I am studying circumbinary planets and I made a python model to calculate certain properties of the orbits, and when I calculate the inclination I get a sine wave.

I do not see a way to predict the shape of the curve in advance, despite it being a rather simple curve. Furthermore, when I plot the eccentricity of the orbit the frequencies of the curves do not align.

Would anyone know why it oscillates at this rate and amplitude?

And a side question, would anyone know some literature on the subject of circumbinary planets?

The inclination curve The eccentricity curve

  • $\begingroup$ The frequency of your inclination curve looks pretty close to twice the frequency of your eccentricity curve. Maybe you should add some more details to your question about this circumbinary system. What are the masses of the stars & planet? What are their orbital periods & radii? How big is your time step? Are you using a library to calculate the motion, or are you doing it yourself? Which integration method are you using? $\endgroup$ – PM 2Ring Jun 29 '19 at 1:10
  • $\begingroup$ Thank you for accepting my answer! By the way if you find out more either through reading or simulation it is always okay to post another answer to your own question. It might be helpful to future readers and it's a quick way to pick up some more reputation (points). Once you reach 50 you can post comments on other peoples posts for example. Welcome to Stack Exchange! $\endgroup$ – uhoh Jun 30 '19 at 13:44
  • $\begingroup$ Ive been trying to look up how to determine the J2 term, and have been working on understanding arxiv.org/pdf/1609.00915.pdf, but in 7.2.5 it says the inclination doesnt change. because of this I thought I could use the torus shape to determine a static gravitational field, and then letting it act on the planet. Would this a a viable way, or is the static element of this torus shape problematic? $\endgroup$ – daan Jul 2 '19 at 19:14

The period of the stars' orbits is likely so much shorter than the planet's orbit that from the planet's perspective the pair looks like two donuts, or toroidal-shaped masses, possibly overlapping depending on the masses and sizes.

In orbital mechanics you can think of this roughly as a single star with a really huge J2 term (gravitational quadrupole moment).

In addition to apsidal and nodal precession (which you would have seen) J2 is well known to induce oscillations in inclination. For example see J2 long-period perturbations in the inclination.

Also from Two-Body, J2 Perturbation & J4 Perturbation Propagators:

The solutions produced by the J2 Perturbation and J4 Perturbation propagators are approximate, based upon Keplerian mean elements. In general, forces on a satellite cause the Keplerian mean elements to drift over time (secular changes) and oscillate (usually with small amplitude). In particular, the J2 and J4 terms cause only periodic oscillations to semi-major axis, eccentricity and inclination, while producing drift in argument of perigee, right ascension and mean anomaly.


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