Estimating visibility amplitudes with the PRIMA fringe trackers

Transcription

1 Estimating visibility amplitudes with the PRIMA fringe trackers Nuno Gomes a, b, c, Christian Schmid b, Joahnnes Sahlmann d, Sérge Ménardi b, Roberto Abuter b, Antoine Mérand e, Françoise Delplancke b a Laboratório de Sistemas, Instrumentação e Modelação em Ciências e Tecnologias do Ambiente e do Espaço (SIM)/Faculdade de Engenharia da Universidade do Porto (FEUP), Rua Dr. Roberto Frias, s/n, Porto, Portugal; b European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Straße, Garching bei München, D München, Germany; c Faculdade de Ciências da Universidade do Porto (FCUP), Rua do Campo Alegre, s/n, Porto, Portugal; d Observatoire de Genève, 5 Chemin Des Maillettes, 9 Sauverny, Switzerland; e European Southern Observatory, Alonso de Córdova 37, Casilla 9, Santiago 9, Chile; ABSTRACT The quality of the tracking performed by the fringe sensor units(fsus) of PRIMA, the ESO s dual feed facility for the VLTI, is affected by the angular separation between the two objects being observed simultaneously, the detector integration time (DIT) and the atmospheric observational conditions. We describe the algorithm we developed to compute visibilities from the FSU data and discuss their validity for the study on the angular anisoplanatism measured through the dependence of the visibility amplitudes on the angular separation. Keywords: PRIMA, FSU, visibility amplitude. INTRODUCTION The Phase Referenced Imaging and Micro-arcsecond Astrometry(PRIMA) instrument is the dual feed facility of the VLTI. It includes two twins unit fringe sensors the FSUs which provide on- and off-axis fringe tracking and narrow angle astrometry[ ] (the FSU was also devised to provide phase referenced imaging when combined with AMBER or MIDI instruments, but the V imaging mode of PRIMA has been postponed by ESO since April ). The FSUs were designed to retrieve in real time the fringe phase but, with proper data reduction, they can also provide estimates of the visibilities. In the following sections we describe a possible approach to compute the visibility of the sources when both FSUs are used in the astrometric mode.. ESTIMATING VISIBILITIES WITH THE PRIMA FSU. The visibility amplitude and phase In order to compute visibilities from FSU data, we use the fact that the FSU measures intensities at four points spaced by λ/4 and we apply a variation of the so-called ABCD method [ ] to the PRIMA case. We start from the relation V = I corr I phot, () where I corr stands for the correlated flux and I phot for the photometric or total flux [ 3 ]. Further author information: send correspondence to Nuno Gomes. nunogomes@fe.up.pt.

2 In the ideal case, where the phase difference between the four beams emerging from the Polarising Beam Splitter (PBS) is π/, the raw intensities measured in each quadrant of the FSU s PICNIC are: I i (t) = I (t)+i corr (t)cos(ϕ i ), () where I corresponds to the non-corrected photometric flux, i = A,B,C,D, and ϕ A = ϕ,ϕ B = (ϕ+ π ),ϕ C = (ϕ+π) and ϕ D = (ϕ+ 3 π). The raw intensities have to be corrected for bias introduced by the readout process dark counts are removed and the flux is normalised by the photometric flat mean value:[ ] S i (t) = I i(t) G i P i G i, (3) where S i stands for the corrected A, B, C and D intensities at time t, I i accounts for the aforementioned instantaneous raw pixel intensities, G i is the dark mean value for each quadrant and P i the photometric mean value (sometimesreferredasflat meanvalue)foreachquadrant thefactorofinthedenominatorarisesfromthe fact that the photometric frames are taken in two steps, implying two detector integrations. The denominator of equation (3) is called photometric factor and, in fact, it corresponds to a photometric calibration. Since PRIMA does not use spacial filtering before the beam combination, it is acceptable to normalise the flux by using only one set of photometric frames acquired before the observation. By construction, I G i P i G i I phot, (4) Thus, combining the previous equations, we arrive to the final reduced fluxes: S i = +V cosϕ i, (5) The amplitude of the visibility and the fringe phase can be estimated by means of the fringe quadratures, X = A C and Y = B D, where X corresponds to the real part of the complex visibility and Y to the imaginary part:. Phase Corrections V = X +Y, (6) ϕ = arctan Y X. (7) During real operations, due to imperfections in the system, we are confronted with non-ideal quadrature signals X = A C and Y = B D, which we can account for by means of phase shift errors a, b, c and d, defined as the difference between the real phase shifts and their ideal values, respectively of, π, π and 3 π[ ]: A = I phot +I corr cos(ϕ+a) B = I phot I corr sin(ϕ+b) C = I phot I corr cos(ϕ+c) D = I phot +I corr sin(ϕ+d), where a,b,c,d [,π/[. Developing the trigonometrical identities and referencing the phase (a = ), we can write the ideal quadratures X and Y as a function of the experimental ones (X and Y ): X = C(γX αy ) Y = C(δX +βy ), where α = sinc, β = +cosc, γ = (cosb+cosd) and δ = (sinb+sind) are the phase shift error coefficients and C = βγ αδ. The phase shift error coefficients can be retrieved from the header of the FITS file corresponding to the observation. They are stored in a keyword (one for each FSU), which is named KWPHAS in the case of the white pixel of the detector. Here we adopt a different notation for the photometric frame (we call it P instead of F) to avoid confusion with flat field denomination common amongst astronomers. (8a) (8b) (8c) (8d) (9a) (9b)

