Abstract

A simple differential analysis of stellar scintillations measured simultaneously with two apertures opens the possibility to estimate seeing. Moreover, some information on the vertical turbulence distribution can be obtained. A general expression for the differential scintillation index for apertures of arbitrary shape and for finite exposure time is derived, and its applications are studied. Correction for exposure time bias by use of the ratio of scintillation indices with and without time binning is studied. A bandpass-filtered scintillation in a small aperture (computed as the differential-exposure index) provides a reasonably good estimate of the atmospheric time constant for adaptive optics.

© 2002 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).
  2. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  3. M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).
  4. A. Fuchs, M. Tallon, J. Vernin, “Focusing on a turbulent layer: principle of the generalized SCIDAR,” Publ. Astron. Soc. Pac. 110, 86–91 (1998).
    [CrossRef]
  5. A. A. Tokovinin, “Study of single-star turbulence profilers,” ESO Report VLT-TRE-UNI-17416-0008 (European Southern Observatory, Garching, Germany, 1998).
  6. G. R. Ochs, Ting-i Wang, R. S. Lawrence, S. F. Clifford, “Refractive-turbulence profiles measured by one-dimensional spatial filtering of scintillations,” Appl. Opt. 15, 2504–2510 (1976).
    [CrossRef] [PubMed]
  7. J. Krause-Polstorf, E. A. Murphy, D. L. Walters, “Instrumental comparison: corrected stellar scintillometer versus isoplanometer,” Appl. Opt. 32, 4051–4057 (1993).
  8. A. Ziad, R. Conan, A. Tokovinin, F. Martin, J. Borgnino, “From the grating scale monitor to the generalized seeing monitor,” Appl. Opt. 39, 5415–5425 (2000).
    [CrossRef]
  9. A. A. Tokovinin, “A new method to measure the atmospheric seeing,” Pis’ma Astron. Zh. 24, 768–771 (1998) [Astron. Lett. 24, 662–664 (1998)].
  10. V. G. Kornilov, A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Maidanak,” Astron. Rep. 45, 395–408 (2001).
    [CrossRef]
  11. A. T. Young, “Aperture filtering and saturation of scintillations,” J. Opt. Soc. Am. 60, 248–250 (1970).
    [CrossRef]
  12. A referee has pointed out that ℐ1(ζ) = 1F2(1/2; 3/2; 2; -π2ζ2), where 1F2 is the generalized hypergeometric function; see, for example, I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, London, 1980), Eq. 9.14.1, p. 1045. We verified that our numerical computation agrees with the series expression for 1F2 in the domain where the series converge.
  13. J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign. Cerro Pachon and Cerro Tololo,” Gemini Report RPT-AO-G0094, http://www.gemini.edu/ (2000).
  14. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, Cambridge, England, 1999), p. 15.

2001 (1)

V. G. Kornilov, A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

2000 (1)

1998 (2)

A. A. Tokovinin, “A new method to measure the atmospheric seeing,” Pis’ma Astron. Zh. 24, 768–771 (1998) [Astron. Lett. 24, 662–664 (1998)].

A. Fuchs, M. Tallon, J. Vernin, “Focusing on a turbulent layer: principle of the generalized SCIDAR,” Publ. Astron. Soc. Pac. 110, 86–91 (1998).
[CrossRef]

1993 (1)

1990 (1)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

1976 (1)

1970 (1)

Borgnino, J.

Clifford, S. F.

Conan, R.

Fuchs, A.

A. Fuchs, M. Tallon, J. Vernin, “Focusing on a turbulent layer: principle of the generalized SCIDAR,” Publ. Astron. Soc. Pac. 110, 86–91 (1998).
[CrossRef]

Gradshteyn, I. S.

A referee has pointed out that ℐ1(ζ) = 1F2(1/2; 3/2; 2; -π2ζ2), where 1F2 is the generalized hypergeometric function; see, for example, I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, London, 1980), Eq. 9.14.1, p. 1045. We verified that our numerical computation agrees with the series expression for 1F2 in the domain where the series converge.

Kornilov, V. G.

V. G. Kornilov, A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

Krause-Polstorf, J.

