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Hi gang, and welcome back to Analyzing the Universe.

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

Â Easy, right? You just take a ruler,

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

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

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

Â We need some sort of stellar bootstrap in order to extrapolate

Â our earthly distance measurements into the realm of the cosmos.

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

Â And as usual, our story begins with the Greeks.

Â And, in particular,

Â Aristarchus of Samos around 250 BC.

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

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

Â And hence, was most likely to be at the

Â center of the Solar System rather than our planet.

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

Â Our dilemma is that all we can measure is the angular

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

Â This means that the size of an object is ambiguous.

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

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

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

Â same angular diameter can be either close and small,

Â 2:34

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

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

Â 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.

Â Otherwise we could never have a solar eclipse where

Â the Moon almost exactly covers the surface of the Sun.

Â He also knew that the distance to the Sun

Â was much bigger than the distance to the Moon.

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

Â quarter to third quarter of the lunar cycle was almost

Â the same as from third quarter to first quarter.

Â Let's look at this carefully.

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We imagine that the Earth is here, and that the

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

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

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we have the Sun, here's the Earth,

Â and for the quarter Moons what has to happen is

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

Â 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

Â of the Moon. Now, when the Moon is

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

Â 5:13

from third quarter to first quarter. Now,

Â note that as the distance to the Sun increases, the

Â differences between the two arcs of the circle, this

Â arc and this arc become smaller.

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

Â much, much further away, somewhere off to the,

Â 5:56

And the 90 degree angle will be

Â formed, something like this, in

Â a way that will start making this arc

Â almost exactly the same as the other one.

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

Â become more equal. This is so ingenious, no?

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

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

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

Â be 19 times larger since the angular extent

Â was the same as the Moon in the sky.

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With the Sun much further away from the

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

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

Â Let's look at that part carefully.

Â 7:56

Here is the shadow that must

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

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

Â that is in this region over here.

Â So, this theta two is the

Â angular size of the Earth's shadow.

Â 8:40

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

Â size of the Sun,

Â which is

Â theta 1, is almost

Â the same as this angle in here, the angular

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

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

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

Â objects actually appear as in the sky. Now,

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

Â in detail during a lunar eclipse. Here

Â 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

Â away. The Moon in the sky must be

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

Â 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

Â this region of the sky.

Â But where do we put the Moon? And here is

Â where the observation of the Moon during a lunar eclipse

Â comes in. Because we observe that the time

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

Â 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

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

Â size must be about 3 8ths the

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

Â the Moon in here to

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

Â is such that the Moon's

Â size, right here.

Â Is 3 8ths of the size of

Â the entire diameter of the Earth's

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

Â Okay? We know these angles.

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

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

Â But only when we put the Moon, well, I

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

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

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

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

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

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

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

Â know how big the Earth is. This

Â problem was solved, also ingeniously, by Eratosthenes

Â about 50 years later, around 200 BC.

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

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

Â Let's imagine a light source near the Earth.

Â If the Earth is here,

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

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

Â If you put the object a little further

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

Â 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

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

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

Â Earth are going to be essentially parallel.

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Now,

Â Eratosthenes noted that at Syene, Egypt, which is

Â now the modern city of Aswan, on the

Â first day of summer, light at noon from

Â the Sun struck the bottom of a vertical well.

Â So that meant that Syene was on a direct

Â line from the center of the Earth to the Sun.

Â The picture looks like this.

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

Â center of the Earth. And at this point, if

Â this is the position of

Â Syene on the Earth,

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this line represents not only

Â the zenith direction at Syene, but

Â also the position of the Sun

Â in Syene. At

Â the corresponding time and date in Alexandria, which was

Â 5,000 stadia north of Syene, the Sun was

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

Â seven degrees to the vertical. So, here's

Â the vertical in Alexandria

Â pointing this way. So, this is

Â the zenith in Alexandria.

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And that makes an angle of seven

Â degrees to the Sun. Okay,

Â here is an angle theta,

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

Â 5,000 stadia.

Â The distance of Alexandria from Syene is 5000 stadia,

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

Â in Alexandria is about seven degrees.

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Here we have Syene. Here we have Alexandria.

Â And since the Sun's rays are essentially parallel,

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

Â is the same as if, this seven

Â degrees was subtended from the center of

Â the Earth. Now you see, ingeniously, that this

Â 5,000 stadia can be extended to measure

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

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is subtended by the circle, right over here.

Â So, you can see that, this angle theta is

Â to 360 degrees as the distance

Â to Alexandria from

Â Syene, is to the circumference

Â of the Earth.

Â Right? Here is a segment of a circle.

Â D is to the whole circumference of the Earth, as

Â seven degrees is to the whole 360

Â degrees, that makes the circle complete.

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Thus, the circumference

Â of the Earth must be about 50

Â times 5,000 stadia or about

Â 250,000 stadia. Seven into

Â 360 is about 50, right? So,

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

Â But what was a stadium?

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

Â There is actually much debate about this.

Â But there is no doubt that Eratosthenes got very close.

Â Ranging from 80% to 99% of the

Â true value for the Earth's circumference.

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

Â A significantly better estimate was not forthcoming until

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

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The various ingenious experiments designed to

Â measure this elusive number are fascinating

Â to study, and more than just of academic interest.

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

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

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

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

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On the surface, it would appear that

Â the situation seems hopeless for even greater distances.

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

Â How can we possibly extend our reach to the stars?

Â Well, let's do a little experiment.

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

Â Just like this.

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This represents the orbit of the Earth. So the Earth

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

Â And the Earth in December might appear here

Â in its orbit. And a nearby star

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relative to the background that might exist,

Â populated by other more distant objects. The

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

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

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

Â sides of its path, are positively miniscule.

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

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

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

Â really did orbit the Sun, the distances to the stars

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

Â that the first stellar parallax was successfully measured.

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

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

Â imagine a golf ball. If you place

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

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

Â No wonder it was so difficult to measure this.

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

Â orbit. We define the parallax in terms of the

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

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

Â this angle P prime,

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

Â P1 and P2, P2 is definitely smaller than P1.

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

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

Â D equaling 1 over

Â the parallax that is measured. So this distance

Â here, in terms of the parallax

Â is defined as 1 over P.

Â And if P is measured in arc seconds, this

Â distance defines a unit of distance

Â called the parsec. If P is 1 arc

Â second, the distance is 1 parsec.

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

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

Â more distant objects? Amazingly, fortuitously,

Â there are a class of very bright stars called Cepheid

Â variables that pulsate with different periods depending

Â on their intrinsic brightness or luminosity.

Â What a stroke of good forture.

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

Â 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-

Â Cephei, and the period-luminosity relationship for many

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

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

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

Â About 30 mega-parsecs, or 30 million

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

Â Don't you need, at least initially, an

Â independent measurement of the distances to these

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

Â And right you are.

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

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

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

Â but also provide us with fundamental data concerning the nature

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

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

Â