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+.help pertue Jun99 "Slalib Package"
+.nf
+
+      SUBROUTINE slPRTE (DATE, U, JSTAT)
+
+     - - - - - - -
+      P R T E
+     - - - - - - -
+
+  Update the universal elements of an asteroid or comet by applying
+  planetary perturbations.
+
+  Given:
+     DATE     d       final epoch (TT MJD) for the updated elements
+
+  Given and returned:
+     U        d(13)   universal elements (updated in place)
+
+                (1)   combined mass (M+m)
+                (2)   total energy of the orbit (alpha)
+                (3)   reference (osculating) epoch (t0)
+              (4-6)   position at reference epoch (r0)
+              (7-9)   velocity at reference epoch (v0)
+               (10)   heliocentric distance at reference epoch
+               (11)   r0.v0
+               (12)   date (t)
+               (13)   universal eccentric anomaly (psi) of date, approx
+
+  Returned:
+     JSTAT    i       status:
+                          +102 = warning, distant epoch
+                          +101 = warning, large timespan ( > 100 years)
+                      +1 to +8 = coincident with major planet (Note 5)
+                             0 = OK
+                            -1 = numerical error
+
+  Called:  slPLNT, slUEPV, slPVUE
+
+  Notes:
+
+  1  The "universal" elements are those which define the orbit for the
+     purposes of the method of universal variables (see reference 2).
+     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.
+
+  2  The universal elements are with respect to the J2000 equator and
+     equinox.
+
+  3  The epochs DATE, U(3) and U(12) are all Modified Julian Dates
+     (JD-2400000.5).
+
+  4  The algorithm is a simplified form of Encke's method.  It takes as
+     a basis the unperturbed motion of the body, and numerically
+     integrates the perturbing accelerations from the major planets.
+     The expression used is essentially Sterne's 6.7-2 (reference 1).
+     Everhart and Pitkin (reference 2) suggest rectifying the orbit at
+     each integration step by propagating the new perturbed position
+     and velocity as the new universal variables.  In the present
+     routine the orbit is rectified less frequently than this, in order
+     to gain a slight speed advantage.  However, the rectification is
+     done directly in terms of position and velocity, as suggested by
+     Everhart and Pitkin, bypassing the use of conventional orbital
+     elements.
+
+     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 slPLNT.  Two
+     levels of interpolation are used, to enhance speed without
+     significantly degrading accuracy.  At a low frequency, the routine
+     slPLNT 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.
+
+  5  This routine is not intended to be used for major planets.
+     However, if major-planet elements are supplied, sensible results
+     will, in fact, be produced.  This happens because the routine
+     checks the separation between the body and each of the planets and
+     interprets a suspiciously small value (0.001 AU) as an attempt to
+     apply the routine to the planet concerned.  If this condition is
+     detected, the contribution from that planet is ignored, and the
+     status is set to the planet number (Mercury=1,...,Neptune=8) as a
+     warning.
+
+  References:
+
+     1  Sterne, Theodore E., "An Introduction to Celestial Mechanics",
+        Interscience Publishers Inc., 1960.  Section 6.7, p199.
+
+     2  Everhart, E. & Pitkin, E.T., Am.J.Phys. 51, 712, 1983.
+
+  P.T.Wallace   Starlink   18 March 1999
+
+  Copyright (C) 1999 Rutherford Appleton Laboratory
+  Copyright (C) 1995 Association of Universities for Research in Astronomy Inc.
+
+.fi
+.endhelp
+
+.fi
+.endhelp