2. General assumptions

 

We will now give a brief description of the conical scan technique and of our plans to implement this for the upgraded Arecibo telescope. Let us start with the following simplifying assumptions (but see Sect. 4.3 (click here)):

(i)

the antenna is commanded to point its beam axis at a certain location, referred as the "target'' throughout the remainder of this article. Due to many inherent pointing errors the antenna boresight does not point to the desired location on the sky. Instead, the instrumentally tracked point is the center of the scanning circle defined below;

(ii)

we define the optical axis (sometimes called also the RF axis or beam axis) as the best-fit axis of the entire deformed reflector, as distinct from the geometrical axis of the undeformed theoretical reflecting system;

(iii)

the beam is Gaussian and depends only on the angular distance (r, see below) from the axis. Beams with elliptical cross-sections are considered in Appendix A;

(iv)

the beam remains unchanged with elevation;

(v)

we neglect for the moment the problem of the inner sidelobes, which will be discussed in Sect. 4.2 (click here);

(vi)

we shall not consider the problem of beam-squint between opposite circularly polarized beams, i.e. in the case of polarized signals this is equivalent to considering a single polarization direction.

Given a point source located at angular great-circle offset r from the beam center, a suitable representation of the value of the normalised antenna power pattern at position r can be written as:
 
where represents the measured receiver output power when the source is at position r, is the maximum recorded output power when the source is at the center of the beam, and represents the measured power when no source is within the beam. For r expressed in units of the full beamwidth at half maximum, FWHM (see Fig. 1 (click here)), ; all other angular distances will hereafter be expressed in these units, unless explicitly stated otherwise. Equation (1 (click here)) implies that at some point during the procedures described in the next sections a measurement is taken of the power OFF-source.

  
Figure 1: Plane of the sky where the antenna beam FWHM contour (solid circle) and the path of rotation of the beam center (dashed circle), are shown. and are defined in the text

During the conical scan the antenna beam moves in a circular path of radius centered on the optical axis. In general, is frequency-dependent. This fact can actually be used as a further aid in the practical implementation of the conical scan technique, but we shall not discuss it in this work, and we shall thus drop the frequency dependence and treat as a constant. Choosing a reference frame whose origin is the center of the scanning circle, the position of the beam center, , can be written as:
 
where we define and to be the positions in elevation and cross-elevation, respectively, is the conical scan angle (see Fig. 1 (click here)), and t is the time.

We assume that a pointing source, or target, having a negligible angular size with respect to the 32'' beam at 10 GHz, is located at position with respect to the center of the scanning circle, (0,0). Then from Fig. 1 (click here) it is easily shown that
 
where and are the angular distances, from the pointing direction (or, equivalently, the optical axis), of the target and the center of the beam, respectively.

The antenna temperature, , when the source is at an angular distance r from the optical axis can be measured in the usual way:
 
where is the on-source system temperature and W is the measured power. Also, by definition
 
is the expected antenna temperature of the pointing source. From Eqs. (1 (click here)), (4 (click here)) and (5 (click here)) we then obtain the known result that or, by Eq. (3 (click here)):
 
where is implicitly assumed to be . Examples of Eq. (6 (click here)) are shown in Fig. 3 (click here), and are discussed in Sect. 4.1.1 (click here).