Atmospheric Intensity Scintillation of Stars. III. Effects for Different Telescope Apertures

The early literature contains a number of photographs, showing flying shadows on telescope mirrors (Gaviola 1949; Mikesell, Hoag, & Hall 1951; Protheroe 1955a). However, these images convey a false impression of the shadow pattern being elongated into "bands," an effect that was caused by the longish exposure times required, during which the imaged pattern was stretched out along the direction of the prevailing winds. During a typical exposure of 25 ms, with a typical wind speed of 20 m s⁻¹, the pattern is displaced by half a meter, a distance much greater than characteristic shadow scales (given by the Fresnel‐zone size r_F = √λh, on the order of 5–10 cm; Paper I). To freeze the pattern requires exposure times of no more than a few milliseconds, not feasible with the then available photographic emulsions. Nevertheless, these early records showed that the turbulence layers causing the shadow patterns were multiple and unstable, and that the fine structure in the patterns could change very rapidly, in mere fractions of a second.

More quantitative studies followed, using photomultiplier detectors and various double and multiple entrance apertures. Thus, Keller (1955), Protheroe (1955b), Barnhart, Protheroe, & Galli (1956), Keller et al. (1956), and Protheroe (1961, 1964) deduced spacetime autocorrelation functions and other statistical properties of the shadow structure.

Modern studies, analyzing the two‐dimensional spatiotemporal power spectra of shadow patterns, clearly reveal the multilayer structure of winds and turbulence structures in the upper atmosphere (Vernin & Roddier 1973; Vernin & Azouit 1983; Caccia, Azouit, & Vernin 1987; de Vos 1993). Images of the telescope pupil, recorded in very short exposure times of a few milliseconds, no longer show any evidence for the "banded" appearance of early photographic recordings; the frozen shadow pattern cast by a star now appears isotropic, as theoretically expected.

During some tens of seconds before and after solar‐eclipse totality, when the almost‐eclipsed Sun constitutes a source of very small angular width, the flying‐shadow pattern can be perceived also with the unaided eye (although the intensity fluctuations amount to only a few percent). For introductory reviews, see Marschall (1984) and Codona (1991). Over long horizontal light paths, analogous phenomena may be seen projecting onto vertical walls, in the light cast by the low Sun when it is almost occulted by sharp features on the horizon, in the light cast by Venus, or from powerful artificial‐light beams.

Such flying shadows were noted by visual observers a long time ago. However, their transient nature made photographic recordings very difficult, and the paucity of precise observations led to a proliferation of exotic theories for their origin. More quantitative photoelectric measurements from recent decades (Young 1970b; Hults et al. 1971; Quann & Daly 1972; Klement 1974; Marschall, Mahon, & Henry 1984; Jones & Jones 1994) demonstrated how the shadow bands become narrower and more closely spaced as totality approaches and how their power spectra are quite similar to those for stellar scintillation.

The scintillation theory of eclipse shadows was very well presented by Codona (1986). The primary difference to the stellar case is that the illumination source is not pointlike but rather crescent‐shaped. While shadow patterns from stars are statistically isotropic, solar shadow bands are not, because of the anisotropic brightness distribution of the uneclipsed solar crescent.

Further differences exist because of the varying spatial extent of the solar crescent. For times greater than about 20 s before (or after) totality, the solar shadows are dominated by turbulence near the ground. Turbulence at the tropopause (important in the stellar case) has almost no impact on shadow bands until 2–3 s from totality, when the uneclipsed solar portions start to become pointlike. Only during these short time intervals does a wavelength dependence develop (cf. Paper II). Shadow bands are related to the same turbulence responsible for seeing; good seeing is predicted to cause a poorer shadow‐band contrast.

For the next several chapters, we will calculate synthetic power spectra and autocovariance functions to compare with our observations. The purpose is not to model the precise atmospheric conditions above La Palma but rather to obtain scintillation functions from a model that is sufficiently simple for the comprehension and traceability of effects, yet capable of identifying subtle differences between different types of aperture, and of allowing an extrapolation of scintillation properties to much larger telescopes.

Simple effects from aperture averaging were discussed in Paper I, e.g., a geometric‐optics approximation predicts the intensity variance σ²_I in the focal plane to obey

where C²_n(h) is the refractive index structure coefficient, h is altitude in the atmosphere, and Z is the zenith distance. This approximation (neglecting wavelength‐dependent diffraction) is valid for circular aperture diameters D (much) greater than the Fresnel‐zone size r_F = √λh (in the zenith; otherwise √λhsec Z). Then the amount of scintillation is independent of wavelength and merely decreases with increasing D.

Our modeling follows Young (1969, 1970a); see also Roddier (1981). The Kolmogorov spectrum of (three‐dimensional) isotropic turbulence throughout the atmosphere is used to compute the spatial intensity spectrum of the (two‐dimensional) flying‐shadow pattern falling onto the ground. Approximate allowances are made for diffraction, atmospheric dispersion, and seeing, integrated through an (assumed) exponential atmosphere with a certain turbulence scale height. The spatial‐filter function of the telescope aperture is multiplied by the two‐dimensional power spectrum of the shadow pattern, and the product is numerically integrated in both spatial‐frequency dimensions to give the total modulation power.

