Seeing The Earth’s atmosphere: seeing, background, All ground-based observatories suffer from a major absorption & scattering problem: Observational Astronomy 2019 light from distant objects must pass through the Part 8 Earth’s atmosphere Prof. S.C. Trager recall from our discussion of optics that the atmosphere has refractive power Seeing Seeing top of atmosphere turbulent atmosphere The refraction itself isn’t the problem: it’s the variation in the index of refraction that causes problems near ground Wind, convection, and land masses cause turbulence, mixing the layers of different indices of This non-uniform, continuous variation in the index of refraction cause refraction in non-uniform and continuously varying the initially plane-parallel wavefronts to tilt, bend, and corrugate ways This causes the twinkling we see when we look at stars with our eyes Any degradation of images by the atmosphere is called seeing Seeing Seeing To understand seeing, we need to examine the index of The dominant terms are refraction and its dependence of temperature and 6 1 n 1 = 77.6 10− pT − pressure (and humidity, too) air − ⇥ Cauchy’s formula, extended by Lorentz to account for Seeing — perturbations of the transiting wavefront — is humidity, gives the excess (over vacuum) index of caused by changes in nair differentially across the refraction of air: wavefront due to turbulence 6 3 77.6 10− 7.52 10− v n 1= ⇥ 1+ ⇥ p + 4810 air − T λ2 T Note that water vapor has little effect on n in the ✓ ◆ where p is the pressure in mbars, T is the ⇣temperature ⌘ optical (except for fog), but can affect columns of air in K, and v is the water vapor pressure in mbars by causing convection: wet air is lighter than dry air! Seeing Thermal turbulence Modern observatories are built at very dry sites, so the overall Thermal turbulence in the effect of humidity is typically unimportant for seeing (but atmosphere is created on a important for differential refraction) variety of scales by Ignoring v above, then, for an ideal gas and adiabatic convection: air heated undisturbed turbulence conditions, by conduction with the wind flow dn p warm surface of the Earth dT / T 2 becomes buoyant and first rises, displacing cooler air mountain Therefore the primary source of index of refraction variations ridge in the atmosphere is thermal variations humid (wet) air also rises and displaces dry ocean We care (mostly) about small-scale variations in T: thermal air turbulence Thermal turbulence Thermal turbulence wind shear: high winds — in particular the jet stream — generate wind shear undisturbed turbulence disturbances: large undisturbed turbulence and eddies at various wind flow land-form variations — wind flow scales, creating a first like mountains — can first turbulent interface mountain create turbulence mountain between other layers in ridge ridge laminar (non-turbulent) ocean ocean flow Thermal turbulence Thermal turbulence Turbulence occurs when Turbulence is created by eddie transfer inertial forces dominate over viscous forces in a kinetic energy is deposited on large scales, which creates fluid eddies undisturbed turbulence This means that turbulent wind flow these in turn create smaller eddies, and so on... motions — eddies and vortices — dominate over first ...until the shears are so large compared to the eddy smooth (laminar) flows mountain scale that the viscosity of air takes over and the kinetic ridge energy is turned into heat This is true for typical wind speeds and length ocean This stops the turbulent cascade and smooths out the scales temperature fluctuations Thermal turbulence Thermal turbulence Kolmogorov showed that When turbulence occurs in an the energy spectrum of atmospheric layers with a such a turbulent cascade temperature gradient differing has a characteristic shape from an adiabatic gradient... –5/3 –5/3 inertial rangeκ one in which no heat is inertial rangeκ The inertial range transferred (i.e., the entropy log E external log E external follows a spectrum cutoff does not increase) cutoff 5/3 internal internal E − cutoff ...it mixes air of different cutoff / temperatures at the same where κ is the log κ altitude, producing temperature log κ wavenumber or inverse fluctuations following a length Kolmogorov spectrum Atmospheric layers Atmospheric layers contributing to turbulence contributing to turbulence surface or ground layer: ~few to 10’s of meters — up to few arcminutes seeing! turbulence generated by wind There are effectively four shear due to friction and layers of the atmosphere topographic effects that affect the seeing, with site geometry, trees, gradual transitions of boulders, etc. properties between them best sites have surface layers <10 m build telescope above this layer Atmospheric layers Atmospheric layers contributing