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I am currently following a class of observational astronomy lab. I will present a brief description of the method in first and then proceed to the question, so anyone is welcomed to read the entire post or move directly to the problem. The current question is about the Moving-Cluster method, for which I quote from Wikipedia

In astrometry, the moving cluster method and the closely related convergent point method are means, primarily of historical interest, for determining the distance to star clusters. They were used on several nearby clusters in the first half of the 1900s to determine distance. The method is now largely superseded by other, usually more accurate distance measures.*

An image for the Convergence point of a Cluster

One can find more here http://pages.uoregon.edu/soper/Stars/movingcluster.html . A brief description of the method is like that:

From photographs taken, say, 10 years apart, we can see that a star of the cluster has moved. (It has proper motion.) Let us suppose that the proper motion is an angle μ. Then the angle θ is the angle between the convergent point and the star or equally the angle between the radial and tangential speed of the moving star

enter image description here

If one knows the angle θ, he/she will be able to find the movement of the star and in the end, calculate it' s distance, through deduced relations beginning with $ v_t =v_r \tan θ.$

Question I know the declination and the right ascension of both the star and the convergent point.

My problem as is to find the angle θ. In the exercise I have it is requested to find the angle between the star and the convergent point, that is θ , using spherical trigonometry. My mind goes on to using the cosine relationship:

$$ \cos a = \cos b \cos c + \sin b \sin c \cos A ,$$ for the triangle

enter image description here

But, although it may be simple, I cannot understand where this triangle is to be implemented. What are it' s angles in my problem. Wherever I have looked, the angle θ is taken as granted.

So, should I use something like the triangle? If yes, what the triangle should be? Should the Sun be on one angle as the image above with the convergent point? Should I define another triangle? In the end, for whatever triangle I have thought, how will I calculate the needed sides? Note that I know the right ascension and the declination of both the convergent point and the star's.

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  • $\begingroup$ Note: I have also posted this question to physics stack exchange, but with little attention and zero responses. That' 's the reason- besides that this is Astronomy SE- for posting here. $\endgroup$ – Constantine Black May 11 '16 at 10:11
  • $\begingroup$ Crossposting is not usually a good idea. Perhaps you should chose just one of them to keep? $\endgroup$ – Hohmannfan May 11 '16 at 12:47
  • $\begingroup$ @Hohmannfan Since I got an answer herem I shall delete my post at Physics SE. But could I ask you why so? What is exactly the problem in posting the same question at two different SE sites. Thank you. $\endgroup$ – Constantine Black May 11 '16 at 19:26
  • $\begingroup$ It is discussed a lot here. $\endgroup$ – Hohmannfan May 11 '16 at 19:43
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The angle you are looking for $$\cos \theta = \sin \delta_1 \sin \delta_2 + \cos \delta_1 \cos \delta_2 \cos (a_1-a_2),$$ where $(a_1, \delta_1)$ are the RA and Dec of the star and $(a_2,\delta_2)$ the RA and Dec of the convergent point.

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