|
telescopeѲptics.net
▪
▪
▪
▪
▪▪▪▪
▪
▪
▪
▪
▪
▪
▪
▪
▪ CONTENTS ◄ 5.1. Atmospheric turbulence ▐ 5.2. Low-level turbulence, tube currents... ► 5.1.2. seeing error: limiting resolution, strehl, otf Due to a different nature of the seeing induced error, the Strehl approximation for non-random aberrations, given by Eq. 56 becomes inaccurate for larger errors, considerably more than it is for conventional aberrations (FIG. 51). The effect is mainly due to the roughness error component, whose increase reduces the average relative size of wavefront irregularity, while at the same time increasing its average RMS error. Similarly to ordinary surface roughness, narrow zones and turned edge, the increase in the P-V (and thus RMS) error after a certain point begins to drain less energy out of the Airy disc, with the main effect gradually becoming merely spreading the energy out wider. As a result, the increase in the RMS (surface or wavefront) error is followed by relatively slower Strehl degradation, as given by the relation derived by Racine for long-exposure "seeing Strehl":
where Φ=2πω is the phase variance across the pupil, with ω=0.162(D/r0)5/6 being the time-averaged (long-exposure) RMS wavefront error induced by seeing. Note that exp(...) denotes the exponent (value in brackets) over the natural logarithm base e(...), with e~2.718, rounded off to three decimals. So, for D/r0=1, the average RMS wavefront error ω=0.162 (Eq. 53), giving Φ=1.02, and the seeing Strehl S=0.445; for identical nominal RMS error ω of non-random aberrations the Strehl is 0.357. Simpler long-exposure Strehl approximation, in terms of D/r0, is S~1/[1+1.23(D/r0)2]. A quick but rough long exposure seeing Strehl approximation, empirically fitted for (D/r0) around 1 and smaller is S~1-0.55(D/r0); for larger (D/r0) values, it is S~1/1.5(D/r0)2.
FIGURE 51: Strehl ratio for wavefront error induced by atmospheric turbulence. For the ratio values bellow ~0.5, or RMS wavefront errors in units of the wavelength greater than ~0.13, long-exposure (time-averaged) Strehl becomes increasingly better than with the comparable RMS wavefront error of balanced aberrations. Strehl approximation (red line), is within 3% of the true value for balanced 3rd order spherical aberration at 0.5, 10% at 0.3 and within 20% at 0.1 Strehl. For instance, for D/r0=2, the long-exposure Strehl is at the level of 0.2 wave RMS error of balanced spherical aberration, while the actual averaged RMS seeing error (Eq. 53) is some 40% larger (~0.28 wave). The short-exposure Strehl plot is for the roughness wavefront error alone, smaller by a factor of 0.37 than long-exposure error. As a result, it degrades with the seeing much more gradually than long-exposure Strehl. Visual Strehl as a function of D/r0 is approximated as described in Fig. 50. It is roughly at the level of "diffraction-limited" (~0.80 Strehl) if D is similar in size to atmospheric coherence length r0. Due to increased effect of the tilt error component in visual observing, either with the increase in aperture (for given seeing level), or in the seeing error (for given aperture), resulting visual Strehl gradually shifts from being near identical to the short-exposure Strehl (for low seeing error, or in very small apertures in average or better seeing), to be closer to the long-exposure Strehl (for large seeing errors). While the plot on Fig. 51 is a gross approximation, it does illustrate basic relations between the three main forms of seeing-induced error. Size of r0 determines angular resolution of large telescopes as ~λ/r0 in radians (λ being the wavelength of light), as opposed to the "standardized" diffraction resolution given by ~λ/D. While the size of r0 results from a complex function, it can be expressed much more simply in terms of the resolution limit it imposes to large telescopes. For the seeing imposed stellar resolution α in radians, and the wavelength λc centered at 0.00055mm, it is given by: r0= 1.27lc/α (54) Hence, an average ~2 arc second seeing at the zenith, for λ=0.00055mm, results from r0~72mm (2 arc seconds in radians is 2/206,265). Since r0 is changing in proportion to λ1.2 and (cosγ)0.6, γ being the zenith angle in degrees, a more complete expression for its size is given by:
