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UDC 539.12
Electrogasodynamic Model of Charged Particle Acceleration in Flares of the Main Sequence Dwarf
Stars
Yu.S. Kopysov *, Yu. I. Stozhkovt
* Institute of Nuclear Research of RAS 7a, Prospect of the 60th October Anniversary, Moscow, 117312, Russia t P. N. Lebedev Physical Institute of RAS 53, Leninsky Prospect, Moscow, 119991, Russia
Dwarf stars of the main sequence generate charged particles in stellar flares. In our Galaxy the total number of such flash stars is about 1011. The power and frequency of stellar flares are enough to populate all cosmic ray spectrum up to particle energy 1015 eV. Electrogasodynamic model of charged particle acceleration in flares of dwarf stars is considered. The physical mechanism includes the acceleration of particles by electric field behind plasma shock wave originating by stellar flare.
Introduction
Supernova explosions are considered as the main sources of galactic cosmic rays (GCR) [1]. However, our Sun and the stars in the low part of the Main Sequence also accelerate particles in flare processes. The bulk of star population in our Galaxy makes up dwarf stars. The most of them with a later spectral class than the Sun (spectral class G2) belongs to flare-stars. These stars with frequent and powerful flares are much more active in comparison with the Sun [2]. Below it is shown that these flare-stars can provide the necessary energy of GCRs. Maximum energy of particles, which can be obtained in explosion processes, is defined by acceleration mechanism of charged particles. Mechanism of particle acceleration up to ~ 1015 eV in the coronal electric fields of flare-stars is considered.
1. Dwarf Stars as GCR Sources
The stellar population of our Galaxy contains ~ 2 x 1011 stars. More than 90% of them belong to dwarf stars. These stars are in the low part of the Main Sequence. The most part of them is much more active than our Sun. In other spiral galaxies many dwarf stars are observed too. For example, the galaxy NGC4565 has halo of numerous red dwarf stars (it was observed in the infrared radiation) [3]. It is probable that our Galaxy has such halo too. The number of dwarf stars in Galactic disc is more than 1011. As these stars have low luminosity, they can be observed at the distances less than (20-30)pc. The red dwarfs have spots like sunspots. The ratio of total area of spots on red dwarf to star surface can get tens of percents (sometimes till 90%). For comparison, this ratio for the Sun is < 0.05%. Weak global magnetic fields and strong fields of active regions define the solar and dwarf star activity. The strength of the solar global magnetic field is < 1G and in active regions it is ~ 3 kG. These values for red dwarfs are (10-20) G and several tens kG, respectively. Numerous starspots, strong magnetic fields and their complex structure create favorable situations for frequent and powerful stellar flares. Hydrogen emission (e.g., Ha line) observed during flares is the direct proof of the existence of flare activity on these objects [2].
This work was supported in part by the Neutrino Physics Program of the Russian Academy of Sciences (RAS).
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2. Flares on Red Dwarfs and Energy of Cosmic Rays
in Galaxy
Our Galaxy has the following characteristics: radius rGal ~ 10 kpc, thickness of optical disc d « 0.3 kpc. The Galactic disc volume Vdisc ~ 2.8 x 1066 cm3. The GCR energy in Galactic disc plus halo is WCRs ~ 0.6 x 1068 eV (the density of GCR energy was taken wCR « 0.5 (eV/cm)3). If GCR contain in Galactic disc only, their energy is WcCRdisc ~ 1.4 x 1066 eV. Let us consider stellar flares. We take that the number of active red dwarfs in our Galaxy as Ndwarf « 5 x 1010. The energy of one powerful stellar flare is Wdwarf « 3 x 1036 erg = 1.9 x 1048 eV [2]. Let the energy of each stellar flare is Wdwarf « 2 x 1047 eV (in 10 times less than maximum value) and such flares occur each (3-4) days. About 10% of flare energy is transferred to cosmic rays [2]. During one year the energy released to GCRs from all active red dwarfs is about 1060 eV. In our Galaxy cosmic rays spend ~ (107-108) years. Taking into account the values given above the total energy released to GCRs from stellar flares equals to WGCRs ~ (1067-1068) eV. As the values discussed are the evaluations, we think that active dwarf stars can provide the total energy of cosmic rays in our Galaxy and halo, WGCRs ~ 1068 eV. If GCRs are in Galactic disc only, the energy released to GCRs from dwarf stellar flares is all the more enough to provide its total energy in Galaxy.
