Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mrao.cam.ac.uk/~bn204/mk2/publications/2006/ptcspn47.pdf Äàòà èçìåíåíèÿ: Wed Nov 25 23:11:09 2009 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 19:08:19 2012 Êîäèðîâêà: IBM-866 Ïîèñêîâûå ñëîâà: angular size

Precision Telescope Control System

PTCS Project Note 47.1

Out-Of-Focus Holography at the Green Bank Telescope, The Gain-Elevation Curve and Absolute Efficiency
Bojan Nikolic, Dana S. Balser & Richard M. Prestage

27 June 2006
GBT Archive: PR059 File: PROJECTS Keys: PTCS, OOF

Abstract Out-Of-Focus (OOF) holography was performed using the GBT Q-band (43.1 GHz) receiver over a range of elevations using astronomical sources to measure possible large-scale, repeatable errors due to gravitational effects. Observations were taken at night when the thermal gradients are minimized and during excellent weather conditions (e.g., low winds). The efficacy of the OOF holography technique was demonstrated by comparison with the finite element model and closure experiments. The aberrations measured by the OOF technique are expressed in terms of Zernike polynomials. Because we expect Hooke's law to apply each Zernike coefficient is fit as a function of elevation using a sin() + b cos() + c, where is the elevation. The active surface control system allows input of Zernike coefficients to adjust the primary surface. The model was validated by measuring the aperture efficiency as a function of elevation using astronomical flux density calibration sources. The validation confirmed the OOF holography results that a significant improvement of the surface could be made at elevations below 35 degree with no improvement at higher elevations. Using the OOF holography model the gain-elevation curve is approximately flat with an aperture efficiency of 0.45.


PTCS/PN/47.1
Contents 1 Introduction 2 OOF holography observational strategy 3 Data reduction 3.1 Preparation of the raw data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 OOF analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Repeatability of the OOF technique 5 Modelling the gravitationally-induced aberrations 6 Aperture Efficiency 6.1 Method . . . 6.2 Observations 6.3 Results . . . . Measurements ....................................... ....................................... .......................................

2

3 3 4 4 6 9 11 16 16 19 23 23

7 Discussion and Summary

History
47.1 Original draft (Bojan Nikolic) 47.1 Add abstract; minor corrections (Dana S. Balser)


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1. Introduction

3

The accuracy of the surface of the primary reflector and the overall collimation of the telescope are two of the factors currently limiting the performance of the 100-m diameter Green Bank Telescope (GBT) at high frequencies, as of course may be expected for a telescope of this size. In this note we describe the use of phase-retrieval, or Out-Of-Focus (OOF), holography using astronomical sources and receivers to measure the large-scale wavefront errors present in the GBT and interpretation of these data to derive a model for the errors as a function of telescope elevation. The wavefront errors we are measuring are the on-axis wavefront errors caused by imperfections in the combined optics of the telescope rather than field aberrations which would occur even if the optics were perfect. The OOF holography technique is appropriate for this task for a number of reasons: § Since astronomical sources are used, it is possible to measure the aberrations at the full range of elevations normally used for science observations. § The OOF technique measures the total aberrations in the complete science receiver system. For example, these deformations may arise due to deformations of the primary reflector surface, the secondary reflector surface, relative mis-alignment of the primary and secondary surfaces and the receiver, and phase response of the science receiver. § The time required for a full measurement set is of the order of the timescales for thermal effects. The primary limitation of the OOF technique is that it constrains only the low spatial frequencies in the aperture plane, that is, it is only sensitive to the large scale structure. For example, as it will be explained later, for the measurements presented here we use a basis set of only 18 independent elements. Although the basis vectors are not localized (we use Zernike polynomials) a very rough estimate of the resolution achieved is about 20 ½ 20 m. The implication of this is that the OOF holography can not, by a long way, be used as the only antenna metrology technique. It is nevertheless useful because the techniques used to measure the small scale structure may not be sensitive to the large scale structure and, because the effects of gravity, miscollimation and possibly thermal effects can all be expected to produce mostly large scale structure. For example, due to its off-axis design, the GBT is non-homologous, that is, the gravity induces deformations of the primary surface that can not be compensated for by a simple re-focusing the telescope. For this reason, during normal high-frequency observations, the primary surface is continuously adjusted using a pre-computed finite element model (FEM) of the GBT. During initial testing of the OOF holography technique at the GBT (Nikolic et al. 2002), this adjustment was intentionally turned off, and it was found that the technique is able to recover aperture maps which coincided with the FEM, indicating that the OOF technique is sensitive to spatial scales required to measure gravitationally induced deformations.

