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<H2><A NAME="SECTION000512000000000000000">
Mean Place to Apparent Place</A>
</H2>
The <I>geocentric apparent place</I> of a source, or <I>apparent place</I>
for short,
is the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> if viewed from the centre of the Earth,
with respect to the true equator and equinox of date.
Transformation of an FK5 mean <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$">, equinox J2000,
current epoch, to apparent place involves the following effects:
<P><UL>
<LI> Light deflection - the gravitational lens effect of
the sun.
<LI> Annual aberration.
<LI> Precession/nutation.
</UL>
The <I>light deflection</I> is seldom significant. Its value
at the limb of the Sun is about
<P> <IMG WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img294.gif"
ALT="$1\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.74$"> ; it falls off rapidly with distance from the
Sun and has shrunk to about
<P> <IMG WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img295.gif"
ALT="$0\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.02$"> at an elongation of <IMG WIDTH="26" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img296.gif"
ALT="$20^\circ$">.<P>
As already described, the <I>annual aberration</I>
is a function of the Earth's velocity
relative to the solar system barycentre (available through the
SLALIB routine
sla_EVP)
and produces shifts of up to about
<IMG WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
SRC="img265.gif"
ALT="$20\hspace{-0.05em}^{'\hspace{-0.1em}'}\hspace{-0.4em}.5$"> .
<P>
The <I>precession/nutation</I>, from J2000 to the current epoch, is
expressed by a rotation matrix which is available through the
SLALIB routine
sla_PRENUT.
<P>
The whole mean-to-apparent transformation can be done using the SLALIB
routine
sla_MAP. As a demonstration, here is a program which lists the
<I>North Polar Distance</I> (<IMG WIDTH="54" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
SRC="img297.gif"
ALT="$90^\circ-\delta$">) of Polaris for
the decade of closest approach to the Pole:
<P><PRE>
IMPLICIT NONE
DOUBLE PRECISION PI,PIBY2,D2R,S2R,AS2R
PARAMETER (PI=3.141592653589793238462643D0)
PARAMETER (D2R=PI/180D0,
: PIBY2=PI/2D0,
: S2R=PI/(12D0*3600D0),
: AS2R=PI/(180D0*3600D0))
DOUBLE PRECISION RM,DM,PR,PD,DATE,RA,DA
INTEGER J,IDS,IDE,ID,IYMDF(4),I
DOUBLE PRECISION sla_EPJ2D
CALL sla_DTF2R(02,31,49.8131D0,RM,J)
CALL sla_DAF2R(89,15,50.661D0,DM,J)
PR=+21.7272D0*S2R/100D0
PD=-1.571D0*AS2R/100D0
WRITE (*,'(1X,'//
: '''Polaris north polar distance (deg) 2096-2105''/)')
WRITE (*,'(4X,''Date'',7X''NPD''/)')
CALL sla_CLDJ(2096,1,1,DATE,J)
IDS=NINT(DATE)
CALL sla_CLDJ(2105,12,31,DATE,J)
IDE=NINT(DATE)
DO ID=IDS,IDE,10
DATE=DBLE(ID)
CALL sla_DJCAL(0,DATE,IYMDF,J)
CALL sla_MAP(RM,DM,PR,PD,0D0,0D0,2000D0,DATE,RA,DA)
WRITE (*,'(1X,I4,2I3.2,F9.5)') (IYMDF(I),I=1,3),(PIBY2-DA)/D2R
END DO
END
</PRE>
<P>
For cases where the transformation has to be repeated for different
times or for more than one star, the straightforward
sla_MAP
approach is apt to be
wasteful as both the Earth velocity and the
precession/nutation matrix can be re-calculated relatively
infrequently without ill effect. A more efficient method is to
perform the target-independent calculations only when necessary,
by calling
sla_MAPPA,
and then to use either
sla_MAPQKZ,
when only the <IMG WIDTH="42" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img3.gif"
ALT="$[\,\alpha,\delta\,]$"> is known, or
sla_MAPQK,
when full catalogue positions, including proper motion, parallax and
radial velocity, are available. How frequently to call
sla_MAPPA
depends on the accuracy objectives; once per
night will deliver sub-arcsecond accuracy for example.
<P>
The routines
sla_AMP
and
sla_AMPQK
allow the reverse transformation, from apparent to mean place.
<P>
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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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