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<H2><A NAME="SECTION0004132000000000000000">SLA_PERTUE - Perturbed Universal Elements</A>
<A NAME="xref_SLA_PERTUE"> </A><A NAME="SLA_PERTUE"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>Update the universal elements of an asteroid or comet by
applying planetary perturbations.
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_PERTUE (DATE, U, JSTAT)</TT>
<P> </DL>
<P> <DL>
<DT><STRONG>GIVEN:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>DATE1</EM></TD>
<TH ALIGN="LEFT"><B>D</B></TH>
<TD ALIGN="LEFT" NOWRAP>final epoch (TT MJD) for the updated elements</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>GIVEN and RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>U</EM></TD>
<TH ALIGN="LEFT"><B>D(13)</B></TH>
<TD ALIGN="LEFT" NOWRAP>universal elements (updated in place)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(1)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>combined mass (<I>M</I>+<I>m</I>)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(2)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>total energy of the orbit (<IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img24.gif"
ALT="$\alpha$">)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(3)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>reference (osculating) epoch (<I>t<SUB>0</SUB></I>)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(4-6)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>position at reference epoch (<IMG WIDTH="17" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img102.gif"
ALT="${\rm \bf r}_0$">)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(7-9)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>velocity at reference epoch (<IMG WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img103.gif"
ALT="${\rm \bf v}_0$">)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(10)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>heliocentric distance at reference epoch</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(11)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP><IMG WIDTH="39" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img104.gif"
ALT="${\rm \bf r}_0.{\rm \bf v}_0$"></TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(12)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>date (<I>t</I>)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="CENTER" NOWRAP COLSPAN=1>(13)</TD>
<TD></TD>
<TD ALIGN="LEFT" NOWRAP>universal eccentric anomaly (<IMG WIDTH="14" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img105.gif"
ALT="$\psi$">) of date, approx</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>JSTAT</EM></TD>
<TH ALIGN="LEFT"><B>I</B></TH>
<TD ALIGN="LEFT" NOWRAP>status:</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> +102 = warning, distant epoch</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> +101 = warning, large timespan
(>100 years)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> +1 to +8 = coincident with major planet
(Note 5)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> 0 = OK</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -1 = numerical error</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>NOTES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>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) <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img24.gif"
ALT="$\alpha$">, 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 <IMG WIDTH="14" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img105.gif"
ALT="$\psi$">, the ``universal eccentric anomaly'' at a
given date and (v) that date.
<DT>2.
<DD>The universal elements are with respect to the J2000 equator and
equinox.
<DT>3.
<DD>The epochs DATE, U(3) and U(12) are all Modified Julian Dates
(JD-2400000.5).
<DT>4.
<DD>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.
<P>
The <I>f</I>(<I>q</I>) 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.
<P>
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.
<P>
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.
<P>
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.
<P>
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 sla_PLANET. Two
levels of interpolation are used, to enhance speed without
significantly degrading accuracy. At a low frequency, the routine
sla_PLANET is called to generate updated position+velocity ``state
vectors''. The only task remaining to be carried out at the full
frequency (<I>i.e.</I> 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.
<P>
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.
<P>
In the interests of simplicity, the background calculations for
the major planets are carried out <I>en masse.</I>
The mean elements and
state vectors for all the planets are refreshed at the same time,
without regard for orbit curvature, mass or proximity.
<P> <DT>5.
<DD>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.
</DL></DL>
<P> <DL>
<DT><STRONG>REFERENCES:</STRONG>
<DD><DL COMPACT>
<DT>1.
<DD>Sterne, Theodore E., <I>An Introduction to Celestial Mechanics,</I>
Interscience Publishers, 1960. Section 6.7, p199.
<DT>2.
<DD>Everhart, E. & Pitkin, E.T., Am. J. Phys. 51, 712, 1983.
</DL></DL>
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<ADDRESS>
<I>SLALIB --- Positional Astronomy Library<BR>Starlink User Note 67<BR>P. T. Wallace<BR>12 October 1999<BR>E-mail:ptw@star.rl.ac.uk</I>
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