Dust Cloud near the Sun

Dust Cloud near the Sun

Icarus 146, 568–582 (2000) doi:10.1006/icar.2000.6419, available online at http://www.idealibrary.com on Dust Cloud near the Sun Ingrid Mann Max-Plan...

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Icarus 146, 568–582 (2000) doi:10.1006/icar.2000.6419, available online at http://www.idealibrary.com on

Dust Cloud near the Sun Ingrid Mann Max-Planck-Institut f¨ur Aeronomie, Katlenburg-Lindau, Germany E-mail: [email protected]

Alexander Krivov Max-Planck-Institut f¨ur Aeronomie, Katlenburg-Lindau, Germany; and Astronomical Institute, St. Petersburg University, Russia

and Hiroshi Kimura Max-Planck-Institut f¨ur Aeronomie, Katlenburg-Lindau, Germany Received September 28, 1998; revised March 6, 2000

1. INTRODUCTION General structure and composition of the near-solar dust cloud are investigated. Based on estimates for sources and transport of dust to the near-solar region, we derive a representative set of trajectories of dust grains by numerical integrations and obtain the spatial distribution of different dust populations within 10 solar radii (R¯ ) from the Sun. For the radial structure, we find the dust number density to be enhanced by a factor of 1 to 4 in a typical heliocentric distance zone with a width of 0.2R¯ in the sublimation region—the formation of a dust ring—depending on the materials and porosities considered. The excess density in the ring increases with increasing initial size for porous grains and decreases for compact ones. Non-zero eccentricities of the dust orbits decrease the enhancement. Moderate enhancements that we predict are consistent with eclipse observations, most of which have not shown any peak features in the F-corona brightness at several solar radii. We describe typical features of β-meteoroids formed by the sublimation of particles near the Sun and estimate the total mass loss due to this mechanism to range between 1 and 10 kg s−1 . For the vertical structure of the dust cloud we show that grains larger than ∼10 µm in size keep in a disk with a typical thickness of tens degrees; grains with radii of several µm fill in a broader disk-like volume which is tilted off the ecliptic plane by a variable angle depending on the solar activity cycle; submicrometer-sized grains form a nearly spherical halo around the Sun with a radius of more than 10R¯ . From our present knowledge we cannot exclude the existence of an additional spheroidal component of larger grains near the Sun, which depends on how effective long-period comets are as sources of dust. Estimates of absolute number densities and local fluxes of dust show that simple extrapolation of the interplanetary dust cloud into the solar vicinity does not describe the dust cloud near the Sun properly. A complex latitudinal dependence of the fluxes of micrometer-sized grains, as well as variability of these fluxes with the solar activity phase, are predicted. The fluxes and their time variations depend on the physical and chemical properties of dust. °c 2000 Academic Press

The overall dust cloud in the Solar System is classically assumed to be in a steady state of interplanetary dust particles (IDPs) drifting toward the Sun under the Poynting–Robertson (P-R) effect. A radial slope of the dust number density n close to n ∝ r −1 can be expected from the radiative and corpuscular P-R drift of particles in circular or low-eccentricity orbits. The solar environment is therefore the region of highest dust concentration in the interplanetary dust cloud and should yield a good description of the dust complex, still bearing information about its parent bodies. However, the dust particles near the Sun are affected by a wide set of forces and effects, some of which are not featured in more distant parts of the zodiacal cloud: much stronger radiation and corpuscular pressure, influence of a strong and time-variable magnetic field, erosion, heating and sublimation, and others. In this paper we model the local features of the near-solar dust cloud, seen as the solar F-corona—both those that reflect local processes near the Sun and those that are determined by the evolution of the overall dust cloud. We start with a brief sketch of observational and theoretical results for the dust cloud from 1 AU inward. The observational results mainly describe large particles—grains in the 1- 100-µm size interval make the main contribution to the zodiacal light at 1 AU. On the basis of the zodiacal light observations by the Helios 1 and 2 space probes from 1 to 0.3 AU in the ecliptic plane, the determined radial increase in the number density is n ∝ r −1.3 , assuming that the volume scattering function is independent of the heliocentric distance r in the infinitely extended zodiacal cloud (Link et al. 1976, Leinert et al. 1981). However, this radial slope describes the variation of size and number density distribution and optical properties of particles, and a clear separation of the values is not possible (Mann 1998). The vertical

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distribution of dust derived from zodiacal light observations is consistent with orbital inclinations of particles mainly below 30◦ (Kneissel and Mann 1991). An additional component of dust in orbits with random inclinations, presumably arising from longperiod comets, is expected especially for most of the models that are based on visible light observations as opposed to infrared observations (Kneissel et al. 1990). These two components may have different optical properties and, due to their different initial orbits, they may also have different distributions of orbital elements. The existence of two different dust populations in the ecliptic plane with different bulk densities and different relative velocities to the spacecraft has been revealed from the Helios in situ measurements and explained with a component of cometary dust and a component of asteroidal dust particles (Gr¨un et al. 1980, Fechtig 1989). The study of the innermost dust cloud (r < ∼ 0.3 AU) mainly relies on the brightness observations of the F-corona, made during total solar eclipses or from satellites equipped with coronagraphs. The brightness, however, arises not only from the dust near the Sun, but also from dust near the observer and along the line of sight (LOS)—see Kimura and Mann (1998) for a review. Brightness observations and their analysis are limited by the high straylight level typical for corona observations at large elongations of the LOS from the Sun and by the influence of the increasing brightness from sunlight scattered by free electrons (K-corona) at small elongations of the LOS from the Sun. Also, the interpretation is hampered by ambiguities of the LOS inversion. The zodiacal light seems to extend smoothly into the F-corona for small elongations of the LOS. There have been, however, some indications that a narrow zone of enhanced dust concentration exists at several solar radii from the Sun, when Peterson (1967) and MacQueen (1968) independently reported the detection of hump features in the radial slope of the equatorial F-corona brightness in the near infrared. Theoretical studies to explain the observational data have been performed by many authors (see, e.g., Mukai and Yamamoto 1979, Kimura et al. 1998). Krivov et al. (1998) investigated the single-particle dynamics of dust near the Sun, based on dust models with different chemical compositions and porosities and taking into account the solar gravity, direct radiation pressure, the P-R effect, the Lorentz force, and sublimation. The dynamics of near-solar dust have been shown to depend on the material composition, morphology, and size of dust grains, which therefore should necessarily be taken into account in model calculations of the near-solar dust cloud. In this paper, we attempt to answer the following questions. What are the conditions for the formation of dust rings and what does their detection or non-detection tell us about the dust cloud? To what extent do the dynamics of dust near the Sun change the latitudinal distributions of dust? What is the expected size distribution of particles and what is its influence on the solar F-corona brightness? To what extent are the spatial distributions and fluxes of dust near the Sun controlled by the initial properties of dust from different sources and to what extent are the dust

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fluxes altered in the solar environment? To answer these and related questions, we start with reasonable assumptions about dust orbits in the inner Solar System and then check their further evolution within the solar corona. In Section 2, we briefly discuss sources and transport of dust to the F-corona, which is needed for us to set necessary initial conditions for our model calculation, and then describe the calculation technique. In Section 3, we calculate and discuss the radial and vertical profiles of the near-solar cloud, their changes with time, and their dependence on the initial distributions of dust and grains’ properties. In Section 4, we estimate absolute number densities of dust and local dust fluxes. In Section 5 we apply our model calculation to the possible production of β-meteoroids near the Sun. Section 6 contains a summarizing discussion and our conclusions. 2. DESCRIPTION OF THE MODEL CALCULATIONS

