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Overview of Astronomical Seeing Refraction in the atmosphere produces: Image motion Turbulence-induced astronomical “seeing” Image motion is the large-scale, highest power effect of refraction Turbulence modifies the index of refraction along a specific line of sight The optical path length, OPL = n*d => Observatories are built on mountaintops!

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Schematic Layout of a Deformable Mirror AO System Claire Max - UCSC

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Optical Layout of and AO System telescope primary mirror Science camera Pair of matched off- axis parabola mirrors Wavefront sensor (plus optics) Beamsplitter Deformable mirror collimated

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Bet you didn’t know about speckles, eh? K-Band (2.2 μm) speckle pattern with integration time 140 ms and pixel resolution 0.0206” made at the Keck I 10m telescope. Each speckle is a diffraction-limited image of the object. This pattern changes significantly ~ 1000 times per second. 1”

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Techniques for the Sharper Image Speckle interferometry Shift-and-add Lucky imaging Optical/infrared interferometry Tip-tilt correction Adaptive optics

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AO in action These IR images are taken at about 100 frames per second. Note that AO corrects the incoming wavefront for atmospheric-induced turbulent modification of the index of refraction.

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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)

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Turbulence arises in several places stratosphere Heat sources w/in dome boundary layer ~ 1 km tropopause 10-12 km wind flow around dome

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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. To control mirror temperature: dome air conditioning (day), blow air on back (night), send electric current through front Al surface-layer to equalize temperature between front and back of mirror credit: M. Sarazin convective cells are bad

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Top View - Flow pattern around a telescope dome Computational fluid dynamics simulation (D. de Young)

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Turbulent boundary layers has largest effect on “seeing” Wind speed must be zero at ground, must equal v wind several hundred meters up (in the “free” atmosphere) Adjustment takes place in boundary layer, via viscosity Where 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 rising from hot ground. Quite turbulent. Night-time: boundary layer collapses to a few hundred meters, is stably stratified. See a few “gravity waves.” Perturbed if winds are high.

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Boundary layer is much thinner at night: Day ~ 1 km, Night ~ few hundred meters Credits: Stull (1988) and Haggagy (2003) Surface layer: where viscosity is largest effect

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Sometimes clouds show great Kelvin-Helmholtz vortex patterns

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Temperature profile in atmosphere Temperature gradient at low altitudes wind shear will produce index of refraction fluctuations

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Kolmogorov turbulence in a nutshell Big whorls have little whorls, Which feed on their velocity; Little whorls have smaller whorls, And so on unto viscosity. Big whorls have little whorls, Which feed on their velocity; Little whorls have smaller whorls, And so on unto viscosity. L. F. Richardson (1881-1953)

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

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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

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How large is the Outer Scale? Dedicated instrument, the Generalized Seeing Monitor (GSM), built by Dept. of Astrophysics, Nice Univ.)

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Outer Scale ~ 15 - 30 m from Generalized Seeing Monitor measurements F. Martin et al., Astron. Astrophys. Supp. v.144, p.39, June 2000 http://www-astro.unice.fr/GSM/Missions.html

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The Kolmogorov turbulence model, derived from dimensional analysis v = velocity, = energy dissipation rate per unit mass, = viscosity, l 0 = inner scale, l = local spatial scale Energy/mass = v 2 /2 v 2 Energy dissipation rate per unit mass ~ v 2 / = v 2 / ( l / v) = v 3 / l v ~ ( l ) 1/3 Energy v 2 ~ 2/3 l 2/3

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Kolmogorov Power Spectrum 1-D power spectrum of velocity fluctuations: k = 2 / l (k) k v 2 ( l ) 2/3 2/3 k 2/3 or, dividing by k, (k) k 5/3 (one dimension) 3-D power spectrum: 3D (k) / k 2 3D (k) k 11/3 (3 dimensions) For rigorous calculation: see V. I. Tatarski, 1961, “Wave Propagation in a Turbulent Medium”, McGraw-Hill, NY

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( cm -1 ) Slope = -5/3 Power (arbitrary units) Lab experiments agree Air jet, 10 cm diameter (Champagne, 1978) Assumptions: turbulence is homogeneous, isotropic, stationary in time Slope -5/3 l0 l0 L0 L0 Credit: Gary Chanan, UCI

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Structure functions are used a lot in AO discussions. What are they? Mean values of meteorological variables change over minutes to hours. Examples: T, p, humidity If f(t) is a non-stationary random variable, F t ( ) = f ( t + ) - f ( t) is a difference function that is stationary for small . Structure function is measure of intensity of fluctuations of f (t) over a time scale D f ( ) = =

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Structure function for atmospheric fluctuations, Kolmogorov turbulence Scaling law we derived earlier: v 2 ~ 2/3 l 2/3 ~ r 2/3 Heuristic derivation: Velocity structure function ~ v 2 here C v 2 = constant to clean up the “look” of the equation

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What about temperature and index of refraction fluctuations? Temperature fluctuations are carried around passively by the velocity field (for incompressible fluids). So T and N have structure functions similar to v: D T ( r ) = = C T 2 r 2/3 D N ( r ) = = C N 2 r 2/3

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How do you measure index of refraction fluctuations in situ? Refractivity Index fluctuations So measure T, p, and T; calculate C N 2

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Simplest way to measure C N 2 is to use fast- response thermometers D T ( r ) = = C T 2 r 2/3 Example: mount fast-response temperature probes at different locations along a bar: X X X X X X Form spatial correlations of each time-series T(t)

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Typical values of r 0 Fried’s parameter, r 0, is used to characterize seeing – it is the circular aperture over which a wavefront is coherent. 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 Mauna Kea in Hawaii, r 0 at = 0.5 micron is 10 - 30 cm. But there is a big range from night to night, and at times also within a night.

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Seeing statistics at Lick Observatory (Don Gavel and Elinor Gates) Left: Typical shape for histogram has peak toward lower values of r 0 with long tail toward large values of r 0 Huge variability of r 0 within a given night, week, or month Need to design AO systems to deal with a significant range in r 0

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