5. Results

Analysis of the performance of the servo system has been made over eight months through October 1995. The results reported here concern "in situ'' power spectra of angle of arrival fluctuations, estimates of image tracking accuracy, and seeing parameters.

5.1. Power spectra of the angle of arrival fluctuations

In this section, we compare spectra predicted by Kolmogorov theory with experimental spectra. Influence of aperture size on the position of the knee frequency is also analyzed.

  
Figure 10: Example of experimental spectral densities W_y(f) and , and transfer function

5.1.1. Comparison of the Kolmogorov model with the experimental spectra

The available pieces of information are the position signal of the mirror y(t) and the residual error . The transfer function in open loop, or gain, defined as the ratio of their spectral densities , leads to the bandwidth at 0 dB . At this frequency, the residual error equals the correction signal (Demerlé et al. 1994), so that, beyond this frequency the correction process is no longer effective. Figure 10 (click here) shows an example of experimental spectral densities W_y(f) and , and the gain . This transfer function is relative to the movements about the horizontal axis of the Southern mirror during tracking of the star Cygni ( photons/sec), on October 7, 1995. W_y(f) follows a f^-2/3 power law at low frequency and a f^-8/3 power law at high frequency. The knee frequency is approximately 6 Hz, yielding a wind speed of nearly 5 m/s from the empirical formula (Eq. (10)). is fitted by the expected 1/f² curve. The frequency corresponding to 0 dB gain is approximately 15 Hz.
From the bulk of recorded data, we have selected 91 temporal spectra of angle of arrival fluctuations. Selection was made on the basis of observation conditions considered as optimal (adjusted gains, cooled detectors).
Our observed spectra do not completely confirm the Kolmogorov prediction, but agreement with theory is quite satisfactory: all the spectra are in agreement with the f^-2/3 power law in the low frequency domain, while 67% of the spectra exhibit a f^-8/3 or f^-9/3 power law in the high frequency domain. The high frequency domain is influenced by noise causing it to rise above the expected f^-11/3 law. Moreover, on the mirror position signal the resonance of the closed loop transfer function also contributes to the presence of the f^-8/3 law. Up till now, in view of our results, we tend to consider that Kolmogorov predictions are relevant. Let us note that on the Sidney University Stellar Interferometer (SUSI), the same f^-8/3 behavior has been observed (ten Brummelaar & Tango 1994).

5.1.2. Knee frequency versus aperture size

Since the diameter of the telescope entrance pupil can be reduced by using a mask, we can check the dependency of the knee frequency on the aperture size. To that aim, we analyzed files obtained on the same nights, with a 12 m baseline and two different pupil diameters, 26 cm (full pupil), and 11.5 cm. The Northern mirror axis considered is the X-axis. Table 1 (click here) reports the aperture diameter D, the power laws followed by the considered spectra in both low frequency (LF) and high frequency (HF) domains, the knee frequency , and the wind velocity derived from Eq. (10).

 

Date UT D power laws

h:min cm LF HF Hz m/s

07/21/95 21:20 26 -2/3 -8/3 7 6.0

22:13 11.5 -2/3 -8/3 13 5.0

23:47 11.5 -2/3 -8/3 12 4.6

07/22/95 00:07 26 -2/3 -8/3 6 5.2

06/25/95 23:37 26 -2/3 -10/3 10 8.7

06/26/95 00:30 11.5 -2/3 -11/3 17 6.2

00:52 11.5 -2/3 -11/3 15 5.8

06/08/95 01:05 26 -2/3 -8/3 5 4.3

01:07 26 -2/3 -9/3 6 5.2

01:41 11.5 -2/3 -10/3 12 4.6

 

Knee frequency is observed to grow by the expected factor of roughly 2, when going from full aperture (26 cm) to masked aperture (11.5 cm). Since factor is quite stable through some given nights, as is the wind speed derived from Eq. (10), we can consider that the expected dependency is verified.

5.1.3. Spectra obtained with internal source

To verify that the measurements made on the sky are not affected by a spurious phenomenon originating on the table itself, we have recorded sequences of mirror position when an internal source on the table is observed.

