A&A 382, 1125-1137 (2002)
DOI: 10.1051/0004-6361:20011671
P. F. Lazorenko^{1}
Main Astronomical Observatory, National Academy of Sciencies of the Ukraine, Golosiiv, 252680 Kyiv-127, Ukraine
Received 25 May 2001 / Accepted 4 October 2001
Abstract
A description of image motion characteristics is given for
a general case of atmospheric turbulence. For this purpose
the parameter p used in the expression
for a
2-D power spectrum of a phase is considered as a variable term with a
typical 1/3-2/3 range of fluctuations. Dependence on p of the
variance
of differential image motion was studied
for stellar configurations with one, two, and a set of reference stars
located in a circular area, for a circular entrance pupil and
interferometers. It was shown that
for non-Kolmogorov turbulence,
is subject to
significant variations even when Fried's
parameter r_{0} and mean square absolute image motion are fixed.
The method allowing us to reduce atmospheric errors
by transition to 1-D
measurements of differential coordinates is considered.
Orientation of coordinate axes in this case depends on
the direction of wind and intensity of turbulence in atmospheric
layers. Atmospheric limitations of large ground-based telescopes
are discussed.
Key words: atmospheric effects - instrumentation: interferometers - astrometry
The distorting influence of the Earth's atmosphere on light wave propagation essentially restricts opportunities for ground-based telescopes and considerably reduces their efficiency in comparison with telescopes working in Space. In differential astrometry atmospheric factors affect apparent relative positions of closely located stars so that they are subject to random displacements.
Theoretical deductions of atmospheric influence on the process of star image formation are usually based on some model of atmospheric turbulence. Most frequently, the Kolmogorov type of the turbulence is used. With this approximation basic relations for fluctuations of a phase and amplitude of the light wave propagating in a turbulent medium have been obtained, and a theoretical description of various atmospheric effects in optical astronomy is given (e.g. Tatarsky 1961; Fried 1965; Martin 1987; Sarazin & Roddier 1990). Lindegren (1980), in particular, has found dependence of the mean error of differential measurements on parameters of turbulent layers, angles between stars, and exposure. Temporal spectra and asymptotic power laws for a phase difference and differential angle of arrival were studied by Conan et al. (1995).
Meteorological data (e.g. Vinnichenko et al. 1976; Bull 1967) and theoretical studies (Weinstock 1980; Dalaudier & Sidi 1987), however, testify that the real atmosphere is not always in that physical state of fully developed 3-D turbulence which is described by the Kolmogorov theory. Strong evidence of deviations from the Kolmogorov model of turbulent spectrum have been frequently detected by astronomical observations, for example, with optical interferometers (Bester et al. 1992; Buscher et al. 1995).
In the present paper, we study differential image motion suggesting a non-Kolmogorov type of the turbulence, and compare results with those known for a classical case. Description of differential displacements of star images is based on the model of a phase screen representing statistical properties of the wavefront. Spatial power spectral density of phase fluctuations is modeled by a power law with a spectral slope which depends on the type of turbulence in a layer. Tentative limits of the spectral index variations are estimated. A spectral approach to the problem allows one to take into account various cases of star relative positions and directions of image centroid measurements by introducing or modifying some necessary filters. Further, we derive expressions for the variance of differential image motion, either averaged, or non-averaged with respect to the model spatial parameters, and discuss the difference in performances of a filled single-aperture telescope and interferometer in very narrow fields.
Direct and remote sensing of an atmosphere shows that intensive clear air turbulence exists only in a discrete set of narrow turbulent layers isolated from each other by laminar flows. A typical depth of turbulent layer is much less than its height h above the surface, and is in the limit of 10-100 m (Barletti et al. 1977; Redfern 1991) to 100-800 m (Vinnichenko et al. 1976). According to Chunchuzov (1996), existence of the layered structure of turbulence is associated with features of a vertical profile of the Brunt-Vaisala frequency N. He also argues that near local maxima in the frequency N, vertical profile gravitational internal waves can create thin horizontal layers of a turbulence whose positions are independent of the type of internal wave source.
Small values of
allow one to use the model of a thin phase
screen (Goodman 1985) for each turbulent layer. In the model,
a layer thickness and extinction of light transmitting through a layer
are neglected, only change of a phase is
considered. A plane wave entering the screen gains
some random phase
on exit, where x, y are
rectangular coordinates of a point in the screen plane.
