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SUBROUTINE slUEPV ( DATE, U, PV, JSTAT )
*+
* - - - - - -
* U E P V
* - - - - - -
*
* Heliocentric position and velocity of a planet, asteroid or comet,
* starting from orbital elements in the "universal variables" form.
*
* Given:
* DATE d date, Modified Julian Date (JD-2400000.5)
*
* Given and returned:
* U d(13) universal orbital elements (updated; Note 1)
*
* given (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
* returned (12) date (t)
* " (13) universal eccentric anomaly (psi) of date
*
* Returned:
* PV d(6) position (AU) and velocity (AU/s)
* JSTAT i status: 0 = OK
* -1 = radius vector zero
* -2 = failed to converge
*
* Notes
*
* 1 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.
*
* 2 The companion routine is slELUE. 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 slELUE followed by one call to the present routine;
* for convenience, the two calls are packaged as the routine
* slPLNE. Multiple predictions may be made by again
* calling slELUE once, but then calling the present routine
* multiple times, which is faster than multiple calls to slPLNE.
*
* It is not obligatory to use slELUE to obtain the parameters.
* However, it should be noted that because slELUE performs its
* own validation, no checks on the contents of the array U are made
* by the present routine.
*
* 3 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).
*
* 4 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
* slELUE routine in fact produces coordinates with respect to the
* J2000 equator and equinox.
*
* 5 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.
*
* P.T.Wallace Starlink 22 October 2005
*
* Copyright (C) 2005 Rutherford Appleton Laboratory
*
* License:
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program (see SLA_CONDITIONS); if not, write to the
* Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
* Boston, MA 02110-1301 USA
*
* Copyright (C) 1995 Association of Universities for Research in Astronomy Inc.
*-
IMPLICIT NONE
DOUBLE PRECISION DATE,U(13),PV(6)
INTEGER JSTAT
* Gaussian gravitational constant (exact)
DOUBLE PRECISION GCON
PARAMETER (GCON=0.01720209895D0)
* Canonical days to seconds
DOUBLE PRECISION CD2S
PARAMETER (CD2S=GCON/86400D0)
* Test value for solution and maximum number of iterations
DOUBLE PRECISION TEST
INTEGER NITMAX
PARAMETER (TEST=1D-13,NITMAX=25)
INTEGER I,NIT,N
DOUBLE PRECISION CM,ALPHA,T0,P0(3),V0(3),R0,SIGMA0,T,PSI,DT,W,
: TOL,PSJ,PSJ2,BETA,S0,S1,S2,S3,
: FF,R,FLAST,PLAST,F,G,FD,GD
* Unpack the parameters.
CM = U(1)
ALPHA = U(2)
T0 = U(3)
DO I=1,3
P0(I) = U(I+3)
V0(I) = U(I+6)
END DO
R0 = U(10)
SIGMA0 = U(11)
T = U(12)
PSI = U(13)
* Approximately update the universal eccentric anomaly.
PSI = PSI+(DATE-T)*GCON/R0
* Time from reference epoch to date (in Canonical Days: a canonical
* day is 58.1324409... days, defined as 1/GCON).
DT = (DATE-T0)*GCON
* Refine the universal eccentric anomaly, psi.
NIT = 1
W = 1D0
TOL = 0D0
DO WHILE (ABS(W).GE.TOL)
* Form half angles until BETA small enough.
N = 0
PSJ = PSI
PSJ2 = PSJ*PSJ
BETA = ALPHA*PSJ2
DO WHILE (ABS(BETA).GT.0.7D0)
N = N+1
BETA = BETA/4D0
PSJ = PSJ/2D0
PSJ2 = PSJ2/4D0
END DO
* Calculate Universal Variables S0,S1,S2,S3 by nested series.
S3 = PSJ*PSJ2*((((((BETA/210D0+1D0)
: *BETA/156D0+1D0)
: *BETA/110D0+1D0)
: *BETA/72D0+1D0)
: *BETA/42D0+1D0)
: *BETA/20D0+1D0)/6D0
S2 = PSJ2*((((((BETA/182D0+1D0)
: *BETA/132D0+1D0)
: *BETA/90D0+1D0)
: *BETA/56D0+1D0)
: *BETA/30D0+1D0)
: *BETA/12D0+1D0)/2D0
S1 = PSJ+ALPHA*S3
S0 = 1D0+ALPHA*S2
* Undo the angle-halving.
TOL = TEST
DO WHILE (N.GT.0)
S3 = 2D0*(S0*S3+PSJ*S2)
S2 = 2D0*S1*S1
S1 = 2D0*S0*S1
S0 = 2D0*S0*S0-1D0
PSJ = PSJ+PSJ
TOL = TOL+TOL
N = N-1
END DO
* Values of F and F' corresponding to the current value of psi.
FF = R0*S1+SIGMA0*S2+CM*S3-DT
R = R0*S0+SIGMA0*S1+CM*S2
* If first iteration, create dummy "last F".
IF ( NIT.EQ.1) FLAST = FF
* Check for sign change.
IF ( FF*FLAST.LT.0D0 ) THEN
* Sign change: get psi adjustment using secant method.
W = FF*(PLAST-PSI)/(FLAST-FF)
ELSE
* No sign change: use Newton-Raphson method instead.
IF (R.EQ.0D0) GO TO 9010
W = FF/R
END IF
* Save the last psi and F values.
PLAST = PSI
FLAST = FF
* Apply the Newton-Raphson or secant adjustment to psi.
PSI = PSI-W
* Next iteration, unless too many already.
IF (NIT.GT.NITMAX) GO TO 9020
NIT = NIT+1
END DO
* Project the position and velocity vectors (scaling velocity to AU/s).
W = CM*S2
F = 1D0-W/R0
G = DT-CM*S3
FD = -CM*S1/(R0*R)
GD = 1D0-W/R
DO I=1,3
PV(I) = P0(I)*F+V0(I)*G
PV(I+3) = CD2S*(P0(I)*FD+V0(I)*GD)
END DO
* Update the parameters to allow speedy prediction of PSI next time.
U(12) = DATE
U(13) = PSI
* OK exit.
JSTAT = 0
GO TO 9999
* Null radius vector.
9010 CONTINUE
JSTAT = -1
GO TO 9999
* Failed to converge.
9020 CONTINUE
JSTAT = -2
9999 CONTINUE
END
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