The dependence of the proton velocity and concentration, electric field, electric charge and electron density on a distance from the centre of the Sun is derived within the framework of hydrodynamical two-zoned model

Efimenko, V.M.

Tokiy, N.V.

Tokiy, V.V.

Kinematika i fizika nebesnyh tel (Online)

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

STATIONARY ELECTRICAL POLARIZING FIELDAND CHARGE IN PLASMA OF THE SOLAR ATMOSPHEREV. M. Eﬁmenko1, V. V. Tokiy2, N. V. Tokiy21Astronomical Observatory, National Taras Shevchenko University of Kyiv3 Observatorna Str., 04053 Kyiv, Ukrainee-mail: efim@observ.univ.kiev.ua2A. A. Galkin Donetsk Physical and Technical Institute, NAS of Ukraine72 R. Luxemburg Str., 83114 Donetsk, UkraineThe dependence of the proton velocity and concentration, electric ﬁeld, electric charge and electrondensity on a distance from the centre of the Sun is derived within the framework of hydrodynamicaltwo-zoned model.INTRODUCTIONA system of four equations including the continuity, ion motion, local electron equilibrium equations andequation of quasineutrality of stationary spherically symmetric plasma ﬂow under the homogeneous temperatureof components is presented in [1]. The dependence of the proton velocity and concentration, electric ﬁeld,electric charge and electron density on a distance from the centre of the Sun is derived in [1–3 ] for isothermalsolar corona. The present work is based on these same simplifying assumptions, excepting the assumptionof isothermal solar corona. The purpose of the present work is the consideration of a stationary expansion ofplasma in vacuum within the framework of hydrodynamical two-temperature-zoned model (T is the temperatureof high-temperature zone and T ′ is the temperature of low-temperature zone).BASIC EQUATIONSWe denote the proton density of high-temperature zone by n(r), and its radial velocity by V (r). We shall supposethe conditions at boundary of high-temperature zone at r = a to be given n0. If as well as [1] the electronmass me is neglected, the equation of motion for electrons for a stationary task turns to a condition of localequilibrium:0 = −knd(Tn)dr− eE. (1)The equation of motion for protons for a stationary task takes the form:mVdVdr= −knd(Tn)dr+ eE −mMSGr2, (2)where G is the gravitational constant, MS denotes the mass of the Sun, E is radial component of an electricﬁeld, V is radial component of ions velocity, m denotes the mass of a proton, −e is the charge of an electron.The condition of charge neutrality of plasma is designatedne = np = n. (3)The continuity equation is:1r2ddr(nV r2) = 0. (4)The Eqs. (1), (2), (3), and (4) form a system of equations for stationary expansion of plasma with homoge-neous distribution of electrons and protons temperatures allowing one to ﬁnd spherical symmetric distributionselectrons ne(r), protons np(r), velocity of protons V (r), and electrical ﬁeld E(r) under given boundary condi-tions.In low-temperature zone, in Eqs. (1)–(4) we substitute T , n, E, V , ne, np for T ′, n′, E′, V ′, n′e, n′p.Density particles, n′0, at r = a in low-temperature zone was deﬁned from the condition of the continuity ofthe ﬂow at aspiration of width of a transitive zone between high and low temperatures to zero.c© V. M. Eﬁmenko, V. V. Tokiy, N. V. Tokiy, 2004169SOLUTION FOR HIGH-TEMPERATURE ZONEEqs. (1) – (4) are so simple that they may be integrated analytically to give V (r) implicitly from(VVc)2− ln(VVc)2= 4 lnrrc+4rcr+ Const. (5)Five types of solution are presented in [4], depending on the value of Const. The solar wind solution (typeIV) corresponds to the value Const=–3, obtained by putting V = Vc and r = rc in Eq. (5).An approximate expression in a solar corona with locations near rc can be derived for V , n, E, and q.At distances r < rc the dependence of the proton velocity on a distance from the centre of the Sun is:V = Vc[1−√2√lnrrc+rcr− 1]. (6)The dependence of the proton concentration on a distance from the centre of the Sun is:n =n0a2r2[1−√2√ln arc +rca − 1][1−√2√ln rrc +rcr − 1] . (7)The dependence of the electric ﬁeld