Though orientation of the coordinate frame x, y is arbitrary,
we assume for definiteness that its axes are parallel to
the axes of the equatorial coordinate system. By
statistical properties,
corresponds to a
homogeneous and isotropic random field, so 2-D
power spectral density
of a phase has circular
symmetry:
.
Here u, v are spatial frequencies corresponding
to x, y, and
is a circular frequency.
For the function
with a fairly complicated
dependence on q (Sect. 3) we use approximation by a simple function
(1) |
The factor
in (1) is determined by the intensity of turbulent
motions and is proportional to the coefficient C^{2}_{n} of the structure
function of refractive index n fluctations
D_{n}(r)=C^{2}_{n}r^{p} , | (2) |
(3) |
(4) |
(5) |
(6) |
(7) |
Note that at greater scales isotropy is not maintained (Sect. 3) and Eqs. (4) and (6) are not valid.
In addition to the already introduced parameters, the model (1) is supplemented with data on module V and the angle that wind velocity vector forms with the x-axis.
The spatial spectrum of phase fluctuations, besides being directly related to that of temperature t fluctuations in a turbulent layer, depends also on localization of a turbulence which can be either 3-D or 2-D, concentrated in a narrow horizontal quasi-2-D space.
To begin, let's refer to results of meteorological studies of a turbulent temperature spectrum under clear sky conditions on scales 1-10^{5} m, of interest for this study.
By the data of airborne measurements in the troposphere and lower stratosphere (Vinnichenko et al. 1976), a 1-D temperature spectrum F'_{t} (q) in a low-frequency spectral range usually follows a dependence which in the higher frequency range is transformed to . The position of the frequency q_{B} separating these spectral ranges is not permanent and depends on stratification. In diagrams presented by Vinnichenko et al. (1976), q_{B} corresponds to scales of 200-1000 m. Spectral slopes intermediate between -5/3 and -3 are also frequently found, and sometimes spectra of unregular shapes with local maxima and minima are detected.
Observed features of the function F'_{t} (q) in general are consistent with modern theories of atmospheric turbulence. According to Dalaudier & Sidi (1987), in low frequencies (buoyancy range) temperature fluctuations are caused by vertical displacements of air particles in a stably stratified medium, some part of kinetic energy is converted here to the potential form. A spectral flux of potential energy (temperature) in the buoyancy subrange is directed towards larger scales. In this subrange both F'_{t} (q) and the kinetic energy spectrum follow a dependency . In higher frequencies (passive range of temperature fluctuations corresponding to an inertial range of kinetic energy) temperature fluctuations are converted to heat in the usual way described by the Kolmogorov theory. Here the spectral slope is -5/3. Dalaudier & Sidi (1987) theoretically have substantiated a possibility of the so-called "spectral gap" occurrence nearby q_{B}, with an adjacent peak at higher frequency. The slope of the spectral curve to the right of the gap is much less than -5/3. Weinstock (1980) suggested some other mechanisms leading to the formation of similar gaps in the kinetic energy spectrum on scales of 30-1000 m.
spectral | spatial form of | structure | functions | power | spectrum | ||
spatial scales | subrange | the turbulence | n and t | n and t | p | ||
from 1 m to | |||||||
10-10^{3} m | passive | 3-D | r^{2/3} | r^{5/3} | q^{-5/3} | q^{-11/3} | +2/3 |
from 10-10^{2} m | |||||||
to 10^{2}-10^{4} m | passive | 2-D | r^{2/3} | r^{2/3} | q^{-5/3} | q^{-8/3} | -1/3 |
from 10^{2}-10^{3} m | |||||||
to 10^{5} m and more | buoyancy | 2-D | r^{2} | r^{2} | q^{-3} | q^{-4} | +1 |
In Table 1, typical scales for buoyancy and passive subranges of turbulence, corresponding theoretical power laws for function F'_{t} (q) and structure functions of t and n quantities are given. It should be noted that "theoretical" power laws are deduced under the classical assumption of no additional sources and sinks of energy (wind shear, gravitational waves), and therefore are rather approximate.
