Re: Why can rainbows only be seen at specific angles?

Differentiating a function and setting it equal to zero is a common method
in calculus that allows one to calculate minima and maxima on a curve. 
When you do this to the function of angle of incidence vs. angle of
refraction for light on a droplet of water, you are solving for the angle
at which the change in angle of refraction is minimum per change in angle
of incidence.  In doing so, you calculate the angle near which an observer,
standing parallel and away from the incident beam of light, capture the
most refracted light.  A different, and perhaps more instructive treatment
of calculating the rainbow angle can be found at:
 http://paos.colorado.edu/~fasullo/pjw_class/rainbows.html

However, what does the rainbow angle really mean?  Just because it's called
the "rainbow angle," one might conclude that it describes the angle above
the ground in which one might expect to see a rainbow.  But common
experience tells you this is not true.  Even the arc of common rainbows
span large vertical angles in your field of vision.  The important thing to
understand about the rainbow angle is that it is a physical description of
a microscopic interaction between a source of light and its path through a
spherical droplet of water.  The macroscopic phenomenon of a rainbow is the
cumulative effect of these light-water interactions, combined with the
relative position of the sun, the viewer, and the geospatial distribution
of water droplets.  The "42" degrees that is commonly calculated in
textbooks brushes over the true phenomenon of rainbows with an artificial
situation in which the sunlight is shining directly from behind the viewer
(parallel to the ground), and that viewer is looking at the
reflection/refraction of the light on water droplets that exist on a plane
orthogonal to the ground and the direction of the light.  It doesn't
consider the distance and depth of the rain or the time of day.  If this
were indeed the case, then you might in fact see a small patch of rainbow
coming from 42 degrees above the horizon.  So while calculating the rainbow
angle is an informative mathematical treatment of the interaction of light
with droplets of water, the calculation should not be mistaken with any
kind of direct and simple relationship with the experience of rainbows in
the atmosphere.  


One interesting and often overlooked aspect of the rainbow angle is that
you can calculate a different value for different colors.  This is due to
the changing index of refraction as a function of wavelength.  I calculate
violet light (index of refraction 1.3422 at about 400nm) to have a rainbow
angle of 40.76 degrees and red light (index of refraction 1.3269 at about
700nm) to have a rainbow angle of 42.97 degrees.  The 2.21 degree
difference between these two colors would correspond to the angular width
of the rainbow band.  Although you will rarely see a rainbow at 42 degrees
above the horizon, the width of the primary rainbow band between 400-700nm
should occupy the same angular width in just about every rainbow you see.