The Dynamics of Dust Particles Near the Sun Geofísica Internacional, Vol

Geofísica Internacional ISSN: 0016-7169 [email protected] Universidad Nacional Autónoma de México México

Maravilla, Dolores The dynamics of dust particles near the Sun Geofísica Internacional, vol. 38, núm. 3, july-september, 1999, p. 0 Universidad Nacional Autónoma de México Distrito Federal, México

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The dynamics of dust particles near the Sun

Dolores Maravilla Instituto de Geofísica, UNAM, México, D.F., México.

Received: August 27, 1998; accepted: March 10, 1999.

RESUMEN El propósito de este trabajo es estudiar la dinámica de las partículas de polvo (granos) que se encuentran cerca de la corona solar partiendo de un modelo tridimensional que describe la ecuación de movimiento de las partículas tomando en cuenta las fuerzas gravitacional y de Lorentz. Las soluciones del modelo tridimensional proporcionan las superficies de reflexión dentro de las cuales los granos pueden quedar confinados. Cuando sólo hay movimiento en un plano, las soluciones tridimensionales son reducidas a soluciones bidimensionales. En particular, se estudia la dinámica de los granos cerca de la corona para dos mínimos solares consecutivos (i.e. antes y después del máximo solar) incluyendo en la ecuación de movimiento la fuerza de presión de radiación. Las soluciones bidimensionales proporcionan las regiones de confinamiento del polvo alrededor del Sol, así como aquellas regiones en donde el polvo escapa de la heliosfera. La magnitud tanto de las regiones de confinamiento como de las zonas de escape depende en gran medida del tamaño de los granos, de la relación carga/masa y de la radiación solar.

PALABRAS CLAVE: regiones de confinamiento, superficie de reflexión, relación carga/masa, granos de polvo.

ABSTRACT A three dimensional (3D) model describes the equation of motion for dust particles taking into account the gravitational and Lorentz forces. Solutions of the model give the reflection surfaces where grains can be confined. In particular, the dynamics of dust grains near the corona is studied when solar activity is between two consecutive minima (i.e. before and after a maximum), including the radiation pressure into the equation of motion. 2D solutions yield the confinement regions for dust close to the Sun, and for regions where grains can escape from the heliosphere. The size of confinement and escape regions depends basically on grain size, charge/mass ratio and radiation pressure.

KEY WORDS: confinement regions, reflection surface, charge/mass ratio, dust grains.

INTRODUCTION We will take into account solar radiation pressure to solve the equation of motion in 2D for fine charged dust in- The dynamics of dust particles in planetary magneto- jected into the heliosphere. The solutions yield the confine- spheres has been studied in the last forty years. Störmer ment regions of the dust, and of particles injected into the (1957) used two integrals of the equation of motion of rela- heliosphere at 10 solar radii. tivistic charged particles injected into the terrestrial magnetosphere to characterize their orbits. These integrals were gen- THE MODEL eralized by Artem’ev (1969) to include the effects of gravitational force and Mendis and Axford (1974) included the The solar magnetic field is approximated by a dipole co-rotational electric field in the gravitoelectrodynamic whose axis is parallel or anti-parallel to the spin axis of the motion of charged dust in planetary magnetospheres. Sun for two consecutive minima (Mendis and Axford, 1974, Artem’ev 1979, Horanyi M. 1993, Maravilla et al., 1995, We use Mendis and Axford’s model to study the dy- 1996). We assume that dust particles are instantly charged at namics of dust particles (grains) close to the solar corona at the point of injection and carry a constant electric charge. 4 solar radii. Such grains were detected during the solar Otherwise it would be necessary to take into account cur- eclipse of June 30, 1973 from infrared observations (Mukai rents of ions, electrons, secondary electron emission and pho- et al., 1974; Mukai and Giese, 1984; Lamy 1974). The equa- toionization picked up by the grains. tion of motion which describes the dust dynamics will be analyzed in three dimensions (3D). We derive integrals for In a solar-centric inertial frame (Figure 1) the motion energy and angular momentum from the 3D equation of of a dust grain is governed by the equation: charged dust within the heliosphere including both solar grav- ity and magnetospheric rotation. The solutions may be re- Q0 GMm mrÇÇ =−××[]( rrBÇ Ω) −r , (1) duced to two dimensions after some physical considerations. c r3

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Ω Integrating equation (3) between the point of injection r0 and r , we obtain the energy equation. Considering that 2 ==2 KGM r=r and vv0 at the point of injection, the en- P(r,φ,θ) 0 r0 ergy equation is expressed by

2AΩsin2 φ 2AΩsin2 φ v2 −−=−−2GM KGM 0 2GM r r r r r . φ 0 00 (5)

