Abstract

We have measured power spectra of atmospheric phase fluctuations with the Mark III stellar interferometer on Mt. Wilson under a wide variety of seeing conditions. On almost all nights, the high-frequency portions of the temporal power spectra closely follow the form predicted by the standard Kolmogorov–Tatarski model. At lower frequencies, a variety of behavior is observed. On a few nights, the spectra clearly exhibit the low-frequency flattening characteristic of turbulence with an outer-scale length of the order of 30 m. On other nights, examination of individual spectra yields no strong evidence of an outer scale less than a few kilometers in size, but comparison of the spectra on different interferometer baselines shows a saturation of the spatial structure function on long baselines. This saturation is consistent with the assumption of an outer-scale length similar to that derived for the nights when low-frequency flattening of the spectra is clearly seen. We discuss possible explanations of this behavior and conclude that power spectra from a single interferometer baseline are a poor diagnostic for the effective outer scale compared with multiple-baseline spectra.

© 1995 Optical Society of America

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References

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  1. C. E. Coulman, J. Vernin, Y. Coqueugniot, J. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
    [CrossRef] [PubMed]
  2. N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155 (1991).
  3. M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Ann. Rev. Astron. Astrophys. 30, 457–498 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. F. Roddier, J. E. Graves, “Seeing monitor based on curvature sensing,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 474–479 (1990).
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    [CrossRef]
  8. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961).
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  10. D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Tech. Rep. RADC-TR-74-19 (Rome Air Development Center, Griffiss Air Force Base, New York, 1974).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  19. M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
    [CrossRef]
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    [CrossRef]
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  23. G. D’Auria, F. S. Marzano, U. Merlo, “Model for estimating the refractive-index structure constant in clear-air intermittent turbulence,” Appl. Opt. 32, 2674–2680 (1993).
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  24. C. R. Masson, “Seeing,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds., Int. Astron. Union Symp.158, 1–9 (1993).
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1993 (2)

1992 (3)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

T. S. McKechnie, “Atmospheric turbulence and the resolution limits of large ground-based telescopes,” J. Opt. Soc. Am. A 9, 1937–1954 (1992).
[CrossRef]

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Ann. Rev. Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

1991 (2)

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155 (1991).

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

1990 (1)

J. Dainty, M. J. Northcott, D.-N. Qu, “Measurements of the temporal correlation of images at La Palma,” J. Mod. Opt. 37, 1247–1254 (1990).
[CrossRef]

1988 (3)

M. M. Colavita, M. Shao, “Atmospheric phase measurements with the Mark III stellar interferometer,” Appl. Opt. 26, 4106–4112 (1988).
[CrossRef]

C. E. Coulman, J. Vernin, Y. Coqueugniot, J. Caccia, “Outer scale of turbulence appropriate to modeling refractive-index structure profiles,” Appl. Opt. 27, 155–160 (1988).
[CrossRef] [PubMed]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

1987 (1)

1985 (1)

C. E. Coulman, “Fundamental and applied aspects of astronomical seeing,” Ann. Rev. Astron. Astrophys. 23, 19–57 (1985).
[CrossRef]

1978 (1)

1973 (1)

1971 (1)

Ackley, M. H.

Azouit, M.

Bester, M.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Bouricius, G. M. B.

Bowers, P. F.

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

Bufton, J. L.

Buscher, D. F.

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155 (1991).

Caccia, J.

Caccia, J. L.

Clifford, S. F.

Colavita, M. M.

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Ann. Rev. Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

M. M. Colavita, M. Shao, “Atmospheric phase measurements with the Mark III stellar interferometer,” Appl. Opt. 26, 4106–4112 (1988).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

M. M. Colavita, “Atmospheric limitations of a two-color astrometric interferometer,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

Coqueugniot, Y.

Coulman, C. E.

D’Auria, G.

Dainty, J.

