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In practice there will be a residual pointing error, i.e. a difference between the true viewing direction and that indicated by the encoders, caused by imperfections in the geometry and the finite structural stiffness of the antenna. These imperfections fall under the categories of misalignment of the axes, gravitational bending of the structure as function of elevation angle, errors in the zero point and linearity of the encoders and refraction by the earth's atmosphere. It will be necessary to apply corrections to the commanded position to ensure that the antenna beam is directed precisely in the desired direction. These corrections can be established by measuring the apparent position of a large number of sources with accurately known celestial position and distributed over the entire visible sky. From these observations we can determine the numerical values of the coefficients of the so-called pointing model of the antenna. The pointing model is created in the context of a pointing theory, where we seek to establish the analytical relationship between the true coordinates of the source, the target, and the read-out of the telescope encoders using a set of physically reasonable relations representing the known or expected geometrical and structural imperfections of the antenna system. Once we have set-up these relations, the coefficients of their terms can be found from the set of pointing observations by a least squares treatment. We call these parameters the pointing constants. They might vary over time due to aging effects in the structure, variable wind forces and diurnal or seasonal temperature variations. Thus for reliable antenna pointing it will be necessary to regularly check the constants for their best value.

In the following discussion we follow essentially the treatment by P. Stumpff (1972). We assume that the antenna has an altazimuth mounting, although this is not necessary for the theory to be discussed. It is most convenient to transform the celestial source coordinate right ascension (a) through the known local time tl to hour angle (h = a - tl) and thence hour angle and declination to apparent azimuth A and elevation (e) angle, whereby the local latitude (f) must be known. Azimuth and elevation are given by the following equations cos e cos A = -sin f cos d cos h + cos f sin d cos e sin A = -cos d sin h

sin e

= cos f cos d cos h + sin f sin d, where azimuth A is reckoned from north through east (0° < A < 360°) and elevation e from horizon to zenith (0° < e < 90°). For a source with precisely known position and a telescope position of exactly known latitude and a perfect clock, making the encoder readings equal to the source (A, e)-values would put the antenna beam exactly on the source, provided the antenna coordinate system (A,, e,) is strictly orthogonal without rotation or offset errors with respect to the astronomical system (A, e). It is the deviation between the instrumental and celestial coordinate system which necessitates the establishment of the pointing model and application of pointing errors in actual practice. Although latitude and clock errors can be incorporated in the least squares solution of the pointing constants, we shall here assume the errors in these to be zero, similarly to the assumption of perfect source coordinates. Normally, these errors will indeed be negligible compared to the pointing errors to be established.

We are thus forced to apply pointing corrections DA and De to the source coordinates (A, e) to make the beam direction coincide with the source for the indicated encoder positions (A,, e,). The first order pointing theory to be described now establishes a functional relationship between the observed errors, the derived pointing constants and the basic coordinates. We write the corrections as "indicated minus commanded" values as follows:

From the theory of the universal astronomical instrument (see e.g. K. Stumpff, 1955 or Smart, 1962) the following expressions, valid to first order, can be derived

DA = A0 + c1 sec e{ - c2 tan e{ - za sin (A, - Aa) tan e

De = e0 + b cos e, - r cot e, - Za cos (A, - Aa), (5.45)

where the constants have the following meaning

A0 and e0 = zero point offset of azimuth and elevation encoder, respectively b = the gravitational bending constant of the elevation section of the antenna c1 = collimation error of the beam (non-perpendicularity of elevation axis and c2 = collimation error of mount (non-perpendicularity of azimuth and elevation beam)

axes)

za = zenith distance (= 90° - elevation) of azimuth axis (azimuth axis tilt)

Aa = azimuth of the azimuth axis r = refraction constant.

Note that Aa and za can also be replaced by the angles between the azimuth axis and the north-south and east-west planes.

If we assume the errors to be small we may replace the "indicated" coordinates (Ai , ei) in Eq. (5.45) by the "true" source coordinates (A, e). In order to point the antenna at the true position of the source, the indicated coordinates must be increased by the values of Eq. (5.45). We now introduce the following set of

"pointing constants"

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