3 .3 The algorithm The algorithm comprises two main steps: the data reduction and the calculation of the visibility amplitude. In the following we briefly describe the tasks involved in each of them and explore the possibility of selecting part of the fluxes before the actual visibility estimation. Dark counts Time lapse (ms) Figure : Dark counts for PICNIC A present in a raw dark frame. It is visible that the exposure time is shorter than the recording time. The remainder of the time stamps are removed..3. Data reduction Firstly, the A, B, C, D fluxes of the dark frame corresponding to the observation are averaged out for non-nil pixels when the recording of a file takes longer than the exposure time, the remainder of the time stamps are removed (see figure ) and for each quadrant of the detector of both FSUs. A representative dark mean value is thus obtained for each quadrant of the PICNIC detector. Accompanying the observation file there are two photometric calibration frames with imprinted fluxes for FSUA and FSUB. For each of these files and one quadrant at a time, the same aforementioned averaging procedure is applied and the averaged fluxes are added up, quadrant by quadrant, resulting in an averaged flux for each quadrant, for both FSUs. The mean dark values are then subtracted twice from the corresponding quadrants (see Section.). The result is a combined photometric flux value for each quadrant of FSUA and FSUB. Finally, the calibrated fluxes are computed for each quadrant of the PICNIC detector as described by equation (3). From this point, two approaches can be used to prepare the data before the calculation of the visibility amplitude: a subset of fluxes is selected from the histogram of the distribution prior to visibility calculations, or data is left as is. In the former, the visibility amplitude is computed directly from the subset of fluxes, while in the latter, only after plotting the histogram of the visibilities a subset of values is selected in order to compute the final averaged visibility amplitude. We explore these two possibilities in the following subsection..3. Flux pre-selection vs visibility selection After data reduction and before visibility estimation, a flux selection can be performed according to some criteria, aiming the improvement of the quality of the results. Having that in mind, for both FSUs, a mask is created for each quadrant of the detector in order to select fluxes within a certain interval, that can be defined as [m c σ, m c +σ] or as [m c + σ, + [, where m c is a weighted arithmetic mean of the distribution with weight h y, given by the equation m c = (hx h y ) hy, () (h x is the array of abscissae, in units of flux, and h y is the array of histogram points) and σ is the standard deviation of the distribution (see Figure ). The former case has the apparent advantage of eliminating less Known as flat-fields in the VLTI nomenclature.

4 Histogram (FSUA) Histogram (FSUA) Histogram of fluxes (FSUA) Flux in PICNIC_A Flux in PICNIC_D Flux in PICNIC_B (ADU) (c) Figure : Histograms of the fluxes in three quadrants of the PICNIC detector for FSUA, corresponding to the second (a and b) and fifteenth (c) observations of the night 8//6 from PRIMA commissioning #. The object is, in this case, HD55. In histograms a and b fluxes were selected around the mean value, while in histogram c fluxes higher than m c +σ were picked. common points of the distribution, while the latter allows the selection of points with high flux values, which in principle would lead to a better SNR. However, two caveats became apparent from Figure : we are modifying the shape of the distribution of fluxes and, specifically for the first case, sometimes not all the pixels with the most common flux values are selected (as can be seen in Figure b). In fact, a careful analysis will lead us to the conclusion that neither approach is good. In figure 3 we plot the fluxes relevant to each fringe quadrature, one versus the other, and in figure 4 we plot the ratio of those fluxes against time. The yellow ellipse encompasses the most common points (considering only fluxes A and C or B and D), the magenta box contains the most bright pixels, and the green ellipses include points where one of the quadrants dominates in flux. Points within the green ellipses correspond to the peaks of figure 4, where the flux was measured on the fringe (correlated flux), although some points inside the yellow ellipse might correspond to correlated flux as well. In figure 4, values around correspond to points of photometric flux, where correlated flux is very low or completely absent. Selecting the most common fluxes from the histogram corresponds to pick points within the yellow ellipse in figure 3, while choosing the most bright fluxes is equivalent to take points within the magenta box in the same plot. Therefore, it is clear from these figures and equation (5) that all points are relevant for the calculation of the visibility. Tests were performed with the data set described in section.4. and we got visibility estimates roughly 7% 9% inferior of what was expected if any flux selection was made before computing the visibility arrays. We conclude that none of the approaches is valid for a good estimation of the visibility amplitude and one should compute the latter directly from the reduced fluxes..3.3 Calculation of the visibility amplitude Using the quantities obtained from the data calibration, together with the phase shift error coefficients retrieved from the FITS header, equation (6) is applied to create visibility arrays for both FSUs. The visibility amplitude is then computed only for measurements on the fringe, i.e., with either the OPD controller (OPDC) and/or the differential OPD controller (DOPDC) in tracking controlling state (see figure 5), averaging out all points of the arrays. The averaging can be done using directly a bootstrapping technique or by selecting firstly the most common visibility points from the histogram of visibilities (see figure 6), as they both produce similar results.

