Sunset Science. II. A useful diagram

Some theorems on "arctic mirages": hillingar effect and superior mirages

In a stratified or a spherically symmetric atmosphere, we consider some cases where a curved light ray connects two points, $A$A and $S$S, on the sea or the ground, at the same height as in some cases of superior mirages or "arctic mirages." We study the range $D$D of this ray (i.e., the geodesic distance between $A$A and $S$S) as a function of ${\alpha _S}$α_S, the ray inclination with respect to the vertical at $A$A or $S$S. The fundamental expression of $D$D is an improper integral. We show that a change of variable, also used to calculate the astronomical refraction, usually gives a well-behaved integral. In a spherically symmetric atmosphere, this happens when the refraction coefficient exceeds unity on the entire ray, producing a hillingar effect (alias strong looming for an erect image) or superior mirages. We then find a nice expression of ${{{\rm d}D}/{{\rm d}{\alpha _S}}}$dD/dα_S, which is usually negative. Inverted images (i.e., mirages in the strict sense) show up when ${{{\rm d}D}/{{\rm d}{\alpha _S}}}$dD/dα_S is positive; for this, we find a necessary condition in each atmospheric model. The strength of these results comes from not requiring precise knowledge of the ray paths.

Tropospheric haze and colors of the clear twilight sky

At the earth's surface, clear-sky colors during civil twilights depend on the combined spectral effects of molecular scattering, extinction by tropospheric aerosols, and absorption by ozone. Molecular scattering alone cannot produce the most vivid twilight colors near the solar horizon, for which aerosol scattering and absorption are also required. However, less well known are haze aerosols' effects on twilight sky colors at larger scattering angles, including near the antisolar horizon. To analyze this range of colors, we compare 3D Monte Carlo simulations of skylight spectra with hyperspectral measurements of clear twilight skies over a wide range of aerosol optical depths. Our combined measurements and simulations indicate that (a) the purest antisolar twilight colors would occur in a purely molecular, multiple-scattering atmosphere, whereas (b) the most vivid solar-sky colors require at least some turbidity. Taken together, these results suggest that multiple scattering plays an important role in determining the redness of the antitwilight arch.

Refraction near the horizon-an empirical approach. Part 1: terrestrial refraction of the dip

This study aims at providing improved closed-form refraction estimates for observations near the horizon. In this first part, over 1800 previously published direct measurements of the horizon's depression (dip) over the sea are reanalyzed using a nonconventional robust procedure for coping with numerous real, large, and asymmetric outliers from abnormal dips. The derived 1-parameter function agrees with those proposed in modern almanacs and for land surveying. It is found that the dips of warmer and colder sea surfaces vs. air are best described with two different functions. The two proposed 3-parameter functions, also using temperature difference between air and sea and wind speed, reduce the estimated error of the 1-parameter function by ∼⅓ and the number of outliers by ∼⅔.

35 minute green flash observed at Little America on 16 October 1929: a retrospective study

On 16 October 1929 five members of the Byrd Expedition 1 observed an intermittent 35 min green flash at the Little America station (latitude -78.57°) in Antarctica. The flash was the result of strong atmospheric refraction, likely associated with a subcritical Novaya Zemlya mirage. This paper examines the constraints placed on the observation by the Earth-Sun orbital kinematics. It is found that the length of the observation cannot be explained solely by the slowness of the setting rate of the Sun, nor the time required just before the beginning of the Antarctic summer for the top of the Sun to set, reach its relative minimum position at the horizon, and then rise back up again. The observed length of the effect, however, is consistent with the Sun effectively setting twice and rising twice during the observation, with the first effective rising being the result of the observers climbing up the radio towers at the Little America station in order to keep the top of the Sun in view.

Measuring and modeling twilight's Belt of Venus

The Belt of Venus (or antitwilight arch) is a reddish band often seen above the antisolar horizon during clear civil twilights, and immediately beneath it is the bluish-gray earth's shadow (or dark segment) cast on the atmosphere. Although both skylight phenomena have prompted decades of scientific research, surprisingly few measurements exist of their spectral, colorimetric, and photometric structure. Hyperspectral imaging of several clear twilights supplies these missing radiometric details and reveals some common spectral features of the antisolar sky at twilight: (1) color differences between the dark segment and the sunlit sky above the antitwilight arch are small or nil; (2) antisolar color and luminance extremes usually occur at different elevation angles; and (3) the two twilight phenomena are most vivid for modest aerosol optical depths. A second-order scattering model that includes extinction by aerosols and ozone provides some preliminary radiative transfer explanations of these twilight features' color and brightness.

Inferior mirages: an improved model

A quantitative model of the inferior mirage is presented, based on a realistic temperature profile in the convective boundary layer, using Monin-Obukhov similarity theory. The top of the inverted image is determined by the logarithmic part of the profile; the bottom is the apparent horizon, which depends on optical obstruction by roughness elements. These effects of surface roughness are included in the model, which is illustrated with a simulation. The vertical magnification varies throughout the mirage, becoming infinite at Minnaert's ill-named "vanishing line"-which makes green flashes apparent to the naked eye.

Sunset science. III. Visual adaptation and green flashes

Photographs of green flashes do not preclude a role for physiological effects in these phenomena. While green flashes are certainly not after-images, there is compelling evidence that adaptation in the visual system strongly affects the perceived color of most sunset green flashes. Furthermore, the retinal image of the setting Sun is usually bright enough to bleach most of the red-sensitive photopigment in a few seconds, making the yellow stage of a sunset flash appear green. Even in air so hazy that no green light reaches the eye, a yellow flash may occur and appear green. Many, but not all, visual observations of sunset green flashes are of this yellow flash. The yellow portion of sunset green flashes helps explain their reported durations, which exceed those expected for the appearance of green light alone.