Astronomical Mapping

``No matter how tricky you are, you cannot pave a sphere with square tiles.''
(Attributed to A. Einstein)

About $101 \%$ of the difficulties encountered in astronomical mapping comes from this very profound evidence. The remaining few percents (within the errors) come from the multiplicity of coordinate systems used by the astronomers. For small field mapping, the sphere can reasonably be approximated by its tangent plane at the field center, but for fields larger than a few degrees (depending on the required positional accuracy), curvature effects become important.

As spherical plotters are not easily commercially available (and their output hardly accepted by journal editors), the big problem is to represent part of a sphere on a plane sheet. GREG offers several facilities to deal with this problem, by means of commands PROJECTION, CONVERT and GRID in the GREG2 language.

The current philosophy to handle this problem is to make all plotting in relative coordinates on the projection plane. No drawing is done in absolute coordinates on the sphere. This was decided because the plot page is plane indeed. Another apparently restrictive convention in GREG projections is that all angles are internally in the natural angular unit Radian. Accordingly, when you give a map to be contoured, the map coordinates should be offsets in radians from the projection center. For user convenience, it is possible to specify limits in Degrees, Minutes or Seconds using the command SET ANGLE_UNIT, but this command has no effect on internal conversion formulae. Again, the rationale behind this convention is that we are in fact working in the projection plane, where angles have no meaning but only their projections remain.

However, it is always possible to bypass these apparent restrictions provided your understand what a conversion formula and a projection are. As an example is always much better than lengthy discussion, assume you want to overlay a contour map to the Equatorial and Galactic grids relevant to the mapped area. Unfortunately, the map coordinates are in Degrees from a point situated at (0.5,0.5) (``degrees'' in projection) from the projection center (note that this sentence is complete nonsense, because you cannot have ``degrees'' in the projection plane...)

SET ANGLE_UNIT/DEFAULT                          ! (1)
LIMITS 5 -5 -5 5                                ! (2)
RGDATA MYMAP                                    ! (3)
LEVELS -1 1 TO 5                                ! (3)
RGMAP/BLANKING -2 .1                            ! (3)
!
SET SYSTEM EQUATORIAL                           ! (4)
PROJECTION 6:25:30 35:40.5 0.0/TYPE GNOMONIC    ! (5)
SET ANGLE_UNIT DEGREES                          ! (6)
LIMITS 4.5 -5.5 -5.5 4.5                        ! (7)
SHOW LIMITS                                     ! (7)
PEN 1                                           ! 
GRID 2 2                                        ! (8)
PEN 2                                           !
GRID 2 2/ALTERNATE                              ! (9)

1. Make sure to work as if you had no problem at all
2. Define the map limits. As you are not worrying about the angular units, the internal limits will be just what you type.
3. Read your map and draw your contours. This is the usual process for any map.
4. Here start the specific astronomical problem. You know that your coordinate system is Equatorial. Specify it.
5. Define the projection in this system. Note that Right Ascension is in Hours, and Declination in Degrees as usual for Equatorial coordinates. The projection type depends on your problem of course.
6. Specify that your will now be giving limits in Degrees
7. Give the limits of the map with respect to the projection center. The system automatically converts the values typed in to internal limits in radians, as you can check by the SHOW command. Note how you handle the shift between the (0,0) of your map and the projection center which is always the (0,0) of the projection plane (compare with command (2))
8. Everything is now correct to plot the grid of meridians and parallels over your map.
9. You may even plot a Galactic grid just by specifying the /ALTERNATE option in the GRID command.

As shown in this example, the only effect of the SET ANGLE_UNIT command is to force automatic conversion of the typed limits to radians. The same result would have been obtained by typing LIMITS 4.5 -5.5 -5.5 4.5 DEGREE
instead of commands (6) and (7).

You can work in three different systems :

• UNKNOWN, for which GREG makes no assumption at all about the meaning of the unprojected coordinates. In particular, there is no alternate system in this case.
• EQUATORIAL, which GREG assumes to be at epoch 1950.0 . The alternate system is GALACTIC of course.
• GALACTIC, with EQUATORIAL 1950.0 system as alternate.

When you measure positions with the cursor in the last two systems, both galactic and equatorial coordinates are given. For user convenience, sexagesimal notations in Hours Minutes Seconds for Right Ascension and Degrees, Minutes Seconds for Declination are used, while decimal notations with angle in degrees are used for L and B (as well as if the system is UNKNOWN).

Approximate absolute labelling in Hours and Degrees (sexagesimal notation) can be obtained for the Box by specifying the option /ABSOLUTE to command BOX if the system is EQUATORIAL. If the system is GALACTIC or UNKNOWN, this option produces absolute labelling in Degrees. Note that it is only meaningful for small fields of view. Without this option, BOX produces relative labelling in current angle units.

In addition, command CONVERT can convert absolute positions or projected coordinates from a different projection (in any angular units), to projected coordinates (in radians) in the current projection. This command may be used to plot star positions on a map, and so on.