We assume a single wind speed (in the x‐direction) and consider a monochromatic optical wavelength of 500 nm. Further, following Taylor's hypothesis, the temporal variation of the illumination pattern is given by a linear shift at constant velocity equal to the wind speed. The modeling does not provide normalizing factors for either the strength of the turbulence or for the calibration of the temporal frequency scale (which depends on the actual wind speed). Both these parameters are obtained by fitting the synthetic functions to actual data, as recorded during representative summer conditions on La Palma.

Through appropriate integrations, one may obtain different statistical descriptions of the shadow pattern (autocorrelation, power spectrum, structure function). For example, by integrating the spatial power spectrum over the spatial‐frequency coordinate perpendicular to the wind direction, and using the projected wind speed to transform the spatial frequency into temporal frequency, we obtain the temporal power spectrum of scintillation, P(f).

Consider a circular telescope aperture of radius a. Its response to modulation at spatial frequency k is found by integrating the sinusoidal intensity fluctuation over the aperture (k = 2π/d, where d is a spatial period in the shadow pattern; cf. § 4.3 in Paper I). The result is that the detected modulation power is diminished by the factor

which is called the aperture filter function (Young 1969; Roddier 1981). J₁ is the Bessel function, familiar from diffraction theory. Equation (2) has the same form as the diffraction pattern of a circular aperture, and the filter function for any aperture indeed has the form of its diffraction pattern, normalized to unity at k = 0. For example, an annular aperture of outer radius a and inner radius qa has the filter function (Young 1967)

and for a rectangular aperture of width 2a_x in the x‐direction and 2a_y in the y‐direction, one has (Young 1969)

Similarly, filter functions for apodized apertures are calculated. Apodizing a zone of width w at the edge of a circular aperture gives an amplitude transmission function that can be approximated by the convolution of a circular aperture of radius (a - w/2) (where a is the clear‐aperture radius) with a small circular aperture of diameter w/2 (radius w/4). This operation makes the amplitude‐transmittance function very nearly (but not exactly) sinusoidal. Convolution in the spatial domain corresponds to multiplication of the corresponding functions in the spatial‐frequency domain, and the spatial‐filter function of such an apodized aperture is

If w/4≪(a - w/2), the second factor is nearly unity, and apodizing a narrow outer zone has practically the same effect as reducing the clear aperture by half the width of the zone. We may regard a_eff = a - w/2 as the effective radius of the apodized aperture. The benefits of apodizing do not appear until the second factor in equation (5) begins to decrease appreciably.

More complex apertures, e.g., annular ones with inner and outer apodized edges, are computed by suitably combining various clear and apodized portions.

Once the aperture filter function F_apert(k ) is known,k = (k_x,k_y), its effect is obtained by multiplying the (two‐dimensional) shadow‐pattern power spectrum P₂(k ) by F_apert(k ):

Thus, the frequency spectrum, autocorrelation function, and other descriptors are obtained by using P^*₂(k ), the filtered spectrum of the smeared shadow pattern, instead of P₂(k) itself.

We now examine the main trends of the functions. Some apparent complexity comes from the oscillations in the aperture‐filter functions, in particular the "beats" between the oscillations of the Bessel functions in obscured apertures. These originate from the idealized model assumptions, and we shall neglect these details, concentrating on the general trends.

The three‐dimensional isotropy of the assumed Kolmogorov turbulence causes a two‐dimensionally isotropic shadow pattern with power spectrum P₂(k ). If the telescope aperture has circular symmetry,F_apert(k ) is also isotropic. This makes P^*₂(k ) in equation (6) isotropic as well. At low frequencies, where all the filter functions are nearly unity,P^*₂ is proportional to k^1/3. Eventually, k becomes large enough to make any filter function begin to decrease rapidly. Since all our filter functions have (oscillating) tails whose envelopes fall off as some substantial power of k, this decrease effectively truncates the integration.

For example, the aperture filter for a radially symmetric aperture with sharp edges falls off (apart from the oscillations) as k^-3 (eqs. [2] and [3]). This applies to apertures much larger than the Fresnel‐zone size; for smaller ones, diffraction effects cause a more rapid cutoff as k^-4 (Young 1969, 1970a). The filter for an apodized aperture has two Bessel‐function factors. It falls as k^-3 when the main factor (associated with the outer radius a_eff) begins to decrease rapidly, and as k^-6 when the second factor (associated with the apodizing‐zone width w) becomes effective. In every case, the falloff begins where the argument of a filter is of order unity. The rectangular aperture (eq. [4]) has asymmetric properties; its filter drops as k^-2 along the x or y edges, but as k^-4 along its diagonal.

The total modulation power, i.e., the intensity variance σ²_I, is found by integrating P₂ up to the point where the first filter truncates the integration. As the lowest turnover frequency corresponds to the largest spatial dimension, this is usually set by the telescope aperture. The aperture filter function turns down when ak ≈ 1, or k ≈ 1/a. The integral is thus

if we use the isotropy to write d k as k dk dø. The 2π comes from integrating over ø.

