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<H2><A NAME="SECTION000511000000000000000">
Mean Place Transformations</A>
</H2>
Figure 1 is based upon three varieties of mean <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> all of which are
of practical significance to observing astronomers in the present era:
<UL>
<LI> Old style (FK4) with known proper motion in the FK4
system, and with parallax and radial velocity either
known or assumed zero.
<LI> Old style (FK4) with zero proper motion in FK5,
and with parallax and radial velocity assumed zero.
<LI> New style (FK5) with proper motion, parallax and
radial velocity either known or assumed zero.
</UL>
The figure outlines the steps required to convert positions in
any of these systems to a J2000 <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> for the current
epoch, as might be required in a telescope-control
program for example.
Most of the steps can be carried out by calling a single
SLALIB routines; there are other SLALIB routines which
offer set-piece end-to-end transformation routines for common cases.
Note, however, that SLALIB does not set out to provide the capability
for arbitrary transformations of star-catalogue data
between all possible systems of mean <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$">.Only in the (common) cases of FK4, equinox and epoch B1950,
to FK5, equinox and epoch J2000, and <I>vice versa</I> are
proper motion, parallax and radial velocity transformed
along with the star position itself, the
focus of SLALIB support.
<P>
As an example of using SLALIB to transform mean places, here is
a program which implements the top-left path of Figure 1.
An FK4 <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> of arbitrary equinox and epoch and with
known proper motion and
parallax is transformed into an FK5 J2000 <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> for the current
epoch. As a test star we will use <IMG WIDTH="30" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img285.gif"
ALT="$\alpha=$"><IMG WIDTH="104" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img286.gif"
ALT="$16^{h}\,09^{m}\,55^{s}.13$">,<IMG WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img287.gif"
ALT="$\delta=$"><IMG WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
SRC="img288.gif"
ALT="$-75^{\circ}\,59^{'}\,27^{''}.2$">, equinox 1900, epoch 1963.087,
<IMG WIDTH="38" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img289.gif"
ALT="$\mu_\alpha=$"><IMG WIDTH="61" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
SRC="img290.gif"
ALT="$-0^{\rm s}\hspace{-0.3em}.0312$">/<I>y</I>, <IMG WIDTH="36" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
SRC="img291.gif"
ALT="$\mu_\delta=$"> <IMG WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
SRC="img292.gif"
ALT="$+0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.103$"> /<I>y</I>,
parallax =
<IMG WIDTH="39" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img293.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.062$"> , radial velocity = -34.22 km/s. The
epoch of observation is 1994.35.
<P><PRE>
IMPLICIT NONE
DOUBLE PRECISION AS2R,S2R
PARAMETER (AS2R=4.8481368110953599D-6,S2R=7.2722052166430399D-5)
INTEGER J,I
DOUBLE PRECISION R0,D0,EQ0,EP0,PR,PD,PX,RV,EP1,R1,D1,R2,D2,R3,D3,
: R4,D4,R5,D5,R6,D6,EP1D,EP1B,W(3),EB(3),PXR,V(3)
DOUBLE PRECISION sla_EPB,sla_EPJ2D
* RA, Dec etc of example star
CALL sla_DTF2R(16,09,55.13D0,R0,J)
CALL sla_DAF2R(75,59,27.2D0,D0,J)
D0=-D0
EQ0=1900D0
EP0=1963.087D0
PR=-0.0312D0*S2R
PD=+0.103D0*AS2R
PX=0.062D0
RV=-34.22D0
EP1=1994.35D0
* Epoch of observation as MJD and Besselian epoch
EP1D=sla_EPJ2D(EP1)
EP1B=sla_EPB(EP1D)
* Space motion to the current epoch
CALL sla_PM(R0,D0,PR,PD,PX,RV,EP0,EP1B,R1,D1)
* Remove E-terms of aberration for the original equinox
CALL sla_SUBET(R1,D1,EQ0,R2,D2)
* Precess to B1950
R3=R2
D3=D2
CALL sla_PRECES('FK4',EQ0,1950D0,R3,D3)
* Add E-terms for the standard equinox B1950
CALL sla_ADDET(R3,D3,1950D0,R4,D4)
* Transform to J2000, no proper motion
CALL sla_FK45Z(R4,D4,EP1B,R5,D5)
* Parallax
CALL sla_EVP(sla_EPJ2D(EP1),2000D0,W,EB,W,W)
PXR=PX*AS2R
CALL sla_DCS2C(R5,D5,V)
DO I=1,3
V(I)=V(I)-PXR*EB(I)
END DO
CALL sla_DCC2S(V,R6,D6)
:
</PRE>
<P>
It is interesting to look at how the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> changes during the
course of the calculation:
<PRE><TT>
<TT>16 09 55.130 -75 59 27.20</TT> <I>original equinox and epoch</I>
<TT>16 09 54.155 -75 59 23.98</TT> <I>with space motion</I>
<TT>16 09 54.229 -75 59 24.18</TT> <I>with old E-terms removed</I>
<TT>16 16 28.213 -76 06 54.57</TT> <I>precessed to 1950.0</I>
<TT>16 16 28.138 -76 06 54.37</TT> <I>with new E-terms</I>
<TT>16 23 07.901 -76 13 58.87</TT> <I>J2000, current epoch</I>
<TT>16 23 07.907 -76 13 58.92</TT> <I>including parallax</I>
</TT></PRE>
<P>
Other remarks about the above (unusually complicated) example:
<UL>
<LI> If the original equinox and epoch were B1950, as is quite
likely, then it would be unnecessary to treat space motions
and E-terms explicitly. Transformation to FK5 J2000 could
be accomplished simply by calling
sla_FK425, after which
a call to
sla_PM and the parallax code would complete the
work.
<LI> The rigorous treatment of the E-terms
has only a small effect on the result. Such refinements
are, nevertheless, worthwhile in order to facilitate comparisons and
to increase the chances that star positions from different
suppliers are compatible.
<LI> The FK4 to FK5 transformations,
sla_FK425
and
sla_FK45Z,
are not as is sometimes assumed simply 50 years of precession,
though this indeed accounts for most of the change. The
transformations also include adjustments
to the equinox, a revised precession model, elimination of the
E-terms, a change to the proper-motion time unit and so on.
The reason there are two routines rather than just one
is that the FK4 frame rotates relative to the background, whereas
the FK5 frame is a much better approximation to an
inertial frame, and zero proper
motion in FK4 does not, therefore, mean zero proper motion in FK5.
SLALIB also provides two routines,
sla_FK524
and
sla_FK54Z,
to perform the inverse transformations.
<LI> Some star catalogues (FK4 itself is one) were constructed using slightly
different procedures for the polar regions compared with
elsewhere. SLALIB ignores this inhomogeneity and always
applies the standard
transformations irrespective of location on the celestial sphere.
</UL>
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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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