Page 1 Adaptive Optics in the VLT and ELT era Atmospheric Turbulence François Wildi Observatoire de Genève Credit for most slides : Claire Max (UC Santa Cruz)

Atmospheric Turbulence Essentials We, in astronomy, are essentially interested in the effect of the turbulence on the images that we take from the sky. This effect is to mix air masses of different index of refraction in a random fashion The dominant locations for index of refraction fluctuations that affect astronomers are the atmospheric boundary layer, the tropopause and for most sites a layer in between (3-8km) where a shearing between layers occurs. Atmospheric turbulence (mostly) obeys Kolmogorov statistics Kolmogorov turbulence is derived from dimensional analysis (heat flux in = heat flux in turbulence) Structure functions derived from Kolmogorov turbulence are  r 2/3

Fluctuations in index of refraction are due to temperature fluctuations Refractivity of air where P = pressure in millibars, T = temp. in K, in microns n = index of refraction. Note VERY weak dependence on Temperature fluctuations  index fluctuations (pressure is constant, because velocities are highly sub-sonic. Pressure differences are rapidly smoothed out by sound wave propagation)

Turbulence arises in several places stratosphere Heat sources w/in dome boundary layer ~ 1 km tropopause km wind flow around dome

Within dome: “mirror seeing” When a mirror is warmer than dome air, convective equilibrium is reached. Remedies: Cool mirror itself, or blow air over it, improve mount To control mirror temperature: dome air conditioning (day), blow air on back (night) credit: M. Sarazin convective cells are bad

Local “Seeing” - Flow pattern around a telescope dome Cartoon (M. Sarazin): wind is from left, strongest turbulence on right side of dome Computational fluid dynamics simulation (D. de Young) reproduces features of cartoon

Boundary layers: day and night Wind speed must be zero at ground, must equal v wind several hundred meters up (in the “free” atmosphere) Boundary layer is where the adjustment takes place, where the atmosphere feels strong influence of surface Quite different between day and night –Daytime: boundary layer is thick (up to a km), dominated by convective plumes –Night-time: boundary layer collapses to a few hundred meters, is stably stratified. Perturbed if winds are high. Night-time: Less total turbulence, but still the single largest contribution to “seeing”

Real shear generated turbulence (aka Kelvin- Helmholtz instability) measured by radar Colors show intensity of radar return signal. Radio waves are backscattered by the turbulence.

Kolmogorov turbulence, cartoon Outer scale L 0 ground Inner scale l 0 h convection solar h Wind shear

Kolmogorov turbulence, in words Assume energy is added to system at largest scales - “outer scale” L 0 Then energy cascades from larger to smaller scales (turbulent eddies “break down” into smaller and smaller structures). Size scales where this takes place: “Inertial range”. Finally, eddy size becomes so small that it is subject to dissipation from viscosity. “Inner scale” l 0 L 0 ranges from 10’s to 100’s of meters; l 0 is a few mm

Outer Scale ~ m, from Generalized Seeing Monitor measurements F. Martin et al., Astron. Astrophys. Supp. v.144, p.39, June

Atmospheric structure functions A structure function is measure of intensity of fluctuations of a random variable f (t) over a scale  D f (  ) = With the assumption that temperature fluctuations are carried around passively by the velocity field (for incompressible fluids), the structure function of the index if refraction N is: D N ( r ) = = C N 2 r 2/3 C N 2 is a “constant” that characterizes the strength of the variability of N. It varies with time and location. In particular, for a static location (i.e. a telescope) C N 2 will vary with time and altitude

Typical values of C N 2 Night-time boundary layer: C N 2 ~ m -2/ Paranal, Chile, VLT

Turbulence profiles from SCIDAR Eight minute time period (C. Dainty, Imperial College) Siding Spring, Australia Starfire Optical Range, Albuquerque NM

Atmospheric Turbulence: Main Points The dominant locations for index of refraction fluctuations that affect astronomers are the atmospheric boundary layer and the tropopause Atmospheric turbulence (mostly) obeys Kolmogorov statistics Kolmogorov turbulence is derived from dimensional analysis (heat flux in = heat flux in turbulence) Structure functions derived from Kolmogorov turbulence are  r 2/3 All else will follow from these points!

