## 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

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### Equations (17)

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(1)
$$\begin{array}{ll}\hfill sin(-h)& =sin\text{\u2211}cos2\alpha +cos\text{\u2211}sin2\alpha cos\varphi ,\hfill \\ \hfill sin(-b)& =sin\varphi sin2\alpha sin\text{\u2211}+{sin}^{2}\alpha cos\text{\u2211}sin2\varphi .\hfill \end{array}$$
(2)
$$I\propto d\omega /d{\omega}^{\prime}.$$
(3)
$$d\omega =sin\alpha d\alpha d\varphi .$$
(4)
$$d{\omega}^{\prime}=coshdhd{b}^{\prime},$$
(5)
$$sin(b/2)=coshsin({b}^{\prime}/2).$$
(6)
$$d{\omega}^{\prime}=cosh{\left[1-{\left(\frac{sinh}{cos(b/2)}\right)}^{2}\right]}^{-1/2}dhdb,$$
(7)
$$\frac{d\omega}{d{\omega}^{\prime}}=\frac{sin\alpha}{cosh}{\left[1-{\left(\frac{sinh}{cos(b/2)}\right)}^{2}\right]}^{-1/2}\frac{d\alpha d\varphi}{dhdb}.$$
(8)
$$\begin{array}{ll}\frac{dhdb}{d\alpha d\varphi}\hfill & =\frac{\partial h}{\partial \alpha}\frac{\partial b}{\partial \varphi}-\frac{\partial h}{\partial \varphi}\frac{\partial b}{\partial \alpha}\hfill \\ \hfill & =\frac{4sin\alpha F}{coshcosb},\hfill \end{array}$$
(9)
$$\begin{array}{ll}F\hfill & =(cos\text{\u2211}cos2\alpha cos\varphi -sin\text{\u2211}sin2\alpha )\hfill \\ \hfill & \times (cos\varphi cos\alpha sin\text{\u2211}+sin\alpha cos\text{\u2211}cos2\varphi )\hfill \\ \hfill & -cos\text{\u2211}cos\alpha {sin}^{2}\varphi sinh.\hfill \end{array}$$
(10)
$$\frac{d\omega}{d{\omega}^{\prime}}=\frac{cosb}{4F}{\left[1-{\left(\frac{sinh}{cos(b/2)}\right)}^{2}\right]}^{1/2}.$$
(11)
$$I\propto P(d\omega )d\omega /d{\omega}^{\prime},$$
(12)
$$P(d\omega )\propto {(sin\alpha )}^{-1},$$
(13)
$$R={\left(\frac{tan(r-i)}{tan(r+i)}\right)}^{2},$$
(14)
$$cosi=cos\alpha sin\text{\u2211}+sin\alpha cos\text{\u2211}cos\varphi $$
(16)
$${C}_{r}\propto cosi,$$
(17)
$$I\propto P(d\omega /d{\omega}^{\prime})R{C}_{r}.$$