, all the way, a long way

away. The Moon in the sky must be

within the Earth's shadow. And how is that going to look?

Well, we know that the angle that the Moon has in the sky is

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And the Moon can be anywhere in

this region of the sky.

But where do we put the Moon? And here is

where the observation of the Moon during a lunar eclipse

comes in. Because we observe that the time

it takes for the Moon to traverse the Earth's shadow,

let's get that shadow depicted like this.

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Is about 8 3rds of the time it takes for the Moon to

move its own diameter in the sky. So the Moon's

size must be about 3 8ths the

size of the shadow. So the only place we can put

the Moon in here to

meet the requirements of the data, the requirements of the observation,

is such that the Moon's

size, right here.

Is 3 8ths of the size of

the entire diameter of the Earth's

shadow, which we can delineate as the line A, A prime.

Okay? We know these angles.

They're about half a degree. We can put the Moon anywhere

in this cone and still have it have the right angular size.

But only when we put the Moon, well, I

probably didn't put it in exactly the right spot.

If we put it a little bit further on, it'll

match a little bit better, but you get the idea.

There's only one place that the Moon can fit so that it is in the proper

proportion of the Earth's shadow in size. So now, we have

the relative sizes of the Sun, the Earth, and the Moon

but in terms of the Earth's diameter, we still don't

know how big the Earth is. This

problem was solved, also ingeniously, by Eratosthenes

about 50 years later, around 200 BC.

To understand how he did this, we have to realize that the Sun is so far away,

that essentially, all the rays that arrive at the Earth are parallel.

Let's imagine a light source near the Earth.

If the Earth is here,

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and you put a light source over here, the rays from that light

source will diverge like this to the top and bottom of the Earth.

If you put the object a little further

away, the rays, don't diverge quite as much.

If you put the object over here,

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the rays diverge even less. And if you put the objects, such as the

Sun, so far away that you can't even really tell the

difference between these rays, all of the rays that will come in to the

Earth are going to be essentially parallel.

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Now,

Eratosthenes noted that at Syene, Egypt, which is

now the modern city of Aswan, on the

first day of summer, light at noon from

the Sun struck the bottom of a vertical well.

So that meant that Syene was on a direct

line from the center of the Earth to the Sun.

The picture looks like this.

Here's the surface of the Earth. Here's the

center of the Earth. And at this point, if

this is the position of

Syene on the Earth,

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this line represents not only

the zenith direction at Syene, but

also the position of the Sun

in Syene. At

the corresponding time and date in Alexandria, which was

5,000 stadia north of Syene, the Sun was

slightly south of the zenith. So, its rays made an angle of about

seven degrees to the vertical. So, here's

the vertical in Alexandria

pointing this way. So, this is

the zenith in Alexandria.

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And that makes an angle of seven

degrees to the Sun. Okay,

here is an angle theta,

that because we have gone along the surface of the Earth, about

5,000 stadia.

The distance of Alexandria from Syene is 5000 stadia,

and at that position, the angle that the Sun makes with the zenith direction

in Alexandria is about seven degrees.

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Here we have Syene. Here we have Alexandria.

And since the Sun's rays are essentially parallel,

the angle of seven degrees between the solar direction and the zenith

is the same as if, this seven

degrees was subtended from the center of

the Earth. Now you see, ingeniously, that this

5,000 stadia can be extended to measure

the circumference of the Earth, because we know this angle that

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is subtended by the circle, right over here.

So, you can see that, this angle theta is

to 360 degrees as the distance

to Alexandria from

Syene, is to the circumference

of the Earth.

Right? Here is a segment of a circle.

D is to the whole circumference of the Earth, as

seven degrees is to the whole 360

degrees, that makes the circle complete.

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Thus, the circumference

of the Earth must be about 50

times 5,000 stadia or about

250,000 stadia. Seven into

360 is about 50, right? So,

250,000 stadia must be the circumference of the Earth.

But what was a stadium?

