The following is an extract from an e-mail communication with a Senior Analyst at the Solar System Dynamics Group of the Jet Propulsion Laboratory. I sent him the following question concerning the differences between the 'orbital period' values given by different Solar System Dynamics-tools:
- "I am fact-checking some of the information in the article "Eris (dwarf planet)" and was wondering about its orbital period.
- The JPL Small-Body Database Browser's "Orbital Elements" give a "period" of 205135.2487190889 (= 561,63). It is obviously using the Julian year (= 365.25 days/yr). This is also the orbital period listed in the respective data on the "Minor Planet Center" web-site.
- The "Object Data Page" of Horizons' ephemerides lists "PER = 557.0777399999999". According to the Horizons-tutorial this is the orbital period in years. And this is the orbital period used by Wikipedia.
- A difference of 4.5 years seems rather hefty. As your data certainly is correct, I am wondering what I am missing, especially as a systematic error of "orbital period"-data in the Wikipedia articles cannot be ruled out."
I received the following reply:
- "Both numbers are correct but need to have the relevant epoch of osculation specified.
- For the JPL Small-Body Database Browser's "Orbital Elements" (and probably MPC) the epoch of osculation for the displayed elements is 2011-Aug-27. That is, the parameters (including period) describe the shape and orientation of the orbit ellipse at that one moment.
- Every 6 months, both sites numerically integrate the orbit forward in time as an aid to those users who take the elements for use in independent low-accuracy two-body Keplerian propagators. Having an epoch close to the presumed time of interest (the present) helps reduce the errors in the common method, hence this advancement.
- Numerical integration takes into account perturbations due to others planets, Keplerian two-body propagation does not.
- For Horizons, the epoch of osculation is always the epoch of the orbit solution, currently 2006-Nov-5. To maintain accuracy, only the actual solution orbital elements are display -- the output epoch is not advanced.
- If people want orbital elements for a different time, the assumption is they would use the precision numerical integration capability of Horizons to obtain them at the desired time.
- If this is done, and the initial 2006 elements are propagated from 2006 to 2011, Horizons shows ...
2455800.500000000 = A.D. 2011-Aug-27 00:00:00.0000 (CT) EC= 4.339082250937790E-01 QR= 3.853435449939457E+01 IN= 4.381801616416351E+01 OM= 3.606467470391839E+01 W = 1.509537705231710E+02 Tp= 2546044.049227256794 N = 1.754939739954980E-03 MA= 2.016280091865038E+02 TA= 1.895440834854001E+02 A = 6.807086095850481E+01 AD= 9.760736741761504E+01 PR= 2.051352486947701E+05
- ... or an orbit period ("PR") of 205135.2486947701 days at the same epoch.
- The 4 year variation in period comes from the fact osculating orbital elements only represent position and velocity (the state vector, hence ellipse) at one instant in time. Over time, gravitational perturbations pull on the ellipse, so that by 2011, Eris has a slightly different ellipse than it did in 2006, hence different orbit period.
- So both orbit periods are correct for their respective moment of validity.
- Because Eris has such a large multi-hundred year period, fractional percent variations in the ellipse due to perturbations manifest as "years" of change in orbit period, which is major on a human scale, but not so much for the whole orbit.
- The actual statistical uncertainty in orbit period at any one epoch or moment in time is about two weeks at present. If you want to see how period varies as function of time, you could use Horizons to numerically integrate the equations of motion and then plot the resulting curve."