<p>There's one mistake in your drawing:</p>
<p>The center of mass of a two-body system lies on the straight line between their centers. So, the 90° angles should be placed between D and distA (and D and distB). Then the tangent applies instead of the sine.</p>
<p>But, as Antispinwards said, it doesn't matter for small angles.</p>
<p>What matters is, whether our viewing direction is really perpendicular to their orbiting plane. Even circular orbits will look elliptical if not viewed exactly from &quot;above&quot;. But that might be beyond the scope of exercise from your book.</p>

CC BY-SA 4.0
2020-09-23T19:59:50.660Z
If it wasn't perpendicular to viewing direction, how would you be able to calculate it ? Yeah the book doesn't go beyond basics of it.
WDUK
CC BY-SA 4.0
2020-09-23T20:03:51.260Z
@WDUK As long as you assume circular orbits, at their maximum separation you will have a perpendicular view.
Ralf Kleberhoff
CC BY-SA 4.0
2020-09-23T20:44:39.243Z
I thought circular orbits were pretty uncommon unless you mean "close enough to perfect circles" ?
WDUK
CC BY-SA 4.0
2020-09-23T13:08:25.547Z
2020-09-23T13:08:25.547Z
Ralf Kleberhoff
<p>I am confused on how to calculate the distances for each star in a binary system to their centre of mass. I am trying to solve it from a book I am reading which shows the solution but I do not understand the solution.</p>
<p>So the info I am given is:</p>
<p>A system is located at a distance <strong>D (in parsecs)</strong>. The two stars orbit around their centre of mass. Star A has an angular separation of <strong>a (in arcsecs)</strong> and star B has an angular separation of <strong>b (arcsecs)</strong>.</p>
<p>So now I am trying to find the physical separation of the stars to their centre of mass.</p>
<p>The book's solution (once converting to correct units) is:</p>
<pre><code>Distance for star A = D tan(a)
Distance for star B = D tan(b)
</code></pre>
<p>I don't fully understand why they use tangent here because when I draw out a diagram of this situation i seem to need to use sine not tangent.</p>
<p>Here is how I drew it:</p>
<p><a href="https://i.sstatic.net/jNKiw.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jNKiw.png" alt="enter image description here" /></a></p>
<p>Why do they use tangent function and not the sine function? Have I drawn my diagram correctly?</p>

<distances><binary-star><mathematics>
Calculating distance from center of mass of two stars in a binary system
WDUK
# Calculating distance from center of mass of two stars in a binary system

(Asked by: username_0 on 2020-09-22)

I am confused on how to calculate the distances for each star in a binary system to their centre of mass. I am trying to solve it from a book I am reading which shows the solution but I do not understand the solution.
So the info I am given is:
A system is located at a distance D (in parsecs). The two stars orbit around their centre of mass. Star A has an angular separation of a (in arcsecs) and star B has an angular separation of b (arcsecs).
So now I am trying to find the physical separation of the stars to their centre of mass.
The book's solution (once converting to correct units) is:
Distance for star A = D tan(a)
Distance for star B = D tan(b)

I don't fully understand why they use tangent here because when I draw out a diagram of this situation i seem to need to use sine not tangent.
Here is how I drew it:

Why do they use tangent function and not the sine function? Have I drawn my diagram correctly?

**Tags:** distances, binary-star, mathematics

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## 1 answers

### Answer by username_1 on 2020-09-23. Score: 1

There's one mistake in your drawing:

The center of mass of a two-body system lies on the straight line between their centers. So, the 90° angles should be placed between D and distA (and D and distB). Then the tangent applies instead of the sine.

But, as Antispinwards said, it doesn't matter for small angles.

What matters is, whether our viewing direction is really perpendicular to their orbiting plane. Even circular orbits will look elliptical if not viewed exactly from "above". But that might be beyond the scope of exercise from your book.

> **Comments:**
> **username_0**: If it wasn't perpendicular to viewing direction, how would you be able to calculate it ? Yeah the book doesn't go beyond basics of it.  
> **username_1**: @WDUK As long as you assume circular orbits, at their maximum separation you will have a perpendicular view.  
> **username_0**: I thought circular orbits were pretty uncommon unless you mean "close enough to perfect circles" ?  

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