← R → B Thus

2φ 2φ    2=+2 Ωsin −sin 0− 11− vv0 22 A GM . (6) rr00 rr

Integrating equation (4) between r0 and r and assuming that

Fig. 1. Three dimensional (3D) model configuration. Asin2φ rrsinφφθ ( sin ) =+k© r where where m and Q are the mass and charge of the grain, B is 2 0 Asin φ the solar magnetic field, c is the velocity of light, G is the k =+k© , r gravitational constant, M is the solar mass and Ω is the solar angular velocity. Asin2 φ k =+0 k© 0 r , The vectorial component of the magnetic field in the 0 3D case is Asin2φ kr©=−=22 sin φθ constant 3 3 r BB=−2 ()RcosφφiBÃ − ()Rsin iÃ , (2) 0r r 0r φ and kr(= 22 sin φθÇ) is the specific angular momentum of the where B0 is the solar surface magnetic field at the equator grains about the dipole axis and k0 is the angular momentum Ã Ã at the point of injection, the equation for the angular mo- and ir and iφ are the radial and transverse unit vectors respectively. mentum is Asin2φ Asin2φ Ç kk−= − 0. Taking the dot product of (1) with r , we get 0 r r Thus sin2 φ  d ()1 vA2 =Ωd −GM dr 2φ 2φ  dt 2 dt r  2 dt , (3) =+sin −sin 0 r kk0 A  . (7) rr0 

QBR3 If where A=- 00 and vvv2=⋅. The suffix ‘o’ represents mc r x = (8) r0 initial conditions. R is the solar radius and v0 is the speed of the grain at the point of injection. and sin2 φ y = Taking the dot product of (1) with iÃθ , the unit vector in 2 φ , (9) sin 0 the azimuthal direction, we find the energy equation becomes Asin2φ  d ()r 22sinφθÇ =−A () sin2 φrrÇ −2 sin φ cos φφÇ = d. dt 2 dt r r  2φ 2=−KGM 2GM −1−−Ωsin 0y v ()121A (10) (4) r00rx r0 x

2 The dynamics of dust particles near the Sun

and organizing all terms, we finally obtain 2 − ≥−k0 2pk0xy 22 2 33 GML R sin φ ΩLR x 2φ − 0 0 0 2=−KGM⊕2GM x −1−Ωsin 0xy v ()2A  .(11) 2 2 r00r x r0x sin φ  xy−   + p2GM 0  Ω2 44 x . (20) LR0  From equation (7), the angular momentum is expressed by Rewriting this expression we obtain Asin2φ  y kk=−01 − 0 r  x; (12) {}−+−− 2φ 0 xy[] x( K222 ) ( x y ) p sin 0 thus k2 2pk xy− 2φ − ≥−0 0 =−Asin 0xy 2φ Ω 22x kk0   (13) GML0 Rsin 0 LR0 r0 x where sin2φ  xy−  2 + p2GM 0 Ω2 33 x . (21) v = k LR0 θ rsinφ (14) Multiplying by x2 and noting that is the azimuthal velocity. Substituting equation (13) into (14), we obtain = φ kr00( sin 0 ) vθ,0, k2 2Ak xy−A22sin φxy−2 v2=−10 0 +0 . θ 2 22φ 3x4 xkr≤=sinφφ vr sinKGM = sin φKGMRL xyr0sin 0 r0 r0 00 000 0 00, r0 (15) and From equations (6) and (7) and noting that =≤βφ β k00sin KGMRL 0 with 1 .

2 Equation (21) can be written 22≥ k vvθ  (16) r22sin φ 32{}−+−− φ≥ xy[] xK(222 ) ( x yp ) sin 0 we obtain βφsin Kxβ 22−−2 pxxy() 0 KGMRL ΩLR22 0 2φ − 0 KGM 2GM x −1sin 0xy −()−2AΩ 2 r r x r x sin φ 00 0 +−≥p2GM 0()xy2 Ω2 33 (22) LR0 k22Ak xy−A22sin φ xy−2 ≥−100 +0  222φ 3x4 x. 22 KGM xyr0sin 0 r0 r0  Kxββφ−−2 pxxy( ) sin 0 Ω23RL3 (17) 0 +−psin2 φ GM (xy )2 0 Ω233 . In equation (17), L0 is the magnetic parameter defined as RL0

r Let us define a parameter |α| as the relationship be- L = 0 , (18) 0 R tween the gravitational constant G, the solar mass M, the and solar angular velocity Ω and the solar radius R AΩ p = . (19) GM⊕ 2 GM α = . (23) Ω23R Thus we obtain Thus xK()−+22 sin2φxy− 1 −2p 0 LR x  LR x 32{}−+−− φ≥ 0 0 xy[] xK(222 ) ( x yp ) sin 0

