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I wanted to calculate the distance between the end points of the Orion's belt (Alnitak and Mintaka).

I searched in the internet and I found out their angular separation to be 2.736°.

Then I searched for the actual distances of the two stars from Earth; and it turned to be 1260 light years for Alnitak and 1200 light years for Mintaka.

Then I used the cosines law to find the actual distance between the two stars. I reached 84 light years as the answer.

Questions:

  • Is the cosines law a good way to find the actual distance between two celestial objects in such far distances?
  • Is there another way to find the actual distance between two celestial objects?
  • Can we find the actual distance between two celestial bodies if we don't know the angular distance between them?
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    $\begingroup$ How accurate and precise do you need these distances to be? According to Wikipedia, the margin of error on Alnitak's distance from earth is ±180 light-years, and for Mintaka, ±90 light years or more, so the real distance between the two stars is probably larger than 84 light-years. $\endgroup$
    – notovny
    Commented May 28, 2023 at 15:00

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  1. If you know the distances of both objects to the Earth and the angular separation between them, using the law of cosines is the only reasonable option.

  2. Suppose you know the distances of both objects to the Earth, but not the angular separation. Since you probably also know the coordinates of both stars $\alpha_1, \ \delta_1$ and $\alpha_2, \ \delta_2$, first calculate the angular separation $\theta$ between the two stars using:

$$\cos \theta = \sin\delta_1 \sin\delta_2 + \cos\delta_1 \cos\delta_2 \cos(\alpha_1 - \alpha_2)$$

Now you know the cosine of the angle of separation $\boldsymbol{\cos \theta}$ and you can use the law of cosines to calculate the distance between the two objects.

$$d=\sqrt{d_1^2+d_2^2-2 \ d_1 d_2 \cos \theta \, \, }$$

  1. Of course, in general, if only the distances to the Earth are known, but neither the angular separation nor the coordinates, it is not possible to calculate the distance between them.

Best regards.

PD. A related question, when objects are at cosmological distances: Comoving distance between two points [(RA1, Dec1, z1) and (RA2, Dec2, z2)]

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  • $\begingroup$ (Of course), your equation for the angular separation is just the law of cosines in spherical trigonometry. $\endgroup$
    – PM 2Ring
    Commented May 31, 2023 at 10:41

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