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- In the last video we talked about how parallax is the apparent change
- in position of something based on your line of sight.
- And if you experience parallax kind of in your everyday life
- if you look outside of your car window while its moving you see that
- nearby objects seem to be moving faster than far away objects.
- So in the last video we measured the apparent displacement of the star at different
- points in the year relative to straight up.
- But you can also meausre it relative to things in the night sky
- at that same time of year that same time of day that don't
- appear to be moving. And they won't appear to be moving because they are
- way way way farther away than this star over here there may be
- other galaxies or maybe even clusters of galaxies or who knows.
- Things that are not changing in position, so that's another option.
- And that's another way of making sure you are looking at
- the right part of the universe.
- So you could measure relative to straight up if you know
- based on the time of year or time of day that you are looking
- at the same direction of the universe.
- Or you could just find things in the universe that are way
- far back that their apparent position isn't changing.
- So just to visualize this again
- I'll visualize it in a slightly different way.
- Let's, let's say this is the night field of vision.
- Let me scroll to the right a little bit. Let's say our night field of vision looks like this.
- I'll do it in a dark color because it's at night.
- So your night feild of vision looks like that.
- And let's say that this right over here is straight up.
- This right over here is striaght up.This right here is if
- we are looking straight up in the night sky.
- And just to make the convention, in the last video
- I changed our orientation a little bit.
- I'll reorientate us in kind of a traditional orientation.
- So, if we make this North, this is South, this will be West
- and this will be East.
- So if we look at the star in the summer
- what will it look like?
- Well, first of all the Sun is just begining to rise.
- So, if you can think about it, this North
- this North direction, we are looking at the sun
- from, we are looking at the Earth from above
- so the North will be the top of this sphere over here.
- And the South will be the bottom of the sphere, the
- other side of the sphere that we're not seeing.
- The east will be this side of the sphere, where
- the Sun is just begining to rise.
- right over there.
- So what will be the apparent position of the star?
- Well it's going to be towards the East, it's going
- to be towards the direction that the Sun is rising.
- So this angle right over here will be right
- right over there.
- So this will be the angle.
- So this will be the angle theta.
- So this is in the summer. And what about the winter?
- Well in the winter, in order for straight up to be that
- same point in time, or that same direction in the universe
- I should say, then the Sun will just be setting.
- So we're rotating that way, so we're just going to be
- capturing the last glimpse of sunlight.
- So in that situation, the sun is going to be setting.
- So this is our winter sun, our winter sun, I'll do that
- in a slightly different color.
- The sun will be setting on the West.
- And now, the apparent direction of that star is going to be in
- the direction of the Sun again, but it's going to be
- shifted away from center. So it is going to be
- to the right of center, sorry, to the LEFT of center.
- So it's going to be right over here, it is going to be
- right over here. And it's a little bit unintuitive the way
- I drew it. In the last video, well I won't make any judgement on the way
- the last one is easier to visualize or this one is.
- Over here, I just wanted to make the convention that North is
- up and South is down. But I just want to be clear,
- over here the Sun is, well the sun always sets to the West.
- So in the Winter, the sun will be right over there, this will be shifted
- from center in the direction of the Sun, so it will
- be at an angle theta just like that.
- in the winter.
- Now, that's all review from the last video. I just
- reorientated how we visualized it. What I want
- to do in this video, given that we can measure theta, how can
- we figure out how far this star actually is.
- So, let's just think about it a little bit before I actually
- give you a theta value.
- If we know theta, then we know, we know, that his angle is
- right over here because this is a right angle. We're
- going to know that this angle right over here is 90 degrees
- minus theta. We also know, we also know the distance from
- the Sun to the Earth. And let's say we're just going to approximate
- here, its 1 astronomical unit (AU) it changes a little bit
- over the course of a year, but the mean distance is 1 AU.
- So we know the angle, we know a side adjacent to the angle
- and what we're trying to do is figure out a side opposite
- the angle, this distance right here.
- The distance from the sun to the star.
- And this is of course a right trianle.
- And you can see it right here, this is the hypotenus.
- So now, we just need to break out some relatively
- basic trigonometry
- so if we know this angle, what trig ratio deals with an adjacent
- side and an opposite side?
- So let me right now, my famous sohcahtoa, I didn't come up
- with it so, the famous sohcahtoa.
- Soh-cah-toa.