  
Figure 11: Internal spectrum (X-axis of the Southern mirror)

Figure 11 (click here) shows a spectrum of internal fluctuations obtained on the horizontal axis of the Southern mirror. Exhibited spectrum is not completely flat and reveals low frequency contributions. The level of energy is 100 to 1 000 times smaller than the one occurring in the spectrum of Fig. 10 when tracking a star. Therefore, we can assume that no additional perturbations occur on the table itself. However, since starlight has to travel in telescopes and delay lines (which is not the case for the light from internal source) we cannot conclude that the data recorded on a star are free from instrumental tip-tilt instabilities.

To interpret the low frequency contribution in the laboratory spectrum, we may invoke local instabilities in optical index of the air (temperature gradient, residual air flows), residual vibrations, or internally induced drift in beam orientation (an effect noticed a number of times in the course of adjusted procedures).

5.2. Image tracking accuracy of the system

The image tracking accuracy is sensitive to optimization of the bandwidth, to the turbulence conditions and to the signal to noise ratio.

5.2.1. Tracking accuracy as a function of r₀

Measurements of the servo-system performance have been made to evaluate influence of r₀ on the tracking accuracy using Cygni, and baseline from 13 m to 17 m. The flux conditions during the various observations were similar (flux ranging from to photons/sec Northward, and from to photons/sec Southward), but these measurements were carried out for different r₀ values. The latter was measured in open loop by ASSI and is relative to the star direction. Southward, the flux is lower than Northward, essentially due to the throughput of the external delay line. In addition, the Northern telescope is partly sheltered by the I2T building when working with small baselines. This implies that the signal to noise ratio, the turbulence conditions, and thus the servo system performance is usually better in the Northern channel. The tracking gains were optimized, and the PMTs cooled to a temperature close to . Observations were carried out at full aperture (D = 26 cm).

 

date UT r₀ (OL) r₀ (CL)

h:min X Y X Y X Y cm cm X Y

09/28/95 18:28 17 15 15 15 13 13 4.0 3.8 0.438 0.459

09/27/95 20:25 10 10 12 - 10 - 5.0 4.8 0.256 0.252

10/08/95 19:43 20 25 20 - 17 - 8.0 11.8 0.282 0.288

10/08/95 18:45 8 7 6 6 5 5 9.0 11.5 0.175 0.233

10/07/95 18:41 12 10 6 6 5 5 11.0 18.4 0.127 0.153

09/21/95 20:17 10 10 11 11 9.5 9.5 12.0 16.4 0.171 0.174

 

 

date UT r₀ (OL) r₀ (CL)

h:min X Y X Y X Y cm cm X Y

09/28/95 18:28 10 12 12 15 10 13 3.0 3.6 0.572 0.487

09/27/95 20:25 8 10 6 6 5 5 3.5 6.1 0.365 0.289

10/08/95 19:43 8 25 10 20 9 17 6.0 11.5 0.414 0.293

10/08/95 18:45 5 12 5 6 4 5 8.0 14.2 0.343 0.109

10/07/95 18:41 15 6 - 6 - 5 9.0 20.2 0.073 0.138

09/21/95 20:17 8 10 - 10 - 9 10.0 16.6 0.249 0.262

 

Tables 2 (click here) and 3 (click here) report the 0 dB gain frequency (open loop transfer function), the knee frequency (Eq. (10)), the wind velocity using Eq. (10), open loop Fried parameter , closed loop Fried parameter , and tracking error (rms) on each axis. Some values are not reported in the Tables since the following selection for this parameter is considered: disregarded are those for which part of the recording was taken during the motion of one telescope axis to keep the active mirror well within its dynamic range. Care must also be taken when . The measurements of are made from at least 10-second data sequences.
Some discrepancy exists between and . Southward and Northward alike, is always larger than in these files. This result is partly due to the residual image motion being disregarded during image tracking. Northward, the discrepancy is lower than Southward where is usually more affected by a spurious drift of the Southern telescope. This effect is noticed on the image monitor. During the observations, we noticed a better reproducibility of Northward, and this parameter is therefore taken as being the most representative in the seeing process. However, we have to keep in mind that it also includes instrumental drifts and vibrations present in the Northern path.
Usually, tracking accuracy is more often better Northward than that Southward (lower than or equal to 0.2 arcsec). We can correlate this result with the measurement of the Fried parameter, which is also better in the Northern channel than in the Southern one.
We also notice that has to be larger than 10 Hz if optimal accuracy has to be reached. For example, October 8, at 18:45, Southward accuracy on the Y-axis reaches 0.11 arcsec at nearly 12 Hz , as against 0.34 arcsec on the X-axis with a 5 Hz .