To obtain results for the time domain, we must write d k as dk_xdk_y and integrate over dk_y. At low frequencies, P₂ is proportional to k^1/3, and the integration extends from k_y = 0 to k_y ≈ 1/a. Then

if we use the fact that k_y ≈ k in the region where the integrand is large. We could also have obtained this result by noting that the total intensity variance σ²_I is spread over the region from k_x = 0 to k_x ≈ 1/a, so that the low‐frequency power must be proportional to σ²_I/(1/a)∝ a^-7/3/a^-1 = a^-4/3.

At high frequencies, the integrand falls off as k^1/3 times the tail of the filter function, which is a large negative power of k. Thus, the function P^*₂(k) looks somewhat like a circular crater, with its lip at the value of k where the filter turns down. The outer slopes, apart from the oscillations, fall off as k^{-3 + 1/3} = k^-8/3 for large sharp‐edged apertures,k^{-4 + 1/3} = k^-11/3 for small ones, and as k^-17/3 for our apodized apertures. The region where the integrand is appreciable extends for a distance in k_y comparable to k_x. Thus, the tail of the one‐dimensional power spectrum is

where p = 3 for sharp‐edged ones, 4 for apertures smaller than the Fresnel‐zone scale, and 6 for apodized ones. Thus, the tails fall off as k^-5/3,k^-8/3, and k^-14/3, respectively, in these three cases.

The turnover for a given aperture begins near k_x = 1/a, or temporal frequency f = V_⊥/2πa, where V_⊥ is the projected wind speed. For 30 m s⁻¹, that corresponds to about 10 Hz for a 1 m aperture. Diffraction becomes important when λhk²/4π ≈ 1, or k ≈ (4π/λh)^1/2, where a typical value of h is a scale height or 8 km. For λ = 500 nm, this corresponds to about 150 Hz near the zenith. At higher frequencies, the diffraction filter reduces the tail exponent by an additional 4 units, making these tails fall off as the −17/3 power of frequency for sharp‐edged apertures and f^-20/3 for apodized ones.

As a final caveat for this chapter, we point out some limitations to the scintillation modeling applied here.

At the highest frequencies, there are limits in the approximations used, e.g., for the diffraction cutoff. Assuming no inner‐scale cutoff, one could just extrapolate the tails of the power laws: a slope of −5/3 up to around 100 Hz and −17/3 beyond that. However, there is evidence (§ 6.6 in Paper I) for a more rapid decrease, at shadow scales of some millimeters. This implies that very high frequency scintillation will be rather weaker than a simple power‐law extrapolation would indicate at frequencies above about 1 kHz. In this range also small structures like the supporting vanes of the secondary mirror could introduce high frequencies into the aperture filter.

The calculated scintillation spectra refer to the logarithm of the intensity, not the intensity itself. Since the scintillation amplitudes almost always are small, the practical difference should be negligible. However, as the intensity is lognormally distributed, the nonlinear transformation from logs to antilogs means that the intensity distribution will, at least in principle, contain harmonics and intermodulation (sums) of the frequencies in the log intensity spectra. These additional high frequencies could dominate the tail of the intensity spectra for small apertures, especially apodized ones.

The Kolmogorov power law is used to predict the spatial spectrum of the shadow pattern. A conclusion from Papers I and II was that the observations generally do support the validity of this Kolmogorov law for turbulence. However, the agreement is not perfect, and there could exist locations or times when atmospheric conditions might deviate. There exist alternative theories for the power spectrum of turbulence, taking a finite outer scale into account, e.g., the von Kármán spectrum (Tatarski 1961).

It would be valuable to test the predictions for very large telescopes, through actual observations. One issue that probably cannot be answered without such measurements concerns effects from the outer scale of atmospheric turbulence, i.e., the largest geometrical scales where a "turbulent" description of the atmosphere still is valid. Quite possibly, its order of magnitude is close to 10 m, the size of the currently largest optical telescopes.

In Figure 3, there are three curves for the same circular (fully transmitting) aperture of 20 cm diameter. One is for observing in the zenith, and two are at different wind‐azimuth angles at zenith distance Z = 35^° (1.22 air masses).

Note the azimuth effect away from the zenith. When looking along the projected wind vector, the shadow pattern's motion is foreshortened. This makes the projected motion slower than the wind speed. When looking at right angles to the wind, we see the full wind speed in the motion of the shadow pattern. In either case, the total variance in the shadow pattern is independent of the speed of motion, so the slower apparent motion along the wind corresponds to a greater power density at low frequencies; the integral of the power spectrum (the total variance) depends only on zenith distance.

The two curves at Z = 35^° are simply separated by the projection factor for the effective wind speed; this scales as sec Z. The difference between the along‐wind and cross‐wind plots is due to this difference in projected wind speed (and not, e.g., saturation effects).

