Adv. Space Res. Vol. 29, No. 9, pp. 1265-1270.2002 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0273-l 177/02 $22.00 + 0.00 PII: SO273-1177(02)00189-8
DYNAMICS IN THE SOLAR RING D. Maravilla
Instituto de Geofisica, UNAM, Circuit0 Exterior C. U. 04510 Coyoacan, D.F, Mexico.
ABSTRACT A two-dimensional model is analyzed in order to study the dynamical behavior of dust particles which are localized near the Sun, in particular at 4 solar radii. The equation of motion which describes the dynamics includes several forces; gravitational, Lorentz and radiation pressure. Also sublimation and gas drags are taken into account. Electrostatic charge is not constant in this model and both collisional and Coulomb drags are included in the equation of motion. It has been assumed that the solar magnetic field is radial outside the source surface. Some numerical results of the equation of motion are presented. 0 2002 COSPAR. Published by Elsevier Science Ltd. All rights reserved.
INTRODUCTION The first work on the solar dust ring formation was done by Belton (1966) who predicted that a matter concentration near the Sun was creating a ring-like zone. According to the author dust arise from the interplay between the Poynting-Robertson force and the sublimation effect. Few years later MacQueen (1968) and Peterson (1967, 1969) made observations of the F-corona brightness enhancement near 4 solar radii in the near infrared during the total solar eclipse of 1966 supporting the Belton’s hypothesis of the solar dust ring. Since then, several works on this topic have been done considering grains are dielectrics or conductors (Mukai and Yamamoto, 1979; Mukai, 1981; Mukai, 1983). In particular, Mukai et al. (1974) computed that graphite grains will concentrate in a sublimation zone near 4 solar radii, creating a dust ring. As the grains sublimate and their radii shrink, B (radiation pressure/gravitational force ratio) will increase and the remnant grain will be accelerated outward. In 1979, Mukai and Yamamoto proposed a two-component model computing both graphite and pobsidian grains form a dust ring at 4-5 solar radii, where both components sublimate rapidly. Krivov et al. (1998) studied the dust ring formation analyzing the dynamics of dust particles located at 4 solar radii and considering gravitational and Lorentz forces, radiation pressure and sublimation. They concluded that dynamics of near-solar grains depends radically on their sizes, chemical composition and structure (porosity). In this work we present the analysis of a similar model that includes both Coulomb and direct collisional drags. We describe the model including all forces and present the results from the numerical analysis. THE MODEL The dynamical behavior of dust particles near the Sun, are dominated by gravitational and electromagnetic forces, sublimation effect, radiation pressure and several drags. In this model are only
both direct collisions
and the Coulomb
drag. The equation
(1) where Fz is the gravitational
c is the Lorentz force, F, is the radiation
the force related with direct and Coulomb
In Eq. (2), G is the gravitational
pressure force and FjrazS is
constant, M is the solar mass and m is the grain mass.
In Eq. (3), Q(t,,
is the electric
charge .which is a function
of time and several
electronic, ionic, photoemission and thermoionic are involved in its evaluation. B is the magnetic field and it is assumed radial outside the solar source surface where this is assumed to be located at 2.5 solar radii.
F, = H(a,r)i^, -H(a,$ c
The first term on the right-hand side represents the radiation pressure term. It is due to the initial interception by the particle of the incident momentum in the beam. The second term is the PoyntingRobertson radiation drag.
From Eq. (l), dC _dm m-jy-g=-
GMm -I r2
If isotropic mass loss is only considered (y=O), the second term on the left side is neglected last equation. The second term on the right side (Eq. 6) includes the charge defined as:
(7) and C is the capacitance
(= 4z~,+z) . Drags are defined as follows: (8)
Dust Particles Dynamics
in the Solar Ring
=$ , I=,/=
and /2, =,/z
RESULTS AND DISCUSSION In this preliminary calculation I have included only electron and ion collection currents. The photoemission and thermoionic emission currents which are important will be included in a more detailed study. The equation of motion has been solved for a two-dimensional case considering that solar wind is a Maxwellian plasma with electron temperature T, , ionic temperature q , electronic density n, and ionic density it, . A Runge-Kutta algorythm has been used in the numerical analysis in order to obtain solutions of Eq. (6). Three different values of grain radii 6.1, 1.O and 10.0 microns) were taken into account. Data of drags calculated at 4 solar radii are shown in Table 1. Charge variation and direct collisional and Coulomb drags are presented in Figures 1 to 4 considering both conductor and dielectric materials for a very small grain (0.1 microns).
