Evaluation of the ALMA Prototype Antennas¹

VertexRSI antenna: The antenna was delivered with a nominal surface error of 80 μm rms, as determined from a photogrammetric measurement. Our first holography map showed an rms of approximately 85 μm. A first setting of the surface resulted in an rms of 64 μm. In four more steps of holographic measurement followed by adjustment, the surface error decreased to 19 μm rms.

The sequence of surface error maps, along with the rms and the error distribution, is shown in Figure 4. As allowed in the specification, we have applied a weighting over the aperture that is proportional to the illumination pattern of the feed. This essentially diminishes the influence of the surface errors in the outer areas of the reflector. The white areas in the surface error maps are the quadripod, the optical pointing telescope, and a few bad panels that could not be set accurately. All of these structures were left out of the calculation of the final overall rms value.

With increasing accuracy, the presence of an artifact in the outer area of the aperture became apparent. There was a wavy structure in the outer section, with a "period" that was too large to be inherent in the panels. Experiments with absorbing material showed that it was not caused by multiple reflections. The effect can be described by a DC offset in the central point of the measured antenna map, i.e., some saturation on the point with the highest intensity. By adjusting this offset in the analysis software, most of the artifact could be removed. This has been done with the final data. The additional set of follow‐up holography measurements in the period of 2004 December to 2005 February did not suffer from this signal saturation, and no artifact was observed in these follow‐up maps. Checks of the holography hardware indeed suggest that the 2003 holography measurements did experience a small amount of signal saturation.

The adjustments were done with a simple tool. Two people on a man‐lift approached the surface from the front, where the adjustment screws are located (see Fig. 5). The time needed for an adjustment of the total of 1320 adjusters was 8 hr, as required by the specification.

The best surface maps were obtained at night. During the spring 2003 period, they consistently show an rms of about 20 μm. Daytime maps tend to be somewhat worse; typical values of the rms lie between 20 and 25 μm. Part of this is certainly due to the atmosphere, even over the short path length of 315 m.

AEC antenna: The apex structure of the AEC antenna does not enable us to mount the holography receiver inside the apex structure cylinder, as is the case for the VertexRSI antenna. Thus, in this case the receiver was bolted to the flange on the outside of the apex structure. Consequently, the feedhorn was brought to the required position by a piece of waveguide of about 500 mm length. This caused significant attenuation in the received signal from the reflector to the mixer. Considering the available transmitter power, we concluded that this would not significantly jeopardize our measurement accuracy.

The AEC antenna surface was set by the contractor with the aid of a Leica laser tracker. The rms of the surface was reported by the contractor to be 38 μm. After this measurement, an accident caused the elevation structure to run onto the hard stops at high speed. The contractor decided to repeat the surface measurement and obtained an rms of 50 μm, with some visible astigmatism in the surface.

Our first holography map indicated an rms of 55 μm with a clearly visible astigmatism. We could identify the high and low regions with those on the final AEC measurement. With two complete adjustments, we surpassed the goal of 20 μm. A third partial adjustment improved the surface rms to about 14 μm. There is no indication of the artifact seen in the VertexRSI antenna. There is one panel with a large deviation over part of the area. This is believed to have been caused during the measurement and setting procedure by the contractor. We have not included this panel in the computation of the final rms value. The results of the consecutive adjustments are summarized in Figure 6. The last panel in this figure shows the final map after the repeated measurement and setting in 2005 January.

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The adjustments were done with a tool provided by the contractor. It was similar to the one used by us on the VertexRSI antenna, but it was calibrated in "turns" rather than in microns. Again, two people on a man‐lift approached the surface from the front, where the adjustment screws are located. The time needed for an adjustment of the total of 600 adjusters was 7 hr, well within the specification of 8 hr.

VertexRSI antenna: To estimate the accuracy and repeatability of the measurements, we produced difference maps between successive measurements throughout the measurement period. The rms difference between consecutive maps is typically less than 10 μm, and usually ∼8 μm. An example of a difference map is shown in Figure 7. The map of measurement number 307 is shown on the left, while the right panel shows the difference between map 307 and 308, made about 1 hr later.

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We have made many maps after the final setting of the surface, changing the orientation of the antenna with respect to the Sun. Maps taken over a few consecutive days were obtained at widely different temperatures and while the wind conditions varied over time. The measured rms error during a 30 hr period in early 2003 May varied between 20 and 23 μm. During this period, the wind was calm (<5 m s⁻¹) and temperature changes of 15°C were encountered. A later 5 day series in mid‐June gave rms errors from 22 to 26 μm, with temperature variation up to 20°C, wind speeds up to 10 m s⁻¹, and periods of full sunshine. Most of this increase is believed to be due to the deteriorating atmospheric conditions at the VLA site during summer, when the humidity was significantly higher than normal. The much better results of 17 μm obtained during the cold and dry winter period also point to a significant atmospheric component in the spring and summer results.

However, some of the changes will be caused by temperature and wind. To increase the rms from 20 to 22 μm, the "additional" component needs a magnitude of 9 μm rms. Such a contribution can be expected from the calculated values of 4 μm each for wind and temperature for the panels, and 5 μm for wind and 7 μm for temperature for the BUS. These numbers are all within the specification. Actually, the measured differences are close to those expected from the estimated accuracy of the holography measurement and the measured rms differences in consecutive maps of about 8 μm.