Figure 3 also illustrates the effects of size and central obstruction, comparing the previous curves with a larger annulus; a 2.5 m telescope with a 90 cm central obscuration (secondary mirror) is shown. The 20 cm aperture is essentially in the geometric‐optics regime, so diffraction effects are negligible; the change in the shape of the curve, in going to the annular aperture, is due to the central obstruction. The amount of scintillation in the latter case is much less but also shows somewhat complex time structure, caused by the secondary mirror.

Examining the features for annular apertures, one notes that they show a shoulder, or (for narrower annuli) even a secondary maximum. As a patch in the shadow pattern drifts across first one side of the annulus and then the other, the autocovariance of the scintillation time series ought to show features rather like the autocorrelation of the aperture with itself—namely, its modulation transfer function (MTF). There is also some similarity to the MTFs of annular apertures; some differences arise because the spatial‐frequency content of the shadow pattern introduces a certain weighting. Since the scintillation power spectrum is a smeared version of the telescope's MTF (diffraction pattern), a central stop increases the high‐frequency scintillation.

For further discussions of the effects of central obscurations, see Young (1967). The central stop in typical telescopes obscures about 1/3 of the aperture and approximately doubles the high‐frequency contribution. Larger obscurations cause greater effects. Such increased high‐frequency scintillation may complicate the comparison of scintillation data between different telescopes.

Also, other obstructions in the telescope pupil, such as the spider vanes commonly holding the secondary mirror, will affect the amount of scintillation at some accuracy level (as well as the image quality through diffraction). For a discussion of the effects of secondary‐mirror spiders on the diffracted image, see Harvey & Ftaclas (1995); the relevance for scintillation follows from the relationship of the aperture‐filtering function with the telescope diffraction pattern.

Since the purpose of apodization is to match the spatial crossing of the flying shadows across the aperture edges, and the shadows often have a preferred direction of motion, apodization in only one coordinate, i.e., in the direction of the flying‐shadow motion, might be sufficient (Young 1967). In the case of a rectangular aperture, and wind along x, there will be no benefit from apodizing also perpendicular to x. However, the mask must be aligned very closely along the wind direction, which thus should consist of a single component. Otherwise, there enters some projection of the unapodized coordinate along the wind, and the tail of the temporal power spectrum eventually assumes its unapodized form.

In the case of a single wind component, fluctuating only in angle, one could even conceive of adaptive apodization filters, correcting for the changing angle in real time (cf. § 7.4); however, the required stable and simple wind patterns might not be encountered very often. Measured wind vectors typically fluctuate several degrees over a few minutes (§ 6), and in practice rotationally symmetric apodizing masks appear more useful, being an "insurance" against wind‐direction variations.

The origin of scintillation in high‐level turbulence and its correlation with high‐level winds suggests that slower scintillation can be expected at sites with relatively slow high‐level winds. Many major observatories (Canary Islands, Chile, Hawaii) are located at latitudes with rapid overhead jet streams. This appears to imply rapid scintillation and (with more kinetic energy available to produce turbulence) a greater amplitude. On dimensional grounds, the turbulence strength should be proportional to the kinetic energy in the wind, as confirmed for at least some combined measurements of (low‐frequency) scintillation and wind speed (Young 1969, his Fig. 20).

Parameters that relate to scintillation are those of the lifetime and temporal correlation of speckles inside focused images. Such quantities are being studied to assess a site's suitability for speckle‐ and other types of interferometry. Those timescales are coupled with the time for the phase and intensity distribution in the telescope aperture to change significantly, and thus are related to scintillation parameters. Often, two distinct timescales are seen: one short (≃2–10 ms) associated with the boiling of the speckles within the image, and a much longer component of image motion, i.e., the random motion of the speckle‐image envelope.

On Mauna Kea, the variability of speckle lifetimes during and between nights appears to be significantly higher (factors of 5 or more: Dainty et al. 1982; O'Donnell et al. 1982) than at other sites (factors of 2–3): Herstmonceux, England (Scaddan & Walker 1978; Parry, Walker, & Scaddan 1979), Haute‐Provence (Aime et al. 1986), La Palma (Dainty, Northcott, & Qu 1990), or La Silla (Vernin et al. 1991). Also, at Paranal there are indications for rapid variability of speckle lifetimes (M. Sarazin 1997, private communication). These differences could be due to the proximity of Mauna Kea and Paranal to main jet streams, one reason for the otherwise excellent weather these sites are enjoying.

For finding regions with low wind speeds for scintillation, global maps for, e.g., the 200 mb atmospheric level (≃11 km altitude) may be examined (Vernin 1986). The wind maxima are along ±30° latitude (where many major observatories are located, due to the large number of clear nights), while the minima are near the equator and near the poles. In response to concerns about the short timescales, and their ensuing problems for interferometry and adaptive optics, some site testing has been done on candidate sites with expected low wind speeds, such as Réunion in the Indian Ocean. However, equatorial sites can be problematic because of frequent or seasonal clouds, but perhaps Antarctic sites might be feasible.

Observatory "sites" with the most rapid wind conditions are found on board airplanes. The effective wind speed of ≃250 m s⁻¹ means that the shadow pattern crosses the telescope an order of magnitude faster than usual. By itself, this does not change the total amount of scintillation, but rather has the effect of shifting the scintillation spectrum to much higher temporal frequencies, and spreading its power over a wider frequency band.

