Paul Goldsmith October 31, 2002 Revised Nov. 17, 2002
It would obviously be desirable to make a quantitative comparison of the antenna performance and surface errors. This requires a number of assumptions. The following is a first step in this effort. It is hoped that this motivates the continuation of efforts to measure and adjust the primary reflector, the secondary and tertiary reflectors, and to push forward to improve the pointing which will be essential for meaningful operation at the higher frequencies.
1. We assume that the performance of L-narrow system at 1400 MHz defines the performance of telescope at a wavelength where the surface is perfect. This is reasonable, and it is self-consistent in that the derived surface error produces only about a 2 percent reduction in performance.
2. We use the L-band performance to normalize the performance at all other frequencies. This ignores several real-life complexities such as the different illumination of the surface by the different feedhorns used.
3. At each frequency, we will use the maximum gain as a function of elevation angle from the data on Phil Perillat's web page. This is justified by the fact that we believe that the gain changes are due to systematic problems with the focus and dome tilt, both of which produce aperture plane phase errors. Any errors will reduce the antenna gain, and in fact we are not certain that at the highest frequencies we have achieved the optimum performance possible, limited only by the surface errors. As we shall see in what follows, the suggestion is that at low zenith angles we are not far from that situation, however.
4. We will analyze the errors using the simplest Ruze formula for gain reduction, assuming randomly distributed errors of zero correlation length. The relevant expression for the gain G, relative to the gain of antenna with identical illumination and no phase errors, Go, is
G/Go = exp(-x^2),
x = 4 pi epsilon/lambda.
We can use as a surrogate for the gain, the sensitivity S (K/Jy) since we are going to be normalizing to the value at lambda = 21cm, the surrogate for zero frequency, for which we have sensitivity S0, as discussed above. Epsilon is the normal rms surface error and lambda is the wavelength. Defining y = G/Go, we see that for a measurement at any other wavelength than that which defines Go, we can solve for the rms error, and denoting this wavelength and associated determination of epsilon by the subscript i, we see that
epsilon_i = [lambda_i/(4*pi)]*sqrt[-ln(y_i)] .
5. We adopt the value for the antenna sensitivity at L-band to be 11 K/Jy. This is consistent with the approached suggested in section 3 above.
The following table gives the results. The two last columns are the projected performance that should be achievable after some further work on the surface, and these will be explained later. I here use rss to indicate that the primary, secondary, and tertiary all make contributions to the overall surface error.
|f (MHz)||lambda (cm)||(K/Jy)||(cm)||(K/Jy)|
There are a number of caveats here. First, the 2380 MHz point has a very large uncertainty in the derived epsilon because the efficiency is so close to that at the reference wavelength. Second, as mentioned above, we have not made any correction for feedhorn illumination differences. Can this be a significant effect? The only suspect frequency is 9200 MHz, as we know our current X-band horn is underilluminating the tertiary and hence the whole antenna system. Thus, the sensitivity should be a little higher than indicated, and the derived rms would drop somewhat.
Bearing this in mind, I feel that the CONSISTENCY of the derived aggregate surface rss error is impressive. It is so good, you might worry that it is fortuitous, but I do not see that a conspiracy theory can explain the data. But of course, you must remember that the sensitivities are the BEST values, and we still have a lot of work to do to make these apply over all azimuth and zenith angles.
But this said, there appears to be a very strong case to be made that the aggregate rss surface error at the present time is close to 0.24 cm. The next question, then, is how this compares with the actual surface measurements.
This is not a straightforward comparison, since we have three reflector surfaces, and each has a contribution from the surface panels themselves, and another from their setting inaccuracies. However, we must push ahead, and to do so we first assume that the tertiary reflector is "perfect". This is at least consistent with panel fabrication results and reasonable assumption about setting. The situation for the secondary reflector is complicated as the setting error varies a lot over its surface (as last measured) and furthermore, it may have changed significantly since it was measured, as the collar truss may have been distorted by alignment procedures that been implemented since the secondary was aligned. The secondary panels were measured to have rms surface error equal to 0.025 cm, and the overall setting error (per Lynn Baker) is 0.123 cm.
The primary reflector panels were said to have once had rms error (from fabrication) on the order of 0.07 cm. There is general agreement that time has not been kind to the telescope surface, and some number of panels is surely far worse than this (as can be ascertained by looking at the panels, especially near the center of the antenna). For the moment, I'll adopt a panel surface rms equal to 0.1 cm.
