Abstract

Using a technique for CCD polarimetry, we have obtained stellar polarization data with the 0.9-m telescope on Kitt Peak and with the Tek 2048 CCD camera. Measurements of stars viewed through holes in polarizing filters serve to correct for variations in atmospheric transparency. For the brightest stars the uncertainties in the measured polarization during a single, hour-long, polarimetric sequence are 0.3–0.5%, because of the residual effects of the variations in transparency. For fainter stars we are Poisson noise limited. Our technique provides some advantages over other techniques for CCD polarimetry, primarily because it can be easily integrated into existing photometric systems.

© 1995 Optical Society of America

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References

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  1. J. S. Hall, “Polarization of light from distant stars by interstellar medium,” Science 109, 166–167 (1949).
    [CrossRef] [PubMed]
  2. W. A. Hiltner, “Observations of the polarized light from stars,” Science 109, 165 (1949).
    [CrossRef] [PubMed]
  3. L. Davis, J. L. Greenstein, “The polarization of starlight by aligned dust grains,” Astrophys. J. 114, 206–240 (1951).
    [CrossRef]
  4. P. C. Myers, A. A. Goodman, “On the dispersion in direction of interstellar polarization,” Astrophys. J. 373, 509–524 (1991).
    [CrossRef]
  5. R. H. Hildebrand, “Magnetic fields and stardust,” Q. J. R. Astron. Soc. 29, 327–351 (1988).
  6. K.-W. Hoddap, “A polarimetric study of the Mon R2 star forming region,” Astron. Astrophys. 172, 304–310 (1987).
  7. K.-W. Hoddap, “The magnetic field in star-forming large globules,” Astrophys. J. 319, 842–849 (1987).
    [CrossRef]
  8. K. Serkowski, “Polarization techniques,” in Methods of Experimental Physics, N. Carleton, ed. (Academic, New York, 1974), Vol. 12A, pp. 361–414.
    [CrossRef]
  9. T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
    [CrossRef]
  10. Hoya Corp. USA, San Jose, Calif. 95131.
  11. O. J. Eggen, A. R. Sandage, “New photoelectric observations of stars in the old galactic cluster M67,” Astrophys. J. 140, 130–143 (1964).
    [CrossRef]
  12. K. A. Janes, G. H. Smith, “The giant branch of the old open cluster M67,” Astron. J. 89, 487–495 (1984).
    [CrossRef]
  13. D. C. B. Whittet, Dust in the Galactic Environment (Institute of Physics, Philadelphia, 1992), p. 87.
  14. K. Serkowski, D. S. Mathewson, V. L. Ford, “Wavelength dependence of interstellar polarization and ratio of total to selective extinction,” Astrophys. J. 196, 261–290 (1975).
    [CrossRef]
  15. D. S. Mathewson, V. L. Ford, “Polarization observations of 1800 stars,” Mem. R. Astron. Soc. 74, 139–182 (1970).
  16. F. J. Vrba, S. E. Strom, K. M. Strom, “Optical polarization measurements in the vicinity of nearby star-forming complexes. I. The Lynds 1641 giant molecular cloud,” Astron. J. 96, 680–694 (1988).
    [CrossRef]
  17. H.-J. Röser, “Photographic polarization survey with a Savart plate,” Astron. Astrophys. 103, 374–381 (1981).
  18. S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).
  19. H. Dekker, S. D’Odorico, “A Wollaston prism for the ESO faint object spectrograph and camera,” ESO Messenger 46, 21–22 (1986).
  20. A. M. Magalhaes, W. F. Velloso, “Opitcal multi-mode instruments at the University of Sao Paulo,” in Instrumentation for Ground-based Optical Astronomy: Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 638–646.
    [CrossRef]
  21. H.-J. Röser, K. Meisenheimer, “CCD photo-polarimetry of the jet of 3C273,” Astron. Astrophys. 154, 15–24 (1986).
  22. K. Meisenheimer, H.-J. Röser, “CCD polarimetry using a Savart plate,” in ESO-OHP Workshop on the Optimization of the Use of CCD Detectors in Astronomy, J.-P. Baluteau, S. D’Odorico, eds. (European Southern Observatory, Garching bei Munchen, Germany, 1987), pp. 227–236.
  23. H.-J. Röser, K. Meisenheimer, The synchroton light from the jet of 3C273,” Astron. Astrophys. 252, 458–474 (1991).
  24. D. P. Clemens, R. W. Leach, “Charge-coupled-device-based imaging polarimeter for astronomy,” Opt. Eng. 26, 923–929 (1987).
  25. B. T. Jannuzi, R. F. Green, H. French, “An optical polarization survey for BL lacertae objects and highly polarized quasars,” Astrophys. J. 404, 100–111 (1993).
    [CrossRef]
  26. D. Clarke, B. G. Stewart, “Statistical methods of stellar polarimetry,” Vistas Astron. 29, 27–51 (1986).
    [CrossRef]

