Positions on the celestial sphere may be specified by using
a spherical polar coordinate system, defined in terms of
some fundamental plane and a line in that plane chosen to
represent zero longitude. Mathematicians usually work with the
co-latitude, with zero at the principal pole, whereas most
astronomical coordinate systems use latitude, reckoned plus and
minus from the equator.
Astronomical coordinate systems may be either right-handed
(*e.g.* right ascension and declination ,Galactic longitude and latitude )or left-handed (*e.g.* hour angle and
declination ). In some cases
different conventions have been used in the past, a fruitful source of
mistakes. Azimuth and geographical longitude are examples; azimuth
is now generally reckoned north through east
(making a left-handed system); geographical longitude is now usually
taken to increase eastwards (a right-handed system) but astronomers
used to employ a west-positive convention. In reports
and program comments it is wise to spell out what convention
is being used, if there is any possibility of confusion.

When applying spherical trigonometry formulae, attention must be paid to rounding errors (for example it is a bad idea to find a small angle through its cosine) and to the possibility of problems close to poles. Also, if a formulation relies on inspection to establish the quadrant of the result, it is an indication that a vector-related method might be preferable.

As well as providing many routines which work in terms of specific
spherical coordinates such as , SLALIB provides
two routines which operate directly on generic spherical
coordinates:
sla_SEP
computes the separation between
two points (the distance along a great circle) and
sla_BEAR
computes the bearing (or *position angle*)
of one point seen from the other. The routines
sla_DSEP
and
sla_DBEAR
are double precision equivalents. As a simple demonstration
of SLALIB, we will use these facilities to estimate the distance from
London to Sydney and the initial compass heading:

IMPLICIT NONE * Degrees to radians REAL D2R PARAMETER (D2R=0.01745329252) * Longitudes and latitudes (radians) for London and Sydney REAL AL,BL,AS,BS PARAMETER (AL=-0.2*D2R,BL=51.5*D2R,AS=151.2*D2R,BS=-33.9*D2R) * Earth radius in km (spherical approximation) REAL RKM PARAMETER (RKM=6375.0) REAL sla_SEP,sla_BEAR * Distance and initial heading (N=0, E=90) WRITE (*,'(1X,I5,'' km,'',I4,'' deg'')') : NINT(sla_SEP(AL,BL,AS,BS)*RKM),NINT(sla_BEAR(AL,BL,AS,BS)/D2R) END

(The result is 17011 km, .)

The routines sla_PAV and sla_DPAV are equivalents of sla_BEAR and sla_DBEAR but starting from direction-cosines instead of spherical coordinates.

Starlink User Note 67

P. T. Wallace

12 October 1999

E-mail:ptw@star.rl.ac.uk