English: Luminous intensity versus phase angle for a sphere with a Lambertian or specular surface.

The y-axis gives the number of candelas emitted toward a distant observer per lumen of light reflected in all directions from a distant source. The number of lumens emitted is equal to the number hitting the sphere if the reflectivity is 1 (that is, if it is perfectly white in the Lambertian case). The phase angle is the angle between a line from the sphere to the source and a line from the sphere to the observer. In the Lambertian case the light hitting a spot is reflected in all directions according to Lambert's cosine law, so that the luminance of the spot is the same viewed from any angle. In the specular case, the light reflects in one direction, with angle of incidence equal to angle of reflexion. The formula for the Lambertian case is:

${\displaystyle {\frac {2}{3\pi ^{2}}}[(\pi -\phi )\cos \phi +\sin \phi ]}$

where ${\displaystyle \phi }$ is the phase angle. For the specular case, the value is the constant ${\displaystyle 1/(4\pi ).}$ (If the moon had a surface like a smooth mirror, then the amount of moonlight would not depend on the phase of the moon, but only on the Earth-moon and moon-sun distances.) In both cases, multiplying by ${\displaystyle 2\pi \sin \phi }$ and integrating over the whole range gives 1, the total amount of light reflected.

The graphs also apply for the ratio of radiant intensity to radiant flux.