5 Figure 3: Plot of reduced fluxes C vs A (3a) and D vs B (3b). The yellow ellipse corresponds to points with the most common fluxes, the magenta box to points with the highest fluxes and the green ellipses encompass points with fluxes measured on the fringes. Figure 4: Plot of the ratio of the reduced fluxes C/A (4a) and D/B (3b) vs time. The peaks correspond to points of flux measured on the fringe (correlated flux) while values around correspond to points of photometric flux or where the correlated flux is almost absent. Comparing figures 4 and 5a we realise that the peaks in the latter correspond in fact to points of flux where the FSU was fringe tracking (FTK) and, thus, they are compatible to fluxes measured on the fringe. Figure 7 is similar to figure 3, but it depicts points of flux acquired during the first 4 seconds of recording, just before the fringe was found for the first time. As expected, the flux concentrates roughly in the line bisecting the graph, indicating that this is a photometric flux. As soon as the FTK begins, points near the axis start to be present, indicating fringes were found. This is consistent with figure 5, where it is easy to see that the fringe was found for the first time around 4s after the beginning of the recording. It is interesting to note from figures 3b and 7b that there is an imbalance of flux between quadrants B and D of FSUA, as the ratio of photometric fluxes is not around the slope D/B.9. This situation was caused by a defective fibre in the cold optics of PRIMA. The optical fibre was already replaced and, in principle, no imbalance should be detected with more recent data.

6 Visibility (black)/opdc State (red) PACMAN_OBS_GENERIC33_5.fits 5 time (s) Visibility (black)/opdc State (red) PACMAN_OBS_GENERIC33_5.fits time (s) (c) Figure 5: OPDC controlling state (red lines) over-plotted on the visibility array (black lines) vs time for the first observation of the set, corresponding to an integration of s over the object HD889. Figure 5a represents the full visibility array after elimination of non-tracking points, while figures 5b and 5c depict two different levels of zoom of the former. The OPDC state values were scaled down for clarity:.7 corresponds to fringe tracking,.5 to system waiting and. to searching. Clearly, the visibility points from figure 5a match the peaks of flux of figure 4, confirming that the latter correspond to points of flux measured on the fringe. Visibility (black)/opdc State (red) 3 PACMAN_OBS_GENERIC33_5.fits time (s) 5 No. of occurrences visa PACMAN_OBS_GENERIC33_5.fits Figure 6: Example of histogram plotted for an array of visibilities. The grey shadow highlight the region ±σ around the mean value of the visibility. In this case, the computed visibility is roughly.4 with σ.7. PACMAN_OBS_GENERIC33_5.fits 35 Flux_C, FSUA 3 Flux_D, FSUA Flux_A, FSUA Flux_B, FSUA Figure 7: Similar to figure 3, but only for the first 4s of recording, just before the fringe was found for the first time. Only photometric flux is identified.

