Intensity Distribution in the Heads of Comets

In optical wavelengths, the light emitted by comets consists of sunlight, reflected by dust, and the emission features of atoms and molecules. In his paper from 1957, L. Haser develops the mathematical framework to describe the density distribution of cometary comas. When comet scientists refer to the Haser model , they mostly point to the density distribution of parent and fragment molecules in the coma. However, in his paper, Haser (1957) presented the analytic solution over the line of sight (column density) and considers two particular observation configurations of a nonhomogeneous coma. This translation of the paper, originally in French, stays as close to the source as possible, including using the original figures. Minor typesetting corrections were introduced and are annotated in the text. The Haser model is foundational in cometary atmosphere science, but Haser’s paper was not widely read due to language barriers and a lack of digital availability. This translation is presented in an effort to make Haser’s work accessible to the wider community. We would like to thank Drs Emmanuel Jehin (Univ. Liège) and Kumar Venkataramani (Auburn Univ.) for critically reading this translation.


Introduction
The radial distribution of molecules in the head of a comet is investigated theoretically. Formulae for comparison with the observations are given (Haser 1957).
Taking Whipple's cometary model (Whipple 1951) as a basis, we have calculated the radial distribution of molecules in the head of a comet and the distribution of observable intensity that results.
We adopt the following hypotheses: 1. The comet has a spherical nucleus of radius r 0 , and its matter evaporates as a result of absorbing solar radiation. The molecules leave the surface of the nucleus in every direction with a radial speed v 0 . 2. The molecules emit light according to their resonant frequencies after being excited by sunlight. 3. The molecules are disintegrated by photodissociation following the law · = t n n e , 0 t 0 with n 0 being the number of molecules present at time t=0, and τ 0 measures the average lifespan of a molecule. 4. Near the nucleus, the molecular density is D(r 0 ) molecules per cm 3 .
At a distance r from the center of the nucleus, the molecular density D(r) is given by the equation 3 with the term r 1 2 being the dilution factor, andt e t t 0 0 the disintegration factor. With the substitution = t valid for rr 0 . Notice that if the emission mechanism responsible is fluorescence, we may interpret D(r) and D(r 0 ) as the number of quanta emitted per unit time and volume at a particular wavelength.

Distribution of Observable Intensity
We define the distribution of observable intensity as the projection of the radial distribution (Equation (1)) onto the celestial sphere along the line of sight ( Figure 1): Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
3 Note: modified from the original, which read Where the observable intensity distribution can be written in the form  Müller (1939). In the case of comets, the distribution in the range ρ<r 0 has little effective importance (though it may be important in analogous problems like the solar chromosphere). We limit the line of sight to the nucleus as in Figure 2.
The expression in Equation (6) is replaced by the following: It seems that the result cannot be expressed in terms of known functions, but we may instead expand it in a series: The numerical values of the functions E n (X) are found in Mocknatsch (1938).

Distribution of Dissociation Products
Let N(r 0 ) be the number of molecules that escape the nucleus per second per cm 2 , and N(X) the same quantity at distance X from the nucleus. The total number of molecules that cross a sphere of radius X is therefore It is assumed that the molecules are produced by successive disintegrations of parent molecules. When the molecules produced at a distance X from the nucleus reach the distance r, their number is reduced by a factor molecules arriving at the distance r that originate from a layer of thickness dX at a distance X from the nucleus. The total number of molecules arriving per cm 2 at a distance r is therefore Taking into account the results of the preceding section, .

Nonspherical Density Distribution
If the nucleus of the comet is not rotating and its heat conductivity is small, there is evaporation only on the illuminated part (Figure 3). The distribution of energy is then given by j = I I cos . 0 Where j=0 corresponds to the comet-Sun axis.
The density distribution is not spherical and the appearance of the comet is a function of the Sun-comet-Earth phase angle α. We consider the cases a = p 2 and α=0.

Case 1
We first consider the case a = p 2 . The radial distribution of molecules is again   To express this integral in terms of functions with known numerical values, we consider the general case Using this expression, the observable intensity distribution may be written in the form The profiles we have deduced may be compared with the intensity distribution from photos or spectra to determine the relevant physical parameters. Our calculations differ from those published by Mocknatsch (1938). They also seem to differ from those of L. V. Wallace, of which only a short summary has been published (Wallace 1956).