The seeing conditions for such observing were examined by Zwicky (quoted in Oosterhoff 1957), but more quantitative data were obtained by Dunham & Elliot (1983). Comparing ground‐based data to photometric measurements made with a 90 cm telescope on board the Kuiper Airborne Observatory, they found the major difference to be "the absence of scintillation" and large (≃5 ^'') images. The latter is probably caused by the boundary layer of the aircraft. The low level of scintillation is consistent with it originating from distant sources: when already flying at the tropopause level, the optical effects from the turbulence remaining at very high levels are tiny.

The generally lower scintillation level, combined with its shift to higher frequencies, can be exploited for various photometric tasks, including the observing of lunar or planetary occultations of stars. Larger‐size airborne telescopes, such as the 2.5 m one currently being built for SOFIA, are likely to offer very low scintillation levels.

Theoretical considerations provide an approximate scaling law for the rms error due to the low‐frequency component of scintillation (Young 1967, 1974):

where D is the aperture diameter in centimeters, sec Z is the air mass, h is the observer's height above sea level,h₀ ≃8000 m is the atmospheric scale height, and T is the integration time in seconds. The air mass exponent of 1.75 is an approximate value not far from the zenith; it equals 2 when looking along the wind direction and 1.5 perpendicular to it. The rms deviation σ is then obtained in units of relative intensity ΔI/I.

This expression applies for timescales longer than those for flying shadows to cross the telescope aperture, i.e., integrations on scales of seconds and longer. (More strictly, for frequencies below V_⊥/πD, where V_⊥ is the speed at which turbulence crosses the line of sight and D the diameter of the telescope, ≃10 m s⁻¹/10 m = 3D 1 Hz for telescopes in the 2–3 m class.) When incorporating also the high‐frequency part of scintillation, the aperture dependence is steeper (eq. [1]). However, it is the low‐frequency power in equation (10) that is perceived as noise in ordinary photometry, and "limiting" accuracies for ground‐based photometry can be estimated from this expression (Brown & Gilliland 1994; see also Young et al. 1991; Gilliland & Brown 1992; Kjeldsen & Frandsen 1992; Heasley et al. 1996).

From various observing campaigns it is possible to compare the scintillation levels at different observatory locations. Although the precise amount of scintillation of course changes from night to night, the scaling law (eq. [10]) appears to hold quite well for apertures up to at least 4 m, and for quite different sites (Kjeldsen 1991; Gilliland & Brown 1992; Gilliland et al. 1993).

Our data for the power spectral density measured on La Palma can be used to calculate the scintillation noise expected in photometric integrations. The intensity variance for a given integration time T can be computed from the power‐density spectrum P(f) through the integral

In the case T = 0, one obtains

which is the relation used earlier to normalize P(f).

For large T, the expression may be approximated by setting P(f) = P(0) for f<1/T, whereupon P can be taken outside the integral:

Applying this for our power spectra in Figure 1, we find P(0) ≃1.5 × 10^-5 when interpolating to a 4 m aperture. This is for Z = 45^°; scaling to the zenith by the factor (sec Z)³ gives ≃5 × 10^-6. For an integration T = 60 s, we then find a σ on the order of ≃200 μmag, quite comparable to near‐zenith values measured on 4 m class telescopes on Kitt Peak and Mauna Kea (Gilliland et al. 1993).

There might conceivably exist some ground sites with exceptional scintillation properties. Sites in Antarctica appear to be particularly promising, although these have as yet been only incompletely examined.

Potentially unique scintillation properties might exist at high‐altitude sites (≳4000 m) on the plateau in East Antarctica (see, e.g., Burton 1995). Here the equivalent pressure altitudes in winter are in excess of 5000 m. The geomagnetic latitude of some potential sites (e.g., Dome C at 73° S, 127° E) places them close to the center of the auroral oval, making the night sky much darker than at the South Pole.

Of probable relevance for scintillation is the circumstance that during most of the year, the tropopause at these sites, in effect, reaches the ground. The ground temperature is already very low and (except for a few summer months) the atmospheric temperature increases monotonically from the ground into space. The lack of significant thermal gradients leads to low wind speeds at all atmospheric heights; the wind systems are dominated by slow laminar air flows downwelling from the stratosphere, without any jet streams. The low kinetic energy content in the air mass limits the energy available to generate turbulence and mixing. Seeing data from various Antarctic sites suggest that there is significant turbulence only over the lowest hundreds of meters. While this causes some angular seeing, it should not much affect scintillation, since its dominant contributions come from distant (focusing) atmospheric layers.

Balloon‐borne microthermal sensors, measuring C²_nup to some 40 km altitude, have verified the exceptional stability of the high‐Antarctic air mass. Heasley et al. (1996) discuss scintillation limits, and special sites, including the South Pole. They quote radiosonde data, from which C²_nvalues are deduced, concluding that the intensity variance σ²_I should be a factor of 5 smaller at the South Pole than at Mauna Kea, two sites at about equal equivalent pressure altitude. It appears likely that sites such as Dome C, at still higher altitude and with a quieter atmosphere, could be even better.