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α2 2  ββαφφ22−−K +−222 2α2342 Kx2 xxy( )3 sin0ppxy ( ) sin 03 −+−222xp sinφφxxpy sin L L. 30 0 0 0 L0  (24) α2 +≤px22 sin2φ 0 L3 0 , (27) In terms of y, this yields 0

with the solutions  2   2 2 φ2 α− 32− 2φ2α sin0  p3 22px y x sin 0 p 3  L0   L0 ≤ ≤ p1 (y) y p2 (y) . (28)  −+−+−βαK φ4342 φ  2222x3sin0pxK ( ) x xp sin 0 y DUST GRAINS IN THE INNERMOST REGIONS OF L  0 THE HELIOSPHERE  α2 +−Kxββαφ2220 x 2 Ksinppx +22 sin 2 φ≤ 3 0 0 3 . In addition to electromagnetic and gravitational forces  L0 L0 we now include radiation pressure. The equation of motion (25) is Q 2 Q0 GMm pr Laor From equation (25), we obtain a quadratic equation ex- mrÇÇ =−×[]()rrBÇ Ω×−+r , (29) c r3 c 4r3 pressed in dimensionless quantities:

where the last term is the radiation pressure, m is the grain p22α  Fxypk( , , ; ,βφ , , L ; α )=− sin2 φ 2px32 y mass, c is the light velocity, G is the gravitational constant, 00 0L2  0 M is the solar mass, Qpr is the radiation pressure coefficient, ρ L° is the solar luminosity, is the density and a is the grain  22αφ  12/  radius. 2xp sin −βαK 2 φ −  2x   psin 0   L3 L3  −0 0 y 4342−+ − φ  Equation (29) describes the situation of dust grains in- xK(22 ) x 2 xp sin 0   jected at L =10 into the heliosphere considering two con-   0 secutive solar minima (1 and 2), before and after a solar maximum (Figures 2 a, b, c). Dust particles are moving in 12/2  sin φ two dimensions (Figure 3). We select L0=10 in order to find ββαφα22− 2 K +22 2 0≤ + Kx20 x 3ppxsin 0 3 , whether dust particles from the interplanetary medium can LL 0 0 be trapped by the Sun near the corona. (26)

3 =− RÃ If the solar magnetic field is a dipole, BB0()iz β KGM r where K and are defined through v0= and r0 3 = RÃ β φ β ≤ for solar minimum 1 and BB0()iz for solar minimum k0= (r0sin 0)v0 (where | | 1). r 2 (22 years later) and ΩΩ=iÃ. Then β φ α z Thus F(x,y,p; K, , 0,L0; )=0 taken at specific values β φ of the parameters p, K, , 0 and L0 is the equation of the reflection surface (i.e. where v=v so that v = 0=v ). This equa- AGM(φÇ −−Ω)© θ r θ ÇÇrr−=φÇ 2 for solar minimum 1 (30) tion is quadratic in y, which is convenient to discuss the re- r2 gion occupied by the allowed orbits in the x-y plane. This is ≤ ≤ given by y1 y y2, where y1 and y2 are the roots of F=0. For and the special case when the particle is projected on the meridi- Ω−−φÇ onal plane with the local gravitational speed K=2, the 3D −=φÇ 2 AGM( )© ÇÇrr 2 for solar minimum 2 , (31) equation is reduced to two dimensions as r

where  p22α  sin2 φ − 2px32 y d 2φÇ=− rÇ 0  3  (rA)2, L0 dt r

4 The dynamics of dust particles near the Sun

B Ω

current sheet ←→

←→

L0=10R0

Fig. 2a. Magnetic field configuration at solar minimum 1.

Ω

equatorial plane

Fig. 2b. Magnetic field configuration at solar minimum.

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B Ω

current sheet ←→

←→

L0=10R0

Fig. 2c. Magnetic field configuration at solar maximum 2.

Ω

P(r,φ,θ)

Fig. 3. Two dimensional (2D) model configuration.

6 The dynamics of dust particles near the Sun

QBR3 2L3 y3 2L3 y3 A =− 00 p2−0 p −0 =−()ppyppy()()−()≤0 mc αα21−y 2()1−y 2 12 and (34) 3QL  GG© =−1 pr o 1 16πρGM c a. (32) and the physically allowed regions for dust orbits are