The slopes of observed spectral functions F'_{t} (q) frequently and in a random way deviate from the "theoretical" ones. This was noticed, for example, by Bull (1967) who studied temporal spectra of n index measured with a microwave refractometer. While an average spectral slope was -1.68 at scales 2-100 m (temporal frequencies transformed into spatial scales with V=5 m/s), which is very close to the Kolmogorov's -5/3, sampled values of the spectral index varied from -1.2 to -2.5. Additional, modifying influences on the theoretically predicted temperature spectrum were affected by internal gravitational waves (Hogstrom et al. 1998).
The indicated details of the function F'_{t} (q) shape directly affect the phase power spectral density . Function , besides, depends on the type of spatial (2-D or 3-D) localization of the turbulent temperature field, that is, on the dimensionality of space where the temperature fluctuations can be considered as isotropic. On small scales not exceeding the thickness of a turbulent layer or outer scale of turbulence L (these value estimates range from 3-5 m (Coulman et al. 1988), to 30-100 m (Busher et al. 1995) and to 100-1000 m (Vinnichenko et al. 1976), fluctuations are 3-D and basic functions of a phase are determined by expressions (1) and (4). Therefore, with a "theoretical" spectrum we have here (first line in Table 1) , and p=2/3.
On scales and the turbulence is essentially 2-D. The power spectral densities and structure functions of n and t are described here (in a horizontal plane) by the same power laws as for 3-D turbulence. Corresponding functions for a phase, however, are modified. Really, on scales one can neglect changes of n and t across a layer. Accordingly, a phase difference between two points of the screen and is proportional to the difference , hence the phase structure function is proportional to the structure functions of n and t. So, in a passive range , while in the buoyancy range is expected (Table 1). Now, using Eq. (5) it is easy to find spectral densities of a phase (buoyancy range) and (passive range) corresponding to these functions. Two values of parameter p for 2-D turbulence are therefore valid: p= + 1 at very large scales q^{-1}>q_{B}^{-1}, and p=-1/3 for scales shorter than q_{B}^{-1}. Thus, the function relevant even to a very simplified theoretical description of a turbulence, on scales 1-10^{5} m depends on frequency in a rather complicated way.
Discussion of astronomical sources of data of the parameter p, in view of astronomical applicability of this analysis, is of a special value.
Diagrams of phase structure functions, temporal spectra, and Allan functions are very typical for path differences from a star to two telescopes of infrared interferometer with baselines 4 and 13 m (Bester et al. 1992). The peculiarity of these diagrams over spatial scales m (at wind velocity V=10 m/s corresponding to temporal scales 2-1000 s and frequencies 10-0.5 Hz) is in a strongly varying form of observed functions from sample to sample. For scales indicated, the parameter p ranged from 0.2 to 1.0 with an average p=0.5.
Similar results in the low-frequency spectral domain were obtained by Busher et al. (1995) with a MARK-III interferometer. Temporal spectra of phase differences at 0.001-0.1 Hz (scales 10^{2}-10^{4} m at V=10 m/s) had distinct deviations from the Kolmogorov type of the spectrum, displayed mainly as the decrease of p. In 16% of cases the flat spectra relevant to p=0 were obtained. In the high-frequency 3-40 Hz region (scales 0.25-3 m) the dispersion of individual values of p was smaller, 90% of which did not exceed the limits 0.4-0.7. On average p=0.55.
Coulman & Vernin (1991) analysed observations of stars with NRAO's Very Large Array of radio telescopes (Armstrong & Sramek 1982) at baselines from 50 m up to 22 km.
At moderate baselines 100-1200 m they found that represents both autumn and winter observations, and does so for a spring and summer period. On scales considered, the turbulence is intermediate between 2-D and 3-D, thus logarithmic slopes of the phase structure functions in the 2/3-5/3 range are expected (Table 1). Observed slopes, however, are consistent with the expected range only for the spring to summer observations. Corresponding mean values of p are -0.57 and +0.11 for the two periods.
It should be noted that the well-known Hog's (1968) model of the frequency spectra of absolute image motion based on observational data of various types has a parameter corresponding to p=0.5.