There could still be other problems, such as effects from blowing snow. Atmospheric transparency noise was measured at the South Pole by Duvall, Harvey, and Pomerantz (Harvey 1988). For frequencies ≳0.1 Hz, that is found to be very much smaller than scintillation noise, although it starts to dominate for fluctuations on scales of minutes and longer.

When discussing Antarctic sites, the difference between sites for good angular seeing and for low scintillation should be understood. Sites with excellent seeing could be exploited already with moderate‐sized telescopes in the 1 m class. Even if the seeing would be poor (presence of low‐altitude turbulence), a site could still be excellent for scintillation (Paper I). In contrast to telescopic image quality, however, scintillation noise significantly improves with increased telescope size; an Antarctic site would have to be enormously much better in order for a 1 m telescope, say, to compete with a very large telescope at an ordinary location. While moderate‐sized telescopes may be realistic to operate at these remote sites, the logistic problems suggest that very large telescopes will have to await the possible development of better infrastructure in the more distant future.

The potential of sites in Greenland has also been voiced, though it is not clear how good such sites could be, compared to Antarctica. Although elevations reach above 3000 m, it appears that Greenland is not big enough to create its own stable weather systems, as opposed to Antarctica.

As a general caveat about the scintillation to be expected at "exotic" sites, we want to stress that the atmospheric processes causing scintillation should be understood. As stressed in § 4.6 of Paper I, thermal stratification alone does not cause scintillation; there must be a wind shear to produce turbulence and mixing of air across, e.g., inversion layers (by themselves, such layers tend to suppress turbulence). Monographs introducing the relevant topics of atmospheric dynamics and boundary‐layer meteorology include Lumley & Panofksy (1964), Panofsky & Dutton (1984), Oke (1987), and Stull (1988). No matter how "promising" a site might appear, its quality should be verified by optical scintillation measurements.

Irrespective of geographical location, scintillation effects can be significantly reduced by combining the signals from several telescopes. At low frequencies, the scintillation‐noise amplitude σ decreases with the telescope diameter only as D^−2/3 (eq. [10]). The telescope area is proportional to D², so the noise amplitude decreases only as the cube root of this area. However, if instead of one large telescope, one uses several little ones with the same total area, the noise amplitude decreases with the square root of the number of telescopes (hence, the square root of the area).

To reduce (low‐frequency) scintillation noise by a factor of 10, using a single aperture, thus requires an increase of its area by a factor of 1000, or its diameter by over 30 times. The same gain can be obtained with an array of 100 small telescopes, whose signals are averaged together. If the construction of new telescopes is considered for projects where photometric precision is an important element, it must be noted that—since the cost of large telescopes increases at least as fast as their collecting area—a given capital investment will be much more efficient in terms of photometric noise if distributed over many small telescopes, rather than over a few larger ones.

To optimize the spatial distribution of the elements in such an array, the possible correlations among their individual apertures have to be evaluated. A calculation of the covariance shows that the scintillation in neighboring telescopes is weakly anticorrelated if they are within the atmospheric outer scale (believed to be on the order of 10 m, although deduced values differ greatly; § 6.7.2 in Paper I). The anticorrelation follows from the circumstance that, next to a bright structure in the atmospheric shadow pattern, there must follow a darker one, also manifest in the negative autocovariances in Figure 3. Thus, one can actually cancel out slightly more of the scintillation by placing the telescopes side by side. However, such gains can be only slight, because several telescopes placed side by side will in effect start making up one single large aperture. Time‐delayed correlations will enter for telescopes along the wind vector (partially) observing the same flying shadows (cf. Fig. 8). Probably, an optimum configuration has to be dilute enough to be pseudorandom, i.e., without regular spatial features—perhaps corresponding to a minimum‐redundancy array in ordinary interferometry. Since the likely gains from precise optimization are likely to be modest, we have not pursued these studies further.

This discussion applies to low‐frequency scintillation. Its high‐frequency component obeys other relations and will not be equally reduced by the measures discussed in this section (cf. eq. [1], Fig. 1).

With the possible exception of exotic Antarctic sites, and air‐ and spaceborne telescopes, the reported differences in scintillation amplitudes among different sites are quite modest. The decrease of scintillation noise in very large telescopes, coupled with the logistic problems of operating them at either exotic sites or off the ground, suggests that (at least among current facilities) very large ground‐based telescopes (and perhaps medium‐sized airborne ones) are the most useful for critical photometric studies. In the future, clusters of smaller telescopes (ideally spread out over different sites) could offer superior performance, at least regarding the lower frequencies of scintillation.

Since the mere increase of telescope size does not yield great gains, other methods have been tried, e.g., simultaneously observing one or more reference stars near the target. Although such schemes reduce effects from transparency fluctuations (as required for time series photometry), they will normally not reduce the component from scintillation proper, because that is uncorrelated over even small angular distances (Paper I, § 9); such procedures may even enhance that particular noise component! Thus, a successful attack on the problem appears to require the use of the target star itself as a reference (Harvey 1988).

