Dust Spatial Distribution and Radial Profile in Halley's Inner Coma

ICARUS

126, 78–106 (1997) IS965599

ARTICLE NO.

Dust Spatial Distribution and Radial Profile in Halley’s Inner Coma B. GOIDET-DEVEL UFR Sciences et Techniques, 16 route de Gray, 25030 Besanc¸on Cedex, France E-mail: [email protected] AND

J. CLAIREMIDI, P. ROUSSELOT,

AND

G. MOREELS

Observatoire de Besanc¸on, B.P. 1615, 25010 Besanc¸on Cedex, France Received November 7, 1994; revised August 9, 1996

be reached if an efficient fragmentation process at R , 1000 km with a(R) 5 0.87 is introduced, with the fragmentation coefficient decreasing to zero at R 5 6000 km. Different values of the density r and complex index (n 2 ik) of the grains are used. Two good fits with the model results are obtained for r 5 2.2 2 1.4 a/(a 1 1) g cm23, where a is expressed in micrometers (Lamy, P. L., E. Gru¨n, and J. M. Perrin, 1987, Astron. Astrophys. 187, 767–773) and both indices 1.387 2 0.031 i (Mukai, T., S. Mukai, and S. Kikuchi, 1987, Astron. Astrophys. 187, 650–652) or 1.7 2 0.02 i (Khare, B. N., C. Sagan, E. T., Arakawa, F. Suits, T. A. Callcott, and M. W. Williams, 1984, Icarus 60, 127–137). However, if a color index variation with p is considered, the agreement is much better when the complex index of tholin (1.7 2 0.02 i) is adopted.  1997 Academic Press

Mosaic images of the brightness distribution of solar radiation scattered by dust in Halley’s coma are constructed using the spectra obtained by the three-channel spectrometer TKS during the approach phase of the Vega 2 spacecraft. They cover a sector having a radius of 40,000 km centered at the nucleus with an angular extent of 508. The dust scattered brightness is plotted as a function of the distance p between the nucleus and the line of sight. This distance p is also called impact parameter. The brightness varies as the inverse of p in the inner coma when p is less than 3000 km and larger than 7000 km. In the 3000–7000 km distance range, the brightness varies as p21.52. At distances larger than 7000 km, two dust jets are clearly visible with a contrast comparable to the gaseous jets which appear in the OH, NH, CN, C2, and C3 images (Clairemidi, J., G. Moreels, and V. A. Krasnopolsky, 1990, Astron. Astrophys. 231, 235–240; 1990, Icarus 86, 115–128). In the inner coma, the spatial distribution of dust seems to be more isotropic and less contrasted than the distribution of gaseous emissive species. A model is developed to calculate the scattered intensity integrated along a line of sight at a projected distance p from the nucleus. The model is based upon Mie theory and uses the data of the impact particle counter SP-2 on board the Vega spacecraft (Mazets, E. P., R. Z. Sagdeev, R. L. Aptekar, S. V. Golenetskii, Yu. A. Guryan, A. V. Dyachkov, V. N. Elyinskii, V. N. Panov, G. G. Petrov, A. V. Savvin, I. A. Sokolov, D. D. Frederiks, N. G. Khavenson, V. D. Shapiro, and V. I. Shevchenko, 1987, Astron. Astrophys. 187, 699–706), extrapolated from 8030 to 440 km. The model takes into account the fountain effect due to the competition between solar gravitation and radiation pressure, the variation of phase function with scattering angle, and fragmentation processes. A simple method is used to simulate fragmentation: a particle in a mass decade class splits into particles of the lower mass decade, assuming that the mass conservation law is fulfilled. A single a(R) fragmentation coefficient is introduced, R being the distance between the dust particle and the nucleus. A good agreement with the measured dust-scattered intensity radial profile can

I. INTRODUCTION

Among the numerous scientific objectives of the missions to Comet Halley, the analysis of the components of the coma, dust and gas, was expected to provide major information about the composition of the nucleus, which is one of the most exciting goals in cometary research. Therefore, the interplanetary spacecraft were directed at distances to Halley’s nucleus as short as possible. Vega 1 and Vega 2 approached the nucleus at a minimum distance of 8890 and 8030 km, respectively (Sagdeev et al. 1986a), and Giotto at a much shorter distance, 600 km (Reinhard et al. 1986). The spacecraft carried two types of instruments to collect data on the physical and chemical properties of Halley’s dust: in situ and remote-sensing instruments. • In situ instruments were DIDSY (McDonnell et al., 1986) and PIA 5 (Kissel et al. 1986a) on board Giotto and SP-1 (Vaisberg et al. 1986), SP-2 (Mazets et al. 1986), DUCMA (Simpson et al. 1986), and PUMA (Kissel et al. 1986b) on board Vega. They collected data on the spatial distribution of dust particles as a function of the spacecraft 78

0019-1035/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

distance to the nucleus. They also measured the mass distribution in the 10216 –1026 g range and showed the presence of a large quantity of sub-micrometer-sized particles (Mazets et al. 1987). Furthermore, particles with a high fraction of light elements (Kissel et al. 1986a,b, Langevin et al. 1987), later called CHON grains (Clark et al. 1987) were detected. Since then Mukhin et al. (1991) and Fomenkova et al. (1992) have shown that a population of Mgrich grains, that probably contain magnesium carbonates, is present in the dust of comet Halley. • Remote-sensing optical instruments were video cameras (HMC for Giotto and TVS for Vega (Keller et al. 1986, Sagdeev et al. 1986b)), the photopolarimeter OPE on board Giotto (Levasseur-Regourd et al. 1986), the infrared spectrometer IKS (Combes et al. 1988), and the near UV– visible spectrometer TKS (Moreels et al. 1987). The OPE photopolarimeter measured the linearly polarized brightness of the light scattered by dust particles crossing the instrument line of sight, while the IKS and TKS spectrometers of Vega supplied a large number of spectra of the gas fluorescence emissions and of the dust-scattered radiation. This paper aims at presenting and analyzing the experimental results obtained by the three-channel spectrometer TKS, with an emphasis on the model of the mass distribution of dust and of the brightness due to scattering. In the first part of this paper, the geometry of the encounter and a description of the two sets of experimental results used for the model are given. In the second part, these results are used to show evidence of dust jets and discuss the dependency of the dust scattered brightness with p, the distance between the nucleus and the line of sight. As shown by Jewitt and Meech (1986a,b) and Jewitt (1991), the intensity and color of dust continuum provide valuable information on the physical and optical properties of cometary grains. In the third part of the paper, a model is constructed in which only one source of dust—located at the nucleus—is supposed to feed the coma. Its results are presented in the fourth part where they are discussed and compared with the observational data. Finally, the validity and the limits of the model are discussed.

