The Physics of Astronomical Seeing
Overview
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Atmospheric Turbulence : Why and When
Quantifying Turbulence : Kolmogrov (1941) theory
Tatarski (1961) Model : Wave Propagation in Turbulent medium
Observational measures of seeing
Observed Effects of Seeing
Overcoming Atmospheric Seeing
Observing through Earth’s Atmosphere
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Why so Turbulent?
• Strong temperature gradient in troposphere, at low altitudes
• Wind shear mixes layers with different temperatures
• resulting in turbulent temperature fluctuations T
When does Turbulence arise?
• The properties of fluid flows are characterized by the Reynold's
number, a dimensionless quantity relating inertial to viscous forces
Re = V L / nu = (Inertial forces) / (Viscous forces)
• This determines whether the flow will be:
Re low - dominated by viscosity, smooth / laminar / constant; or
Re high - dominated by inertial forces, turbulent, full of vortices
and eddies.
• For air, viscosity is very small: nu = 1.5 X 10-5 m2/s.
• Thus, for typical wind speeds and length scales of meters to kilometers,
Re > 106 and the air is moving turbulently.
Kolmogrov Theory of Turbulence in a Nutshell
solar
Outer scale L0
Inner scale l0
h
Wind shear
convection
Big whorls have little whorls,
Which feed on their velocity;
Little whorls have smaller whorls,
And so on unto viscosity.
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ground
Kolmogrov Theory of Turbulence : Eddy Cascade
• Assume energy is added to system at largest scales - “outer scale” L0
• 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” l0
• L0 ranges from 10’s to 100’s of meters; l0 is a few mm
Kolmogrov Theory of Turbulence : The Spectrum
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For the inertial range the energy
spectrum shape is : k-5/3
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When turbulence occurs in an
atmospheric layer with a temp
gradient it mixes air of different
temperatures at the same
altitude and produces
temperature fluctuations.
•
Hence, the spectrum also
describes the expected variation
of temperature in turbulent air.
•
Aside: The 3-D version follows
k-11/3.
Kolmogrov Theory : A Dimensional Approach
• Let u be velocity , outer scale L , then
energy ~ u2
energy dissipation rate, E ~ u2 /t ~ u3 /L
u2 ~ (EL)2/3
Hence :
• Taking its Fourier transform gives the power spectrum L ~ k-1
f(k)dk ~ u2 ~ k-2/3
or
f(k) ~ k-5/3
Turbulence : In the context of Seeing
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To understand turbulence in the context of seeing, we need to
translate the above thermal turbulence spectrum into a spatial
context i.e., how the atmosphere is going to transform wavefronts
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The wavefront perturbations are brought about by variations in the
refractive index of the atmosphere. These refractive index
variations lead directly to :
•
Phase fluctuations (Dominant)
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Amplitude fluctuations (second-order effect, negligible)
Wave Propagation in Turbulent medium
• For monochromatic plane
waves arriving from a distant
point source with wave-vector
k, we have
• The Turbulent layer
1. Scatters light
2. Perturbs Phase of the
wave
3. Causes fractional
Amplitude change
with effect:
Tatarski (1961) Structure function
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The phase fluctuations in Tatarski's model are assumed to have a Gaussian
random distribution with the following second order structure function:
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D ( ) is the atmospherically induced variance between the phase at two
parts of the wavefront separated by a distance r in the aperture plane,
and  represents the ensemble average.
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The structure function of Tatarski (1961) can be described in terms of a
single (Fried) parameter r0:
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r0 indicates the “strength" of the phase fluctuations as it corresponds to
the diameter of a circular telescope aperture at which atmospheric phase
perturbations begin to seriously limit the image resolution. Fried (1965)
and Noll (1976) noted that it also corresponds to the aperture diameter
for which the variance of the wavefront phase averaged over the aperture
comes approximately to unity.
Link with Observational parameters I
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Modify the structure function as
which is profile of the turbulence strength as a function of altitude and is
also referred to as index of refraction structure constant.
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Connection with Fried Parameter :
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The quality of an image depends on how much turbulence there is along a
line of sight. A larger Fried parameter is better, hence the inverse
dependence on Cn2.
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Note the direct dependence on cos(gamma), which shows that the Fried
parameter gets smaller as one looks through more atmosphere with
increasing zenith angle.
Link with Observational parameters II
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Another parameter t0 is proportional
to r0 divided by the mean wind speed
and corresponds to the time-scale over
which the changes in the turbulence
become significant. t0 determines the
correction speed required to
compensate for the effects of
the atmosphere.
