Stellar Parallax

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Stellar Parallax : Another version of the stellar parallax introduction

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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.