Ye. Grynko - K. Jockers - R. Schwenn
Max-Planck-Institut für Sonnensystemforschung,
Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany
Received 23 January 2004 / Accepted 20 July 2004
Abstract
We have analyzed brightness and polarization data of comet 96P/Machholz, obtained with the
SOHO-LASCO C3 coronagraph at phase angles up to 167
and 157
,
respectively. The polarization data are characteristic of a typical dusty comet.
Within error limits the corresponding trigonometric fit describes the new data
measured at larger phase angles as well as those of the previously known range.
In the phase angle range from 110
to 167
the brightness increases almost linearly by about two orders of
magnitude. The gradient is independent of wavelength. From the absence of
a diffraction spike we conclude that the grains contributing significantly to
the scattered light must have a size parameter
,
i.e.
have a radius larger than 1
m. Fits of the data with Mie
calculations of particles having a power law distribution of power
index
2.5 provide a best fit refractive index
+
0.004. In the framework of effective medium theory
and on the assumption of a particle porosity P= 0.5 this leads to a
complex refractive index of the porous medium
= 1.43 +
0.009.
A higher refractive index is possible for more porous grains with very
low absorption. The large particle sizes are
in qualitative agreement with findings derived from
the analysis of the motion of cometary dust under solar radiation pressure (Fulle and
coworkers, see Fulle et al. 2000; Jockers 1997) and with the in-situ measurements of the
dust of Halley's comet.
Key words: comets: individual: 96P/Machholz 1 - techniques: photometric - techniques: polarimetric
Light scattering models of cometary dust suffer from the fact that measurements
of brightness and polarization of the scattered light at large phase angles
(forward scattering) are almost nonexistent. If a comet is seen from the Earth
at large phase angles, most often its solar elongation is very small and
observations are impossible. Up to now the largest phase angle at which
the brightness of a comet was measured was equal to 149
(comet C/1980 Y1
(Bradfield) = 1980 XV (Gehrz & Ney 1992). For polarization the maximum angle
is still smaller. For comets C/1989 X1 (Austin) = 1990 V and C/1996 B2
(Hyakutake) (Kiselev & Velichko 1998) the degree of polarization was measured
at =111
.
Polarization data of comet C/1999 S4 exist at a maximum phase angle
of 121
(Kiselev et al. 1998, Hadamcik & Levasseur-Regourd 2003), but these data were obtained during the
disruption of the nucleus and therefore may not be representative. In January 2002 Comet
96P/Machholz 1 was photographed with the C3
coronagraph on board SOHO. On the images suitable for measurements the maximum
phase angle is equal to 157
for polarization and 167
for brightness.
In this
paper we make an attempt to derive brightness and polarization of Comet
96P/ Machholz 1 from the SOHO C3 coronagraph observations.
![]() |
Figure 1: LASCO C3 color filter data (dashed lines) and a cometary spectrum (solid line). Spectrum of comet 109P/Swift-Tuttle curtesy A. Cochran, Univ. of Texas. |
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The Large Angle and Spectrometric Coronagraph (LASCO) instrument is one of 11 instruments operating on the joint ESA/NASA SOHO (Solar and Heliospheric Observatory) spacecraft. The spacecraft was launched in 1995 and was placed in an orbit about the first Lagrangian, or L1, libration point, on the Earth-Sun line where the gravitational attraction of the spacecraft to the Earth is just balanced by its attraction to the Sun.
The three coronagraphs comprising LASCO are designated C1, C2 and C3.
C1 covers a field from 1.1 to 3.0
(solar radii from Sun center), C3 spans the outer
corona from about 3.7 to 32
,
and C2 extends from 2.0 to 6.0
,
i.e. overlaps with both C1 and C3. In this work we have used
only C3 data as the comet did not enter the field of view of the other instruments.
C3 has a coronagraph aperture of 9.6 mm diameter, an effective
focal length of 77.6 mm and an effective f-number f/9.3
(Brueckner et al. 1995). Its optical design takes into account the
necessity for blocking direct sunlight and obtaining absolutely
minimal scattering. The filter and polarizer wheels are installed
in front of the 21.5 mm square
pixel CCD detector. The 21
m
CCD pixel size subtends an angle of 56 arcsec. The CCD square
accommodates a 30
radius image, so that portions of the 32
optical field of view are lost off the top, bottom, left, and right
edges of the CCD surface. The image becomes unvignetted beyond 10
.
C3 was not designed for narrowband observations
and therefore does not have narrowband, spectroscopic quality filters.
