Apparent magnitude of earthshine: a simple calculation

Abstract

The Sun illuminates both the Moon and the Earth with practically the same luminous fluxes which are in turn reflected by them. The Moon provides a dim light to the Earth whereas the Earth illuminates the Moon with somewhat brighter light which can be seen from the Earth and is called earthshine. As the amount of light reflected from the Earth depends on part of the Earth and the cloud cover, the strength of earthshine varies throughout the year. The measure of the earthshine light is luminance, which is defined in photometry as the total luminous flux of light hitting or passing through a surface. The expression for the earthshine light in terms of the apparent magnitude has been derived for the first time and evaluated for two extreme cases; firstly, when the Sun’s rays are reflected by the water of the oceans and secondly when the reflector is either thick clouds or snow. The corresponding values are and respectively. The earthshine value reported by Jackson lies within these apparent magnitudes. This paper will motivate the students and teachers of physics to look for the illuminated Moon by earthlight during the waning or waxing crescent phase of the Moon and to reproduce the expressions derived here by making use of the inverse-square law of radiation, Planck’s expression for the power in electromagnetic radiation, photopic spectral luminous efficiency function and expression for the apparent magnitude of a body in terms of luminous fluxes.