2. OOF holography observational strategy Each observing block consisted of a pointing and focus optimisation measurement using the standard procedures for the GBT and a set of three beam maps, one in-focus and the other two out-of-focus. This


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set of three beam maps was then used as the input to the OOF holography software which produces a map of the total aberrations as a function of position in the aperture plane. The de-focusing of the telescope was carried out by moving the sub-reflector in its `Y' direction, that is, the direction along which its motion does not move the position of the beam on the sky. The convention used at the GBT is that a positive displacement correspond to moving the subreflector away from the primary. The receiver used for these observations was the facility Q-band receiver operating at 43.1 GHz and the bandwidth used was between 80 MHz and 320 MHz. This receiver consists of two feeds, each with two polarizations, and thus has two beams on the sky, separated by 58 arcseconds. To minimise the effect of sky-brightness variations on our measurements the output of these two receivers was differenced so that the effective response of the telescope was the real beam convolved by two delta functions separated by 58 arcseconds in the azimuth direction. The aberrations due to both of the feeds being off (and on opposite sides of) the optical axis are negligible in this instance and were not taken into account in the further analysis. The beam maps were obtained by the on-the-fly map technique (in the Alt/Az frame), that is, the telescope was continuously in motion and a high sampling rate (10 Hz) used to avoid excessive smearing in the in-scan direction. The maps consisted of seventeen azimuth rows each 350 arcseconds in length and separated by eight arcseconds in the elevation direction. The scanning speed of the telescope was fifty arcseconds per second of time and ten seconds was allowed for the telescope to turn around at the end of each row; hence, the total time required for each map was just under five minutes and the integration time per Nyquist sample was 0.3 seconds. The entire trajectory was optimised to avoid jerking the telescope during end-of-row turn around. Figure 1 shows a typical scan pattern as executed by the telescope, that is, as indicated by the telescope encoders. The time required to carry out the entire block consisting of a pointing, focusing and OOF runs is normally 25 minutes.

3. Data reduction 3.1. Preparation of the raw data The first stage of data analysis was to associate an on-the-sky position, relative to the source, with each sample from the receiver and to remove instrumental baselines which are mostly due to residual sky-brightness variations and gain fluctuations of the receivers. Data were obtained directly from the GBT engineering FITS files using PYTHON scripts. First, the readouts of the antenna encoders were linearly interpolated to obtain the actual (rather than the commanded) position of the antenna at the time of each data sample. The offset of the antenna from the expected position of the source was calculated via: = (MNT Az - OBSC Az + ) cos(MNT El), = MNT Az - OBSC Az + (1) (2)


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Fig. 1.-- The actual scan pattern executed by the telescope for one of the beam maps used in the OOF analysis (scan 114 on 11th April 2005).

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where MNT Az, etc., are columns in the antenna FITS files, and and are the pointing corrections derived from header values as follows: = SOBSC Az - SMNTC Az, = SOBSC El - SMNTC El. Figure 1 shows a typical OOF map raster extracted from the antenna FITS files in this way. The two beams of the Q-band are read out sequentially, and so in order to form an accurate differenced data stream (given that the telescope is continuously scanning) it is necessary to interpolate, for each measurement of beam A, the adjacent measurements of beam B and vice versa. The values are then differenced to obtain a data streams which typically look like those shown in the top panel of Figure 2. The resulting time-series typically still have some instrumental structure, which were mitigated by removing linear baselines from each row. Finally, the first and last few seconds of the time series were removed since, as can be seen in Figure 1, the telescope was not stable during these times. This is due to the well known trajectory generation problem. The final data series are shown in the bottom panel of Figure 2. The top row of Figure 3 shows observed beam maps produced by griding (using portions of the OOF analysis software) the data streams obtained in this way. (3) (4)

3.2.

OOF analysis

The time series data, consisting of azimuth offset (), elevation offset (, both offsets are relative to source position) and antenna temperature (T A ), for the set of three out-of-focus maps were then analysed using the custom OOF software to produce a map of the total optical aberrations in the telescope. We stress that the gridded maps shown in Figure 3 are not a suitable input data set since the griding process significantly decreases the information content of the data. After some experimentation it was decided to only fit for the aberration coefficients, overall normalisation of all three maps and the relative gains of the beams (see Table 1). Some other parameters that were fitted for in some previous analyses were altogether dropped, including a mean offset of the maps from zero and differences in pointing centres of the individual beam maps. Other parameters were fixed after investigation that they do not appear to vary in a significant way. The most important of these was the aperture plane amplitude distribution, that is, the illumination of the primary surface. This was approximated as a well-centred and circular Gaussian with a width (in radius-normalised units) defined by = 0.3, which corresponds to 14.5 dB of illumination taper at the edge of the dish. All of the fixed parameters and their values are shown in Table 2. The maps used for the fitting were 256 squared pixels in size and the physical scale was set so that the pixels oversampled the perfect beam size by a factor of two. It is also necessary to specify the full-width half-maximum and extent of the kernel used to interpolate the model beam maps to the positions of the individual data samples. These parameters were set to 1 and 2 pixels respectively. These variables which control the behaviour of the fitting procedure are summarised in Table 3. The total number of terms used in fitting for the aperture phase distribution was 20, i.e., the first five radial orders of Zernike polynomials. In order to avoid the possibility of finding local solutions, the


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Fig. 2.-- Example of a data series measured for an in-focus beam map. The vertical axis is the antenna temperature (in cal-signal units), the horizontal axis is time offset (MJD). The top panel shows the raw data series, the bottom panel shows the data series after removal of atmospheric baseline and the first, and last, few seconds of the observation.