II. EXPERIMENT AND DATA PROCESSING

II. 1. Experiment Layout and Line of Sight Geometry The Vega 2 probe went through Halley’s inner coma with a relative velocity of 76.78 km sec21 on March 9, 1986. The trajectory of the spacecraft was inclined at 1118 with respect to the Sun–nucleus line (Fig. 1). The spacecraft was stabilized with a three-axis control system, as explained in the Venus–Halley mission manual book (CNES, Intercosmos 1985). The closest approach distance dm was equal to 8030 km (Sagdeev et al. 1986a). Fifteen experi-

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ments were carried by the spacecraft that was launched on December 21, 1984. Among the data provided by these instruments we shall primarily use here the results produced by the three-channel spectrometer TKS (Moreels et al. 1987, Krasnopolsky et al. 1987) along with inputs from the SP-2 impact particles counter results (Mazets et al. 1986, 1987). TKS was composed of a Cassegrain telescope (focal length 350 mm, f number 3.5) and a spectrometer with holographic gratings and intensified linear photodiode arrays. The optical beam was separated into three parts by three slits located at the telescope focal plane. Each part was directed to a dispersive UV, VIS, or NEAR-IR channel. As a result, each spectrum consisted of a sequence of 700 consecutive pixels, each one having a spectral width of 0.6 nm. Three specific wavelengths, 377, 482, and 607 nm, were chosen to measure the dust-scattered brightness. For each wavelength, the bandwidth is limited to two adjacent pixels, i.e., Dl 5 1.2 nm in order to keep the molecular emissions at a negligible level in comparison with the dustscattered brightness. In a further step, monochromatic mosaic maps of the field of view scanned by the spectrometer are presented for the previously chosen wavelengths. Then, the color of dust was calculated by computing the pixelto-pixel ratio of the monochromatic maps. The extended sensitivity range of the three-channel spectrometer was obtained by using an intensified array detector which had a controlled gain, adjustable according to the intensity level at 516 nm. It was necessary to use a detector of this type because, due to the short duration of the encounter, the exposure time had to be very short in comparison with the usual exposure conditions in astronomy. The spectrometer worked as a spectro-imager because the secondary mirror inside the telescope was sequentially tilted around two axes. The rotation of this secondary mirror allowed the optical line of sight of the instrument to scan the coma around the central position of the axis which was an angle f P 38 off the Vega–Halley straight line (Fig. 1). There were two scanning modes. In mode A, rectangular 28 3 1.58 wide areas were scanned in an array of 7 successive rows of 15 positions each, i.e., 105 positions (Fig. 2).This mode was in operation during the encounter session which lasted for 2 hr, except in the final 33 min before closest approach. During this period (mode B), the optical axis only scanned a row, the 4th row of the preceding rectangular areas in a back and forth motion. During the period of 2 hr before the encounter with the comet, 7 rectangular maps, i.e., 735 spectra, and 28 successive scanning rows, i.e., 420 spectra, were obtained. Due to the relative approach motion of the Vega probe, the scanned area, projected on a plane perpendicular to the optical axis and containing the nucleus became progressively smaller, producing a ‘‘zoom’’ effect. In the present paper, special

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FIG. 1. Geometry of the encounter of Vega with Halley’s comet. The coordinate system is a relative frame having its origin at the nucleus. The plane of the figure is the comet orbital plane. The angle between the spacecraft trajectory and the comet’s orbital plane is equal to 8.928. As represented, the Vega probe is below the figure plane. The VJ line is the projection of the spacecraft trajectory on the comet’s orbital plane.

attention will be given to a series of 88 spectra for which the projected distance p decreases from 38,759 to 440 km. Among the 1155 available spectra, these 88 spectra are emitted by regions in the coma where it may be assumed that the dust particles mainly originate from the nucleus (Fig. 2, lower box). The geometric parameters for these spectra are listed in Table I where the distance d between the probe and the nucleus, the projected distance p between the nucleus and the optical axis, and the scattering angle u are given for each individual spectrum. The scattering angle slowly increases from 65.58 to 76.68 when the distance Vega–nucleus decreases from 516,032 to 43,742 km, a period that lasts 1h50min. It rapidly increases from 898 to 1208 during the last 5 min period, i.e., for the 5 or 6 spectra of Table I having the smallest p. The variation of u with p is depicted in Fig. 3. Additional information concerning the 5 min period before encounter is given in Sagdeev et al. (1986b). II. 2. Data Processing

FIG. 2. Geometrical configuration of the field of view scanned by the three-channel spectrometer. The distance between the spacecraft and the nucleus decreased from 516,000 to 8030 km during the encounter session on March 9, 1986. Two modes were used to record data. In mode A, the scanned field of view is rectangular and consists of 7 lines of 15 positions. At the end of the session, 33 min before the closest approach, the instrument commutes to mode B which consists in a back and forth motion of the central line. The mosaic monochromatic images constructed with the data take into account the zoom effect. Their frames are delimited by a triangle with its sharp summit located at the nucleus.

After subtraction of dark current and division by the calibration curve, spectral intensities are obtained in kiloRayleighs per nanometer (the Rayleigh is a unit of 106 photons emitted in 4f steradians by an emissive column of section 1 cm2). The optical spectrum of the coma consists of the usual OH, NH, CH, NH2, CN, C3, and C2 molecular bands superimposed on a continuum background due to solar radiation scattered by dust grains. This continuum radiation is very intense in the inner coma. It is typically ten times brighter than the discrete molecular emissions at distances from the nucleus shorter than 1000 km. To extract the molecular contribu-

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TABLE I List of the 88 Values of the Probe–Nucleus Distance d, Projected Distance between the Nucleus and the Optical Axis p, and Scattering Angle u, for the Individual Spectra Used to Draw the Radial Profiles of Dust-Scattered Intensity at Several Wavelengths (Fig. 7)

tions, the cometary spectrum called Il is divided by a solar spectrum called Fl in a given (l1, l2) wavelength range. The solar spectrum is taken from Labs et al. (1987) and is convolved with a 10-pixel FWHM function to have the same resolution as the experimental spectrum. The ratio is called h (l1, l2). This operation assumes that the color of dust is ‘‘solar’’ between l1 and l2 . Figure 4 shows a spectrum of Halley’s inner coma having a p parameter equal to 421 km. The cometary spectrum Il is depicted in A. The solar spectrum Fl is shown in B. It can be seen that A and B coincide around 482 nm. The difference A 2 B 5 C is mainly composed of molecular emissions plus a small excess of dust-scattered radiation in the parts of the spectrum where the color of dust-scattered radiation is not exactly ‘‘solar’’.

III. MONOCHROMATIC MOSAIC IMAGES

III. 1. Construction of the Maps As previously shown in Section II.1, advantage was taken of the fact that the telescope of the spectrometer was equipped with a secondary mirror which could be rotated around two perpendicular axes to construct monochromatic 2D maps of the inner coma. The approach motion of the probe toward the nucleus was employed as a zooming effect, since the angle f and the scanned field of view (28 3 1.58 in mode A and 28 in mode B) remained constant during the session. As a result, the rectangular maps progressively shrink in size while containing the same number of spectra. At a given wavelength, a monochromatic map was con-

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shown in Fig. 2. Any point of this monochromatic image corresponds to one, and only one, pixel value taken from a spectrum obtained when the spacecraft was located on its trajectory at a point defined by a set of d and p distance values. With the exception of the lower part of the image, where distances p , 1200 km and d , 24,000 km (Table I), the monochromatic images can be compared to monochromatic photographs of a restricted area of the inner coma taken from a distance d 5 516,000 km. The plane of the map is the plane containing the nucleus that is perpendicular to the average position of the line of sight as shown in Fig. 1. FIG. 3. Variation of the scattering angle u as a function of the projected distance p, also called impact parameter. The scattering angle is the angle between the dust/Sun and the observer/dust lines.

structed in the following way: the first rectangle of 7 3 15 relative positions of the spectrometer slit was displayed on the computer monitor. The second rectangle, smaller than the first, was also displayed, removing the common area previously displayed, and taking into account the diminution of the scanned field of view due to the reduction of the probe–nucleus distance. The following rectangles were successively displayed in the same manner. The whole monochromatic image, including the 7 rectangular maps of mode A and the 28 rows of mode B, has the shape of a sector of angle 508 with its apex at the comet nucleus as

III. 2. General Characteristics of the Maps Monochromatic mosaic maps constructed as explained above provide a 2D description of the area scanned by the instrument that shows evidence of dust and gas jets. In addition, ratios of monochromatic maps can be used to obtain a 2D representation of the color of dust. Furthermore these monochromatic maps can be constructed for any wavelength corresponding to a single pixel or to a group of pixels inside the wavelength range of TKS. For example, monochromatic maps of the OH (309 nm) and CN (388 nm) emissions can be constructed as shown in Fig. 5a and 5b. Two well defined gaseous jets are present in both maps. The jets can be identified on any of the seven individual rectangles defined in Section III.1, but their extension is clearly visualized on the mosaic images.