Lazorenko (1992), analyzing photographic and visual data on star image tracks obtained for site testing programmes, has shown that the shape of autocorrelation functions and power spectra of image motion is usually highly biased in consequence of a limited duration of observational series. The absolute image motion spectrum corrected for this effect should have a maximum at 0.01-0.015 Hz (scales 600-1000 m at V=10 m/s). It was found that p=0.97 at higher frequencies, and at lower frequencies p< 0, which agrees well with the data in Table 1.
This short review of meteorological, astronomical, and theoretical data allows us to make following conclusions:
1) the samplings of p have an intrinsic scatter caused by changes of physical conditions in the atmosphere,
2) on scales 1-300 m giving a main contribution to the variance of differential image motion, p varies at least from 1/3 up to 2/3, with a mean value of about 0.5,
3) on scales 200-1000 m the function has a smaller slope, with p decreasing sometimes to negative magnitudes.
The magnitude of differential displacement is a function of the phase screen parameters , p, h, Vand , and depends on location of the program object with respect to a reference frame. The direction along which the differences of coordinates of star image centroids are measured is also of importance. Taking into consideration a vast variety of possible configurations of reference stars, we shall examine only those which are most typical.
a) Two stars, located in the sky at angular distance . At some points A and B a phase screen is crossed by light beams passing from these stars to the centre of the telescope pupil. Here and further, point A refers to the target star while point B refers to the geometrical centre of the reference frame (in this instance, to star B). The line AB is the baseline of configuration.
b) Two reference stars B_{1} and B_{2}, seen at an angle . The program star A is on a common arc with reference stars shifted some angle off their middle point B. B_{1}B_{2} is the baseline of configuration.
c) A set of some reference stars, located uniformly in a circular area. In the limit stars completely fill in the area forming a circle of angular diameter with the target star A displaced some angle off the circle centre B.
Each configuration projected to the phase screen is characterized by its linear size , the distance between the geometrical centres of star systems (for the elementary configuration "a" S=S' and is assumed), and angle the baseline formed with the x-axis.
With light rays propagating perpendicularly to a wavefront,
the star image displacements
in the focal plane are proportional to the mean gradient of the
phase taken over the screen section involved
in image formation at exposure T. It should be noted that
axes of the coordinate
system ,
in reference to which positions and
displacements of stars are
measured, may not necessarily be parallel to the x,y axes fixed by
the celestial frame. The angle between the axes of these
coordinate frames is here denoted .
Differential displacement
in the direction of the
-axis
is equal to the difference of displacements of the
target star and a system of reference stars. An
efficient approach for studying statistical
properties of various parameters relevant to the turbulent
wave-front phase, and based on the use of
convolution operators, was described by Conan et al. (1995).
Using his formalism, it is easy to derive the following
expression which we give in polar coordinates
:
(8) |
- calculation of the phase gradient in the direction of the -axis rotated by from the x-axis;
- calculation of the running average over a straight 1-D
line VT oriented along the wind
(9) |
- averaging over the telescope aperture D
(10) |
- evaluation of a difference between two values of a function
in points A and B, which for configurations "a", "b",
and "c" is correspondingly:
(11) |
Expressions of the type (11) may be easily modified for other stellar configurations. Note that Eq. (11) for configuration "b" is valid even if the star A is not in line with reference stars.
The use of convolution operators in Eq. (8) allows direct
proceeding to the analysis in
the spectral domain. In this case the spatial power
spectral density
of
in polar coordinates
is expressed as a sequential
product of
and four filters
originating from the corresponding functions in Eq. (8)
and their Fourier transforms:
(12) |
(13) |
(14) |
The effect of averaging over is created in a natural way by fluctuations and changes with height of the wind direction. Though there are a few possible ways and sources of performing averaging of the variance (16) over angular parameters, further consideration shows that in some instances the resulting quantities follow a similar dependency on T and angular size , the details of reference star distribution in the field being of less importance. This Section deals with the variance which, owing to averaging, does not depend on angular parameters.
Let's consider a case of some randomly distributed pairs of stars
(configuration "a") with equal
baseline length .
The
direction along which the angular distance between stars of the pair is
measured usually either coincides with the direction of the baseline
(
),
is perpendicular to that of (
), or directed along
the x-axis ()
or y-axis (
).
