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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

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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 .

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  • $\begingroup$ Thanks, good answer. How would I go about calculating the semi major axis of the relative orbit? $\endgroup$
    – d1ep
    Dec 15, 2019 at 22:15
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    $\begingroup$ 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 $\endgroup$
    – Tosic
    Dec 15, 2019 at 22:26
  • $\begingroup$ 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? $\endgroup$
    – d1ep
    Dec 15, 2019 at 22:29
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    $\begingroup$ 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 $\endgroup$
    – Tosic
    Dec 16, 2019 at 8:24
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    $\begingroup$ @Tosic Is this right? (feel free to incorporate into your answer if it helps) i.stack.imgur.com/298tF.png $\endgroup$
    – uhoh
    Dec 17, 2019 at 2:01

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