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Fig. 3.-- Top row: A sample set of out-of-focus maps taken on 11th April 2005 (scan numbers 114ff). The left-most map is in-focus, the other two are both 35 mm out of focus, with the sense of the defocus such that in the centre map the subreflector is further away from the primary than the optimal focus and in the right-most map the subreflector is closer to the primary than optimal. Middle row: simulated beam maps for the GBT with no wavefront errors. Bottom row: simulated beam maps of the best-fitting model (using Zernike polynomials up to fifth radial order inclusive) to the observed maps in the top row. Angular scale is indicated in the units of radians and contours in all maps are at (min, ma x) ½ 0.5 i .


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number of radial orders fitted was iteratively incremented from one to five. The piston (zeroth order) term was of course not fitted for, and the coefficients of the first and second Zernike polynomials (which correspond to tilts of the telescope) were fitted for but not set to zero when calculating RMS values and plotting aperture phase maps . A typical quality of the fit to the observed beams obtained in this way is shown if the bottom row Figure 3.

4. Repeatability of the OOF technique One of the ways of investigating the accuracy of the OOF holography technique is to measure the best closure that can be achieved using the technique, that is, measuring the aberrations, correcting them and repeating the measurements to evaluate the effect of the correction on the aberrations. There are some obvious sources of error with this approach, most notably that the actual aberrations are likely to be changing continuously with time and so correcting for the aberrations at the time of the first measurement can not be expected to fully remove the aberrations at the time of the validation measurement. There is also a possibility that the procedures for correcting the telescope optics given a measurement of the aberrations are themselves not entirely accurate. Nevertheless, since there is no significant chance that any of these effects will accidently make the aberrations smaller, a measurement of the closure does provide a good upper limit on the random errors associated with the OOF technique. It is of course essentially insensitive to systematic errors. We have performed a few such measurements, most recently on 11th September 2005. This last measurement, which is shown in Figure 4, was done at an elevation close to the rigging angle and under very good conditions. The measured large scale aberrations (left panel of Figure 4) were quite low, corresponding to an illumination-weighted half-path RMS of around 150 ²m. These measurements were processed, applied to the telescope within 30 minutes, and another set of OOF maps with these corrections applied was taken. The right panel of Figure 4 shows the aberrations derived from this second set of OOF maps, that is the residual aberrations after correcting for what we measured in the top panel. The estimated weighted half-path RMS derived from this second set of OOF maps is 100 ²m. This indicates that the random error of the OOF technique, operating at a wavelength of 7 mm and using a basis set consisting of the first five radial orders of Zernike polynomials is around, or slightly smaller than, 100 ²m illumination-weighted half-path RMS. In terms of the observing wavelength, the accuracy of the technique (at the scales being probed) is better than /50. Parameter Name z1,z2 z3íz20 amp beamgainf Description Telescope tilts Coefficients of Zernike polynomials describing the aberrations of the telescope. Overall normalization of the intensity of model maps The difference in gains of the two beams

Table 1: List of parameters that are fitted for in the OOF analysis presented here.


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Parameter sigma hchop vchop

Value 0.3 -0.000280 0

Explanation The illumination taper (corresponds to 14.5 dB at dish edge). The separation of the two beams in azimuth direction in units of radians. The separation of the two beams in elevation direction in units of radians.

Table 2: List of parameters which were fixed rather than fitted for.

Parameter nzmax npix oversample ds fwhm ds extent

Value 5 256 4 1 2

Explanation Maximum radial order of Zernike polynomials to fit for. Number of pixels to use in the maps representing the aperture and model sky. Make model sky pixel size this smaller than would be required from the Nyquist criterion. The full-width half-maximum (in pixels) of the convolution kernel used to interpolate model sky to data position points. The maximum extent (in pixels) of the kernel used to interpolate model sky to data positions. (The kernel is truncated outside this radius.)

Table 3: Parameters to the fitting procedure adopted for analysis of OOF data presented here.

Fig. 4.-- Closure of the OOF technique at the GBT. Left: an aperture phase distribution derived from OOF measurements with phase RMS corresponding to weighted surface RMS of 150 ²m. Right: the measured aperture phase distribution after applying corrections derived from the measurement shown in the top panel. The weighted RMS is reduced to 100 ²m.


