What is meant by refractive angle for a rainbow is the angle between the light ray that enters a rain droplet and the light ray leaving that droplet. This angle determines the angle at which the top of the rainbow appears in the sky; that angle being the difference between the refractive angle and the elevation angle of the sun in the sky, as is shown in the diagram below. (For reflected rainbows a different relation applies.)
The deviation angle resulting from a ray of light entering a rain droplet and being reflected k times bebore leaving the droplet is given by the formula:
where i is the incident angle of the ray relative to the perpendicular to the droplet surface (measured in degrees) and r is the refracted angle of the ray relative to the perpendicular to the droplet surface. Δ is the deviation of the ray leaving the droplet from the ray entering the droplet.
The rainbow is formed from the light rays of incidence angle i that produce the maximum value for Δ. The Δ maximizing value of i is found from the first order condition
Let n denote the index of refraction of water. The value of dr/di is found from Snell's Law; i.e.,
The above relation for (dr/di) and the firtst order condition together imply that
Snell's Law gives
This relationship combined with the previous relationship derived from the first order condition gives:
This determines i and with i and Snell's Law r can be determined and hence the value of Δ. For an index of refraction of 1.33, a roughly average value for visible light in water and a value of k of 1:
For k=2 the values are:
Since the index of refraction is dependent upon the wavelength of the light it is of interest to determine dΔ/dn.
From the equation for Δ
From Snell's Law
This means that
From Snell's Law
From the condition for determining i we have
The angular width of a rainbow is proportional to dΔ/dn so the above formula says that the secondary rainbow (k=2) should be wider than the primary rainbow (k=1). According to the formula the ratio of the widths should be about 1.8.