μ ¼ cos φ M cos λ M
ð2:77Þ
and
μ 0 ¼ cos φ M cos α À λ M
ð
Þ
ð2:78Þ
where λ M is the photometric longitude, φ M is the photometric latitude and
0 < k M < 1 is a parameter. A M is the Minnaert albedo and typically one would
adjust this value to match observational data. The Minnaert function has little basis
in theory, is not derived from first principles, is restricted in validity to certain classes
of scattering geometries, and, perhaps most importantly, cannot be used to interpret
photometric behaviour in terms of the physical properties of the reflecting surface
(Meador and Weaver 1975).
The Lunar-Lambert law used by McEwen (1991) takes the form
ρ F α, e, i
ð
Þ¼A N 2k E
μ 0
μ 0 þ μ
þ 1 À k E
ð
Þμ 0
!
ð2:79Þ
where A N is the normal albedo and k E is a parameter. It has some uses as a simple
empirical fit to photometric data especially where there are limited numbers of
observations.
As pointed out by Shepard (2017), the definition of the normal albedo has some
ambiguity. In some texts the normal albedo is defined as being the ratio of the
radiance from a surface seen at zero phase angle and an emission angle of zero (I
(0,0,0)) to that of a perfectly diffusing Lambertian surface in the same geometry
(I L (0,0,0)), i.e.
A N ¼
I 0, 0, 0
ð
Þ
I L 0, 0, 0
ð
Þ
ð2:80Þ
whereas other texts define the quantity at zero phase but with variable emission
angle, i.e.,
A N ¼
I e, e, 0
ð
Þ
I L 0, 0, 0
ð
Þ
ð2:81Þ
There are subtle differences and it should be noted that techniques such as laser
altimetry lead to use of the latter (more imprecise) definition because surface slopes
may not be known a priori. However, in the case of McEwen’s formula, the
difference in definition would usually be of little significance.
Fitting of the bidirectional reflectance over the hemisphere allows the derivation
of the directional-hemispherical albedo, A H , introduced in Eq. (2.4), by using its
defining equation
64
2 The Nucleus
ð2:77Þ
and
μ 0 ¼ cos φ M cos α À λ M
ð
Þ
ð2:78Þ
where λ M is the photometric longitude, φ M is the photometric latitude and
0 < k M < 1 is a parameter. A M is the Minnaert albedo and typically one would
adjust this value to match observational data. The Minnaert function has little basis
in theory, is not derived from first principles, is restricted in validity to certain classes
of scattering geometries, and, perhaps most importantly, cannot be used to interpret
photometric behaviour in terms of the physical properties of the reflecting surface
(Meador and Weaver 1975).
The Lunar-Lambert law used by McEwen (1991) takes the form
ρ F α, e, i
ð
Þ¼A N 2k E
μ 0
μ 0 þ μ
þ 1 À k E
ð
Þμ 0
!
ð2:79Þ
where A N is the normal albedo and k E is a parameter. It has some uses as a simple
empirical fit to photometric data especially where there are limited numbers of
observations.
As pointed out by Shepard (2017), the definition of the normal albedo has some
ambiguity. In some texts the normal albedo is defined as being the ratio of the
radiance from a surface seen at zero phase angle and an emission angle of zero (I
(0,0,0)) to that of a perfectly diffusing Lambertian surface in the same geometry
(I L (0,0,0)), i.e.
A N ¼
I 0, 0, 0
ð
Þ
I L 0, 0, 0
ð
Þ
ð2:80Þ
whereas other texts define the quantity at zero phase but with variable emission
angle, i.e.,
A N ¼
I e, e, 0
ð
Þ
I L 0, 0, 0
ð
Þ
ð2:81Þ
There are subtle differences and it should be noted that techniques such as laser
altimetry lead to use of the latter (more imprecise) definition because surface slopes
may not be known a priori. However, in the case of McEwen’s formula, the
difference in definition would usually be of little significance.
Fitting of the bidirectional reflectance over the hemisphere allows the derivation
of the directional-hemispherical albedo, A H , introduced in Eq. (2.4), by using its
defining equation
64
2 The Nucleus