1.5 Radiometric Relations

The imperfections of a telescope, either due to systematic or random errors, produce beam deformations, a loss in gain, and focus and pointing errors. These effects must be taken into account when mapping and measuring a source. Information on the beam pattern obtained from a map (for instance holography map) of a strong point-like source; information on the sensitivity [Jy/K] and calibration of the telescope is obtained from absolute power measurements of, for instance, the planets, of which the brightness temperatures are quite well known. This information is usually collected by the observatory staff, and provided to the observer (30-m Telescope Manual; observation protocols of Plateau de Bure measurements).

We summarize the influence of random deformations, at least as far as the main beam is concerned, since for this case the RUZE equation provides sufficient precision for an understanding of the telescope behaviour; also for the astronomer observer without going into complicated radio optics detail. This relation appears in the expression of the diffraction beam $\cal F_{\rm c}$ (see Eq.1.20) and shows clearly the fact that the degradation of the telescope, in particular for power measurements, increases exponentially with wavelength.

  

Figure 1.12: Illustration of the RUZE relation as function of wavelength (frequency) and surface error values as indicated (in mm).

Aperture Efficiency:

$${\epsilon}_{\rm ap} = {\epsilon}_{o}\,{\bf exp}[-{\sigma}_{\v... ...{\epsilon}_{o}{\bf exp}[-(4{\pi}{\rm R}{\sigma}/{\lambda})^{2}]$$

Antenna Gain:

$${\rm S/T}^{*}_{\rm A} = 2 ({\rm k/A}){\rm F}_{\rm eff}/{\epsi... ...{\rm R}/{\lambda})^{2}]/ {\epsilon}_{\rm ap}\ \ \ {\rm [Jy/K]}$$

Beam Efficiency:

$${\rm B}_{\rm eff} = 0.8899\,[{\Theta}_{\rm b}/({\lambda}/{\rm D})]^{2} / {\epsilon}_{\rm ap}$$

$${\Theta}_{\rm b} = {\alpha}{\lambda}/{\rm D},\ \ 1\ {\leq}\ {\alpha}\ 1.2\ \ \ {\rm [radians]}$$

$${\rm B}_{\rm eff}\ {\approx}\ 1.2\,{\epsilon}_{o}{\bf exp}[-(4{\pi}{\rm R}{\sigma}/{\lambda})^{2}]$$

The quantities in these expressions are $\epsilon_{o}$: aperture efficiency of the perfect telescope (usually of the order of $\sim 75 - 90$% ; see Table 1.1); $\epsilon_{\rm ap}$: effective aperture efficiency at the wavelength $\lambda$, including all wavefront / telescope deformations; $\sigma$: rms-value of the telescope optics deformations; R $\approx$ 0.8: reduction factor for a steep main reflector (N = F/D $\approx$ 0.3); S: flux density of a point source [Jy]; T $_{\rm A}^{*}$: measured antenna temperature [K]; A: geometric surface area of the telescope [m²]; F $_{\rm eff}$: forward efficiency, measured at the telescope for instance from a sky dip; $\Theta_{\rm b}$: main beam width (FWHP).