AEC antenna: In Figure 8 we show one of the final results and a difference map of this and the following measurement, made 1 hr later. The difference map indicates a repeatability of ∼5 μm rms. There is no indication of the wavy structure in the outer region of the aperture, as was the case for the VertexRSI antenna. We ascribe this to the lower signal level due to the long piece of waveguide between feed and mixer.

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We made a series of 16 maps over a period of more than 2 days in early 2004 February. Temperatures ranged from +2°C to −10°C, while the wind was mostly calm, with some periods of speeds up to 10 m s⁻¹. During one day, there was full sunshine. The measured rms error is very constant, with a peak‐to‐peak variation of less than 2 μm on an average of 14 μm. The differences are fully consistent with the allowed errors under environmental changes, and they are also of the same order as the measurement accuracy. We believe that the significantly better overall result is mainly due to the much drier and more stable atmosphere during these measurements as compared with the summer data from the VertexRSI antenna.

VertexRSI antenna: Further analysis of the VertexRSI time‐series data presented in § 5.1.2 has been made in order to extract the dependence of the antenna surface on changes in ambient temperature. The analysis entailed the following:

• 1.  
  For each series, the average of all maps in the series was subtracted from all the maps in the series. This suppresses dependencies on any long‐term effect.
• 2.  
  In each series, the maps were divided into several temperature ranges; in each temperature range, the average map was computed.
• 3.  
  Finally, the differences between the coldest and warmest range averages were computed for each series. From these data, we derived the surface deformation caused by a temperature change of 10°C, as shown in Figure 9.

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From this analysis, we note that

• 1.  
  Many of the samples were for temperatures outside the primary operating range of −20°C to +20°C.
• 2.  
  The results for the spring 2003 and winter 2005 periods are very consistent with each other.
• 3.  
  The magnitude of the rms deformations is ∼0.6–0.7 μm K⁻¹. This value seems high. It is not clear to which degree it should be split into panels and BUS and between absolute and gradient terms in the surface error budget. Assuming all effects to be linear with temperature, and antennas set at 0°C, we would expect rms temperature contributions of ∼13 μm for the VertexRSI antenna at either end of the operational temperature range (+20°C or −20°C).
• 4.  
  The thermal deformations for the VertexRSI antenna appear to be a mixture of BUS (the BUS sector edge sticks out at higher temperature) and panels, not all panels deforming equally.
• 5.  
  From the BUS overlay drawing (Fig. 9), it appears that there is a print‐through of the connections of the BUS with the Invar cone (the turnbuckles). The small red islands on (or just outside) the second stippled ring are at the diameter of the outer support of the BUS on the cone. We are not certain whether they are at or between the connections to the cone. In both cases, such a print‐through might reasonably be expected.
• 6.  
  The BUS is also supported on the cone toward the center, outside the inner stippled circle by about one‐fifth of the distance between the two stippled circles. Here we also see clear red islands of print‐through. These effects are less pronounced in the lower section of the dish; actually, they seem most pronounced in the left and right quadrants. We do not understand why this would be the case.
• 7.  
  There are clear, narrow "valleys" along the radials where the sectors of the BUS are joined, especially in the outer half of the radius. They are not equally strong all around, but are visible in most cases. The inner part is less clear, presumably because of the already existing effects of the support points.
• 8.  
  From the map with the panel outline, and ignoring the islands connected to the BUS, as argued above, there is some but not very much evidence for individual panel deformations. It is most likely the case that individual panel deviations are caused more by forces originating in the stiffer BUS than from the panel itself.

AEC antenna: Further analysis of the AEC time‐series data presented in § 5.1.2 has been made in order to extract the dependence of the antenna surface on changes in ambient temperature. The analysis entailed the same sequence of calculations as those presented for the VertexRSI antenna (see § 5.1.3). Figure 10 shows the temperature span assuming a uniform 10°C temperature difference.

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From this analysis, we note that

• 1.  
  The sampled temperature range is completely inside the primary operating condition ambient temperature range of −20°C to +20°C.
• 2.  
  The magnitude of the rms deformations is ∼0.8 μm K⁻¹. This value seems high. It is not clear to which degree it should be split into panels and BUS and between absolute and gradient terms in the surface error budget. Assuming all effects to be linear with temperature, and antennas set at 0°C, we would expect rms temperature contributions of ∼16 μm for the AEC antenna at either end of the operational temperature range (+20°C or −20°C).
• 3.  
  The large‐scale deformation at a 45° position angle (which looks like astigmatism), along with the additional deformation of the inner ring, are hard to explain. These deformations are perhaps due to the temperature gradients in the cabin and the BUS, coupled to the quadripod connection points. The vertical stripes on the edges of the AEC maps are an artifact of the measurement that is not related to temperature.
• 4.  
  In principle, there should be little print‐through structure in the difference maps, due to the fact that the BUS design is more homogeneous, being all CFRP, and it should have a continuous transfer of forces to the cabin, which is also of the same material.
• 5.  
  We do not see any individual panel deformation.

If a telescope or antenna is pointed at a star, there is generally a difference between the true direction of the incoming light and the demands to the mount servos, due to mechanical misalignments, flexures, and other imperfections. By observing stars all over the sky and logging the actual and commanded positions (or data from which these angles can be deduced), a mathematical model can be developed that estimates the servo inputs required to point the antenna accurately in a given direction. The TPOINT pointing analysis software reads a file containing the logged observations and provides graphics and modeling tools that allow a mathematical description of the pointing errors to be created. The resulting pointing model then has to be built into the telescope or antenna control system so that the demands to the servos are appropriately adjusted and the accurate dead‐reckoning acquisition of celestial targets is brought about.