Thus, the extracted stellar resolution α=5.7λ1.2(cosγ)0.6/r0, for given γ (zenith distance) and r0, also changes in proportion to λ1.2. From Eq. 55.1 it is obvious that this stellar resolution is not the actual resolution at the zenith angle γ, rather the zenith resolution limit in terms of λ, γ and r0. That makes it proportional to (cosγ)0.6, i.e. diminishing with larger zenith distances. Actual stellar resolution at the zenith angle γ is inversely proportional to (cosγ)0.6 which, as expected, makes it directly proportional to the atmospheric coherence length r0. So, for instance, at 30° from zenith (γ=30°) and the wavelength λ=0.0005mm, atmospheric coherence length r0=100mm corresponds to the zenith seeing α=0.000005716 radians, or 1.18 arc seconds (after multiplying by 206,265). The seeing at the zenith angle is numerically larger by a factor of 1/(cosγ)0.6, or
with r0 calculated from Eq. 55.1. An interesting aspect of seeing-limited telescope resolution is that in imaging applications, either with photo-emulsion or CCD chip as a detector. The limit to resolution is determined by the diameter of first-order speckle structure. With the number of first-order speckles approximated by N~(D/2r0)2, and speckle angular center separation of ~2λ/D, in radians (thus with the speckle diameter ~2λF linearly, after multiplying the angular value with focal length), average linear radius of the stationary seeing FWHM for moderately bright stars, for (D/r0)≥3, can be roughly approximated as √NλF, or:
In the average 2 arc seconds seeing, with r0~72mm, a 200mm aperture with (D/r0) of ~2.8, average seeing FWHM radius is nearly 1.4λF linearly, still not significantly larger than the Airy disc radius, but larger than system's FWHM λF by a (D/r0) factor or, in this case, 2.8 times. This is approximation for the stationary speckle structure, i.e. for subsecond exposures. For longer exposures (couple of seconds and up), the speckle movement caused by tilt wavefront error further enlarges the blur. If, for the purpose of obtaining a general idea of the magnitude of long-exposure enlargement, the tilt RMS wavefront error - from Eq. 52.1-2, rounded to 0.1(D/r0)5/6, in units of the wavelength - is assumed to be ~1/3 of the P-V tilt error W on the wavefront, angular displacement of the speckle structure around its center due to the tilt wavefront error is roughly αS~Wλ/D~0.3(D/r0)5/6λ/D. Corresponding average linear displacement from the center in the image plane is αSƒ, ƒ being the focal length, or 0.3(D/r0)5/6λF. Since the linear radius of the bright core of the stationary speckle structure for (D/r0)≥3 is, from Eq. 55, approximated by (D/2r0)λF, it indicates that long-exposure radius of the bright central half of the blur is larger by roughly a factor 1+[0.6(D/r0)5/6/(D/r0)] than that of the stationary (short-exposure) image. For relatively good seeing conditions, with (D/r0)<3, the ratio of the seeing versus actual (optics') FWHM is roughly 1+0.6(D/r0)5/6. So, for instance, the long-exposure bright core blur radius (i.e. seeing FWHM radius) for (D/r0)=1 would be ~1.6 times the system FWHM, while for (D/r0)=3 it would be 1.5 times its short-exposure blur radius of 1.5λF, or about 4.5 times the system FWHM λF.
Table bellow summarizes approximate relation between optic's and
seeing FWHM.
In terms of contrast transfer, the effect of atmospheric error is given by atmospheric OTF (Optical Transfer Function), which is for long-exposure given by:
and for short-exposure it is given by
In both, ν is the normalized spatial frequency. Combined atmospheric and telescope contrast transfer is given by the product of their respective OTF's. It is the final OTF in the image plane of the objective. Obstructed apertures will have lower combined OTF due to the contrast-lowering effect of central obstruction, but not more. In fact, the larger obstruction, the larger relative size of r0 vs. annulus, resulting in a slower increase in phase variance over the annulus area and, consequently, slower decrease in the average Strehl (as a measure of wavefront quality, independent of the effect of central obstruction).
Closer to the observer, it is terrain topography, immediate
surroundings and the very parts of a telescope itself that can
create layers of unsteady air resulting in
wavefront deformations.
|
