Juno's original orbit around Jupiter - is this apsidal precession? If so, need expression

Question

Below is a plot of Juno's originally planned orbit around Jupiter, extracted from JPL Horizons. It's shown in J2000 ecliptic coordinates, centered on the Jupiter barycenter. It turns out the orbit is essentially polar (inclination of about 90 degrees) and almost completely within a $yz$ plane in those coordinates. (Other plots on the internet look different because they rotate the coordinates to keep the solar direction constant.) The black dots represent approximate Jovian apoapses.

The plot shows the same orbit viewed side-on (along x axis) and front-on (along y axis), the big red dot is Jupiter.

While the inclination of the orbit stays about 90 degrees, the plot shows what looks like a very pronounced apsidal precession. It seems that during the close flyby of Jupiter's equatorial bulge the extra attraction beyond the overall $1/r^2$ causes the orbit to advance substantially. The second plot shows the movement of the apoJove over time, showing an apsidal precession of about 31.2 degrees in 477 days, or about $1.3 \times 10^{-8}$ rads/sec.

My question is: Is this motion actually apsidal precession due to Jupiters non-spherically-symmetric gravitational potential, or is it something else, or even a spacecraft maneuver? If it is indeed precession, where can I find a mathematical expression for the apsidal precession rate?

above x2: plots of Juno's orbit around Jupiter as described above, data from JPL Horizons.

Answer 1

My question is: Is this motion actually apsidal precession due to Jupiters non-spherically-symmetric gravitational potential?

Yes, that apsidal precession results from Jupiter's oblateness. Gravitationally, this oblateness effect is expressed in terms of a planet's second dynamic form, or $J_2$. Jupiter's $J_2$ is over ten times that of the Earth due to Jupiter's high rotation rate. Note that Juno's orbit also suffers a bit of nodal precession, as seen in various diagrams of the orbit from above. The average apsidal and nodal precession rates over the course of an orbit due to a planet's oblateness are $$\begin{aligned} \dot\omega &= \phantom{-} \frac 3 4 J_2 \left(\frac R p\right)^2 n\,(5 \cos^2 i -1) \\ \dot\Omega &= - \frac 3 2 J_2 \left(\frac R p\right)^2 n \cos i \end{aligned}$$ where $R$ is the equatorial radius of the planet in question, $J_2$ is the planet's second dynamic form, $p=a(1-e^2)$ is the semi-latus rectum, $a$ is the semi-major axis length of the orbit, $e$ is the eccentricity of the orbit, $n$ is the mean motion, and $i$ is the inclination of the orbit.

See Effect of terrestrial oblateness on artificial satellite orbits for a derivation of these expressions. This derivation can be found in many other places. The key search terms needed to find these derivations are "Lagrange's planetary equations" and "oblateness". The key concepts needed to understand these derivations are perturbation techniques and Lagrange's planetary equations.

Answer 2

This probably isn't helpful, but it has an image, so can't be a comment. Try looking at the elliptical elements directly, instead of plotting the path:

Of course, the first "few" days, Juno is traveling to Jupiter, so the results won't make much sense.

If you do plot the path, consider using the IAU_JUPITER frame, since Juno is effectively in orbit around Jupiter, and J2000 coordinates really aren't appropriate.