Colour Brightness & Angular Distance

Rainbow colour refers to the colours seen in rainbows and other situations where visible light separates into its component wavelengths and the corresponding hues become visible to the human eye.

• Rainbow colour (also called spectral colour) is a colour model.
• A colour model is a theory of colour that establishes terms, definitions, rules and conventions for understanding and describing colours and their relationships with one another.
• A spectral colour is a colour evoked in normal human vision by a single wavelength of visible light (or by a narrow spread of adjacent wavelengths).
• When all the spectral colours are mixed together in equal amounts and at equal intensities, they produce white light.
• In order of wavelength, the rainbow colours (ROYGBV) are red (longest visible wavelength), orange, yellow, green, blue and violet (shortest visible wavelength).
• It is the sensitivity of the human eye to this small part of the electromagnetic spectrum that results in our perception of colour.
• Whilst the visible spectrum and its spectral colours are determined by wavelength (and corresponding frequency), it is our eyes and brains that interpret these differences in electromagnetic radiation and produce colour perceptions.
• Naming rainbow colours is a matter more closely related to the relationship between perception and language than anything to do with physics or optics.
• Even commonplace colour names associated with rainbows such as yellow or blue defy easy definition. These names are concepts related to subjective impressions.
• Modern portrayals of rainbows show six colours – ROYGBV. This leaves out other colours such as cyan and indigo.
• Atmospheric rainbows actually contain millions of spectral colours. Measured in nanometres there are around 400 colours between red and violet, measured in picometres there are 400,000.

Angular distance is the angle between the rainbow axis and the direction in which an observer must look to see a specific colour within the arcs of a rainbow.

• Angular distance, viewing angle and angle of deflection all produce the same value measured in degrees.
• Angular distance is a measurement on a ray-tracing diagram that represents the Sun, an observer and a rainbow in side elevation.
• Think of angular distance as an angle between the centre of a rainbow and its coloured arcs with red at 42.4⁰ and violet at 40.7⁰.
• Angular distances for different colours are constants determined by the laws of refraction and reflection.
• The elevation of the Sun, the location of the observer and exactly where rain is falling are all variables that determine where a rainbow will appear to an observer.
• The coloured arcs of a rainbow form the circumference of circles (discs or cones) and share a common centre.
• The angular distance to any specific colour is the same whatever point is selected on the circumference.
• The angular distance for any observed colour in a primary bow is between 42.4⁰ and violet at 40.7⁰.
• The angular distance for any observed colour in a secondary bow is between 53.4⁰ and 50.4⁰ from its centre.
• The angular distance can be calculated for any specific colour visible within a rainbow.
• Considered from an observer’s viewpoint, it is clear that all incident rays seen by an observer run parallel with each other as they approach a raindrop.
• Most of the observable incident rays that strike a raindrop follow paths that place them outside the range of possible viewing angles. The unobserved rays are all deflected towards the centre of a rainbow.

Viewing angle, angular distance and angle of deflection

• The term viewing angle refers to the number of degrees through which an observer must move their eyes or turn their head to see a specific colour within the arcs of a rainbow.
• The term angular distance refers to the same measurement when shown in side elevation on a diagram.
• The angle of deflection measures the degree to which a ray striking a raindrop is bent back on itself in the process of refraction and reflection towards an observer.
• The term rainbow ray refers to the path taken by the deflected ray that produces the most intense colour experience for any particular wavelength of light passing through a raindrop.
• The term angle of deviation measures the degree to which the path of a light ray is bent back by a raindrop in the course of refraction and reflection towards an observer.
  • In any particular example of a ray of light passing through a raindrop, the angle of deviation and the angle of deflection are directly related to one another and together add up to 180⁰.
  • The angle of deviation is always equal to 180⁰ minus the angle of deflection. So clearly the angle of deflection is always equal to 180⁰ minus the angle of deviation.
  • In any particular example, the angle of deflection is always the same as the viewing angle because the incident rays of light that form a rainbow are all approaching on a trajectory running parallel with the rainbow axis.