FIG. 4. Spectrum of Halley’s comet taken from Vega 2 at d 5 9257 km. Projected distance is p 5 421 km. A is the measured spectrum. B is the solar-type spectrum adjusted below A. C is the difference spectrum showing the molecular emissions.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

Since grains are carried away by the gas flow, we try now to visualize the presence of dust jets in the coma. For this purpose, monochromatic maps of the intensity scattered by dust grains are constructed as explained in Section III.1, for three wavelengths (377, 482, and 607 nm) chosen to keep the contamination by molecular emissions to a negligible level. The dust-scattered intensity decreases as 1/p or faster, with increasing p. This decrease for dust is more rapid than the decrease for molecular emissions OH and CN depicted in Figs. 5a and 5b. In order to show in Fig. 5 the 2D distribution of intensity with an acceptable contrast, two fields of view are presented for each wavelength. Figures 5c, 5e, and 5g show the overall field of view scanned by the spectrometer working successively in mode A and mode B. The nucleus is located on the lower right part of the image. The radial extent from the nucleus to the upper left corner is pmax p 42,000 km. Figures 5d, 5f, and 5h show the field of view restricted to mode B with a different half-tint scale. On each chart, four domains are clearly visible as shown in the lower box of Fig. 2. The first is a triangular sector of high intensity extending from the nucleus to about p 5 3000 km (Figs. 5d, 5f, and 5h). The second domain is constituted by two well contrasted jets visible in Figs. 5c, 5e, and 5g. The third domain is the so-called ‘‘valley,’’ which extends from about 7000 to 40,000 km and is located between these two jets. The fourth domain is the intermediate region between about p 5 3000 and p 5 7000 km where the jets are not yet apparent and where the dust distribution progressively becomes anisotropic. The dust intensity is higher on the right side of the scanned area (Figs. 5d, 5f, and 5h), and the jet becomes more contrasted with increasing p. However, the limits between the above mentioned domains are not well defined. III. 3. Dust Jets versus Gas Jets Monochromatic images obtained with ground-based telescopes (A’Hearn et al. 1986, Cosmovici et al. 1988) complemented a detailed description of the inner coma obtained with the spacecraft. These images, as well as the ones taken of comet Austin (Suzuki et al. 1990) and P/Swift-Tuttle (Jorda et al. 1994) when properly processed, revealed the existence of gaseous jets in the CN and C2 . The presence of gas jets in Halley’s coma has been reported both by Vega observers (Krasnopolsky et al. 1986, Clairemidi et al. 1990a, 1990b) and by ground-based observers (A’Hearn 1986, Cosmovici 1988, Schulz et al. 1994). The distance scale at which they were observed, several ten thousand kilometers from the nucleus, was unexpectedly high. The images of the extended field of view scanned by TKS (Figs. 5c, 5e, and 5g) in the dust spectral bandpasses show that dust jets are present in the same region as the gas jets: 7000–40,000 km. Iso-intensity contour lines show the extension of the jets.

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Considering Figs. 5a–5c, 5e, and 5g, we may draw circular arcs of given radii p centered on the nucleus and plot the intensity value as a function of azimuthal angle. The resulting diagrams are presented in Fig. 6 for two values of p, p 5 5000 km in Fig. 6a and p 5 25,000 km in Fig. 6b. This figure shows that the jets are not distinguishable at p , 5000 km, but are present, well contrasted at p 5 25,000 km. At that distance, the maximum-to-minimum ratio between the jets and the valley is equal to 1.6 for the gas and 2 for the dust. Present observations show that jets in the coma gradually appear at cometocentric distances longer than 7000 km. There is no direct connection with the collimated jets shown in the Giotto and Vega photographs as ‘‘springing’’ out of the nucleus surface, rather these jets rapidly vanish and merge in the dusty environment of the inner coma (Keller 1994). III. 4. Dust Brightness Profiles Radial profiles of dust scattered intensity are plotted in Fig. 7 as a function of distance p in a log–log representation of the 440–40,000 km range. The radius along which the spectra are chosen is located in the ‘‘valley,’’ as shown in the lower box of Fig. 2. Along this radius, it may be assumed that the contribution from the jets to the total dust flux is negligible in comparison with the dust flux from the nucleus. Three radial profiles are presented in Fig. 7 for the respective wavelengths 377, 482, and 607 nm. The error bars depend upon the instrumental mode, either the 2D scan mode at p . 9000 km (mode A), or the 1D scan mode (mode B) at p , 9000 km during the close encounter period. The brightness reaches about 800 kR/nm very close to the nucleus, at p 5 440 km, but falls to a value smaller than 100 kR/nm at p . 4000 km. Three sections can be distinguished in the diagram of Fig. 7. The intensity can be described by a power law I Y p2s( p), where 2s( p) is the slope measured in Fig. 7. For values of p less than 3000 km and higher than 7000 km, the slope is close to 21. In the 3000–7000 km distance range, the slope is steeper. A least square fit to a straight line yields the following values of the slopes: 21.43 6 0.30 at 377 nm, 21.52 6 0.22 at 482 nm, and 21.49 6 0.24 at 607 nm. The measured value at 482 nm may be compared with the radial gradient deduced from the OPE photometric experiment of Giotto (Nappo et al. 1988). The OPE intensity at the wavelength of 442 nm, Dl 5 4.7 nm in the 1990–7200 km range decreases with a slope of 21.6. This value is in good agreement with the 21.52 slope of our intensity radial profile at 482 nm in the intermediate 3000–7000 km distance range. Now the following question is to be addressed: Is it possible from the TKS measurements described above to obtain information about the physical properties of dust (dust size distribution, refractive index, and density)? Before answering this question, we describe in the next sec-

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FIG. 5. Monochromatic mosaic images of the OH (309 nm) and CN (388 nm) emissions and of the dust scattered intensity measured by the three-channel spectrometer at 377, 482, and 607 nm. The distance scale, shown in the upper part of each figure is 29,131 km for a, b, c, e, and g and 8414 km for the B mode d, f, and h. The images in a, b, c, e, and g describe the entire field of view scanned by the instrument optical axis in the successive A and B modes. Its extent is 40,000 km. The nucleus is located on the lower right. The Sun is located on the left. The white outlines are intensity contour plots to the jets and the valley. The images in d, f, and h describe the field of view scanned during the 33 min period preceding closest approach when the instrument mode was B. On the photograph, the symbol PIX means Pixel numbers. There is a bijective relation between the pixel number and the wavelength. (a) Monochromatic image of the OH gaseous emission at l 5 309 nm, Dl 5 1.2 nm, after subtraction of the dust continuum. (b) Monochromatic image of the CN gaseous emission at l 5 388 nm, Dl 5 1.2 nm, after subtraction of the dust continuum. (c, d) l 5 377 nm, Dl 5 1.2 nm; (e, f ) l 5 482 nm, Dl 5 1.2 nm; (g, h) l 5 607 nm, Dl 5 1.2 nm.