PTCS/PN/47.1
5. Modelling the gravitationally-induced aberrations

11

The main aim of this campaign was to determine the gravitationally induced aberrations to the GBT optics. We approached this by collecting data taken at a wide range elevations and looking for signatures of gravitational effects. It is of course desirable to minimise any other, non-repeatable, source of aberrations when performing this so we restricted ourselves to the measurements obtained under most stable thermal conditions. In total 36 separate OOF measurements, listed in Table 4, were used in the gravitational analysis. The distribution of these measurements over elevation is shown as a histogram in Figure 6. Typically during these observations, corrections derived from a previous analysis were applied and this was taken into account during the analysis to derive the total aberration of the telescope at the elevation of each measurement. One way of summarising these results, shown in Figure 5, is to calculate the effect the large-scale aberrations (derived from each set OOF maps) have on the aperture efficiency of the telescope. As is clear from this figure, the large-scale structure becomes significant at elevations around 40 degrees and below, giving rise to a approximately 20% decrease in overall aperture efficiency at 40 degrees and as much as 45% decrease at elevations around 20 degrees. It can also be seen from Figure 5 that at elevations around 60 degrees, the measured large scale can not have a dramatic effect on the overall aperture efficiency. Comparison of the spread in measured LS S with its difference from unity at these elevations suggests that it is not realistic to expect large improvements in aperture efficiency by making corrections at these elevations. The observed spread is most likely explained by non-repeatable aberrations induced by, for example, residual thermal gradients in the telescope structure. The results of Section 4 show that the expected random error on measurements is 100 ²m RMS, which, using the Ruze formula1 , corresponds to only about 3% drop in the aperture efficiency. Therefore, the observed spread is somewhat too large to be explained by random errors only. In so far as the telescope can be considered a linear elastic structure that is symmetric in the plane of the feed arm, any gravitationally induced aberrations should depend on the two resolved components of gravity only, that is, g sin and g cos , where is the telescope elevation. On the basis of such an assumption, we found the best-fitting models of the form: zi () = a sin() + b cos() + c (5)

to each (non-trivial) coefficient of Zernike polynomials (zi ) using the simple least-squares iterative optimisation routine SCIPY.OPTIMIZE.LEASTSQ2 . The coefficients of a, b and c from the best fitting models are shown in Table 5. The best-fitting models are plotted together with observed values for each of the coefficients in Figure 7. These coefficients define a continuous function correction to the existing finite element model the GBT. The current implementation however degree elevation intervals and simply applies to
1 2

in elevation which can be regarded as a relatively small which is currently routinely used for observations with makes use of the corrections evaluated at discrete fivethe active surface the closest pre-evaluated correction.

= e-(

4 2

) where is the aperture efficiency, is the half-path RMS and the wavelength.

Available from http://www.scipy.org/


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Date 050911 050911 050911 050911 050911 050911 050911 050911 050911 050911 050911 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050912 050411 050411 050411 050411 050411 050411 050411 050411 050411

Scan number 6 14 29 37 45 53 69 77 85 93 101 19 27 35 51 59 67 75 83 91 99 107 115 123 131 139 147 155 114 141 156 183 198 225 240 348 375

Elevation 65 67 67 66 64 61 55 52 48 44 40 12 15 18 25 28 28 24 21 17 50 54 57 61 65 69 73 77 41 40 45 58 40 74 27 59 48

Table 4: List of scans used in the gravitational deformation analysis.


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1 0. 9 0. 8
LSS



0. 7 0. 6 0. 5 0. 4 0 20 40 (degrees) 60 80

Fig. 5.-- Predicted decrease in on-axis gain due to the large-scale structure ( measurements listed in Table 4 as a function of elevation ().

LSS

) derived from OOF

8

6

4

N

2

0 0 20 40 (deg) 60 80

Fig. 6.-- A histogram of mean elevations of OOF beam map sets for the measurements used in the analysis of gravitation aberrations.


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n 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 5

l -2 0 2 -3 -1 1 3 -4 -2 0 2 4 -5 -3 -1 1 3 5

c 0.1 -0.3 1.5 4.0 0.1 0.4 -0.8 0.6 -0.5 -0.2 0.1 -0.1 -1.0 0.4 -0.9 -0.5 -0.5 -0.2

a -0.0 0.2 -1.1 -3.2 0.5 -0.2 0.5 -0.5 0.7 0.5 -0.4 0.2 1.3 -0.5 0.2 0.3 0.3 0.3

b 0.3 -0.1 -0.1 -2.4 -1.1 -0.1 0.7 -0.4 0.5 0.0 0.0 1.1 0.5 -0.1 1.2 -0.1 0.3 0.3