Phase structure function, spatial coherence and r 0

Definition – Phase Structure Function Definition - Spatial Coherence Function Spatial coherence function of field is defined as Covariance for complex fn’s C  (r) measures how “related” the field  is at one position x to its values at neighboring positions x + r. Do not confuse  the complex field with its phase 

Definitions - Spatial Coherence Function Spatial coherence function of field is defined as Covariance for complex fn’s C  (r) measures how “related” the field  is at one position x to its values at neighboring positions x + r. Do not confuse  the complex field with its phase 

Now evaluate spatial coherence function C  (r) For a Gaussian random variable  with zero mean, So So finding spatial coherence function C  (r) amounts to evaluating the structure function for phase D  ( r ) !

Next solve for D  ( r ) in terms of the turbulence strength C N 2 We want to evaluate Remember that the Structure function is defined as: But for a wave propagating vertically (in z direction) from height h to height h +  h: This means that the phase is the product of the wave vector k (k=  [radian/m]) x the Optical path. Here n(x, z) is the index of refraction.

Finally we have the spatial coherence function C  (r) For a slant path you can add factor ( sec  ) 5/3 to account for dependence on zenith angle  Concept Question: Note the scaling of the coherence function with separation, wavelength, turbulence strength. Think of a physical reason for each. After some algebric development we have:

Given the spatial coherence function, calculate effect on telescope resolution Define optical transfer functions of telescope, atmosphere Define r 0 as the telescope diameter where the two optical transfer functions are equal Calculate expression for r 0

Define optical transfer function (OTF) Imaging in the presence of imperfect optics (or aberrations in atmosphere): in intensity units Image = Object  Point Spread Function I = O  PSF   dx O(r - x) PSF ( x ) Take Fourier Transform: F ( I ) = F (O ) F ( PSF ) Optical Transfer Function is Fourier Transform of PSF: OTF = F ( PSF ) convolved with

Examples of PSF’s and their Optical Transfer Functions Seeing limited PSF Diffraction limited PSF Intensity   Seeing limited OTF Diffraction limited OTF / r 0 / D r 0 / D / r 0 / D /  -1

Now describe optical transfer function of the telescope in the presence of turbulence OTF for the whole imaging system (telescope plus atmosphere) S ( f ) = B ( f )  T ( f ) Here B ( f ) is the optical transfer fn. of the atmosphere and T ( f) is the optical transfer fn. of the telescope (units of f are cycles per meter). f is often normalized to cycles per diffraction-limit angle ( / D). Measure the resolving power of the imaging system by R =  df S ( f ) =  df B ( f )  T ( f )

Derivation of r 0 R of a perfect telescope with a purely circular aperture of (small) diameter d is R =  df T ( f ) = (  / 4 ) ( d / ) 2 (uses solution for diffraction from a circular aperture) Define a circular aperture r 0 such that the R of the telescope (without any turbulence) is equal to the R of the atmosphere alone:  df B ( f ) =  df T ( f )   (  / 4 ) ( r 0 / ) 2

More about r 0 Define r 0 as telescope diameter where optical transfer functions of the telescope and atmosphere are equal r 0 is separation on the telescope primary mirror where phase correlation has fallen by 1/e (D/r 0 ) 2 is approximate number of speckles in short- exposure image of a point source D/r 0 sets the required number of degrees of freedom of an AO system Timescales of turbulence Isoplanatic angle: AO performance degrades as astronomical targets get farther from guide star

After some algebra we have: Using what we know: We obtain

Scaling of r 0 r 0 is size of subaperture, sets scale of all AO correction r 0 gets smaller when turbulence is strong (C N 2 large) r 0 gets bigger at longer wavelengths: AO is easier in the IR than with visible light r 0 gets smaller quickly as telescope looks toward the horizon (larger zenith angles  )

Typical values of r 0 Usually r 0 is given at a 0.5 micron wavelength for reference purposes. It’s up to you to scale it by -1.2 to evaluate r 0 at your favorite wavelength. At excellent sites such as Paranal, r 0 at 0.5 micron is cm. But there is a big range from night to night, and at times also within a night. r 0 changes its value with a typical time constant of 5 to10 minutes

r 0 sets the number of degrees of freedom of an AO system r 0 sets the number of degrees of freedom of an AO system Divide primary mirror into “subapertures” of diameter r 0 Number of subapertures ~ (D / r 0 ) 2 where r 0 is evaluated at the desired observing wavelength Example: Keck telescope, D=10m, r 0 ~ 60 cm at =   m. (D / r 0 ) 2 ~ 280. Actual # for Keck : ~250.