Analysis of Arecibo 305m antenna performance and surface errors

Paul Goldsmith

October 31, 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

appears to be a reasonable first step in this effort.

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 is only

about 2 percent.

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),

where

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 i, we see

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.

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.

Table 1

______________________________________________________________________________

Rss surface errors derived from 305m antenna measurements using astronomical sources

______________________________________________________________________________

freq(MHz) wavelength (cm) S(K/Jy) S/So epsilon (cm) S/So* S*(K/Jy)

f lambda

2380 12.61 10.5 0.955 0.22 0.98 10.8

4500 6.67 9.0 0.818 0.24 0.92 10.2

5400 5.56 8.0 0.727 0.25 0.89 9.8

8500 3.53 5.0 0.455 0.25 0.75 8.3

9200 3.26 4.0 0.364 0.26 0.72 7.9

----------------------------------------------------------------------------------------

12000 2.50 0.15 0.57 6.2

15000 2.00 0.15 0.41 4.5

______________________________________________________________________________

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 erived 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.

Table 2

______________________________________________________________________________ ________

Known contributions to aggregate surface error of 305m antenna

______________________________________________________________________________ ________

Contribution rms (cm)

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 1mm 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% imrovement!

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 (1mm). 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 inceasing

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.

______________________________________________________________________________