Classical "brute‐force" approaches by reducing scintillation by averaging over larger apertures, and integrating for longer times, do not seem especially promising for any order‐of‐magnitude reduction of scintillation noise. Instead, we will now examine other schemes for circumventing scintillation, doing exactly the opposite to classical methods, avoiding all averaging by instead using very small (sub)apertures and integrating over very short timescales. We want to treat scintillation as a signal to be accurately determined and physically modeled, not as a random noise that is averaged in larger data bins in the hope that it will eventually go away. Following its accurate measurement, the photometric segregation of the scintillation signal from that of the astrophysical source should become feasible. Further, for imaging purposes (if a high photometric stability is not required), a reduction of scintillation effects appears feasible in the sense of real‐time compensation for the inhomogenous pupil illumination cast by the flying shadows.

Using the data from such a system, the signatures of scintillation‐induced intensity fluctuations, ΔI_scint, may be segregated from that of the true stellar intensity (unperturbed by atmospheric fluctuations),I_*(λ). Scintillation has the effect of modifying the intensity (photon flux per unit area) falling onto the telescope to I(t,λ,x,y) = I_*(λ) + ΔI_scint(t,λ,x,y), where x and y are the spatial coordinates in the pupil plane. A detector system such as that in Figure 10 records this signal for different times, wavelengths, and spatial positions. To segregate I_*(λ), a physical modeling of ΔI_scint(t,λ,x,y) is required. The constraints in such modeling include (but need not be limited to) the following :

• 1.  
  During a short observing period (10 s, say), the true stellar intensity can often be regarded as constant:I_*(λ) = const, or else modeled as slowly varying with time.
• 2.  
  I(t) = I_*(λ) + ΔI_scint(t,λ,x,y) must follow a lognormal distribution for all λ, x, and y (Paper I).
• 3.  
  ΔI_scint(x,y) must follow a relation such that the spatial power spectrum is consistent with that predicted from a Kolmogorov spectrum of turbulence (Paper I).
• 4.  
  I(λ) must follow a relation such that the intensity variance σ²_I at different wavelengths shows a λ^−7/6 dependence (or perhaps somewhat weaker; Paper II).
• 5.  
  I(t,λ) must follow a relation such that the timescales for σ²_I at different wavelengths follow a √λ‐dependence (or perhaps somewhat weaker; Paper II).
• 6.  
  I(λ,x,y) must follow a relation such that the spatial scales for ΔI_scint(x,y) follow a √λ‐dependence (or perhaps somewhat weaker; Paper II).
• 7.  
  I(t,λ,x,y) must show a behavior such that the velocity of the flying‐shadow pattern across the pupil is independent of λ (Paper II).

Using such types of constraints,I(t,λ,x,y) can be modeled in terms of its components I_*(λ) and ΔI_scint(t,λ,x,y), perhaps best by maximum likelihood fitting of the scintillation relations to observed data, taking the photon‐counting and detector noise into account. The amount of data produced in this scheme is considerable; already a short observation of 10 s, say, with a 2 m class telescope (300 spatial pixels, three wavelength channels, 10 ms time resolution) produces 10⁶ intensity measurements of I(t,λ,x,y), from which I_*(λ) is to be deduced. A full night's observation with a very large telescope would then produce more than 10¹⁰ measurements, a significant but not unreasonably large data volume by today's standards. Since the types of computations are repetitive in nature, and data can be handled sequentially in blocks of short observation periods, this appears well suited to automated data analysis schemes.

Unfortunately, it appears not straightforward to precisely quantify which components of this scheme can contribute what fraction of scintillation reduction (e.g., chromatic vs. temporal information; number of spatial channels vs. sampling rates). To obtain credible numbers appears to require numerical modeling and extensive simulations on a level that will make it a research project in its own right.

Although the physical concepts may appear simple, there are several degrees of freedom in parameterizing and modeling the flying‐shadow behavior. A higher spatial resolution across the telescope pupil on one hand improves the resolution of the flying shadows, but at the cost of making the imaging photometrically less precise due to increased photon noise. Similar trade‐offs go for the temporal and spectral resolutions. Utilizing chromatic cross‐correlations in scintillation may, away from zenith, permit a certain "early warning" since a disturbance in one color may appear earlier than that in another (Paper II). But how reliably can such a prediction be utilized? In some cases, perhaps apodization of the telescope aperture can be beneficial (by digitally weakening the signal from the aperture edges). Can this be further improved by adaptively changing the degree of apodization in real time, in response to, e.g., varying wind velocities?

Is there some additional information to be had from, e.g., imaging a pupil higher in the atmosphere, where there is less intensity variation? With a beam splitter, such a pupil could be incorporated (but at a cost in photon noise). If there is a known (average) height from which scintillation comes, that would simplify the modeling. The physical modeling might be improved if the actual (not idealized) properties of scintillation are better understood, e.g., the apparent deviation in its wavelength dependence from an idealized λ^−7/6 law (Paper II).

