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

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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 Available in: http://www.redalyc.org/articulo.oa?id=56838306 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Geofísica Internacional (1999), Vol. 38, Num. 3, pp. 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 magneto- sphere to characterize their orbits. These integrals were gen- THE MODEL eralized by Artem’ev (1969) to include the effects of gravi- tational 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 1 D. Maravilla 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 ()RcosφφiBà − ()Rsin ià , (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 à à at the point of injection, the equation for the angular mo- and ir and iφ are the radial and transverse unit vectors re- spectively. 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 2 =Ωd −GM dr 2 2 ()vA 2 , (3) sin φ sin φ dt 2 dt r r dt =+ − 0 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 0y 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 2pk0xy 22 2 33 GML R sin φ ΩLR x 2φ − 0 0 0 2=−KGM⊕2GM x −1−Ωsin 0xy 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 0xy 2φ Ω 22x 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 φxy−2 v2=−10 0 +0 . θ 2 22φ 3x4 xkr≤=sinφφ vr sinKGM = sin φKGMRL xyr0sin 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 0xy −()−2AΩ 2 r r x r x sin φ 00 0 +−≥p2GM 0()xy2 Ω2 33 (22) LR0 k22Ak xy−A22sin φ xy−2 ≥−100 +0 222φ 3x4 x. 22 KGM xyr0sin 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 sin2φxy− 1 −2p 0 LR x LR x 32{}−+−− φ≥ 0 0 xy[] xK(222 ) ( x yp ) sin 0 3 D. Maravilla 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) .
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