The Physics of Astronomical Seeing - PowerPoint Presentation

Slide1

The Physics of Astronomical Seeing

Talk for the course Astro Tech. I (2010)underDr. Yogesh Wadadekar

Suprit SinghSlide2

Overview

Atmospheric Turbulence : Why and WhenQuantifying Turbulence : Kolmogrov (1941) theoryTatarski (1961) Model : Wave Propagation in Turbulent medium Observational measures of seeing Observed Effects of Seeing Overcoming Atmospheric SeeingSlide3

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 NewtonSlide4

Why so Turbulent?

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

T Slide5

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.

When

does

Turbulence arise?Slide6

Kolmogrov Theory of Turbulence in a NutshellBig whorls have little whorls, Which feed on their velocity; Little whorls have smaller whorls,

And so on unto viscosity.

Outer scale

L

0

ground

Inner scale

l

0

h



convection

solar

h



Wind shearSlide7

Kolmogrov Theory of Turbulence : Eddy Cascade

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 mmSlide8

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

. Slide9

Kolmogrov Theory : A Dimensional Approach

Let u be velocity ,

outer scale L

, then

energy

~

u

2

energy dissipation rate, E

~

u

2

/t

~

u

3

/L

Hence : u

2

~

(EL)

2/3Taking its Fourier transform gives the power spectrum L ~ k-1

f(k)dk ~ u2 ~ k

-2/3

or f(k) ~ k-5/3 Slide10

Turbulence : In the

context of SeeingTo 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)Slide11

Wave Propagation in Turbulent mediumFor monochromatic plane waves arriving from a distant point source with wave-vector k, we haveThe Turbulent layer

Scatters light

Perturbs Phase of the wave

Causes fractional Amplitude change

with effect:Slide12

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 r

0

:

r

0

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.

Slide13

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

C

n

2

.

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.

Slide14

Link with Observational parameters II

Another parameter t

0

is proportional

to r

0

divided by the mean wind speed

and corresponds to the time-scale over

which the changes in the turbulence

become significant. t

0

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.

Slide15

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/

r

o

, 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).Slide16

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. Slide17

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:

theta

o

~ 0.6

r

o

/

h

where

h

is the altitude of the primary turbulence layer over the telescope.Slide18

Images with Seeing

Image = Object  Point Spread Function I = O  PSF  

d

x

O(

x

-

r

) PSF

(

x

)

2 r

0

7 r

0

20 r0 Slide19

Observed Seeing Effects I

ScintillationScintillation ("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. Slide20

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 diffraction-limited Airy disk (FWHM ~ lambda / D). These individual Airy spots are called

speckles

.

Seen together, the speckles give a shimmering blur. Slide21

Observed Seeing Effects IIISlide22

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. Slide23

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. Slide24

Overcoming atmospheric seeing

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.Slide25

Thanks*

* The distortion of above text is not due to seeing !!!