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In the last video we talked about how parallax is the apparent change

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in position of something based on your line of sight.

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And if you experience parallax kind of in your everyday life

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if you look outside of your car window while its moving you see that

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nearby objects seem to be moving faster than far away objects.

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So in the last video we measured the apparent displacement of the star at different

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points in the year relative to straight up.

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But you can also meausre it relative to things in the night sky

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at that same time of year that same time of day that don't

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appear to be moving. And they won't appear to be moving because they are

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way way way farther away than this star over here there may be

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other galaxies or maybe even clusters of galaxies or who knows.

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Things that are not changing in position, so that's another option.

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And that's another way of making sure you are looking at

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the right part of the universe.

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So you could measure relative to straight up if you know

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based on the time of year or time of day that you are looking

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at the same direction of the universe.

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Or you could just find things in the universe that are way

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far back that their apparent position isn't changing.

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So just to visualize this again

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I'll visualize it in a slightly different way.

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Let's, let's say this is the night field of vision.

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Let me scroll to the right a little bit. Let's say our night field of vision looks like this.

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I'll do it in a dark color because it's at night.

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So your night feild of vision looks like that.

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And let's say that this right over here is straight up.

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This right over here is striaght up.This right here is if

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we are looking straight up in the night sky.

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And just to make the convention, in the last video

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I changed our orientation a little bit.

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I'll reorientate us in kind of a traditional orientation.

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So, if we make this North, this is South, this will be West

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and this will be East.

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So if we look at the star in the summer

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what will it look like?

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Well, first of all the Sun is just begining to rise.

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So, if you can think about it, this North

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this North direction, we are looking at the sun

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from, we are looking at the Earth from above

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so the North will be the top of this sphere over here.

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And the South will be the bottom of the sphere, the

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other side of the sphere that we're not seeing.

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The east will be this side of the sphere, where

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the Sun is just begining to rise.

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right over there.

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So what will be the apparent position of the star?

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Well it's going to be towards the East, it's going

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to be towards the direction that the Sun is rising.

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So this angle right over here will be right

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right over there.

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So this will be the angle.

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So this will be the angle theta.

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So this is in the summer. And what about the winter?

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Well in the winter, in order for straight up to be that

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same point in time, or that same direction in the universe

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I should say, then the Sun will just be setting.

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So we're rotating that way, so we're just going to be

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capturing the last glimpse of sunlight.

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So in that situation, the sun is going to be setting.

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So this is our winter sun, our winter sun, I'll do that

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in a slightly different color.

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The sun will be setting on the West.

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And now, the apparent direction of that star is going to be in

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the direction of the Sun again, but it's going to be

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shifted away from center. So it is going to be

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to the right of center, sorry, to the LEFT of center.

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So it's going to be right over here, it is going to be

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right over here. And it's a little bit unintuitive the way

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I drew it. In the last video, well I won't make any judgement on the way

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the last one is easier to visualize or this one is.

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Over here, I just wanted to make the convention that North is

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up and South is down. But I just want to be clear,

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over here the Sun is, well the sun always sets to the West.

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So in the Winter, the sun will be right over there, this will be shifted

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from center in the direction of the Sun, so it will

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be at an angle theta just like that.

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in the winter.

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Now, that's all review from the last video. I just

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reorientated how we visualized it. What I want

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to do in this video, given that we can measure theta, how can

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we figure out how far this star actually is.

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So, let's just think about it a little bit before I actually

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give you a theta value.

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If we know theta, then we know, we know, that his angle is

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right over here because this is a right angle. We're

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going to know that this angle right over here is 90 degrees

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minus theta. We also know, we also know the distance from

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the Sun to the Earth. And let's say we're just going to approximate

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here, its 1 astronomical unit (AU) it changes a little bit

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over the course of a year, but the mean distance is 1 AU.

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So we know the angle, we know a side adjacent to the angle

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and what we're trying to do is figure out a side opposite

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the angle, this distance right here.

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The distance from the sun to the star.

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And this is of course a right trianle.

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And you can see it right here, this is the hypotenus.

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So now, we just need to break out some relatively

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basic trigonometry

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so if we know this angle, what trig ratio deals with an adjacent

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side and an opposite side?

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So let me right now, my famous sohcahtoa, I didn't come up

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with it so, the famous sohcahtoa.

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Soh-cah-toa.