RMS(zi ) 0.40 0.29 0.81 0.43 0.45 0.20 0.23 0.16 0.30 0.23 0.22 0.82 0.32 0.20 0.27 0.35 0.16 0.26

RMS(zi - z 0.20 0.15 0.27 0.25 0.13 0.10 0.22 0.16 0.07 0.07 0.11 0.18 0.17 0.16 0.15 0.06 0.11 0.19

i,model

)

Table 5: Best-fitting Hooke's models to coefficients of Zernike polynomials. The column headings have the following meanings: n and l are respectively the radial and angular orders of the Zernike polynomial corresponding to the coefficient being fitted for; c, a, b are the best-fitting coefficients of the Hooke model as shown in Equation 5; RMS(zi ) is the root-mean-square of the coefficients over the entire sample; RMS(zi - zi,model ) is the root-mean-square of the residual of the coefficients after taking away the best-fitting model as a function of elevation. The units of c, a, b, RMS(z i ) and RMS(zi - zi,model ) are all radians of phase at the aperture edge.


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0.25

2

1. 5 1 0. 5

0

1. 5

-0.25 -0.5 -0.75

z3

z4

z6 0 -0.5

1

z5

0. 5

0
20 40 Elevation (deg) 60 80

20

40 Elevation (deg)

60

80

20
1

40 Elevation (deg)

60

80

-1

20

40 Elevation (deg)

60

80

1

0.6

0. 5

0.5

0.4

0.5
z10

0.25

0 -0.5 -1

0.2

0 -0.5 -1

z7

z8

z9

0 -0.25 -0.5

0 -0.2

20

40 Elevation (deg)

60

80

20

40 Elevation (deg)

60

80

20

40 Elevation (deg)

60

80

20

40 Elevation (deg)

60

80

0. 6

0.6

0.2

1. 5 1.25 1

0. 4 z11
z12

0.4 z13

0 -0.2 -0.4 -0.6

0.2

z14 0.75 0. 5 0.25
20 40 Elevation (deg) 60 80 -0.1 -0.2 z18 -0.3 -0.4

0. 2
0

0 20
1

40 Elevation (deg)

60

80

-0.2

20

40 Elevation (deg)

60

80

20

40 Elevation (deg)

60

80

0.6 0.4 0.2

1

0.5 z15 z16

0.5 z17

0 -0.5 -1

0 -0.5 -1

0 -0.2 20 40 Elevation (deg) 0.2
0.75 0.5 0.25

-0.5 20 40 Elevation (deg) 60 80 -0.6 20 40 Elevation (deg) 60 80

60

80

-0.4

20

40 Elevation (deg)

60

80

0 z19 -0.2 -0.4 -0.6
z20

0 -0.25

20

40 Elevation (deg)

60

80

-0.5

20

40 Elevation (deg)

60

80

Fig. 7.-- Measured values of the coefficients of Zernike polynomials as a function of elevation (crosses) and the best fitting model of the form shown in Equation 5 (solid line). The coefficients are shown as follows. Top row, left to right: (n = 2, l = -2), (n = 2, l = 0), (n = 2, l = 2), (n = 3, l = -3). Second row, left to right: (n = 3, l = -1), (n = 3, l = 1), (n = 3, l = 3), (n = 4, l = -4). Third row, left to right: (n = 4, l = -2), (n = 4, l = 0), (n = 4, l = 2), (n = 4, l = 4). Fourth row, left to right: (n = 5, l = -5), (n = 5, l = -3), (n = 5, l = -1), (n = 5, l = 1). Fifth row, left tor right: (n = 5, l = 3), (n = 5, l = 5). Units throughout are radians of phase at the aperture edge.


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Figure 8 shows how the model discussed here varies with elevation, while Figures 9 and 10 compare the best-fitting model with some individual observations taken at similar elevations.

6. Aperture Efficiency Measurements 6.1. Method The on-axis gain of a radio telescope is the constant of proportionality (assuming a completely transparent Earth's atmosphere) between the flux density of a point source, S , and the measured antenna temperature, T A , when the point source is on the optical axis of the telescope. The aperture efficiency, a , is the ratio of the actual on-axis gain of the telescope and the maximum possible gain given the physical size of the aperture of the telescope. In the case of the GBT, the following relation for the aperture efficiency holds: 8kT
A

a = where

D2 S

= 0.351

TA S

(6)

T A = T A exp(o / sin )

(7)

is the antenna temperature now corrected for the effect of atmospheric opacity, k is the Boltzmann's constant, D is diameter of the aperture of the telescope, o is the atmospheric opacity at zenith, and is the telescope elevation. A traditional method for quickly determining the aperture efficiency of a telescope is to drive the telescope back and forth along its cardinal directions so that the scans cross a (usually relatively strong) source with a well measured and constant intrinsic flux density (a Peak observation). The cardinal directions are typically those of the telescope drive system, which in the case of the GBT are azimuth and elevation. Such Peak observations not only provide a measure of the peak antenna temperature used to derive the aperture efficiency but also provide some information about the shape of the telescope beam.