  
Figure 12: Image tracking error versus open loop Fried parameter r₀ (at 600 nm)

Figure 12 (click here) illustrates the observed tracking error as a function of Fried parameter for each mirror and each axis. For both motions, we note a significant degradation as becomes smaller than 8 cm. Best accuracies reached on North mirror for the X and Y axes respectively are 0.13 arcsec and 0.15 arcsec. Similar figures for South mirror are 0.07 arcsec and 0.11 arcsec. Nevertheless, while this performance is currently met Northward, accuracies larger than 0.2 arcsec are more frequently estimated Southward.
From Fig. 12 (click here), the decrease of the error with the increase of is quite apparent but there is not a similar rule with wind velocity (see Tables 2 (click here) and 3 (click here)). Nevertheless, as accuracies obtained with a 12 cm Northward and 8 cm Southward are not as good as those obtained with lower , this can be ascribed to wind velocity (9 m/s to compare with 5 m/s).

5.2.2. Tracking accuracy as a function of flux

In a search for a thread effect on fluxes, several stars with magnitude up to 2.2 have been observed. Table 4 (click here) gives the visible magnitude m_v and the spectral type of these stars.
Tracking errors of both mirrors versus flux are shown in Tables 5 (click here) and 6 (click here), and are plotted in one diagram (Fig. 13 (click here)) (open loop r₀ estimates larger than 10 cm Northward and 8 cm Southward). Also presented is the frequency . For the Northern mirror, accuracy is roughly constant down to a flux of  photons/s. Southward accuracy is lower than that Northward for a flux of photons/s and degradation is quite rapid. Even though the flux range in common for North and South is limited, it is apparent that accuracy does not behave similarly on the two channels. Moreover, the nearly permanent deficit in image quality undergone Southward increases the noise in the quad cell measurements (Eqs. (6) and (7)).

  
Figure 13: Tracking error of Southern and Northern mirrors versus incident flux (r₀ in Tables 5 (click here) and 6 (click here))

 

star m_v spectral type

2.2 F8

2.1 K0

2.0 M0

1.8 F5

1.3 A2

 

 

Date UT flux r₀(OL) (Hz) (arcsec)

h:min 10³ photons/s cm X Y X Y

09/26 19:32 10 10 15 0.232 0.198

09/27 01:05 12 7 10 0.211 0.186

10/20 22:13 14 8 10 0.140 0.167

10/21 01:36 11 8 10 0.197 0.202

10/07 18:41 11 10 10 0.127 0.153

 

 

Date UT flux r₀(OL) (Hz) (arcsec)

h:min 10³ photons/s cm X Y X Y

09/26 19:32 8 10 10 0.544 0.609

09/27 01:05 8 7 10 0.463 0.442

10/20 22:13 8 7 7 0.325 0.431

10/21 01:36 8 7 10 0.513 0.379

09/21 20:17 10 8 10 0.249 0.262

 

As discussed in Sect. 4.3, for 1 arcsec seeing, the theoretical tracking error after partial compensation for turbulence is on the order of 0.3 arcsec and not very sensitive to the noise level. This explains the behavior versus flux observed in Table 5 (click here) for the Northern mirror. For the Southern mirror, both the Signal-to-Noise Ratio and the bandwidth are lower than for the Northern mirror and this could justify the worse performance as shown in Table 6 (click here). Moreover, the contribution of the noise as deduced from Eqs. (6) and (7) is not sufficient to explain the difference in performance. More observations are necessary if progress is to be made on this subject.
In summary, we can say that a typical accuracy of 0.2 arcsec is achievable when r₀ is larger than 8 cm, and the flux greater than photons/s.

5.3. Seeing measurements

A set of observations recorded on 11 nights distributed between July and October 1995, and another set of 8 observations distributed between February and July 1995, enable statistic analysis of parameters r₀ and t₀.

5.3.1. Fried parameter

For the first set, we have selected observations for which estimates of r₀ in open loop mode and in closed loop mode are available both Northward and Southward. The open loop mode is considered as reliable as soon as it yields reproducible estimates. Besides, in order to somewhat enrich the statistic, we used the second set of observations for which only estimated r₀ values obtained in closed loop mode are available. Figure 14 (click here) shows occurrences of open loop (solid lines) and closed loop (dashed lines) r₀ Northward and Southward.

