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- Let's say I'm walking along some trail and there's some trees on the side of the road,
- and let's just say that these are some plants and these are the barks of the trees,
- and maybe I should do it in brown, but you get the idea.
- These are some plants that are along the side of the road,
- or at least the stem of the plant or the bark of the tree.
- And in the background I have some mountains,
- maybe those mountains are several miles away,
- and so I have some mountains in the background.
- We know just from experience that if I'm walking
- (let me draw myself over here)
- we know that if I'm walking this way,
- the trees look like they're going past me much faster,
- much faster than the mountains.
- Like if I'm going past just one tree after another then
- they'll just whiz pasts me, maybe if I'm running
- but the mountains don't seem to be moving that quickly
- And this idea that if you change your position,
- the things that are closer to you seem to move more
- than the things that are further than you
- This idea, or this, I guess, this property, is called parallax.
- And what we're going to do
- in this video, and maybe it's especially obvious if
- you're driving in a car, then the things closer to you
- are whizzing by you, maybe the curb of the street
- or whatever, while the things that are further away
- don't seem to be whizzing by as fast
- What I want to do in this video is think about how
- we can use parallax to figure out how far certain stars are.
- And what I want to emphasize, is that
- this method is only good for relatively close stars.
- We don't have instruments sensitive enough yet
- to use parallax to measure stars that are really, really, far away
- But to think about how this is done, how we use
- stellar parallax (so let me write stellar up here)
- How we use stellar parallax, the parallax of stars
- to figure out how far away they are
- Let's think a little bit about our solar system
- So here is our Sun in the solar system
- And here is Earth at one point in the year
- And what I want to do is, and let's just say that
- this is the north pole kind of popping out of the screen right here
- and so the Earth is rotating in that direction
- And I also want to think about the star that is
- obviously outside of our solar system
- and I'm really underestimating the distance to this star
- As we'll see, or as you might already know
- the distance to the nearest star from our solar system
- is 250,000 times the distance between the earth and the sun
- So if I wanted to draw this to scale, first of all
- the Earth would be this unnoticeable dot here,
- but you would also, whatever this distance is
- you would have to multiply this by 250,000 to get
- the distance to this nearest star.
- Anyway, with that said, let's think about what
- that star would look like from the surface of the Earth.
- So let me pick a point on the surface, maybe we're
- talking about North America, or right there in the
- Northern Hemisphere.
- So let's take that little patch of land, and think about
- how the position of that star would look.
- So that's the patch of land, maybe this is my house
- right over here, jutting out the side of the Earth
- maybe this is me standing, I'm drawing everything sideways
- because I'm trying to hold this perspective, so
- this is me looking up.
- And, let's say at this point in time, the way I've drawn
- this patch, the Sun will just be coming over the horizon.
- So the Sun, this is essentially at sunrise. So let me
- do my best to drawing the Sun from my point of view.
- It looks like, remember the Earth is rotating in this way
- it's rotating, the way I've drawn it, it's rotating counterclockwise.
- But from the surface of the Earth, it would like the
- Sun is coming up here, it's rising in the East.
- But right at that dawn, on this day, when the Earth
- is right over here, what would that star look like?
- Well the star, so if we look at this version of the Earth
- these stars are kind of skewed a little bit.
- Not straight up, straight up would be this direction
- from the point of view of my house. It is now skewed
- a little bit closer to the Sun, so if you go in this
- zoomed in version, straight up would look something
- like that. And maybe, based on my measurement, it
- would look like the star is right over there. So it's
- a little bit skewed towards where the Sun is rising,
- towards the East, relative to straight up. Now, let's
- fast forward six months, so that the Earth is on the
- other side of its orbit from the Sun. So let's fast
- forward six months. We're over here, and let's wait
- for a time of day, where we are essentially, that
- little patch of the Earth is pointed in the same direction
- at least in our galaxy, maybe. So it's pointing in the
- same direction, and if you think about it, if we go back
- to this patch of Earth, now the Earth is still rotating
- on that direction, but now the Sun is on the West, the
- Sun is going to be right over here, maybe I'll do it
- like this, just to make it clear. I'll draw this side
- of the Sun with this greenish color, obviously the Sun
- is not green, but it'll make clear that the sun is going
- to be over here. The patch is going to be turning away
- from the Sun, so it'll look to that observer on Earth like
- the Sun is setting, so it will look like the Sun is going
- down over the horizon. But the important thing is, once
- we're at this point in the year, what will that star
- look like? Well if we have this large diagram, we see
- that the star is now, relative to straight up, a little
- bit to the West now, a little bit more now on the side
- of that setting sun. So the star would now look like
- it's right there. And if we have good enough instruments,
- we can measure the angle between where the star was
- six months ago, and where it is now. And let's call
- that angle, well I'll call that angle, two times theta.
- And the reason why I call it two times theta, we could
- call that angle relative, we could call theta the angle
- between the star and being straight up. So this would
- be theta, and that would be theta. And I care about that,
- because if I know theta, and if I know the distance from
- the Earth to the Sun, I can then use a little bit of
- trigonometry to figure out the distance to that star
- Because if you think about it, this theta right over
- here is the same as this angle. So if this is straight up
- that is looking straight up into the night sky, this
- would be the angle theta. If you know that angle,
- from basic trigonometry, or even basic geometry, if
- you say this is a right angle right over here, this
- would be 90 minus theta. And then you could use some
- basic trigonometry. If you know this distance right
- here, and you're trying to figure out this distance
- the distance to that nearest star. So this is what we're
- trying to figure out. We could say, "Look, we need
- a trigonometric function that deals with the opposite
- angle, the opposite angle of what we know (we know
- this thing right over here) and the adjacent angle
- (we already know this thing right over here). So let
- me call this the Earth-Sun distance, or let me just
- call this d. And we want to figure out x. So with some
- basic trigonometry, and you might want to do this if
- you forget the basic trigonometric function.
- SOH CAH TOA. Sin is opposite over hypotenuse,
- cosine is adjacent over hypotenuse, tangent is opposite
- over adjacent. So the tangent function deals with the
- two sides of this right triangle that we can now deal with.
- So we can say that the tangent of 90 minus theta, this
- angle right over here, tangent of that angle right over here,
- Let me write it over here, tangent of ninety minus theta,
- that angle right over there, is equal to the opposite
- side, is equal to x, over the adjacent side, over "d".
- Or another way, if you assumed that we know
- the distance to the sun you multiply both side times that distance,
- you get "d" times the tangent of ninety minus theta is equal
- to "x". And you can figure out the distance from our
- solar system to that star. Now I want to make it very,
- very, very clear. These are huge distances. I did not
- draw this to scale. The distance to the nearest star
- is actually 250,000 times the distance to our sun.
- So this angle is going to be super, super, super super small.
- So you need yo have very good instruments even to
- measure, even to observe the stellar parallax to the nearest
- stars. And we're constantly launching, or having better
- instruments, and actually the Europeans right now are
- in the process of a mission called GAIA to measure these
- with enough accuracy that we can start to measure the
- accurate distance to stars several tens of thousands of
- light years away. So that'll start to get us a very
- accurate map of a significant chunk of our galaxy, which
- is about a hundred thousand light years in diameter.

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