1994 (1)

T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
[CrossRef]

1993 (1)

B. T. Jannuzi, R. F. Green, H. French, “An optical polarization survey for BL lacertae objects and highly polarized quasars,” Astrophys. J. 404, 100–111 (1993).
[CrossRef]

1991 (2)

H.-J. Röser, K. Meisenheimer, The synchroton light from the jet of 3C273,” Astron. Astrophys. 252, 458–474 (1991).

P. C. Myers, A. A. Goodman, “On the dispersion in direction of interstellar polarization,” Astrophys. J. 373, 509–524 (1991).
[CrossRef]

1988 (2)

R. H. Hildebrand, “Magnetic fields and stardust,” Q. J. R. Astron. Soc. 29, 327–351 (1988).

F. J. Vrba, S. E. Strom, K. M. Strom, “Optical polarization measurements in the vicinity of nearby star-forming complexes. I. The Lynds 1641 giant molecular cloud,” Astron. J. 96, 680–694 (1988).
[CrossRef]

1987 (3)

K.-W. Hoddap, “A polarimetric study of the Mon R2 star forming region,” Astron. Astrophys. 172, 304–310 (1987).

K.-W. Hoddap, “The magnetic field in star-forming large globules,” Astrophys. J. 319, 842–849 (1987).
[CrossRef]

D. P. Clemens, R. W. Leach, “Charge-coupled-device-based imaging polarimeter for astronomy,” Opt. Eng. 26, 923–929 (1987).

1986 (3)

D. Clarke, B. G. Stewart, “Statistical methods of stellar polarimetry,” Vistas Astron. 29, 27–51 (1986).
[CrossRef]

H. Dekker, S. D’Odorico, “A Wollaston prism for the ESO faint object spectrograph and camera,” ESO Messenger 46, 21–22 (1986).

H.-J. Röser, K. Meisenheimer, “CCD photo-polarimetry of the jet of 3C273,” Astron. Astrophys. 154, 15–24 (1986).

1984 (1)

K. A. Janes, G. H. Smith, “The giant branch of the old open cluster M67,” Astron. J. 89, 487–495 (1984).
[CrossRef]

1983 (1)

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

1981 (1)

H.-J. Röser, “Photographic polarization survey with a Savart plate,” Astron. Astrophys. 103, 374–381 (1981).

1975 (1)

K. Serkowski, D. S. Mathewson, V. L. Ford, “Wavelength dependence of interstellar polarization and ratio of total to selective extinction,” Astrophys. J. 196, 261–290 (1975).
[CrossRef]

1970 (1)

D. S. Mathewson, V. L. Ford, “Polarization observations of 1800 stars,” Mem. R. Astron. Soc. 74, 139–182 (1970).

1964 (1)

O. J. Eggen, A. R. Sandage, “New photoelectric observations of stars in the old galactic cluster M67,” Astrophys. J. 140, 130–143 (1964).
[CrossRef]

1951 (1)

L. Davis, J. L. Greenstein, “The polarization of starlight by aligned dust grains,” Astrophys. J. 114, 206–240 (1951).
[CrossRef]

1949 (2)

J. S. Hall, “Polarization of light from distant stars by interstellar medium,” Science 109, 166–167 (1949).
[CrossRef] [PubMed]

W. A. Hiltner, “Observations of the polarized light from stars,” Science 109, 165 (1949).
[CrossRef] [PubMed]

Axon, D. J.

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

Bingham, R. G.

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

Clarke, D.

D. Clarke, B. G. Stewart, “Statistical methods of stellar polarimetry,” Vistas Astron. 29, 27–51 (1986).
[CrossRef]

Clemens, D. P.

D. P. Clemens, R. W. Leach, “Charge-coupled-device-based imaging polarimeter for astronomy,” Opt. Eng. 26, 923–929 (1987).

D’Odorico, S.

H. Dekker, S. D’Odorico, “A Wollaston prism for the ESO faint object spectrograph and camera,” ESO Messenger 46, 21–22 (1986).

Davis, L.

L. Davis, J. L. Greenstein, “The polarization of starlight by aligned dust grains,” Astrophys. J. 114, 206–240 (1951).
[CrossRef]

Dekker, H.