Northward, both histograms show a predominance appearing for r₀ ranging from 10 cm to 12 cm. The open loop estimates do not exceed 14 cm, although the closed loop ones rise up to 20 cm and present one value at 25 cm.
Southward, the open loop estimates are continuously distributed from 2 cm to 12 cm. They are generally lower Southward than Northward, probably due to the spurious drift of the Southern telescope, usually stronger than the Northern one. The closed loop estimates show a predominance around 10 cm. Same as Northward, the latter can reach values up to 20 cm, unlike the open loop ones which never do.
As we already mentioned, we could suspect the system to over-estimate the closed loop r₀ as the image tracking is not perfect and residual error remains. But let us consider the case for which closed loop r₀ reaches 25 cm Northward and 30 cm Southward. The respective open loop r₀ are 11 cm and 9 cm. The values of respective parameters are as follows: tracking accuracy at 0.19 arcsec and 0.15 arcsec (included X and Y axes) and 0 dB-frequency at 12 Hz and 15 Hz. Since these parameters are favorable for the relevance of closed loop r₀ estimates, we can thus expect the latter to be realistic. So, in this case, the tracking imperfection does not seem to be the main cause of the discrepancy between closed loop and open loop r₀. In fact, over-estimation of closed loop r₀ due to imperfect tracking and under-estimation of open loop r₀ due to telescope drifts should jointly contribute to these discrepancies. In order to know which one is the best estimate, it would be very interesting to perform a cross calibration with measurements made by the Nice University team, currently in charge of the Grating Scale Monitor (GSM) experiment on one of the rails of I2T. This experiment is intended to measure temporal and spatial characteristics of the wavefront (Ziad 1993; Agabi 1994). Their observations, carried out on few short periods (of a few nights each) distributed over the year, were not simultaneous to ours. An alternative way to estimate r₀ could be to sum the mirror position and the quad-cell error, but with a required proper calibration of the photon noise. This could be envisaged for future work.

  
Figure 14: Occurrences of open loop (solid lines) and closed loop (dashed lines) estimates of Northward and Southward r₀ (at 600 nm)

  
Figure 15: Occurrences of open loop (solid lines) and closed loop (dashed lines) estimates of Northward and Southward t₀ (at 600 nm)

5.3.2 Coherence time

From r₀ and wind speed estimates, we can deduce coherence time t₀ (Eq. (11)). We have built histograms as shown in Fig. 15 (click here), for which the whole set of observations was used. Coherence times obtained from open loop r₀ and from closed loop r₀ have been distinguished. Northward, we see that t₀ stands near 3 ms most of the time in the open loop case and near 5 ms in the closed loop case. Southward, the behavior of t₀ in the closed loop case is similar to that Northward but the predominance in the open loop case is only 1 ms. The open loop case is more pessimistic than the closed loop one due to the low r₀ values. Though these values should be reviewed after calibration of r₀, they tend to confirm that our records of dispersed fringes (Robbe et al. 1994), using a 20 ms exposure time, are not exactly entered into the short-exposure mode. Nevertheless, the histogram shows that t₀ longer than 10 ms sometimes happen in the closed loop case Northward as Southward.

These values can be compared with those measured by other instruments. Nightingale & Buscher (1991) give median values of 5 ms and 10 ms at 500 nm, obtained during two different nights at La Palma Observatory in the Canary Islands. These values correspond to 6.2 ms and 12.4 ms at 600 nm. A mean value of 5 ms at 492 nm is given by Davis et al. (1995) at the Paul Wild Observatory, Australia, as equivalent to 6.3 ms at 600 nm. On the same site, Davis & Tango (1995) report values from less than 1 ms to 7 ms at 443 nm, equivalent to less than 1.4 ms to 10 ms at 600 nm. Colavita et al. (1987) at Mt Wilson, U.S.A., give T₀ values from 11 ms to 18 ms at 550 nm, but as their definition is a factor 2.58 larger than ours (Eq. (11)) their measurements are equivalent to t₀ from 4.7 ms to 7.7 ms at 600 nm. Our values are as good as or lower than the ones measured at other observatories. Anyhow, this modest statistic confirms the need for servo-control systems on the I2T interferometer, even though typical conditions of turbulence on the site are not quantified on a large temporal basis.