For any two‐axis gimbal mount, the basic a priori model consists of seven terms: two encoder index errors, two nonperpendicularities, two azimuth‐axis tilt components, and Hooke's law flexure. All of these terms have a clear mechanical interpretation, and in general, all will be present to some degree. This basic model usually performs well, typically leaving only a small remaining systematic residue to be mopped up using the repertoire of other TPOINT terms. Most of the latter set out to represent plausible mechanical effects—for example, harmonic terms to deal with runouts and mechanical deformation errors—although it is also possible to introduce unashamedly empirical terms such as polynomials, should there be no alternative.

The functional form of the basic seven‐term model is given in Table 3. The following should be noted:

• 1.  
  The sign of the correction terms is such that the expressions should be subtracted from the true direction in order to generate mount demands.
• 2.  
  The coefficient values in service will either be those from the most recent set of observations or averages of several recent runs.
• 3.  
  The expressions are approximate, which is adequate when the coefficients are small or the target is not too close to the zenith. TPOINT in fact fits rigorous vector formulations that do not have these limitations.

The a priori seven‐term model (Table 3) is also that specified by ALMA for internal implementation in the low‐level antenna controllers, principally as an aid to commissioning. Once the final TPOINT model is available, the optimum coefficients to be used with this basic seven‐term model can be estimated by subtracting the full model from a set of points distributed evenly over the sky, and then fitting the seven‐term model. The final rms from such an experiment represents the maximum achievable performance of the ALMA internal model for the antenna in question, while the residual plots illustrate the antenna's pointing idiosyncrasies. This experiment was carried out for both antennas, using OPT data, for illustrative purposes. For the AEC antenna, the best‐fit sky rms was 246, while that for the VertexRSI antenna was 325. The limitations of this simple seven‐parameter model are illustrated in Figure 14.

• 1.  
  The optical seeing at the VLA site is generally poor. To quantify this seeing contribution to the OPT measurements, a sequential multiple star measurement, in which each star measurement was repeated three times, was used to derive seeing and short‐timescale antenna positioning errors. These measurements suggest a seeing/vibration contribution of ∼15, in line with the results from tracking tests. With the OPT's 100 mm aperture, the seeing manifests itself as large and rapid movements of the image, making optical measurement of the all‐important tracking and offsetting performance almost impossible.
• 2.  
  The OPT design, which was always going to be a major challenge, is itself marginal for this application, given the impressive performance of the antennas. The comparatively short focal length means that the image is undersampled, and in some of the tracking tests, quantization effects are evident, making performance limitations much below 1^'' difficult to detect. It is far from obvious how to deal with this; a refracting OPT of, say, twice the aperture would have been prohibitively expensive and also very large, whereas a reflector (a commercial Schmidt‐Cassegrain, for example) would probably have had insufficient optical stability.

The pointing measurement data logging procedure evolved during the antenna evaluation process, ultimately leading to a very self‐consistent and robust scheme for position calculation and logging. We refer to this final position calculation and logging scheme as PMDR (pointing model done right).

The two schemes (pre‐PMDR and PMDR) are illustrated in Figure 15. In the pre‐PMDR mode, the inputs to the TPOINT pointing analysis are (1) the "observed" azimuth and elevation (i.e., the true line of sight) and (2) the encoder readings from the mount axes. The TPOINT analysis produces updated pointing coefficients for the ALMA internal seven‐term model, calculated to minimize the offsets required to peak up on a new source and, equivalently, the drifts during subsequent tracking. However, note that coefficients obtained in this way presume zero metrology input; they are not suitable for use with metrology enabled, and the fitting statistics (e.g., rms) do not reflect any benefits coming from the metrology system. Moreover, if the control system supplied by the manufacturer were to be used, the TPOINT model would be limited to the seven standard terms (from Table 3), and additional forms of correction would not be available. This final point proved to be a major handicap, hampering the pointing performance characterization of the ALMA prototype antennas and limiting the ultimate performance (see Fig. 14).

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These shortcomings were addressed by the PMDR scheme (Fig. 15, green items). In this scheme, the quantities logged for input to TPOINT are (1) the observed azimuth‐elevation (or az‐el) as before, but this time including (2) the commanded inputs to the manufacturer's model and metrology systems, the latter constituting a "smart encoders" interface. The link between the two is a separate TPOINT model that can contain any required terms, not just the seven basic terms. In fact, it is advantageous to disengage the manufacturer's model so that the TPOINT model provides the whole correction, apart from the metrology inputs: the end result is the same, and the TPOINT model then represents absolute physical estimates (nonperpendicularities, misalignments, flexures, etc.). All measurements quoted in the following analysis used the PMDR positioning and logging scheme.

The respective data samples for each prototype antenna are listed below. For both samples, we note that

• 1.  
  The short time span available for evaluating pointing performance unfortunately precluded any studies of the long‐term stability of the pointing terms.
• 2.  
  A handful of these pointing tests were carried out with contractor‐delivered metrology systems. We describe the results of these few metrology tests in § 6.6.
• 3.  
  To obtain the total sample of measurements for both antennas, individual runs were for the most part accepted or rejected on the basis of notes made at the time they were carried out, rather than whether they gave better or worse pointing residuals. However, certain runs were clearly aberrant. These obviously atypical runs were rejected.