Was it a Fenway Park stadium? Was it a Yankees stadium or what?

There is actually much debate about this.

But there is no doubt that Eratosthenes got very close.

Ranging from 80% to 99% of the

true value for the Earth's circumference.

So now we have a crude estimate of the distance to our nearest star, the Sun.

A significantly better estimate was not forthcoming until

the invention of the telescope almost 2,000 years later.

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The various ingenious experiments designed to

measure this elusive number are fascinating

to study, and more than just of academic interest.

For our knowledge of the distances to the remote stars, which are so far

away that we cannot even measure directly their angular diameters, depend

crucially on our ability to perform measurements in our own backyard.

Namely, to determine the distance to the closest star, our Sun.

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On the surface, it would appear that

the situation seems hopeless for even greater distances.

I mean, it's almost a miracle that we can determine the solar distance.

How can we possibly extend our reach to the stars?

Well, let's do a little experiment.

Hold your finger up in front of your eyes, and blink your eyes alternately.

Just like this.

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This represents the orbit of the Earth. So the Earth

in June might appear over here, relative to the Sun.

And the Earth in December might appear here

in its orbit. And a nearby star

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relative to the background that might exist,

populated by other more distant objects. The

nearest stars then, should move relative to the backdrop of the further stars.

But the distances involved are so great, relative to the diameter of the Earth's

orbit, that changes over the six month span as the Earth traverses opposite

sides of its path, are positively miniscule.

Indeed, they are so small that many of the ancient Greeks used the lack of measurable

parallax to conclude that the Earth was really at the center of the solar system.

Aristarchus, himself, was forced to admit that if the Earth

really did orbit the Sun, the distances to the stars

must be vast, indeed. It wasn't until 1838

that the first stellar parallax was successfully measured.

The displacement was less than 2 3rds of an arc second.

To give you some idea of how small that angle is, let's

imagine a golf ball. If you place

this ball about six miles or ten kilometers away from you,

it would subtend an angle in the sky of one arc second.

No wonder it was so difficult to measure this.

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Okay? And here is the radius of the Earth's

orbit. We define the parallax in terms of the

radius of the Earth's orbit instead of the diameter.

But the idea is basically the same. Notice that if the star gets further away,

this angle P prime,

I'd better not call it a prime, because you'll think that that's an arc minute.

P1 and P2, P2 is definitely smaller than P1.

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So, the smaller the parallax, the greater the distance.

Indeed, we can define a new unit of distance, by

D equaling 1 over

the parallax that is measured. So this distance

here, in terms of the parallax

is defined as 1 over P.

And if P is measured in arc seconds, this

distance defines a unit of distance

called the parsec. If P is 1 arc

second, the distance is 1 parsec.

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Unfortunately, only the nearest few hundred stars have measurable parallaxes,

at least from the ground. Can we ever hope to get measurements of

more distant objects? Amazingly, fortuitously,

there are a class of very bright stars called Cepheid

variables that pulsate with different periods depending

on their intrinsic brightness or luminosity.

What a stroke of good forture.

This means that just by measuring how long it takes for the brightness of these stars

to change, we get for free, a measurement of their intrinsic luminosity.

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What we see here, in the following picture, shows the light curve of delta-

Cephei, and the period-luminosity relationship for many

similar stars. And the fact that they are so bright, with

some being over 10,000 times the luminosity of the Sun, means

that they are visible out to very, very far distances.

About 30 mega-parsecs, or 30 million

parsecs, but wait a second, I hear you cry.

Don't you need, at least initially, an

independent measurement of the distances to these

objects, in order to figure out what their luminosity is in the first place?

And right you are.

So the story, while fascinating, is not that simple.

But we will touch upon this matter in the coming lectures.

Which not only will allow us to use ordinary stars to determine distances,

but also provide us with fundamental data concerning the nature

of stellar evolution, and the role that this plays in our understanding of the

incredibly hot X-ray sources in our galaxies and beyond.