Equations for Albedo

  • Albedo A presents geometric albedo at a specific phase (scattering) angle as defined in [1]. For a single material, each filter and each size of particles, it was calculated following the approach described in [2]: \[ A(\theta) = \pi Q_{\text{sca}}\Phi_{\text{S}}(\theta) \] where \[ \Phi_{\text{S}}(\theta) = \frac{S_{11}(\theta)\lambda^2}{4 \pi^3 r^2_{\text{eff}} Q_{\text{sca}}}\] normalized such that \[ 2 \pi \int_0^\pi \Phi_{\text{S}}(\theta)\sin{(\theta)} \text{d}\theta = 1\] $S_{11}$ is the 11 element of the Mueller scattering matrix
    $Q_{\text{sca}}$ is the scattering efficiency (averaged over the size distribution)
    $\lambda$ is the wavelength of light (red or blue filter)
    $r_{\text{eff}}$ is the effective radius of particles
    In the database the phase angles equal to 180 - 0 are used
  • For a mixture of two materials, albedo was calculated as \[A(\theta) = \frac{r1_{\text{eff}}^2 A1(\theta)r + r2_{\text{eff}}^2 A2(\theta)(1-r)}{r1_{\text{eff}}^2 r + r2_{\text{eff}}^2 (1-r) }\] where $A1$ and $A2$ correspond to the albedo of material 1 (e.g., water ice) and material 2 (e.g., olivine), $r1_{\text{eff}}$ and $r2_{\text{eff}}$ are effective radii or particles for material 1 and 2, and $r$ is the fraction of the particles of the material 1 in the mixture, e.g., 0.1 means that each 10 particles in the mixture contain 1 particle made of material 1 and 9 particles made of material 2; 0.9 means that among 10 particles 9 are made of material 1 and 1 is made of material 2.

References

[1]. Hanner, M. S., Giese, R. H., Weiss, K., & Zerull, R., 1981, On the definition of albedo and application to irregular particles. Astronomy & Astrophysics,104, 42-46.

[2]. Bertini, I., Thomas, N. & Barbieri, C., 2007. Modeling of the light scattering properties of cometary dust using fractal aggregates. Astronomy & Astrophysics, 461(1), 351-364.