It has broadband color filters for the purpose of separation of the F
from the K corona and polarizers for polarization analysis. The filter
wheel contains the blue, orange, deep red, and infrared filters, and a
clear glass position. Their bandwidths are indicated in Fig. 1. The
polarizer wheel contains three polarizers at 120 degrees (their positions
are designated as "
'',"0
'', and "+60
''), the H-alpha filter,
and a clear glass position.
The SOHO mission was interrupted in June 1998, and due to excessive
cold the "0
'' polarizing filter was damaged. After that the only thing
we could do was to use the clear filter to calculate a synthetic "0
'' image,
according to the technique applied in the polarization routines of the
LASCO IDL software library.
Between January 5 and 10, 2002, comet 96P/Machholz 1 passed through the field
of view of the C3 coronagraph. Several sequences of images were obtained
at different positions of the filter and polarizer wheels. Four sets of
them were suitable for further processing and only two sets could be used
to determine the degree of polarization. There is also a long set of images
taken with clear glass filter and short exposure where the comet is not
oversaturated. The part of the comet orbit covered by the C3 coronagraph
is shown in Fig. 2. Although the projection of the cometary orbit in the
plane of vision appears to be very close to the Sun, the comet is not a
sungrazer. Its highly inclined (i = 60.18)
orbit has a perihelion distance
q = 0.1241 AU, and an eccentricity e = 0.96. The orbital period is 5.34
years. The comet went through perihelion on January 8, 15 UT. During the
period of useful observations from Jan. 8, 08 UT to Jan. 10, 11 UT the
phase angle a of the comet decreased from 167
to 114
and
the heliocentric distance increased from 0.124 to 0.155 AU.
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Figure 2: Comet 69P/Machholz1 orbit near perihelion in 2002. |
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For the polarization measurements we also used C3 images of the comet from
its apparition in October 1996. At that time the maximum phase angle was 113.
As at that time the corresponding polarizing filter was not yet damaged we had an
opportunity to test the 0
image reconstruction procedure.
The position of the comet at the time of the exposures is shown in Fig. 2.
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Figure 3: LASCO C3 images of 96P/Machholz 1 near perihelion. |
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We had so-called Level 0.5 images at our disposal. This is raw data with brightness expressed as CCD detector counts. The database is available at http://ares.nrl.navy.mil/database.html. Figure 3 shows three representative images. Following the steps realized in the data reduction routines of the LASCO IDL library we subtracted the bias value, corrected for vignetting, and multiplied with the calibration factors (different for different color filters) to translate the data into physical units (mean solar disk brightness). No flatfields were applied to the data. Unfortunately no information about the level of instrumental polarization is available. Thus we do not make such a correction in our analysis, which in principle can give noticeable errors.
On those images where the comet is close to the Sun the bright solar corona
is superimposed on the image of the comet. The irradiance of the corona
could affect the result of the measurements. In a first step the
influence of background was reduced by subtracting two images taken close
in time. As the comet moves in the field, if the corona does not change
between the two exposures the background would be fully eliminated in this
step. To take into account possible changes of the corona between the two
exposures, in a second step we marked an area around the comet and
constructed an artificial background in this area by means of radial
interpolation between the outer and inner sides of the marked area.
We decided on radial interpolation because the structures of the solar
corona are oriented mainly radially. This artificial background was then
subtracted from the image. The remaining unevenness of the background field
was of the order of 1% of maximum cometary brightness. Then, in the third
step, the comet photometric center (maximum brightness in the cometary image)
was searched in each frame and the values of the integrated flux were
extracted within a square window of 33 pixels (168
168 arcsec) centered
on the photometric center. We made one measurement of the flux from the
comet and 5 additional measurements of background flux close to the comet.
Then we calculated the average value for the background and subtracted it
from the comet count. In this way the background subtraction was further
improved and, from the standard deviation of the 5 measurements, an estimate
of the error in the background determination was obtained. The resulting
fluxes are plotted in mean solar flux units in Fig. 5. The errors connected with the
uncertainty of the background are
smaller than the plotting symbols. The main error shown in
Figs. 5-7 is caused by the statistical uncertainty of the CCD detector
counts. For each data point this error is equal to the square root of
the corresponding number of counts multiplied by the CCD quantization step (i.e. the
number of electrons per count) expressed in mean solar flux
units. We note, however, that the absence of a useful flat field may cause
systematic errors.
Up to here the described data reduction applies to photometry as well as polarimetry. Two additional steps were taken to obtain polarization values.
A synthesized image I
was calculated with the following formula:
![]() |
(1) |
After calculation of I
we had 3 measurements of brightness
I
, I
, I
at 3 polarizer positions. Then the linear polarization P was deduced by
applying the following formula:
![]() |
(2) |
In order to test the accuracy of the synthesis procedure one can compare
I
values measured in 1996 with the ones derived from clear images.