AEC: A selection of 63 pointing test runs were carried out during an intensive period of testing between 2004 March 6, when the antenna first became available, and May 12. There were a total of 7411 observations in this sample. Following some modifications by the antenna contractor in 2004 December, an additional series of measurements were made during the period of 2005 January 7 to March 2. As these additional measurements are not directly comparable to those made during the 2004 March 6 through May 12 period considered here, we defer their analysis to § 6.3.5.

VertexRSI: A selection of 34 pointing test runs that utilized the PMDR positioning system (see § 6.3.1) were carried out between 2004 March 10 and May 3, comprising 4059 observations.

In the final optical‐pointing–derived models, we have distinguished a set of floating basic pointing terms (IA, IE, CA, AN, and AW) from a fixed "core" set of terms that are not only considered to have a specific mechanical cause, but can also be assumed to be fixed in size. Moreover, for most of these core terms, there is no reason to suppose that they will be affected by the transition from optical to radio pointing.

The core terms for both prototype antennas are listed in Table 5. For each term, we list the nominal mechanical deformation that produces the term. Note the following:

• 1.  
  The sign of the correction terms is such that the expressions should be subtracted from the true direction in order to generate mount demands.
• 2.  
  The smallest terms are of order of 1^'' and 05 in size for the AEC and VertexRSI models, respectively. Their significance can be gauged by eliminating the smallest of them, namely HESA2 (AEC) and HECA3 (VertexRSI), fitting the full model to take up the slack, and plotting the residuals. The result of this experiment shows a distinct sin (2A) (AEC) and cos (3A) (VertexRSI) systematic residual in Δelevation versus azimuth for the respective antennas. Figure 16 shows the graphical results for the AEC antenna.

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In principle, all of the terms of an antenna's pointing model should maintain constant amplitude for long periods unless something on the antenna has been adjusted. This supposition was studied by taking the individual models for the series of pointing tests on each antenna and plotting the coefficient values for the five terms thought most likely to be affected by mechanical adjustments and drifts. The stability of these coefficient values was then studied by plotting them versus sequence number (in effect vs. time).

For the AEC antenna, the stability of the five basic pointing model terms over the period of 2004 March 6 through May 12 was (ΔIA, ΔIE, ΔCA, ΔAN, ΔAW) = (±3^'', ±5^'', ±3^'', ±1^'', ±1^''), with very little drift of the average values for each term. This is very good basic pointing term stability for an antenna. The VertexRSI antenna pointing term stability over a much longer measurement period (2003 October 16 to 2004 March 1) was (ΔIA, ΔIE, ΔCA, ΔAN, ΔAW) = (±3^'', ±7^'', ±5^'', ±5^'', ±3^''), with very little drift of the average values for most terms (the AN term showed a moderate ∼5^'' drift of the average during the measurement period). The basic pointing‐model‐term stability for the VertexRSI antenna was notably worse than that of the AEC antenna. Part of this poorer performance is likely attributed to the failure of the tiltmeter metrology system (see § 6.6), which was designed to correct for variations in the azimuth axis tilt (AN and AW).

Following the all‐sky OPT measurements described above, the antenna contractor made some modifications designed to improve the all‐sky pointing performance. In particular, the attachment points of the antenna base to its foundation were tightened, and a tiltmeter‐based metrology correction was enabled in an attempt to correct the suspected instability in the azimuth bearing. An additional series of measurements were made during the period of 2005 January 7 to March 2 following these modifications. From these measurements, we note that

• 1.  
  Two terms (HVSA2 and HVCA2) were no longer significant and need not exist in the pointing model solution. The terms HVSA2 and HVCA2 represent changes in the az‐el nonperpendicularity as the mount rotates. The insignificance of these terms suggests that the azimuth axis had become less wobbly.
• 2.  
  Figure 17 shows the cumulative AEC optical pointing model results for this period. For each total rms value shown, we also show the contributions to this total rms due to cross‐elevation and elevation pointing residual.
• 3.  
  A large observed shift in the IA and AW pointing terms relative to the previously derived pointing models reported in P. Wallace et al. (2004; see footnote 4) was ultimately diagnosed as a timing error of approximately 13 s in the monitor and control system. This timing error does not affect the resulting pointing rms values derived. On 2005 January 20, this timing error was corrected.
• 4.  
  A timing signal cable problem was fixed on 2005 January 23.

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The ALMA pointing specification permits monthly recalibration of the pointing model. To investigate this, two representative all‐sky OPT runs taken a month apart were chosen. The first was used to simulate the periodic recalibration of the five "noncore" terms of the AEC model (IA, IE, CA, AN, and AW). This simulated operational model was then applied to the second data set, and the resulting pointing performance was assessed with various terms refitted, as might be done in a quick daily recalibration.

Operationally, the frequent observation of standard stars means there is an opportunity to adjust at least the boresight offsets more frequently, and this was simulated by allowing the terms IE and CA to float. For AEC (VertexRSI), they changed by 29 (146) and 17 (15), respectively, and gave sky rms values of 244 (305), still somewhat outside the specification.

If the tilt terms AN and AW were also allowed to vary, the changes for AEC (VertexRSI) were 13 (18) and 00 (24), respectively, and the sky rms values were 222 (185), somewhat better and well within the specification for the VertexRSI antenna. This is an indication of the improvement that correctly working metrology (base tiltmeters) would offer.

With the azimuth zero‐point term IA added, completing the standard five floating terms, the sky rms reduced only slightly, to 216 for the AEC antenna, and was unchanged for the VertexRSI antenna. However, a significant improvement in the fit to the AEC measurements, to 195 rms, occurred when the az‐el nonperpendicularity term NPAE was included in the fit. The changes in the IA and NPAE terms for the AEC antenna were 112 and 126, respectively, and may be evidence of the suspected azimuth bearing instability in this antenna.

