Parameters for the viewing condition and their dependence on turbulence
The viewing conditions at an observatory site relate to two major aspects. The first aspect is the resolution attainable with long exposures and a telescope of given mirror diameter, D. For small telescopes the angular resolution is entirely limited by diﬀraction: ε = 0.976 λ/D (1.2).
For a 0.1 m-telescope, the angular resolution at wavelength λ = 0.5 m equals 4.9 10−6 rad, i.e. 1”. With increasing mirror sizes the resolution tends to be increasingly aﬀected by the wavefront distortions due to atmospheric turbulence. To characterize the magnitude of this influence, it is convenient to refer to the mirror diam-eter where the eﬀect of diﬀraction becomes just equal to the degradation of resolution due to the atmosphere. This diameter is termed Fried parameter, r0, where the somewhat uncommon use of the letter r for a diameter needs to be noted. Under poor conditions – for example even during clear nights at sea level – the Fried parameter may rarely be better than about 1 cm. Under optimum conditions, such as on clear nights at Paranal, Chile, the value may go up to 1 m for wavelengths between 0.5 to 1 m. Large telescope mirrors can then profitably be used, and they will attain excellent resolution, when adaptive optics are being used that correct for phase diﬀerences by acting on individual mirror segments.
In practice one tends to refer to the seeing, ε0, rather than the Fried pa-rameter, r0, but the two quantities are essentially equivalent, the seeing being the angular resolution of a telescope with mirror diameter r0: ε0 = 0.976 λ/r0 (1.3).
Thus, the Fried parameters 1 cm and 1 m correspond, at the wavelength 0.5 m, to seeing values of 10” and 0.1”.
If the exposure time is shorter than the characteristic time of turbulent mo-tions, say below 5 ms, the image of a star consists of individual speckles inside the spot formed by the long-exposure image. Speckles result from the inter-ference between those parts of the wavefront where the turbulence-induced wavefront-inclinations are the same; and these small interference patterns con-tain information at angular resolutions up to the diﬀraction limit. Accordingly, the angular resolution of large telescopes approaches the diﬀraction limit with decreasing exposure times – the price being a considerable loss in sensitivity.
Assessing time scales of turbulence at Dome C, Antarctica
Dome C is a 3235 m high summit (75◦06′ S, 123◦23′ E) on the Antarctic plateau. Because of its elevation, the location does not experience the winds that are typical for the coastal regions of Antarctica. This has led to the assumption, that the atmospheric conditions might be particularly advantageous. In 2005, Concordia, a French-Italian station opened on Dome C, for research in astron-omy, glaciology, earth-science, etc. Aristidi et al.  and Lawrence et al.  determined the size of the turbulent cells, as measured 30 m above ground, to be 2 to 3 times larger than at the best mid-latitude sites. The latter au-thors concluded, that an interferometer built on Dome C could potentially work on projects that would otherwise require a space mission. This is a clear possibility, but it needs to be confirmed by measurements of the coherence time.
Chapter 2 presents an analysis of the first interferometric fringes recorded at Dome C, Antarctica. Measurements were taken between January 31st and February 2nd 2005 at daytime. The instrumental set-up, termed Pistonscope, aims at measuring temporal fluctuations of the atmospheric piston, which are critical for interferometers and determine their sensitivity. The characteristic time scales are derived through the motion of the image that is formed in the focal plane of a Fizeau interferometer. Although the coherence time of piston could not be determined directly – due to insuﬃcient temporal and spatial sampling – a lower limit was, nevertheless, determined by studying the decay rate of correlation between successive fringes. Coherence times in excess of 10 ms were determined in the analysis, i.e. at least three times higher than the median coherence time measured at the site of Paranal (3.3 ms).
To test the validity of the results derived in terms of the pistonscope, mea-surements with this instrument have subsequently been obtained at the ob-servatory of Paranal, Chile, in April 2006 with high temporal and spatial resolution. In Chapter 3 the observations are analyzed, and it is found that the resulting atmospheric parameters are consistent with the data from the astronomical site monitor, if the Taylor hypothesis of “frozen flow” is invoked with a single turbulent layer, i.e. if the atmospheric turbulence is taken to be displaced along a single direction. This has permitted a reassessment of our preliminary measurements – recorded with lower temporal and spatial resolu-tion – at the Antarctic site of Dome C, and it was seen that the calibration in terms of the new data sharpened the conclusions of the first qualitative examination in Chapter 2.
