The Rainbow

The applet is simulating a rainbow and based on geometrical optics only, using Snell's law of refraction, named after Dutch astronomer Willebrord Snellius (1580–1626):
Snell's law of refraction
with angles α (incident) and β (refracted) measured from the normal of the boundary, and
with the refractive index n of the respective medium.

For air, 10 °C, pressure 100 kPa, 100% humidity, 520 nm, (o):
n = 1.000283
For water the following values are used (**):
violet, 400 nm: n = 1.3443
green, 520 nm: n = 1.3366
red,    750 nm: n = 1.3302
In a primary rainbow, the arc shows red on the outer part and violet on the inner side. This rainbow is caused by light being refracted when entering a droplet of water, then reflected once inside on the back of the droplet and refracted again when leaving it.

In a double rainbow two inner reflections occur, and a second arc is seen outside the primary arc, and has the order of its colours reversed, with red on the inner side of the arc.

The Rainbow Applet is using Snell's law of refraction and the law of reflexion only. The  impact parameter of minimum/maxium deflexion was computed using 5000 rays.

Applet Results Primary Rainbow:

index n
parameter b/R
violet 1.3443 40.52° 139.48° 0.8552 58.78°
1.3366 41.61° 138.39° 0.8592 59.23°
1.3302 42.54° 137.46° 0.8624 59.59°
width of rainbow:  2.02°

Applet Results Secondary rainbow:

index n
parameter b/R
1.3443 53.76° 126.24° 0.9482
1.3366 51.78° 128.22° 0.9494 71.70°
1.3302 50.10° 129.90° 0.9506 71.92°
width of rainbow:   3.66°
For impact parameters b/R smaller than at extremal deflection the rays extended backwards are convergent (gray), otherwise they are divergent (check the box "Extend").
extend box

convergent divergent

The cone of rainbow is centered on the line to the antisolar point, opposite the Sun from the viewpoint of an observer. The primary rainbow is visible only when the altitude H of the sun is less than 42°. :

rainbow cone

rainbow angles
obsever: O
antisolar point: S
altitude of the Sun: AOS = H
horizon: CD
axis of rainbow cone: OS
rainbow angle: BOS = φ = 42°
visible part of rainbow: CBD
angle of rainbow above horizon:  ∠AOB = h
angle of bow ends on the horizon: AOD = α

φ = H + h

rainbow angles

The angle α as a function of the Sun's altitude H is shown in the following diagram:

rainbow angle horizon

The dark area of unlit sky lying between the primary and secondary bows is called Alexander's band, after Alexander of Aphrodisias who first described it (200 AD).

Incident angle and deflction angle

Incident angle and deflction angle

Angular distribution

For the primary rainbow, using 5000 rays, they are distributed like this. The rays are concentrated at the extreme angle of deflection (41.61°, or 138.39°, for green):

angular distribution of
                      deflected rays

Distribution for secondary rainbow:

angular distribution of deflected rays

The interval of time for observation, if any, can be evaluated for any location and date of the year by my Azimuth and Elevation Applet, eg. Berlin (52.51° N, 13.41° E):

time of observation 42°

Select "Elevation/AM" or "Elevation/PM" from the menu: e. g. on May 10, the altitude of the Sun is less than 42° from sunrise to about 10:10 and from about 15:50 to sunset (local time CEST) :

rainbow AM  
altitude 42° PM

The intensity ratio of primary and secondary rainbow can be calculated using the interval of the impact parameter b/R for angles ±1° below or above the the minimum or maximum deflexion angle:

intensity primary

intensity primary

intensity primary
                      secondary rainbow

The intensity ratio secondary/primary can by aproximated by the areas of the arcs

(0.9712 - 0.9202)/(0.9122 - 0.7902) = 0.0964/0.208 = 0.464

For the primary rainbow the rainbow angle can be calculated:

At extreme deflection (primary and secondary rainbow) the incident angles α and relative impact parameters are:
incident angle
                          primary rainbow         incident angle secondary
relative impact parameter

The equations above are deduced here:       

The Mathematics of the Rainbow

My rainbow experiment using a
Glass Ball

Estimating the rainbow angle

M. Minnaert (Google Books)

In his book, Minnaert presents three equations to estimate the rainbow angle:

Minnaert rainbow

The first equation is obvious, the second can be derived easily.
In the 1993 edition of his book, and in the German edition, the third equation has a printing error:
tan(r)=1 - [1-cos(α)cos(h)]/[cos(α)cos(h)] is wrong.

Refractive index of water:

refractive index of water

Refractive index of water, based on a table  (*)
and approximated by a 3rd order polynom.

The refractive indices used in my applet (at 400 520, 750 nm) were computed by the formula in (**)
for 10° C, mass desity of water 0.999702 g/cm^3 (***)


's law (1621) was a precondition for a quantitative theory.

René Descartes was responsible for the first detailed study of the rainbow in Europe. He showed by using geometric construction and the law of refraction that the angular radius of a rainbow is 42 degrees ("Dioptrics", 1637).

Isaac Newton discovered by his experiences with prisms that white light was composed of the light of all the colours of the rainbow ("Opticks", 1704).

The wave theory of light was established by Thomas Young (1773-1829) by his interference experiments.

George Biddell Airy applied the wave theory, taking into account interference and diffraction ("On the intensity of light in the neighbourhood of a caustic", 1838).  He showed that the intensity of light in a rainbow could be modelled using a cubic wave-front and calculated a table of intensity values by his "Airy Integral".

Gustav Mie worked on the scattering of an electromagnetic wave by a homogeneous dielectric sphere applying Maxwell's equations ("Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", 1908).

Web Links
Refractive Index of Air Calculator (NIST)

(*) The Mathematics of Rainbows (American Mathematical Society)

http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6TVP-4442FHH-2-25J&_cdi=5540&_user=650619&_orig=browse&_coverDate=01%2 The mathematical physics of rainbows and glories (John A. Adam)


Mie theory, Airy theory, and the natural rainbow (Raymond L. Lee, Jr.)

Refractive index of water

Optical properties of water and ice (Wikipedia)

Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region Measurement of the refractive index of distilled water from the near-infrared region to the ultraviolet region

The International Association for the Properties of Water and Steam (**) Release on the Refractive Index of Ordinary Water Substance as a Function of Wavelength, Temperature and Pressure (The International Association for the Properties of Water and Steam)

(***) Water Density Calculator (Frostburg State University)

Light dispersion (Universitat de Barcelona)

Regenbogen (Peter Heiß)

The Calculus of Rainbows (James Stewart)

The Calculus of Rainbows (Jesse Amundsen)

Airy theory and rainbows (Philip Laven)
The International Association for the Properties of Water and Steam
Minnaert, Marcel: The Nature of Light and Colour in the Open Air; Dover Pubn Inc; revised ed. 1973; ISBN 978-0486201962.

Minnaert, Marcel: Light and Color in the Outdoors; Springer; 1st ed. 1993. Corr. 2nd printing 1995; ISBN 978-0387979359.
Extract: Google Books

2016-2023 J. Giesen

updated: 2023, Oct 06