## Kjd

in which the integration is extended over a solid angle ©S, equal to that of the source. For a known source size, Eq. (5.34) would be the best representation of the coupling of the extended antenna pattern to the source. For a uniformly bright source of size ©S, the measured antenna temperature would now be TA = hB TB (Gordon et al, 1992).

We can obtain an impression of the effective beam efficiency as function of the angle 0 by measuring sources of different angular size, as the planets (a few arcsec-onds to one arcminute) and the Moon (30 arcmin) and interpolate in between. We shall illustrate below how this can be used to obtain an indication of the sidelobe level of the antenna. The interpolation from 1 to 30 arcminutes is of course prone to error, but it might still be better than trying to make a full theoretical calculation of the effective beam efficiency. For very large objects, beyond 30' size, we might include the "forward beam efficiency" (over 2p steradian) in the interpolation. This is sometimes determined from the measurement of the atmospheric emission over the full 90° elevation range - the so-called "sky-dip" (Kutner and Ulich, 1981). Its value will be very close to one.

At millimeter wavelengths, especially in the observation of large molecular clouds, procedures have been proposed by Kutner and Ulich (1981), which describe observations in terms of the parameter TR* , the observed antenna temperature corrected for all telescope-dependent parameters except the "coupling" of the antenna to the source brightness distribution. Under the assumption of a uniformly bright source, we find that the effective beam efficiency, defined above is related to these "coupling" (hc) and "extended source" (hs) efficiencies of Kutner and Ulich by the relation hB = hc hs. Unfortunately, hc generally cannot be measured and can be calculated only under simplifying assumptions. It appears preferable to use the hB, introduced above in these cases. If a reasonable estimate of hB (©) for a source of solid angle © is available, one could correct the measured antenna temperature at each point of the map into a "main beam" value by multiplying by hmB /hB. The intensities of the resulting map would then appear as to have been observed with a "clean" beam of efficiency hmB.

A useful summary of beam efficiency measurements, along with practical data on planetary brightness temperatures and applied to the Caltech Submillimeter Telescope, is presented by Mangum (1993).

■ 5.4. Sidelobe level and error pattern

### 5.4.1. Diffraction beam sidelobes

The availability of sources over a range of angular size, from real "point-like" to significantly larger than the antenna beamwidth, can be used to obtain an estimate of the sidelobe level near the main beam without the need to measure these individually, which would require a very high signal to noise ratio. By the same token, it is possible to estimate the level of the "error pattern" in the case of significant random errors in the reflector surface profile.

The radiation pattern of a perfect reflector consists of a central lobe surrounded by ring shaped sidelobes of decreasing amplitude (the Airy pattern, see Ch. 3). Assuming circular symmetry, we represent the pattern, as function of the radial angular coordinate 0, by g(0) = gm (0) + / gli (0) .

From the representation of the beam in terms of the Lambda function (see Ch. 3), we can derive the following reasonably good approximation for the half-power width 0li and the radius of maximum intensity ri of the ith sidelobe as 0li = 0A / 2 and ri = 0A (1 + i). Using again the gaussian approximation for the main beam, we can now write for the pattern solid angle up to and including i sidelobes

1.133 0a2 + / p0a2(1 + i) gli(max) = Wm {1 + (f g1 + { g2 + ..)}

We now apply the concept of effective beam efficiency hs, as introduced in Eq. (5.34). Thus by measuring hS on an extended source and hmBon a point source, we obtain an estimate of the average level of those sidelobes covering the source from the relation

We can also use the curve of Fig. 3.5, based on the Bessel function representation of the radiation pattern. From that curve and the numbers in Table 3.1 we derive values for the ratio hs / hmB which are very close to those found above. Remember that we assume a uniform brightness distribution over the source of a known size. If we take as an example a 30 m diameter telescope operating at 1.2 mm wavelength, we have a HPBW = 10 arcseconds. With planets in size between a few arcseconds (Uranus, Neptune) and 1 arcminute (Jupiter, Venus) this method can be used to estimate the average near-in sidelobe level.

### 5.4.2. Error pattern due to random surface errors

In the case, where the sidelobe level is dominated by the scattering from random errors in the reflector profile (the so-called "Ruze-error"), we can use the concept of

"effective" beam efficiency to estimate the size and level of the error pattern. We have already treated the theory of these errors and the resulting radiation pattern characteristics in Chapter 4.6. There we described the beam pattern as the sum of the diffraction pattern gD of the perfect reflector and the error pattern gE, caused by the scattered radiation from the random errors. The errors cause an rms phase fluctuation over the aperture of s = 4 p e / l, where the surface error e can be weighted by the aperture illumination function, if desired. Also, we repeat that e is the deviation parallel to the reflector axis, i.e. half the total pathlength error. It is related by Eq. (4.43) to the error normal to the reflector surface, which is the quantity usually measured or calculated from structural analysis.

We repeat Eq. (4.42) for the relative change in aperture efficiency caused by the random errors,

Normally the second term has a negligible contribution to the aperture efficiency and can be dropped for this purpose, leading to the widely known "Ruze" formula for the loss of gain due to random surface errors. If we measure hA at a number of sufficiently separated wavelengths, we can make a good estimate of both hA0 and e. This is illustrated in Fig. 5.8, which shows the logarithm of the measured aperture efficiency as function of the reciprocal square of the wavelength [Mat.5.4]. The slope of the fitting line gives the rms surface error from < slope >— (4 pe)2 and the intercept

1/A2

Fig. 5.8. Measured aperture efficiency at 6 wavelengths. Plotted is the logarithm of the efficiency against the reciprocal square of the wavelength. The slope gives the rms error as 92 mm.

1/A2

Fig. 5.8. Measured aperture efficiency at 6 wavelengths. Plotted is the logarithm of the efficiency against the reciprocal square of the wavelength. The slope gives the rms error as 92 mm.

for delivers hA0- These are measurements made early in the operation of the

IRAM 30-m millimeter telescope. The derived surface error is 92 mm, which is in reasonable agreement with a holographic surface measurement of 85 mm. Since then the surface has been significantly improved. The value hA0 = 0-6 agrees with the calculated value from the illumination function-

Also, a change in measured hA as function of elevation angle can be used to compute the relative change in e and give an impression of the gravitational deformations of the reflector surface, which can be compared with structural finite element analysis. An example is shown in Fig. 5.9, based on efficiency measurements with the IRAM 30-m mm-telescope at 1.2 mm wavelength, shown as the solid line. The dashed line is the predicted change in efficiency from the finite element analysis. On the right hand axis is a scale with the inferred increase in the reflector rms surface deviation- At 80° elevation angle the predicted deformation is 40±10 mm, while the measurement indicates 52±5 mm. It should be noted that it is essential to accurately correct for the considerable atmospheric attenuation at these sort wavelengths as function of elevation, not a simple task because of the variable nature of the troposphere-

If a measurement has sufficient signal to noise ratio to determine the error pattern reliably, we can derive the correlation length c from the ratio of the peaks of the error pattern and the diffraction pattern (see Eq. (4.44)

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