5.6 Modeling Stellar Positions and Characteristics

David M. Gottlieb

The value of using star observations for attitude determination lies in the high degree of accuracy that can be obtained. This accuracy derives from the point source nature of stars. Identifying observations with catalogued stars is, however, difficult (see Section 7.7). To alleviate star identification problems, and to obtain as much precision as possible from available star data, it is crucial to have an accurate and complete star catalog.

5.6.1 Star Catalog Data Required for Attitude Determination

Each star in any catalog used for attitude determination should have an identifying number to facilitate checkout of computer software and to aid the investigation of anomalous results. Unfortunately, there are many identification systems in use and few catalogs cross-reference more than one or two of them. Four major systems are in common use:

BD/ CD/ CPD. The most widely used and extensive system, generated from three positional catalogs: the Bonner Durchmusterung (BD) [Argelander, 1859— 1862 and Schonfeld, 1886], the Cordoba Durchmusterung (CD) [Thome, 18921914 and Perrine, 1932], and the Cape Photographic Durchmusterung (CPD) [Gill and Kapteyn, 1896-1900]. Unfortunately, regions of the sky covered by these systems overlap, resulting in nonunique numbers. HD. TTie Henry Draper number [Cannon, 1918-1924], also widely used. Because the catalog is virtually complete to eighth visual magnitude, most stars visible to present star sensors have HD numbers.

HR. Number from the Catalog of Bright Stars [Hoffleit, 1964], frequently cross-referenced in the literature. However, few stars dimmer than sixth visual magnitude have HR numbers.

SA O. The Smithsonian Astrophysical Observatory number [Smithsonian Institute, 1971], relatively new and used principally in the SAO catalog itself. It covers approximately as many stars as the HD, but it is seldom cross-referenced.

All catalogs also contain star positions, given as right ascension and declination at some epoch. The accuracy of the stated catalog position depends on the accuracy of the original observation and the time between the epoch of observation and the epoch of the catalog position. Star positions reported in the SAO or AGK-3 [Astronomisches Rechen Institut, 1975] catalogs are accurate to approximately 1 arc-sec. For about 2% of the stars brighter than eighth magnitude, and 15% from eighth to ninth magnitude, only the nineteenth-century HD positions exist with typical inaccuracies of 1 arc-min (one standard deviation) in both right ascension and declination.

Because star catalogs give positions at an epoch (typically 1900.0 to 1950.0) that differs from the time of the spacecraft observations, the star positions must be updated to the observation time. Corrections are usually required for the precession of the equinoxes (Section 2.2) and the proper motion, or space motion, of each individual star. Proper motion can be applied linearly for periods of several hundred years when the rates are available in the star catalog. For 95% of the stars brighter than ninth magnitude, proper motion is less than 10 arc-sec per century, and for 99.9%, it is less than 1 arc-min.

An additional correction may be required for aberration—the apparent shift in the position of a star caused by the motion of the spacecraft, Thv original observation of aberration by Astronomer Royal Bradley in 1728 was one of the first confirmations of Roemer's postulate that the speed of light was finite. For Earth-orbiting spacecraft, the motion of the Earth around the Sun causes a maximum aberration of about 20 arc-sec; the motion of the spacecraft about the Earth accounts for less than 5 arc-sec of additional aberration. The aberration, A0, may be computed from the spacecraft velocity relative to the Sun, v, by:

where c is the speed of light and 9 is the angular separation between v and the star vector, s. The star appears shifted toward v in the v-s plane.

Star intensity is included in most catalogs and is measured by magnitude, z logarithmic quantity defined by m= —2.5 log (F) + m0> where m0 is constant and I is the brightness or flux density. Note that brightness decreases as magnitudi increases. Magnitudes are usually reported in one of two systems. The UB^ (Ultraviolet, Blue, Visual) system of Johnson and Morgan [1953] is the mor modern and accurate of the two. Commonly, only the V magnitude and sometime the B magnitude are available. Figure 5-13 defines these magnitudes in terms c sensitivity versus wavelength. Some catalogs list V and the difference, B - V. Th second system is the photographic-photovisual magnitude used in the Henr Draper Catalog [1918-1924]. These are more frequently available, but are far le; accurate than the UBV system. No sensitivity-wavelength plots exist for them, only photographic and photovisual magnitudes are available for a star, these ca be used analogously for B and V, respectively.

Observed B and V magnitudes have errors of about 0.02 magnitude (oi standard deviation). Only about 20% of the stars brighter than eighth magnituc have observed B and V magnitudes, and very few fainter than eighth magnitud Photographic and photovisual magnitudes are uncertain to about 0.3 and C

Fig.- 5-13. Curves Defining B and V Magnitude Scales

magnitude, respectively. Conversion of these to B and V adds an additional error of 0.1 magnitude. Because sensor responses do not normally coincide with the wavelength sensitivities of either B or V magnitudes, some combination of these will be required to accurately represent star magnitudes on an instrumental scale (see Section 7.6).

Because star sensors detect light from the entire segment of the sky covered by their apertures, an additional requirement for modeling stellar magnitudes is the integrated intensity of faint background stars. Table 5-14 summarizes mean star densities and background level for the entire sky and for regions near the galactic plane. The background level is the integrated contribution of all stars fainter than the limiting magnitude, expressed in terms of stars of brightness equal to the limiting magnitude per square degree. Only 0.5% of the stars in the sky brighter than ninth magnitude have magnitudes known to vary with time by more than 0.1 magnitude. Some catalogs flag these stars and give values of the maximum and minimum magnitude to be expected. Brighter stars are more likely to be known variables, because dimmer ones are not observed as frequently.

Finally, those components of a multiple star system which are separated by about 1 to 5 arc-min may cause misidentifications and position errors. Many star

Table 5-14. Star Densities (Adapted from Allen [ 1973D

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