3. Electric Fields in Solar and Dwarf Star Coronas
As it is known from observations, our Sun accelerates particles up to energies E < 1011 eV. The solar magnetic fields of active regions could keep and accelerate particles up to E < 1015 eV [4]. One of the possible ways of particle acceleration is Syrovatsky's mechanism in which the reconnection of magnetic field lines in the flare regions takes place and neutral current sheet is formed [5]. Below we present another mechanism, in which the process of stellar heat energy transformation into electric energy plays the main role. Experiments with laser beams are known. When powerful laser beam strikes crystal, the hot electrons are evaporated from crystal and dense electron cloud is formed. These clouds produce strong electric fields, which, in turn, accelerate protons and nuclei up to energies of several tens of MeV and more [6-8]. In this mechanism up to 50% of laser beam energy can be transformed into electric field accelerating particles [8]. One can expect that such kind of mechanisms can work in solar and stellar flares where the electron temperature is much higher than the temperature of environmental medium. Such electric fields can arise in a quiet stellar atmosphere when electrons have got energy, for example, from acoustic waves or ascending magnetic fields.
Let us start the investigation of solar atmospheric electricity from the simple events when electric charges have spherically symmetric distribution. In this case the Pois-son's equation has the form
1A (r
r2 dr V dr
) = —4nq(r), (1)
where q(r) = e^Zana(r) is the charge volume density, na(r) is the concentration of particles with electric charge Z in the units of electron charge (Ze = 1). We will look for the solution of the equation (1) as
= const at r < r0 , (2)
(r) = eZ[Ui(r)/rj; Ui(r) = Csin[k(r — r0) + at r0 <r < п , (3) (r) = eZ[U2(r)/r] ; U2(r) = D exp[—x(r — r1)] at r1 < r< œ , (4)
qfr) - - — ^El - — - ifc2C/l(r) at ro<r^ri,
4-ЛТ dr2 4-ЛТ (r) at n < r < oo ,
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where Z is positive electric charge of the solar sphere concentrated at the photosphere level with radius r0 « R0 (R0 — radius of the Sun), n « 2.3R0 is the radius of spherical surface at the boundary between K- and F-corona. Let us take the boundary conditions as
1) Condition of logarithm derivatives U and U2 equality (at r = n)
1 dUi (r)
1 dU2(r)
U2 dr
kctg[k(ri — ro) + 6] = ~x , <5 = nir — k(r\ — ro) — arctg (—, n = 0,±l,±2,
\xJ
Ui dr
It gives the following condition of interfacing two solutions
'Ï
x
2) Continuity condition for potential 0 at r = ri
Ui(ri) = U2(r2) . It gives the relation between constants C and D
D = ±Ck(k2 + x2)1/2.
3) Homogeneity condition for electric field at r ^ r0
d2 0(r)
= 0.
r=r 0
dr2
This additional condition gives the relation between k and x
2kr0 k + xtg[fc(n -r0)]
+ 7—rTZnJZ—rrr = 0 • (6)
2 — k2r2 x — k tg[k(r1 — r0 )]
From these equations we can get the discrete values of taking the scale of height in the external F-corona with the use of the parameter chi. These discrete values of km correspond to different configurations of electric charge distribution in the inner part of the solar atmosphere. In this case the energy field decrease with the increase of km at the same values of x and Z and the conservation of electro-neutrality condition of the Sun as a star. 4) Electro-neutrality condition.
The density of volume charge q defined by formulas (4) depends on the constant C, which can be found from the electro-neutrality condition. Let us write this condition
as
eZ + y dr 4nqr2 = 0. (7)
ro
After integration one can find
C = (sin Ô — kr0 cos ¿)-1 .
The solutions do not depend on the value of Z and can be used to describe the electric fields of solar and red dwarf coronas, and of the Earth's atmosphere too. As an example we calculated electric field configuration similar to that of electric field of the Earth. The important parameter of this configuration is the decrement of the field x at r ^ œ because x can be used to define main sizes of the field configuration: (r1 — r0) and km. For r0 = R0 and r1 = 2.3R0 we took x = 0.3543. Then the first 3 values of km equal to k0 = 0.8276, k1 = 2.1301, k2 = 4.0823. The state with m = 0 corresponds to monotonic decrease of the field with the distance increase. At m = 1 the electric field strength has minimum at r « 2R0 and also decreases with the distance increase. At this point the maximum values of potential barrier and electron volume density take place.