While the scintillation amplitude of course is independent of stellar brightness, the quality of its segregation will depend on the stellar magnitude; the brighter the star, the less photon noise will affect the determination of scintillation parameters. Also, for measuring the wavelength dependence, a "white" star with adequate flux throughout the spectrum will be easier to measure than pronounced blue or red ones.

Perhaps one could apply some of the mathematical formalism developed for signal extraction in other fields of study. For example, considerable efforts have been invested in developing algorithms for optimally filtering photographic images against the noise in the emulsion (see, e.g., Katsaggelos 1991). When measured on a microphotometer, that noise usually appears approximately Gaussian in photographic density, corresponding to the logarithm of intensity. The mathematical problem of segregating (two‐dimensional) linear‐intensity images against a logarithmic grain noise in photography is thus of a similar character to segregating (one‐dimensional) stellar linear‐intensity variations versus time, against a background of logarithmic scintillation noise.

Even at lower resolutions and slower sampling, some scintillation characteristics remain visible. Therefore, one can envision also simpler schemes that should be able to tangibly reduce scintillation noise, despite not fully resolving the flying‐shadow signatures.

Using only one temporal channel, one measures I(t) = I_* + ΔI_scint(t), i.e., as a conventional high‐speed photometer that records the intensity with ≲10 ms time resolution. The distribution of measured values, I_i, closely follows a lognormal distribution, and there will appear frequent "spikes" in the intensity record data, with amplitudes rather higher than expected, had the distribution been normally Gaussian (Paper I). Because these "spikes" may reach quite high values, they contribute a disproportionate fraction of the "noise" to a classically recorded photometric signal. If more spatial channels (resolving the entrance pupil) are available, the lognormal character of the signal will be better resolved.

Such time‐resolved data can be handled with different levels of sophistication. Even a temporal median filter would cut out events in the lognormal tail of the intensity distribution and the variance they contribute.

A better way would be to compute the logarithm of each intensity sample, and to average these logarithms, instead of the linear intensities. This is equivalent to taking the geometric mean, exp (〈ln I_i〉), of the time series (as opposed to the arithmetic mean,〈I_i〉, in classical photometry). In the limit of negligible photon noise, that is the quantity on which the mathematically optimum estimate of the average value of a lognormally distributed sample is based (see, e.g., Kendall & Stuart 1979, p. 74). However, the gains from this procedure alone will be modest. The variance of the estimated mean intensity will change by approximately a factor σ²_I/[exp (σ²_I - 1)], or ≃0.9 if we take σ²_I = 0.2 as representative for fully resolved scintillation (Paper I). The geometric mean is smaller than the arithmetic one and requires a correction factor that in the lognormal case is equal to exp (σ²_I/2). (The variance σ²_I must therefore be estimated as part of the procedure.) In the limit of photon noise only, the counting statistics obey a Poisson distribution, and the optimum intensity estimate is simply the arithmetic mean; there are no gains from subdividing the measuring sequence.

For photometric light curves requiring high time resolution (i.e., not many orders of magnitude longer than scintillation timescales), it might perhaps be worthwhile to apodize the sampling time window, to depress noise components originating from occasional intensity "spikes" at the beginning or end of the sample time intervals.

Such schemes, involving high sampling rates, are related to efforts made elsewhere for reducing transparency (extinction) fluctuations for astronomical photometry. Those are caused by changes in the pressure, humidity and turbidity of the atmosphere blowing across the path between the observer and a star. Gains in the signal‐to‐noise ratio by factors ≃1.5 are reported when the data are initially recorded at high speed, and then suitably filtered, instead of merely pursuing an ordinary boxcar integration (Poretti & Zerbi 1993; Zhilyaev, Romanyuk, & Svyatogorov 1994).

The scheme of Figure 10 provides a further reduction of photometric noise. In ordinary photometers, a field (Fabry) lens is used to image the entrance pupil onto the detector. Its purpose is to avoid output variations caused by the focused stellar image moving across the surface of the nonuniform detector. Instead, the flying shadows in the entrance pupil are imaged onto the detector, which thus still experiences a variable illumination. Even if the shadow contrast is low, and their large numbers largely average out the fluctuations, this still does contribute to the apparent scintillation, and should also be eliminated. This is solved by spatially resolving the flying shadows, i.e., the atmospheric speckle pattern; there are no further spatial fluctuations inside each resolved speckle element.

To circumvent this noise source is, in principle, also feasible in more ordinary photometry, by bringing the light to an integrating cavity instead of imaging the pupil on the detector. However, such schemes are wasteful in terms of photons.

Once the atmospheric sources of intensity fluctuation can be contained, telescopic sources of analogous noise must also be tracked down. Telescope mechanics often imply tracking irregularities, vibrations, and other effects, which introduce a spatial modulation of the stellar image with high temporal frequency, possibly mimicking scintillation effects (Jenkins et al. 1996; Erm 1997; St‐Jacques et al. 1997).

Also, some use could be made of the experience gained in related laboratory experiments, where scintillation noise has been reduced by exploiting correlation between perturbations at different wavelengths (Kjelaas & Nordal 1982).