2.1. Grain Model Since the properties of actual dust grains near the Sun are vaguely known, we choose two models for the shape and the structure of grains: irregular-shaped porous grains, represented by Ballistic Particle-Cluster Aggregates (BPCAs) that are built up of single constituents of ∼0.01-µm size (see, e.g., Mukai et al. 1992), and compact spherical particles. We also choose two model materials, glassy silicate as an example of highly refractive grains and glassy carbon as an example for highly absorbing ones (cf. Kimura et al. 1997), and apply their mechanical and optical properties, as well as expected charging, as described in Krivov et al. (1998). For grains’ radii s > ∼ 0.5 µm, the four models are ranked, from lowest to highest β-ratios (the ratio of the radiation pressure to the solar gravity), as follows: silicate BPCAs, silicate compact spheres, carbon compact spheres, carbon BPCAs (Krivov et al. 1998). Our model grains are somewhat extreme choices as compared with the asteroidal and old cometary dust grains modeled by Wilck and Mann (1996). Model calculations of Mann et al. (1994) for the temperature profiles of dust near the Sun have also shown that composite grains can be expected to have properties between these extreme cases. 2.2. Initial Conditions We place the outer boundary of the region under study at a distance of 10R¯ from the center of the Sun. Our calculation requires the assumption of initial distributions of the orbital elements of dust grains at this distance. To derive plausible assumptions for these distributions we start at 1 AU, a distance where the distributions of dust are established better than in the inner Solar System, and consider the dust transport from 1 AU to 10R¯ , taking into account the collisional evolution of dust. Dust from asteroids and short-period comets. As was shown by Ishimoto and Mann (1999) by calculating the collisional evolution of the dust drifting towards the Sun under the P-R force,

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the majority of interplanetary dust particles with s < ∼ 20 µm at distances ∼0.1 AU from the Sun are fragments of collisions that took place inside 1 AU. Larger particles with sizes s > ∼ 20 µm at ∼0.1 AU are collisionally depleted. Hence we consider the following scenario. A typical zodiacal dust particle tens of micrometers in size starts from an Earth-crossing orbit and drifts toward the Sun, decreasing its semimajor axis and eccentricity. At a certain instant, it experiences collision with another particle, creating smaller fragments. The orbital elements of a typical fragment differ from those of the parent particle for two reasons. First, the fragment gets a velocity increment with respect to the parent grain from the projectile (Mann et al. 1996). Second, the typical fragment is smaller and has a different β-ratio than the parent grain and hence instantaneously gets into a new orbit (Burns et al. 1979). If the fragment is large enough not to be ejected as a β-meteoroid, it spirals toward the Sun again and reaches the perihelion distance q = 10R¯ . There are two major effects to change the distribution of inclinations of dust grains in this scenario. The first, and the most important for big grains, is the broadening of the latitudinal distribution induced by changes of the orbital velocity at the moments of destructive collisions just discussed. The second effect that largely determines the latitudinal distribution of micrometer-sized dust near the Sun is the Lorentz force. At distances larger than several tenths of AU, the motion of dust across the sectors with alternative magnetic field polarities leads to near-cancellations of the Lorentz force contributions (Morfill and Gr¨un 1979, Consolmagno 1980). Closer to the Sun this effect becomes more important. Using a kinetic approach based on a simple Parker’s field, Barge et al. (1982) have studied the dispersion of the orbital elements of the dust particles between 60 and 20R¯ and have found an appreciable scattering of the orbital inclinations in the vicinity of the Lorentz resonances, i.e., at the distances where the orbital period of a grain is commensurable with the rotation periods of the magnetic field sectors. Assuming the 4-sector Parker field, the most important is the primary (1 : 1) resonance at 23R¯ . Wallis and Hassan (1985) have undertaken a similar study and found that diffusion outside the Lorentz resonances makes a sensible contribution as well. Other effects are much less significant. Although gravitational interactions with planets are very important outside the Earth orbit (Jackson and Zook 1992, Kortenkamp and Dermott 1998), they do not contribute much inside 1 AU (see Gustafson and Misconi 1986 for further details). The modification due to latitudinal components of the solar wind (Banaszkiewicz et al. 1994) can be neglected at a reasonable accuracy level as well. To find the distribution of semimajor axes a, eccentricities e, and inclinations i at q = 10R¯ we use the following algorithm. 1. Distributions of dust at 1 AU are selected. Jackson and Zook (1992) report simulated distributions of three populations of dust—coming from asteroids, comets with q > 1 AU, and comets with q < 1 AU—at Earth-crossing orbits. Kortenkamp and Dermott (1998) give similar quantities on the base of a more elaborate set of simulations. They take special care of resonant

trapping of dust by Jupiter and show that cometary grains previously trapped in such resonances (about 20% of all cometary IDPs) have lower e and i, almost as small as e and i of asteroidal particles. They also concluded that the asteroidal and cometary populations have a considerable overlap in distributions of e and i at 1 AU. We assume that the particles have Gaussian distributions of both e and i and take the parameters of these distributions from Jackson and Zook (1992). Two cases considered are: asteroidal dust (eccentricity distribution with the expectation e¯ = 0.10 and standard deviation σe = 0.05 and the inclination distribution with i¯ = 7.2◦ and σi = 6.6◦ ) and dust coming from comets with q < 1 AU (¯e = 0.72, σe = 0.17 and i¯ = 17◦ , σi = 11◦ ). 2. We launch a grain from an Earth-crossing orbit with elements a0 , e0 , and i 0 , where e0 and i 0 are generated randomly in accordance with selected distributions, and a0 is chosen randomly from the condition of Earth-crossing: a0 (1 − e0 ) ≤ 1 AU ≤ a0 (1 + e0 ). The grain drifts toward the Sun, changing its (orbit-averaged) elements in such a way that (Wyatt and Whipple 1950) ae−4/5 (1 − e2 ) = constant.

(1)

Note that Breiter and Jackson (1998) have recently found a solution to the non-averaged P-R problem, which predicts, in contrast to Eq. (1), that close to the Sun the eccentricity starts to grow. However, the effect is very small (see Fig. 1 in the cited paper). 3. We assume that the particle experiences one and only one catastrophic collision between 1 and 0.1 AU. This is an assumption that bears some resemblance to a single-scattering assumption in the radiation transfer theory. The collisional probability p(r ) dr at [r, r + dr ] is written as p(r ) dr = σ n(r )vi (r )/vr (r ) dr , where σ is the collisional cross section, n(r ) ∝ r −1 is the number density, vi ∝ r −1/2 is the impact velocity, and vr = dr/dt ∝ r −1 is the radial (P-R) velocity of the grains, so that p(r ) ∝ r −1/2 . The distance where the collision occurs is generated randomly in accordance with this distribution. Interestingly, the results are not very sensitive to p(r ); even for p(r ) ∝ r −2 the resulting distributions are similar. 4. We then trace the orbital evolution of the biggest fragment of the collision. Given the elements (a1 , e1 , i 1 ) of the parent grain, we calculate the initial elements (a2 , e2 , i 2 ) of the fragment as follows. First we estimate its velocity increment, vb . Let m t and m p be the masses of two grains, the target and the projectile, respectively, and st and s p their radii (st ≥ s p ). In impact experiments with basalt (see Dohnanyi 1978 and references 2 therein), a catastrophic collision occurs if m p > ∼ m t /(250vi ) and the mass of the biggest fragment is estimated as m b ≈ 2m t /vi2 . Here vi , the impact velocity in km s−1 , is given by vi ≈ 20r −1/2 , where r is the heliocentric distance of collision in AU (Gr¨un et al. 1985). To a high probability, the parent grain is destroyed by a projectile with the mass just above the minimum limit, because smaller projectiles are much more abundant, whereas the collisional cross section σ = π (st + s p )2 ≥ πst2 , no matter how small the impactors are. Denoting by f = (m b /m p )(vb /vi )2 a fraction of a projectile’s kinetic energy imparted by the largest