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FIG. 5 —Continued

tion the dust model constructed to tentatively explain the main features of the observed radial profile of Fig. 7. The interaction between solar radiation and dust particles will be calculated using Mie theory. In the absence of a detailed knowledge of the shape and composition of the dust particles, this theory is of great help in characterizing dust properties from polarization measurements (Dollfus and Suchail 1987, Le Borgne et al. 1987, Lamy et al. 1987) and

spectrophotometric observations (Jewitt and Meech 1986a, Hoban et al. 1989). IV. A MODEL OF THE RADIAL DISTRIBUTION OF DUST

Before the flights to Comet Halley, models of solar radiation scattered by dust particles of the coma were intensively

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FIG. 5 —Continued

used to design the spacecraft and instruments (Divine et al. 1986, Moroz 1985). However, compared to in situ dustparticle measurements using particle counters and flux analyzers, measurements of the scattered brightness have the disadvantage of providing an intensity integrated along the optical axis and over the size-range where the particles are optically efficient. First, the geometric integral corresponding to the integration along the line of sight can only be

inverted by using crude approximations. Second, the scattering properties of dust particles are most efficient when the ratio 2fa/ l p 1, where a is the ‘‘effective radius’’ of the particle and l is the radiation wavelength. As a consequence, since the optical observation wavelength is generally between 300 and 1000 nm, the optical remote sensing method for measuring dust particles is most useful for observing a whole family of particles of radii ranging from 50

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FIG. 5 —Continued

nm to several micrometers. Greenberg (1982) suggested that these particles were constituted of aggregates of small precometary grains like a bird’s nest or, possibly a small pack of cooked rice grains. More recently, Greenberg and Hage (1990) gave a more detailed description of their particle model, suggesting that they have a silicate core covered by an organic refractory mantle. Our ambition is not to deal with the optical properties of Greenberg’s particles, but to investigate whether the spectrophotometric observa-

tions of dust by TKS are consistent with calculations based on the widely employed Mie theory for spherical homogeneous particles. This model enables comparison with the results of other authors. The effects of fragmentation processes, modeled here in a simple manner, will be investigated and analyzed. By using Mie theory we restricted the number of essential parameters being discussed to three: the n and k values of the (n 2 ik) complex index of the grains, and the density of dust particles, hereafter called r.

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FIG. 6. Variation of intensity with azimuthal angle along two arcs of a circle of radii 5000 (a) and 25,000 km (b). The intensities show the OH and CN emissions and the dust-scattered intensities at 377, 482, and 607 nm. The scale for the OH and CN intensities is shown on the right, and the scale for the dust-scattered intensities is shown on the left.

The intensity integrated along an optical axis defined by its direction, the projection point H of the nucleus N, and a scattering angle uH at point H (Fig. 8) is expressed by relation (1). Il 5 Dl

E E y

a2

z1

a1

Fl Qsca(n, k, a, l) n9(a, R) fa2 S(a, l, u) da dz (1)

In this equation: • Fl is the solar intensity expressed as a number of photons cm22 sec21. • Qsca (n, k, a, l) is the scattering efficiency. Coefficients n and k are the real and imaginary parts of the effective complex index n 2 ik of the dust particles. • n9(a, R) is the number density of dust particles with an effective radius comprised between a and a 1 da. R is the distance from the nucleus to the considered point in the coma. • fa2 is the geometric cross section of an isolated particle. • S(a, l, u) is the phase function where u is the scattering angle, i.e., u 5 f-phase angle (Van de Hulst 1957).

FIG. 7. Radial distribution, in a log–log scale graph, of the dust intensity at 377 (squares), 482 (crosses), and 607 (triangles) nm. The horizontal coordinate is the logarithm of the projected distance p between the nucleus and the optical axis. The data of the Giotto OPE instrument, taken from Nappo et al. (1989) have been reported to allow a comparison of the slope of the curves between 2000 and 7000 km. The curve for OPE data has been translated vertically for readability. The intensity scale in kR nm21 does not apply to this curve.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

FIG. 8. Geometric configuration of the spectrometer line of sight. The figure shows the projections of the line of sight and the Sun–nucleus line on the comet orbital plane. The specific closest approach period was a very short time about 10 min. In (a) the spacecraft is located at a distance from the nucleus d . 43,700 km. The projected distance to the line of sight is p . 2300 km. In (b), the spacecraft is located at a distance from the nucleus d , 43,700 km. The c angle rapidly increases from 1.98 to 54.38, which corresponds to an increase of the scattering angle uH from 76.68 to 120.78 at the moment of closest approach.

The solar intensity Fl is taken from Neckel and Labs (1981, 1984). The scattering efficiency Qsca (n, k, a, l) is calculated by using the Mie code of Herman (Laboratoire d’optique atmosphe´rique. Lille, 1992, personal communication). Three parameters need to be defined: n and k in the complex index n 2 ik and the size parameter x 5 2fa/ l. Two complex indices were used in our calculation. The first index was calculated by Mukai et al. (1987) to fit the actual polarization curve measured in Halley (m 5 1.38 2 0.03i at l 5 484 nm). This hypothetical material

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was called ‘‘silicate B’’ by Gru¨n and Jessberger (1990) in their review paper. The second material is tholin (Khare et al. 1984). This is an organic residue produced in a laboratory-simulated atmosphere of Titan (Sagan and Khare 1979). Choosing the index of tholin is a step toward trying simulating the dust particles that are perhaps covered by a mantle of organic material (Greenberg 1982). The average complex index for tholin is m 5 1.70 2 0.02i in the visible domain. The scattering efficiencies Qsca calculated for both sets of complex index values for radiation of wavelength 377, 482, and 607 nm are presented in Figs. 9a and 9b. The curves are expanded toward the right side of the abscissa axis because the efficiency is a function of a/ l. The first maximum of Qsca occurs for a p l. The ratios of scattering efficiencies at two wavelengths, Qsca,607 /Qsca,377 and Qsca,607 /Qsca,482, have been also calculated (Fig. 10). Both ratios show important variations as a function of a when a/ l , 5, which corresponds to dust particles of micrometer or sub-micrometer size. Particles with a radius a . 3 em will not produce any color effect in the visible range. An ‘‘ideal’’ calculation of the scattered intensity would require a 3D description of the dust particle size distribution at any point of the inner coma. This is of course out of reach. Since one of our main goals is to obtain a reasonable agreement between the radial profiles of intensity and color with a model computation based on parameterized processes, we restricted the number of variables to the sole dust grain density r. The input data are the dust particle flux (cumulative) distribution curves of Mazets et al. (1987) for Vega 2. The mass-range 10216 –1026 g is divided in 10 decades that are numbered using an index i running from 1 to 10 as shown in Table II (1 and 10 for, respectively, the lowest and the highest mass decades). This method, compared to the use of a polynomial approximation of the log–log dust particle flux distribution, presents the advantage of allowing a comparison of the relative importance of the different mass (or size) ranges of particles. An extrapolation problem arises at R , 8100 km because the dust impact experiments provided in situ measurements only for R $ 8100 km, whereas the optical axis of TKS scanned the coma to shorter distances. Consequently, to compute the integral (1), we need some data for the dust distribution at R , 8100 km. First, we have tried to use the data given by the DID experiment on GIOTTO spacecraft (McDonnell et al. 1987) that provided the dust distribution at R 5 3400 km. In fact, these data appear to be a factor 2–3 lower than the extrapolated data of Vega 2, which shows that the dust flux on March 14 was less intense than on March 9 (Fig. 11). This point is only partially supported by the time variability curves of Schleicher et al. (1987) who report for the same type of comparison a decrease of the continuum at 365 nm by a smaller factor that is only

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FIG. 9. Variation of the scattering efficiency Qsca as a function of the particle radius a for three wavelengths: 377, 482, and 607 nm. 9(a) Complex index is 1.387 2 0.031i at l 5 482 nm. This index was proposed by Mukai et al. (1987) to describe a material that would have the same polarimetric properties as Comet Halley in the visible. (b) Complex index is 1.700 2 0.020i in the visible domain. This index value was measured for tholin by Khare et al. (1984).

