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Let's say I'm walking along some trail and there's some trees on the side of the road,

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and let's just say that these are some plants and these are the barks of the trees,

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and maybe I should do it in brown, but you get the idea.

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These are some plants that are along the side of the road,

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or at least the stem of the plant or the bark of the tree.

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And in the background I have some mountains,

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maybe those mountains are several miles away,

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and so I have some mountains in the background.

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We know just from experience that if I'm walking

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(let me draw myself over here)

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we know that if I'm walking this way,

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the trees look like they're going past me much faster,

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much faster than the mountains.

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Like if I'm going past just one tree after another then

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they'll just whiz pasts me, maybe if I'm running

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but the mountains don't seem to be moving that quickly

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And this idea that if you change your position,

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the things that are closer to you seem to move more

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than the things that are further than you

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This idea, or this, I guess, this property, is called parallax.

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And what we're going to do

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in this video, and maybe it's especially obvious if

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you're driving in a car, then the things closer to you

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are whizzing by you, maybe the curb of the street

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or whatever, while the things that are further away

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don't seem to be whizzing by as fast

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What I want to do in this video is think about how

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we can use parallax to figure out how far certain stars are.

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And what I want to emphasize, is that

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this method is only good for relatively close stars.

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We don't have instruments sensitive enough yet

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to use parallax to measure stars that are really, really, far away

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But to think about how this is done, how we use

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stellar parallax (so let me write stellar up here)

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How we use stellar parallax, the parallax of stars

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to figure out how far away they are

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Let's think a little bit about our solar system

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So here is our Sun in the solar system

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And here is Earth at one point in the year

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And what I want to do is, and let's just say that

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this is the north pole kind of popping out of the screen right here

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and so the Earth is rotating in that direction

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And I also want to think about the star that is

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obviously outside of our solar system

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and I'm really underestimating the distance to this star

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As we'll see, or as you might already know

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the distance to the nearest star from our solar system

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is 250,000 times the distance between the earth and the sun

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So if I wanted to draw this to scale, first of all

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the Earth would be this unnoticeable dot here,

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but you would also, whatever this distance is

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you would have to multiply this by 250,000 to get

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the distance to this nearest star.

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Anyway, with that said, let's think about what

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that star would look like from the surface of the Earth.

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So let me pick a point on the surface, maybe we're

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talking about North America, or right there in the

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Northern Hemisphere.

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So let's take that little patch of land, and think about

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how the position of that star would look.

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So that's the patch of land, maybe this is my house

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right over here, jutting out the side of the Earth

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maybe this is me standing, I'm drawing everything sideways

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because I'm trying to hold this perspective, so

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this is me looking up.

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And, let's say at this point in time, the way I've drawn

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this patch, the Sun will just be coming over the horizon.

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So the Sun, this is essentially at sunrise. So let me

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do my best to drawing the Sun from my point of view.

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It looks like, remember the Earth is rotating in this way

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it's rotating, the way I've drawn it, it's rotating counterclockwise.

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But from the surface of the Earth, it would like the

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Sun is coming up here, it's rising in the East.

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But right at that dawn, on this day, when the Earth

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is right over here, what would that star look like?

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Well the star, so if we look at this version of the Earth

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these stars are kind of skewed a little bit.

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Not straight up, straight up would be this direction

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from the point of view of my house. It is now skewed

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a little bit closer to the Sun, so if you go in this

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zoomed in version, straight up would look something

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like that. And maybe, based on my measurement, it

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would look like the star is right over there. So it's

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a little bit skewed towards where the Sun is rising,

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towards the East, relative to straight up. Now, let's

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fast forward six months, so that the Earth is on the

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other side of its orbit from the Sun. So let's fast

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forward six months. We're over here, and let's wait

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for a time of day, where we are essentially, that

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little patch of the Earth is pointed in the same direction

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at least in our galaxy, maybe. So it's pointing in the

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same direction, and if you think about it, if we go back

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to this patch of Earth, now the Earth is still rotating

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on that direction, but now the Sun is on the West, the

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Sun is going to be right over here, maybe I'll do it

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like this, just to make it clear. I'll draw this side

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of the Sun with this greenish color, obviously the Sun

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is not green, but it'll make clear that the sun is going

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to be over here. The patch is going to be turning away

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from the Sun, so it'll look to that observer on Earth like

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the Sun is setting, so it will look like the Sun is going

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down over the horizon. But the important thing is, once

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we're at this point in the year, what will that star

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look like? Well if we have this large diagram, we see

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that the star is now, relative to straight up, a little

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bit to the West now, a little bit more now on the side

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of that setting sun. So the star would now look like

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it's right there. And if we have good enough instruments,

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we can measure the angle between where the star was

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six months ago, and where it is now. And let's call

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that angle, well I'll call that angle, two times theta.

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And the reason why I call it two times theta, we could

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call that angle relative, we could call theta the angle

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between the star and being straight up. So this would

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be theta, and that would be theta. And I care about that,

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because if I know theta, and if I know the distance from

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the Earth to the Sun, I can then use a little bit of

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trigonometry to figure out the distance to that star

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Because if you think about it, this theta right over

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here is the same as this angle. So if this is straight up

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that is looking straight up into the night sky, this

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would be the angle theta. If you know that angle,

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from basic trigonometry, or even basic geometry, if

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you say this is a right angle right over here, this

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would be 90 minus theta. And then you could use some

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basic trigonometry. If you know this distance right

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here, and you're trying to figure out this distance

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the distance to that nearest star. So this is what we're

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trying to figure out. We could say, "Look, we need

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a trigonometric function that deals with the opposite

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angle, the opposite angle of what we know (we know

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this thing right over here) and the adjacent angle

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(we already know this thing right over here). So let

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me call this the Earth-Sun distance, or let me just

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call this d. And we want to figure out x. So with some

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basic trigonometry, and you might want to do this if

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you forget the basic trigonometric function.

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SOH CAH TOA. Sin is opposite over hypotenuse,

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cosine is adjacent over hypotenuse, tangent is opposite

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over adjacent. So the tangent function deals with the

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two sides of this right triangle that we can now deal with.

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So we can say that the tangent of 90 minus theta, this

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angle right over here, tangent of that angle right over here,

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Let me write it over here, tangent of ninety minus theta,

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that angle right over there, is equal to the opposite

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side, is equal to x, over the adjacent side, over "d".

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Or another way, if you assumed that we know

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the distance to the sun you multiply both side times that distance,

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you get "d" times the tangent of ninety minus theta is equal

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to "x". And you can figure out the distance from our

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solar system to that star. Now I want to make it very,

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very, very clear. These are huge distances. I did not

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draw this to scale. The distance to the nearest star

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is actually 250,000 times the distance to our sun.

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So this angle is going to be super, super, super super small.

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So you need yo have very good instruments even to

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measure, even to observe the stellar parallax to the nearest

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stars. And we're constantly launching, or having better

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instruments, and actually the Europeans right now are

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in the process of a mission called GAIA to measure these

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with enough accuracy that we can start to measure the

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accurate distance to stars several tens of thousands of

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light years away. So that'll start to get us a very

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accurate map of a significant chunk of our galaxy, which

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is about a hundred thousand light years in diameter.