Lawrence, R. S.

Martin, F.

Murphy, E. A.

Ochs, G. R.

Roddier, F.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, Cambridge, England, 1999), p. 15.

Ryzhik, I. M.

A referee has pointed out that ℐ1(ζ) = 1F2(1/2; 3/2; 2; -π2ζ2), where 1F2 is the generalized hypergeometric function; see, for example, I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, London, 1980), Eq. 9.14.1, p. 1045. We verified that our numerical computation agrees with the series expression for 1F2 in the domain where the series converge.

Sarazin, M.

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Tallon, M.

A. Fuchs, M. Tallon, J. Vernin, “Focusing on a turbulent layer: principle of the generalized SCIDAR,” Publ. Astron. Soc. Pac. 110, 86–91 (1998).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

Tokovinin, A.

Tokovinin, A. A.

V. G. Kornilov, A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

A. A. Tokovinin, “A new method to measure the atmospheric seeing,” Pis’ma Astron. Zh. 24, 768–771 (1998) [Astron. Lett. 24, 662–664 (1998)].

A. A. Tokovinin, “Study of single-star turbulence profilers,” ESO Report VLT-TRE-UNI-17416-0008 (European Southern Observatory, Garching, Germany, 1998).

Vernin, J.

A. Fuchs, M. Tallon, J. Vernin, “Focusing on a turbulent layer: principle of the generalized SCIDAR,” Publ. Astron. Soc. Pac. 110, 86–91 (1998).
[CrossRef]

Walters, D. L.

Wang, Ting-i

Young, A. T.

Ziad, A.

Appl. Opt. (3)

Astron. Astrophys. (1)

M. Sarazin, F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294–300 (1990).

Astron. Rep. (1)

V. G. Kornilov, A. A. Tokovinin, “Measurement of the turbulence in the free atmosphere above Maidanak,” Astron. Rep. 45, 395–408 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Pis’ma Astron. Zh. (1)

A. A. Tokovinin, “A new method to measure the atmospheric seeing,” Pis’ma Astron. Zh. 24, 768–771 (1998) [Astron. Lett. 24, 662–664 (1998)].

Publ. Astron. Soc. Pac. (1)

A. Fuchs, M. Tallon, J. Vernin, “Focusing on a turbulent layer: principle of the generalized SCIDAR,” Publ. Astron. Soc. Pac. 110, 86–91 (1998).
[CrossRef]

Other (6)

A. A. Tokovinin, “Study of single-star turbulence profilers,” ESO Report VLT-TRE-UNI-17416-0008 (European Southern Observatory, Garching, Germany, 1998).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
[CrossRef]

A referee has pointed out that ℐ1(ζ) = 1F2(1/2; 3/2; 2; -π2ζ2), where 1F2 is the generalized hypergeometric function; see, for example, I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, London, 1980), Eq. 9.14.1, p. 1045. We verified that our numerical computation agrees with the series expression for 1F2 in the domain where the series converge.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign. Cerro Pachon and Cerro Tololo,” Gemini Report RPT-AO-G0094, http://www.gemini.edu/ (2000).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, Cambridge, England, 1999), p. 15.

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Figures (8)

Fig. 1
Fig. 1

DSI WFs plotted for three configurations of the two apertures. Calculations are made for λ = 500 nm, ∊ = 0.5, and aperture size d of 16, 8, and 4 cm. Top, circular concentric apertures; middle, WFs for two clear circular apertures with d = 4 cm, ∊ = 0.5, which are concentric (dashed curve) or decentered by Δ = 0.1d; bottom, square concentric apertures.

Fig. 2
Fig. 2

WF for the DSI is given by the integral of a three-term product over spatial frequency [Eq. (3)]. The product of turbulence spectrum f -8/3 and Fresnel factor sin2(πλzf 2) is plotted for layer altitudes of 4 km (solid curve) and 6 km (dotted curve). The aperture filter for differential scintillations with 4- and 2-cm apertures (dashed curve) cuts off low frequencies and averages the oscillations, leading to an altitude-independent WF.

