|DATE1||D||final epoch (TT MJD) for the updated elements|
|U||D(13)||universal elements (updated in place)|
|(1)||combined mass (M+m)|
|(2)||total energy of the orbit ()|
|(3)||reference (osculating) epoch (t0)|
|(4-6)||position at reference epoch ()|
|(7-9)||velocity at reference epoch ()|
|(10)||heliocentric distance at reference epoch|
|(13)||universal eccentric anomaly () of date, approx|
|+102 = warning, distant epoch|
|+101 = warning, large timespan (>100 years)|
|+1 to +8 = coincident with major planet (Note 5)|
|0 = OK|
|-1 = numerical error|
The f(q) part of the full Encke method is not used. The purpose of this part is to avoid subtracting two nearly equal quantities when calculating the ``indirect member'', which takes account of the small change in the Sun's attraction due to the slightly displaced position of the perturbed body. A simpler, direct calculation in double precision proves to be faster and not significantly less accurate.
Apart from employing a variable timestep, and occasionally ``rectifying the orbit'' to keep the indirect member small, the integration is done in a fairly straightforward way. The acceleration estimated for the middle of the timestep is assumed to apply throughout that timestep; it is also used in the extrapolation of the perturbations to the middle of the next timestep, to predict the new disturbed position. There is no iteration within a timestep.
Measures are taken to reach a compromise between execution time and accuracy. The starting-point is the goal of achieving arcsecond accuracy for ordinary minor planets over a ten-year timespan. This goal dictates how large the timesteps can be, which in turn dictates how frequently the unperturbed motion has to be recalculated from the osculating elements.
Within predetermined limits, the timestep for the numerical integration is varied in length in inverse proportion to the magnitude of the net acceleration on the body from the major planets.
The numerical integration requires estimates of the major-planet motions. Approximate positions for the major planets (Pluto alone is omitted) are obtained from the routine sla_PLANET. Two levels of interpolation are used, to enhance speed without significantly degrading accuracy. At a low frequency, the routine sla_PLANET is called to generate updated position+velocity ``state vectors''. The only task remaining to be carried out at the full frequency (i.e. at each integration step) is to use the state vectors to extrapolate the planetary positions. In place of a strictly linear extrapolation, some allowance is made for the curvature of the orbit by scaling back the radius vector as the linear extrapolation goes off at a tangent.
Various other approximations are made. For example, perturbations by Pluto and the minor planets are neglected, relativistic effects are not taken into account and the Earth-Moon system is treated as a single body.
In the interests of simplicity, the background calculations for the major planets are carried out en masse. The mean elements and state vectors for all the planets are refreshed at the same time, without regard for orbit curvature, mass or proximity.
SLALIB --- Positional Astronomy Library