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### The Physics of Astronomical Seeing

Suprit Singh

Talk for the course Astro Tech. I (2010)

under

Dr. Yogesh Wadadekar

Overview

- 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

- "If the Theory of making Telescopes could at length be fully brought into Practice, yet there would be certain Bounds beyond which telescopes could not perform

- For the Air through which we look upon the Stars, is in perpetual Tremor;
- as may be seen by the tremulous Motion of the Shadows cast from high
- Towers, and by the twinkling of the fix'd Stars...

- The only Remedy is a most serene and quiet Air, such as may perhaps be
- found on the tops of the highest Mountains above the grosser Clouds."

Isaac Newton

Why so Turbulent?

- Strong temperature gradient in troposphere, at low altitudes
- Wind shear mixes layers with different temperatures
- resulting in turbulent temperature fluctuations T

- 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 scalel0

h

Wind shear

convection

h

ground

Big whorls have little whorls,

Which feed on their velocity;

Little whorls have smaller whorls,

And so on unto viscosity.

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

- For the inertial range the energy spectrum shape is : k-5/3
- 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
- Hence : u2 ~ (EL)2/3
- 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 thecontext of Seeing

- 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
- 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)
- 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
- Scatters light
- Perturbs Phase of the wave
- Causes fractional Amplitude change
- with effect:

Tatarski (1961) Structure function

- The phase fluctuations in Tatarski's model are assumed to have a Gaussian random distribution with the following second order structure function:
- 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 á...ñ represents the ensemble average.
- The structure function of Tatarski (1961) can be described in terms of a single (Fried) parameter r0:
- 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

- 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.
- Connection with Fried Parameter :
- 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.
- 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

- 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.
- 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".
- 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

- 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".
- 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:
- 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

- 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.
- 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.
- 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

- 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 .
- 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.

Image = Object Point Spread Function

I = O PSF dx O( x - r ) PSF( x ) 2 r0 7 r0

- 20 r0

- 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.

- 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 diffraction-limited Airy disk (FWHM ~ lambda / D). These individual Airy spots are called speckles.
- Seen together, the speckles give a shimmering blur.

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.

Seeing quality from Gemini South (Chile, left) and Gemini North (Mauna Kea, right).

From http://www.gemini.edu/metrics/seeing.html.

- Speckle imaging : which allows bright objects to be observed with very high resolution.
- Working outside the Atmosphere : Hubble Space Telescope and thus not have any seeing problems
- Adaptive optics : Systems that partially solve the seeing problem. Observations are usually limited to a small region of the sky surrounding relatively bright stars.
- 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 !!!

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