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MODELLING OF POLARIZATION PROPERTIES
OF COMETARY DUST GRAINS
D.A. Semenov and N.V. Voshchinnikov
Astronomy Department and Sobolev Astronomical Institute, St. Petersburg University,
St. Petersburg, 198504 St. Petersburg­Peterhof, Russia
ABSTRACT
Observational data for dusty comets are summarized
and systemized. The dependence of the linear polariza­
tion on various parameters (the phase angle, wavelength,
etc.) is analyzed. An ensemble of aligned spheroidal par­
ticles having various sizes and chemical compositions is
used to fit the linear and circular polarization observed for
the comet Halley.
1. INTRODUCTION
Except for a few space missions to comets our
information about cometary dust grains comes from the
analysis of their electromagnetic radiation. Like atoms
and molecules, dust particles have their own spectra.
Most spectral features are located in the infrared (IR)
wavelength range.
The optical properties of dust grains depend not
only on their chemical composition but also on their size
distribution, shape, the degree and direction of alignment.
This creates additional difficulties in interpretation of
observations of cometary dust.
The luminosity of comets is tightly connected with
the solar radiation. Dust grains scatter or absorb it, the
absorbed energy is reradiated in the IR wavelength range.
Here, we paid our attention to the scattered part of the
solar radiation.
The electromagnetic waves are characterized by their
intensity and polarization. For atomic and molecular emis­
sions the degree of linear polarization may be theoretically
predicted (Le Borgne and Crovisier, 1987). For a dust
grain of unknown shape and composition, the polarization
may be additional diagnostic tool for their investigations.
Usually Mie theory for spherical particles is used
for fitting the cometary observations. This contradicts
with several direct indicators of a non­sphericity of
cometary grains: for instance, non­zero degree of the
linear polarization at the phase angle   = 0 ô (in this case
the scattering angle  = 180 ô   = 180 ô and the Sun,
the Earth and a comet are placed on one line).
In this paper, for the first time we interpret observa­
tions of the linear polarization for the comet Halley at dif­
ferent wavelengths using an ensemble of homogeneous
oriented spheroids of various sizes, chemical compositions
but a fixed shape. The degree of the circular polarization is
also calculated.
2. OBSERVATIONAL DATA
2.1 Direct (in situ) measurements
The comet Halley was investigated by the Giotto and
Vega 1, 2 spacecraftes. Chemical analysis of the cometary
dust has shown that the grains mainly consist of silicates
(chondrites) and carbonaceous materials. The dust experi­
ments discovered the presence of both large and small par­
ticles in the cometary dust. We used the power law size
distribution provided by the experiments
dn  r 2:5 dr; (1)
where dn is the number of particles in the size range
[r; r + dr] (Mazets et al., 1986).
2.2 Polarimetric observations
The observed degree of the linear polarization is de­
termined by the two processes: scattering of the solar ra­
diation by cometary dust and resonance fluorescence of
molecules
P = P sca
dust F sca
dust + P em
mol F em
mol
F sca
dust + F em
mol
; (2)
where P sca
dust , F sca
dust , P em
mol , F em
mol are the degree of polariza­
tion and flux of the dust and gaseous components, respec­
tively. In Eq. (2)
P = P (  ; ; D; r H ); (3)
where   is the phase angle,  the wavelength, D the
aperture, r H is the heliocentric distance of a comet.
The phase dependence of the polarization shows
features common to all comets: it has negative branch at
  < 20 ô , changes the sign at phase angle  20 ô (the
inversion angle  
0 ) and reaches the maximum around 90 ô
(see Fig. 1 in Jockers, 1998). The maximum degree of
polarization is about 25%--30% for the dusty comets and
8%--15% for the gaseous ones.

The dusty comets are characterized by strong con­
tinuum, weak gas emissions and a high dust­to­gas ratio.
The phase curves for these comets are similiar within
the accuracy of observations. Before the appearance of
the comet Hale­Bopp, the dusty comets were believed
to follow a common phase curve of polarization. The
group of the dusty comets includes the comets: 1P/Halley,
C/Hale­Bopp (1995 O1), C/Hyakutake (1996 B2), etc.
The phase curve of polarization observed for the comet
Halley is typical of the comets of this type (see Figs. 1--3).
The gaseous comets have weak continuum, numerous
emission lines observed almost everywhere in the visible
spectrum and a low dust­to­gas ratio. The group of the
gaseous comets includes the comets: P/Austin (1982 VI),
Kobayashi­Berger­Milon (1975 IX), Tabur (1996 Q1), etc.
The degree of polarization grows with an increase of
the wavelength from the visual to IR. The main observed
parameters describing the polarization of the comet Halley
are presented in Table 1.
Table 1: Parameters of the phase curve for the comet Hal­
ley at different wavelengths. Errors are about 0:5 ô for the
angles and 0:25% for the polarization degree
  