The conclusions were as follows:

• 1.  
  Some form of rapid recalibration (say, daily) will be needed in addition to any relatively infrequent full calibration.
• 2.  
  Frequent recalibration of the boresight offsets IE and CA is essential.
• 3.  
  Recalibration of the tilt terms AN and AW is also essential for the VertexRSI antenna performance, unless correctly functioning base tiltmeters can render this superfluous.
• 4.  
  Even with frequent recalibration of the boresight offsets, the AEC antenna performance would not quite reach the 2^'' specification.

A pointing run consisting of observations of a single star for about 45 minutes was used to test antenna tracking performance. Because only a small region of sky was observed (a line about 10° long), the data set is not suitable for fitting a pointing model. Although it turns out that doing so produces a sensible model, there is a danger of fitting out some of the tracking error that the experiment is designed to expose. Accordingly, an all‐sky pointing run acquired during the following day was used instead. Applying this model to the tracking test and allowing only a zero‐point correction (i.e., the TPOINT terms IE and CA) produced a sky rms of 146 and the ΔAcos E and ΔE tracking residuals shown in Figure 18 (the x‐axis is sequence number and hence time).

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Note the following:

• 1.  
  The scatter up‐down is larger than left‐right and is not normally distributed; the larger deviations tend to be downward on the plot.
• 2.  
  Beneath the noise, both axes show slow drifts. However, these are very small, barely 1^'' peak to peak. They might be the underlying errors in the pointing model, in its variance from day to day and (if temperature plays a role) from hour to hour, or they could be anomalous refraction or changing wind pressure. They happen too slowly to affect the ability to offset accuracy over short timescales, but do perhaps suggest that small adjustments to the pointing model on timescales of a few tens of minutes may be a wise precaution.
• 3.  
  The noise is of such high frequency that it is not what would usually be thought of as tracking error. This high‐frequency noise is more of a "jitter" and is likely caused by the seeing‐induced movements of the optical image, rather than the smoothness of the antenna motion.
• 4.  
  There is a fairly abrupt change in the left‐right tracking near the end of the run, and hints of other relatively high frequency artifacts. If such events can happen during offsetting maneuvers, the 06 in 2° science requirement might be in jeopardy. However, if the typical tracking were an illustration of the offsetting performance, then the 06 figure would be easily achievable. In other words, in a typical 2° portion of the tracking plot, the drift is within that figure. The largest slope on the elevation plot, for example, corresponds to about 2^'' in the total of 11° of track, or about 04 per 2°. The slopes on the ΔAcos E track are somewhat steeper if we overlook the high‐frequency events.

• 1.  
  The rapid slewing of the ALMA antennas leads to data sets of ample size compared with the five or six variable terms of the model, and so the rms figure is a good guide to the population standard deviation.
• 2.  
  All‐sky optical pointing measurements made over ambient temperatures from −10°C to +20°C and for wind speeds ranging up to 17 m s⁻¹ show no dependence on either of these site conditions.
• 3.  
  For the AEC antenna, the all‐sky rms pointing accuracy from the 2004 March 6 through May 12 period (Fig. 19) is typically 24, with the better results in the first half of the test period, which was when the more stable IE and CA values were seen.
• 4.  
  In Figure 19, the AEC sky rms is resolved into two components, left‐right and up‐down. It is evident that the left‐right component (i.e., ΔAcos E) is worse than the up‐down component (i.e., ΔE). This worse azimuth behavior is fairly constant over the test period, whereas the unexplained decline in elevation performance in the second half accounts for most of the deterioration in overall rms.
• 5.  
  Figure 17 shows the AEC all‐sky rms pointing accuracy for the period of 2005 January 7 and March 2. The mechanical modifications made by the antenna contractor have produced a moderate improvement to the all‐sky pointing performance.
• 6.  
  For the VertexRSI antenna, Figure 20 shows that if we neglect the earlier, pre‐PMDR period, the sky rms is not much affected by the passage of time and changing conditions. A typical result is 20 rms, and most of the PMDR runs delivered better than this, in some cases much better. Figure 20 also shows the sky rms resolved into two components, left‐right and up‐down. It is evident that the up‐down component (i.e., ΔE) is worse than the left‐right component (i.e., ΔAcos E). The performance variations in the two axes seem to be somewhat correlated, perhaps evidence that variations in optical seeing were involved, while the occasional poor sky rms happened when there was a sudden worsening in elevation performance.
• 7.  
  A sequential multiple star measurement, where each star measurement was repeated three times, was used to derive seeing and/or short‐timescale antenna positioning errors. These measurements suggest a seeing/vibration contribution of ∼15, in line with the results from radiometric tracking tests. Extending the interpretation to account for the seeing contribution suggests that the true antenna pointing residual could be as low as 17 for the AEC antenna and 08 for the VertexRSI antenna.

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Essentially the same procedure was used for both the OPT and the radio data. Special TPOINT scripts were used to mass‐reduce the final selection of files, with automatic removal of outliers and with no manual intervention. The residuals from the different runs were standardized and superimposed, first to look for hitherto undetected terms and then to determine the coefficient values for a fixed "core" group of pointing coefficients.

