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I was reading about what is involved in making an ephemeris and wondered about this. Feeling I'd read pieces that extrapolated orbits over many thousands of years, I looked for an example and found this rather interesting one at phys.org:

Astronomy & Astrophysics is publishing a new study of the orbital evolution of minor planets Ceres and Vesta, a few days before the flyby of Vesta by the Dawn spacecraft. A team of astronomers found that close encounters among these bodies lead to strong chaotic behavior of their orbits, as well as of the Earth's eccentricity. This means, in particular, that the Earth's past orbit cannot be reconstructed beyond 60 million years.

The Wikipedia article on JPL's Development Ephemeris gives a sense of how tremendously complex these calculations are, and how many observations are used. This quote gives a bit of a sense:

DE418[23] was released in 2007 for planning the New Horizons mission to Pluto. New observations of Pluto, which took advantage of the new astrometric accuracy of the Hipparcos star catalog, were included in the fit. Mars spacecraft ranging and VLBI observations were updated through 2007. Asteroid masses were estimated differently. Lunar laser ranging data for the Moon was added for the first time since DE403, significantly improving the lunar orbit and librations. Estimated position data from the Cassini spacecraft was included in the fit, improving the orbit of Saturn, but rigorous analysis of the data was deferred to a later date. DE418 covered the years 1899 to 2051, and JPL recommended not using it outside of that range due to minor inconsistencies which remained in the planets' masses due to time constraints

How quickly does uncertainty creep into such calculations, and what sort of margin of error is involved? Does the computing ability of modern computers make a difference - that is, can all known objects and forces be entered into a program that crunches the numbers, or do the smaller things have to be dropped out of the formula? Does this affect mission planning to asteroids?

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    $\begingroup$ Chaotic systems diverge exponentially over time. So better knowledge of initial conditions only get you a slight (non-linear) increase in predictive ability. (See Lyapunov exponent) Adding in a few more objects is almost useless. The smaller things don't drop out for long term effects, but they can be (nearly) ignored in the short term (such as for asteroid mission planning). $\endgroup$ – BowlOfRed Oct 30 '14 at 22:17
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    $\begingroup$ In a different wikipedia page, you can see that the JPL ephemerides are updated rather more often than once every 20 years. $\endgroup$ – Mark Adler Oct 31 '14 at 16:00
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    $\begingroup$ That 48,479 figure is the number of observations used for DE102, not the number of objects. An observation comprises the time at which the observation was made, where it was made, what body was observed, and the observation itself. $\endgroup$ – David Hammen Oct 31 '14 at 16:51
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    $\begingroup$ This question is too vague to really answer. Though almost any question could be answered with "it depends", this one deserves a giant IT DEPENDS. It depends on what you mean by "accurate", what body (Mars is known much, much better than Pluto), what component of its motion (e.g. you might know the shape of the orbit well, but the downtrack position in the orbit poorly). Also as noted, the degradation in accuracy is not always steady. Singular events can cause a sudden, large amplification of the propagation errors. $\endgroup$ – Mark Adler Oct 31 '14 at 17:11
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    $\begingroup$ You might want to read about Jack Wisdom (MIT prof)'s work on extremely long (>100Myr) integrations of solar system dynamics: web.mit.edu/wisdom/www/measurements.pdf web.mit.edu/wisdom/www/longterm.pdf en.wikipedia.org/wiki/… tinyurl.com/q2klvuo $\endgroup$ – pericynthion Nov 1 '14 at 1:38
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The current accuracy of the JPL Developmental Ephemerides released in September of 2013 is given by its authors in their article The Planetary and Lunar Ephemerides DE430 and DE431, quoted below:

The present-day lunar orbit is known to submeter accuracy through fitting lunar laser ranging data with an updated lunar gravity field from the Gravity Recovery and Interior Laboratory (GRAIL) mission. The orbits of the inner planets are known to subkilometer accuracy through fitting radio tracking measurements of spacecraft in orbit about them. Very long baseline interferometry measurements of spacecraft at Mars allow the orientation of the ephemeris to be tied to the International Celestial Reference Frame with an accuracy of 0′′.0002. This orientation is the limiting error source for the orbits of the terrestrial planets, and corresponds to orbit uncertainties of a few hundred meters.The orbits of Jupiter and Saturn are determined to accuracies of tens of kilometers as a result of fitting spacecraft tracking data.The orbits of Uranus, Neptune, and Pluto are determined primarily from astrometric observations, for which measurement uncertainties due to the Earth’s atmosphere, combined with star catalog uncertainties, limit position accuracies to several thousand kilometers.

Because the gravity of each body in the solar system affects all the others continuously, there is no way to calculate the future motion of bodies accurately for long periods of time. This is known as the n-body problem. Perturbation theory is used to find approximate projections of future behaviour. The Lyapunov time is the time it would take for the trajectory projections of the bodies of the system to diverge by a factor of e. For the solar system as a whole, this time is approximately 50 million years. Functionally this means that no matter how precise and complete the initial measurements, nothing can really be said about where anything will be in 50 million years.

In the near term, error accumulates slowly. DE431 projects orbits for 17,000 years into the future. DE430 uses a more complex model - it included a calculation of how the movement of the moon's core with respect to its mantle would affect its orbit. For this reason its accuracy was only considered appropriate for projections over the years 1550 to 2650.

The authors of DE430 and DE431 did not attempt to quantify the probable error margins for their ephemeris projections. They did carefully note the ranges of residual errors in all the observations used in the calculations. It must not be possible to state the probable error margins in such predictions, for the following reasons in descending order of importance:

  • Such systems are inherently chaotic
  • Measurements and models used have inaccuracies
  • There are factors known to influence outcomes which have not been quantified enough to include in the modelling (such as the orbit of the solar system around the galaxy)
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