The Satellite Mass: Newsletter Challenge of 02/01/11 created some interest in the age-old two-body orbital problem. A scan on the internet showed various treatments^(a) that were mostly confusingly complex, probably done by mathematicians. One source for the confusion seems (to me) that they use the barycenter of two masses as origin for the reference frame. Even when they referred to r, the distance between the two masses, they still use a 'reduced mass' and place it at the barycenter. In a sense, this constitutes the mixing of two coordinates systems in the same calculation.

Another point is that it is uncommon in solar system calculations to use barycentric coordinates. We normally use heliocentric coordinates for describing the orbits of planets and geocentric coordinates to describe the orbits of satellites around Earth. Because the mass of an artificial satellite is negligible relative to Earth's mass, the barycenter of the two and the geocenter are practically the same. However as soon as we think in terms of natural satellites like the Moon with one eightieth of Earth's mass, it is no longer the case. The said newsletter challenge is also a case like that: the answer has to be some one eighth of Earth's mass.

Fortunately, working in mass-centric coordinates for two non-negligible masses is very simple. You simple sum the two masses and then consider it as a single mass with a 'lightweight' satellite orbiting at a radial coordinate r equal to the distance between the mass centers of the individual bodies. The proof of this is simple. Referring to the figure, the centrifugal acceleration of each body in orbit around the barycenter must permanently equal the gravitational acceleration between the two bodies.^(b)

This then gives us the barycentric orbital speeds:

Now, since the mass-centric relative orbital speed v = v₁ + v₂, you can easily check that we have this simple relation:

This is just the sum the two masses considered as a single mass with a 'lightweight' satellite orbiting at a radial distance r. You can choose any one of the two masses as the origin of the new coordinate system and you will have a the same circular orbit around the origin (same r and same angular speed ω).

One way of reality-checking equation (4) is to realize that if m1 = m2, then the relative speed must be √2 times the relative speed between Earth and a lightweight body (m₂ << m1). Makes some sense, because the centrifugal acceleration must be double that for a lightweight orbiting body, since the centripetal gravitational acceleration doubles.

Equations (3) are sometimes written as:

equivalent to the 2-body equation in Wikipedia's "Orbital speed" article. Although correct, it can be very confusing, because the equations use r in geocentric coordinates, while yielding speeds in barycentric coordinates.

[Edit] I have now completed the PDF referred to in note (b) below, including non-circular orbits. My conclusion is that, as long as the situation does not become relativistic, the two-body equations are exactly as per the particle orbiting a massive object with the combined mass of the two bodies at the same relative distance. It is a bit lengthy to include here, so at the risk of infuriating some, I have to refer interested readers to the PDF.

-J

Notes

(a) A limited list of internet articles on two-bodies:

http://en.wikipedia.org/wiki/Orbital_speed

http://scienceworld.wolfram.com/physics/Two-BodyProblem.html

http://en.wikipedia.org/wiki/Gravitational_two-body_problem

http://www.applet-magic.com/2body.htm

(b) I've written a short (5-pager) PDF on circular two-body orbits, but cannot attach it here, so I've posted it on my website. The equation numbers are those in the PDF, hence there may be gaps in the numbers in this post. You can download the PDF here for more detail, but please do not 'run ahead' to elliptical orbits for now - we will get to that.