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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 57, Iss. 1 — Jan. 1, 1967
  • pp: 44–47

Optics InfoBase > JOSA > Volume 57 > Issue 1 > How Large is a Point Source?

How Large is a Point Source?

ARTHUR C. HARDY  »View Author Affiliations

JOSA, Vol. 57, Issue 1, pp. 44-47 (1967)

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A source of light having many times the angular subtense of the sun or moon can usually be treated as a point when making photometric calculations based on the inverse square law. When viewed by the eye, such sources are not perceived as points (like stars) but rather as extended surfaces. In this paper, some typical data on contrast thresholds for targets, both large and small, are converted into threshold stellar magnitudes. This is done for two important cases which, unless carefully distinguished, can lead to erroneous results.

ARTHUR C. HARDY, "How Large is a Point Source?," J. Opt. Soc. Am. 57, 44-47 (1967)

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  1. H. Richard Blackwell, J. Opt. Soc. Am. 36, 624 (1946).
  2. Seibert Q. Duntley, J. Opt. Soc. Am. 38, 237 (1948).
  3. Charles Fabry, Trans. Illum. Eng. Soc. 20, 12 (1925).
  4. N. N. Sytinskaya, Russ. A. J. 34, 899 (1957).
  5. Initially, the Tiffany experiments were undertaken as part of a study of ship camouflage; the problem was to establish the ranges at which military targets would almost never be sighted by an enemy vessel. For this and other valid reasons, Blackwell's threshold contrasts represent a 50% probability that a target will be sighted under virtually ideal conditions of observation. Blackwell discusses his criterion of visibility at some length, and mentions that if we desire to increase the probability of sighting from 50% to 90%, the threshold contrasts should be multiplied by 1.62.I have not done this in my conversion of his data to threshold stellar magnitudes, but the reader can, in effect, multiply Blackwell's threshold contrasts by 1.62 by simply subtracting 0.52 from all the threshold magnitudes listed in Table I. Some authors have multiplied Blackwell's threshold contrasts by 2 to bring the probability of detection to about 98%. In this case, 0.75 should be subtracted from the listed threshold magnitudes.Obviously, multiplying Blackwell's threshold contrasts by 2.51 is effected by subtracting 1.00 from all the listed threshold magnitudes.
  6. The conversion of the ratio E0/E into the equivalent stellar magnitude can usually be done satisfactorily with a log—log slide rule. The trick is to set 5 on the C scale opposite 100 on one of the log scales. Thereafter, by moving only the hairline, magnitude values are read on the C scale for E0/E ratios on any of the three log scales, attention being given, of course, to the position of the decimal point. If the ratio E0/E is less than unity, the ratio is merely inverted; and the magnitude becomes negative. Anyone who undertakes to convert ratios into stellar magnitudes will promptly discover other tricks. For example, if the ratio E0/E is 3909, the corresponding magnitude can be read directly from the C scale. On the other hand, greater accuracy is achieved by imagining that the ratio is 39.09 and remembering to add 5 to the resulting magnitude. It is even more accurate to imagine the ratio to be 3.909 and then to add 7.5 to the magnitude read on the C scale.
  7. The visibility of stars in the daylight sky was discussed by R. Tousey and E. O. Hulburt, J. Opt. Soc. Am. 38, 886 (1948). These authors erroneously assumed that the targets viewed by the Tiffany observers were opaque disks, but their Eq. 4 is nevertheless correct.
  8. A. Ricco, Annali d'Ottalmol. 6, 373 (1877).
  9. Blackwell, in Part II of the aforementioned reference discusses the case when the target is darker than its background and the contrast is therefore negative. For the purpose of this paper, it seemed best to consider positive contrasts only. Actually, in the framework of my approach to the problem, stellar magnitudes could be assigned to a completely black target (C= ~1), but I shall spare the reader the mental gymnastics involved.
  10. W. E. Knowles Middleton, Vision Through the Atmosphere (University of Toronto Press, Canada) 1958 ed., p. 96.
  11. In footnote 5, I described how the entries in Table I could be corrected to correspond to a higher probability of etection. I was not able to devise an equally simple procedure for Table II.

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