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fragment, we obtain vb /vi ∼ ( f /500)1/2 . With a plausible assumption that f ∈ [0.001, 0.1] (Mann et al. 1996) this results in vb /vi ∼ 10−3 to 10−2 . We adopt more conservatively a random value vb /vi ∈ [0, 0.1] and assume a random direction of vb . Then we calculate the elements (a2 , e2 , i 2 ) of the fragment. Finally, we correct these for the β-ratio effect by applying the formulae of Burns et al. (1979). 5. We check whether the fragment is large enough not to be ejected as a β-meteoroid. If so, it spirals toward the Sun and we apply Eq. (1) again to find the elements (a, e) the grain has when reaches the perihelion distance q = 10R¯ . The modification of inclination is calculated from the diffusion formulae of Barge et al. (1982) in the magnetic resonance zone a ∈ [21R¯ , 25R¯ ]. After straightforward rescaling from materials used by Barge et al. (1982) to materials used here (see Krivov et al. 1998) one gets: µ ¶ µ ¶µ ¶ ¡ ◦ ¢ 8 1 2.37 g cm−3 1 µm 2 1t ◦ 3 |1i| ≈ 1.2 + 3.5 e2 , 6V α ρ s 10 yr (2) where 8 is the electrostatic potential (+5.0 to +5.5 V for silicate and +3.4 V for carbon); the fitting factor α = 1 and 0.11 respectively for compact and BPCA grains; ρ = 2.37 and 1.95 g cm−3 is the material density for silicate and carbon grains; 1t is the diffusion time—the P-R time it takes for a grain to diminish its semimajor axis from 25 to 21R¯ . The time interval 1t should not be less than several years, otherwise the diffusion in a sectored magnetic field does not take place. This means that Eq. (2) is applied to grains with s > ∼ 1 µm. For smaller grains, actual Lorentz perturbations in i are larger than predicted by Eq. (2). Equation 2 shows that grains bigger than approximately 1–2 µm in size (the exact numerical value depends on the material and porosity) drift through these zones with scattering within several degrees. Using formulae by Wallis and Hassan (1985) instead of Eq. (2) would lead, for micrometer-sized grains and for semimajor axes a < 0.1 AU, to comparable or somewhat larger scattering, depending on the material composition assumed. The numerical outcome of this algorithm is shown in Fig. 1. Each panel presents grain distributions in (log e, i)-plane. Two top panels show initial distributions at Earth-crossing orbits for asteroidal (left) and cometary (right) dust. The other panels depict distributions of final inclinations against final eccentricities of the grains at q = 10R¯ . Eight cases are included: asteroidal and cometary populations, silicate and carbon, s = 10 and 1 µm. A general result is that final eccentricities are typically small, whereas the inclination distributions are only moderately broadened compared to distributions at 1 AU. The other orbital elements—argument of perihelion ω and longitude of the ascending node Ä (mean anomaly is out of interest)—are assumed to be randomized. This is confirmed by zodiacal light observations to a level that is sufficient for our models as well as being widely accepted that gravitational per-

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turbations by major planets cause the randomization of these orbital elements. Dust from long-period comets. So far we have not considered dust from long-period comets. Delsemme (1976) has shown that, even for parabolic orbits of the comets, small non-zero ejection velocities with an isotropic angular distribution would cause half of the emitted dust grains to get into hyperbolic orbits, while half of the emitted grains would stay in bound ones. The influence of radiation pressure will decrease the portion of dust ejected in bound orbits. But still it will be a significant amount, especially if the dust production rate from long-period comets is high compared to short-period comets (Fulle 1987). Assuming ejection at the perihelion, using Eq. (1), and noting that the eccentricity immediately after ejection is close to unity while the final eccentricity is close to zero, we get an approximate expression e ≈ [q/(2q0 )]5/4 , where q0 is the perihelion distance of the parent body and q is the final perihelion distance of the grain. With q = 10R¯ and q0 ≥ 0.2 AU, this yields e ≤ 0.08. Therefore, at the outer boundary of the F-corona region the eccentricities e of dust grains from long-period comets are expected to be comparable with those of grains coming from short-period comets. The inclinations of long-period cometary grains are distributed randomly between 0◦ and 180◦ regardless of their sizes, because the inclination of the comets themselves are so distributed (Rahe 1981). These estimates should be applied with caution, however—most of the grains are initially ejected in very eccentric orbits and experience a long and complex dynamical evolution and collisional reprocessing, particularly changing their angular distribution, before they get in low-eccentricity orbits inside 1 AU (Fulle 1987). Distributions of dust at 10R¯ . On the basis of our estimates, we expect the following picture. Dust grains arriving at 10R¯ have radii predominantly from ∼20 µm down to ∼0.5 to 1 µm. Bigger grains are collisionally destroyed (this is not true for long-period comets which may inject grains with s > ∼ 20 µm as well). Smaller grains are blown away by radiation pressure (except for very transparent and extremely porous silicate BPCAs, the β-ratio of which is always <0.1). For many materials and compositions, very small grains with s ≤ 0.1 µm have β < 0.5 and might also be present. However, a possibility for such small grains to travel smoothly toward the Sun is questionable—small stochastic forces caused by fluctuations in solar wind parameters, electric charges, and magnetic field may overwhelm the P-R drag (Wallis 1986). In addition, inside 10R¯ there exist internal mechanisms, such as rotational bursting, that produce tiny grains in this size regime at much higher rates than the income rate from a zodiacal cloud (Paddack and Rhee 1975, Misconi 1993). Thus we do not need to consider a supply of such small grains to the F-corona from outside. The distributions of orbital elements of asteroidal, shortperiod cometary, and long-period cometary dust are significantly overlapped. Still, some differences are expected. Dust of asteroidal origin has typical eccentricities up to ∼0.01 (s ∼ 10 µm)

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FIG. 1. Modeled distributions of orbital eccentricities and inclinations of IDPs at 1 AU (top) and at 10R¯ (other panels): Top left, Asteroidal dust; top right, dust from short-period comets. The other panels present eight cases: asteroidal and short-period cometary dust populations, silicate and carbon compact spheres, s = 10 and 1 µm (see legends in the panels).