Table 1. Mapnitude
and Direct Collisional
Radius Coulomb drag (dynes) (microns)
Draw for Small Grains
Direct collisional drag (dynes)
From Table 1, meanwhile the magnitude of Coulomb drag is almost the same for all cases considered here, the magnitude of direct collisional drag is much larger for smaller silicate grains (0.1 microns) than for the other sizes. It is important to note here that for dielectric grains, this drag is two orders of magnitude weaker for smaller ones than for larger grains (10 microns) indicating that chemical composition and grain size are two important parameters involved in the evaluation of these forces At the
same time, heliospheric conditions change with the distance from the Sun and perhaps direct collisional drag is negligible under them. On the other hand, for dielectric material both forces are positive and grow as a function of time (see Figures 2 and 4) until they stabilize although for conductors, direct drag presents a little negative variation in the first 8.46 x 10’ seconds possibly as a result of charging processes.
Time (e5 s)
Fig. 1. Charge variation
of a silicate grain with radius=O. 1 microns.
Coulomb and Direct Drags 1.6 ~.........._..______._._._.....................-...__
Time (e5 s) Fig. 2. Coulomb and direct drags of a small silicate grain (0.1 microns). dotted line represents the direct drag.
Dust Particles Dynamics
in the Solar Ring
Charge variation 0
-5 3i? -10 Ig,-15 8-h 2 -20 6
-25 -30 Time (e5 s)
Fig. 3. Charge variation
of a carbon grain (radius=O. 1 microns).
Coulomb and Direct Drags 1.6 1.4
1.2 1 0.8 0.6 0.4 0.2 0
Time (e5 s) Fig. 4. Coulomb
and direct drags of a carbon grain (0.1 microns).
to the charge,
it is changing
The dotted line shows the direct drag.
particular, for conductors, it has a small variation at - 15 x 1O4 (esu) where the slope becomes positive. In both cases, grains are negatively charged because electronic velocity is higher than ionic velocity and more electrons than ions are deposited on grain surface. As the other currents were not included in the numerical results , it is not possible to know now how the other currents are affecting the dynamical behavior of small dust grains.
CONCLUSIONS This is a very preliminary investigation where it has been found that the magnitude of drags is different for all considered cases. The Coulomb drag being larger by several orders of magnitude than the direct collisional drag. The magnitude of Coulomb drag for several size grains is similar in both conductor and dielectric materials. A more complete analysis including photoemission and thermoionic emission, while also using a more realistic model of the solar wind and solar magnetic field will be presented in a future paper. REFERENCES Belton M. J. S., Dynamics of interplanetary dust, Science, 151,3706,35-44, 1966. Krivov A., I. Kimura, and I. Mann, Dynamics of dust near the Sun, Icarus, 134,31 l-327, 1998. MacQueen R. M., Infrared observations of the outer solar corona, Astrophys. J., 154, 1059-1076, 1968. Mukai T., and T. Yamamoto, A model of the circumsolar dust cloud, Publ. Astron. Sot. Jpn., 31, 585- 595,1979. Mukai T., T. Yamamoto., H. Hasegawa., A. Fujiwara, and C. Koike, On the circumsolar dust materials, Publ. Astron. Sot. Japan, 26,445-458, 1974. Mukai T., and T. Yamamoto, A model of the circumsolar dust cloud, Publi. Astron. Sot. Japan, 31, 585-595, 1979. Mukai T., On the charge distribution of interplanetary grains, Astron. Astrophysics, 99, l-6, 198 1. Mukai T., Mass loss rate of the zodiacal dust cloud, Moon and Planets, 28,305-309, 1983. Mann I., and I. Kimura, Dust Near the Sun, in Advances in Dustv Plasmas, eds. P. K. Shukla, D. A. Mendis and V. W. Chow, p. 346-351, World Scientific Publishing Co., Singapore, 1997. Peterson A.W., Experimental detection on thermal radiation from interplanetary dust, Astrophys. J., 148, L37-L39,1967. Peterson A. W., The coronal brightness at 2.23 microns, Astrophys. J., 155, 1009-1015, 1969.