H. Dekker, S. D’Odorico, “A Wollaston prism for the ESO faint object spectrograph and camera,” ESO Messenger 46, 21–22 (1986).

Eggen, O. J.

O. J. Eggen, A. R. Sandage, “New photoelectric observations of stars in the old galactic cluster M67,” Astrophys. J. 140, 130–143 (1964).
[CrossRef]

Ford, V. L.

K. Serkowski, D. S. Mathewson, V. L. Ford, “Wavelength dependence of interstellar polarization and ratio of total to selective extinction,” Astrophys. J. 196, 261–290 (1975).
[CrossRef]

D. S. Mathewson, V. L. Ford, “Polarization observations of 1800 stars,” Mem. R. Astron. Soc. 74, 139–182 (1970).

French, H.

B. T. Jannuzi, R. F. Green, H. French, “An optical polarization survey for BL lacertae objects and highly polarized quasars,” Astrophys. J. 404, 100–111 (1993).
[CrossRef]

Goldsmith, P. F.

T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
[CrossRef]

Goodman, A. A.

P. C. Myers, A. A. Goodman, “On the dispersion in direction of interstellar polarization,” Astrophys. J. 373, 509–524 (1991).
[CrossRef]

Green, R. F.

B. T. Jannuzi, R. F. Green, H. French, “An optical polarization survey for BL lacertae objects and highly polarized quasars,” Astrophys. J. 404, 100–111 (1993).
[CrossRef]

Greenstein, J. L.

L. Davis, J. L. Greenstein, “The polarization of starlight by aligned dust grains,” Astrophys. J. 114, 206–240 (1951).
[CrossRef]

Hall, J. S.

J. S. Hall, “Polarization of light from distant stars by interstellar medium,” Science 109, 166–167 (1949).
[CrossRef] [PubMed]

Hildebrand, R. H.

R. H. Hildebrand, “Magnetic fields and stardust,” Q. J. R. Astron. Soc. 29, 327–351 (1988).

Hiltner, W. A.

W. A. Hiltner, “Observations of the polarized light from stars,” Science 109, 165 (1949).
[CrossRef] [PubMed]

Hoddap, K.-W.

K.-W. Hoddap, “A polarimetric study of the Mon R2 star forming region,” Astron. Astrophys. 172, 304–310 (1987).

K.-W. Hoddap, “The magnetic field in star-forming large globules,” Astrophys. J. 319, 842–849 (1987).
[CrossRef]

Janes, K. A.

K. A. Janes, G. H. Smith, “The giant branch of the old open cluster M67,” Astron. J. 89, 487–495 (1984).
[CrossRef]

Jannuzi, B. T.

B. T. Jannuzi, R. F. Green, H. French, “An optical polarization survey for BL lacertae objects and highly polarized quasars,” Astrophys. J. 404, 100–111 (1993).
[CrossRef]

Jarrett, T. H.

T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
[CrossRef]

Leach, R. W.

D. P. Clemens, R. W. Leach, “Charge-coupled-device-based imaging polarimeter for astronomy,” Opt. Eng. 26, 923–929 (1987).

Magalhaes, A. M.

A. M. Magalhaes, W. F. Velloso, “Opitcal multi-mode instruments at the University of Sao Paulo,” in Instrumentation for Ground-based Optical Astronomy: Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 638–646.
[CrossRef]

Mathewson, D. S.

K. Serkowski, D. S. Mathewson, V. L. Ford, “Wavelength dependence of interstellar polarization and ratio of total to selective extinction,” Astrophys. J. 196, 261–290 (1975).
[CrossRef]

D. S. Mathewson, V. L. Ford, “Polarization observations of 1800 stars,” Mem. R. Astron. Soc. 74, 139–182 (1970).

Meisenheimer, K.

H.-J. Röser, K. Meisenheimer, The synchroton light from the jet of 3C273,” Astron. Astrophys. 252, 458–474 (1991).

H.-J. Röser, K. Meisenheimer, “CCD photo-polarimetry of the jet of 3C273,” Astron. Astrophys. 154, 15–24 (1986).

K. Meisenheimer, H.-J. Röser, “CCD polarimetry using a Savart plate,” in ESO-OHP Workshop on the Optimization of the Use of CCD Detectors in Astronomy, J.-P. Baluteau, S. D’Odorico, eds. (European Southern Observatory, Garching bei Munchen, Germany, 1987), pp. 227–236.

Myers, P. C.

P. C. Myers, A. A. Goodman, “On the dispersion in direction of interstellar polarization,” Astrophys. J. 373, 509–524 (1991).
[CrossRef]

Novak, G.

