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1.00 10.00 100.00 1000.00 Particle Radius |>m

Fig. 53. Hypothetical dust size distributions computed from (45) for different constants M and N

1.00 10.00 100.00 1000.00 Particle Radius |>m

Fig. 53. Hypothetical dust size distributions computed from (45) for different constants M and N

measurements and by studying the dynamical distribution of grains in the dust tail .

A method to estimate the size distribution using IR measurements is to determine the so-called "super-heat" parameter. At thermal equilibrium the grain temperature is balanced by the absorbed flux at UV and visible wavelengths and the re-radiated flux in the infrared, and for a spherical particle:

'Qabs(A, a>a2dA = ^ tvB(A, T(a, r))Qabs(A, a)4™2dA (46)

To obtain the emitted flux, we need to integrate over the size distribution of the grains.

The temperature of very small grains can deviate from the equilibrium temperature, because small grains absorb solar light very efficiently at optical wavelengths with absorption coefficient Q™s, but because of their low infrared emissivity, Q™, it is difficult to reradiate this energy again. They therefore can heat up to relatively high temperatures. For particles larger than a few micron, Q™s and Q™ are about equal and the particle temperature is closer to the black body temperature [79]. The so-called "superheat" parameter has been defined as the ratio of the grain temperature, Tg, to the equilibrium black body temperature, Tb, at the same heliocentric distance. A high superheat parameter, therefore, is an indication for small grains.

For the spatial distribution of dust grains in the tail and the relation to their size and other material parameters, see Sect. 3 and the references therein. A detailed overview on how to derive the material properties of dust grains including the use of polarization measurements is given in [133].

In general, observations suffer from the fact that very large dust particles represent only a small total cross-section because they are very rare. Therefore we gain little information about them, resulting in large uncertainty of the total dust mass. Very small dust particles are not efficient light scatterers and also remain undetected. If they are very numerous, they could also significantly increase the error of the dust mass determination. Therefore, the determination of a good dust mass production rate is a difficult task.

7.3 The Dust Production Rate

To be able to derive the dust-to-gas ratio of comets, we need to determine the dust mass production rate. To quantitatively determine the dust production rate from observations, we need to know the particle size distribution, their density and mass, their optical properties, as well as their dynamics. These quantities are difficult to derive observationally, as outlined above. Therefore, often a simplified approach is made based on observational quantities, such as the spatial dust distribution or the scattered optical light.

Reference [4] introduced a quantity called "Afp"-parameter to determine the dust content of a cometary coma. We discuss this parameter here, because it is often used as an estimate of the cometary dust content.

The Afp parameter is given by:

Here, A is the average grain albedo, f is the filling factor of the grains in the FOV of the observations, p is the projected aperture radius used for the photometry at the comet, and F is the observed photometric continuum flux. Fq is the solar flux measured in the same filter bandpass used to measure F. Therefore, the parameter can be directly derived from photometric observations. For isotropic outflow with constant velocity (and without dust fragmentation), the Afp parameter is independent of the aperture size p and the geocentric distance, A, of the comet. It is therefore an ideal quantity if different comet observations are to be compared.

The Afp parameter is a measure for the effective dust scattering area in the FOV. This can be seen when looking at the filling factor which is given as:

N(p) is the number of grains in the FOV and a denotes the grain scattering cross-section.

Although the Afp parameter is related to the amount of dust in the aperture and scattered solar light, it is not a direct measure of the dust production rate. To obtain a production rate, the dust expansion velocity, v, the scattering phase function, D(0), the dust size distribution, f (a), and the density of dust particles, pdust, must be taken into account. A method to derive the dust production rate, Qdust, from Afp was given, for example, by [127]:

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