- Sine is opposite over hypotenus. Those aren't
- the two we care about.
- Cosine is adjacent over hypotenus, we don't know
- what the hypotenus is and we don't
- care about it just yet. But the
- tangent is the opposide over the adjacent.
- Opposite over the adjacent. So if we take the tangent of
- the angle, if we take the tangent of 90 minus theta,
- if we take the tangent of 90 minus theta, this
- is going to be equal to the distance to the star.
- This distance right over here.
- The distance to the star, or the distance to the Sun to the star.
- We can later figure out the distance from the Earth to the star,
- it's not going to be too different, because the
- star is so far away.
- But the distance from the sun to the star dividied by
- the adjacent side, divided by 1 astronomical unit.
- And I'm asumming everything is in astronomical units.
- So you can multiply both sides by 1, and you'll get the
- units in astronomical units. The distance is equal to
- the tangent of 90 minus theta. Not too bad.
- So let's figure out what a distance would be
- based on some actual meaurments.
- Let's say you were to measure some star
- measure this change in angle right here.
- And let's say you got the total change in angle
- right over here, from 6 months apart the
- biggest spread, and you're making sure you're
- looking at a point in the universe relative to straight up.
- You can do it other ways, but this is just simplifing
- our visualization and simplifies our math.
- And you get to be 1.5374 arcseconds.
- And I want to be very clear,
- this is a very very very very small angle.
- Just to visualize it, or another way to think about it is
- there are 60 arcseconds per arcminute,
- and there are 60 arcminutes per degree.
- Another way to think about it is a degree is like
- an arc hour. So if you want to
- convert this to degrees you have 1.5374 arcseconds
- times 1 degree divided by 3600 arcseconds.
- The dimentions cancle out. And you get this
- equal to, let's get the calculator out, this is being
- equal to, 1.5374/3600, so it's 4.206, I'll round, because we
- only want 5 significant digits.
- This is an infinite precision right here, because
- that's an absolute quantity, it's a definition.
- So let me write this down.
- So this is going to be 4.2707 x 10^-4 degrees,
- you can write it just like that.
- Now, let me be clear, this is the total, this is the
- total angle, this angle that we care about is going
- to be half of this, so
- we could divide this by 2,
- let's just do just our significant digits,
- 4.2706 x10^-4 divided by 2, is going to be
- 2.1353 x10^-4. So that's this angle right over here.
- This angle, or this shift from center we could vizualize it
- is going to be 2.135 x 10^-4 degrees.
- So now that we know that, we already figured out
- how to find the distance, we could just
- apply this right over here.
- So let's just take, let's just take the tangent
- which make sure your calculator is in 'degree' mode,
- I made sure about that before I started this video,
- tangent of 90 minus this angle right here (2.1359x10^-4)
- so instead of retyping it, I'll just type the last answer
- So 90 minus this angle, and we get this large
- number, 268326. Now remember, what were our units?
- This distance right here, this distance right here
- is 268326, I should just round because I only have
- 5 significant digits here.
- Although with the trig, trig number of significant digits
- get a little bit shaddier. But I'll just write
- the whole number here, 268326 astronomic units.
- So it's this many, it's this many distances between
- the Sun and the Earth.
- Now, if w wanted to calculate that into lightyears,
- we just have to know, and you can calculate this
- multiple ways, you could just figure out how far
- an AU is versus a lightyear. But, there are
- so this is AUs, 1 lightyear is equivalent to 63,115AU,
- give or take a little bit.
- So this is going to be equal to, AU's cancle out,
- this quantity divided by that quantitiy is lightyears.
- Let's do that, so let's take this number that we just
- got divided by 63,115 and we have it in lightyears.
- So it's about 4.25 lightyears.
- I'm messing with the significant digits here, but
- just a round about answer, 4.25 lightyears.
- Now remember, that's about how far the
- closest star to the Earth is. And so the closest
- star to the Earth has this very very very apparent
- a very small change in angle. You can imagine
- as you go farther and farther stars from this,
- that angle, this angle right here is going to
- get even smaller and smaller and all the way
- until you get really far stars and it would be even with
- our best instruments you wouldn't be able to measure that.
- angle. Anyway, hopefully you found that cool
- because you just figured out a way to
- use trigonometry in a really good way to measure angles
- in the night sky, with the night sky, to actually figure out
- how far we are from the nearest stars.
- I think that's pretty neat.

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