1.3. We finally used the experimental dust distribution of Mazets et al. (1987) extrapolated at R , 8000 km by a 1/R2 law. The fluxes F(.m) of dust particles are presented in their paper as a function of the particle mass m for eleven different masses 102161i (i 5 0, . . . , 10) g. For each mass value, a linear regression line of slope 22 was drawn in the log F(.m) versus log R diagram. The different lines were extrapolated to the low R values (440 km) as shown in Fig. 11. The flux F(.m) data are used to compute the local dust particle distribution function n9(a, R) according to (2) n9(a, R) 5

dF(.m, R) dm dF(.m, R) 5 4fa2r. V dm da V dm

(2)

In this expression, n9(a, R) is the number density of dust particles that have a radius between a and a 1 da as a function of distance R from the nucleus, r is the density of the particles, and V is the relative speed of the spacecraft. At this point, it is supposed that the particle speed in the cometocentric frame is negligible compared to V. The dust particle density is a key parameter. Constant values, independent of size (0.30, 1.00, and 1.45 g cm23) will be used following other authors (Krasnopolsky et al. 1987, Hoban et al. 1989). A value of r that is a decreasing function of the effective radius a in micrometer will also be used, such as the following function, proposed by Lamy et al. (1987),

r(a) 5 2.2 2 1.4

a . a11

(3)

Such a variation of r with a provides a means of taking into account the porous, fluffy, aggregated, or filamentary structure of the dust particles that may fragment into smaller grains of higher density. The last term needed for computing the intensity (1) is the phase function S(a, l, u), which is defined as S(a, l, u) 5

FIG. 10. Scattering efficiency ratios Qsca,607 /Qsca,482 and Qsca,607 / Qsca,377 as a function of the particle radius a. The ratios presented in this figure are calculated by using the complex index proposed by Mukai et al. (1987).

a2 (uS1u2 1 uS2u2), 2x2

(4)

with x 5 (2fa)/ l. The functions S1 and S2 are two complex amplitude functions which describe the scattering of radiation with electric fields perpendicular to and parallel to the plane of scattering (Van de Hulst 1957). Depending upon the size of the scattering particles, the variation of S(a, l, u) with the scattering angle u may be important.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

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TABLE II Radii of the Dust Particles in the Ten Mass Decade Classes

For instance, Fig. 12a shows the variation of S as a function of u for l 5 377 nm and for the 10 mass classes defined in Table II. Due to the operation mode of TKS, during the 10 min period of time before closest approach (43,700 km . d . 9884 km), u rapidly increased from 76.68 to 1208, while it had only increased from 65.68 to 76.68 before.

As a result, the presence of the phase function S(a, l, u) depending upon a and u in the integral (1) modifies the intensity profile only in the distance range that corresponds to the 10 min period of closest approach (Fig. 3, Table I). For the sake of comparison, we represented in Figs. 12b and 12c the 10 phase functions by groups of 5, with a u scale restricted to the range of interest (658 to 1208). Particles of the first decade (a , 0.1 em) have a Rayleigh-type phase function (Figs. 12a and 12b), whereas the particles of decades 2–6 have a phase function decreasing with u, and the biggest particles (decades 7–10) have almost the same relatively flat phase function (Figs. 12a and 12c). Consequently, the smallest particles (decades i 5 1, 2, 3) provide only a minor contribution to the total intensity because the product S(a, l, u)n9(a, R)fa2 is small in these decades compared to same quantity in the other decades. Next we analyze the results of computational runs of our model under various conditions to simulate the principal physical mechanisms occurring in the coma. V. INTERPRETATION OF THE INTENSITY RADIAL PROFILE

FIG. 11. Extrapolation of the cumulative flux distribution F(.m) of Vega 2 (Mazets et al. 1987) to cometocentric distances R less than 8100 km. The different lines refer to masses 102161i (i 5 0, 10). They are lines having a slope 22 shifted to match the experimental points of Vega 2 (at distances R 5 8100, 14,000, 27,000–57,000, and 57,000–95,000 km). The points on the left at 3430 km are taken from the Giotto data (McDonnell et al. 1987). They are located under the extrapolated Vega 2 data, which shows that the dust flux on March 14 was less intense than on March 9, dates of closest approach of Giotto and Vega 2.

The main characteristics of the scattered intensity radial profile measured by the three-channel spectrometer and shown in Fig. 7 may be summarized as follows. The intensity profile obeys a p2s( p) law with s 5 1 at p , 3000 km and p . 7000 km, and s 5 1.52 at 3000 km , p , 7000 km, where p is the distance between the nucleus and the optical axis. This shows that the quantity 4fR2 n(R), where

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

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FIG. 12. Variation of the phase function with the scattering angle u in the ten mass range decades considered in the model. Decade 1 means 10216 –10215 g mass range. Decade 10 means 1027 –1026 g mass range. The assumed density here is r 5 0.3 g cm23, wavelength l 5 377 nm, complex index m 5 1.387 2 0.031i (Mukai et al. 1987). (a) The phase function is calculated using two different codes: Mie Tab (MT; solid line) and M. Herman (MH, crosses). The curves are close to each other. (b) The five phase functions for the five lower decades used in the model are presented in this figure under the same vertical scale. The scattering angle range 658–1208 is chosen to be adapted to the line of sight rotation. In this range, the phase function, for a given scattering angle, strongly varies with the particle size, but the contribution of the smaller particles to the total intensity is not very important. (c) The five phase function for the higher decades are presented with the same vertical scale, the curves for decades i 5 7, 8, 9, 10 are superposed. Their variation with scattering angle is small.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

95

R is the cometocentric distance and n(R) is the number of particles per unit volume, is not constant with R. The observations obtained with the OPE photopolarimeter (Nappo et al. 1989) and with the HMC camera of Giotto (Thomas and Keller 1989) also show an intensity that does not follow an 1/p law in the inner coma. Given the fact that the brightness integral (1) cannot be inverted in the present case, we adopted the reverse process and calculated the expected integrated intensity in order to identify and evaluate quantitatively the respective efficiencies of the three processes: (i) radiation pressure and subsequent fountain effect, (ii) phase function influence due to the rapid variation of u in the inner coma, and (iii) fragmentation processes. The computation runs were first made by using the complex index recommended by Mukai et al. (1987) with various densities r. Then, additional runs showed that the complex index of tholin (Khare et al. 1984) leads to a better agreement, especially when the radial variations of the dust color are considered. V.1. Calculated Brightness Radial Profile First, the following integrated intensity is calculated:

Il 5 Dl

E

y

z1

O SE 10

i51

mi11

mi

D

Fl Qsca(n, k, a, l)n9(m, R) fa2 S dm dz. (5)

The terms have the same meaning as in (1) with m being the mass of a dust particle and a its radius with m 5 4/3 r fa3. The calculation is performed for three values of the density r: 0.3 and 1.0 g cm23 and r defined in Eq. (3). The phase function S is supposed to be a fixed given number in this first step. The scattering efficiency Qsca is calculated by using the complex index 1.387 2 0.031i of Mukai et al. (1987). The results are presented in Figs. 13a–13c. Subscripts a, b, and c refer to wavelengths 377, 482, and 607 nm. The intensity Il follows an 1/p law that shows only an approximate agreement with the experimental radial profile. The main difference is the existence of an inflexion point at p p 6000 km where the slope s( p) is equal to 1.5–1.6.

FIG. 14. Schematic representation of the particle trajectories creating the fountain effect.