J. Dainty, M. J. Northcott, D.-N. Qu, “Measurements of the temporal correlation of images at La Palma,” J. Mod. Opt. 37, 1247–1254 (1990).
[CrossRef]

Danchi, W. C.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Degiacomi, C. G.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Di Benedetto, G. P.

J. M. Mariotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer,” in Very Large Telescopes, Their Instrumentation and Programs, M. H. Ulrich, K. Jaer, eds., Int. Astron. Union Colloq.79, 257–265 (1984).

Dutton, J. A.

H. A. Panofsky, J. A. Dutton, “Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley, New York, 1984).

Gaume, R.

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

Graves, J. E.

F. Roddier, J. E. Graves, “Seeing monitor based on curvature sensing,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 474–479 (1990).

Greenhill, L. J.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Greenwood, D. P.

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Tech. Rep. RADC-TR-74-19 (Rome Air Development Center, Griffiss Air Force Base, New York, 1974).

Hershey, J. L.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Hines, B. E.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Hughes, J. A.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Hutter, D. J.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

Johnston, K. J.

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Kaplan, G. H.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Mariotti, J. M.

J. M. Mariotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer,” in Very Large Telescopes, Their Instrumentation and Programs, M. H. Ulrich, K. Jaer, eds., Int. Astron. Union Colloq.79, 257–265 (1984).

Marzano, F. S.

Masson, C. R.

C. R. Masson, “Seeing,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds., Int. Astron. Union Symp.158, 1–9 (1993).

McKechnie, T. S.

Merlo, U.

Mozurkewich, D.

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Nightingale, N. S.

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155 (1991).

Northcott, M. J.

J. Dainty, M. J. Northcott, D.-N. Qu, “Measurements of the temporal correlation of images at La Palma,” J. Mod. Opt. 37, 1247–1254 (1990).
[CrossRef]

Ochs, G. R.

Panofsky, H. A.

H. A. Panofsky, J. A. Dutton, “Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley, New York, 1984).

Qu, D.-N.

J. Dainty, M. J. Northcott, D.-N. Qu, “Measurements of the temporal correlation of images at La Palma,” J. Mod. Opt. 37, 1247–1254 (1990).
[CrossRef]

Roddier, F.

F. Roddier, J. E. Graves, “Seeing monitor based on curvature sensing,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 474–479 (1990).

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

Scaddan, R. J.

Shao, M.

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Ann. Rev. Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

M. M. Colavita, M. Shao, “Atmospheric phase measurements with the Mark III stellar interferometer,” Appl. Opt. 26, 4106–4112 (1988).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Simon, R. S.

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Staelin, D. H.

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Tarazano, D. O.

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Tech. Rep. RADC-TR-74-19 (Rome Air Development Center, Griffiss Air Force Base, New York, 1974).

Tatarski, V. I.

Townes, C. H.

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

Vernin, J.

Walker, J. G.

Zavorotny, V. U.

Ann. Rev. Astron. Astrophys. (2)

M. Shao, M. M. Colavita, “Long-baseline optical and infrared stellar interferometry,” Ann. Rev. Astron. Astrophys. 30, 457–498 (1992).
[CrossRef]

C. E. Coulman, “Fundamental and applied aspects of astronomical seeing,” Ann. Rev. Astron. Astrophys. 23, 19–57 (1985).
[CrossRef]

Appl. Opt. (6)

Astron. Astrophys. (1)

M. Shao, M. M. Colavita, B. E. Hines, D. H. Staelin, D. J. Hutter, K. J. Johnston, D. Mozurkewich, R. S. Simon, J. L. Hershey, J. A. Hughes, G. H. Kaplan, “The Mark III stellar interferometer,” Astron. Astrophys. 193, 357–371 (1988).