T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
[CrossRef]

Pallister, W. S.

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

Röser, H.-J.

H.-J. Röser, K. Meisenheimer, The synchroton light from the jet of 3C273,” Astron. Astrophys. 252, 458–474 (1991).

H.-J. Röser, K. Meisenheimer, “CCD photo-polarimetry of the jet of 3C273,” Astron. Astrophys. 154, 15–24 (1986).

H.-J. Röser, “Photographic polarization survey with a Savart plate,” Astron. Astrophys. 103, 374–381 (1981).

K. Meisenheimer, H.-J. Röser, “CCD polarimetry using a Savart plate,” in ESO-OHP Workshop on the Optimization of the Use of CCD Detectors in Astronomy, J.-P. Baluteau, S. D’Odorico, eds. (European Southern Observatory, Garching bei Munchen, Germany, 1987), pp. 227–236.

Sandage, A. R.

O. J. Eggen, A. R. Sandage, “New photoelectric observations of stars in the old galactic cluster M67,” Astrophys. J. 140, 130–143 (1964).
[CrossRef]

Scarrott, S. M.

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

Serkowski, K.

K. Serkowski, D. S. Mathewson, V. L. Ford, “Wavelength dependence of interstellar polarization and ratio of total to selective extinction,” Astrophys. J. 196, 261–290 (1975).
[CrossRef]

K. Serkowski, “Polarization techniques,” in Methods of Experimental Physics, N. Carleton, ed. (Academic, New York, 1974), Vol. 12A, pp. 361–414.
[CrossRef]

Smith, G. H.

K. A. Janes, G. H. Smith, “The giant branch of the old open cluster M67,” Astron. J. 89, 487–495 (1984).
[CrossRef]

Stewart, B. G.

D. Clarke, B. G. Stewart, “Statistical methods of stellar polarimetry,” Vistas Astron. 29, 27–51 (1986).
[CrossRef]

Strom, K. M.

F. J. Vrba, S. E. Strom, K. M. Strom, “Optical polarization measurements in the vicinity of nearby star-forming complexes. I. The Lynds 1641 giant molecular cloud,” Astron. J. 96, 680–694 (1988).
[CrossRef]

Strom, S. E.

F. J. Vrba, S. E. Strom, K. M. Strom, “Optical polarization measurements in the vicinity of nearby star-forming complexes. I. The Lynds 1641 giant molecular cloud,” Astron. J. 96, 680–694 (1988).
[CrossRef]

Velloso, W. F.

A. M. Magalhaes, W. F. Velloso, “Opitcal multi-mode instruments at the University of Sao Paulo,” in Instrumentation for Ground-based Optical Astronomy: Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 638–646.
[CrossRef]

Vrba, F. J.

F. J. Vrba, S. E. Strom, K. M. Strom, “Optical polarization measurements in the vicinity of nearby star-forming complexes. I. The Lynds 1641 giant molecular cloud,” Astron. J. 96, 680–694 (1988).
[CrossRef]

Warren-Smith, R. F.

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

Whittet, D. C. B.

D. C. B. Whittet, Dust in the Galactic Environment (Institute of Physics, Philadelphia, 1992), p. 87.

Xie, T.

T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
[CrossRef]

Astron. Astrophys. (4)

K.-W. Hoddap, “A polarimetric study of the Mon R2 star forming region,” Astron. Astrophys. 172, 304–310 (1987).

H.-J. Röser, “Photographic polarization survey with a Savart plate,” Astron. Astrophys. 103, 374–381 (1981).

H.-J. Röser, K. Meisenheimer, “CCD photo-polarimetry of the jet of 3C273,” Astron. Astrophys. 154, 15–24 (1986).

H.-J. Röser, K. Meisenheimer, The synchroton light from the jet of 3C273,” Astron. Astrophys. 252, 458–474 (1991).

Astron. J. (2)

F. J. Vrba, S. E. Strom, K. M. Strom, “Optical polarization measurements in the vicinity of nearby star-forming complexes. I. The Lynds 1641 giant molecular cloud,” Astron. J. 96, 680–694 (1988).
[CrossRef]

K. A. Janes, G. H. Smith, “The giant branch of the old open cluster M67,” Astron. J. 89, 487–495 (1984).
[CrossRef]

Astrophys. J. (7)

K. Serkowski, D. S. Mathewson, V. L. Ford, “Wavelength dependence of interstellar polarization and ratio of total to selective extinction,” Astrophys. J. 196, 261–290 (1975).
[CrossRef]

