2.9: Spheres - Bond Albedo, Phase Integral and Geometrical Albedo

Originally defined for a sphere, the Bond albedo is defined as the ratio of the total power P_r scattered by the sphere to the total power P_i intercepted by it.

If we let the intensity of the sphere as a function of solar phase angle α be I(α) watts per steradian, then the total scattered flux may be obtained by multiplying by 2π sin α dα and integrating over α from 0 to π

\[ P_r = 2 \pi \int_0^{\pi} I ( \alpha ) \sin \alpha d \alpha,\]

which can be expressed in terms of the normalised phase law ψ(α) = I(α)/I(0)

\[ P_r = 2 \pi I(0) \int_0^{ \pi} \phi ( \alpha ) \sin \alpha d \alpha.\]

For a sphere of radius α, the intercepted flux is simply P_i = π α² F, so that the Bond albedo may be expressed as

\[ A = \frac{I(0)}{ \alpha^2 \textbf{F}} \times 2 \int_0^{ \pi} \phi ( \alpha ) \sin \alpha d \alpha = p \times q\]

in which it may be seen as the product of two factors, the second of which,

\[ q = 2 \int_0^{ \pi} \phi ( \alpha ) \sin \alpha d \alpha,\]

is called the phase integral, which depends only on the directional reflecting properties of the planet. The first factor

\[ p = \frac{I(0)}{ \alpha^2 \textbf{F}}\]

depends only on the geometrical and photometric properties of the planet when observed at full phase. The quantity p is itself a (kind of) albedo since α²F can be seen as the intensity, scattered back towards the source, of a normally irradiated lossless (ϖ₀=1) Lambertian disc of the same radius as the planet. The factor p is called the geometrical albedo. [When albedo is used without qualification in the context of the photometry of asteroids it (usually) means geometrical albedo, in particular that observed in the Johnson V-band, p_V, the visual geometrical albedo].

For the reflectance rules we have considered so far, i.e. Lambert’s law and the Lommel-Seeliger law, analytical expressions for A, p and q are readily found, as summarised in Table II.

Table II. Properties of Spheres

	Lambertian	Lommel-Seeliger
q	\( \frac{3}{2}\)	\( \frac{16}{3} (1 - \ln 2)\)
p	2ϖ0/3	ϖ0/8
A	ϖ0	\( \frac{3}{2} \varpi_0 (1 - \ln 2)\)

More complicated reflectance laws, in particular those which address the problem of the opposition effect for atmosphereless bodies do not readily lend themselves to analytical solutions. In general, such laws exhibit a BRDF which depends on phase angle α and a possible set of reflectance parameters, symbolised by the ellipsis, so that the BRDF would be generally expressed in the form

\[ f_r = f_r \left( \mu_0, \mu, \alpha; ... \right),\]

where the dependence on φ and φ₀ has been replaced by α, the angle between the incident and scattered radiation, i.e. α does not always mean solar phase angle.