Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
Seeing Analysis Theory
ATST Science Working GroupNov. 17, 2003
The data
High Low
Shabar scintillation cross-correlationraw data
Average over day for 2 seeing layer heights
Basic ideas 1
• Atmospheric turbulence creates variations in n(index of refraction)
• Assuming turbulence in form of a “thin” layer, then wave passing through layer suffers phase shift Φ due to variations in n.
• Important assumptions: Φ depends only on horizontal position and Φ « 1
• Can then represent continuous turbulence as superposition of contributions from thin layers
Basic ideas 2
• Phase fluctuation at base of layer is observed at ground as intensity fluctuations related to amplitude of the wave
• The amplitude has a spatial power spectrum determined by the turbulence in the layer.
• Assuming Kolomogorov turbulence, the spectrum is
• where λ is the wavelength, f is the spatial frequency, h is the height of the layer, δh is the layer thickness, and Cn
2 is the structure function.
Basic ideas 3
• For continuous turbulence, the phase fluctuations add linearly ( ).
• For statistically independent layers, the power spectra add linearly.
• Then for continuous turbulence,
Φ «1
• The scintillation is 4x the amplitude fluctuations.• The real part of the Fourier transform of the power spectrum is the
spatial covariance, B(x) where x is a distance. B(x) is measured by the Shabar.
Single-layer star at zenith example
Extended source away from the zenith
• Equations must be modified to account for fact that Sun is not a point source and not at the zenith.
• Incorporation of non-zenith case is simple: replace height h by line-of sight distance to layer z, with z = h sec ζ
• Then, power spectrum is
Detector geometry
• Effect of extended source depends on detector geometry.
• θ is the solar angular diameter.
• At height h, the rays from the sun to the detector pass through an area defined by the cone in the drawing.
Averaging over the solar disk
• Amplitude from point source case must be averaged over the beam area changing power spectrum to
• Where is the spatial power spectrum of the beam cross-section:
The Kernel
Can replace sin term by its argument, so
The function Qs: detector separation in units of cone diameter at height h and zenith angle ζ
The Kernels for each Shabar detector
Another view of the Kernels
From the theory to the observations
• This is a classic inversion problem• Use method of regularized least-squares
Determining the coefficients
• The numerics for doing this is non-trivial, next talk
• Once the coefficients ai are determined, the function Cn
2 can be computed.• Finally, get r0(h) from
A few limitations
• The Shabar analysis can provide information only near the ground.
• The height range of the current Shabarinstrument is at best 1000 m, but typically much less at non-zero zenith angles.
• In order to correctly constrain the high-altitude contribution to the seeing, it is necessary to include the S-DIMM measurement of r0 .