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      SUBROUTINE slPRTE (DATE, U, JSTAT)

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      P R T E
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  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.

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