Astron. J. (1)

D. Mozurkewich, K. J. Johnston, R. S. Simon, P. F. Bowers, R. Gaume, “Angular diameter measurements of stars,” Astron. J. 101, 2207–2219 (1991).
[CrossRef]

Astrophys. J. (1)

M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, C. H. Townes, “Atmospheric fluctuations: empirical structure functions and projected performance of future instruments,” Astrophys. J. 392, 357–374 (1992).
[CrossRef]

J. Mod. Opt. (1)

J. Dainty, M. J. Northcott, D.-N. Qu, “Measurements of the temporal correlation of images at La Palma,” J. Mod. Opt. 37, 1247–1254 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Mon. Not. R. Astron. Soc. (1)

N. S. Nightingale, D. F. Buscher, “Interferometric seeing measurements at the La Palma Observatory,” Mon. Not. R. Astron. Soc. 251, 155 (1991).

Other (9)

H. A. Panofsky, J. A. Dutton, “Atmospheric Turbulence: Models and Methods for Engineering Applications (Wiley, New York, 1984).

J. M. Mariotti, G. P. Di Benedetto, “Pathlength stability of synthetic aperture telescopes: the case of the 25 cm CERGA interferometer,” in Very Large Telescopes, Their Instrumentation and Programs, M. H. Ulrich, K. Jaer, eds., Int. Astron. Union Colloq.79, 257–265 (1984).

F. Roddier, J. E. Graves, “Seeing monitor based on curvature sensing,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1236, 474–479 (1990).

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 2.

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Tech. Rep. RADC-TR-74-19 (Rome Air Development Center, Griffiss Air Force Base, New York, 1974).

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

C. R. Masson, “Seeing,” in Very High Angular Resolution Imaging, J. G. Robertson, W. J. Tango, eds., Int. Astron. Union Symp.158, 1–9 (1993).

M. M. Colavita, “Atmospheric limitations of a two-color astrometric interferometer,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

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Figures (14)

Fig. 1
Fig. 1

Spatial structure functions of the wave-front perturbations for various turbulence models. Solid line, infinite-outer-scale model; dashed curve, von Karman model with an outer scale of 10 m; dashed–dotted curve, Greenwood–Tarazano model with the same outer scale.

Fig. 2
Fig. 2

Theoretical power spectra of the phase fluctuations in an interferometer. The solid curve is for Kolmogorov turbulence with an infinite outer scale. The dashed and the dotted curves are for spectra with outer scales of the magnitude of 100, 10, 1, 0.1, and 0.01 times the interferometer baseline; dashed curves, von Karman spectrum in the input range; dotted curves, Greenwood–Tarazano spectrum. The nominal values of the system parameters used to set the vertical and the horizontal scales were r 0 = B = 1 m, λ0 = 500 nm, and V/B = 1 Hz. P.S.D., power spectral density.

Fig. 3
Fig. 3

Schematic of the simplified power spectrum with a low-frequency cutoff f 0. The hatched area indicates the region that contains the neglected fraction of the power. (Clearly, the whole region cannot be shown on a log–log plot, because the true region extends to zero in both coordinates.)

Fig. 4
Fig. 4

Power spectrum of the internal seeing fluctuations on the Mark III interferometer. The spectra were taken (a) on the 12-m north–south astrometric baseline on the night of 5 March 1991, (b) on the 31.5-m imaging baseline on the night of 19 July 1991.

Fig. 5
Fig. 5

Two power spectra of stellar fringe motion taken by the Mark III interferometer. The upper solid curve is from 20 February 1991 and was measured on the 31.4-m baseline. The lower solid curve is from 3 July 1989 and was measured on the 12-m north–south astrometric baseline. The dashed curves are fitted atmospheric models with parameters t 0 = 5.5 ms, r 0 = 14 cm and t 0 = 12.3 ms, r 0 = 57 cm, respectively.

Fig. 6
Fig. 6

Histogram of the spectral index of the high-frequency temporal spectrum of fringe motion on the June–July 1989 nights.

Fig. 7
Fig. 7

Power spectrum of the atmospheric phase fluctuations measured on the 31.5-m baseline on 3 July 1989 (solid curve). The dashed curve corresponds to that predicted from an infinite-outer-scale model with t 0 = 17.4 ms and r 0 = 1.2 m.