and ∼0.1 (s ∼ 1 µm). The eccentricities of most short-period cometary grains are less than ∼0.1, but there exists a fraction of grains with much larger eccentricities. For long-period cometary grains, eccentricities ∼0.1 are assumed. The expected distributions of orbital inclinations are i ∈ [0, 30◦ ] for asteroidal grains and i ∈ [0, 45◦ ] for short-period cometary ones. Inclinations of grains supplied by long-period comets are assumed to be distributed evenly between 0◦ and 180◦ . We do not discuss here the relative abundances of grains in asteroidal and cometary populations, because this problem, despite a long discussion, has not been given a satisfactory solution even at 1 AU (Liou et al. 1995). Ratios from 1 : 4 to 1 : 1 between the asteroidal and cometary dust (see, e.g., Fechtig 1989, Dermott

et al. 1992) could be used for crude estimates. The contribution of long-period comets is even more uncertain. Assuming the total dust production rate to replenish the zodiacal cloud to be 104 kg s−1 (Leinert et al. 1983), the fraction of dust supplied by long-period comets may range from 4 to 15% (Fulle 1987, 1990), but can be as small as 0.3% (Mukai 1989). 2.3. Modeling Technique We choose the initial orbital elements at a perihelion distance of 10R¯ in accordance with our calculations and at appropriately selected initial time instants. We then solve the equation of motion for single particles allowing for solar gravity, direct

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radiation pressure and P-R effect, the sublimation, and the Lorentz force. The force model used here is the same as in Krivov et al. (1998). In particular, the Lorentz force is computed on the base of the Potential Field–Source Surface model (Hoeksema and Scherrer 1986) with the magnetic field coefficients representing the actual magnetic field of the Sun from 1976 to 1998. The time step of this field model is one Carrington rotation period; note that there are no data with better time resolution available, but we do not expect the calculation results to change significantly under a more detailed model of the magnetic field. Thus, to calculate the effect of the Lorentz force on the near-solar dust cloud, we use a deterministic approach as opposed to the previous diffusion approach of Barge et al. (1982) and Wallis and Hassan (1985) which is not applicable inside 10R¯ because of the much stronger magnetic field and the short time scale involved. We neglect mutual collisions in the near-solar cloud, because short typical lifetimes of grains in this region (from 1 to 100 yr inside 10R¯ ) and smaller collisional cross sections (recall that grains with s > ∼ 20 µm are depleted after destructive collisions farther out from the Sun) make this process less efficient inside 10R¯ . By integrating numerically trajectories of several hundred like-sized particles with a given porosity and chemical composition (see Krivov et al. 1998 for further detail), we get ∼105 modeled instantaneous positions of grains, which is sufficient to derive spatial distributions of the grains by counting the occurrences of grains in various spatial bins (see Krivov and Hamilton 1997 for more details about this technique). The results of these modeling calculations are presented in the subsequent sections. 3. MORPHOLOGY OF THE NEAR-SOLAR DUST CLOUD

3.1. Radial Structure Dust rings. Since the orbital eccentricities are close to zero for dust particles at small distances from the Sun, the radial drift of particles under the P-R effect will cause a radial number density variation proportional to 1/r . The interplay between the P-R drag and the radiation pressure force, increasing due to sublimation, may yield an enhancement of number density of the dust grains (“a dust ring”) in the region just exterior to the dustfree zone (Belton 1966, 1967). Mukai et al. (1974) calculated the dust number density for 1-µm-sized graphite grains initially on near-circular orbits to obtain an enhancement of the number density over the r −1 law by factors of 30 and 20 for e = 0 and e = 0.02, respectively. Mukai and Yamamoto (1979) reported an enhancement factor to range from 5 for p-obsidian to 10 for graphite grains. Kimura et al. (1997) found the much lower excess of 1.5 for irregular silicate particles contaminated by carbon, and the absence of an enhancement for very fluffy dust, assuming circular orbits and an initial size of 10 µm. We calculate the radial distribution of four types of dust grains (silicate and carbon BPCAs and compact spheres) for two typical sizes (10 and 1 µm) and for two maximum initial eccentricities (0.01 and 0.1). The two choices of eccentricity crudely corre-

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spond to asteroidal and short-period cometary dust, respectively. Differences in the inclinations have little influence on the radial structure (see Krivov et al. 1998). The resulting radial profiles of number density are shown in Fig. 2. Plotted is the ratio of the dust number density n(r ) to its smooth level C/r , where the constant C is found from the fitting of the number density outside 5R¯ . For some materials and porosities, the function n(r ) goes to infinity at the edge of the dust-free zone. It happens if the sublimation effect exactly counterbalances the P-R drift, provided that orbits are exactly circular. This is, for instance, the case for carbon BPCAs and carbon compact spheres at a certain distance from the Sun. In such cases, the enhancement factor that we get from the histograms shown in Fig. 2 depends on the binning of distance. We discuss the excess in the number density in a zone with the typical width of 0.2R¯ . Return to a discussion of Fig. 2 which shows that 10-µm carbon BPCAs produce an enhanced number density at the sublimation zone by a factor of 4 (at most) for particles with initial e = 0.01, i.e., for some asteroidal grains. For dust in orbits with initial e = 0.1 (i.e., typical cometary motes), the peak is even a bit smaller (by a factor of 2) and broader. For carbon compact grains with e = 0.01, the excess is less than two-fold if s = 10 µm and three-fold if s = 1 µm; the enhancements reduce for e = 0.1. For silicate solid spheres, the excess is in most cases not noticeable; only for 1-µm-sized grains with e = 0.01 is there an enhancement by a factor of two. Finally, 10-µm-silicate BPCAs show a small excess, up to a factor of two; interestingly, it is the same for both values of eccentricity. However, the enhancement almost disappears for s = 1 µm. These results show that the formation of a ring-like structure depends not only on the material composition and structure of dust particles, but also on the eccentricity of initial orbits. Clearly a dust ring can only be formed in the case of particles that initially have very small eccentricities, whereas larger eccentricities (already on the order of 0.1) tend to smear out the dust ring structure. The sublimation distances and hence the formation of a dust ring are expected between 3 and 4R¯ for carbon particles considered here and between 2 and 3R¯ for silicate ones. These values respond strongly to the composition, porosity, and initial size of the grains. For instance, a silicate contaminated with carbon does not exhibit a sublimation zone at an intermediate distance between those typical for pure silicate and pure carbon, as one might expect, but the sublimation zone displaces to a more distant region between 10 and 4 solar radii from the Sun (Mann et al. 1994). A strong dependence of both the enhancement factor and the location of a dust ring on the materials, porosities, initial sizes, and initial orbital eccentricities of dust grains has the consequence that a dust ring would probably be broader and even less pronounced than what we derive from model calculations. Smooth radial profiles. As far as the spatial distribution of dust (aside from the formation of dust rings) is concerned, the analysis of near-infrared observation of the solar corona during

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FIG. 2. Radial profiles of the dust number density in the solar corona. The profiles show the calculated number density divided by the smooth number density profile, which is proportional to 1/r . Panels from top to bottom present the profiles for carbon BPCAs, carbon compact spheres, silicate compact spheres, and silicate BPCAs (the order corresponds to decreasing β-ratios). Left and right panels depict calculation results for initial eccentricities e ≤ 0.01 and e ≤ 0.1, representing particles produced by asteroids and short-period comets, respectively. Each panel contains radial density profiles for 10-µm (solid lines) and 1-µm (dashed lines) grains, except for carbon BPCA 1-µm grains, for which β > 1 and that is why they are absent.