We shall denote the relevant quantities of
as ,
,
,
and ,
and introduce their average
Figure 1: The function I(z) (solid), its approximation by Eq. (21) (long dashed), and the function (short dashed). | |
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Figure 2: Dependence of the variance on geometrical size of stellar configurations described in Sect. 4: a) (binary star), b) triple star, and c) (circular distribution of reference stars). Solid lines - filled circular aperture D=0.78 m; dashed lines - interferometer of equal baseline d. The case of a target star displaced off the center of reference frame (configuration c)) is shown by triangles. | |
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The values of h/V and V in (22) are approximately equal to the weight-average parameters used by Lindegren (1980) for differential effect evaluation at normal conditions of multi-layered atmosphere. In this sense parameters (22) represent "typical" atmospheric conditions, though they are used in this paper exceptionally for the purpose of qualitative interpretation of the expressions obtained. A normalizing coefficient is introduced in (22) implicitly, through . Such an approach to the calibration of in our opinion has some advantages. So, the quantity unlike has a clear astronomical sense and a rich history of determinations. Besides, is equal to the accuracy of meridian observations at T_{30} =30 s, and is known to be in the limits 0.12-0.14'', which follows also from the model of absolute motion power spectrum derived by Hog (1968). The use of instead of for calibration of the expression (19) is also more convenient, as in some cases it reduces the number of model parameters. For example, while two parameters h and V are used in Eq. (24), (25) contains only their ratio h/V which can be considered as a single parameter.
The expression relating
and is found from (19) in the limit
,
:
For a single-layer model of an atmosphere with parameters (22),
p=2/3, and ,
the use
of Eqs. (25) and (26)
allows one to find
,
where Tis expressed in seconds. At
arbitrary atmospheric parameters and
A similar power law of
dependence on geometric
size
of a stellar group is seen for configurations
"b" and "c" (Fig. 2). Calculations were
carried out using Eqs. (12) and (16)
with the same parameters at which the curve "a" was plotted,
with a central
position of the target star (), and with
averaging over two (one in the case "c") angles,
the function Q^{2}(q) taken from (15).
Both curves at wide
follow a power law
,
are parallel to, and below the curve "a". The most efficient,
of course, is filtration of phase distortions for configuration "c".
Equation (16) under the condition
in this case transformes to
Thus, a symmetric circular distribution of reference
stars, in comparison to one reference star,
ensures a decrease in the variance equivalent to a 5-9 times longer
exposure. If p=2/3, then
E(2/3)=0.8458, and
The use of configuration "b" with two reference stars
is also efficient. To find the relevant variance,
it is useful to take advantage of the similarity of the function Q^{2}(q)for configurations "a" and "b".
The comparison of relevant expressions in Eq. (15) allows us to write
The value of
is larger when the position of
a target star is displaced off the geometrical centre B of
reference frame. Estimation of this effect requires a term
with non-zero S' of Q^{2}(q) function (13) to be included into
expression (29). The expression obtained for S'<Sand
is easily transformed to
Differential motion for small has already been studied by Lindegren (1980) who found
We now derive analytical expressions for the filled aperture.
An approximate expression for
(configuration "a")
in small
domain is found
from Eq. (19). Assuming
for
,
and
for
,
we find an expression of the type (35)
Concerning configurations "b" and "c", the use of
Eq. (29) with approximation
at z>1,
or
results in
The power law in Eqs. (38)-(41) differs
from that in expression
(36) again by a factor S/D. Such a difference
is consistent with that in the second term of Eqs. (35)
and (37).
This may be explained using Lindegren's (1980) results.
Considering an arbitrary distribution of n stars, he derived
Using Eq. (44), and a filter (9) describing wind
(time) averaging, a power spectral density of image motion
is found
Evaluation of Eq. (46) for large separations S with
an accuracy to the leading term gives an
expression that exactly coincides with Eq. (24).
For very narrow angles
an expression obtained
Similarly, replacing a term
in Eq. (46)
by a proper function Q^{2}(q) allows us to obtain expressions
for other stellar configurations at small :
conf.-n | offset | interferometer, d | filled pupil, D | ||
type | 10 m | 100 m | 10 m | 100 m | |
"a" | - | 1820 | 390 | 1530 | 330 |
"b" | 0 | 180 | 18 | 70 | 2.3 |
"c" | 0 | 150 | 15 | 30 | 1.0 |
"b" | 15'' | 490 | 100 | 380 | 83 |
Table 2 shows the relative performance of an interferometer and
filled aperture of large d=D=10 m and very large d=D=100 m
size, ,
moderate T=100 s, Kolmogorov turbulence,
and atmospheric layer (22). Due to a strong dependence
of
on h, the data given are only illustrative.
