Abstract

An analytic sun pillar model is developed which indicates that large hexagonal ice columns can cause sun pillars. The model shows that the Stuchtey method for explaining sun pillars is not incorrect, only incomplete. The model uses an expression for the intensity of the sun pillar that considers both the optical mapping of light for a single crystal orientation and for the probability of having that particular orientation. The model, which is extended to a sun of finite size, clearly shows that the portion of the pillar that is seen above the horizon is brightest when the sun is below the horizon.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), pp. 201–202.
  2. Karl Stuchtey, “Subsuns and Light Pillars on Sun and Moon,” Ann. Phys. IV, (59), 33–55 (1919).
    [Crossref]
  3. Robert G. Greenler and et al., “The Origin of Sun Pillars,” Am. Sci.  60, (3), 292–302 (1972).
  4. For nomenclature buffs, it should be noted that this use of the word intensity is correct in that what will be calculated is a (relative) power per unit solid angle that leaves an infinitesimal volume containing ice crystals. Because it is the relative intensity, it can also be interpreted as the relative radiance from the sky, or if the reader wants to multiply by the response function of the eye, the relative luminance.
  5. This is the only crystal type that is considered so that nothing is deduced here either for or against the ability of plates to cause sun pillars.
  6. It was noticed while the paper was in press that only one polarization component is represented by Eq. (6). A quick calculation showed, however, that no qualitative difference is produced by the proper inclusion of both components. The quantitative difference is very small.

1972 (1)

Robert G. Greenler and et al., “The Origin of Sun Pillars,” Am. Sci.  60, (3), 292–302 (1972).

1919 (1)

Karl Stuchtey, “Subsuns and Light Pillars on Sun and Moon,” Ann. Phys. IV, (59), 33–55 (1919).
[Crossref]

Greenler, Robert G.

Robert G. Greenler and et al., “The Origin of Sun Pillars,” Am. Sci.  60, (3), 292–302 (1972).

Minnaert, M.

M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), pp. 201–202.

Stuchtey, Karl

Karl Stuchtey, “Subsuns and Light Pillars on Sun and Moon,” Ann. Phys. IV, (59), 33–55 (1919).
[Crossref]

Am. Sci (1)

Robert G. Greenler and et al., “The Origin of Sun Pillars,” Am. Sci.  60, (3), 292–302 (1972).

Ann. Phys. (1)

Karl Stuchtey, “Subsuns and Light Pillars on Sun and Moon,” Ann. Phys. IV, (59), 33–55 (1919).
[Crossref]

Other (4)

M. Minnaert, Light and Colour in the Open Air (Dover, New York, 1954), pp. 201–202.

For nomenclature buffs, it should be noted that this use of the word intensity is correct in that what will be calculated is a (relative) power per unit solid angle that leaves an infinitesimal volume containing ice crystals. Because it is the relative intensity, it can also be interpreted as the relative radiance from the sky, or if the reader wants to multiply by the response function of the eye, the relative luminance.

This is the only crystal type that is considered so that nothing is deduced here either for or against the ability of plates to cause sun pillars.

It was noticed while the paper was in press that only one polarization component is represented by Eq. (6). A quick calculation showed, however, that no qualitative difference is produced by the proper inclusion of both components. The quantitative difference is very small.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

FIG. 1
FIG. 1

Geometric relations on the celestial sphere where Σ = solar elevation, α = tip angle of the normal to the reflecting surface, ϕ = azimuth angle of the normal, S = sun position SS = subsolar position, S′ = location of the reflected light, h = elevation angle of S′, and b = great-circle angular distance of S′ from solar vertical.

FIG. 2
FIG. 2

Lines of constant intensity for a point source reflecting off large columns, [P ∝ (sinα)−1] as specified by proportionality (4). By also turning the graph upside down, solar elevations of −8, −4, 0, 4, and 8 degrees are illustrated. Contours of Pdω/dω′ equal to 16, 32, 64, 128, and 256 are shown. The patterns are identically symmetric with respect to the horizon. The ordinate is h in degrees and although the abscissa b is plotted to the same scale, it is unlabeled. The horizon is at h = 0 and the sun is marked by a +.

FIG. 3
FIG. 3

Lines of constant intensity for a point source reflecting off large columns when the reflection coefficient and the geometric reflection cross section are included. The patterns are no longer horizon symmetric but show a flaming out with distance from the sun. Contours of P(dω/dω′)RCr equal to 0.5, 1, 2, 4, and 8 are shown. As before, the ordinate is h in degrees.

FIG. 4
FIG. 4

Same as Fig. 3 only for a simulation of the finite sun obtained by calculating the envelope of all the intensity contours of the sun.

Equations (17)

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

sin ( h ) = sin cos 2 α + cos sin 2 α cos ϕ , sin ( b ) = sin ϕ sin 2 α sin + sin 2 α cos sin 2 ϕ .
I d ω / d ω .
d ω = sin α d α d ϕ .
d ω = cos h d h d b ,
sin ( b / 2 ) = cos h sin ( b / 2 ) .
d ω = cos h [ 1 ( sin h cos ( b / 2 ) ) 2 ] 1 / 2 d h d b ,
d ω d ω = sin α cos h [ 1 ( sin h cos ( b / 2 ) ) 2 ] 1 / 2 d α d ϕ d h d b .
d h d b d α d ϕ = h α b ϕ h ϕ b α = 4 sin α F cos h cos b ,
F = ( cos cos 2 α cos ϕ sin sin 2 α ) × ( cos ϕ cos α sin + sin α cos cos 2 ϕ ) cos cos α sin 2 ϕ sin h .
d ω d ω = cos b 4 F [ 1 ( sin h cos ( b / 2 ) ) 2 ] 1 / 2 .
I P ( d ω ) d ω / d ω ,
P ( d ω ) ( sin α ) 1 ,
R = ( tan ( r i ) tan ( r + i ) ) 2 ,
cos i = cos α sin + sin α cos cos ϕ
sin i = n sin r .
C r cos i ,
I P ( d ω / d ω ) R C r .