Fig. 8
Fig. 8

Plots of the derived value of r 0 for (a) the June–July 1989 data set, (b) the November 1992 data set. The r 0 values are derived with Eq. (8) and the values for the variance of the fringe motion in the bandpass, 2 × 10−3 to 8 Hz. Triangles, r 0 values from the 4-m baseline; squares, r 0 values from the 12-m baseline; circles, r 0 values from the 31.5-m baseline. It should be noted that these are only nominal r 0 values because they take into account neither the possibility of a finite outer scale nor the bias that is due to the finite bandpass (see text).

Fig. 9
Fig. 9

Histograms of the two-point spatial structure function spectral index (see text) for (a) the June–July 1989 data set, where the baseline was switched between 12 and 31.5 m, (b) the November 1992 data set, where the baseline was switched between 4 and 12 m.

Fig. 10
Fig. 10

Two examples of stellar fringe-motion spectra showing distinct low-frequency flattening, characteristic of a finite outer scale. The power spectra are from the night of 28 June 1989 on (a) the 31.5-m baseline, (b) the 12-m baseline.

Fig. 11
Fig. 11

Fit of (a) a Greenwood–Tarazano model with an outer scale of 100 m, (b) an infinite-outer-scale Kolmogorov model to a power spectrum measured on the 31.5-m baseline on the night of 2 July 1989. The dashed curves show the best-fit curves, with parameters of (a) t 0 = 11 ms, V = 14 m/s, and r 0 = 47 cm, (b) t 0 = 11 ms, V = 24 m/s, and r 0 = 83 cm, respectively.

Fig. 12
Fig. 12

Fit of three turbulent layers, each with an outer scale of 30 m, to a stellar power spectrum that shows no low-frequency flattening (solid curve). The lower dashed–dotted and dotted curves show the spectra due to each of the layers individually, while the upper dashed curve is the sum of these spectra. The lower curves have been offset by 2 decades for clarity. The individual layers have values for t 0 of 17, 30, and 300 ms, wind speeds of 25, 4.4, and 0.31 m/s, and r 0 values of 135, 42, and 29 cm.

Fig. 13
Fig. 13

Variation of derived r 0 value as a function of the model L 0 value for the June–July 1989 data set. (a)–(f) each show one night’s data, with the values derived from each 1-h data set on the 12-m baseline (dashed curves) and those from the 31.5-m baseline (solid curves). The intersections of the solid and the dashed curves give an indication of the range of L 0 values that reconcile the data on different baselines.

Fig. 14
Fig. 14

Radiosonde measurements made at Mt. Wilson of wind speed versus altitude for the night of 28 June 1989. There are no measurements between 11 and 15 km because of a loss of telemetry. (Courtesy of Don L. Walters, U.S. Naval Postgraduate School.)

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

D ϕ ( r ) ϕ ( r ) - ϕ ( r + r ) 2 = 6.88 ( r / r 0 ) 5 / 3 ,
Φ N ( κ ) [ κ 2 + ( 1 / L 0 ) 2 ] - 11 / 6 .
Φ N ( κ ) ( κ 2 + κ / L 0 ) - 11 / 6 .
D ϕ ( t ) ϕ ( t ) - ϕ ( t + t ) 2 = ( t / t 0 ) 5 / 3 ,
t 0 = 0.31 r 0 / V .
Φ d ( f ) = 2.84 × 10 - 4 λ 0 2 t 0 - 5 / 3 f - 8 / 3 ,
f 1 0.2 V / B .
r 0 = B [ D ϕ ( B ) / 6.88 ] - 3 / 5 .
P measured / P tot = 1 - / 6 5 ( f 0 / f 1 ) 1 / 3 1 - / 6 5 ( B / 0.2 V T ) 1 / 3 .
α = log P 1 - log P 2 log B 1 - log B 2 ,

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