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

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.

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.

• OK that's great - I'll do a quick numerical check to make sure it's in the ballpark of what I estimate from the drift of the apoapsis throughout 2017.
– uhoh
Nov 28, 2016 at 15:19
• @uhoh -- Assuming a pure polar orbit, the above expression for $\dot\omega$ yields a change of $270\,J_2 (R/p)^2$ degrees per orbit. Given Jupiter's $J_2$ of 0.014733 and the spacecraft's semi-latus rectum of about 2.12, the above becomes about 0.9 degrees per orbit, or about 32 degrees in 36 orbits. Nov 28, 2016 at 15:33
• yep! You've obviously had more experience. I did a manual plug and chug using estimates for $a$ and $\epsilon$ of 1.68E+06 km and 0.955, and got -1.3E-08 rad/sec which matches just what I got from the drift. This is great, thanks!
– uhoh
Nov 28, 2016 at 16:01

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.

• Thanks! I'm writing BC (before coffee) but I'll mention 1) When I say I'm using J2000 ecliptic coordinates I've shifted the origin to move with the Jupiter barycenter by downloading both Juno and Jupiter-barycenter vectors and subtracting. This leaves the directions of the x, y, z axes are unchanged (non-relativistically at least) and seems to give the same answer as using Jupiter's frame. I'll confirm again and add a note in the question.
– uhoh
Nov 28, 2016 at 0:20
• 2) I mean to ask for an analytical expression for precession of the apses around an oblate body to compare to the data. The de-facto apoapsis (actual location of greatest distance) moves steadily from one orbit to the next, the instantaneous best fit periapsis $\omega$ (omega) shown in ELEMENTS as the fifth item "W" varies wildly and is not good to extract a precession rate, and...
– uhoh
Nov 28, 2016 at 0:44
• 3) I am looking at the OLD reference data with the 14 day orbit, as you saw in this question but now ELEMENTS reflects the 53 day orbit.
– uhoh
Nov 28, 2016 at 0:44
• I checked. Subtracting the two state vectors (Juno - Jupiter barycenter) gives the same position as Juno with Coordinate Center: Jupiter System Barycenter [500@5] within about 20km, and this difference may be because they use different sources (DE431mx vs DE434), so I think what I did is fine here.
– uhoh
Nov 28, 2016 at 1:53
• Actually, I mean the inclination. You note "It turns out the orbit is essentially polar (inclination of about 90 degrees) and almost completely within a yz plane in those coordinates". However, Jupiter's "north pole" is quite different from ours. Probably not a big deal, though.
– user7073
Nov 28, 2016 at 2:33