the 1991 solar eclipse at 2.12 µm (Hodapp et al. 1992) has led to a model of thermal emission and scattered light brightness of the solar F-corona. Mann and MacQueen (1993) explain the data with a radial slope r −1 for the number density distribution and a radial slope of the albedo ∼r −0.15 to r −0.25 . The data show no evidence for a further increase of the number density from 9R¯ toward the Sun and the beginning of the dust-free zone is not seen in the data at distances >3R¯ . We assume that such a constant number density may simply be a result of the subsequent sublimation of dust particles with different composition, mentioned above. Another likely possibility is the rotational bursting of grains (Paddack and Rhee 1975, Misconi 1993) that occurs starting from ∼8R¯ inward at different distances for grains with different sizes and compositions. The local color temperature

of particles derived from the observations is about 10% lower than black body temperature (Mann and MacQueen 1993). The model calculations for the irregular dust particles with mixed composition have also shown that the equilibrium temperature of such dust aggregates can explain the derived temperature profile (Mann et al. 1994). Hence knowledge of the heliocentric distance at which the grains with given mechanical and chemical properties intensively sublimate enables a study of the dust temperature profiles. Temporal variations. The question arises whether a possible dust ring around the Sun may be influenced by transient processes that cause temporal fluctuations of the number density. If such processes exist, this might help to explain the fact

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that in 1960s–1980s many observers (e.g., MacQueen 1968) reported the presence of a peak feature in the near-infrared Fcorona brightness which was not confirmed by most of the observations made in the 1990s (see Kimura and Mann 1998 for a review of the brightness observations). Some models were discussed to explain a solar cycle dependence in the observation of infrared features but were rejected by detailed analysis of the F-coronal brightness (see Kimura and Mann 1998). Previous studies have also shown (Mann 1992) that brightness data do not depend markedly on the presence of small particles, which are especially affected by the Lorentz force. The time-variable Lorentz force induced by the variable solar magnetic field most likely does not influence the radial structure of the cloud, because the magnetic field up to 10R¯ is essentially radial (Krivov et al. 1998). The variation of the magnetic field is not the only manifestation of the solar activity, however. Coronal mass ejections (CMEs), for instance, cause temporary enhancements of solar wind velocity and density. They are too weak to sweep the grains out of the near-solar region, but they are strong enough to increase momentarily the pseudo P-R drift rate of the grains. This should cause a formation of dust ringlets at several solar radii from the Sun (Misconi and Pettera 1995). Besides, proton fluxes associated with CMEs may drastically enhance windmill rotation of the near-solar dust grains, causing their intensive rotational bursting and therefore enhanced production of small grains (Misconi 1993). However, CMEs typically occur with the frequency ranging from one event per several days to several events per day (Bird and Edenhofer 1990), so that the time scales for the effect in question are too short to explain any temporal variations in the F-corona brightness observed so far. Possible time variations in the near-solar dust cloud may also be influenced by Sun-grazing comets. MacQueen and St. Cyr (1991) reported the discovery of 10 Sun-grazers during 6 years of observation by the Solar Maximum Mission coronagraph, and tens of Sun-grazing comets per year are being observed by SOHO/LASCO (see, e.g., http://lasco-www.nrl.navy.mil/lasco. html). These Sun-grazing comets belong to the Kreutz group with i = 140◦ (see, e.g., Marsden 1989). Assuming a typical mass of a cometary nucleus of 1011 to 1013 kg and a mass loss rate in the form of dust per one perihelion passage of 10−3 to 10−2 of the nucleus mass (Rahe 1981), an individual passage of a comet near the Sun would produce ∼108 to 1011 kg of dust. The number of grains with sizes s > 10 µm (masses m > 10−11 kg) is on the order of N ∼ 1019 to 1022 . Were this material distributed evenly inside a 10R¯ sphere with the volume V ∼ 1030 m3 , the enhancement of the number density would be n ∼ N /V ∼ 10−11 to 10−8 m−3 . This would be comparable with, or even greater than, the typical mean number density in the same region due to the regular influx of IDPs, which we find in Section 4 to be ∼10−8 m−3 . Note that sometimes the temporary excess of the number density might be even larger. For instance, in the case of the Great Comet of 1729, the nucleus mass is estimated as 1018 kg, which translates to the number densities up to n ∼

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10−3 m−3 . Of course, these estimates are extremely sensitive to the shape and dimensions of the temporary cloud. In fact, most of the dust released from a Sun-grazer near the perihelion will be moving in similar orbits which implies rapid loss of material either by entering the close proximities of the Sun or by fast hyperbolic ejection out of the system in time scales of several hours (Sekanina 1982). Still, some observational consequences might take place if a fresh meteoric swarm produced by a Sun-grazer were occasionally projected along the line of sight. However, such a geometry is impossible for the Kreutz group comets because of their typical inclinations of 140◦ relative to the ecliptic plane. We conclude that the dust produced by Sun-grazers most likely will not affect the equatorial brightness of the solar Fcorona. Indeed the LASCO observations of the Sun-grazers do not show any correlation with pronounced dust features. Nevertheless, Michels et al. (1982) reported the observations of a Sun-grazing comet which caused a major change in the coronal brightness distribution, which persisted for more than one full day in August 1979. Apart from the comets with small perihelion distances, Farinella et al. (1994) suggest that close encounters with, or falls to, the Sun are likely to be common events for certain populations of asteroids, as a result of their chaotic dynamical evolution. It is questionable, however, that the collisional evolution of the asteroidal population close to the Sun is effective enough to keep producing a sufficient amount of dust in the near-solar region. Also, it looks hardly probable that an appreciable amount of dust is produced by these asteroids via IDP bombardments. For logical completeness, we mention also a hypothesis by Brecher et al. (1979) of the possible existence of a set of kilometer-sized bodies, at several R¯ from the Sun. Not claiming that such bodies really exist, they have formally shown that kilometer-sized bodies could survive over very long (even cosmogonical) time scales against all known destructive physical effects. In addition, Evans and Tabachnik (1999) have recently found that circular, non-inclined orbits inside 0.19 AU are very stable dynamically—primordial bodies would have survived there over 5 Gyr. Such a ring, if it existed in reality, would inevitably act as a permanent local source of dust through a variety of mechanisms (impacts of IDPs, mutual collisions between the ring bodies, etc.). Unfortunately, there seem to be no tools at present either to confirm or to reject this idea. 3.2. Vertical Structure Assuming a mutual compensation of radial forces in the sublimation zone, Rusk (1988) has shown that the magnetic field tends to scatter particles toward the polar regions, with a strong effect arising when the dipole components of the field become strong (which is the case at least 50% of the time). Krivov et al. (1998) have performed a similar study for a more extended force model, including not only the solar gravity and Lorentz force, but also the radiation pressure, the P-R effect, and sublimation—still at the level of single-particle dynamics.

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FIG. 3. Snapshots of the near-solar dust cloud, as seen edge-on (i.e., in X Z projection of the heliocentric ecliptic coordinate system). From top to bottom: “partial” clouds formed by compact silicate particles of 3, 0.5, and 0.1-µm size.

