ρ is the radial size of a circular aperture on the sky in units of [km] and is thus
equivalent to the impact parameter, b, in the nomenclature used here. f is a filling
factor which represents the ratio of the geometric cross-section of dust grains within
the aperture to the area of the aperture itself. A is an “albedo”, which we place in
inverted commas to denote that it is sometimes described as such but is actually the
product of the scattering efficiency, Q sca , and the scattering function Φ s i.e.
A ¼ 4π Φ s α
ð ÞQ sca
ð4:54Þ
where Φ S is the single scattering particle phase function and is again normalized such
that
Z
4π
Φ S dΩ ¼ 1
ð4:55Þ
Φ S is in units of [sr
À1 ]. (We repeat that caution should be exercised because there
are alternate normalizations for Φ S . Here we follow Divine et al. 1986.)
The key point of this definition is that while A and f cannot be determined
independently, the product, Af, is an observable by comparing the observed flux
from the comet to the flux from the Sun at the comet’s distance. Mathematically, this
is
Af ¼
2Δr h
b
2 F coma
F ⨀
ð4:56Þ
in which we have substituted our symbol definitions. The resulting Afρ quantity has
units of length. Free radial outflow would suggest that f decreases with 1/b and hence
Afρ should be independent of aperture size for distances where other effects (such as
radiation pressure) can be ignored.
The convenience of Afρ lies in the relatively straightforward comparison of dust
production rates with time and between comets. However, the dependence of Af on
several physical parameters should not be forgotten.
The filling factor can be related to the production rate through the column density,
N d (b),
f ¼
R b max
0
N d b
ð Þσ x 2πb db
R b max
0
2πb db
:
ð4:57Þ
The geometric cross-section, σ x , can be used for the particles in this equation if
other effects (such as the scattering efficiency and the phase function) are taken into
account in the computation of A as has been performed in Eq. (4.54). Substituting for
N d (b) in the equation for f gives
4.3 Afρ (“Afrho”)
307
equivalent to the impact parameter, b, in the nomenclature used here. f is a filling
factor which represents the ratio of the geometric cross-section of dust grains within
the aperture to the area of the aperture itself. A is an “albedo”, which we place in
inverted commas to denote that it is sometimes described as such but is actually the
product of the scattering efficiency, Q sca , and the scattering function Φ s i.e.
A ¼ 4π Φ s α
ð ÞQ sca
ð4:54Þ
where Φ S is the single scattering particle phase function and is again normalized such
that
Z
4π
Φ S dΩ ¼ 1
ð4:55Þ
Φ S is in units of [sr
À1 ]. (We repeat that caution should be exercised because there
are alternate normalizations for Φ S . Here we follow Divine et al. 1986.)
The key point of this definition is that while A and f cannot be determined
independently, the product, Af, is an observable by comparing the observed flux
from the comet to the flux from the Sun at the comet’s distance. Mathematically, this
is
Af ¼
2Δr h
b
2 F coma
F ⨀
ð4:56Þ
in which we have substituted our symbol definitions. The resulting Afρ quantity has
units of length. Free radial outflow would suggest that f decreases with 1/b and hence
Afρ should be independent of aperture size for distances where other effects (such as
radiation pressure) can be ignored.
The convenience of Afρ lies in the relatively straightforward comparison of dust
production rates with time and between comets. However, the dependence of Af on
several physical parameters should not be forgotten.
The filling factor can be related to the production rate through the column density,
N d (b),
f ¼
R b max
0
N d b
ð Þσ x 2πb db
R b max
0
2πb db
:
ð4:57Þ
The geometric cross-section, σ x , can be used for the particles in this equation if
other effects (such as the scattering efficiency and the phase function) are taken into
account in the computation of A as has been performed in Eq. (4.54). Substituting for
N d (b) in the equation for f gives
4.3 Afρ (“Afrho”)
307