Fig. 3
Fig. 3

SI measurement in the generalized SCIDAR mode. The aperture, instead of being placed in the exit pupil (H = 0), is optically conjugated to some other altitude H. A turbulent layer gives no scintillation if aperture is conjugated to it, but gives increased scintillation if H < 0, as though an extra propagation path were added. The aperture must be smaller than the exit pupil to avoid diffraction at the pupil edge.

Fig. 4
Fig. 4

Exposure time bias R(τ, z) [Eq. (16)] for normal and differential scintillation as a function of single-layer wind velocity (solid curves). The suggested bias corrections are plotted as dotted (correction 1) and dashed (correction 2) curves. Top, normal SI, R(τ) for a 10-cm aperture, a layer at z = 15 km, and exposure times of 5 and 10 ms. Bottom, R(τ, z) for DSI with 4- and 2-cm apertures, layer at z = 5 km, exposure times of 1 and 2 ms. The bias depends only on the Vτ product, hence the curves for different exposure times can be obtained from each other by horizontal stretching.

Fig. 5
Fig. 5

For true turbulence and wind profile with 0.5-km resolution, the relative contribution of layers to the DSI as measured with 4- and 2-cm apertures is plotted as a solid curve. This contribution is proportional to C n 2(z)Q(z). The dashed curve represents the reduced contribution with a 1-ms exposure time. Exposure time bias R (1 ms) is plotted as a dotted curve.

Fig. 6
Fig. 6

WFs for the DESI are plotted versus wind speed for exposure times of 1 and 3 ms (solid curve) and 0.5 and 1.5 ms (dashed curve). A single layer at 10 km and a circular aperture of 1-cm diameter are assumed. The dotted curve represents the V 5/3 law.

Fig. 7
Fig. 7

WFs for the DESI: exposure times of 1 and 3 ms; 1-cm aperture diameter; and wind speeds of 40 m/s (solid curve), 20 m/s (dashed curve), and 10 m/s (dotted curve).

Fig. 8
Fig. 8

Atmospheric time constant τ a compared with its estimate τde for the five real turbulence profiles measured by balloons at Cerro Pachon on nights with different conditions (asterisks). We processed the same profiles by increasing or decreasing the wind velocities by two times (pluses and diamonds, respectively).

Tables (1)

Tables Icon

Table 1 DSI Exposure Time Bias for Five Profiles at Cerro Pachon

Equations (22)

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σI2ln I-ln I2,
σI2=0ZmaxdzCn2zQz,
Qz=9.62λ-20d ff-8/3 sin2πλzf2Af.
Af=2π-102πdϕ|W˜f, ϕ|2.
Acircf=2J1πdfπdf2,
σd2=lnI1I2-lnI1I22.
Adf=2π-102πdϕ|W˜1f, ϕ-W˜2f, ϕ|2,
W˜1f=11-22J1πdfπdf-22J1πdfπdf,W˜2f=2J1πdfπdf,
Adf=1-2-22J1πdfπdf-2J1πdfπdf2.
Adf=W˜12f+W˜22f-2W˜1fW˜2fJ02πΔf.
W˜1f, ϕ=1-2-1sincdf cos ϕsincdf sin ϕ-2 sincdf cos ϕsincdf sin ϕ,W˜2f, ϕ=sincdf cos ϕsincdf sin ϕ,
Adf=1-2-22π-102πdϕ×sincdf cos ϕsincdf sin ϕ-sincdf cos ϕsincdf sin ϕ2.
W˜τfx, fy=sincfxVτ.
Aτf=2π-102πdϕ sinc2fVτ cos ϕ=1Vτf,
1ζ=2π-102πdϕ sinc2ζ cos ϕ.
Rτ, z=σI2τ, z/σI20, z=Qτz/Qz
σ2τ=02σ2τ-σ22τ=σ2τ2-r.
σ2τ=0σ2τ/1.4-0.4/r.
τa=0.314r0/V¯=0.057λ6/5×0ZmaxdzCn2zV5/3z-3/5.
σde2=lnIτ1Iτ2-lnIτ1Iτ22.
2ζ=2π-102πdϕsinc3ζ cos ϕ-sincζ cos ϕ2.
τde=Kσde2-3/5.

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