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let's say I'm walking along some trail and there are some trees on the side of the road and these are let's just say the deserve are some plants and these are the these are the Bark's 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 so then 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 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 I'll just going past one tree after another and they'll just whiz past me maybe if I'm running but the mountains don't seem to be moving that quickly and this idea that as you [ __ ] as you change your position the things that are closer to you seem to be moved more the things that are further from you this idea or this I guess this property is called parallax parallax and what we're gonna do in this video and maybe it's especially obvious if you're driving in a car then things close to you're 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 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 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 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 I 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 it's 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 under estimating the distance to this star 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 for thinking 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 where 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 at drawing the Sun from my point of view it looks like remember the earth is rotating in this way is rotating it's rotating the way I've drawn it it's rotating it 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 bright 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 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 zoomed-in version straight up would look something like that and maybe based on my measurement it would look like the star it would look like the star is 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 then the earth is on the other side of its orbit from the Sun so let's fast forward six months six months we're over here and let's wait for a time of day where we are essentially the that little patch of the earth is pointed in the same direction in at least in our galaxy maybe so let's 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 Sun 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'll make clear that now we're talking about the Sun is going to be over here the Sun is going to be over here the patch is going to be turning away from the Sun so it looked to that observer on earth like the Sun is setting so it looked 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 we if we have this large diagram we see that the star is now the star is now it's now relative to straight up it is now a little bit a little bit to the west now a little bit more on the side of that Setting Sun so the star would now look the star would now look like it is right there and if we have good enough if we have good enough instruments we can measure the angle between where the Sun where that 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 2 times theta we could call that angle relative we could call theta the angle relative or 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 we can 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 this would be 90 minus theta and then you could use some basic trigonometry if you know this distance 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 well 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 let me call this the earth-sun distance or let me just say let me just call this let me just call this D and we want to figure out X so some basic trigonometry and I know you might want to do this if you forget the basic trigonometric functions so Toa sine 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 could say that the tangent tangent of 90 of 90 minus theta this angle right over here so this angle right over here tangent of that angle right over there let me write it tangent of 90 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 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 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 that 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 to 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 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 a hundred thousand light-years in diameter