A H ¼
1
μ 0
S ⨀
r 2
h
Z
2π
I α, e, i
ð
Þcos edΩ s
ð2:82Þ
where I(α,e,i) is again the radiance from the surface element typically in units
[W m
À2 sr
À1 ]. The integral is over all solid angles, Ω s . Note that the integral on
the right-side is the flux from the surface as given by Eq. (2.59). A more general form
would also include the wavelength dependence of reflectance
A H ¼
r
2
h
μ 0
Z 1
0
1
F ⨀
Z
2π
I α, e, i, λ
ð
Þcos edΩ s dλ
ð2:83Þ
with I expressed in [W m
À2 sr nm
À1 ] and the solar flux is now F ⨀ in units
[W m
À2 nm
À1 ] as shown in Eq. (2.5). If the surface is Lambertian, then I(α,e,i) is
independent of the viewing geometry (i.e. it is constant) and Eq. (2.82) reduces to
A H ¼
π I α, e, i
ð
Þ
μ 0
S ⨀
r 2
h
ð2:84Þ
which is itself equal to the radiance coefficient, ρ C , as noted above. This illustrates
that A H expresses the ratio of scattered to incident light at a surface element with the
scattered light being integrated over all directions.
We note in passing that this quantity is occasionally referred to casually as the
“Bond albedo”. This is actually erroneous. The Bond albedo is the fraction of power
incident on an entire body, integrated over all wavelengths, that is scattered back out
into space and is used to determine, for example, a predicted value for the equilibrium temperatures of the giant planets. The concept here is similar because we are
seeking the total energy reflected over all wavelengths. However, here, it is being
applied locally on a surface element at an incidence angle specific to that element.
Furthermore, A H can vary with the angle of incidence because of the asymmetry of
the phase curve. Hence, the use of the term Bond albedo should be avoided.
Figure 2.20 shows the reflectance factor for observations of the nucleus of
1P/Halley at the time of the Giotto encounter. Unlike the observations of 67P, the
Giotto imaging system (HMC) only observed the nucleus from one geometry
(a phase angle of 107
) and hence it is difficult to compare the results. However,
the geometric albedo of 1P/Halley was estimated to be about 0.04 and the reflectance
near the terminator in Fig. 2.20 is around a factor of 10–12 less than this and so it
agrees roughly with the Rosetta observations. There seems little reason to doubt that
the photometric behaviour of the two surfaces are broadly similar.
2.6 Surface Reflectance
65
1
μ 0
S ⨀
r 2
h
Z
2π
I α, e, i
ð
Þcos edΩ s
ð2:82Þ
where I(α,e,i) is again the radiance from the surface element typically in units
[W m
À2 sr
À1 ]. The integral is over all solid angles, Ω s . Note that the integral on
the right-side is the flux from the surface as given by Eq. (2.59). A more general form
would also include the wavelength dependence of reflectance
A H ¼
r
2
h
μ 0
Z 1
0
1
F ⨀
Z
2π
I α, e, i, λ
ð
Þcos edΩ s dλ
ð2:83Þ
with I expressed in [W m
À2 sr nm
À1 ] and the solar flux is now F ⨀ in units
[W m
À2 nm
À1 ] as shown in Eq. (2.5). If the surface is Lambertian, then I(α,e,i) is
independent of the viewing geometry (i.e. it is constant) and Eq. (2.82) reduces to
A H ¼
π I α, e, i
ð
Þ
μ 0
S ⨀
r 2
h
ð2:84Þ
which is itself equal to the radiance coefficient, ρ C , as noted above. This illustrates
that A H expresses the ratio of scattered to incident light at a surface element with the
scattered light being integrated over all directions.
We note in passing that this quantity is occasionally referred to casually as the
“Bond albedo”. This is actually erroneous. The Bond albedo is the fraction of power
incident on an entire body, integrated over all wavelengths, that is scattered back out
into space and is used to determine, for example, a predicted value for the equilibrium temperatures of the giant planets. The concept here is similar because we are
seeking the total energy reflected over all wavelengths. However, here, it is being
applied locally on a surface element at an incidence angle specific to that element.
Furthermore, A H can vary with the angle of incidence because of the asymmetry of
the phase curve. Hence, the use of the term Bond albedo should be avoided.
Figure 2.20 shows the reflectance factor for observations of the nucleus of
1P/Halley at the time of the Giotto encounter. Unlike the observations of 67P, the
Giotto imaging system (HMC) only observed the nucleus from one geometry
(a phase angle of 107
) and hence it is difficult to compare the results. However,
the geometric albedo of 1P/Halley was estimated to be about 0.04 and the reflectance
near the terminator in Fig. 2.20 is around a factor of 10–12 less than this and so it
agrees roughly with the Rosetta observations. There seems little reason to doubt that
the photometric behaviour of the two surfaces are broadly similar.
2.6 Surface Reflectance
65