where we have indicated the quantities that are affected by the choice of particle type
(Divine et al. 1986). Once the scattered radiance has been found, the reflectance is
computed according to Eq. (2.62).
There are obviously ways in which this equation can be simplified if assumptions
can be made. For example, assuming a single particle type and size, we can remove
the integrals and, by substitution for Q sca we can obtain
I ¼ N d σ ext
1
Q ext
p d
π
Φ s θ
ð Þ
Φ s π
ð Þ
F ⨀
r 2
h
ð4:44Þ
where N d is the column density and
ρ F ¼ N d σ x p d
Φ s θ
ð Þ
Φ s π
ð Þ
:
ð4:45Þ
These equations can be useful as long as the importance of the assumptions is
appreciated.
4.2.9.2 Optical Thickness Effects
For an active comet such as 1P/Halley or comets that make close approaches to the
Sun (e.g. 96P/Machholz), the dust production rate can be sufficiently large that
optical depth effects need to be accounted for. The optical depth through a part of the
dust coma can be expressed as
τ d ¼
Z s 2
s 1
Z 1
0
n d a
ð Þ σ ext a
ð Þ da ds
ð4:46Þ
where n d (a) is a local dust density of particles of radius, a, and s is a distance along a
line of sight. This equation is valid for a single particle type (i.e. no variation in m ref )
but can be summed over all particle types as shown in Divine et al. (1986) to give
τ d ¼
Z s 2
s 1
X
k T
Z 1
0
n k T a
ð Þ σ ext a, k T
ð
Þda ds
ð4:47Þ
where n k is the density of each particle type. It should be clear here that the optical
depth seen by an observer viewing the dust coma is not the only optical depth of
importance in the computation of radiances. The solar irradiance of the dust coma
can also be affected by optical depth effects. This is particularly true along the
Sun-nucleus line because the peak dust production is usually related to the maximum
insolation and thus is often assumed to be from the sub-solar point. Hence, the
maximum optical depth towards the nucleus is, to first order, from the direction of
4.2 Scattering of Light by Dust
303
(Divine et al. 1986). Once the scattered radiance has been found, the reflectance is
computed according to Eq. (2.62).
There are obviously ways in which this equation can be simplified if assumptions
can be made. For example, assuming a single particle type and size, we can remove
the integrals and, by substitution for Q sca we can obtain
I ¼ N d σ ext
1
Q ext
p d
π
Φ s θ
ð Þ
Φ s π
ð Þ
F ⨀
r 2
h
ð4:44Þ
where N d is the column density and
ρ F ¼ N d σ x p d
Φ s θ
ð Þ
Φ s π
ð Þ
:
ð4:45Þ
These equations can be useful as long as the importance of the assumptions is
appreciated.
4.2.9.2 Optical Thickness Effects
For an active comet such as 1P/Halley or comets that make close approaches to the
Sun (e.g. 96P/Machholz), the dust production rate can be sufficiently large that
optical depth effects need to be accounted for. The optical depth through a part of the
dust coma can be expressed as
τ d ¼
Z s 2
s 1
Z 1
0
n d a
ð Þ σ ext a
ð Þ da ds
ð4:46Þ
where n d (a) is a local dust density of particles of radius, a, and s is a distance along a
line of sight. This equation is valid for a single particle type (i.e. no variation in m ref )
but can be summed over all particle types as shown in Divine et al. (1986) to give
τ d ¼
Z s 2
s 1
X
k T
Z 1
0
n k T a
ð Þ σ ext a, k T
ð
Þda ds
ð4:47Þ
where n k is the density of each particle type. It should be clear here that the optical
depth seen by an observer viewing the dust coma is not the only optical depth of
importance in the computation of radiances. The solar irradiance of the dust coma
can also be affected by optical depth effects. This is particularly true along the
Sun-nucleus line because the peak dust production is usually related to the maximum
insolation and thus is often assumed to be from the sub-solar point. Hence, the
maximum optical depth towards the nucleus is, to first order, from the direction of
4.2 Scattering of Light by Dust
303