Up: The gainelevation correction of
Subsections
The experimental determination of ,_{0}), and
(), is preferentially made at short wavelengths
despite the higher atmospheric noise and less precise atmospheric correction,
in particular at low elevations. Figure 1 shows the gainelevation dependence
at 1.22 mm wavelength obtained from onoff measurements (30''
wobbler throw) with the MPIfR bolometer of the quasar 0923+392 and the
extended source Mars (14'' diameter). The bolometer has a taper of
10 dB, a bandwidth of 60 GHz, and for
observations of the planets an effective wavelength
1.22 mm (245 GHz), as calculated from the spectral energy distribution
of the planets and the wavelength dependent atmospheric attenuation. Figure 1
shows also the gainelevation dependence calculated from Eq. (2) and Eq. (3)
for the bestfit values (0) = 0.085 mm and (90) = 0.075 mm
derived from the lsqfit of the 245 GHz bolometer measurements, using R =
0.9. The figure shows furthermore the gainelevation dependence derived from
the FE calculations. Comparison of these curves shows that the rmsvalues
() predicted from the FE calculations are 10
 15% lower than those derived from the radiometric measurements.
Inserted in Fig. 1 is the gainelevation dependence for 3 mm (100 GHz), 2
mm (150 GHz), and 0.86 mm (350 GHz) wavelength, derived from scaling of the
empirical 1.22 mm gainelevation dependence by using the bestfit values
(0) and (90) in Eq. (2) and Eq. (3).

Figure 1:
Gainelevation dependence measured with the MPIfR bolometer at
1.22 mm (245 GHz) on the quasar 0932+392 [open circles] and Mars (14'')
[dots] (Feb. 1995). The heavy line shows the bestfit gainelevation
dependence Eq. (3) derived from these measurements; the dashed line shows the
corresponding gainelevation dependence derived for the surface deformations
predicted from the finite element calculations. The other lines show the
gainelevation dependence at 3 mm (100 GHz): a), 2 mm (150 GHz):
b), and 0.86 mm (350 GHz): c), derived from scaling of the 1.22
mm gainelevation dependence (heavy line) 

Figure 2:
Beam pattern
(,,) of the IRAM
30m telescope calculated for = 1.30 mm (230 GHz) and the elevation
angle 0 a), 22.5 b), 43 c), 67.5 d), and
90 e). The reflector is optimized at _{0} = 43 so
that at this elevation the calculated reference beam is the perfect beam
pattern. Heavy contours: levels at 3 dB, 6 dB, and 9 dB; thin contours:
levels at 12 dB, 15 dB, 18 dB, and 21 dB 
In a diffraction calculation we derived the illumination weighted beam patterns
(,,)
( = azimuth direction, = elevation
direction) degraded by the wavefront deformation of the homology deformations
() predicted from the FE calculations^{}, which
were scaled in amplitude by 10  15 to match the empirical values
(). As example, Fig. 2 shows for several elevation
angles the calculated beam pattern at = 1.30 mm (230 GHz), i.e. at
the wavelength of the ^{12}CO(21) line. From these and similar beam
patterns we calculated for disklike and constant surface brightness sources
of diameter
[(,) = 1 for
^{2} + ^{2}
(/2)^{2} and (u,v) = 0 for (/2)^{2}
< ^{2} + ^{2}] the
received power
 

 (5) 
and the corresponding gainelevation dependence
 
(6) 
Table 2 summarizes the onaxis gainelevation dependence
, = 0) G_{0}() derived from
the diffraction calculations, and to be applied on the IRAM 30m telescope
for observations at = 3 mm (100 GHz), 2 mm (150 GHz), 1.30 mm (230
GHz), and = 0.86 mm (350 GHz). For an estimated
error of 10 in the values (), the accuracy
of the values G_{0} is 2 at long wavelengths (3 mm 
2 mm) and 5 at short wavelengths (1.3 mm  0.8 mm). As
evident from Fig. 1 and Table 2, the gainelevation dependence at 3 mm
wavelength is only a few percent and usually below the accuracy of the
measurements. The small difference between the curves of Fig. 1 and the
data of Table 2 is due to the empirical determination and the calculation of
,_{0}); however, the difference is below the accuracy
of the measurements.
Table 2:
Onaxis gainelevation dependence
G_{0}(,_{0}) for = 3 mm (100 GHz), 2 mm
(150 GHz), 1.3 mm (230 GHz), and 0.86 mm (350 GHz) [receivers of 13
dB edge taper]. The accuracy of and R is
10

An extended source is covered by a larger fraction of the beam than a
pointsource and hence should show a smaller gainelevation dependence. To
quantify this effect, we use the loss in gain
 
(7) 
and display in Fig. 3 the relative loss in gain
(,) between a pointsource (,0))
and an extended source (,)) calculated from
 
(8) 
The calculations show as important result that the quantity
() is nearly independent of the wavelength and the
elevation of the observation^{}. Using the data of Fig. 3 and the entries of Table 2,
the actual value ,_{0},) for an extended
source is
 
(9) 
Figure 3 shows that 0.95 () for sources not exceeding
2 so that from Eq. (9) for these sources
) G_{0}, within 5  10. The
gainelevation dependence is weaker for more extended sources since for these
() < 0.9. The gainelevation dependence is negligible
for sources of diameter 8 since for
these () 0.1 and ) 1.
We explain in the following example how to correct observations with the 30m
telescope. Suppose a source of = 30'' diameter, observed at
1.3 mm wavelength and 20 elevation, has a flux corrected for atmospheric
attenuation of S'() =
(2 k/A)/ = 10 Jy. The diameter
of the source is 30''/ 3 beamwidth. From Fig. 3
we find (3) = 0.70, so that from Eq. (9) and the value
G_{0}(20) = 0.92 of Table 2 we have G(30'') =
1  0.70(1  0.92) = 0.94. The homologycorrected telescopeindependent
flux is S = S'(20)/ 0.94 = 10.6 Jy. A 10 Jy pointsource observed
under similar conditions has the intrinsic flux S = 10/0.92 = 10.9 Jy.

Figure 3:
Relative gainelevation dependence
(_{S}) (Eq. (8)) as function of source size
_{S} (in units of beamwidth _{b}), for = 2.0
mm (150 GHz): triangles, = 1.30 mm (230 GHz): squares, =
0.86 mm (350 GHz): circles. The dashed vertical line indicates the diameter
of the full beam 2.4_{b}, i.e. the position of the
first minimum 
Up: The gainelevation correction of
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