Peak observations at Q-band (43.1 GHz) have in our experience produced estimated peak antenna temperatures that vary by as much as 25% even under benign weather conditions. For example, Figures 11í 12 show four typical Peak total power scans that vary in intensity by 15% with an average intensity of 12.7 ‘ 0.92 K.1 Similar results are obtained for beam-switched data.
One of the problems with Peak observations arises from the fact that the telescope's optical axis is within a beam full-width at half-maximum (FWHM) of the source for a time of only about two seconds. Any significant pointing fluctuations on timescales comparable to, or longer than, these two seconds will cause significant random errors in the measured peak antenna temperature and consequently the
1 The intensity scale is set by injecting a known noise into the signal path with a rate of 10 Hz. We have used the engineering noise calibration temperatures measured in 2004 where T cal (L1) = 9.81 K; T cal (R1) = 9.37 K; T cal (L2) = 5.73 K; and T cal (R2) = 5.13 K.


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Fig. 8.-- The derived model for gravitationally induced total large-scale aberrations, covering the range from 10 (top left) to 70 (bottom row) degrees in elevation and calculated at intervals of ten degrees. The units of maps are radians of phase, the contours are at intervals of 0.5 radians and the assumed wavelength is 7 mm.

Fig. 9.-- A comparison of the aberration model evaluated at an elevation of 25 degrees (left panel) and two sample OOF measurements of the aberrations close to this elevation (centre and right panels). Units in these plots are as in Figure 8.


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estimated aperture efficiency. Pointing fluctuations on shorter scales will lead to a systematic error in the estimate of these quantities. Half-power-point tracking observations have been used to measure the pointing stability while tracking celestial objects at the sidereal rate (Balser et al. 2002; Constantikes 2003; Balser & Prestage 2003). These observations consist of tracking for about 15-30 minutes at the sidereal rate while a strong astronomical point-source is at the half-power point of the beam. The fluctuations in the measured antenna temperature during such observations correspond to pointing errors, fluctuations in the receiver, and any changes in the atmosphere. Therefore, these observations provide upper limits to the pointing stability of the telescope. Under benign weather conditions the root mean square (RMS) pointing stability was measured to be 1í2 arcseconds. The pointing fluctuations are dominated by servo errors that correspond to temporal frequencies of 0.15 and 0.30 Hz in the elevation direction and 0.2 Hz in azimuth direction. RMS fluctuations of 1í2 arcsec correspond to peak-to-peak fluctuations of 2.8í5.6 arcseconds. Since the half-power beam-width (HPBW) of the GBT at 43.1 GHz is about 18 arcseconds, such pointing fluctuations of 2.8í5.6 arcseconds can decrease the measured peak antenna temperature by between 5í 25%. This is consistent with the range of intensities measured with Peak data. Given these uncertainties in estimating the intensity using Peak observations, we rely primarily on Nod observations to determine the aperture efficiency. Nod observations consist of a single scan during which the telescope is driven to `ON' and `OFF' positions, that is to positions such that the source is either in the centre of the beam or far from it. Since the Q-band receiver has two beams on the sky, we move the telescope so that one of the beams is always ON source. The telescope spends 20 seconds in either position before being commanded to move to the other position using a trajectory which was designed to minimise the magnitude of the first derivative of the acceleration (jerk) of the telescope. In order to allow the telescope the time to settle on the source, the first few seconds on each position are not considered when analysing the data. Typical total power at the receiver as a function of time during a Nod observation is shown in Figure 13. A best-fitting linear polynomial model to the OFF portion of the scan has been removed from the data. It is noticeable that the antenna temperature fluctuations while the telescope is ON source are significantly larger than the fluctuations when the telescope is OFF source. This suggests that the fluctuations when

Fig. 10.-- As Figure 9, but for observations and model at 65 degrees elevation. Left: the derived bestfitting model model evaluated at 65 degrees; centre: aberrations inferred from observations on 20050911 at 65 degrees; right: aberrations inferred from observations on 20050912 at 65 degrees.


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ON source are not due to atmospheric effects or receiver instabilities but almost certainly due to pointing errors. The average antenna temperature is 12.5 ‘ 0.61 K with peak-to-peak variations of about 20%, which is consistent with our earlier estimates given the pointing accuracy estimated from the half-powerpoint tracking experiments. Since the pointing errors can only cause a decrease in the measured antenna temperature, we normally only use the 10% of the data with the highest antenna temperature (the mean of these data is 13.6 K in this case) to estimate the aperture efficiency. Although the effective aperture efficiency for point sources will be less and can be estimated by using the average over all the samples. Analysis of the beam-switched Nod data produces similar results, except that the measured intensities are systematically lower by about 5%. This can be attributed to uncertainties in the calibration since the calibration for both beams is being applied for beam-switched data, while the calibration for only the first beam is being applied for total power data. This is consistent with the uncertainty in the calibration (Balser et al. 2005).

