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<H2><A NAME="SECTION000478000000000000000">SLA_EL2UE - Conventional to Universal Elements</A>
<A NAME="xref_SLA_EL2UE"> </A><A NAME="SLA_EL2UE"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>Transform conventional osculating orbital elements
into ``universal'' form.
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_EL2UE (
DATE, JFORM, EPOCH, ORBINC, ANODE,
PERIH, AORQ, E, AORL, DM,
U, JSTAT)</TT>
<P> </DL>
<P> <DL>
<DT><STRONG>GIVEN:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>DATE</EM></TD>
<TH ALIGN="LEFT"><B>D</B></TH>
<TD ALIGN="LEFT" NOWRAP>epoch (TT MJD) of osculation (Note 3)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>JFORM</EM></TD>
<TD ALIGN="LEFT"><B>I</B></TD>
<TD ALIGN="LEFT" NOWRAP>choice of element set (1-3; Note 6)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>EPOCH</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>epoch of elements (<I>t<SUB>0</SUB></I> or <I>T</I>, TT MJD)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>ORBINC</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>inclination (<I>i</I>, radians)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>ANODE</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>longitude of the ascending node (<IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$">, radians)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>PERIH</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>longitude or argument of perihelion
(<IMG WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img100.gif"
ALT="$\varpi$"> or <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img101.gif"
ALT="$\omega$">,</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> radians)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>AORQ</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>mean distance or perihelion distance (<I>a</I> or <I>q</I>, AU)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>E</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>eccentricity (<I>e</I>)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>AORL</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>mean anomaly or longitude
(<I>M</I> or <I>L</I>, radians,</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> JFORM=1,2 only)</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>DM</EM></TD>
<TD ALIGN="LEFT"><B>D</B></TD>
<TD ALIGN="LEFT" NOWRAP>daily motion (<I>n</I>, radians, JFORM=1 only)</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>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 orbital elements (Note 1)</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>
<TR VALIGN="TOP"><TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT"> </TD>
<TD ALIGN="LEFT" NOWRAP> </TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>JSTAT</EM></TD>
<TD ALIGN="LEFT"><B>I</B></TD>
<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> 0 = OK</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -1 = illegal JFORM</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -2 = illegal E</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -3 = illegal AORQ</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -4 = illegal DM</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -5 = 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).
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 companion routine is sla_UE2PV. 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 sla_UE2PV;
for convenience, the two calls are packaged as the routine
sla_PLANEL. Multiple predictions may be made by again calling the
present routine once, but then calling sla_UE2PV multiple times,
which is faster than multiple calls to sla_PLANEL.
<DT>3.
<DD>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).
<DT>4.
<DD>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.
<DT>5.
<DD>The universal elements returned in the array U are in canonical
units (solar masses, AU and canonical days).
<DT>6.
<DD>Three different element-format options are supported, as
follows. <BR>
<P>
JFORM=1, suitable for the major planets:
<P> <PRE><TT>
EPOCH = epoch of elements <I>t<SUB>0</SUB></I> (TT MJD)
ORBINC = inclination <I>i</I> (radians)
ANODE = longitude of the ascending node <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$"> (radians)
PERIH = longitude of perihelion <IMG WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img100.gif"
ALT="$\varpi$"> (radians)
AORQ = mean distance <I>a</I> (AU)
E = eccentricity <I>e</I> <IMG WIDTH="83" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img106.gif"
ALT="$( 0 \leq e < 1 )$">
AORL = mean longitude <I>L</I> (radians)
DM = daily motion <I>n</I> (radians)
</TT></PRE>
<P>
JFORM=2, suitable for minor planets:
<P> <PRE><TT>
EPOCH = epoch of elements <I>t<SUB>0</SUB></I> (TT MJD)
ORBINC = inclination <I>i</I> (radians)
ANODE = longitude of the ascending node <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$"> (radians)
PERIH = argument of perihelion <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img101.gif"
ALT="$\omega$"> (radians)
AORQ = mean distance <I>a</I> (AU)
E = eccentricity <I>e</I> <IMG WIDTH="83" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img106.gif"
ALT="$( 0 \leq e < 1 )$">
AORL = mean anomaly <I>M</I> (radians)
</TT></PRE>
<P>
JFORM=3, suitable for comets:
<P> <PRE><TT>
EPOCH = epoch of perihelion <I>T</I> (TT MJD)
ORBINC = inclination <I>i</I> (radians)
ANODE = longitude of the ascending node <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$"> (radians)
PERIH = argument of perihelion <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img101.gif"
ALT="$\omega$"> (radians)
AORQ = perihelion distance <I>q</I> (AU)
E = eccentricity <I>e</I> <IMG WIDTH="91" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img107.gif"
ALT="$( 0 \leq e \leq 10 )$"></TT></PRE>
<DT>7.
<DD>Unused elements (DM for JFORM=2, AORL and DM for JFORM=3) are
not accessed.
<DT>8.
<DD>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.
</DL></DL>
<P> <DL>
<DT><STRONG>REFERENCE:</STRONG>
<DD>Everhart, E. & Pitkin, E.T., Am. J. Phys. 51, 712, 1983.
</DL>
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<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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