V.2. Effect of Solar Radiation Pressure and Fountain Effect In order to improve the agreement between the experimental and calculated profiles, the next step was to add the fountain effect mechanism in the simplest way possible, as shown in Fig. 14. As explained in Section III.4, the experimental radial profile was chosen in the valley (Fig. 2) where the possible contribution of dust particles coming from the jets is negligible. Therefore, the presence of the jets will not be taken into account in the following. The release of dust is supposed to be isotropic, radial from the nucleus, with a velocity depending upon the size and the mass of the grains. Let us consider a cometary grain that is ejected from the nucleus. Two main forces act upon the particle: the gravitational force due to the Sun and the radiation pressure force in the opposite direction. This second force has the form

FIG. 13. Radial distribution, in a log–log scale graph, of the dust intensity at 377 (boxes on the left), 482 (boxes in the middle), and 607 nm (boxes on the right). In each box, the horizontal coordinate is the logarithm of the projected distance p between the nucleus and the optical axis. In the whole set of figures, the assumed complex index is 1.387 2 0.031i. The calculation is performed with three different densities: r 5 0.3, 1.0 g cm23 and r 5 2.2 2 1.4a/(a 1 1) (Eq. (3)). (a, b, c) The intensity is calculated by integrating along the line of sight the locally scattered radiation. (d, e, f ) The fountain effect is taken into account. (i, j, k) The phase function is introduced into the model. This results in a change of the intensity profile mainly at p , 6000 km where the scattering angle increases from 708 to 1208. (l, m, n) A fragmentation mechanism is added to the model in the inner coma. A good fit with the observational profile can be obtained under these conditions.

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FIG. 15. Variation of the radiation pressure quantum efficiency Qpr(a, l) with dust particle radius for three wavelengths: 377, 482, and 607 nm. Qpr p 1 for a . 1 em, but Qpr drops rapidly for a , 0.3 em.

Fpr 5 Qpr (a, l)

Ls fa2, 4fr 2H c

(6)

where Qpr (a, l) is the radiation pressure quantum efficiency, Ls is the solar luminosity, rH is the heliocentric distance of the considered particle, and c is the velocity of light. The variation of Qpr with a is given in Fig. 15. It is computed using the classical expressions given by Van de Hulst (1957) and Bohren and Huffman (1983) for Mie theory. This figure shows that Qpr P 1 for a . 1 em. For values of a , 1 em, the radiation pressure efficiency rapidly drops to very low values. The ratio of the radiation pressure force to the gravitational force is generally called e. It is expressed as e5

Fpr 3 Ls Qpr 5 , Pgr 16 fcGMs ra

(7)

where G is the gravitation constant and Ms is the Sun mass. The ratio e is inversely proportional to the product ra. Small grains with a . 0.3 em will be efficiently decelerated. As a result, the trajectories of small grains are approximately parabolic and, for identical grains, they are defined only by the direction of the initial velocity vector vo(a). The envelope of all the trajectories is also a parabola (Massonne 1985, 1990). It defines a partition of the cometary environment since no small grain of the considered size will

be present outside the parabolic envelope. The distance between the nucleus and the apex is written as Rm(a, r) 5

v2o(a) 2ucu

(8)

with the acceleration c defined by c5

S

D

GMs GMs 3Lc Qpr 12 . 2 (1 2 e) 5 2 rH rH 16fcGMs ra

(9)

The same function for vo(a) which was used in the dust model for the mission (CNES-Intercosmos, p. 14, Gombosi et al. 1983, also Gombosi et al. 1986) is employed in our model vo(a) 5

7.44 3 104 1 1 10

2

!

r a 3 4 3 106 Md

.

(10)

In this expression, vo(a) is expressed in cm sec21, the density r in g cm23, and the radius a in cm. Md 5 107 g sec21 is the total mass of dust ejected per second by the nucleus. The values of the dust velocity vo(a), of the acceleration and of the apex cometocentric distance for different values of r are given in Table III. The results of the intensity radial profile are presented in Figs. 13d–13f which show that the calculated intensity

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

is slightly higher, by a factor 1.3, when the fountain effect is introduced into the model. An intermediate conclusion can be drawn at this point: the additional emission due to the dust particles moving in the anti-solar direction is not sufficient to explain the intensity excess at p , 6000 km. We next calculate the resulting effect of taking into account the phase function S(a, l, u). V.3. Phase Function S(a, l, u) The calculated intensity is now Il 5 Dl

E

y

z1

O SE 10

i 51

mi11

mi

Fl Qsca(n, k, a, l)

D

(11)

? n9(m, R) fa2 S(a, l, u) dm dz. The results are given in Figs. 13i–13k. The slope of the intensity radial profile is 21 at p . 6000 km. At smaller projected distances, the slope is smaller than 20.6. This is a result of the variation of the phase function with u, studied in Section IV. In Figs. 13i–13k, the intensity profiles for r 5 1 g cm23 and for r 5 f (a) (Eq. (3)) do not show parallel variations. The differences will be more accentuated when the color of the scattered intensity is considered. The comparison between the calculated and the experimental profiles shows that the first one does not reproduce the excess of intensity measured at p , 6000 km. In order to obtain a better agreement, we next introduce a fragmentation process considering the fact that a spheri-

97

cal particle of radius a that splits into particles of radius a/n has its total geometrical cross section multiplied by n. V.4. Fragmentation Effect Once released from the nucleus, dust particles are heated by the solar radiation. Fragmentation processes occur resulting in the production of smaller particles. Fragmentation has been introduced by Thomas and Keller (1989) to interpret the variation with R of the quantity IR2 in the inner coma, where I is the measured intensity and R the cometocentric distance. Such a process is considered by Combi (1994) as being the sole mechanism able to produce the elongated shape of the dust coma in the anti-solar direction on the scale 104 –105 km. We introduce fragmentation using only one parameter called a(R). We suppose that each mass decade indexed by i looses a fraction ai,i21 of its mass to produce particles of the lower decade. It receives a fraction ai11,i of the mass of the upper decade. The ai,j coefficients are calculated in a way that obeys mass conservation. As shown in the Appendix, a matrix is calculated. The unique a(R) parameter to be chosen is the fraction of particles of the mass decade 1027 –1026 g that undergoes fragmentation. The model shows that fragmentation processes are very efficient in increasing the scattered intensity. The type of function a(R) for the fragmentation coefficient that provides the best agreement between the model results and the experimental data is presented in Fig. 16. The fragmentation coefficient a(R) is taken to be equal to 0.87 between 0 and 1000 km. At larger distances, it is supposed to decrease following a law of the type

TABLE III Values of the Dust Terminal Velocity vo(a), Acceleration, and Cometocentric Distance of the Apex as a Function of the Mass of the Particles

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VI. DISCUSSION

The set of complex index and density parameters that provides the best fit with actual data can be determined on a quantitative basis by the calculation of the root-mean square deviation between the observed and calculated profiles for various parameter set values. Let us call xexp 5 log(Iexp,l) and xmod 5 log(Imod,l), where Iexp,l and Imod,l are, respectively, the measured and calculated intensities. We calculate the root-mean square deviation according to rms x 5 FIG. 16. Variation of the fragmentation coefficient a(R) with cometocentric distance R.

a(R) 5 20.175 1 1050/R,

(12)

where the cometocentric distance R is expressed in kilometers (see Appendix). At distances R . 6000 km, the fragmentation coefficient is supposed to be equal to zero. The method used here to simulate the fragmentation emphasizes the importance of this process in the inner coma at R , 1000 km. This point is in total agreement with the extensive work of Combi (1994) who introduces large scale fragmentation of dust particles within 1000 km from the nucleus to obtain an adequate model of the isophotes in the inner coma. The possible effect of dust fragmentation on the dust environment of Comet P/Grigg-Skjellerup was investigated by Goidet et al. (1993) and Fulle et al. (1994) in order to obtain a comprehensive analysis of the measured slope of the radial scattered intensity profile. Fulle et al. (1994) using a Monte Carlo model obtained a good agreement with ground-based images but their comparison is limited to cometocentric distances R . 2000 km and does not include the region at R , 1000 km where fragmentation processes should be taken into account. V.5. Results Obtained with Other Indices When the complex index of silicate B (Gru¨n and Jessberger 1990) 1.4 2 0.03 i was used, the resulting curves looked the same as in the case of Mukai’s indices. The complex index of tholin, 1.7 2 0.020 i (Khare et al. 1984) was also used. The results, obtained under the same conditions as in Figs. 13l–13n are shown in Figs. 17a–17c. Here again, the model was run with several dust grain densities r 5 0.3, 1.0, 1.45 g cm23 and r 5 f (a) (Eq. (3)). A good agreement with the experimental data is obtained for r 5 f (a).

