```      Anusha Sekar               Stephen Taylor
University of Washington    Brigham Young University
anusha.sekar@gmail.com         staylor4@byu.edu
```
This article provides an explanation for an atmospherical phenomena known as the green flash. We use Java, Postscript, Adobe Photoshop, Mathematica, Terragen and Latex as explanatory aides. All Java and Postscript source codes may be found in [1].

### Introduction

A green flash can be described as a hazy green cloud hovering directly above the apex of sun's corona (Figure 1). The flash only occurs just after sunrise or sunset and never lasts more than a few seconds.
We will restrict the discussion to a particular type of green flash known as the inferior-mirage or Omega sunset
[2]. The photograph of the Omega sunset in Figure 1 was taken near sea level seconds before the sun fell behind the horizon.

A green flash is both a rare and spectacular event. One can not help but wonder the cause of such an event, and may initially presume air pollution is responsible for the phenomena. However, we will discover that the opposite is true; a green flash has a higher probability of occurring in an unpolluted area and is the result of basic physics, namely optics. We now lay the groundwork for this explanation by defining several key optical ideas.

### Background

We must first develop some basic notions of what constitutes "light".

##### Light

Light is one of the most mysterious objects in physics due to its dual particle and wave nature. Light demonstrates particle like properties such as scattering and absorbtion while simultaneously expressing the wavelike properties of wavelength, frequency and dispersion.

We define wavelength to be the distance between the maximizers of two consecutive periods of a wave. This is depicted in Figure 2 by the red line which has an approximate length of 7 units per cycle. The wavelengths of light range from Extremely Low Frequency ten million (1010) meters to 1 picometer
(10-12) gamma rays.
The human eye can detect a near infinitesimal subset of the overall range lying between 380 nanometers (violet) to 740 nanometers (red). Such light waves illuminate the world with color.
The blue dots in Figure 2 designate the termination of the wave period that lies on their left. We see that we have 6 complete cycles. Assume that the units on the horizontal axis now correspond to time in seconds. The sixth blue dot occurs at approximately 37 seconds. From this we determine the frequency of the wave to be 6/37 cycles per second. Light waves in the visible spectrum have frequencies of 4.3*1014 cycles per second (red) to 7.5* 1014(violet) cycles per second.

To obtain the velocity for a wave, we simply multiply its frequency by its wavelength. We find the wave in Figure 2 to have a velocity of 42/37 units per second.

Dispersion is the next fundamental wave property of light. The speed of light is constant in vacuum; however, when light enters a medium its speed slows with respect to the properties of the composition of the medium. Such information about a medium is encoded in a quantity known as the refractive index which gives the adjusted velocity of light. For example the refractive index of glass is approximately 1.5, so the velocity of light slows by 3/4 (decreases by 25 percent). Now consider the prism in Figure 3. The refractive index of air on the outside is 1.0003 and 1.50 inside the glass prism.
Note when the white light beam enters the prism its slope decreases. This is described in Snell's law
[3] and is referred to as refraction. When the white beam of light comes into contact with the prism, different wavelengths are refracted at different angles. This causes the rainbow effect seen in the inside of the prism. We call this property of light, dispersion. As the light passes through the interior of the prism and exits, refraction further separates the colors of light.

One can imagine that if light were to pass through several consecutive media of increasing density one would obtain the dispersion depicted in Figure 4.
As the refraction angles become small between consecutive media, light appears to "bend" in a smooth curve as approximated by Figure 4 rather than following a polygonal line. We call this the bending of light.

Finally scattering and transmission of light beams are essential to understanding the green flash. We draw attention to yellow arrows depicting an incident beam of white light travelling in a medium of refractive index n2 in Figure 5. As the beam inters a medium with refractive index of n2 the blue spectrum of the light is scattered while the red is refracted.

Now that we have discussed several fundamental optics concepts, we are prepared to explain why the sun appears red upon sunset.

### The Red Sun

Consider the sun at high noon when it has a bright yellow hue. The sunlight reaches the human eye after travelling through many layers of air of increasing density. Figure 5 is a depiction of what happens when light from the sun enters the Earth's atmosphere. Air molecules and aerosol particles scatter the shortest wavelengths(blue, violet) of light while absorbing the longer wavelengths (red, orange). This is why the strongly scattered blue light appears as the color of the atmosphere.

At sunset, light rays travel the greatest distance through the atmosphere. The bending of light in Figure 4 is more apparent to the eye at this time, so the red wavelengths, being the longest, reach an observer first followed by the rest of the spectrum (see applet).

If there were no particles in the atmosphere an observer of a sunset would witness a red sun with a rainbow-like corona. However, there are air particles which give the sun its red/orange tint. If a large amount of aerosol is present in the air at the horizon, larger amounts of orange light will be absorbed and the sun will appear a dark reddish color. The following applet animates this event.

Also note the apparent elongation of the sun while it is setting. This is also an effect of atmospheric dispersion.

### The Green Flash

The Omega sunset resulting in a green flash is represented in a series of snapshot diagrams in Figure 6. Note that the green hue is present in each diagram; however, it is obscured in reality by the red light rays.

Variations in atmospheric density are the result of a temperature gradient of cold air at lower altitude. Sometimes this gradient is locally inverted from which atmospheric mirages result. Such a mirage can be seen in the first snapshot of Figure 6.

As the sun descends, the mirage and the sun coalesce to form a column-like image at the base of the sun as seen in snapshots two through four. Upon further descent the column collapses into a football shape as seen in snapshots five and six.

At this point, the sun is actually below the horizon, but the bending of light gives the perception of it being above the horizon. Just prior to dusk, the time delay between the different wavelengths of light is maximized. This effect causes the green light to travel "behind" the red light (see applet).

When the light rays are obstructed by the physical horizon, the observer ceases to see the red part of the spectrum, but for a couple of seconds he still sees the green part of the spectrum which lagged behind the red. This causes the green flash seen in the last diagram of Figure 6.

### References

[1] Source Code
[2] Andrew T. Young: San Diego State University
[3] Eric Weisstein's World of Phyics