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Figure 3.6. Distribution of discrete radio sources plotted in equatorial coordinates, in continuum radiation near v = 1420 MHz. Larger diameters are brighter sources. The brightest sources lie in the galactic plane and dramatically delineate the galactic equator. [Courtesy G. Verschuur]

the axis of rotation that transforms one plane to the other. It then becomes apparent that the two equators cross (ascending node) at

«node = 12 h 49.0 m + 6h = 18 h 49.0 m = 282.25° (Node) (3.5)

This specification of the relation between the two coordinate systems leads to formulae for conversion of B1950 equatorial coordinates a, 8 to galactic coordinates l, b and vice versa.

The significance of the galactic coordinate system can be seen from a map of discrete radio sources in equatorial coordinates (Fig. 6). This plot represents a sphere projected onto a flat surface. The great circle of the galactic equator is clearly seen as a sinusoidal curve of bright radio sources. Clearly these sources are associated with the Galaxy. It therefore is often convenient to use galactic coordinates. In such a plot they would lie along the equator as in the cover illustrations. The general background of discrete sources in Fig. 6 represents an isotropic distribution of sources, mostly (distant) extragalactic sources.

Ecliptic coordinate system

A third coordinate system on the celestial sphere is the ecliptic coordinate system. It is used less often but is very convenient for planning observations from an earth-orbiting satellite that must keep its solar panels pointed toward the sun. In this system, the equator is simply the path on the sky that the sun follows, i.e., the ecliptic. The plane of the ecliptic lies 23.4° from the celestial equator (Fig. 1). The ecliptic latitude of a star directly indicates the closest that the sun can come to o o

the source; the ecliptic longitude indicates directly the date (month and day) that the sun is at this closest point.

Reference frames

An important complement to a celestial coordinate system is a catalog of reference stars with precisely measured positions. These are equivalent to surveyors' permanently placed markers (bounds) in your neighborhood which are used for later detailed surveys of particular lots, etc. Without them every survey would have to work all the way from some basic reference possibly miles away. These permanent bounds would not be necessary if there were lines of longitude and latitude painted all over the earth, say every 100 meters or so.

Similarly, the sky is not marked with J2000.0 lines of constant right ascension and declination. Thus a set of well measured star positions and proper motions is invaluable for the determination of precise coordinates of stars in a local region. The proper motions (rate of change of a and 5 in units of milliarcseconds per year (mas/yr) allow one to calculate precise positions for a number of years after the measurements. The systems of well measured stars have become of higher and higher quality as technology improves. The FK4 catalog adopted in 1976 contained the precise equatorial coordinates and proper motions of 3522 stars. The newer FK5 catalog with fainter stars and more accuracy gives its positions and motions in equatorial celestial coordinates J2000.0.

The current (since 1995) reference frame is the International Celestial Reference System (ICRS). It is based on the extremely accurate positions, ~ ± 0.5 mas, of about 250 extragalactic radio sources. Radio observations with widely separated telescopes yield these great precisions. The great distance of extragalactic sources ensures that their angular motions (proper motions) on the sky will be so small as to be undetectable. In contrast the optical stars in the FK4 and FK5 catalogs are stars in our part of the Galaxy; they do show proper motions.

More recently, the Hipparcos earth-orbiting satellite (1989-1993) measured the positions and proper motions of 118 000 optical stars with high precisions, ~0.7 mas and ~0.8 mas/yr respectively. The positions were determined in the Hipparcos Reference Frame. This frame was aligned to the radio ICRS frame with high precision through ground-based optical position measurements of the extragalactic radio sources. Since the entire sky contains ~40 000 sq. deg (see below), the Hipparcos catalog contains, on average, ~3 very well-located stars per square degree on the sky. The positions are in epoch J1991.25 but these can easily be precessed to J2000.0 or any other epoch. If you discover an important optical object in the sky, you can measure its location relative to several nearby Hipparcos stars in order to get its precise celestial coordinates.

Transformations

The several coordinate systems are simply redefinitions of the coordinates on the same celestial sphere. In all cases the stars are defined on a two-dimensional surface with two angles, a latitude and a longitude. One can convert the coordinates of a given star from one coordinate system to another with standard spherical coordinate transformation formulae. As noted the galactic coordinate system was defined in terms of the B1950 equatorial coordinates, a (B1950.0) and 8 (B1950.0). According to this definition, the celestial (B1950.0) to galactic transformations (and vice versa) are cos b cos(1 - 33°) = cos 8 cos(a - 282.25°) (3.6)

cos b sin(1 - 33°) = cos 8 sin(a - 282.25°) cos 62.6° + sin 8 sin 62.6°

sin b = sin8 cos 62.6° - cos 8 sin(a - 282.25°) sin 62.6° (3.8)

cos 8 sin(a - 282.25°) = cos b sin(1 - 33°) cos 62.6° - sin b sin 62.6°

sin 8 = cos b sin(1 - 33°) sin 62.6° + sin b cos 62.6° (3.10)

where the angles in the formulae are exact (not rounded off).

The first two of these equations yield values for b and l, given values of a and 8. However, there remains an ambiguity as to the quadrant of angle b. The third equation resolves this. Recall that b can reside in only two quadrants since it ranges only from -90° to +90°. Similarly, the final two equations yield values for a and 8, given values of b and l. In this case, quadrant ambiguity remains for a which can be resolved with (6). Similar transformations allow one to convert between equatorial coordinate systems for different epochs. For precise telescope pointing, one must further correct the coordinates for parallax and aberration (Chapter 4).

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