HO 1 N

J

(xexp 2 xmod)2

1/2

.

(13)

The results are presented in Table IV where the array cells corresponds to the boxes of Fig. 13 and Fig. 17. The lowest value of rms x corresponds to the best fit. It may be seen that the introduction of the fountain effect and adopting r 5 1.0 g cm23 provides an apparently good fit with the experimental curve. In further steps, the phase angle variation and fragmentation processes are introduced. The best fit for the three wavelengths 377, 482, and 607 nm is obtained when the density of (3) and the complex index of tholin are used. The mean deviations are then as low as 0.07, 0.10, and 0.08 for 377, 482, and 607 nm, respectively. The final results are presented in Fig. 18 where the computed radial profiles in the three colors 377, 482, and 607 nm are drawn to allow a global comparison with the measured distributions. VI.1. Relative Contributions of the Different Mass Decades An important difference between the measured dust particle distribution function (Mazets et al. 1987, McDonnell et al. 1987) and the dust distributions used in models (Divine and Newburn 1984) prior to the comet exploration by the spacecraft appeared to be the presence of an excess of small particles with masses between 10216 –10213 g. Since the radius a of these particles is between 0.02 and 0.4 em, we calculated the relative importance of these small particles as optical emitters. The relative contributions of the 10 mass decades defined in Table II are calculated and presented in Fig. 19 under the same assumptions as in the previous sections: the fountain effect is introduced first (Figs. 19d–19f ), then a phase function (Fig. 19i–19k), and finally fragmentation processes (Figs. 19l–19n) are added. In all cases, the contribution of the first two decades (masses 10216 –10214 g) is negligible. The intensity distributions in Figs. 19a to 19f show that the particles that scatter the maximum number of photons belong to classes 6–9 (masses 10211 –1027 g). When the fountain effect is included, the contribution of class 5 becomes more important, espe-

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99

FIG. 17. (a, b, c) The figures are the same as 13(l, m, n), but the complex index is different. It is here 1.7 2 0.02i for tholin. The processes included in the model are the fountain effect, phase function variation, and fragmentation of dust particles in the inner coma.

cially at 440 km (Fig. 18d). Taking into account the phase function shows that the contribution of class 5 (mass 10212 – 10211 g) is dominant. The effect of fragmentation (Figs. 19l and 19m) is to enhance the contribution of small particles in mass decades 3, 4, 5, 6. At p 5 440 km, the resulting intensity integrated along the line of sight is enhanced by a factor 5. At p 5 1032 km, the integrated intensity is multiplied by 2.5. This result is due to the fact that particles created in the mass ranges 10214 –10210 g are efficient scatterers. VI.2. Color of Dust as a Function of Projected Distance p Two brightness ratios (I607 /I482 and I607 /I377) are used as parameters to characterize the dust color. Their values are

represented in Fig. 20 by false color maps at any point of the field of view scanned by the instrument. As a result, images of the coma region explored by the telescope are constructed to describe the spatial variations of both color parameters. The color parameters I607 /I482 and I607 /I377 depicted in Figs. 20a and 20b can be compared with values calculated in using the model. The radial profile of the color parameter along the radius defined in Fig. 5 as a function of the projected distance p is shown in Fig. 21. In Figs. 21a and 21b, the index 1.387 2 0.031 i of Mukai et al. (1987) was used. In Figs. 21c and 21d, the same calculation was done with the index of tholin. The experimental curve is drawn with a solid line. Both ratios show that the color of dustscattered intensity is progressively reddened between

TABLE IV Values of the rms x Deviations for the Different Fits Shown in Fig. 13a to 13n and in Fig. 17a to 13c

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FIG. 18. Comparison between the observed radial profile of intensity and the results of the model. The complex index is here 1.7 2 0.02i for tholin. The density r varies with particle radius (Eq. (3)). A fragmentation process is introduced in the inner coma at p , 6000 km.

10,000 and 40,000 km along a radius of the valley. As already stated, this area is supposed to contain mainly dust particles coming from the nucleus, so that these ratios should be a very precise test of the parameters of our model. Indeed, as can be seen in Figs. 21a–21d, the theoretical curves obtained with various values for the parameters exhibit very different shapes. The ratio I607 /I377 shows two behaviors nearly opposite between 10,000 and 40,000 km, for the index of Mukai et al. and for the index of tholin. The better agreement is obtained with the index of tholin. Furthermore, when comparing the results calculated with different density values, the best fit is obtained for the density r decreasing with the particle radius a as in Eq. (3). Finally, it can be seen that the overall best fit, obtained for the index of tholin and the variable density, reproduces the data fairly well. Present results are coherent with the data of Hoban et al. (1989) obtained on March 1 and 6, 1986. These authors define a reflecting gradient S9 that may be calculated in our case as

S91 5

(R1 /R1s)21 Dl

and S92 5

(R2 /R2s)21 , Dl

(13)

where R1 5 I607 /I482, R2 5 I607 /I377 for cometary intensities and R1s 5 Is,607 /Is,482, R2s 5 Is,607 /Is,377 for solar intensities. All intensities are expressed in units of photons per time, surface, and solid angle. As shown in Figs. 21c and 21d, the dust-scattered intensity presents an excess of red coloration at p . 25,000 km. The reflectivity gradients are calculated in taking R1s 5 1.11 and R2s 5 2.10 from the solar data of Neckel and Labs (1981). At p 5 40,000 km inside the valley, S91 5 14% per 100 nm and S92 5 11% per 100 nm. At shorter projected distance, the gradients decrease to about 0 at 1000 km. The S9 values reported here are in good agreement with the values measured by Hoban et al. (1989) in the 484.5–684 nm range, S9 5 7% per 100 nm and with the values of Meech and Jewitt (1987) in the 439–682 nm range, S9 5 9% per 100 nm. VII. CONCLUSION

During the Vega 2 encounter session, a triangular sector inside the inner coma of Halley was scanned by the threechannel spectrometer. Mosaic images were reconstructed to show the spatial extension of the dust and molecular components of the brightness measured by the spectrometer. The dust scattered brightness is higher than the molecular emissions in the inner coma at projected distances p , 3000 km. The angular distribution of dust in this region of high density appears fairly isotropic. The radial profile follows a p21 decrease. In the intermediate 3000–7000 km range, the radial profile gradient is steeper, as p21.52, which is in good agreement with the distributions measured in situ by the OPE instrument of Giotto (Nappo et al. 1989). At distances larger than 7000 km, the dust angular distribution becomes anisotropic. It may be compared with the molecular emission distribution. Two jets appear with a contrast of 1.6 for the gas and 2 for the dust. In this distance range, dust shows a noticeable degree of coloring. The color becomes progressively redder along a radius issued from the nucleus and located in the so-called valley, between the jets. The reflectivity gradient S91 in the spectral range 482–607 nm measured at p 5 40,000 km is equal to 14% per 100 nm, which is comparable to values recently measured in the case of other comets. In an attempt to analyze the mechanisms responsible

FIG. 19. Analysis of the contribution of the different mass range decades from 10216 to 1026 g to the total intensity scattered by dust particles along the line of sight. Three projected distances p are considered: p 5 440 (boxes on the left), p 5 1032 (boxes in the middle), and p 5 9803 km (boxes on the right). In each box, the histograms are drawn for three wavelengths: 377, 482, and 607 nm. The mass range classes are defined in Table II. (a, b, c) The intensity is the result of the integration along the line of sight of the locally scattered radiation. (d, e, f ) The fountain effect is taken into account. (i, j, k) Introduction of the phase function into the model. (l, m, n) Addition of a fragmentation mechanism to the model.