Now, we perform a more detailed modeling to derive the vertical distributions of dust inside 10R¯ . Consider the asteroidal or short-period cometary dust first. The typical snapshots of the clouds formed by different-sized grains (for definiteness, silicate compact spheres of asteroidal origin), as seen “edge-on” (in the XZ-projection of the heliocentric ecliptic coordinate system) are shown in Fig. 3. The vertical distribution of grains larger than a few µm keeps almost unchanged near the Sun, as is expected; neither tilts nor warps of the swarm are present (Fig. 3a). That the Lorentz force for these grains is weak enough is also supported by our analysis of inclination histories (which do not get far outside the initial belt of 30◦ ), as well as histories of the ascending nodes (they are nearly constant, except for a few individual trajectories). On the contrary, small particles with s < ∼ 0.1 µm exhibit essential randomization of both i and Ä (Fig. 3c). The swarm has a spheroidal shape. When constructing Fig. 3c for 0.1-µm grains, we adopted the initial inclinations at 10R¯ to be distributed between 0◦ and 30◦ . Note, however, that from our discussion of the transport of dust to the solar vicinity we expect the distribution to be randomized already at 10 or 20R¯ .

For intermediate-sized particles, at a narrow size range around several tenths of a micrometer, the results are more intricate (Fig. 3b). In Fig. 4, we show typical distributions of 0.5-µm silicate compact grains of asteroidal origin. Top panels (a) to (d) depict scatter plots of modeled positions of these particles during two periods, 1982 and 1991, in two projections. Bottom panels picture the distribution of orbital inclinations i (Fig. 4e) and longitudes of ascending node Ä (Fig. 4f) and how they change over a complete 22-year solar activity cycle—from 1976 to 1998. The swarm of the particles is moderately broadened vertically (see how the inclinations are in excess of 30◦ in Fig. 4e) and tilted up to some 10◦ or 20◦ (Figs. 4a, 4b). There are traces of correlation between both the tilt angle and orientation and the 22-year cycle. It is also illustrated by Ä plots (Fig. 4f). This effect stems from the dipole component g10 of the magnetic field which changes sinusoidally with a period of 22 years (see Krivov et al. 1998). Histograms that describe the latitudinal distributions of dust are shown in Fig. 5. Again, we have chosen asteroidal silicate compact grains, with radii in the intervals [0.5, 2.0 µm] and [2.0, 5.0 µm], and plotted their average vertical distributions over a 22-year solar cycle (1976–1998), as well as those for two particular time instants, 1982 and 1991, which can be referred to as periods of stronger and weaker magnetic field (see Krivov et al. 1998 for explanations). The distribution of 0.2- to 0.5-µm grains (not shown in the figure) is nearly uniform and looks almost independent of time, because even in the minimum phase the Lorentz force is strong enough to scatter the grains over the whole range of inclinations, from 0◦ to 180◦ . For particles with s > 0.5 µm, the picture is quite different. On the average, the number density over the ecliptic poles is by 2 or even 3 orders of magnitude smaller than near the ecliptic plane, but the concentration of grains at high latitudes changes strongly with time. For instance, the 0.5- to 2-µm grains were more confined to the ecliptic plane in 1991 and more scattered in 1982. Larger particles with s > 2 µm are weakly affected by the solar cycle. We note finally that all these conclusions, made for silicate compact spheres, hold true for silicate BPCAs. The only difference is that various patterns of behavior just described take place for somewhat larger BPCAs. Also, replacement of asteroidal grains with short-period cometary ones (i.e., the particles with different initial orbital eccentricities and inclinations) does not change the general picture. For dust from long-period comets, which forms an initially near-spherical cloud with random inclinations, we were looking after a possible “focusing” by the Lorentz force, which would flatten the cloud. No effects of this kind are noticed. The initially broad distribution of inclinations keeps virtually the same, regardless of the material, porosity, and sizes of the dust grains. These results bring up the question of the possible existence of the out-of-ecliptic component of the dust cloud (a bulge structure in the innermost part of the zodiacal cloud), which can be inferred from zodiacal light observations (Kneissel and

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FIG. 4. A cloud of 0.5-µm compact silicate particles of asteroidal origin and its variation with solar cycle. From top to bottom: snapshots of the cloud in X Z -projection (left) and Y Z -projection (right) in year 1982; the same in year 1991; histories of inclination; histories of longitudes of the ascending node.

Mann 1991). As far as micrometer-sized and larger particles are concerned, we only identify a relatively narrow intermediate size range (several µm), in which an initially flat latitudinal distribution of grains may transform to tilted, time-variable, halolike structures closer to the Sun, starting from several tens solar radii. If large particles exist at high latitudes, they have to be produced by parent bodies in high-inclination orbits. For example, if long-period comets significantly contribute to the dust complex (Delsemme 1976, Fulle 1987, Fulle and Cremonese 1991), this necessarily introduces a spheroidal component of dust with a concentration toward the Sun. 4. ABSOLUTE NUMBER DENSITIES AND FLUXES

So far, in discussing the radial and vertical structure of the dust cloud in Section 3, we dealt with the relative number densities of dust, which allow one to compare the number densities at different heliocentric distances or latitudes, but do not tell us how much dust is expected at a given location near the Sun. We

now describe the assumptions and algorithm of the calculations of the absolute number densities, as well as the local fluxes of dust. 4.1. Number Densities Denote by F + (r0 ) the flux of IDPs at r0 = 1 AU. We use the flux model of Gr¨un et al. (1985) (their Eq. (A3)), which gives the (isotropic) flux of IDPs, observed by one side of a flat detector with an effective sensitive solid angle of π sr. Then the number density at 1 AU is given by n(r0 ) =

k0 + F (r0 ), v0

(3)

where k0 = 4 and v0 = 20 km s−1 is the mean impact speed. The latter quantity can be interpreted as the average random relative velocity between the dust grains with typical eccentricities e0 and inclinations i 0 and the Earth moving with the circular Keplerian velocity v0(k) = 30 km s−1 , through the formula

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FIG. 5. Vertical profiles of the number density of dust particles inside 10R¯ (in arbitrary units). The compact silicate particles are initially (at 10R¯ ) within the ecliptic latitudes of ±30◦ (asteroidal grains). Shown are distributions for particles in the size intervals 0.5 < s < 2 µm and 2 < s < 5 µm: Solid lines, an average profile over the whole solar cycle; dashed lines, the period of a strong magnetic field; dotted lines, the period of a weak field.

(e.g., Greenberg et al. 1991) r v0 = v0(k)

5 2 e + i 2. 8 0 0

(4)

The number density at a heliocentric distance r < r0 can be obtained from the number density at r0 by applying the inverse proportionality law n ∝ r −1 which, as mentioned before, is a reasonable approximation for particles inside 1 AU. We have: n(r ) = n(r0 )(r0 /r ).

(5)

When constructing Fig. 5, we considered narrow size intervals: radii from 0.5 to 2 µm and from 2 to 5 µm. For each interval, we compute N + , the number of grains that cross the sphere r = 10R¯ per second, moving inward due to P-R force: N + (r ) = n(r ) · vr (r ) · S(r ).

