Heliocentric position and velocity of a planet, asteroid or comet, from universal elements
"
universal variables"
form. "
(1) total energy of the orbit (alpha) "
(2) reference (osculating) epoch (t0) "
(3-5) position
at reference epoch (r0) "
(6-8) velocity at reference epoch (v0) "
(9) heliocentric distance
at reference epoch "
(10) r0.v0 returned (11) date (t) "
(12) universal eccentric anomaly
(psi) of date 1 = radius vector zero
2 = failed to converge
The "
universal"
elements are those which define the orbit for the purposes of the method of universal
variables (see reference). They consist of the combined mass of the two bodies, an epoch, and the
position and velocity vectors (arbitrary reference frame) at that epoch. The parameter set used here
includes also various quantities that can, in fact, be derived from the other information.
This approach is taken to avoiding unnecessary computation and loss of accuracy. The
supplementary quantities are (i) alpha, which is proportional to the total energy of the orbit, (ii) the
heliocentric distance at epoch, (iii) the outwards component of the velocity at the given
epoch, (iv) an estimate of psi, the "
universal eccentric anomaly"
at a given date and (v) that
date.
The companion routine is palEl2ue. This takes the conventional orbital elements and transforms them into the set of numbers needed by the present routine. A single prediction requires one one call to palEl2ue followed by one call to the present routine; for convenience, the two calls are packaged as the routine palPlanel. Multiple predictions may be made by again calling palEl2ue once, but then calling the present routine multiple times, which is faster than multiple calls to palPlanel.
It is not obligatory to use palEl2ue to obtain the parameters. However, it should be noted that because palEl2ue performs its own validation, no checks on the contents of the array U are made by the present routine.
DATE is the instant for which the prediction is required. It is in the TT timescale (formerly Ephemeris Time, ET) and is a Modified Julian Date (JD-2400000.5).
The universal elements supplied in the array U are in canonical units (solar masses, AU and canonical days). The position and velocity are not sensitive to the choice of reference frame. The palEl2ue routine in fact produces coordinates with respect to the J2000 equator and equinox.
The algorithm was originally adapted from the EPHSLA program of D.H.P.Jones (private
communication, 1996). The method is based on Stumpff’
s Universal Variables.
Reference: Everhart, E. & Pitkin, E.T., Am.J.Phys. 51, 712, 1983.