Transform conventional osculating elements into " universal" form.

void palEl2ue ( double date, int jform, double epoch, double orbinc, double anode, double perih, double aorq, double e, double aorl, double dm, double u[13], int $\ast$jstat );

date = double (Given)

jform = int (Given)

epoch = double (Given)

orbinc = double (Given)

anode = double (Given)

perih = double (Given)

aorq = double (Given)

e = double (Given)

aorl = double (Given)

dm = double (Given)

u = double [13] (Returned)

jstat = int $\ast$ (Returned)

• 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 palUe2pv. This takes the set of numbers that the present routine outputs and uses them to derive the object’ s position and velocity. A single prediction requires one call to the present routine followed by one call to palUe2pv; for convenience, the two calls are packaged as the routine palPlanel. Multiple predictions may be made by again calling the present routine once, but then calling palUe2pv multiple times, which is faster than multiple calls to palPlanel.

• DATE is the epoch of osculation. It is in the TT timescale (formerly Ephemeris Time, ET) and is a Modified Julian Date (JD-2400000.5).

• The supplied orbital elements are with respect to the J2000 ecliptic and equinox. The position and velocity parameters returned in the array U are with respect to the mean equator and equinox of epoch J2000, and are for the perihelion prior to the specified epoch.

• The universal elements returned in the array U are in canonical units (solar masses, AU and canonical days).

• Three different element-format options are available:

Option JFORM=1, suitable for the major planets:

EPOCH = epoch of elements (TT MJD) ORBINC = inclination i (radians) ANODE = longitude of the ascending node, big omega (radians) PERIH = longitude of perihelion, curly pi (radians) AORQ = mean distance, a (AU) E = eccentricity, e (range 0 to $<$1) AORL = mean longitude L (radians) DM = daily motion (radians)

Option JFORM=2, suitable for minor planets:

EPOCH = epoch of elements (TT MJD) ORBINC = inclination i (radians) ANODE = longitude of the ascending node, big omega (radians) PERIH = argument of perihelion, little omega (radians) AORQ = mean distance, a (AU) E = eccentricity, e (range 0 to $<$1) AORL = mean anomaly M (radians)

Option JFORM=3, suitable for comets:

EPOCH = epoch of perihelion (TT MJD) ORBINC = inclination i (radians) ANODE = longitude of the ascending node, big omega (radians) PERIH = argument of perihelion, little omega (radians) AORQ = perihelion distance, q (AU) E = eccentricity, e (range 0 to 10)

• Unused elements (DM for JFORM=2, AORL and DM for JFORM=3) are not accessed.

• 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.

Everhart & Pitkin, Am.J.Phys. 51, 712 (1983).