Originally Posted by
chornedsnorkack
See my above post:
I use small angle approximation twice.
Designating as A the point behind Proxima, I approximate the triangle A-Proxima-Alpha as a right triangle. Not an exact shape, because the Alpha-Sun-Proxima angle is nonzero. But close enough for the estimation of errors.
Now, taking the external error of Sun-Proxima distance relative to Alpha, I designate as B the point behind Proxima which is the furthest possible possible location of Proxima.
Then I draw a normal from B to Alpha-Proxima line and designate the crossing as C.
Since I above approximated Alpha-Proxima-A as a right triangle (right angle at A), note that Proxima-B-C is a similar right triangle (sharing the angle A-Proxima-Alpha, and with right triangles at A and C). Therefore, I can use the above established distances Proxima-Alpha, Proxima-A, A-Alpha, and the uncertainty distance Proxima-B, to compute the remaining sides Proxima-C (and B-C)
Using small angle approach for the second time, I approximate Alpha-B as equal to Alpha-C. Therefore the distance Proxima-C is the uncertainty of distance Alpha-Proxima.