to turbulence contributing to turbulence planetary boundary layer: <~1 km above sea level — 10’s of arcseconds seeing atmospheric boundary layer: above planetary both frictional effects and boundary layer day-night heating-cooling cycles not dominated by creates convection up convection but by to inversion layer abrupt changes in topography most observatories built above this layer Atmospheric layers Seeing contributing to turbulence free atmosphere: seeing ~0.4″ — best sites mostly see this To understand turbulence in the context of seeing, we layer need to translate the Kolmogorov spectrum into a unaffected by frictional spatial variation influence of ground To do this, we define a structure function: in 1D, large-scale flows 2 tropical trade winds (at DT (r)= [T (x + r) T (x)] lower altitudes), jet stream h − i (at higher altitudes) where T(x) and T(x+r) are the temperatures at two points separated by r mechanical turbulence Seeing Seeing For a Kolmogorov spectrum, 2 2/3 There is a characteristic transverse linear scale over DT (r)=CT r which we can consider the atmospheric variations to 2 where CT is the temperature structure constant and “flatten out” and in which plane-parallel wavefronts are gives the intensity of thermal turbulence transmitted The variation of the index of refraction (of air) is given by over these scales, the images are “perfect” the index of refraction structure constant, 2 2 6 3 2 2 2 This scale is called the Fried parameter r0 C = C [77.6 10− (1 + 7.52 10− λ− )pT − ] n T ⇥ ⇥ Seeing Seeing The Fried parameter is well-described by integrating Cn2 along the line-of-sight through the atmosphere: 2 1 2 3/5 r =[1.67λ− (cos z)− C (z0)dz0]− 0 n The Fried parameter is inversely proportional to the size Z where z is the zenith angle and z′ is the distance along the of the PSF transmitted by the atmosphere line of sight The FWHM of the observed PSF can be predicted if A larger Fried parameter is better! Hence the inverse we know the variation of Cn2 along the line-of-sight: dependence on Cn2 3/5 5 1/5 1 2 note also the dependence on cos z, so the Fried parameter ✓ (00)=2.59 10− λ− (cos z)− C (z0)dz0 FWHM ⇥ n gets smaller as one looks through more atmosphere Z Isoplanaticity and the Fried Seeing model In 1665, Hooke suggested the existence of “small, This is a typical Cn2 profile moving regions of the atmosphere having different Note that FWHM is refractive powers that act like lenses” to explain bigger for larger zenith scintillation, the “twinkling” of stars angles and smaller for longer wavelengths In 1966, Fried showed that this simple model was a reasonable representation of reality... Isoplanaticity and the Fried Isoplanaticity and the Fried model model Assume that at any given moment, the atmosphere acts like an array of small, contiguous, wedge- Thus, each isoplanatic patch transmits a quality image shaped refracting cells of the source These locally tilt the wavefront randomly over the The size of this isoplanatic patch is the Fried parameter size scale of the cells r0, roughly the diameter (not the radius!) of the patch each cell imposes its own tilt on the wavefront... This characterizes the seeing ...creating isoplanatic patches in which the wave The more turbulence there is, the smaller r0 is is fairly smooth and has little curvature Isoplanaticity and the Fried Seeing effects model Typically r0≈10 cm in the optical and varies as λ6/5 Seeing is always better at longer wavelengths! There are primary effects of seeing: The angular size of the isoplanatic patch is Scintillation (“twinkling”) ✓ 0.6r /h Image wander (“agitation”) 0 ⇠ 0 where h is the height of the primary turbulence layer Image blurring (“smearing”) above the telescope Seeing effects Seeing effects Scintillation (“twinkling”) is the result of a varying amount of energy being Image wander (“agitation”) is the motion of an image in received over time the focal plane due to changes in the average tilt of the intensity changes wavefront Variations in the shape of the turbulent layer result in moments when it Because the wavefront is locally plane-parallel, the acts like a (net) concave lens, focussing the light, and other moments when it acts like a (net) convex lens, defocussing the light image can actually be diffraction-limited if the aperture is <~r0 This is only obvious when the aperture is <~r0, like for the human eye For larger apertures, these effects average out Then the Airy disk is imaged, but it will wander Note that planets don’t twinkle, because they’re much bigger than θ0 Larger apertures average over more isoplanatic patches, Because it is an interference effect, scintillation is highly chromatic so this effect is also only present for small apertures Seeing effects Seeing: summary Airy disk Image blurring (smearing)