backscattering peak suggests a higher radiation pressure cross-section. We saw
earlier in the discussion of the fountain model that the large scale structure of the
coma is influenced by the value of β which is related to the scattering through the
radiation pressure cross-section, Q pr . In the past, it has been assumed that β can take
quite large values (see for example the works of Sekanina and co-authors in the
1980s where values of β > 2.0 can be found) and that this is the result of the dust
grains being more porous than solid spheres producing a larger backscattering peak.
Recent work by Silsbee and Draine (2016) using T-matrix and DDA calculations
suggest, however, that this is not an adequate explanation. For fluffy grains with the
properties of astronomical silicates, β < 1.0 is seen for all particle sizes. Modifying
the composition of the material in the particles has some influence on the results but
even here, when integrated over the solar spectrum and using realistic particle
compositions, β is typically less than unity. Hence, there remain significant issues
in resolving inconsistencies between measurement and theory.
The fluffy nature of particles is also important for the dust-gas interaction during
the initial ejection and acceleration of particles because of its relationship to the drag
coefficient, C D (Eq. 4.87) as we shall see in Sects. 4.6.1 and 4.7.1.
4.5 Dust Size Distributions
Within the equations for radiation pressure, we see the particle size, a, appearing. It
will also appear shortly in the equations for gas drag close to the nucleus and we have
already seen it in the equations for the radiance from the dust coma (Eq. 4.43). But
what is the particle size? Incorporation of some form of continuous size-frequency
distribution (SFD) of particle sizes into cometary studies was foreseen well before
the Halley fly-bys and various mathematical forms have been proposed. In general,
these equations are some variant on a power law distribution so that
n d a
ð Þ ¼ k p a
Àb d
ð4:75Þ
where n d is the number density of particles with a size, a, and k p is a proportionality
constant. Note that the power law description can only hold above a certain size
threshold because the density diverges as a ! 0 if b d > 1. In our physical case this
threshold (a min ) must be larger than, for example, molecules. Thus to be accurate we
must say that for this model “the tail of the distribution follows a power law” (Virkar
and Clauset 2014). The power law can also be thought of as a probability distribution
and k p then becomes a normalization constant and can be computed if a min is defined
using the equation
k p ¼ b d À 1
ð
Þa min
b d À1
ð4:76Þ
It is usual to express the number of particles, dn d , within a size interval, da, so that
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4 Dust Emission from the Surface
earlier in the discussion of the fountain model that the large scale structure of the
coma is influenced by the value of β which is related to the scattering through the
radiation pressure cross-section, Q pr . In the past, it has been assumed that β can take
quite large values (see for example the works of Sekanina and co-authors in the
1980s where values of β > 2.0 can be found) and that this is the result of the dust
grains being more porous than solid spheres producing a larger backscattering peak.
Recent work by Silsbee and Draine (2016) using T-matrix and DDA calculations
suggest, however, that this is not an adequate explanation. For fluffy grains with the
properties of astronomical silicates, β < 1.0 is seen for all particle sizes. Modifying
the composition of the material in the particles has some influence on the results but
even here, when integrated over the solar spectrum and using realistic particle
compositions, β is typically less than unity. Hence, there remain significant issues
in resolving inconsistencies between measurement and theory.
The fluffy nature of particles is also important for the dust-gas interaction during
the initial ejection and acceleration of particles because of its relationship to the drag
coefficient, C D (Eq. 4.87) as we shall see in Sects. 4.6.1 and 4.7.1.
4.5 Dust Size Distributions
Within the equations for radiation pressure, we see the particle size, a, appearing. It
will also appear shortly in the equations for gas drag close to the nucleus and we have
already seen it in the equations for the radiance from the dust coma (Eq. 4.43). But
what is the particle size? Incorporation of some form of continuous size-frequency
distribution (SFD) of particle sizes into cometary studies was foreseen well before
the Halley fly-bys and various mathematical forms have been proposed. In general,
these equations are some variant on a power law distribution so that
n d a
ð Þ ¼ k p a
Àb d
ð4:75Þ
where n d is the number density of particles with a size, a, and k p is a proportionality
constant. Note that the power law description can only hold above a certain size
threshold because the density diverges as a ! 0 if b d > 1. In our physical case this
threshold (a min ) must be larger than, for example, molecules. Thus to be accurate we
must say that for this model “the tail of the distribution follows a power law” (Virkar
and Clauset 2014). The power law can also be thought of as a probability distribution
and k p then becomes a normalization constant and can be computed if a min is defined
using the equation
k p ¼ b d À 1
ð
Þa min
b d À1
ð4:76Þ
It is usual to express the number of particles, dn d , within a size interval, da, so that
316
4 Dust Emission from the Surface