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The distortion in an image changes at a high rate, typically more
frequently than 100 times a second and the different distortions average
out as a filled disc called the point spread function or "seeing disc".
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The diameter of the seeing disk is defined as the full width at half
maximum (FWHM) and is a measure of the astronomical seeing
conditions.
Link with Observational parameters III
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The FWHM of the seeing disc (or just Seeing) is usually measured in
arcseconds. At the best high-altitude mountaintop observatories the
wind brings in stable air which has not previously been in contact with
the ground, sometimes providing seeing as good as 0.4".
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The FWHM of the observed PSF can be predicted if one knows how Cn2
varies along the line of sight. From the previous equation, and assuming
now that the seeing is given by the diffraction limit of one Fried cell so
that theta ~ lambda/ro , one gets:
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Again, note the dependence on zenith distance (in the sense that the PSF
is smaller for smaller zenith distance) and wavelength (in the sense that
the PSF is smaller for longer wavelengths).
The Fried Model and Isoplanaticity I
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In 1665, Robert Hooke first suggested the existence of
"small, moving regions of the atmosphere having
different refracting powers which act like lenses" to
explain scintillation. In 1966 David L. Fried showed
that the atmosphere can indeed be modeled in this
fairly simple way.
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One can assume that at any moment the atmosphere
behaves like a compressed, horizontal array of small,
contiguous, wedge-shaped refracting cells. These act
on the plane parallel incoming waves from an
astronomical source by locally tilting the wavefront
randomly over the size scales of the cells.
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Each cell imposes its own tilt to the plane waves,
creating local isoplanatic patches within which the
wavefront is fairly smooth (~ lambda / 17) and has
little curvature. Thus, each isoplanatic patch transmits
a quality (though perhaps shifted) image of the
source.
The Fried Model and Isoplanaticity II
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The critical scale over which images begin to lose quality (i.e., the size of the
isoplanatic patch) is the Fried parameter or Fried length, ro .
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Another important characteristic of seeing is the angular size of the isoplanatic
patch. This isoplanatic angle is the angle on the sky over which the effects of
turbulence are uniform/correlated, given by:
thetao ~ 0.6 ro / h
where h is the altitude of the primary turbulence layer over the telescope.
Images with Seeing
Image = Object  Point Spread Function
I = O  PSF   dx O( x - r ) PSF ( x )
2 r0
7 r0
20 r0
Observed Seeing Effects I
Scintillation
Scintillation ("twinkling") is the result of a varying amout of
energy being received by a pupil over time.
Variations in the "shape" of the turbulent layer results in
moments where it mimics a net concave lens that defocuses
the light and other moments where it is like a net convex lens
that focuses the light. This curvature of wavefront results in
moments of more of less light being received by a fixed pupil
Image Wander
This is the motion of the instantaneous image in the focal
plane due to changes in the average tilt of the wavefront
with time.
Observed Seeing Effects II
Image Blurring
For a large telescope (D > ro), since many
isoplanatic patches will be in the beam of the
telescope, and image blurring or image
smearing dominates.
At any given time, if looking at the image of a
single star in a large telescope, each
isoplanatic patch creates its own diffractionlimited Airy disk (FWHM ~ lambda / D). These
individual Airy spots are called speckles.
Seen together, the speckles give a shimmering
blur.
Observed Seeing Effects III
Natural Site Seeing I
When one discusses "the seeing" one is generally referring to the long exposure FWHM
of the PSF.
One can characterize the typical image quality results by compiling statistics on the
seeing from night to night.
Natural Site Seeing II
Seeing quality from Gemini South (Chile, left) and Gemini North (Mauna Kea, right).
From http://www.gemini.edu/metrics/seeing.html.
Overcoming atmospheric seeing
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Speckle imaging : which allows bright objects to be observed with very high
resolution.
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Working outside the Atmosphere : Hubble Space Telescope and thus not have
any seeing problems
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Adaptive optics : Systems that partially solve the seeing problem.
Observations are usually limited to a small region of the sky surrounding
relatively bright stars.
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Lucky Imaging : The technique relies on the fact that every so often the
effects of the atmosphere will be negligible, and hence by recording large
numbers of images in real-time, a 'lucky' excellent image can be picked out.
This technique can outperform adaptive optics in many cases and is even
accessible to amateurs. It does, however, require very much longer
observation times than adaptive optics for imaging faint targets, and is limited
in its maximum resolution.
* The distortion of above text is not due to seeing !!!