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tl;dr: The orbital period for a Keplerian orbit of a small body around a massive one with standard gravitational parameter $\mu=GM$ and semi-major axis $a$ is given by

$$T = 2\pi\sqrt{\frac{a^3}{\mu}}.$$

I'm asking if for near-circular orbits the rise times of the object will always come earlier than predicted by a Keplerian model when the central body is oblate, i.e. has a positive $J_2$.


Assuming that we want to make a "long-term" prediction of the rise-time wrt any given observation point on a planet, of an object circling said planet, assuming that we want to make a rough estimate using the Kepler model. Is it correct to state that the said rough estimate is ALWAYS lagging because we ignore the oblateness of the planet? By "lagging", I mean we should observe the rise somewhat BEFORE the predicted calculation, but always before?

A side question: Assuming we know the J2 zonal coefficient and the orbital elements at epoch (t0), is it correct to say that the prediction "error" at time t is simply linear with delta_t (t=t0+delta_t)? (the point is that if your answer is yes, then I can use the simplified model w/o making any error compared to a J2-aware model).

Note: For now, I am not interested in a real-life situation, where the orbit is perturbed by other forces than J2.

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    $\begingroup$ By the way, if the orbit is low and near zero inclination (equatorial) then if the reduced $J_2$ is say +0.001 the gravity field near the planet in the orbital plane will always be 0.001 stronger and so the orbit will still be circular with a period will be 0.9995 $T_{Kepler}$. In this condition the small moon will just think the planet is one part per thousand more massive. It of course gets much more complicated for higher inclinations, and the orbit will precess around the planet which Kepler doesn't predict; after a while the Keplerian predictions will not resemble anything like reality. $\endgroup$
    – uhoh
    May 25, 2021 at 22:11
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    $\begingroup$ So I think a better question might be more like your different but related one in Space SE about the period defined by subsequent ascending node crossings rather than tied to rise times a a specific location on the planet's surface, since nodal precession (of the orbital plane) adds a complication, unless that really is the ultimate question you need addresses in which case it's fine! $\endgroup$
    – uhoh
    May 25, 2021 at 22:14
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    $\begingroup$ @uhoh, Great edit. Agreed. $\endgroup$
    – Ng Ph
    May 26, 2021 at 8:25
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    $\begingroup$ @uhoh, Intuitively, when there is symmetry and a single central force, then the Kepler model must apply. Hence, Kepler must apply to an equatorial orbit around an oblate planet, imho. $\endgroup$
    – Ng Ph
    May 26, 2021 at 8:30
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    $\begingroup$ @uhoh, I would say that my related question in Space SE is to validate (or invalidate) a reasoning step. This one here is about the practical use. Indeed, the effect of J2 on the orbit plane brings an additional complication to the effect on the mean motion. However, this could be bounded by clarifying the magnitude of "long-term" in the statement of problem (I hope). If not the simple prediction could be a little bit more sophisticated (which may defeat the ultimate goal, on the other hand). $\endgroup$
    – Ng Ph
    May 26, 2021 at 8:45

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