International Meteor Organization
Gnomonic Atlas Brno 2000.0
The charts of Atlas Brno 2000.0 are, due to its large scale and limiting magnitude, the most suitable ones for meteor plotting known to the IMO. Observers are permitted to photocopy the charts for use in their own observations only. Please take care that the original scale remains unchanged when copying, because photocopiers tend to change the scale somewhat. Therefore always use the originals whenever you are going to make copies, otherwise you may add to the errors. If the length of the long side of a copy differs by more than 3 mm from 350 mm you should use another photocopier. The following paragraphs describe the details of the Atlas and the mathematical formulae to convert cartesian coordinates into equatorial ones and vice versa. You may skip to this section if you are not interested in these details now.
Stars and constellations
The Atlas contains stars down to magnitude +6.5, according to the SAO Catalog, with the addition of a number of stellar objects not included in this catalog. Since it is mainly designed for use by naked eye observers, binary and multiple stars are not marked. Objects separated by more than 3' are recorded separately, while those closer together are depicted as one object, provided their total brightness is over the lower limit of the Atlas. Variable stars are drawn according to their maximum brightness, and where the amplitude of their variations is larger than half a magnitude, they are marked with the letter "V".
Reference stars were selected so as to have a small color index (B-V) in the UBV system; reference stars fainter than +4.5 have (B-V) smaller than half a magnitude, but in view of the lack of suitable bright reference stars, the value of this limit increases to 1.25 for stars brighter than magnitude +1.5. The brightness of the reference stars is expressed in units of 0.1 mag, without a decimal point. Sirius (alpha CMa) is noted as "-15" on the Atlas, which means its magnitude is -1.5. Rigel (beta Ori) is "01" (+0.1). Variable stars were not used as reference stars, though some of them have quite a low amplitude (e.g. Capella).
Star disc diameters on the maps are in units of 0.7~magnitudes, from +6.5 upwards, in nine classes of magnitude. The positions of all stars was calculated for the epoch 2000.0, taking into account their proper motions.
Most constellations are depicted with the customary alignments joining their stars, as an aid to orientation. Their names are represented by the official three-letter abbreviations, in capital letters.
Page arrangement
The Atlas contains 12 pages covering the entire sky. The competing demands for a large-scale atlas, along with an overlap of at least 20° between adjacent maps, has led to a rather unconventional arrangement of pages. The page arrangement is based on a representation of the whole sky as a dodecahedron with one of its apices at the North Pole. The nine "uppermost" faces were then optimized with regard to position and shape, in view of the rectangular form of the individual maps. It was thus possible to achieve a scale of 2.8 mm/1° at the center of the maps, with a map size of 280 mm by 350 mm. The whole set of maps was then turned in right ascension, so that most of the radiants of the main meteor streams were closer to the center of the maps.
Coordinates
The atlas contains guidance marks in the form of a set of coordinates at 7-cm intervals (25° according to the scale at the center of the map). Projection or reading of positions according to these is much more precise than in terms of right ascension and declination. The conversion factors are simple, and, given the present widespread use of calculators and small computers, easy to perform.
From the X and Y coordinates, with their origin at the bottom-left corner of the map (in mm, X-axis to the right, Y-axis upwards), the conversion to standard coordinates (x,y) with regard to the center of the map goes as follows:
where R is the radius of the projection and (X_0,Y_0) the position of the center of the map. With a distance of 70 mm between the marks on the map, R = 160.43 mm. For maps 1 to 3 and 7 to 12, X_0 = 175 mm and Y_0 = 140 mm; for maps 4 to 6, X_0 = 140 mm and Y_0 = 175 mm.
We further define the direction vector of an object by means of the relationships:
where alpha and delta are the right ascension and declination of the object and alpha_0 is the right ascension of the center of the map (given in degrees in Table 8).
------------------------------------------------------------------- Map | 1 2 3 alpha_0 | 30° 150° 270° delta_0 | 55.68° 55.68° 55.68° X_0 | 175 175 175 Y_0 | 140 140 140 ------------------------------------------------------------------- Map 4 5 6 | 7 8 9 alpha_0 90° 210° 330° | 30° 150° 270° delta_0 4.89° 4.89° 4.89° | -4.89° -4.89° -4.89° X_0 140 140 140 | 175 175 175 Y_0 175 175 175 | 140 140 140 ------------------------------------------------------------------- Map 10 11 12 | alpha_0 90° 210° 330° | delta_0 -59.29° -59.29° -59.29° | X_0 175 175 175 | Y_0 140 140 140 | -------------------------------------------------------------------
Table 8 - Right ascension alpha_0, declination delta_0 and cartesian coordinates X_0 and Y_0 of the center of each map in the Gnomonic Atlas Brno 2000.0.
The calculation of the position of an object on the map starts with the calculation of the direction vector using system (2). Then calculate s = p sin delta_0 + r cos delta_0, with delta_0 the declination of the center of the map; the values of delta_0 are also given in Table 8. If s < 0.582, the object cannot be drawn on the map. The standard coordinates of the object are then given by the relationship:
from which it is easy to calculate the coordinates (X,Y), using relation (1).
In the other direction, one has to calculate first the standard coordinates using (1) and the polar radius t = sqrt(1+x²+y²). The components of the direction vector are then given by:
from which the values of alpha and delta can be determined using the system in (2)
Subtract 360° if this value is exceeded by alpha.