We found that the difference in the counts is within the limits of the measurement errors.
![]() |
(3) |
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Figure 4: Polarization observations with different color filters vs. phase angle. The solid curve is a trigonometric fit to the typical cometary dust polarization phase dependence observed at 684 nm. |
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Figure 5: Phase curves for comets 69P/Machholz 1 (circles and diamonds) for corresponding color filters and Bradfield 1980 XV (black stars). The values of comet Bradfield have been adjusted by multiplication with a constant to fit the measurements of comet Machholz 1. |
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Figure 6:
Theoretical best fit curves for the clear glass measurements
at four combinations of ![]() ![]() |
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![]() |
Figure 7:
Dependence of the phase function on the absorption index at ![]() |
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In Fig. 5 the resulting phase dependence of the flux of 96P/Machholz 1, with
the dependence on heliocentric distance reduced to perihelion, is shown in mean
solar flux units for the different wavelength bands and for the clear glass
filter. They are calculated from the nominal calibrations of the C3
coronagraph and may not be sufficiently accurate
to allow the determination of reddening (i.e. the increase of brightness with
wavelength).
In what follows we will only make use of the gradient of brightness with respect
to phase angle. The curves are compared with a measurement of comet C/1980 Y1
(Bradfield) (Gehrz & Ney 1992), which is the only other comet known to us which was
observed at phase angles larger than 120.
The quantity f
/f
given in Gehrz & Ney (1992)
was multiplied with an arbitrary scale factor to produce a
curve close to our measurements of comet 96P. All the curves show a steep
decrease of total brightness with decreasing phase angle.
Cometary dust particles have irregular shapes and, probably, complex structures.
So one should apply a proper light scattering model, which takes into account
the complex structure of the particles. In the forward scattering part of the
phase curve the light scattering of particles comparable to the wavelength is
dominated by diffraction. However, we do not see any evidence for the diffraction
spike in the obtained phase curves. The absence of a diffraction spike can be
used to obtain a lower limit for the size of the particles, as for particles
having a size parameter
(where r is radius of the particle and
is the wavelength) larger than
20 the first forward scattering
diffraction minimum is already
,
i.e. beyond the range of
the observed
phase angles. We can therefore conclude that the particles predominantly
contributing to the observed scattered radiation must have a size parameter
larger than
20, i.e. have a radius larger than 1
m.
Phase curves of particles of sizes larger than 1
m have been measured by
Weiss-Wrana (1983) and Volten et al. (2001). Calculations have been performed by Macke et al. (1995), by Grundy et al. (2000) and more recently by Grynko & Shkuratov (2003). Up to now large particles can only be studied with the
T-matrix method and the geometric optics approximation. The first method is
more accurate but applicable only to rotationally symmetric particles.
The second is more approximate but can be used for arbitrary sizes and
shapes. For particles where both methods can be applied the results agree
for size parameters
-60 (Macke et al. 1995). The phase curves
produced by these authors show a very narrow diffraction spike at small
scattering angles which for particles of size parameter x = 30 ends at about 10
scattering angle. For larger scattering angles the decrease of the phase
curve is linear, in agreement with our observations. Calculations using the
geometric optics approximation are independent of particle size if the
particles are nonabsorbing. The gradient of the phase curves of such
particles strongly increases with decreasing real part of the refractive
index. A similar but weaker dependence exists for rising imaginary part of
the refractive index (Grundy et al. 2000; Grynko & Shkuratov 2003).
According to Grynko & Shkuratov (2003), compared to the backscattering
part, the forward scattering part of the phase function is much less
dependent on particle shape. This is because the forward scattering slope,
at least for the larger particle sizes, is formed mostly by rays transmitted
by the particle without any further interaction. Therefore in this
phase angle domain phase functions for spheres and irregular particles
behave in a similar way (see also Weiss-Wrana 1983). Thus we can apply Mie
theory to extract the complex refractive index
from the measured brightness gradient.
Presently the size distribution of dust grains in comets is a matter of
debate (see e.g. Jockers 1997). Space observations of Halley's comet
(Mazets et al. 1986) have
revealed particle size distributions of the particle fluences
d
=
d
,
where d
is the number of dust grains in a small interval around
radius
,
and
is the power index. For the small submicron particles the
space observations indicate power indices
2 which increase to
3 and more with increasing size.