(6)

Here vr (r ) specifies the mean radial velocity of the grains at distance r . To a good accuracy, we can assume that vr is the radial component of the P-R velocity and calculate it by the formula 1R¯ vr = 1.27β , −1 r 1 km s

(7)

which immediately follows from Eq. (50) of Burns et al. (1979). Another quantity in Eq. (6) to be explained is S, the area of the effective latitudinal belt on the initial sphere. This effective belt is a zone on the initial sphere which is confined by the median ecliptic latitudes of the IDPs. For asteroidal dust, we adopt S = 4πr 2 sin 30◦ . Having computed N + for each size interval, we derived from numerical integrations, for the same size intervals, the lifetimes T of grains inside the sphere 10R¯ . Once N + and T for a certain size interval are known, their product N + T is an estimate of a full steady-state number of dust grains inside the sphere. We therefore obtain a scaling factor: each modeled instantaneous grain’s position corresponds to as many as N + T /Nmodeled real dust particles, where Nmodeled is the number of the positions modeled. Dividing estimated numbers of actual grains by volumes of relevant latitudinal belts, we finally arrive at absolute number densities. For the relative number densities which label the Y -axes in Fig. 5, the scaling factor from relative to absolute number densities (m−3 ), found by the procedure just described, turns out to be unity. It means that the relative densities given in Fig. 5 can be considered as absolute number densities (m−3 ), of course, under the assumption that all dust grains are silicate compact spheres of asteroidal origin. Similar estimates can be obtained for other types of grains. These estimates of number densities are influenced by relative abundances of grains coming from various sources, as well as their material composition. Although the r −1 law used above is consistent with the observations, model calculations of Ishimoto and Mann (1999), assessing collisional processes in the inner solar system, suggest that the number density of −2 grains with 0.5 µm < ∼s < ∼ 20 µm has a steep increase ∝ r toward the Sun, whereas that of the larger grains rapidly decreases with the decreasing distance. The overall cross section of the particle in a broad size range from tenths of micrometer to hundreds of micrometer may be again ∝ r −1 , being consistent with the brightness observations. Thus we may underestimate the number densities of micrometer-sized particles by ≈1 order of magnitude. 4.2. Dust Fluxes To estimate the dust fluxes expected in the near-solar region, we first calculate the number densities n(s, r, φ) of grains with different radii s at different heliocentric distances r and ecliptic latitudes φ, using the algorithm described in Section 4.1. Then we consider a spacecraft moving at a heliocentric distance r in a circular orbit in the ecliptic plane. The typical velocity of the vehicle relative to the dust environment can be expressed as (cf. Eq. (4)) r v=v

(k)

5 2 e + i 2, 8

(8)

where v (k) is the Keplerian velocity in a circular orbit at distance r , and e and i are the typical eccentricity and inclination of the

NEAR-SOLAR DUST CLOUD

TABLE I Expected Dust Fluxesa inside 10R¯ , m−2 s−1 Grain radii, µm

Heliocentric distances, R¯

Ecliptic latitudes, deg

0.5–2

2–10

>10

8 . . . 10

<30 >30

3 × 10−3 6 × 10−5

1 × 10−3 9 × 10−6

7 × 10−4 —

6...8

<30 >30

5 × 10−3 2 × 10−4

1 × 10−3 3 × 10−5

1 × 10−3 —

4...6

<30 >30

8 × 10−3 3 × 10−4

2 × 10−3 8 × 10−5

2 × 10−3 —

<30 >30

1 × 10−2

4 × 10−3

1 × 10−3

2...4 a

1 × 10−3

1 × 10−4



For a spacecraft moving in a circular ecliptic orbit.

dust grains at the same distance. From Eq. (3), the flux detected by the spacecraft can be estimated as v F + (r ) = n(r ), k

(9)

where 1 ≤ k ≤ 4, depending on directionality of the flux, as well as on the solid angle and orientation–rotation of the dust sensor. For the sake of simple estimates, we assume e = 0 and i = 30◦ in (8) and e = 0 and k = 4 in (9). The numerical results for asteroidal silicate compact grains are given in Table I. They correspond to several hits of grains with s > 0.5 µm per day onto a detector with the effective area of 0.02 m2 . The impact speeds range from 60 to 130 km s−1 . Close to the ecliptic the fluxes are several times higher than the given mean fluxes in the belt |φ| < 30◦ . For eccentric and/or inclined trajectories, the maximum count rates can be severalfold higher than those given in Table I because of higher relative velocities between the spacecraft and the dust grains. In addition, the deviation from standard interplanetary flux models due to collisional processes inside 1 AU (see remark at the end of Section 4.1) may lead to higher flux rates along with a change of size distribution. Table I shows that significant modifications of the reference flux are expected in the near-solar region as a direct consequence of both radial and vertical evolution of dust discussed in Section 3. First, these data reflect sublimation of the grains. Interestingly, both number densities and fluxes of ∼10 µm particles, which are expected to dominate the total cross section of dust near the Sun because larger grains are collisionally depleted there, are almost constant (within a factor of 3) inside 10R¯ . Although this is only true for silicate compact grains (see Fig. 2), this falls in agreement with the observational result of a nearly constant number density at 3R¯ < r < 9R¯ (Section 3.1). Second, a complex dependence of the fluxes of micrometer-sized grains on the ecliptic or heliographic latitude is expected. Third, although we give in Table I only the fluxes averaged over the solar cycle, we predict a variability of the fluxes with the solar

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activity phase. Near the maxima the flux of tiny grains at high ecliptic latitudes should be higher than near the minima (see Section 3.2). The amplitude of fluctuations may reach one order of magnitude already for moderate ecliptic latitudes (Fig. 5a). Above the poles, the temporal variations may be even higher. The real picture will be influenced by short-time-scale solar activity phenomena, such as coronal mass ejections. 5. PRODUCTION OF β-METEOROIDS NEAR THE SUN

In situ measurements have shown the existence of so-called β-meteoroids, particles that leave the Solar System in unbound orbits (Zook and Berg 1975). These particles may be produced as a result of direct ejections in hyperbolic orbits from a parent body. In addition, β-meteoroids are produced whenever a particle in a bound orbit is reduced in size which increases the β-value and finally causes liberation of the grain from the gravitational coupling to the Sun. Typical processes for that are mutual collisions, as well as erosion by sputtering or sublimation (Mukai et al. 1974, Whipple 1976, Le Sergeant d’Hendecourt and Lamy 1981). Mechanisms of self-fragmentation were also suggested: catastrophic fracturing of heterogeneous grains with volatile inclusions (Mukai 1984) and spin-up with subsequent rotational bursting of grains in proton flows associated with CMEs (Misconi 1993). Here we address the formation of βmeteoroids by sublimation. From our model calculations we assume that all porous (compact) carbon particles larger than 2.4 µm (0.5 µm) would transform into β-meteoroids. Smaller particles would have β ≥ 1, so they could not reach the near-solar region, being blown away immediately after separation from their parent bodies. From Eqs. (3)–(7), for porous carbon grains with sizes > ∼2.4 µm (so that β ≈ 1) we have the following numerical estimates: F + (1 AU) ∼ 10−5 m−2 s−1 , n(1 AU) ∼ 2×10−9 m−3 , n(10R¯ ) ∼ 5 × 10−8 m−3 , vr ∼ 1.3 × 102 m s−1 , S ∼ 3 × 1020 m2 , and N + ∼ 2 × 1015 s−1 . For compact carbon grains with sizes > ∼0.5 µm, the values above will be several times larger. If we assume a content of 10% carbon particles in the total flux, we finally get a production rate of 1014 to 1015 s−1 for both porous and compact carbon particles, which corresponds to a mass production rate of 1 to 10 kg s−1 . This estimate is to be compared with the identification of β-meteoroids. Earlier detections by Pioneer 8 and 9 spacecraft (Berg and Gr¨un 1973) gave a value of 8 × 10−4 m−2 s−1 (2π sr)−1 for the flux at 1 AU. Concerning identification of β-meteoroids in the dust measurements aboard Ulysses (Gr¨un et al. 1995, Wehry and Mann 1999), two observational phases can be considered: detection near the ecliptic plane (robs =1.0 to 1.6 AU, observed flux Fobs ∼ (1.5 ± 0.3) × 10−4 m−2 s−1 ) and at high ecliptic latitudes (robs = 1.8 to 2.7 AU, observed flux Fobs ∼ (9.0 ± 6.3) × 10−5 m−2 s−1 , but for the northern passage only— no reliable detections during the southern one). If we assumed the detected β-meteoroids to have been produced in the vicinities of the Sun, for the average heliocentric distance of detection

