Statistical description of atmospheric turbulence
In his extensive analysis of the statistical properties of atmospheric turbulence, Roddier  examined the implications of Kolmogorov’s law for the propaga-tion of optical wavefronts. Some of the results that are used in the subsequent chapters can be summarized as follows.
The turbulent mixing of the air creates inhomogeneities of temperature, T , which likewise follow Kolmogorov’s law: WT( f ) ∝ f −5/3 (1.4) ˜ 2 is the power spectrum of temperature fluctuations, where WT( f ) = |T ( f ) | the symbol: ˜ denotes the Fourier transform. In an isotropic medium, the three-dimensional power spectrum WT(f) = WT( fx, fy, fz) is related to the one-dimensional power spectrum through an integration over two directions: WT( f ) = 4πf 2 WT(f) (1.5).
Because it determines the sensitivity of interferometers and the performance of adaptive optic systems, the atmospheric coherence time, τ0, is a parameter of major importance. Several instruments measure τ0 or related parameters, but all current methods have limitations: either the instrument is not well suited for site monitoring, or the method is burdened by intrinsic uncertainties and biases.
– SCIDAR (Scintillation Detection And Ranging) has provided good results on τ0, but it requires large telescopes and is not suitable for monitoring, since it necessitates manual data processing (Fuchs et al. ).
– Balloons provide only single-shot profiles of low statistical significance (Azouit & Vernin ).
– Adaptive-optic systems and interferometers give good results, but are suitable neither for testing projected sites nor for long-term monitoring (Fusco et al. ).
The four subsequent methods all use small telescopes and can, thus, be used for site-testing. They all have their special attractions. However, with regard to the coherence time each has intrinsic problems:
– SSS (Single Star SCIDAR) in essence extends the SCIDAR technique to small telescopes: profiles of Cn(h)2 and V (h) are obtained with less altitude resolution than with SCIDAR, and are then used to derive the coherence time (Habib et al. ).
– The GSM (Generalized Seeing Monitor) measures velocities of prominent atmospheric layers. By refined data processing, a coherence time, τAA – but one with a diﬀerent dependence on the turbulence profile than τ0 – is deduced from the angle-of-arrival fluctuations (Ziad et al ).
– MASS (Multi-Aperture Scintillation Sensor) is a recent, but already well proven, turbulence monitor. One of the measured quantities, related to scintillation in a 2 cm-aperture, approximates the coherence time, but this averaging does not include low-altitude layers and thus gives a biased estimate of τ0 (Kornilov et al. ).
– DIMM (Diﬀerential Image Motion Monitor) is not actually meant to de-termine τ0, but an estimation of the coherence time can nevertheless be obtained by combining the measured r0 with meteorological wind-speed data (Sarazin & Tokovinin ).
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.
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) – II
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