Anyhow it clearly shows the great advantage of symmetric frames like
"b" and "c". To achieve
as level
with a D=100 m telescope, a target star, however, should be exactly at the
center of the reference frame. Analysis of Eq. (13) for a slightly non-central
position of the target star (0<S'<S) allows us to find approximation
Figure 3: Differential image motion of two stars spaced 10' as a function of at some fixed . Horizontal dashed line is the mean variance , Eq. (19). Atmospheric parameters are the same as in Fig. 2, p=2/3, and T=100 s. | |
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Let's consider measurements of coordinate differences for a pair of stars at fixed spatial angles, assuming a single-layer model of an atmosphere. Taking into account that integral (17) is a function of and differences, the dependence of on at fixed , 15, 45, and 90^{0} can be calculated. Figure 3 shows the functions calculated with , T=100 s, and parameters (22). Graphical data in the region are well fitted by a function , where is the variance at and some fixed . The horizontal dashed line marks a level of the mean variance in the sense of Eq. (19), with averaging over and . Figure 3 shows that at some particular orientations of a baseline ( parameter) and -axis ( angle) the level of can be much lower than . There are two cases of interest to be discussed:
1) The baseline AB connecting two stars formes a small angle with the wind direction, with arbitrary;
2) The -axis along which positions are measured is oriented approximately along the wind: , with arbitrary .
Condition (1) essentially limits the number of star pairs which can be measured in the field. For that reason the opportunity (2) that places no limitations on the orientation of baselines is of greater practical value. In the important case of configuration "c", with a central position of a target star, and angle varying over a set of reference stars, this opportunity is unique. Note that improvement in precision expected for -coordinates is achieved at the expense of an equivalent loss in the accuracy of -coordinates in the perpendicular direction. Such a feature actually causes transition to 1-D mode of measurements. However, it is not necessarily a shortcoming, as for example in parallax programs where only 1-D, high precision differential coordinates are practically of use.
The highest gain in precision for a single-layer
atmosphere is expected at
.
In this case evaluation of
the integral (17) averaged over in the limits for a small aperture
yields
For a real multi-layered atmosphere, the transition to measurements in a selected direction is rarely justified, and only under conditions of approximately parallel orientation of wind velocity vectors of all turbulent layers. In this case, to ensure best filtering of a dominant layer i=1 with the greatest index C^{2}_{n}, the -axis is positioned parallel to the vector thus ensuring . It does not however garantee the decrease of the total variance, which is achieved unconditionally only when for each layer i.
An important feature of atmospheric layer distribution in C^{2}_{n} is that at each moment of observations there are only a few layers which greatly dominate over others in the intensity of turbulence (Gendron & Lena 1996; Redfern 1991; Benkhaldoun et al. 1996). This circumstance brings up quasi-single-layer features to the atmospheric behaviour, and is favorable for 1-D techniques of measurements.
layer | h, | V, | , | |||
No. i | km | m/s | degr. | mas | mas | |
1 | 4 | 14 | 0 | 25.8 | 11.7 | 1.31 |
2 | 10 | 59 | -8 | 2.1 | 2.3 | 0.47 |
3 | 12 | 51 | -9 | 3.4 | 3.3 | 0.78 |
4 | 12 | 44 | -13 | 2.1 | 2.8 | 0.91 |
5 | 14 | 33 | -4 | 2.7 | 3.8 | 0.62 |
6 | 16 | 36 | 1 | 1.9 | 3.2 | 0.44 |
7 | 17 | 10 | 16 | 1.8 | 6.1 | 2.70 |
8 | 18 | 17 | 37 | 1.4 | 4.2 | 3.51 |
Caccia et al. (1987) | total: | 41 | 15.5 | 4.8 | ||
1 | 3 | 9 | -45 | 15 | 10.0 | 10.0 |
2 | 6 | 6 | 0 | 28 | 21.5 | 4.3 |
3 | 10 | 4 | 63 | 9 | 17.9 | 21.7 |
Rocca et al. (1974) | total: | 52 | 29.7 | 24.3 |
For discussion we shall consider some observational data. In the upper part of Table 3, Cols. 2-5, are the parameters of atmospheric layers h, V, , and found by Caccia et al. (1987) from spatiotemporal correlation analysis of single-star scintillations. Observations were obtained with the 1.93 m telescope of the Haute-Provence observatory. In the second part of Table 3, the data found by Rocca et al. (1974) from the analysis of star scintillations on a 10 cm telescope are presented. In Col. 6, values for each layer calculated with Eq. (24) at T=100 s, p=2/3, and averaged with respect to and , are given. Column 7 contains image motion measured in the direction of the vector (that is ), and also with averaging over . Total values of and integrated over the depth of an atmosphere are also given. The transformation of into was carried out using Eq. (3) which is valid for 3-D isotropic turbulence.
