Astrometric accuracy depends on having a precise reference frame within which the position of celestial objects can be measured. Using a typical amateur's telescope equipped with a CCD camera, observers routinely make measurements to a precision of 0.2 arcsecond or better. The key concept is that astrometry measures the relative location of a star in an image that covers only a small section of the sky with a precision of 1 part in 5,000 relative to a background of stars whose positions are accurately known.
When astronomers speak of "the plane of the sky," they mean a section of the celestial sphere that is sufficiently small that it can be treated as a plane surface. Standard coordinates take the "plane of the sky" concept literally, defining it as a plane that is tangent to the celestial sphere at a point (a0, S0) on the sky. The X-axis is aligned with right ascension (a), the Y-axis is aligned with declination ( 8), and the origin, i.e., (0, 0), lies at the point of tangency. The coordinates (X, Y) are expressed in terms of the unit radius of the celestial sphere.
It is helpful to think of this sphere as a transparent globe with a tiny bright lamp at its center, and the plane containing standard coordinates as a sheet of stiff cardboard touching the outside of the globe at some point. Star symbols on the globe cast (or "project") shadows on the cardboard, and the locations of the shadows are the positions of the stars in standard coordinates. This geometry is called a central, or gnomonic, projection.
It is straightforward to compute the locations of stars in standard coordinates located on the plane tangent to the celestial sphere at the point (a0, 80). Given the right ascension and the declination of a star, its standard coordinates are:
cos5sin(a - aft) coso0cosocos (a - a0) + sino0smo sin8n cos 8 cos (a - an) - cos8nsin8
Given the standard (X, Y) coordinates for a star, the right ascension and declination are computed from:
8 = arcsin
As we shall see, standard coordinates are useful because they mimic the formation of images in a telescope. When an ideal telescope forms an image of the sky, parallel light from sources on the celestial sphere passes through its optics and comes to focus not on a spherical surface behind the lens, but instead on the flat surface of a photographic plate or a CCD chip (see Figure 9.1).
The image of a star with an angular distance ft from the telescope's optical axis forms an image at a linear distance r from that axis:
where F is the focal length of the telescope objective. Thus, star images are projected from locations on the celestial sphere to locations on a plane tangent to the celestial sphere at the aim-point of the telescope; i.e., as a gnomonic projection. Because the angles involved are small (typically, the field of view in an astromet-ric image is less than 1°), the small-angle approximation tan$ = $ is valid, where $ is measured in radians. For example, at an angle of 0.01 radian, the tangent equals 0.010003333. Thus, we can simplify the relationship to:
stipulating that $ be given in radian measure. Virtually all telescopes used for astronomy approach the ideal so closely that deviations from this relationship are negligible over a 1 ° full field.
When an astrometric image is taken, the astronomer does not know the exact coordinates of the center of the image (a0, 80). After the plate is developed or the CCD read out to a computer, the locations of stars on the image are measured in (jc, y) coordinates. Photographic images have customarily been measured using a precision measuring engine, and the location of each star image recorded in millimeters. With CCD camera, however, the centroid of each star image is measured in units of pixels and, from their known dimensions, converted to millimeters.
In an ideal situation, the x and axes used in measuring the image would be perfectly aligned with its right ascension and declination axes, and the coordinates of the exact center of the image (a0, 80) would be known. If they were known, then the measured (jc, y) coordinates could be converted into standard coordinates by dividing by the focal length of the telescope:
In real life, however, (a0, 80) is inevitably somewhat off the center of the image, and the x and y axes of the detector will be rotated through some angle (see Figure 9.2). If the departure from the aim-point is (xoffset, yoffset) and the detector is rotated through an angle p, the relationships between plate coordinates and standard coordinates become:
y= sinp cosp offset
In addition to displacement and rotation, the detector may be slightly tilted relative to the incoming light; the manufacturer's figures for the pixel dimensions might be slightly inaccurate at the CCD's operating temperature; and in the case of scanned photographs, the axes may not be perfectly orthogonal and equally
scaled. These errors, provided that they are small, can be included as further terms involving x, y, and offset.
However, rather than try to take into account every peculiarity of the telescope and optics, we simply rewrite the relationship between the two coordinate systems as a general linear transformation:
The terms a, b, c, d, e, and/are called plate constants. The plate constants can be determined empirically from the image itself. The trick is to measure the x,y locations of three or more reference stars and to compute the standard coordinates of these stars, and then solve the resulting linear equations for the plate constants. The equations in X for three stars are:
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