But for the large
particles of interest here the statistics are poor and, worse, because of the
high encounter speed at comet Halley no good calibrations of the measuring
device were available. Also it is not clear if comet Halley is representative
for all comets or at least for comet Machholz 1. Additional information comes
from models (mostly by Fulle and coworkers) of dust tails of a number
of comets based on the radiation pressure force.
Power indices of dust production are provided in Table II of Jockers (1997). In a more recent paper (Fulle et al. 2000) the particle data as well
as the optical data on comet Halley obtained by the Giotto probe are
combined into a similar radiation pressure model and a power index of
2.7 is derived. Because we are interested here in fluences along the line of sight we must
correct the power indices of dust production for the fact that smaller particles
have larger emission speeds. After correction these power indices are
lowered by a number between 0 and 0.5, i.e. become significantly less
than 3.
Using Mie theory we have performed calculations of the phase functions for
= 2.5. In our calculations we restricted the range of size parameter
between 10 and 400, since the presence of smaller and larger particles
influences the result very little. The number of large particles decreases
according to the distribution law. On the other hand, small particles do not
have a high enough scattering efficiency to make a significant contribution
to the phase function behavior at
= 2.5. Besides, they introduce a noticeable
wavelength dependence of the scattering, which is absent in our data (see Fig. 5).
In order to determine the real and imaginary part of the refractive index
=
+
we proceeded as follows. For four fixed
values of
we determined
the best fit absorption indices
.
In Fig. 6 one can see that the
best fit
refractive indices
produce gradients close to the observed phase curve
(clear glass filter). Higher refraction is coupled with lower absorption and
vice versa. The best coincidence corresponds to
= 1.2 +
.
Higher and
lower values of n give certain deviations from the measurements at all k. Then we
tried to find the maximum possible k at n = 1.2. Fig. 7 shows that the increase
of the absorption index in the very wide range from
= 0.001 to 0.1 leads to
small changes in the phase dependence, keeping the same gradient in the given
limits of the phase angles.
Simple application of Mie theory gives unrealistically low values for the
refractive index. As all conceivable constituent minerals of cometary dust
have higher refractive indices we assume that the particles are not single
homogeneous grains, but porous aggregates of smaller grains. The calculated
values of the complex refractive index are considered as an
"effective''
refractive index m
= n
+ i k
in terms of effective medium theory
and the porosity
is introduced as an additional free parameter.
The case
of grains composed of a single porous material can be treated with the
Maxwell-Garnett (Bohren & Huffman 1983) and Bruggeman (Bruggeman 1935)
formulae taking vacuum (
= 1) as the "host'' material and filled
subvolumes
as the "inclusion''. Both formulae provide practically equal results.
For example, a porosity rate
= 0.5 changes the value of the complex
refractive index from
= 1.2 +
0.004 to
= 1.43 +
0.009.
Corrigan et al. (1997) found that interplanetary dust particles, which can
perhaps serve as examples for cometary particles, have a porosity which very
rarely exceeds
= 0.3. The maximum real part of the complex refractive
index
that we can obtain with
0.5 using a mixing rule is around
= 1.65,
but in this case the particles must be very transparent. In order to fit the
data using higher absorption of the dust particles one must assume either a
higher porosity rate, which seems to be unlikely, or an unrealistically low
real part of the refractive index.
Water ice has a refractive index
of
1.3 and therefore may seem
well suited to explain the refractive index
derived in our models. Water ice has been detected by the ISO spacecraft
in comet C/1995 O1 (Hale-Bopp) at heliocentric distances
AU by
Lellouch et al. (1998) but is very unlikely to be present at
AU (Hanner 1981).
The accuracy of the polarization measurements proved to be low. In general
the polarization data follow the phase law
characteristic of a typical dusty comet.
The brightness increases linearly by almost two orders of magnitude in the given range of phase angles. The gradient is independent of wavelength.
The forward scattering diffraction spike is absent on the phase curve.
Therefore we conclude that the contributing grains must have a size parameter
larger than
20, i.e. have a radius larger than 1
m. The
best fit refractive
index of the data is
= 1.2 +
0.004 according to Mie
calculations of particles
having a power law distribution of power index
.
On the assumption of a
particle porosity
= 0.5 application of the mixing rules gives a complex refractive
index of particles
= 1.43 +
0.009. At lower porosity particles
must either be
very transparent or have an unrealistically low real part of the refractive
index.
Acknowledgements
We thank the whole LASCO team for making the data used here available to us and in particular Dr. Doug Biesecker for providing the required explanations.The German effort for SOHO has been supported by the DLR (Deutsches Zentrum für Luft- und Raumfahrt e.V. in Bonn). SOHO is a mission of international cooperation between ESA and NASA.