As seen in Chapters 2 and 3, we have, in spite of the current limitations in methodology and instrumentation, been able to infer considerably increased coherence times at Dome C, Antarctica, which is consistent with the earlier ex-tensive determinations of other parameters that demonstrate the superior condition for astronomical observations at this site (Agabi et al.  and Lawrence et al.). The two chapters make it equally clear, however, that a major eﬀort was required for this limited achievement, and that – even with more extensive sampling – the reliability and accuracy of the measured coherence times could not be fully satisfactory because of the influence of the uncertain angle between the instrumental set-up and the wind direction. The tempo-ral variations of the fringe pattern become faster, as the angle between the wind direction and the interferometric axis diminishes. To derive – without continuous assessment of changing wind directions – meaningful values of τ0, a parameter must, therefore, be measured that is independent of the wind direction. To make routine monitoring possible, the measurement would also have to be comparatively simple. The challenge to find such a parameter and to develop an instrument that permits its fast and reliable determination has, thus, become central to this thesis.
A new instrument to measure the coherence time
Since there exists currently no method to measure the coherence time directly and to achieve this with a compact instrument, Andrei Tokovinin and myself have sought a new approach to close the gap. A comparatively simple method has been adopted and an instrument has been designed to shift the image of a star somewhat out of focus, which converts it – due to a suitably enlarged central blind area of the telescope – to a ring. Insertion of a lens with proper spherical aberration sharpens this ring into a narrow circle. Atmospheric tur-bulence causes then distortions which can be conveniently assessed, because, to a first approximation, they appear as ring-radius changes. The strength of the Fast Defocus Monitor, FADE, lies in the fact, that it is insensitive to tip and tilt, which – being jointly caused by telescope vibrations and atmospheric turbulence – can not be meaningful indicators of turbulence alone. Instead we measure the higher order aberration defocus, that causes the radius changes. A relation between the temporal properties of the radius variations and the coherence time has been developed in the framework of the Kolmogorov theory of turbulence.
First measurements with FADE were obtained at Cerro Tololo, Chile, from October 29th to November 2nd 2006. The instrumental set-up is based on a telescope with mirror diameter 0.35 m and a fast CCD detector. Ring images were recorded during five nights with a broad range of instrument settings. The measurements and their uncertainties are analyzed in Chapter 5, and the seeing and coherence-time values obtained in terms of our instrument are compared with simultaneous measurements from the MASS and DIMM site-monitoring instruments.
Astrophysical application: interferometric observations of δVelorum
Chapter 6 presents an example of how research is facilitated, when the influ-ence of the atmospheric fluctuations can be partly overcome. Interferometers have been introduced in astronomy to gain spatial resolution without the need to build extremely large telescopes. To resolve δVelorum in the infrared would require a telescope of about 100 m mirror diameter. In contrast, the VLT In-terferometer Commissionning Instrument, VINCI, installed on Paranal in Chile, allows to resolve the bright, eclipsing binary Aa-Ab in δVelorum with two small 0.4 m siderostats 100 m apart.
Today, interferometric observations are limited to the brightest sources be-cause of turbulence-related rapid motions of the image. In spite of this current limitation, interferometry proves to be a key technique in many astrophysical domains. The study of multiple star systems is an example: to understand the state, evolution and origin of such systems, the results of dynamical studies need to be compared to observations with high angular resolutions.
In 2000, δVelorum had become infamously famous among the engineers of the Galileo spacecraft; δVelorum was used as reference star for the guidance system, but at some point the system failed. While an instrumental defect was assumed at first, it turned out subsequently that the star, not the space probe, was at fault. Galileo had in fact witnessed an eclipse. Since then δVelorum had been classified as a quintuple stellar system and it promised to become a key system for testing stellar evolutionary models: five stars of same age and with diﬀerent masses.
Three years ago, I began analyzing observations that had been obtained with the VINCI recombination instrument. The results were startling, be-cause the diameters of the two eclipsing stars appear to be 2 to 3 times larger than expected for main sequence stars: the two stars are thus probably in a more advanced evolutionary state. In the continued analysis of existing pho-tometric and spectroscopic data we found, that two of the five stars are, in fact, not part of the system. Thus, δVelorum has become more attractive due to the unexpected properties of the eclipsing binary, while, at the same time, it relapsed to the status of a triple stellar system. This work is detailed in Chapter 6.
Estimating coherence time through the evolution of correlation
Although the sampling rates were too low in the present measurements to assess the fastest atmospheric turbulence through Fourier analysis, some in-formative inferences are still possible, because relevant information can be obtained by tracing the decay time of correlation between successive fringe positions. Figure 2.6 represents the following structure function as a function of temporal separation: DΔθ(t) = < |Δθ(τ + t) − Δθ(τ)|2 > (2.7).