0  
min P min Pmax dP
d  j  
0
3650 š
A 17 ô :0 ? ? 15:5% ?
4840 20:5 10 ô :0 1:3% 16:9 0:20%=1 ô
6700 23:5 10:5 1:5 17:1 0:25%=1 ô
We used the observational data from the following
papers: the degree of polarization from Kikuchi et al.
(1987), the positional angle of polarization and circular
polarization from Dollfus and Suchail (1987).
Data obtained with approximately equal projected area
of the aperture centered on the nucleus of the comet were
utilized. This allows us to exclude the influence of varia­
tions of the degree and direction of polarization observed
in outer parts of coma.
The dependence of the degree of the linear polariza­
tion from the heliocentric distance of a comet is masked
by phase angle variations. For group of dusty comets the
degree of the linear polarization is similiar at the same
phase angles while their heliocentric distances can be
different. Apparently, available observations do not allow
us to study this effect.
The observations of circular polarizations of comets
are very hard to do, so they can be done only episodically.
The comet Halley is an exception: the observational data
are availible for various phase angles.
Figure 1: Degree of the linear polarization as a function of
the phase angle   for  = 3650 š
A. All error bars usually
are less than size of signs
Figure 2: The same as Fig. 1 but for  = 4845 š
A
Figure 3: The same as Fig. 1 but for  = 6700 š
A

Figure 4: Deviations of the position angle of the linear po­
larization as a function of the phase angle  
2.3 Non­spherical particles in comets
Single light scattering by the oriented non­spherical
particles produces phenomena which cannot be explained
using the spherical particles, namely:
1. the non­zero circular polarization
(Metz and Haefner, 1987);
2. rotation of the position angle with time,
wavelength and in different parts of a comet
(Dollfus and Suchail, 1987);
3. non­zero degree of the linear polarization at
the phase angle   = 0 ô ;
4. non­zero degree of the linear polarization of
stellar radiation observed during stellar occultations
by a comet (Rosenbush et al., 1994).
3. MODEL
For calculations we used the numerical code based on
the exact solution to the light scattering problem by the
Separation of Variable Method (Voshchinnikov and Fara­
fonov, 1993). The algorithm of calculations was as fol­
lows:
1. calculation of an array of scattering matrices;
2. the averaging over rotation;
3. the averaging over a size distribution function;
4. mixing of scattering matrices for different materials;
5. calculations of Stokes parameters and a comparison
with observations.
Figure 5: Degree of the circular polarization as a function
of the phase angle   . All error bars are presented by ver­
tical lines
Relative errors of the calculations were usually less than
0.1%.
The best model found has the following parameters:
 mixture of 80% astronomical silicate (astrosil)
and 20% amorphous carbon (AC1);
 prolate (50%) and oblate (50%) spheroids;
 aspect ratio a=b = 2:0;
 r V = 0:05 0:25 m for AC1; 1
 r V = 0:05 0:55 m for astrosil;
 = 2:5;

= 75 ô ; 2
 perfect rotational (Davies­Greenstein, DG)
orientation.
The value of and the chemical composition of grains
were chosen on the basis of direct (in situ) measurements
and ground­based observations. Other parameters were es­
timated by fitting the observed quantaties.
4. COMPARISON WITH OBSERVATIONS
The observed and calculated curves are plotted in
Figs. 1--5. A good agreement with the observations is
only for the negative branch of the polarization curve,
when   < 30 ô (Figs. 1--3). The degree of the calculated
polarization is higher than observed at   > 40 ô 50 ô
that is connected with the high polarization efficiency of
the small amorphous carbon grains at these phase angles.
1 r V is the radius of the sphere whose volume is equal to that of a
spheroid.
2
is the angle between alignment direction and direction of light
propagation.

Deviations of the position angle of the linear polar­
ization plotted in Fig. 4 show that the model satisfactorily
explains the observations. The same can be said about the
behaviour of the circular polarization presented in Fig. 5.
A better agreement between the theory and observations of
the circular polarization may be reached if one decreases
the fraction of astrosil in the model.
ACKNOWLEDGEMENTS
The authors are thankful to V.B. Il'in for careful read­
ing of the manuscript. This work was financially supported
by grants from the INTAS (grant 99/652) and Volkswagen
foundation.
REFERENCES
Dollfus, A. and J.­L. Suchail, 1987: Polarimetry of grains
in the coma of P/Halley. I. Observations. Astronomy
abd Astrophysics, 187, 669­688.
Jockers, K., 1997: Observations of scattered light from
cometary dust and their interpretation. Earth, Moon
and Planets, 79, 221­245.
Kikuchi, S. et al., 1987: Polarimetry of Comet P/Halley.
Astronomy and Astrophysics, 187, 689­692.
Le Borgne, J. and J. Crovisier, 1987: Polarimetry of
molecular bands in comets P/Halley and Harley­Good.
In ESA Proceedings of the 20th ESLAB symposium on
the exploration of Halley's comet, ESA­SP­278, 571­
576.
Mazets, E. et al., 1987: Dust in comet P/Halley from Vega
observations. Astronomy and Astrophysics, 187, 699­
712.
Metz, K. and R. Haefner, 1987: Circular polarization near
the nucleus of comet P/Halley. Astronomy and Astro­
physics, 187, 539­542
Rosenbush, V. et al., 1994: Comets Okazaki­Levy­
Rundenko (1989 XIX) and Levy (1990 XX). Po­
larimetry and stellar occultations. Icarus, 108, 81­91.
Voshchinnikov, N. and V. Farafonov, 1993: Optical Prop­
erties of Spheroidal Particles. Astrophysics and Space
Science, 204, 19­86.