Radiometric pointing measurements were obtained during the period of 2004 May 16–25 with the AEC antenna and 2004 March 20–22 with the VertexRSI antenna. Some preprocessing of the data was necessary before conventional TPOINT analysis could begin, to fold the operational model back into the observations. Once treated in this way, the four AEC and three VertexRSI measurement runs, which contained 22, 30, 46, and 57 and 44, 48, and 37 pointing measurements, respectively, proved to be sufficiently consistent to simply concatenate them. Fitting each one separately was less promising, given the relatively small number of observations in each data set.

For both antennas, the first model tried was the OPT model, in which only IE, CA, and HECE were allowed to vary. The first and second terms are simply the radio boresight offset relative to that of the OPT. The third term is the Hooke's law vertical flexure, which will be different for the radio and OPT cases. With this approach, the residuals were clearly systematic.

To deal with the elevation errors as a function of elevation, both HESE (elevation errors proportional to sin E) and TX (elevation errors proportional to tan Z) were tried, and the former worked best. This suggested some asymmetry in the BUS, subreflector supports, or nutator.

There were also signs that the azimuth zero‐point correction (IA) and/or the az‐el nonperpendicularity (NPAE) had changed. The explanation (see § 6.4.2) seems to be that the asymmetric location of the OPT, mainly in the horizontal direction, makes it vulnerable to elevation‐dependent flexures in the BUS, a sideways deflection of the OPT occurring as the BUS "opens out" with increasing elevation. It was therefore decided to allow both of these terms to vary, in addition to CA. For the AEC antenna, these tests showed that a substantial 15^'' change in IA was needed, as well as an additional core term (HESE), but with no significant change in NPAE (under 1^'', in fact). For the VertexRSI antenna, only an 8^'' change in IA was needed, but large changes in HESE (46^'') and NPAE (27^'') were required.

By allowing IA, IE, HESE, HECE, and CA to vary for both antennas (and for good measure, AN and AW; these may change slightly, depending on static wind pressure, and as they are rather uncorrelated with other terms, they are safe to include), satisfactorily nonsystematic residuals were obtained. The final model that was obtained for both prototype antennas using this procedure is listed in Table 6.

For both antennas, a plot of the ΔAcos E and ΔE errors against observation number shows evidence of drifts during each of the pointing runs. For the VertexRSI antenna, the average drift for each of the three runs was just under 02 per observation. If the data from the three VertexRSI radiometric pointing runs are simply concatenated, without individual boresight adjustments, drifts in both ΔAcos E and ΔE can be seen.

To derive the "best radiometric pointing model" from these measurements without any drift corrections, we used an analysis procedure similar to that used for the optical pointing measurement reductions. The HESE, HECE, and NPAE coefficients were iteratively refined, and the individual pointing runs were normalized by freeing them from their preferred IE, CA, AN, and AW values. The normalized data were then combined and fitted for the five AEC and four VertexRSI core terms, plus IA, HESE, and HECE (and NPAE for the VertexRSI antenna). The resulting "best radiometric pointing models" for the AEC and VertexRSI antennas are shown in Table 7.

The model for each antenna was applied individually to each run by fitting only IE (plus drift), CA (plus drift), AN, and AW, and the results were combined (Fig. 21). The overall 32 and 53 rms figures for the AEC and VertexRSI antennas, respectively, should be treated with some caution: superimposing individually fitted runs is a particularly optimistic way to present pointing results.

An analysis of the same 3 days of data was done (R. Lucas et al. 2004; see footnote 7) using the Plateau de Bure pointing model software. The results are in full agreement. No systematic effect was detected: the results are dominated by measurement errors.

The location of the OPT within the backup structure of the antenna requires that one consider how the measurement axes of both systems are related. In the pointing model for the antenna concerned, for an ideal OPT fixed to the radio telescope surface and assuming mechanical symmetry about the axis of the parabola, one would expect only the boresight offset (IE and CA) and vertical flexure (HECE) terms in the pointing model to change. However, on both antennas, preliminary analysis of the radiometric pointing showed signs of changes in the azimuth zero‐point correction (IA) and/or the az‐el nonperpendicularity (NPAE). The likely explanation of such effects is that the asymmetric location of the OPT, mainly in the horizontal direction, makes it vulnerable to elevation‐dependent flexures in the BUS, a sideways deflection of the OPT occurring as the BUS "opens out" with increasing elevation. A deflection proportional to sin E would produce a term that is functionally identical to NPAE, while the cos E phase would produce an apparent change in IA.

For a perfect parabola at an elevation of 45°, the aperture of the antenna will become more elongated in azimuth at the zenith and more elongated in elevation at the horizon. Since the ALMA OPTs are located azimuthally off‐axis, the boresights of the OPTs are expected to be pointed to more positive azimuths at the zenith and to more negative azimuths at the horizon. One would expect then the following additional harmonic elevation and azimuth terms to the OPT pointing correction with respect to the radiometric pointing model:

This indicates that one should expect fixed differences between the radiometric and optical determinations for IA, NPAE, HESE, and HECE.

Taking the VertexRSI antenna as an example, a mechanical verification of this theory can be derived from the measured axial focus change due to BUS deflection (see Greve & Mangum 2006). The VertexRSI BUS axial focus is measured to deflect by +1.7 mm over an elevation range of 0°–90°. Using an 80% radiometric taper leads to a deflection of +1.35 mm. With the constants

we find that for a perfect parabola,

This corresponds well to the combined measured shift in NPAE (+24^'') and IA (−7^'') between our optical and radiometric pointing models for the VertexRSI antenna.

