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How do you find the distance from a star/planet/black hole to another? I know people can calculate the distance from Earth to a star, but what about from one to another?

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A method used to calculate the distance between stars is 3 Dimensional trigonometry. Here we treat stars as points in a 3 dimensional plane instead of 2 dimensional plane as we use in traditional trigonometry.

The co-ordinates of the stars are denoted by the parameters ascension and declination, just like we refer a location on earth with latitude and longitude. The 3rd parameter is the distance is denoted by R, the radius vector (the line drawn from eye to the star to measured).

We imagine the surface to be drawn on the infinitely large sphere as a grid with our eye at the center and the star at a distance R. When then calculate the coordinates for both the stars and find their difference .

Then the distance is obtained from the formula:

enter image description here

Source: http://www.neoprogrammics.com/distance_between_two_stars/

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If you know distance from Earth to both objects and the angle between them viewed from Earth, it is just a matter of trigonometry.

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That's true if the distance are less than few millions of ligthyears. Otherwise you have to account for cosmological effects and possible cosmic curvature –  Francesco Montesano Dec 4 '13 at 9:22
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@FrancescoMontesano It is still trigonometry, just not Euclid's one. –  Envite Dec 4 '13 at 9:29
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its 3 dimensional trigonometry just almost similar to what we use, but far more complex when considering the fact that you are measuring distance of two objects in the vast universe and not on a sheet of paper. –  Mahe Dec 4 '13 at 12:38
    
The key with cosmological distances is using the appropriate distance measure depending on the purpose of the measurement. Transverse comoving distance can be used to measure distances that take the expansion of the Universe into account and don't vary with time. –  Aaron Jun 17 at 22:42

You only need two-dimensional trigonometry if you know the distances to the two stars and their angular separation. Any two dimensional plane can be defined by three points that lie on it so we just use the plane containing the two stars and the Earth.

You can use Earth as the origin and the closest star as a point on the X axis (X1,Y1) where X1 is the distance and Y1 is zero. You then can use the distance of the second star and it's angular separation from the first star (which is on the X axis) to plot a point (X2 = distance*cos(angle), Y2 = distance*sin(angle). The distance between those two points is SQRT((X2-X1)^2 + Y2^2):

enter image description here

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This does not apply to large distances, as noted by Francesco. –  called2voyage Dec 5 '13 at 21:47
    
Please explain more, I don't see why since space has been shows to be flat over most of the universe and the 'cosmological effects' aren't explained. The fact that it does not explain exactly does not make it invalid just like we can still use Newton's equations for approximations and ordinary tasks instead of invoking general relativity (they got us to the moon). –  Jason Goemaat Dec 5 '13 at 22:52
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When dealing with stars in the milky-way or even galaxies in the local group you are right, and simple flat-space (euclid) works just fine. However, for the very long distances you have to account for light travel time. The universe is expanding and the rate of acceleration has changed in the past. Objects at very different (large) distances can then be thought of as being set at a different age of the universe and therefore at a different sized universe. This is what enters in the equation as 'cosmological' effects. –  user494 Dec 9 '13 at 11:03
    
I don't think that was in the question. Also, if that is the case then you have to take into account all motion of the star. For instance SDSS J091759.5+672238 have moved 400 light-years since the light we are seeing left it. I also don't hear assumed movement taken into account since there is no universal 'now' in relativity. For instance I always hear the furthest galaxies are about 13 billion light years away, not 26 billion. –  Jason Goemaat Dec 9 '13 at 17:40

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