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The semi-major axis a here,

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what exactly does it represent? Point it out in the image below.

enter image description here

d1ep
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1 Answers1

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It is the semi major axis of the relative orbit i.e. the ellipse whose major axis is the sum of the two major axes and whose focus is in the barycentre. Essentially, this is like fixing the reference frame at the barycentre and looking at how the stars' position vectors change. It will turn out (try looking up two body problem) that this is equivalent to an object of reduced mass $\mu = \frac{m_1*m_2}{m_1+m_2}$ orbiting the one with total mass.
As uhoh pointed out, this image illustrates semi major axes : https://i.stack.imgur.com/298tF.png .

Tosic
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  • Thanks, good answer. How would I go about calculating the semi major axis of the relative orbit? – d1ep Dec 15 '19 at 22:15
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    The semi-major axis of the relative orbit is the sum of the semi-major axes of two orbits around the barycentre ... that way, the two body problem is reduced to a one body problem (the reduced mass orbiting the total mass in an elliptical path), which is something you can find a lot of information about – Tosic Dec 15 '19 at 22:26
  • OK. Cannot entirely visualize why, but have a better understanding now at least. Is the maximal distance between the stars then 2x semi-major axis of the relative orbit? – d1ep Dec 15 '19 at 22:29
  • I worded it strangely (actually incorrectly, sorry) when I first answered, but I think the full derivation, explanation and visualisation is a bit too long and you ought to find some two body problem resources. The maximum distance is the aphelion of the relative orbit, since the orbit in your question and the reduced mass one are considered equivalent. – Tosic Dec 15 '19 at 22:43
  • I think it was a good description. Any easy way to calculate the maximum distance (aphelion of the relative orbit)? – d1ep Dec 15 '19 at 22:47
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    Due to the mass centre relations, the eccentricities of two orbits are equal, and so is the one of the relative orbit meaning that the maximum distance is a1(1+e) + a2(1+e) = a(1+e) where a is the relative orbit semimajor axis – Tosic Dec 16 '19 at 08:24
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    @Tosic Is this right? (feel free to incorporate into your answer if it helps) https://i.stack.imgur.com/298tF.png – uhoh Dec 17 '19 at 02:01