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<H2><A NAME="SECTION0004174000000000000000">SLA_UE2EL - Universal to Conventional Elements</A>
<A NAME="xref_SLA_UE2EL"> </A><A NAME="SLA_UE2EL"> </A>
</H2>
<DL>
<DT><STRONG>ACTION:</STRONG>
<DD>Transform universal elements into conventional heliocentric
osculating elements.
<P> <DT><STRONG>CALL:</STRONG>
<DD><TT>CALL sla_UE2EL (
U, JFORMR,
JFORM, EPOCH, ORBINC, ANODE, PERIH,
AORQ, E, AORL, DM, JSTAT)</TT>
<P> </DL>
<P> <DL>
<DT><STRONG>GIVEN:</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 (updated; 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>JFORMR</EM></TD>
<TD ALIGN="LEFT"><B>I</B></TD>
<TD ALIGN="LEFT" NOWRAP>requested element set (1-3; Note 3)</TD>
</TR>
</TABLE></DL>
<P> <DL>
<DT><STRONG>RETURNED:</STRONG>
<DD>
<BR>
<TABLE CELLPADDING=3>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM>JFORM</EM></TD>
<TH ALIGN="LEFT"><B>I</B></TH>
<TD ALIGN="LEFT" NOWRAP>element set actually returned (1-3; Note 4)</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>
<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 PMASS</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -2 = illegal JFORMR</TD>
</TR>
<TR VALIGN="TOP"><TD ALIGN="LEFT"><EM></EM></TD>
<TD ALIGN="LEFT"><B></B></TD>
<TD ALIGN="LEFT" NOWRAP> -3 = position/velocity out of allowed range</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 mean equator and
equinox of epoch J2000. The orbital elements produced are with
respect to the J2000 ecliptic and mean equinox.
<DT>3.
<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>4.
<DD>It may not be possible to generate elements in the form
requested through JFORMR. The caller is notified of the form
of elements actually returned by means of the JFORM argument:
<P> <PRE><TT>
JFORMR JFORM meaning
1 1 OK: elements are in the requested format
1 2 never happens
1 3 orbit not elliptical
2 1 never happens
2 2 OK: elements are in the requested format
2 3 orbit not elliptical
3 1 never happens
3 2 never happens
3 3 OK: elements are in the requested format
</TT></PRE>
<DT>5.
<DD>The arguments returned for each value of JFORM (<I>cf</I> Note 5:
JFORM may not be the same as JFORMR) are as follows:
<P> <PRE><TT>
JFORM 1 2 3
EPOCH <I>t<SUB>0</SUB></I> <I>t<SUB>0</SUB></I> <I>T</I>
ORBINC <I>i</I> <I>i</I> <I>i</I>
ANODE <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$"> <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$"> <IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$">
PERIH <IMG WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img100.gif"
ALT="$\varpi$"> <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img101.gif"
ALT="$\omega$"> <IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img101.gif"
ALT="$\omega$">
AORQ <I>a</I> <I>a</I> <I>q</I>
E <I>e</I> <I>e</I> <I>e</I>
AORL <I>L</I> <I>M</I> -
DM <I>n</I> - -
</TT></PRE>
<P>
where:
<PRE><TT>
<I>t<SUB>0</SUB></I> is the epoch of the elements (MJD, TT)
<I>T</I> is the epoch of perihelion (MJD, TT)
<I>i</I> is the inclination (radians)
<IMG WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img99.gif"
ALT="$\Omega$"> is the longitude of the ascending node (radians)
<IMG WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img100.gif"
ALT="$\varpi$"> is the longitude of perihelion (radians)
<IMG WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img101.gif"
ALT="$\omega$"> is the argument of perihelion (radians)
<I>a</I> is the mean distance (AU)
<I>q</I> is the perihelion distance (AU)
<I>e</I> is the eccentricity
<I>L</I> is the longitude (radians, <IMG WIDTH="48" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img143.gif"
ALT="$0-2\pi$">) <I>M</I> is the mean anomaly (radians, <IMG WIDTH="48" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img143.gif"
ALT="$0-2\pi$">) <I>n</I> is the daily motion (radians)
- means no value is set
</TT></PRE>
<DT>6.
<DD>At very small inclinations, the longitude of the ascending node
ANODE becomes indeterminate and under some circumstances may be
set arbitrarily to zero. Similarly, if the orbit is close to
circular, the true anomaly becomes indeterminate and under some
circumstances may be set arbitrarily to zero. In such cases,
the other elements are automatically adjusted to compensate,
and so the elements remain a valid description of the orbit.
</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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