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for the observed profiles, a model was developed to calculate the scattered intensity integrated along a line of sight at a projected distance p from the nucleus. The model uses Mie theory applied to spherical particles. The spatial density function n9(m, R) is calculated using the data of the impact particle counter (Mazets et al. 1987). The data are extrapolated assuming an R22 law from 8030 to 440 km. Two different complex indices are used: 1.387 2 0.031i from Mukai et al. (1987) and 1.7 2 0.02i from Khare et al. (1984). The physical processes taken into account in the model are radiation pressure and subsequent fountain effect, phase function, and fragmentation. A good fit with the observational data is obtained if it is assumed that dust particles suffer fragmentation in the inner coma at R , 6000 km. A fragmentation matrix that satisfies the mass conservation law is defined. The fraction of particles that are fragmented is adjusted to obtain the best fit with the data. The fragmentation coefficient is taken as a 5 0.87 at R , 1000 km which means that fragmentation processes are most efficient in the innermost coma. This result is in total agreement with the conclusions of Combi (1994) obtained with an extensive Monte Carlo model. The best agreement is reached when choosing the following law for the density from Lamy et al. (1987) (Eq. (3)). An analysis of the variation of color with p shows that the use of the 1.387 2 0.031 i index does not provide a satisfying agreement with the data. On the contrary, a good fit is obtained when the tholin index is used. As for the shortcomings of our model, we note that a model simulation based on Mie theory for spherical particles is probably not sufficient to explain the actual radiation–matter interaction at the microscopic scale because cometary particles are far from being compact spheres (Greenberg and Shalabiea 1994). However, it yields quantitative information about the amount of radiation scattered and the variation of color when fragmentation processes are added. It should also be noted that the dustcounter experiment provided a 1D description of the dust content along the spacecraft trajectory, which is insufficient to obtain a precise 2D description of the coma. New data concerning observations of Comet Borrelly show that the color of dust is not uniform in the coma (Goidet-Devel et al. 1996). This color effect is also visible in C2 data. We therefore plan to study this correlation.

particles in the lower decades. Let us call no,i and n1,i the number density of particles in the mass decade i before and after fragmentation occurs. We suppose that the relationship between the n1,i (i 5 1, 10) and no,j ( j 5 1, 10) are linear and are described by the matrix product (n1,i ) 5 (A)(no,j ),

(1A)

where (n1,i ) and (no,j ) are column vectors of dimension 10 and (A) is a 10 3 10 matrix, called the fragmentation matrix, the coefficients Aij of which need to be determined. It first may be noted that the fragmentation matrix is restricted to its upper right triangle including the diagonal, because small grains cannot fragment into particles of larger mass. Aggregation processes are not considered here. The rate of particles in the decade j that are fragmented into particles in the decade i is called aij . The number of particles of size i produced by the fragmentation of the particles of size j is thus aij no,j. The number of particles in the decade i after the fragmentation process is then

S O D O

(n1i ) 5 1 2

i21

k51

aki nio 1

10

j5i11

aij njo.

(2A)

An equation stating that, in a given volume, the total mass of the local cometary dust content is conserved, during fragmentation, should be written next.

On m 5On m . 10

10

1 i

i51

i

o i

i51

(3A)

i

In this equation, mi is the average mass of particles in the decade i. Let us replace n1,i by its value from Eq. (2A). After simplification, it yields

O O a n m 5 O SO a D n m . 9

10

i21

i51

k51

10

ij

i51 j5i11

o j

i

o i

ki

(4A)

i

In order to simplify this schema, we now suppose that if a particle in the decade i is subject to fragmentation, it only will produce particles in the lower adjacent decade i 2 1. This hypothesis means that the probability of two successive collisions is negligible. Equation (4A) takes the form

Oa 9

i51

o i,i11 ni11 mi

5

Oa n m . 10

i

i51

o i

(5A)

i

i21

The sum ok51 ak,i reduces itself to only one term that will be called ai in the following. Due to the mass conservation law, the particles of decade 1 cannot suffer fragmentation. This implies that a1 5 0. On the right side of Eq. (5A), let us replace the current index i by an index i 1 1. As the mass mi11 is equal to 10 mi , it yields

Oa 9

i51

o i,i11 ni11 mi

5

Oa 9

i51

o i11 ni11 10mi .

(6A)

Equation (6A) should be satisfied for any value of mi . Thus ai,i11 5 10ai11.

(7A)

APPENDIX: FRAGMENTATION MATRIX The fragmentation matrix (A) is bidiagonal. It is written in the form This appendix describes the method used to simulate the fragmentation of dust particles in the inner coma. Let us suppose that the particles suffer fragmentation due to collisions only, neglecting sublimation effects. Note that this imposes the conservation of the total mass of the dust. Let us consider a distribution of particles in the mass range 10216 –1026 g that will be divided into 10 mass decade classes numbered using the index i from i 5 1 (decade 10216 –10215 g) to i 5 10 (decade 1027 –1026 g). A particle in a decade i that is subject to fragmentation will produce

1

1

10a2

0 1 2 a2

A5 0 . .

0 . .

0

0

0

...

0

10a3

...

0

1 2 a3 . . . . . . .

0

0

0

a10 1 2 a10

2

.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

103

FIG. 20. Mosaic maps of two color parameters. (a) A display with a gray scale code of the intensity ratio I607 /I482 . (b) A display of the intensity ratio I607 /I377. A very light gray, nearly white, means an excess of red color. On the photograph, the symbol PIX means Pixel numbers. There is a bijective relation between the pixel number and the wavelength. The symbol ‘‘714, 715 SUR 506, 507’’ means intensity ratio I(l 5 607) over I(l 5 482).

In order to reduce the number of adjustable parameters, it is assumed that a2 5 ? ? ? 5 a10 5 a. This parameter a will be called the fragmentation coefficient. In the present model, the fragmentation process is described by only one parameter that is a function of cometocentric distance R. The function a(R) chosen here is a 5 0.87

at R # 1000 km

a 5 20.175 (60.06) 1 1050 (6109)/R at 1000 # R # 6000 km a50

at R $ 6000 km.

The present calculation is quite simple, but appears sufficient to provide a good agreement with the experimental data from the Vega 2 threechannel spectrometer.

ACKNOWLEDGMENTS We thank Professor M. Herman (Laboratoire d’Optique Atmosphe´rique, Universite´ de Lille) for having generously provided us with his Mie scattering computer code. We acknowledge constructive and helpful reviews by M. Fomenkova and an anonymous referee.

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FIG. 21. Color of dust as a function of projected distance p. Two parameters are used to characterize the color of dust scattered radiation. R1 5 I607 /I482 (a, c) and R2 5 I607 /I377 (b, d). (a, b) Complex index 1.387 2 0.031i (Mukai et al. 1987). (c, d) Complex index 1.7 2 0.02i for tholin. The best fit with the observational data (solid line) is obtained when the index of tholin is adopted and the (Eq. (3)) for r (a) is used.

DUST SPATIAL DISTRIBUTION IN HALLEY’S INNER COMA

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