We have investigated the influence of refractive pointing errors, or "radiometric seeing," on our radiometric tracking measurements. Measurements of the differences between radiometrically derived positions and their corresponding encoder positions while tracking a source for 15–20 minutes strongly indicate the existence of a refractive pointing error term (see M. Holdaway et al. 2006, in preparation). These refractive fluctuations result in an ∼06 radiometric positioning jitter. The magnitude of the inferred atmospheric pointing structure function at 1 s (the atmospheric crossing time of a 12 m antenna) is predicted to within 10% by theoretical arguments about refractive pointing jitter and atmospheric phase interferometer data.

The apex structures of both antennas rotate about an axis that is assumed to be aligned with the boresight axis. However, for the AEC antenna, this axis is offset by −1.5 to +1.0 cm, depending on the elevation. Fast switching can excite the rotation mode, which has translation components on‐axis that can amount to 30 μm peak to peak. The rotation and corresponding translation component is at 5 Hz and damps out with a 1/e decay time of 5 s. With the prime‐focus plate scale of 34^'' mm⁻¹, this translates to radio pointing errors of up to 1^'' peak to peak in cross‐elevation, provided that the rotation axis is parallel to the boresight axis, which could not be confirmed with the equipment available.

For the VertexRSI antenna, minor apex structure rotation was observed, which did not affect antenna performance, since the rotation appears to be on‐axis.

All OPT fast‐switching measurements were made on 2004 April 20. Four different star pairs were observed with each antenna. Two star pairs did not yield valid data for the AEC observations, while all four pairs of stars yielded good data on the VertexRSI antenna. Details of these observations are summarized in Table 10.

For each measurement run, the following analysis was performed:

• 1.  
  The 20 Hz sampled encoder az‐el position information was linearly interpolated to match the corresponding 10 Hz sampled OPT az‐el star position information.
• 2.  
  Positions from the OPT az‐el data stream were identified as "target" (on‐source) and "calibrator" (off‐source) positions by assigning a star S/N cutoff to the OPT measurements.
• 3.  
  Each target‐to‐target observation consumed about 5 and 4 s for the VertexRSI and AEC antenna measurements, respectively. Roughly speaking, the first ∼1.5 s are spent slewing to the source, the next second is spent refining the pointing, and the remaining time has a nearly stable pointing level.
• 4.  
  The constant az‐el offset, representing the antenna pointing model, is removed from the OPT star position fits in order to derive the az‐el offset for each measurement.
• 5.  
  For each run, a single fourth‐order polynomial fit to the azimuth and the elevation encoder readings for the last half of all scans was subtracted from the azimuth and elevation encoder readings for that run in order to derive the rms pointing errors.
• 6.  
  For each slew, we are interested in the rms pointing error as a function of the time since the last measurement from the previous slew. Time bins 0.2 s long (twice the basic data sampling rate), which represents the time from the end of the previous scan and from the rms of the azimuth and elevation offsets that fall into that time bin for all data in a 10–20 minute observation, were constructed.
• 7.  
  Typically, about 2% of the data points in each run were determined to be outliers and were removed.
• 8.  
  Optical rms azimuth offsets, rms elevation offsets, and the total rms optical and encoder pointing errors as a function of time for each run and antenna were then derived. Figure 27 shows these results.

Comparing each run az‐el profile (see Fig. 27), one sees general agreement, indicating that these two data streams are reliably aligned in time. However, the optical pointing rms generally exceeds the encoder rms pointing, as expected, as the optical rms includes the effects of optical seeing. The optical seeing (σ_atmos) will be uncorrelated with the mechanical pointing errors, so it can be extracted from the optical pointing contribution by a quadratic subtraction:

Table 10 shows the inferred atmospheric pointing contribution for each observation. In deriving these data, we considered optical and encoder data past the 2.5 s point in the profiles shown in Figure 27, calculated the inferred atmospheric optical pointing for each time bin, and took the average for all time bins past 2.5 s. Note that for the VertexRSI antenna, the ratio of elevation atmospheric pointing to azimuth atmospheric pointing is greater than 1.0, but for the AEC antenna, it is about 1.0. We suspect that this is a further indication of the poorer elevation (VertexRSI) and azimuth (AEC) tracking and pointing performance noted in § 6.

In conclusion, we find the following from our OPT measurements of fast‐switching performance:

• 1.  
  The VertexRSI antenna typically reaches its target source in 1.5–2.0 s.
• 2.  
  The VertexRSI antenna can have a significant pointing bounce in elevation between 2 and 3 s after the slew begins. Presumably, the servo is not well tuned, and this is a correction for overshooting or something similar.
• 3.  
  The AEC antenna arrives at the target source after about 1.5 s and has a much smaller pointing error after acquiring the source.
• 4.  
  The VertexRSI antenna has larger pointing errors than the AEC after the source has been reached, although the encoder data indicate that the specification of 06 is approximately met.

Using radiometric and encoder data, we have investigated the mechanical aspects of the fast‐switching performance of the ALMA prototype antennas.