Δθ = θf − θ0: separation between the central fringe and the combined Airy discs. The coherence time of piston is estimated by comparing the structure func-tion predicted by the Kolmogorov spectrum of fluctuations  with the ob-served data: D(t) t0: coherence time. = D(t ≫ t0) × (1 − exp(−(t/t0)−5/3 )) (2.8).
Optimal setup for coherence time measurements
The observations reported here were made with the equipment available on the site and are subject to the following limitations, which will need to be lifted in future observations:
– The recording speed of the camera was not fully suﬃcient for sampling of atmospheric turbulence. Increased recording rate will permit the precise determination of the actual coherence time. The highest frequencies of the piston should – in line with the earlier (Agabi et al.  and Lawrence et al. ) and the present measurements – be less than 500 Hz. Accordingly a recording rate of 1000 Hz should ensure adequate temporal sampling.
– Caution is required, when the turbulent cells are larger than the distance between the two mask openings (20 cm). In these cases the diﬀerence between piston and tilt becomes too small to infer coherence times with suﬃcient precision. At the time of observations, the seeing varied between 0.5” and 1.0” (cf. Figure 2.5). Thus, at 500 nm wavelength, the turbulent cells had a characteristic size (Fried parameter) between 10 cm and 20 cm, i.e. only just smaller than the baseline. For future measurements, we consider the use of larger baselines up to 2 m.
Table of contents :
1 Introduction: Our screen towards the Universe, the turbulent atmosphere
1.1 Looking through the screen
1.2 Characterizing the screen
1.2.1 The notion of turbulence
1.2.2 Is there a theory of turbulence?
1.2.3 Parameters for the viewing condition and their dependence on turbulence
1.2.4 Statistical description of atmospheric turbulence
1.2.5 Coherence-time measurements
1.3 Constituents of this thesis
1.3.1 Assessing time scales of turbulence at Dome C, Antarctica
1.3.2 A new instrument to measure the coherence time
1.3.3 Astrophysical application: interferometric observations of δVelorum
2 A method of estimating time scales of atmospheric piston and its application at DomeC (Antarctica)
2.2.1 Observational setup
2.2.2 Data description
2.3 Quantifying the motion of the fringe pattern and the Airy discs
2.4 Coherence time
2.4.1 Estimating coherence time through Fourier analysis
2.4.2 Estimating coherence time through the evolution of correlation
2.4.3 Optimal setup for coherence time measurements
3 A method of estimating time scales of atmospheric piston and its application at DomeC (Antarctica)
3.3 Measurements at Paranal
3.3.1 Observational set-up
3.3.2 Derivation of atmospheric parameters
3.3.3 Performance of the piston scope
3.4 Measurements at DomeC
4 Atmospheric coherence times in interferometry: definition and measurement
4.2 Atmospheric coherence time in interferometry
4.2.1 Atmospheric coherence time τ0
4.2.2 Piston time constant
4.2.3 Piston power spectrum and structure function
4.2.4 Error of a fringe tracking servo
4.2.5 Summary of definitions and discussion
4.3 Measuring the atmospheric time constant
4.3.1 Existing methods of τ0 measurement
4.3.2 The new method: FADE
4.5 Appendix A – Derivation of the piston structure function
4.6 Appendix B – Fast focus variation
5 FADE, an instrument to measure the atmospheric coherence time
5.2 The instrument
5.2.1 Operational principle
5.2.4 Acquisition software
5.3 Data analysis
5.3.1 Estimating the ring radius
5.3.2 Noise and limiting stellar magnitude
5.3.3 The response coefficient of FADE
5.3.4 Derivation of the seeing and coherence time
5.4 Analysis of observations
5.4.1 Influence of instrumental parameters
5.4.2 Comparison with MASS and DIMM
5.5 Conclusions and perspectives
5.6 Appendix A – Estimator of the ring radius and center
5.7 Appendix B – Structure function of atmospheric defocus
5.8 Appendix C – Simulations
5.8.1 Simulation tool
6 Interferometric observations of the multiple stellar system δVelorum
6.1 Introductory remarks to the article
6.3 Characteristics of δ Vel A derived from previous measurements .
6.3.1 Orbit orientation and eccentricity
6.3.2 Semi-major axis and stellar parameters
6.4 VLT Interferometer/VINCI observations
6.4.1 Data description
6.4.2 Comparison to a model
6.5 Results and discussion
6.5.1 The close eclipsing binary δVel (Aa-Ab)
6.5.2 The physical association of δVel C and D