7 .4 Applications to real data The commissioning runs of PRIMA already produced enough data to test the algorithm described in the previous sub-sections. However, the amount of steps necessary to produce a mean visibility point is large. This, together with the fact that ESO s FITS internal file structure has suffered several changes during the last four years, led to the necessity of creating a pipeline to reduce FSU data and to compute the visibility of objects observed with PRIMA. VADER (Visibility Estimation from Automated Data-rEduction PRocedures) is a pipeline written in Yorick able to open any FITS file from ESO and to deal with non standard structures of the format. It aims the automatic compilation and execution of all the tasks necessary for the correct data reduction and visibility estimation using PRIMA s FSU data..4. Commissioning run # single-feed data (November 8) A set of FSU data files, with respective dark and photometric frames, from the commissioning night of 8//6 were analysed in order to compute the visibility amplitudes with VADER. The objects are two stars with similar coordinates and magnitudes (HD889, m K = 4., and HD55, m K = 4.5). For each observation, data was acquired for s at a rate of khz in single-feed mode. Figure 8a illustrates the calculated visibility points together with their error statistics. The latter correspond to the standard deviation of the approximately normal distribution of the several thousand visibility estimates. The visibility values are dominated by the error bars, as they are roughly / of the corresponding averaged visibility amplitude. This might be an indication of weaknesses in our algorithm, although the result is not surprising since the FSUs were designed to measure phases, not visibility amplitudes. Nevertheless, estimating the error statistics for the final averaged value of the visibility amplitude of each star by means of the computation of the RMS for the data set, the visibilities for HD889 and HD55 are found to be respectively.3±.3 and.±.3. Although higher than, these values are compatible with non-resolved objects (as expected) and with the results of Sahlmann et al. 9.[ ] FSUA Coherence Time (8//6 7) Seeing (8//6 7) V HD55 HD889 τ (ms) 6 4 Seeing ( ) UTC (h) Visibility points and respective error bars for HD889 (red squares) and HD55 (blue dots). UTC (h) Orange dots: τ at the beginning of the observation; green diamonds: τ at the end of the observation. UTC (h) (c) Cyan dots: seeing at the beginning of the observation; magenta diamonds: seeing at the end of the observation. Figure 8: Plots of the visibilities (V) (left), coherence times, τ, in ms (centre) and seeing, in arcseconds (right) versus hours of UTC for all valid set of data during the night run. Figures 8b and 8c depict respectively the coherence times and seeing during the observations. In order to check for a hypothetical correlation between the evolution of these quantities, we plotted the visibility against the coherence time (figure 9a) and the visibility versus the seeing (figure 9b). From these pictures it is apparent that no correlation exists between them, although the error bars in the visibility points dominate the plot. Regarding the data set of visibilities, the final values computed after the histogram selection are the same as their counterparts calculated without previous selection (see Figure ). This indicates that previous histogram selection does not affect the estimation of the visibility and that the bootstrapping technique is a powerful method to estimate statistical properties when sampling from approximated distributions.

8 FSUA FSUA V (HD889 & HD55) V (HD889 & HD55) τ (ms) Seeing ( ) Figure 9: Plot of the visibility values vs the coherence time, in ms (9a), and vs the seeing, in arcseconds (9b). It is apparent from the graphs that no correlation exists between the quantities.. HD55 HD889.5 V...5. V (after histogram selection) Figure : Plot of visibility amplitudes (V) versus visibility amplitudes computed after histogram selection (V histo ). As in Figure 8a, red points correspond to HD889 and blue ones to HD55. V V histo for all visibility points. 3. CONCLUSIONS We have presented an algorithm to estimate visibility amplitudes from FSU data and showed that, in the present status, the algorithm produces individual visibility points with too large error bars. The results reflect the fact that the instrument was not designed to measure visibilities, although it can be used to roughly estimate the visibility of targets being observed by PRIMA. The visibility estimation data reduction procedure should be revised and investigated for limitations and weaknesses. It would be interesting to study the angular anisoplanatism measured through the dependence of the visibility amplitudes and phase variances on the angular separation between pairs of stars. This was the initial goal of the study and we hope to conclude it in a near future. Finally, it will be important as well to study the relation between the visibility amplitude, the detector integration time and the light tunnel conditions (temperature and seeing). We will be able than to extrapolate the results to the PRIMA+AMBER case and to conclude on the advantages/disadvantages of combining both instruments using the FSU as an off-axis fringe tracker. ACKNOWLEDGMENTS N. G. thanks Rainer Köhler for useful comments to improve the algorithm. This research is partially supported by Fundação para a Ciência e Tecnologia Ph.D. grant SFRH/BD/448/8. All computations and graphics were performed and created with Yorick.

9 REFERENCES [] Sahlmann, J., Ménardi, S., Abuter, R., Accardo, M., Mottini, S., and Delplancke, F., The PRIMA fringe sensor unit, Astronomy & Astrophysics 57, 9 (Sept. 9). [] Shao, M. and Staelin, D. H., Long-baseline optical interferometer for astrometry, Journal of the Optical Society of America (97-983) 67, 8 86 (Jan. 977). [3] Glindemann, A., [Principles of Stellar Interferometry], Astronomy and Astrophysics Library, Springer Berlin Heidelberg New York, Berlin, Heidelberg, astronomy ed. ().