O. J. Eggen, A. R. Sandage, “New photoelectric observations of stars in the old galactic cluster M67,” Astrophys. J. 140, 130–143 (1964).
[CrossRef]

K.-W. Hoddap, “The magnetic field in star-forming large globules,” Astrophys. J. 319, 842–849 (1987).
[CrossRef]

T. H. Jarrett, G. Novak, T. Xie, P. F. Goldsmith, “Magnetic field structure in Monoceros R2,” Astrophys. J. 430, 743–756 (1994).
[CrossRef]

L. Davis, J. L. Greenstein, “The polarization of starlight by aligned dust grains,” Astrophys. J. 114, 206–240 (1951).
[CrossRef]

P. C. Myers, A. A. Goodman, “On the dispersion in direction of interstellar polarization,” Astrophys. J. 373, 509–524 (1991).
[CrossRef]

B. T. Jannuzi, R. F. Green, H. French, “An optical polarization survey for BL lacertae objects and highly polarized quasars,” Astrophys. J. 404, 100–111 (1993).
[CrossRef]

ESO Messenger (1)

H. Dekker, S. D’Odorico, “A Wollaston prism for the ESO faint object spectrograph and camera,” ESO Messenger 46, 21–22 (1986).

Mem. R. Astron. Soc. (1)

D. S. Mathewson, V. L. Ford, “Polarization observations of 1800 stars,” Mem. R. Astron. Soc. 74, 139–182 (1970).

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

S. M. Scarrott, R. F. Warren-Smith, W. S. Pallister, D. J. Axon, R. G. Bingham, “Electronographic polarimetry: the Durham polarimeter,” Mon. Not. R. Astron. Soc. 204, 1163–1177 (1983).

Opt. Eng. (1)

D. P. Clemens, R. W. Leach, “Charge-coupled-device-based imaging polarimeter for astronomy,” Opt. Eng. 26, 923–929 (1987).

Q. J. R. Astron. Soc. (1)

R. H. Hildebrand, “Magnetic fields and stardust,” Q. J. R. Astron. Soc. 29, 327–351 (1988).

Science (2)

J. S. Hall, “Polarization of light from distant stars by interstellar medium,” Science 109, 166–167 (1949).
[CrossRef] [PubMed]

W. A. Hiltner, “Observations of the polarized light from stars,” Science 109, 165 (1949).
[CrossRef] [PubMed]

Vistas Astron. (1)

D. Clarke, B. G. Stewart, “Statistical methods of stellar polarimetry,” Vistas Astron. 29, 27–51 (1986).
[CrossRef]

Other (5)

A. M. Magalhaes, W. F. Velloso, “Opitcal multi-mode instruments at the University of Sao Paulo,” in Instrumentation for Ground-based Optical Astronomy: Present and Future, L. B. Robinson, ed. (Springer-Verlag, New York, 1988), pp. 638–646.
[CrossRef]

K. Meisenheimer, H.-J. Röser, “CCD polarimetry using a Savart plate,” in ESO-OHP Workshop on the Optimization of the Use of CCD Detectors in Astronomy, J.-P. Baluteau, S. D’Odorico, eds. (European Southern Observatory, Garching bei Munchen, Germany, 1987), pp. 227–236.

Hoya Corp. USA, San Jose, Calif. 95131.

K. Serkowski, “Polarization techniques,” in Methods of Experimental Physics, N. Carleton, ed. (Academic, New York, 1974), Vol. 12A, pp. 361–414.
[CrossRef]

D. C. B. Whittet, Dust in the Galactic Environment (Institute of Physics, Philadelphia, 1992), p. 87.

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Figures (2)

Fig. 1
Fig. 1

For each star in the NW field, the standard deviation s qu of the normalized Stokes parameters that are derived from a polarimetric sequence is plotted versus R magnitude. The dashed curve shows the standard deviation expected from Poisson fluctuations of the sky background and the stellar flux; the solid curve shows a two-parameter fit to the s qu .

Fig. 2
Fig. 2

Histogram of the ratio of the s qu shown in Fig. 1 to the values given by the two-parameter fit. The curve shows the distribution expected if the scatter is due to statistical uncertainties that result from the small number of polarimetric sequences used to compute the standard deviation.

Tables (1)

Tables Icon

Table 1 Results for Polarized Calibration Star VSS185

Equations (2)

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

s q u = { s Poission 2 + s 1 2 + [ s 2 N ( R ) ] 2 } 1/2 .
s Poission = { 1 2 N ( R ) + ( 1 + r ) S 2 [ N ( R ) ] 2 } 1/2 ,

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