# JPL ephemerides: effect of Saturn, Uranus and Neptune

According to the Folkner ( Folkner et al, 2014, The Planetary and Lunar Ephemerides DE430 and DE431, IPN Progress Report 42-196 • February 15, 2014), JPL ephemerides consider the following effects:

The modeled accelerations of bodies due to interactions of point masses with the gravitational field of nonspherical bodies include: (a) the interaction of the zonal harmonics of the Earth (through fourth degree) and the point mass Moon, Sun, Mercury, Venus, Mars, and Jupiter; (b) the interaction between the zonal, sectoral, and tesseral harmonics of the Moon (through sixth degree) and the point mass Earth, Sun, Mercury, Venus, Mars, and Jupiter; (c) the second-degree zonal harmonic of the Sun ( J2) interacting with all other bodies.

However, at the JPL Horizons website it is said, that the effects of 8 planets are considered.

Question: Do JPL Horizons ephemerides consider the effects of Saturn, Uranus and Neptune?

• ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf appears to be the link if anyone wants to read the original document – barrycarter Oct 6 at 15:07
• This text appears in a very specific section of the document titled "Point Mass Interaction with Extended Bodies". Because the Earth is nonspherical (it's closer to an ellipsoid), the Moon's gravity is stronger where the Moon is closer, so the lunar gravitational effect can't be modeled by treating the Earth and Moon as point masses. Apparently, they extend this all the way out to Jupiter, but not as far as Saturn. I'm guessing that Saturn is far enough away that it can be treated as a point mass. For the general orbits, all 8 planets + Pluto (+ more) are considered. – barrycarter Oct 6 at 15:13
• For the positions, as page 2 notes "Perturbations from 343 asteroids have been included in the dynamical model.", so it's much more complete. – barrycarter Oct 6 at 15:16
• @barrycarter I didn't understand. For the positions they take the planets as point masses, but up to Jupiter they take into account the oblateness also? – Leeloo Oct 6 at 15:42
• This is a really interesting discussion! In this answer I included very approximate corrections for non-point-mass-to-point-mass effects. I only selectively "turned on" the Sun's J2 for Mercury, and the Earth's J2 for the Moon because my numerical accuracy was low, but these references discuss turning on more J2's and also some higher order multipole moments. – uhoh Oct 7 at 1:01

The equations of motion for how bodies move in the Solar System, which are then fitted to the observational data of positions and ranges to provide the ephemeris, involve a nested set of effects which account for increasingly subtle and smaller effects.

As detailed in the documentation for DE430 and DE431 and the introduction in Section III these are:

1. the basic N-body gravitational attraction between all the bodies, treated as point-masses
2. the effects of the non-spherical oblateness of the Sun (its figure as it is described) on the other bodies of the Solar System
3. the effects of the static non-spherical shape of the Earth and the Moon on each other and on the planets Mercury to Jupiter
4. the effects of the time-varying shape (tides) raised on the Earth by the Sun and the Moon back on the Moon's orbit.

For 1. this is a generalized version of the classical force/acceleration due to 2 bodies $$F=\frac{Gm_1m_2}{r^2}$$ (e.g as in these course notes) but extended to include multiple (N) bodies (Newtonian N-body equations of motion) and generalized beyond the effects of Newtonian gravity to allow general relativity to be included (the so-called parametrized post-Newtonian (PPN) metric). This acceleration on a particular body is summed over everything else: the Sun, the Moon, the planets Mercury through Pluto and the 343 largest asteroids. So this is where the statement you quote

However, at the JPL Horizons website it is said, that the effects of 8 planets are considered.

comes from as all the planets (plus more) are included in the basic equations of force/acceleration.

In addition to the basic equations from 1., the effects of non-spherical, non-point mass bodies are included as detailed in Section III B and which you quote in your question. These effects are:

1. the non-spherical Earth (up to 4th degree in the spherical harmonics expansion of the non-spherical Earth) on the Moon, the Sun, the planets Mercury - Jupiter (all treated as point masses)
2. the non-spherical Moon (up to 6th degree) on the Earth, the Sun, the planets Mercury - Jupiter (all treated as point masses)
3. the effect of the second order oblateness of the Sun on everything else

These effects are going to be much smaller than the main gravitational effect from 1. For example we very rarely need to take into account the $$J_2$$ effect of the Earth when calculating the effects on Near Earth Object trajectories and this is the largest of the non-spherical effects (the higher harmonics are weaker still). An additional issue is that we don't have very good gravity data that would reveal higher harmonics for the outer planets as this can normally only be measured by close-orbiting spacecraft and Uranus, Neptune and Pluto have only received brief distant flybys. (I suspect additional gravity data may be coming out for Saturn based on the 'Grand Finale' orbits of the Cassini spacecraft but this is likely still being worked based on these abstracts)