on a distance from the centre of the Sun is:E(r) =2kTer⎡⎢⎣1 +rcr − 12√2√ln rrc +rcr − 1[1−√2√ln rrc +rcr − 1]⎤⎥⎦ . (8)Using Maxwell’s equation for integral on the closed spherical surface S of radius r, we derive a charge ofhigh-temperature zone of solar corona near the Sun at distances r < rcq(r) =8πε0kTre⎡⎢⎣1−rcr − 12√2√ln rrc +rcr − 1[1−√2√ln rrc +rcr − 1]⎤⎥⎦ , (9)where the designations are entered:Vc =√2kTm, (10)rc =GMSm4kT. (11)At distances r > rc the dependence of the proton velocity is:V = Vc[1 +√2√lnrrc+rcr− 1]. (12)The dependence of the proton concentration is:n =n0a2r2[1 +√2√ln arc +rca − 1][1 +√2√ln rrc +rcr − 1] . (13)The dependence of the electric ﬁeld is:E(r) =2kTer⎡⎢⎣1−rcr − 12√2√ln rrc +rcr − 1[1 +√2√ln rrc +rcr − 1]⎤⎥⎦ . (14)Using (14), we derive a charge of high-temperature zone of solar corona near the Sun at distances r > rc:q(r) =8πε0kTre⎡⎢⎣1 +rcr − 12√2√ln rrc +rcr − 1[1 +√2√ln rrc +rcr − 1]⎤⎥⎦ . (15)170SOLUTION FOR LOW-TEMPERATURE ZONEEqs. (1) – (4) may be integrated analytically as well.By analogy for V ′, n′, E′, and q′ an approximate expression in low-temperature of a solar corona withlocations at distances r  r′c may be derived.At distances r  r′c where V ′  V ′c we shall obtain an approximate expression for distribution of protonsvelocity in stationary spherical symmetric ﬂow in low-temperature zone of the solar coronaV ′(r) = V ′c(r′cr)2exp(32− 2r′cr), (16)where V ′c and r′c we deﬁned from (10) and (11) substitution T for T′.Density particles in low-temperature zone were deﬁned from a condition of continuity of the ﬂow at aspirationof width of a transitive zone between high and low temperatures to zero.n′(r) = n′0 exp(2r′cr− 2r′ca), (17)wheren′0 = n0VcV ′c(ar′c)2 [1−√2√lnarc+rca− 1]exp(2r′ca− 32). (18)We derive an approximate expression for an electrical ﬁeld in low-temperature zone of solar corona nearthe Sun:E =GMSm2er2. (19)Using Maxwell’s equation for the integral on the closed spherical surface S of radius r, we derive a chargeof low-temperature zone of the solar corona near the Sun:q′(r) =2πε0GMSme. (20)CONCLUSION. ELECTRICAL AND GRAVITATIONAL FORCES NEAR THE SUNUsing (7) and (8) let us carry out a comparison of forces of the electron and proton gas pressure, electrical andgravitational forces acting on protons and electrons near layer of the solar corona at r = rc. Electrical forceacting on electron is equal to the force of the electron gas pressure:FEe = −FPe = −12 (kT )2GMSm. (21)The forces of the proton gas pressure, electrical and gravitational forces acting on protons near the solarcorona layer at r = rc are equal to:FEp = −43FPp =−16 (kT )2GMSm. (22)Estimations show that force of the proton gas pressure and electrostatic force acting on proton are the mainaccelerating forces. The ratings of values of electrical ﬁelds and charge of the Sun show that they are very small.However, near the surface of the Sun the electrical force acting on a proton is comparable with gravitationalone. The sum of proton gas pressure force and electrical force acting on a proton is 1.5 times stronger thanforce of the gravitational attraction of a proton to the Sun. The electrical force acting on an electron surpassesgravitational force many times over. All this can play a decisive role in ﬁnding a mechanism of solar activity.[1] Eﬁmenko V. M., Tokiy V. V., Tokiy N. V. The electric ﬁeld and charge in the Solar corona // Kinematics andPhysics of Celestial Bodies.–2004.–20, N 1.–P. 27–34.[2] Geiss J., Hirt P., Leutwyler H. On acceleration and motion of ions in corona and solar wind // Solar Phys.–1970.–12.–P. 458–483.[3] Parker E. N. Dynamics of the interplanetary gas and magnetic ﬁelds // Astrophys. J.–1958.–128.–P. 664–676.[4] Priest E. R. Solar Magneto-hydrodynamics / Ed. B. M. McCormac.–Dordrecht: D. Reidel, 1982.–21.–470 p.171

Stationary electrical polarizing field and charge in plasma of the solar atmosphere

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Stationary electrical polarizing field and charge in plasma of the solar atmosphere

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