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2 robs ∼ 2 AU we would get the production rate ∼Fobs × 4πrobs ∼ 20 −1 10 s , which is far beyond the theoretical prediction made above. Of course, the latter is based on the assumption about the fraction of absorbing grains in the solar vicinity, but this does not explain the big difference. Also, analysis of the orbits of βmeteoroids detected by Ulysses shows that the perihelia are not necessarily within 0.1 AU from the Sun (Wehry and Mann 1999). The mechanism in question may only be responsible for a small part of the β-meteoroid flux detected beyond 1 AU. Hence the β-meteoroids that were detected with Ulysses may have been produced not by sublimation of absorbing grains near the Sun, but predominantly by collisions. Contrary to previous experiments, in the case of Ulysses the β-meteoroids were detected at high ecliptic latitudes. In order to explain the very fact of the appearance of hyperbolic grains at high ecliptic latitudes with the Ulysses measurements, Hamilton et al. (1996) suggested that the submicrometer-sized dust can be efficiently thrown out of the ecliptic plane by the Lorentz forces during relevant (defocusing) 11-yr semi-cycles of the solar activity. Still, some part of the β-meteoroids may have not been driven to high latitudes by the Lorentz force. If a sufficient amount of absorbing dust comes to the solar vicinities from long-period comets, the distribution of orbital inclinations of which is known to be very broad, then the sublimation of these grains near the Sun will transform them to β-meteoroids with both low- and highly inclined orbits. Before the absorbing grains become β-meteoroids, they lose most of their mass as a result of sublimation. Silicate grains are totally sublimated near the Sun. Therefore most of the material (∼103 kg s−1 according to simple estimates) is lost in the form of ions which eventually become part of the solar wind.

6. SUMMARY

Starting with estimates for the production and transport of dust to the vicinities of the Sun and making model calculations for evolution of dust in the near-solar region, we have studied the general structure and properties of the dust cloud near the Sun. The following conclusions can be drawn about the radial structure of the dust cloud. The enhancement factor of the dust number density in a dust ring formed near the Sun is between 1 and 4 for the materials and porosities considered and for a typical heliocentric distance zone with a width of 0.2R¯ . The density in the ring increases with the increasing initial size for porous grains and decreases for compact ones. Non-zero eccentricities of the dust orbits decrease the enhancement. For the vertical structure of the dust cloud we show that assuming that the dust particles are initially in orbits close to the ecliptic plane, grains with s > ∼ 10 µm keep in a disk with a typical thickness of tens degrees; grains with radii of several µm fill in a broader disk-like volume, which shows moderate (and time-variable) tilt with respect to the ecliptic plane and slightly responds to the solar activity cycle; submicrometer-sized grains (only dielectric ones—absorbing particles are absent there) form

a nearly spherical halo around the Sun with a radius of more than 10R¯ . From our present knowledge we cannot exclude the existence of a dust cloud of micrometer-sized and larger grains at high ecliptic latitudes. We have only demonstrated that large grains cannot be brought into high latitudes with the dynamic effects considered in this study. However, they may get there, being supplied by parent bodies with essentially random inclinations, such as long-period comets. Our calculations show that simple extrapolation of the interplanetary dust cloud into the solar vicinity does not yield a proper description of the dust near the Sun. Significant modifications of the reference flux are expected as a direct consequence of both radial and vertical evolution of the dust complex. The fluxes may be several-fold larger in the sublimation zones, but the sublimation of particles with different properties at different distances tends to flatten the overall dust cloud beginning from about 10R¯ toward the Sun. Temporary enhancements of local fluxes may be caused by Sun-grazing comets, and—at the periods of CMEs—by increased corpuscular drag and by rotational breakups of dust grains in the proton flows. We also predict a complex dependence of the fluxes of micrometer-sized grains on the ecliptic latitude, as well as variability of these fluxes with the solar activity phase. The amplitude of fluctuations may reach one order of magnitude already for moderate ecliptic latitudes. The sublimation and resulting diminution of the size of particles can lead to the production of β-meteoroids near the Sun. This process is only expected for absorbing particles (carbon in our model), whereas dielectric particles (silicate) completely sublimate. It would result in a formation of relatively large βmeteoroids: for our model materials, the radii will be ≈2.4 and ≈0.5µm for porous and compact carbon grains, respectively. Assuming that absorbing particles make up 10% of the dust cloud, a production rate would amount to of 1014 to 1015 s−1 , which corresponds to a mass production rate of 1 to 10 kg s−1 . While a part of sublimating grains become β-meteoroids, the others may be a source of pick-up ions that are measured in the solar wind (Gloeckler and Geiss 1998). From our calculations we expect the input from sublimation of dust to the solar wind to take place mainly between 10 and 2R¯ from the Sun. It should be stressed that our modeling does not allow for several physical processes of potential importance, and it is of value to estimate how their inclusion into the model could change the results. The first remark can be made about the influence of the solar wind, to which the dust particles near the Sun are exposed. We have taken into account the charging of dust particles in the solar wind when calculating the Lorentz force. The solar wind drag that leads to a pseudo P-R effect is not explicitly considered, but it would only moderately reduce P-R time scales and would not change our basic results. Mutual collisions of dust grains inside 10R¯ and rotational bursting have not been considered here. Although without detailed consideration, we theorize that our results could be applied to resulting debris as well. These particles are supposed to be in orbits with similar inclinations, but with larger eccentricities than the particles that we take into

NEAR-SOLAR DUST CLOUD

account. Hence our results about the radial distribution of dust in the corona would change in such a way that the formation of dust rings would be even less pronounced. The results about the vertical distribution of dust would be comparable to the results that we present in this study. We derive from our calculations that detailed observations of the solar F-corona would lead to a better understanding of the sources of dust in the inner solar system. The detection of a significant component of large particles at high ecliptic latitudes would support the idea that long-period comets provide an essential source of dust in the inner Solar System. In order to separate the influence of the initial orbital distribution from the influence of local variations of the dust fluxes, the size distribution needs to be estimated either from observational data or from in situ measurements. Estimates of the size distribution from brightness observations are possible by studying the existence or non-existence of dust rings, by estimating the extension of the dust-free zone, and by retrieving average optical properties from the observed brightness. A clearer understanding of the size distribution would result from in situ measurements of dust fluxes. The expected variation of the dust distribution for sizes below several micrometers would yield information about the interactions with the solar radiation and plasma environment. ACKNOWLEDGMENTS We thank Jer-Chyi Liou and Tadashi Mukai for useful and constructive reviews. This work was supported by the Bundesministerium f¨ur Bildung, Wissenschaft, Forschung und Technologie (BMBF). The work was done during short-term stays of A.K. at MPI f¨ur Aeronomie (MPAe), funded by German Space Agency, and a long-term stay at MPAe in the framework of his Alexander von Humboldt fellowship.

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