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# Stellar parallax

## Video transcript

Let's say I'm walking
along some trail and there are some trees
on the side of the road. And let's just say
these are some plants. And these are the
barks of the trees-- 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. 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 than the mountains. Like I'll just be going
past one tree after another. And they'll just whizz past
me-- maybe if I'm running. But the mountains don't seem
to be moving that quickly. And this idea, that as
you change your position the things that
are closer to you seem to move more than the
things that are further from you, 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 close to you are whizzing by you, maybe
the curve of the street or whatever, while the
things that are further away don't seem to be
whizzing by you 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, really far away. But to think about
how this is done, how we use stellar parallax--
just 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 this is the North
Pole, kind of popping out of the screen here. And so the Earth is
rotating in that direction. And I also want to think
about a 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-- well, first of all, the Earth would
be this unnoticeable dot here-- but you would also, whatever
distance this is, you would have to multiply
that 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 if we're thinking
about North America, we're 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 is
essentially at sunrise. So let me do my best at drawing
the sun from my point of view. Remember, the Earth is
rotating in this way-- the way I've drawn it, it's
rotating counterclockwise. But from the
surface of the Earth it would look 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? So if you look at this
version of the Earth, the star is 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
is 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. And if you think about it, if we
go back to this patch of Earth, now the Earth is still
rotating in 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 will make clear that now we're about 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'll 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,
relative to straight up, it is now a little bit to the
west now, a little bit more on the side of that setting sun. So the star would now look
like it is right there. And if we have good
enough instruments, we can measure the angle
between where that star was six months ago and
where it is now. And let's call that
angle 2 times theta. And the reason why
I call it 2 times theta-- 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 actually even basic geometry,
if you say this is a right angle 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,
we could say, look, we need a trigonometric
function that deals with 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 some basic
trigonometry-- and you might want to do
this if you forget the basic trigonometric
functions-- SOHCAHTOA. Sine is opposite/hypotenuse. Cosine is adjacent/hypotenuse. Tangent is opposite/adjacent. So the tangent function
deals with the two sides of this right triangle
that we can now deal with. So we could say that the tangent
of 90 minus theta, this angle right over here--
let me write it-- is equal to the
opposite side-- is equal to x over the
adjacent side, over d. Or another way, if you assume
that we know the distance to the sun, you multiply both
sides times that distance. You get d times the tangent of
90 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 would need to have
very good instruments even to measure-- even to
observe-- the stellar parallax to the nearest stars. And we're constantly
having better instruments-- and actually the
Europeans right now are in the process of a mission
called Gaia to measure these to 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 give
us a very accurate map of a significant chunk
of our galaxy, which is about 100,000 light
years in diameter.