KG© M 2 ≤≤ . (35) If grains are emitted at a distance r with speed v = py() p py1 () 0 0 r0 φ at an angle 0 to the radial direction and if they are instantly charged to a value Q , we obtain a 2-D solution (Figure 3) Taking different values of a, we may find the solutions 0 µ from (30) and (31) as follows: p1(y) and p2(y) shown in Figure 4 (a, b) for a=1 m, in Figure 5 (a, b) for a=5µm and in Figure 6 (a, b) for a=0.6µm. 3  12/  φ =−2 2L0 y ±2αφK FpyL( , ;00 , , K ) p y m sin 0 ≤ ≤ α21−y 3 p The physically allowed region for the orbits p2(y) p  L0  p1(y) is shown as a lightly shaded area. The lower non-shaded ≤ ≤ 32 region also complies with p2(y) p p1(y), but this region is −L0y 22−φ +−()≤ p 22()Ky()sin 021 y y 0 (33) physically spurious because the grain cannot access it at y=1. α()1−y Thus as a decreases, more grains can escape from the for solar minima 1 and 2 respectively. The parameters p, L0 heliosphere; otherwise they are trapped by Lorentz and gravi- α and ’ are defined as follows: tational forces.

Ω p = A GM© Note that the shaded region grows substantially as the grain radius decreases. In this case, trapped grains remain in r L = 0 a narrow region where they are confined by magnetic field 0 R lines because they evolve around these lines. 2 GM© α© = . Ω23R There is a critical radius when G’=0; in this case ac = 0.574 µm. When G’<0, dust grains are ejected by radiation We solve the quadratic equations for grains approach- pressure from the heliosphere. ing in the radial direction to r0=10R and we consider the regions of p-y space where real orbits are possible. In this case, CONCLUSIONS β − p =−(©)aa21 −α G We derive two integrals of the general 3-D where gravitoelectrodynamic motion of charged dust within the co- 3 rotating region of a planetary magnetosphere to study the 3φBRΩ − β ==×0 3. 869 10 7 case of dust grains injected into the heliosphere. 4πρcM and Our approach requires the use of several aproximations, 3QL α ==×pr 0 −5 including the assumption that the magnetic moment and the ©.πρ574 10 . 16 GM spin vectors are strictly parallel and that the grain charge is constant. RESULTS AND DISCUSSION We construct the spatial regions where the grains are confined, at least initially, before evolutionary effects take When L , φ and K are fixed, then for a given value of y, 0 0 over. φ the equation F(p,y; L0, 0, K)=0 is quadratic in the parameter p. When p is real, p must lie between two real roots p1(y) and A particular case of our 3-D model is the 2-D model. p2(y) of this equation. This enables one to find the regions of This model was used to derive the confinement regions where p - y space where real roots are possible. dust can be trapped around the Sun.

φ When K=2 and 0=0 we obtain solutions for solar mini- The 2-D model has now been modified in order to in- mum 1 and solar minimum 2 as follows: clude solar radiation pressure. This model was used to study

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Fig. 4a. Orbits of charged dust grains launched at L0=10. The lightly shaded region is the region in the p-y plane where real orbits can lie. The point of launch is given by y=1 at minimum 1. Positively charged grains correspond to p<0. The size of the injected grains is 1.0 micron.

Fig. 4b. Same as Fig. 4a except that grains are launched when the solar activity is at minimum 2.

8 The dynamics of dust particles near the Sun

Fig. 5a. Same as Fig. 4a except that the size of the injected grains is 5.0 microns.

Fig. 5b. Same as Fig. 4b except that the size of the grains is 5.0 microns.

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Fig. 6a. Same as Fig. 5a except that the size of injected grains is 0.6 microns.

a = 0.6 microns

theta = 0

Fig. 6b. Same as Fig. 5b except that the size of the grains is 0.6 microns.

10 The dynamics of dust particles near the Sun

interplanetary dust particles that are close to the Sun, in- MARAVILLA, D., K. FLAMMER and D. A. MENDIS, cluding the cyclical change of the solar magnetic field. We 1995. On the injection of fine dust from the Jovian mag- find that there is a critical radius for the grains (G’=0), such netosphere. Astrophys. J., 438, 968-974. that grains are dominated by radiation pressure when G’< 0. MARAVILLA, D., K. FLAMMER and D. A. MENDIS, The effect of sublimation will be included in a future 1996. The regions of confinement of charged dust injected model. into the terrestrial magnetosphere, Proceedings of the Sixth Workshop on the Physics of Dusty Plasmas, Mendis, Chow BIBLIOGRAPHY and Shukla (eds).

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HORANYI, M., G. MORFILL and E. GRÜN, 1993b. Mecha- ______nism for the acceleration and ejection of dust grains from Jupiter’s magnetosphere. Nature, 363, 144-146. Dolores Maravilla Instituto de Geofísica, UNAM LAMY, Ph. L., 1974. The dynamics of circum-solar dust Cd. Universitaria 04510, México, D.F. México. grains. Astron. and Astrophys.,33, 191-194. E-mail. [email protected]