Last evening I was laying out on my deck gazing at the sky. It was just past dusk and the stars were becoming visible. Almost directly overhead I spotted three stars forming a triangle. (I know, I know any three points will form a triangle)
Two of the stars were about 5 degrees apart. My question: Assuming that the stars are about the same distance from Earth, what would be the distance from one star to the other? Light minutes or light days or light years?
If D is the distance to the stars then they would be 5piD/180 apart in the same units as D.
The exact answer is 2Dsin(angle/2). The formula in the above post works fine for angles like 5[sup]o[/sup]
Dave I am pretty sure that gives you the arc length of a circle with radius D.
The distance between the stars should be 2*sin(2.5 degrees)*D.
Great minds and so on and so forth.
I think my angular size calculator would come in handy here:
http://www.1728.com/angsize.htm
Let’s suppose the stars are 5 degrees apart and the distance is 20 light years, then their distance from each other would be 1.75 light years.
Certainly, though, what you were seeing was Saturn (the brighter of the three) next to the twin stars of Gemini, Castor and Pollux. Castor and Pollux are 4 1/2 degrees apart, but Castor is 52 lightyears away and Pollux is only 32 lya. That makes them 20 lightyears apart, no matter how close they seem to be.
Saturn was 6 1/2 degrees from Pollux, but it is only a little more than a lighthour away. 
I don’t know about this method. My trig teacher would have had a tizzy!
Funny, in the trig I was taught, you’d NEVER use degrees to determine arclength! I think you’re getting the straight-line “equivalent” approximating the length of arc. And, for small angles, it’s probably nit-picky.
So, shouldn’t you be using radians to find arclength?
- Jinx
Multiplying degrees by the fraction pi/180 converts to radians.
but then you’d just multiply that by D
That is, 5 degrees times pi/180 times D
But, that 5 degrees is off by more than the error in using the sine function, and as I mentioned the stars are not at the same “D”
All I can do is point to my first post in the thread which contains the equation. And point out that the OP said to assume the stars were at “about the same distance” from the observer.
Jinx
But in this case we are not calculating arc length.
Let’s suppose there is a circle 12 inches in diameter. The 2 points at the ends of the diameter are 12 inches apart.
If we viewed this circle “edge-on” from a distance of 50 feet, the diameter would have an angular size of 1.15 degrees. And the distance between the 2 points is still 12 inches.
The arc length between the 2 points would be 18.85 inches.
We are not concerned with the arc length in this type of problem so the value of 18.85 inches is meaningless.
Just so the OP can decide whether or not he wants to look up sines or just use the angle for small angles -
For the 5[sup]o[/sup] of separation at distance D the difference between computing arc length by the formula in my first post and the exact answer by the formula in my second or treis’ first post is 3.17380306291026 parts in 10000 according to my Mathcad.
For angles out to 8.8[sup]o[/sup] the difference is less than 1 part in 1000.
Erhm no it doesn’t. By my calculation they are 20.255 light years apart.
That’s quite a quibble. (Especially since the distances he quoted probably aren’t precise to within 1/1000th of a light year.)
The point, I think, was that close objects in the sky are likely to vary widely in their distance from us, and that these differences probably account for more of their mutual seperation than the angular distance that we can see.
20.255 lightyears is 20 lightyears, in that case
Bah with that attitude I hope you’re the one aiming the super death at Earth. You’ll miss the entire solar system 
What we need are more sig fig debates.
Nah… I’ll just realize that for that kind of application, I’ll probably need more precise figures than earth astronomers have about distant stars, at the moment.
My ‘attitude’ is that sometimes precision is important, and sometimes it ain’t.
Aren’t we in a pretty sparsely endowed region of the galaxy? I recall that the nearest star to the sun id Barnard’s Star (4.5 LY distant?). In the center of the galaxies, are stars a lot closer together? That is where most likely, interstellear travel is taking place.