The vertical structure of an atmosphere which is shown in the first part of the table, at the optimal position of the -axis allows us to reduce the variance of image motion by a factor of 10. In this particular case for each layer. The wind situation described in the second part of Table 3 is much worse and is characterized by a strong change of the wind orientation with a height. As a result, the contribution of a layer i=3even encreases, and the total effect for the whole atmosphere becomes negligable.
Barletti gives an even more unfavorable example of the wind shifting 250^{0} at heights 13-17 km, where most layers of intense turbulence were concentrated. The vertical profile of the wind velocity given by Hogstrom et al. (1998) for heights 500 m-3 km, on the contrary, has very weak variations of the angle, being in the limits .
The above examples show that there is a certain possibility to come across a lucky meteorological situation allowing us to take advantage of anisotropic properties of differential image motion. Detection of atmospheric conditions favourable for a 1-D strategy of observations, however, implies a permanent control of atmospheric parameters using some remote methods of sensing with satisfactory precision of the vertical wind profile determination (Caccia et al. 1987; Rocca et al. 1974).
When the variance of differential image motion is calculated, calibration is performed with some measured quantity that characterizes the intensity of turbulence. This procedure is however biased because there are usually no reliable data on the p quantity available, so p=2/3 is set tentatively. Besides, we note that there are some quantities that specify the turbulence in a different way, by refering to either one-point or integral estimates of one of the phase-related functions , or D_{n}(r). Thus, (measured in units of m^{-p-1}) is numerically equal to the function at some unit frequency (here 1 m^{-1}) that depends on the adopted system of units; a similar remark concerns C^{2}_{n}. More frequently in use are integral estimates such as FWHM, Fried's parameter r_{0}, and rarely, .
Parameter | , all conf.-s; | , conf. "a" | |
, symm. conf.-s | |||
- | |||
- | |||
<0 | |||
r_{0} | >0 |
Let us estimate the bias in
caused by assuming p=2/3when r_{0} and
are used as normalizing factors.
To express r_{0} via
one may compare the expression for a
phase structure function
where
(Fried 1965)
with representation (5). Assuming that the turbulence is
completely developed, the use of Eqs. (4-7) yields
With typical , h=2-15 km, V=5-30 m/s, m, and long T we find that [S/(VT_{30})]<1 and [D/(VT_{30})]<1, therefore the derivative is always negative at any given . On the contrary, when using r_{0}, expressions for contain ratios S/r_{0}>1and D/r_{0}>1, so in this case . Considering that usually 1/3<p<2/3 (Sect. 3), we find , where is the actual value of , and is that calculated with p=2/3and measured . Using r_{0} results in inverse inequality .
Figure 4: The variance as a function of p for binary star "a", , T=100 s, . Calibration based on (circles) or (triangles). Solid lines: atmospheric layer h=2.8 km, V=14 m/s, r_{0}=83 mm (or ); dashed: h=15 km, V=30 m/s, r_{0}=71 mm (or ). | |
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In Table 5 we reproduce data of Table 2 for the non-Kolmogorov spectral index p=1/3 and the former value of r_{0}=83 mm. Comparison of Tables 2 and 5 shows that the largest change of image motion magnitude occures when double-star separations are measured. Owing to the dependence , the effect is stronger for large telescopes. Concerning symmetric stellar groups, one may note comparatively stable estimates of , especially for interferometers.
