Abstract

A method of calculating distribution temperatures by numerically fitting Planck radiation curves to measured spectra is discussed. Numerically generated spectra were used to test the method and to determine the sensitivity to noise and the effects of linear emissivity changes. A comparison with the multiple-pair method of calculating color temperature as described in a previous paper [ Appl. Opt. 27, 4073– 4075 ( 1988)] is presented. It was found that the method described here is ~2 times less sensitive to noise than the previously described method. Nonconstant emissivity (the linear model) produces the same effect on calculated distribution temperatures regardless of the calculating method.

© 1992 Optical Society of America

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References

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  1. Ž. Andreić, “Numerical evaluation of the multiple-pair method of calculating temperature from a measured continuous spectrum,” Appl. Opt. 27, 4073–4075 (1988).
    [CrossRef]
  2. M. Pivovonsky, M. R. Nagel, Tables of Blackbody Radiation Functions (Macmillan, New York, 1961), Chap. 3, p. xiii.
  3. M. A. Bramson, Infrared Radiation (Plenum, New York, 1968), Chap. 5, p. 164.
  4. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1989), Chap. 10, pp. 326–330.
  5. J. Kletzek, Exercises in Astronomy (Reidel, Dordrecht, The Netherlands, 1987), pp. 194–196.
  6. J. Dufay, Introduction to Astrophysics: The Stars (Dover, New York, 1964), pp. 89–90.
  7. The data for emissivity variations in Ref. 1, Fig. 4, are drawn from a mirror reflected around the vertical axis. Figure 2 in Ref. 1 is correct.

1988 (1)

Andreic, Ž.

Bramson, M. A.

M. A. Bramson, Infrared Radiation (Plenum, New York, 1968), Chap. 5, p. 164.

Dufay, J.

J. Dufay, Introduction to Astrophysics: The Stars (Dover, New York, 1964), pp. 89–90.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1989), Chap. 10, pp. 326–330.

Kletzek, J.

J. Kletzek, Exercises in Astronomy (Reidel, Dordrecht, The Netherlands, 1987), pp. 194–196.

Nagel, M. R.

M. Pivovonsky, M. R. Nagel, Tables of Blackbody Radiation Functions (Macmillan, New York, 1961), Chap. 3, p. xiii.

Pivovonsky, M.

M. Pivovonsky, M. R. Nagel, Tables of Blackbody Radiation Functions (Macmillan, New York, 1961), Chap. 3, p. xiii.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1989), Chap. 10, pp. 326–330.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1989), Chap. 10, pp. 326–330.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1989), Chap. 10, pp. 326–330.

Appl. Opt. (1)

Other (6)

M. Pivovonsky, M. R. Nagel, Tables of Blackbody Radiation Functions (Macmillan, New York, 1961), Chap. 3, p. xiii.

M. A. Bramson, Infrared Radiation (Plenum, New York, 1968), Chap. 5, p. 164.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing (Cambridge U. Press, Cambridge, England, 1989), Chap. 10, pp. 326–330.

J. Kletzek, Exercises in Astronomy (Reidel, Dordrecht, The Netherlands, 1987), pp. 194–196.

J. Dufay, Introduction to Astrophysics: The Stars (Dover, New York, 1964), pp. 89–90.

The data for emissivity variations in Ref. 1, Fig. 4, are drawn from a mirror reflected around the vertical axis. Figure 2 in Ref. 1 is correct.

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Figures (3)

Fig. 1
Fig. 1

Effects of noise on the calculated distribution temperature. The solid circles represent the mean distribution temperatures, and the solid lines indicate the absolute spread in values calculated by the method described in this paper. The open circles and dashed lines indicate the color temperatures that were calculated from the same spectra by the multiple-pair method described in Ref. 1. The theoretical temperatures were 2000 and 3000 K for (a) and (b), respectively.

Fig. 2
Fig. 2

When emissivity is a linear function of wavelength, the calculated distribution temperatures are systematically shifted. Both calculating procedures give nearly identical results. Thus only temperatures calculated by the methods in this paper7 are indicated. Theoretical temperatures were 2000 and 3000 K for (a) and (b), respectively.

Fig. 3
Fig. 3

Spectrum of the star HD 154417 as published in Ref. 5 and the Planck radiation curve for the calculated distribution temperature and the scaling factor. Open circles indicate the data points from Ref. 5. The solid curve shows the Planck radiation curve for the calculated distribution temperature (6441 K) and the scaling factor.

Equations (16)

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L λ BB ( λ , T ) = υ 5 ω λ [ exp ( υ ) 1 ] ,
υ = c 2 λ T ,
ω λ = 21 . 20143566 .
m ( λ , T ) = L λ ( λ ) f L λ BB ( λ , T ) ,
M F ( T , λ ) = λ 1 λ 2 [ m ( λ , T ) ] 2 d λ,
L λ G ( λ , T 0 ) = υ 0 5 ω λ [ exp ( υ 0 ) 1 ] ,
υ 0 = c 2 λ T 0 ,
m G ( λ , T ) = L λ G ( λ, T 0 ) f L λ BB ( λ, T ) = υ 0 5 ω λ [ exp ( υ 0 ) 1 ] [ 1 f ( υ υ 0 ) 5 exp ( υ 0 ) 1 exp ( υ ) 1 ] ,
M F G ( T , f ) = λ 1 λ 2 [ m G ( λ , T ) ] 2 d λ .
f = ( υ 0 υ ) 5 exp ( υ ) 1 exp ( υ 0 ) 1 .
υ 1 ,
exp ( υ ) 1 + υ .
f = ( T T 0 ) 4 .
f = ( υ 0 υ ) 5 exp ( υ υ 0 ) .
F ( T , f ) = i = 1 M [ L λ ( λ i ) f L λ BB ( λ i , T ) ] 2 ,
( λ ) = B + c ( λ λ B ) ,

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