Figure: An explication of a two-body system with unequal mass bodies orbiting their mutual center of mass

Caption: A diagram of a gravitational 2-body system with the spherically-symmetric bodies orbiting in elliptical orbits the system center of mass marked by a red cross.

The orbits are physical orbits (in an astro-jargon sense) meaning the center of mass is in free-fall in the external gravitational field (i.e., gravitational field of the rest of the observable universe) and both bodies are orbiting relative to the observable universe (i.e., the bulk mass of the observable universe) which is "absolute rotation" in our modern sense though we do not use the expression "absolute rotation" ordinarily.

The center of mass is, of course, at rest in the center-of-mass (CM) inertial frame defined by the gravitational 2-body system. The center of mass is also the common focus of the elliptical orbits. The other focuses for the elliptical orbits are just empty points in physical space with NO special significance.

Further Explication:

1. The more massive body in a two-body system is called primary (or primary body) and the less massive body is called the secondary (secondary body).

   Since the center of mass of a two-body system is dominated by the more massive body, the primary is always closer to the center of mass than the secondary. If the primary overwhelmingly dominates the center of mass, then it effectively is at the center of mass.

   The mathemtical formulae for the mean orbital radii of the primary and secondary from the center of mass are given below.

2. The elliptical orbits of this gavitational two-body system are determined by Newtonian physics (what is universal about the physical system) and initial conditions (what is peculiar or individual about the physical system).

3. Gravity is, of course, the force that pulls the astro-bodies into orbits.

4. Now in astro jargon, an apsis (plural apsides) is an extreme point in the orbit of an astro-body.

   The two kinds of apsides are periapsis and apoapsis.

5. The periapsis is the arrangement of closest separation and the term is also used for the closest separation distance.

6. The apoapsis is the arrangement of farthest separation and the term is also used for the farthest separation distance.

7. As dictated by Newtonian physics for orbits, the astro-bodies move slowest at apoapsis and fastest at periapsis.

8. The apse line (AKA line of apsides) is drawn through the periapsis and apoapsis.

9. In the analysis of gavitational two-body systems, people usually use the relative orbit of the smaller astro-body to the larger astro-body.

   A relative orbit is also an elliptical orbit if the non-relative orbits are.

   You will have imagine the relative orbit since it is NOT shown in the diagram.

10. By normal convention, the formula for the relative orbit mean orbital radius (AKA relative semi-major axis) is specified as follows:

    The mean radius formula is calculated as one-half multiplied by the sum of the periapsis radius and the apoapsis radius:

    r mean = 1 2 ( r periapsis + r apoapsis ) ,

    where the two radii are, respectively, periapsis separation and apoapsis separation.

11. For the relative elliptical orbit:

    The periapsis radius formula is calculated as r mean multiplied by the quantity one minus e and the apoapsis radius formula is calculated as r mean multiplied by the quantity one plus e:

    r periapsis = r mean ( 1 − e ) and r apoapsis = r mean ( 1 + e ),

    where e is the eccentricity of the relative elliptical orbit as well of that of the orbits relative to their mutual center of mass.

    Note, fiducial values of eccentricity e are:

    1. e = 0 for a circle.
    2. e ∈ (0,1) for a non-circular ellipse.
    3. e = 1 for an ellipse stretched into a line segment.

12. The actual elliptical orbits relative to the center of mass are scaled down versions of the relative elliptical orbit. The scale radii for any epoch are

    r 1 = r ( m2 m ) and r 2 = r ( m1 m )

    where 1 is the index for astro-body 1, 2 is the index for astro-body 2, r is the mean separation distance, and m is the total mass. As you can see, if mass 1 is much greater than mass 2, we have radius 1 nearly 0 and radius 2 nearly r. This just shows that if mass 1 is much greater than mass 2, we effectively have astro-body 2 orbiting astro-body 1 which is effectively at rest at the center of mass as we asserted above.

    The mean orbital radius for the primary (secondary) is the smaller (larger) of the two values given by the above formulae.

13. In most real orbits, astronomical perturbations cause noticeable deviations from exact gavitational two-body system behavior.

14. There are special-case names for the apsides of familiar astronomical objects (see Wikipedia: Apsis: Terminology). But yours truly thinks mostly these are overly fancy, except for the most familiar astronomical objects: e.g.,
    1. Earth: perigee and apogee, where suffix gee is derived from Greek Earth goddess Gaia.
    2. Sun: perihelion and aphelion, where suffix helion is derive from Greek Sun god Helios.
    3. star: periastron and apastron, where suffix astron is ancient Greek language word for star.

    Other than the example cases, if one wants a fancy name for the periapsis and apoapsis of a body, yours truly suggests you just prefix the name of the body by peri- or ap-: e.g., peri-Jupiter and ap-Jupiter.