Stellar Distance Using Parallax

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Stellar Distance Using Parallax

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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.