Hapke parameters

The Hapke parameters are a set of parameters for an empirical model that is commonly used to describe the directional reflectance properties of the airless regolith surfaces of bodies in the Solar System. The model has been developed by astronomer Bruce Hapke at the University of Pittsburgh.

The main parameters are: ^[1]

1. ${\displaystyle {\bar {\omega }}_{0}}$ — Single scattering albedo. This is the ratio of scattering efficiency to total light extinction (which includes also absorption), for small-particle scattering of light. That is, ${\displaystyle K_{s}/(K_{s}+K_{a})}$, where ${\displaystyle K_{s}}$ is the scattering coefficient, and ${\displaystyle K_{a}}$ is the absorption coefficient ^[2]
2. ${\displaystyle h}$ — The width of the opposition surge.
3. ${\displaystyle B_{0}}$ or ${\displaystyle S_{0}}$ — The strength of the opposition surge.
4. ${\displaystyle P_{0}}$ or g — The particle phase function parameter, also called the asymmetry factor.
5. ${\displaystyle \theta }$ — The effective surface tilt, also called the macroscopic roughness angle.

The Hapke parameters can be used to derive other albedo and scattering properties, such as the geometric albedo, the phase integral, and the Bond albedo.

Limitations of the Hapke Model

The Hapke model has been criticized for having a large number of unconstrained parameters that can make the parameters difficult to constrain. ^{[3] [4]} In particular, the model has been shown to often require unphysical parameters (e.g., the asymmetry factor) when modeling the albedo of ices. ^{[5] [6] [7] [8]}

See also

• Albedo
• Geometric albedo
• Bidirectional reflectance distribution function

References

1. P. Helfenstein; et al. (1996). "Galileo Photometry of Asteroid 243 Ida". Icarus. 120 (1): 48–65. Bibcode:1996Icar..120...48H. doi: 10.1006/icar.1996.0036 .
2. Single Scattering Albedo, scienceworld.wolfram.com.
3. Shkuratov, Y.; Kaydash, V.; Korokhin, V.; Velikodsky, Y.; Petrov, D.; Zubko, E.; Stankevich, D.; Videen, G. (2012-12-01). "A critical assessment of the Hapke photometric model" . Journal of Quantitative Spectroscopy and Radiative Transfer . Electromagnetic and Light Scattering by non-spherical particles XIII. 113 (18): 2431–2456. Bibcode:2012JQSRT.113.2431S. doi:10.1016/j.jqsrt.2012.04.010. ISSN   0022-4073.
4. Hapke, Bruce (2013-02-01). "Comment on "A critical assessment of the Hapke photometric model" by Y. Shkuratov et al" . Journal of Quantitative Spectroscopy and Radiative Transfer . 116: 184–190. Bibcode:2013JQSRT.116..184H. doi:10.1016/j.jqsrt.2012.11.002. ISSN   0022-4073.
5. Mishchenko, Michael I. (1994-07-01). "Asymmetry parameters of the phase function for densely packed scattering grains". Journal of Quantitative Spectroscopy and Radiative Transfer . 52 (1): 95–110. Bibcode:1994JQSRT..52...95M. doi:10.1016/0022-4073(94)90142-2. ISSN   0022-4073.
6. Mishchenko, M. I.; Macke, A. (1997-06-01). "Asymmetry parameters of the phase function for isolated and densely packed spherical particles with multiple internal inclusions in the geometric optics limit" . Journal of Quantitative Spectroscopy and Radiative Transfer . 57 (6): 767–794. Bibcode:1997JQSRT..57..767M. doi:10.1016/S0022-4073(97)00012-5. ISSN   0022-4073.
7. Warren, Stephen G.; Brandt, Richard E.; O'Rawe Hinton, Patricia (1998). "Effect of surface roughness on bidirectional reflectance of Antarctic snow" . Journal of Geophysical Research: Planets. 103 (E11): 25789–25807. Bibcode:1998JGR...10325789W. doi:10.1029/98JE01898. ISSN   2156-2202.
8. Khuller, Aditya R.; Emran, Al (2025-05-31). "Quantitative evaluation of the delta-Eddington, Hapke, and Shkuratov models for predicting the albedo and inferring the grain radius of ice" . Icarus 116671. arXiv: 2509.06224 . doi:10.1016/j.icarus.2025.116671. ISSN   0019-1035.