6.2. The This with ON.

Observations

OOF holography models were validated on the three separate occasions summarized in Table 6. was done by performing Peak, Focus, and Nod observations on a flux density calibration source the OOF holography model turned OFF and then the same three observations with the model turned The Nod observations were used to determine the aperture efficiency as discussed in ç 6.1.

The flux density calibrators are shown in Table 7. Listed are the source name, the J2000 coordinates, and the flux density. The primary flux density calibrators 3C286 and NGC 7027 were used to adjust the intensity scale of the secondary calibrators. The flux density for 3C286 is taken from Condon (2003), while for NGC 7027 we use the data from Peng et al. (2000), corrected for angular size by assuming a disk-like distribution (Ott et al. 1994). The secondary calibrators 1256-0547 and 2253+1608 have the best signal-to-noise ratio and should be used to determine the aperture efficiency gain-elevation curve. All observations were at night when the thermal effects are minimized and during benign weather conditions when the skies are clear and air calm (vwind < 5 - 10 km s-1 ). The opacity was determined using the high frequency weather forecasts determined by Ron Maddalena 2 and listed Table 6. To verify that the atmosphere was stable during the validation observations we have used Nod data taken during the entire evening for each Epoch to determine the system temperature as a function of elevation. The OFF
2

See http://www.gb.nrao.edu/$\sim$rmaddale/Weather/index.html

Table 6: Summary of Nod Observations
Project TPTCSOOF 050411 TPTCSOOF 050910 TPTCSOOF 051001


Date 11 April 2005 10 September 2005 01 October 2005

OOF Model 2005SpringV1 2005SpringV1 2005WinterV1

Opacity 0.11 0.16 0.18



Comments Epoch 1 Epoch 2 Epoch 3

Taken from Ron Maddalena's high frequency weather forecasts.


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Fig. 11.-- Azimuth total power pointing scans at Q-band for the L1 channel. The red solid points are the data with intensity in units of Kelvin. The solid curve is a Gaussian fit to the data. The blue curve shows the regions that were used to fit a third-order polynomial model to the baseline. Only the regions in green have been used in the Gaussian fit. The yellow curve corresponds to the residuals. In parentheses are the azimuth and elevation in degrees. In square brackets are the results of the Gaussian fit: height [K], width [arcmin], and center [arcmin].


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Fig. 12.-- Elevation total power pointing scans at Q-band for the L1 channel. The red solid points are the data with intensity in units of Kelvin. The solid curve is a Gaussian fit to the data. The blue curve shows the regions that were used to fit a third-order polynomial model to the baseline. Only the regions in green have been used in the Gaussian fit. The yellow curve corresponds to the residuals. In parentheses are the azimuth and elevation in degrees. In square brackets are the results of the Gaussian fit: height [K], width [arcmin], and center [arcmin].


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Fig. 13.-- Total power Nod scan at Q-band for the L1 channel. The red curve are the data. The green points are samples used for the OFF source (blank sky) level. The blue points are used for the ON source level. A linear polynomial fit has been used to model the baseline and removed from the data. The horizontal (yellow) line is the mean intensity (12.5 ‘ 0.61 K).

Table 7: Flux Density Calibration Sources
Flux Density (43.1 GHz) Epoch 1 Epoch 2 Epoch 3 1.462 -- 15.7 4.0 -- -- -- 4.55 -- -- 12.0 6.5 -- 4.55 -- 5.3 12.7 6.2

Source 3C286 NGC 7027 1256-0547 1642+3948 2253+1608 2148+0657


RA (J2000) 13:31:08.284 21:07:01.60 12:56:11.1665 16:42:58.8099 22:53:57.7479 21:48:05.4586

Dec (J2000) +30:30:32.94 +42:14:10.0 -05:47:21.524 +39:48:36.993 +16:08:53.560 +06:57:38.604

Comments Primary Calibrator Primary Calibrator Secondary Calibrator (3C279) Secondary Calibrator (3C345) Secondary Calibrator (3C454.3) Secondary Calibrator

Corrected for angular size.


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position was used to detemine the cold sky system temperature for each channel. Figures 14, 15, and 16 show the system temperature gain curves for Epochs 1, 2, and 3, respectively. We fit each gain curve using the following equation: T a = T rx + T
atm

(1 - e

-0 A

) + (1 - ) T

spill

+T

fss

+ T

cmb

e

-0 A

,

(8)

where T a is the antenna temperature (i.e., since these are OFF data, the system temperature), T rx is the receiver temperature, T atm is the mean temperature of the atmosphere (determined using the high frequency weather forecasts), T spill is the effective temperature of the real spillover (300 K), T fss is the forward spillover and scattering temperature (1 K), T cmb is the cosmic microwave background temperature (2.7 K), is the rear spillover efficiency (0.99), 0 is the opacity at zenith, and A is the number of air masses relative to zenith (1/ sin ). We solved for T rx and for each channel. Overall, the atmosphere is reasonably stable.

