Lecture 4 Flashcards
wavefronts
are locus of points having the same phase
variations in atmosphere distort
wavefront of incoming signal
wavefront no longer parallel when it arrives at telescope
turbulent layer of the atmosphere
pockets of air, each of varying size
move around erratically
leads to variations in the refractive index
varying refractive index distorts
plane waves entering the atmosphere
the variation of refractive index (formula)
dn/dT ∝ - P/T^2
the phase lag/lead introduced into the wavefront relative to the reference wavefront (formula)
ΔΦ = l Δn/λ ∝ P/λT^2 LΔT
for visible wavelengths and standard pressures and temperatures (formula)
ΔΦ = 2LΔT
Phase structure function
D(Φ)(r) = D(Φ) (|x’-x|)
Long exposure OTF
< H(r) > = H(atm) . H(tel)
< H(r) > - Ensemble average OTF
H(atm) OTF of atmosphere
H(tel) OTF of telescope
Ensemble-average PSF in focal plane
|h|^2 = F^-1 {H(atm) . H{tel)}
For a high quality telescope with a large primary, the effects of the atmosphere dominate and
<H(r)> = e^(-DΦ(r)/2)
Coherence function
B(r) = e^(-DΦ(r)/2)
Kolmogorov Model
Simple model of turbulence in the atmosphere
Wind-shears gives rise to Kelvin-Helmholtz Instabilites
Turbulent energy is generated on a large scale, Lo
These get smaller and smaller until kinetic energy is dissipated through viscosity at a length scale, lo
universal description for turbulence spectrum
inertial range between lo and L0
strength of the turbulence as a function
of the eddy size or spatial frequency
two parameters determine the strength and spectrum of Kolmogorov turbulence
rate of energy generation per unit mass
kinematic viscosity
structure function of velocity field (formula)
D(v) (R1,R2) = α . f (|R1-R2|/β)
Temperature Cells
Turbulence mixes different layers of air carries around cells of air with different temperatures
these cells are in pressure equilibrium have different densities and, therefore, different indices of refraction, n
Turbulent atmosphere can be modelled as
layers of distortion driven by the wind, moving with velocity v.
Light travelling through high refractive index regions is
delayed compared to other regions
Phase Structure function
D(r) = 6.88(r/r0)^(5/3)
Turbulent fluctuations of refractive index are described by the refractive index structure parameter and determine
the Fried parameter
Atmospheric time constant (formula)
τ0 = r0/V(bar)
r0 = turbulence strength or fried parameter
V(bar) is the wind velocity
images exposed on a time longer than τ0 are called
long-exposure images and are dominated by the effects of atmospheric aberrations
images exposed on a time shorter than τ0 are called
short-exposure images and are free from atmospheric aberrations
Atmospheric Seeing
β = 0.98λ/r0
resolution of diffraction-limited telescope
α = 1.03λ/D
Variations in refractive index alead to
Wavefront Distortion
The OTF can be expressed in terms of a
Phase structure function
turbulence generates
temperature cells
the Kolmogorov model describes
the variation of temperature and refractive index over a wide range of scales
the strength of the turbulence is characterised by the
Fried Parameter r0
the long exposure OTF is given by an
integral of the Phase Structure Parameter C(N)^2 with height
The seeing is the
FWHM of the atmospheric PSF and can be interpreted as the resolution given by an equivalent diffraction-limited telescope diameter r0.
