The planets and all other bodies of the solar system only reflect the radiation of the Sun (we may neglect here the thermal and radio wave radiation and concentrate mainly on the visual wavelengths). The brightness of a body depends on its distance from the Sun and the Earth, and on the albedo of its surface. The term albedo defines the ability of a body to reflect light.

If the luminosity of the Sun is LQ, the flux density at the distance r is (Fig. 7.19)


If the radius of the planet is R, the area of its cross section is nR2, and the total flux incident on the surface




Fig. 7.19. Symbols used in the photometric formulas


Fig. 7.19. Symbols used in the photometric formulas

of the planet is





where the integration is extended over the surface of the sphere of radius A. The surface element of such a sphere is dS = A2 da sin a d0, and we have

The normalisation constant C is 4nA2 2

4nr2 4r2

Only a part of the incident flux is reflected back. The other part is absorbed and converted into heat which is then emitted as a thermal emission from the planet. The Bond albedo A (or spherical albedo) is defined as the ratio of the emergent flux to the incident flux (0 < A < 1). The flux reflected by the planet is thus AL R2

The planet is observed at a distance A. If radiation is reflected isotropically, the observed flux density should be


fS 0(a) d S ¡n 0(a) sin a da The quantity n q = if 0(a) sin a da o

is the phase integral. In terms of the phase integral the normalisation constant is

Remembering that Lout = ALin, the equation (7.23) can be written in the form

In reality, however, radiation is reflected anisotropically. If we assume that the reflecting object is a homogeneous sphere, the distribution of the reflected radiation depends on the phase angle a only. Thus we can express the flux density observed at a distance A as

The function 0 giving the phase angle dependence is called the phase function. It is normalised so that 0(a = 0°) = 1.

Since all the radiation reflected from the planet is found somewhere on the surface of the sphere, we must have


CA 1

The first factor is intrinsic for each object, the second gives the phase angle dependence, the third the distance dependence and the fourth, the incident radiation power. The first factor is often denoted by

When we substitute here the expression of C (7.29), and solve for the Bond albedo, we get

4nr 4

Here p = nr is called the geometric albedo and q is the previously introduced phase integral. These quantities are related by

The geometric albedo seems to have appeared as an arbitrary factor with no obvious physical interpretation. We'll now try to explain this quantity using a Lam-bertian surface. A Lambertian surface is defined as an absolutely white, diffuse surface which reflects all radiation, i. e. its Bond albedo is A = 1. Moreover, its surface brightness is the same for all viewing directions, which means that the phase function is

In reality, no such surface exists but there are some materials which behave almost like a Lambertian surface. A wall with a mat white finish is a good approximation; although it doesn't reflect all incident light, the distribution of the reflected light is about right, and its brightness looks the same from all directions.

For a Lambertian surface the constant C is 2

Thus the geometric albedo of a Lambertian surface is


It turns out that p can be derived from the observations, but the Bond albedo A can be determined only if the phase integral q is also known. That will be discussed in the next section.

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