6.3.

Results

The aperture efficiency gain curves determined from the total power Nod data are show in Figures 17, 18, and 19 for Epochs 1, 2, and 3, respectively. The top panel plots the results from only the brightest object, while the lower panel shows all of the data. The green (filled) and red (open) circles correspond to the OOF holography model turned ON and OFF, respectively. Overall there is significant improvement in the aperture efficiency at elevations below 40 degrees. The OOF holography model flattens the gain curve as expected between 10í70 degrees elevation with an aperture efficiency around 0.45. This corresponds to a total surface rms of 370 microns. Fluctuations in the pointing stability caused primarily by the servo errors decrease the effective aperture efficiency by 5í10%.

7. Discussion and Summary The implications of the measurements presented here are discussed to a large extent by Nikolic (2006).

REFERENCES Balser D. S., Prestage R. M., 2003, Analysis of july 2003 Half-power Tracking Data. PTCS Project Note 19.1, NRAO Balser D. S., Anderson G., Norrod R., 2005, GbT Q-band Calibration. PTCS Project Note 41.1, NRAO Balser D. S., Maddalena R. J., Ghigo F., Langston G. I., 2002, GbT X-band (9 ghz): Pointing stability. GBT Com. Memo 20, NRAO Condon J. J., 2003, GbT Efficiency at 43 ghz. PTCS Project Note 31.1, NRAO Constantikes K., 2003, Analysis of quadrant detector, servo monitor, and half power track data from project tdbalser qd 2 (20 august 2002). PTCS Project Note 13.1, NRAO


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Fig. 14.-- System temperature gain curve for Epoch 1 (11 April 2005). The red and green symbols correspond to LCP and RCP, respectively. The filled and open circles correspond to beam 1 and 2, respectively. Top Panel: system temperature on cold sky versus elevation. The solid curves are fits to the data. Bottom Panel: normalized system temperature ratio versus elevation.


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Fig. 15.-- System temperature gain curve for Epoch 2 (10 September 2005). The red and green symbols correspond to LCP and RCP, respectively. The filled and open circles correspond to beam 1 and 2, respectively. Top Panel: system temperature on cold sky versus elevation. The solid curves are fits to the data. Bottom Panel: normalized system temperature ratio versus elevation.


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Fig. 16.-- System temperature gain curve for Epoch 1 (01 October 2005). The red and green symbols correspond to LCP and RCP, respectively. The filled and open circles correspond to beam 1 and 2, respectively. Top Panel: system temperature on cold sky versus elevation. The solid curves are fits to the data. Bottom Panel: normalized system temperature ratio versus elevation.


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Fig. 17.-- Aperture efficiency gain curve for Epoch 1 (11 April 2005). The filled (green) symbols correspond to observations where the OOF holography models have been applied, while the open (red) symbols correspond to observations where the OOF holography model has not been applied. Each source is denoted by a different symbol: circle (3C286), star (1256-0547), triangle (1642+3948). Top Panel: only 1256-0547. Bottom Panel: all sources.


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Fig. 18.-- Aperture efficiency gain curve for Epoch 2 (10 September 2005). The filled (green) symbols correspond to observations where the OOF holography models have been applied, while the open (red) symbols correspond to observations where the OOF holography model has not been applied. Each source is denoted by a different symbol: circle (NGC7027), star (2253+1608), triangle (2148+0657). Top Panel: only 1256-0547. Bottom Panel: all sources.


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Fig. 19.-- Aperture efficiency gain curve for Epoch 1 (01 October 2005). The filled (green) symbols correspond to observations where the OOF holography models have been applied, while the open (red) symbols correspond to observations where the OOF holography model has not been applied. Each source is denoted by a different symbol: circle (NGC7027), star (2253+1608), triangle (2148+0657), and square (1642+3948). Top Panel: only 1256-0547. Bottom Panel: all sources.


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Nikolic B., Richer J., Hills R., 2002, in Proceedings of the General Assembly of U.R.S.I., Maastricht Nikolic B. e. a., 2006, in prep. Ott M., Witzel A., Quirrenbach A., Krichbaum T. P., Standke K. J., Schalinski C. J., Hummel C. A., 1994, A&A, 284, 331 Peng B., Kraus A., Krichbaum T. P., Witzel A., 2000, A&AS, 145, 1

A This preprint was prepared with the AAS L TEX macros v5.2.