Figure 5: The same as in Fig. 4, for interferometer with a baseline d=10 m (dashed) and filled aperture D=10 m (solid), for triple star "b", , T=100 s; lower atmopheric layer is considered. | |
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In this Paper we derive necessary formulations and analyze
some effects which may be useful for
understanding the properties of differential image motion under
non-Kolmogorov distortions of the wave-front.
Non-classic statistics of the turbulent phase are due to three
effects:
non-Kolmogorov type of refractive index n fluctuations in
3-D space, 2-D spatial location of a turbulence, and
wind-dependent anisotropy of the phase. The two first sources
result in the decrease of p; the effect becomes stronger
at large scales. According to observational data, on scales
1-300 m which are effective for differential image motion, p is
typically in the range 1/3-2/3. Deviation of p from its
Kolmogorov value 2/3 affects all phase-related quantities,
in particular, causes divergence of the observed
value
from that calculated at p=2/3. The bias may exceed 50% or more,
depending on which quantity, r_{0} or
,
is taken
as a measure of the intensity of turbulence (Figs. 4 and 5), and which
stellar configuration is observed. Thus, when measuring double-star
separations,
is proportional to the factors
(D/r_{0})^{p} and
(d/r_{0})^{p} which practically do not depend on
p for small instruments. The situation
is quite different for the Keck 10 m telescope and some operating
interferometers with
m, when small 1/3 to 2/3 variations
of p evoke a change of 5-10 in ,
better accuracies
corresponding to small p.
conf.-n | offset | interferometer, d | filled pupil, D | ||
type | 10 m | 100 m | 10 m | 100 m | |
"a" | - | 1230 | 180 | 850 | 120 |
"b" | 0 | 160 | 16 | 48 | 1.5 |
"c" | 0 | 150 | 15 | 25 | 0.8 |
"b" | 15'' | 350 | 48 | 220 | 30 |
In brief, we analyzed a third factor of non-Kolmogorov distortions of the wave-front, a spatial anisotropy of the phase brought in by the wind. We concluded that at certain conditions it may be useful for improving the accuracy of ground-based observations. Examples given in the upper part of Table 3 show that the use of a special 1-D strategy of measurements which takes advantage of anisotropic properties of the turbulent phase may result in a perceptible reduction of .
Rather unexpected results have been obtained when studying the peformance of a classic circular pupil. Expressions derived for do not follow power laws predicted by Lindegren (1980) as his analysis is directly valid for interferometers. For symmetric stellar groups ("b" or "c") measured in very narrow fields, a new power law derived differs by a factor of S/D from Lindegren's dependence . This circumstance results in an order smaller, as compared to current, estimates of for observations with very large m apertures and small angles (Tables 2 and 5).
High-accuracy differential astrometry is required primarily to study double-stars, determination of parallaxes, and in searching for extrasolar planets. These problems can be solved very effectively with future astrometric satellites that will have an expected accuracy of about 1-10 as. The potential of ground-based astronomy also looks rather promising, considering projects of large-scale facilities. The analysis of atmospheric effects made in this Paper allows us to conclude that this source of errors can be reduced approximately to the level expected for space missions, or 1 as per 100 s exposure.
The largest operating 10 m Keck telescope has a potential narrow-field accuracy of 30-70 as per 100 s exposure at the Kolmogorov parameter p (Table 2), and 20-50 as at p=1/3 (Table 5). Due to a strong improvemnt with D, the 1 as per 100 s time level is already expected at D=100 m. For interferometers, baselines of about 1000 m are required because of the slower improvement of . Interferometers of this class should be able to measure simultaneously at least three objects so as to detect a minimal symmetric configuration of stars. Interferometers of the Mark III type with two separate beam channels are restricted to observations of asymmetric double-star systems, and even at d=1000 m baselines will be limited by as (Sect. 5.3).
The above estimates relate to the dimeter of the reference group; in other cases correction allowing for dependences (filled apertures) and (interferometers) should be applied. Very high accuracies, however, require perfect centering of the reference frame on the target object. Even very small offsets will generate, depending on p, errors of 2-5 as for a D=10 m telescope (Sect. 5.3), and about 0.5-1 as with D=100 m. Errors of this type are completely eliminated with the symmetrizing procedure (Sect. 5.4). An extra 1.5-2 fold improvement is expected at astronomical sites with average on spatial scales 1-300 m.