Stellar parallax

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.