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Reducing roundoff errors in numerical integration of planetary ephemeris


Modern lunar-planetary ephemerides are numerically integrated on the observational timespan of more than 100 years (with the last 20 years having very precise astrometrical data). On such long timespans, not only finite difference approximation errors, but also the accumulating arithmetic roundoff errors become important because they exceed random errors of high-precision range observables of Moon, Mars, and Mercury. One way to tackle this problem is using extended-precision arithmetics available on x86 processors. Noting the drawbacks of this approach, we propose an alternative: using double–double arithmetics where appropriate. This will allow to use only double-precision floating-point primitives, which have ubiquitous support.

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The program code and data used for numerical integration in this work are available online:


  1. PECEC stands for “Predict–Evaluate–Correct–Evaluate–Correct” mode in the predictor–corrector scheme that is used to approximately solve the implicit equation of the state of the system on the next step. In PECEC mode, the right-hand part of the system of differential equations is evaluated twice: the first time for the state predicted by the Adams–Bashforth formula and the second time for the state corrected by the Adams–Moulton formula. The state is then again corrected for the next step.

  2. The semimajor axis of the orbit of the Moon in the ephemeris strongly correlates with X-coordinates of retroreflectors; hence, the said coordinates have the uncertainty of about 3 cm at best Pavlov (2019). However, it still makes sense to require better accuracy in lunar ephemeris when geocentric coordinates of the retroreflectors are of interest; in those coordinates, the uncertainties of the X-coordinates of retroreflectors and the semimajor axis of the orbit of the Moon are largely canceled out.

  3. In a multistep scheme, the approximations are slightly different near the ends of the timespan of integration due to the “warm-up” stage, but as we will later see, this difference does not make any noticeable impact.

  4. “BC” stands for “before Chez”, the Racket compiler and virtual machine that existed before they were replaced with Chez Scheme, but are still supported.


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Correspondence to Dmitry Pavlov.

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Subbotin, M., Kodukov, A. & Pavlov, D. Reducing roundoff errors in numerical integration of planetary ephemeris. Celest Mech Dyn Astron 135, 23 (2023).

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