• 1.  
  The VertexRSI antenna basically meets the fast‐switching specification of "1.5 degrees in 1.5 seconds to an accuracy of 3 arcseconds." The antenna achieves pointing errors of about 08 rms if a bit more settle time is permitted. Minor improvements in the software and servo systems could improve the fast‐switching capability. We find that for nighttime observing, the encoders and the radiometrically inferred pointing errors agree remarkably well. After the slew has been completed and the antenna is pointing on‐source, pointing drifts of under 1^'' over 2 s of time, and pointing jitter of amplitude 03 and frequency 3 Hz, can be seen clearly in both the encoder pointing and the radiometrically inferred pointing. On average, the encoders and radiometrically determined pointing differs by 028 rms while slewing, and 015 after settling.
• 2.  
  Radiometric and encoder data from the AEC prototype antenna indicate that the AEC antenna meets the fast‐switching motion requirements. We see that as the AEC antenna comes to rest after a fast slew, a 5 Hz oscillation is excited by the fast‐switching motion, and that the peak amplitude of this oscillation is about 04, which dies out over a few seconds. Accelerometer measurements have identified this oscillation as being due to a rotation of the apex structure (Snel et al. 2006). For the measurements performed in this report, this oscillation does not prevent the AEC antenna from meeting the fast‐switching specification.

For purposes of path‐length prediction, we have searched for correlations of the path‐length changes with the steel temperature of the pedestal and the fork arms, the ambient air temperature, and the wind speed. The correlation of the fork arm path‐length change ΔL₂ is shown for the VertexRSI and AEC antennas in Figures 31 and 32, respectively. As expected, the correlation of ΔL₂ with the change of the fork arm steel temperature is good and usable for prediction. On the VertexRSI antenna, a similarly useful correlation was found for the path‐length of the pedestal (L₁; Greve & Mangum 2006). There is no correlation between path‐length variation and ambient air temperature or wind speed variation.

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A dedicated investigation of the path‐length changes with variations in wind speed was difficult, due to logistical reasons. For one particular day with changes of the wind speed (v) from ∼5 to ∼15 m s⁻¹, the path‐length variation ΔL₂ (fork arm) of the VertexRSI antenna was measured, and it hinted at a slight increase in ΔL₂ with increasing wind speed. However, the measured variation of L₂ remained within the specification, even for the highest observational wind speed of v = 9 m s⁻¹. There are no data for the AEC antenna.

Both prototype antennas have temperature sensors installed on the steel walls inside the fork arms. From these recordings, we have derived the average, maximum, and minimum temperature of each fork arm. The average temperature distribution measured throughout a fork arm was used in a finite element model (FEM) analysis to calculate the corresponding thermal dilatation ΔL₂. Comparing the calculated thermal dilatation with the path‐length variation measured directly with the API5D, we find very good agreement, as shown in Figure 33. From this, we conclude that the path‐length variations of the steel parts can be predicted to a high degree of accuracy from representative temperature measurements used in the FEM, or from an empirical relation.

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The daily temperature variation of the fork arms and the corresponding path‐length variation ΔL₂, both shown in Figure 33, can be approximated with good accuracy by sine functions. This holds also for the ambient air temperature, at least as measured at the VLA site during most of the tests. When adopting a sine function L₂(t) = L₀ + ΔL₂sin (ωt), where ω = 2π per 24 hr and t = time, the path‐length variation ΔL₂ at a 3, 10, and 30 minute time interval can be derived by differentiation of the function L₂(t). The corresponding result is shown by solid lines in Figure 33, which gives an explanation of the double‐peaked form of the measured variations. The minima of ΔL₂ occur around 0:00 and 14:00 UT, where the temperature changes of the ambient air and the steel of the fork are smallest.

As is evident from Figure 33, the total daily path‐length variation of the fork arms is of the order of 100–200 μm, as is fully understandable and unavoidable, given the height of the fork arms, the thermal properties of steel, the actual temperature variation of the steel, and the solar illumination.

The ALMA interferometer will use sidereal tracking, OTF mapping, and FSW motions between source and calibrator. The OTF, and in particular the FSW motions, involve high accelerations of the antenna, which may affect the path‐length stability. The path‐length variations ΔL₁, ΔL₂, and ΔL₃ were measured under the following motions of the antennas: (1) sidereal tracking as a combined motion in azimuth and elevation direction, (2) OTF motion of 1° azimuth by 1° elevation at 005 s⁻¹, and (3) FSW motion of 1° azimuth by 1° elevation at 6° s⁻¹ in azimuth and 3° s⁻¹ in elevation.

For both antennas, the variations of the path lengths L₁, L₂, and L₃ are within ±2 μm for sidereal tracking and OTF motion. For FSW motion, with the highest acceleration at the subreflector position of the quadripod, the path‐length measurements L₃ are shown in Figure 34. Again, for both antennas, the path‐length variation ΔL₃ at the on and off position is within ±3 μm.

Because of gravity‐induced deformations, the path length L₃ on both antennas will change with elevation of the reflector. To measure this effect, the laser emitter of the API5D was installed on a mount (one especially constructed for the VertexRSI antenna, but the laser tracker platform on the AEC antenna) at the reflector vertex, and the retroreflector was installed on the subreflector. The measurements of the L₃ path‐length variation as function of elevation are reliable, because the measurements are insensitive to a small tilt of the laser mount relative to the antenna structure, and hence of the laser beam reflected in the retroreflector. The antennas were tipped in elevation in steps of 15°, between 5° (AEC) or 15° (VertexRSI) and 90° elevation, and the variation of ΔL₃(E) was recorded. These path‐length measurements, along with the path‐length variation predicted from a FEM calculation of the antenna structure, are shown in Figures 35 and 36. The repeatability of the API5D measurements is approximately ±5 μm. Note that in Figure 35, the change in subreflector position as measured by the ALMA Antenna Integrated Product Team using photogrammetry agrees well with the API measurement. Even though there may exist a difference between the measured and calculated variation of L₃ with elevation, the results indicate that the variation of ΔL₃ must be taken into account in the operation of the interferometer. They are easily put into analytic form.

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