The focus of a great deal of work during the past two years has been the measurement and adjustment of the 38,800 primary panels. The distribution of errors of half of the surface has now been determined by Felipe Soberal, Tony Acevedo, and Lynn Baker. The distribution of errors is not Gaussian, but it has a Gaussian component with added "outliers" having errors up to 3 cm. The rms of the error distribution is approximately 1.85 mm for the half of the primary surface measured to date. This gives the following.
|Primary panel fabrication||0.100|
|Primary panel surface setting||0.185|
|Secondary panel fabrication||0.025|
|Secondary panel setting||0.123|
|Aggregate rss surface error||0.245 cm|
There are several things that are striking about this table. From it alone, we see that the secondary panel setting is now a significant contributor to the aggregate surface error. Since this can be measured to a small fraction of a millimeter with videogrammetry in a few hours, using equipment that can be rented from GSI, it is clear that we can achieve a significant improvement in antenna performance by doing this, at an approximate out of pocket cost of $10,000.
Second, we see that there is actually surprisingly good agreement between this combination of known errors and the aggregate surface errors derived from the astronomical measurements. I do not see any reason why this agreement is fortuitous. It is the case that we cannot rule out the actual surface panels being a little better than assumed 0.1 cm rms and the setting a little worse, but clearly we seem to have a good handle on what is going on.
What does this imply for the future? If, as mentioned above, we reset the secondary panels and reduce their setting errors to 0.05 cm rms (this is conservative in my opinion, given its small size and nice environment) we can reduce the aggregate surface error to 0.218 cm. This is indeed very worthwhile, as it would, for example, increase the sensitivity at 8500 MHz from 5 K/Jy to 6 K/Jy, a 20% improvement!
More provocative, as well as difficult, is to estimate what could be achieved with the primary reflector. The current .185 cm rms primary panel setting is what we achieved after one iteration on measurement and adjusting. As I mentioned earlier, there are some targets that are several cm away from their desired position. It is certainly possible (with some money and time) to readjust all the screws using the data we have in hand. It is hard to know exactly what rms will result, but we can (again assuming money and time are available) put out targets and repeat the process. In such efforts as this, there is a certain fraction of adjustments that are made that are basically erroneous. There are all kinds of explanations, but that is what people doing this kind of project have found. The precision of measurements of a single target that we are now achieving is about 0.07 cm. Thus, with perseverance, it is entirely possible that we could achieve a primary panel setting error of 0.1 cm (1 mm). If we can do this, and also reset the secondary we would have an aggregate surface error of 0.15 cm.
This value has been used to compute the values of relative sensitivity and absolute sensitivity in the two right hand columns of Table 1. The values are certainly impressive! I think this is a very reasonable plan to carry out over the next few years. To take full advantage of this, we must complete the Gregorian trolley upgrade and the rail alignment. We also will have to improve the pointing of the Gregorian, through refining our models, using the tertiary actuators, obtaining a ground-referenced laser tracker system, or some combination of the preceding. Given that we will need to achieve a few arcseconds rms pointing, this is not going to be easy, but I think the scientific returns will be significant. Note that at 10 GHz we would have about 7 K/Jy sensitivity, which would be an impressive number to put in calculation of what could be done with a planetary radar system operating at that frequency. Note that this calculation does not include effects of increasing transmission of the primary panels due to their holes. This is only a few percent at frequencies up to 15 GHz, so if we can make the antenna system point, and can adjust the surfaces as discussed here, Arecibo can certainly be the world's most sensitive system at frequencies up to 15 GHz.
The present and expected relative performance is shown in the figure. The dots show the present performance, and the red curve is the Ruze formula with an rms of 0.245 cm. The blue curve shows the expected performance if we were able to reduce the aggregate rms surface error to 0.15 cm. It is striking how much the performance at all frequencies starting at 5 GHz would be improved!
Some final points -
(a) It would be good to do a better analysis of the measured efficiencies. A starting point would be to correct the gain at various frequencies according to measured feedhorn illumination.
(b) We should measure the rms of a sample of the EXISTING SURFACE PANELS. This would be very interesting, but meaningful only if the sample were not too small and were representative. There may be a jig for measuring panels; if not, one should probably be made. I suggest the Facilities Department get involved in this.
(c) We should make some measurements of the transmission and reflection of actual surface panel material. I can see doing this for (i) "clean" aluminum as used to make the panels, and (ii) "real" material that has the interesting biological growth on it. It would be very significant to see whether there is any evidence of absorption at 10 GHz and above, that could be shown to be a result of this extra "skin". I recommend that a modest add on to the network analyzer in Ithaca consisting of a pair of broadband feedhorns be configured to allow this type of measurement. For the reflection measurement, the reference can be a solid aluminum sheet glued to e.g. a piece of wood. For the transmission measurement, one measures the loss relative to no panel present, which is also straightforward.
I thank a number of NAIC staff members who have made suggestions for improving this
write-up, especially A. Deshpande and G. Cortes.
Last updated 2002 November 19