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Sine is opposite over hypotenus. Those aren't

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the two we care about.

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Cosine is adjacent over hypotenus, we don't know

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what the hypotenus is and we don't

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care about it just yet. But the

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tangent is the opposide over the adjacent.

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Opposite over the adjacent. So if we take the tangent of

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the angle, if we take the tangent of 90 minus theta,

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if we take the tangent of 90 minus theta, this

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is going to be equal to the distance to the star.

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This distance right over here.

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The distance to the star, or the distance to the Sun to the star.

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We can later figure out the distance from the Earth to the star,

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it's not going to be too different, because the

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star is so far away.

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But the distance from the sun to the star dividied by

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the adjacent side, divided by 1 astronomical unit.

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And I'm asumming everything is in astronomical units.

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So you can multiply both sides by 1, and you'll get the

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units in astronomical units. The distance is equal to

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the tangent of 90 minus theta. Not too bad.

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So let's figure out what a distance would be

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based on some actual meaurments.

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Let's say you were to measure some star

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measure this change in angle right here.

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And let's say you got the total change in angle

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right over here, from 6 months apart the

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biggest spread, and you're making sure you're

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looking at a point in the universe relative to straight up.

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You can do it other ways, but this is just simplifing

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our visualization and simplifies our math.

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And you get to be 1.5374 arcseconds.

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And I want to be very clear,

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this is a very very very very small angle.

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Just to visualize it, or another way to think about it is

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there are 60 arcseconds per arcminute,

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and there are 60 arcminutes per degree.

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Another way to think about it is a degree is like

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an arc hour. So if you want to

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convert this to degrees you have 1.5374 arcseconds

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times 1 degree divided by 3600 arcseconds.

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The dimentions cancle out. And you get this

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equal to, let's get the calculator out, this is being

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equal to, 1.5374/3600, so it's 4.206, I'll round, because we

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only want 5 significant digits.

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This is an infinite precision right here, because

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that's an absolute quantity, it's a definition.

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So let me write this down.

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So this is going to be 4.2707 x 10^-4 degrees,

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you can write it just like that.

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Now, let me be clear, this is the total, this is the

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total angle, this angle that we care about is going

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to be half of this, so

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we could divide this by 2,

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let's just do just our significant digits,

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4.2706 x10^-4 divided by 2, is going to be

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2.1353 x10^-4. So that's this angle right over here.

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This angle, or this shift from center we could vizualize it

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is going to be 2.135 x 10^-4 degrees.

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So now that we know that, we already figured out

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how to find the distance, we could just

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apply this right over here.

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So let's just take, let's just take the tangent

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which make sure your calculator is in 'degree' mode,

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I made sure about that before I started this video,

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tangent of 90 minus this angle right here (2.1359x10^-4)

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so instead of retyping it, I'll just type the last answer

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So 90 minus this angle, and we get this large

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number, 268326. Now remember, what were our units?

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This distance right here, this distance right here

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is 268326, I should just round because I only have

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5 significant digits here.

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Although with the trig, trig number of significant digits

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get a little bit shaddier. But I'll just write

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the whole number here, 268326 astronomic units.

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So it's this many, it's this many distances between

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the Sun and the Earth.

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Now, if w wanted to calculate that into lightyears,

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we just have to know, and you can calculate this

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multiple ways, you could just figure out how far

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an AU is versus a lightyear. But, there are

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so this is AUs, 1 lightyear is equivalent to 63,115AU,

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give or take a little bit.

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So this is going to be equal to, AU's cancle out,

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this quantity divided by that quantitiy is lightyears.

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Let's do that, so let's take this number that we just

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got divided by 63,115 and we have it in lightyears.

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So it's about 4.25 lightyears.

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I'm messing with the significant digits here, but

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just a round about answer, 4.25 lightyears.

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Now remember, that's about how far the

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closest star to the Earth is. And so the closest

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star to the Earth has this very very very apparent

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a very small change in angle. You can imagine

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as you go farther and farther stars from this,

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that angle, this angle right here is going to

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get even smaller and smaller and all the way

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until you get really far stars and it would be even with

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our best instruments you wouldn't be able to measure that.

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angle. Anyway, hopefully you found that cool

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because you just figured out a way to

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use trigonometry in a really good way to measure angles

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in the night sky, with the night sky, to actually figure out

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how far we are from the nearest stars.

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I think that's pretty neat.