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4. Solar Atmosphere as Thermoemission Converter of Heat Energy into Electric One and into Energy of Accelerated Particles
In the chromosphere at the altitude h ~ 2000 km above the photosphere the sharp increase of temperature takes place and in the corona at h ~ 104 km the temperature reaches 106 K. The heating is made by the dissipation of energy of mechanical and magnetic fluxes arising from the photosphere and by the scattering light quanta on free electrons. Electrons are heated and the high-energy electrons cannot be kept by gravitation field. So, one can expect the flow of such electrons as it takes place in the laboratory conditions. The low energy electrons are kept by gravitational field and produce large-scale electric fields.
Let us write the system of equations for electron gas flow. The continuity equation of "hot" electron flow has the form
4ner nehUeh = Ieh ,
where neh, ueh and Ieh denote volume concentration, velocity and current of electrons, respectively. Equation of momentum has the form
ШеЪПеЪПеЪ-
dUeh dpeh
dr
dr
d^ GM0meneh , r> - nehZee----^--h > Peh,fc
dr
k
(8) 'hot"
(9)
where peh is the pressure of electron gas flow, M0 — mass of the Sun, G — gravitational constant, 0 — electric potential (see formulae (2)), J2 Peh,fc — total momentum transferring to ions and "cold" electrons in the unit volume. Equation of energy conservation is written as
nehUeh
3 dTeh 2 dr
kTeh dneh A ne h dr
) = " \V1lehk (Teh " Tec) '
where Teh and Tec are the temperatures of "hot" and "cold" electrons, Aeh — coefficient of thermal conductivity of fast electrons, v — constant characterizing the velocity of heat transfer from "hot" electrons to "cold" ones. There are two types of Parker's solutions corresponding to electron wind and electron breeze. The large-scale electric fields considered arise from the interaction of several Maxwell's groups of electrons — running electrons of electron wind and returning electrons of electron breeze. Besides these flows, the fluxes of electrons and ions accelerated by electric field are produced. These fluxes compensate negative electric charge which solar wind carries away. Thus, the current circuit of coronal thermoemission converter is closed and such converter is working at the background of neutral solar wind. The source of energy is the thermal energy of "hot" electrons. The existence of be-Maxwell's electron spectrum in the solar wind at r = 1 a. u. (Teh = 6.9 x 105 K and Tec = 1.25 x 105 K) is the confirmation of this model [9]. One can consider two stages of the acceleration process. At the first stage the hot electrons' cloud leads positive ions (process is like the ambipolar diffusion) and electric field is produced between clouds of the hot electrons and ions. Maximum energy which can get a positive ion is Em ^ Mc27, where Mc2 is the rest energy of ion and 7 is the Lorentz-factor of electron cloud. If flare temperature can get ~ 108 K then the value of Em will be several tens of MeV. Second stage of the positive ions acceleration is due to the large-scale electric field arising between two clouds. This field will accelerate ions up to energies E ~ eL, where e is the electric field strength and L is the linear scale of electric field. If we take e ~ 100V/cm and L ~ 1013 cm then one has got Em « 1015 eV. We suppose that the overturning shockwave front and/or ion-sound turbulence could be responsible for ion injection.
5. Red Dwarfs and Problem of Cold Dark Matter
Hypothesis on red dwarfs as CGR sources involves the problem of cold dark matter (CDM) in Galaxy. The signature of CDM can be high-energy 7-quanta. Neutralino,
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as weakly interacting particle, is the widely discussed candidate on the role of CDM particle. The annihilation of neutralino pairs produces high-energy 7-quanta and antiprotons as by-product of this reaction. The new results were obtained by processing data of the detector EGRET [10]. The excess of galactic high-energy 7-quanta (0.110 GeV) over background expected in 7-ray spectrum from GCR was found. The authors of the paper consider this result as the indicator of CDM presence in Galaxy (neutralino annihilation, if the neutralino mass is (50-100) GeV). In this case one can expect the excess flux of antiprotons in the energy range of (10-30) GeV. If this excess of antiprotons exists, it will be soon detected by experiment PAMELA [11]. In the case of negative result of PAMELA experiment, the excess of high-energy 7-quanta will testify that flare red dwarfs can be sources of these 7-quanta, and that averaged 7-spectrum has an universal form as it was observed by EGRET 7-telescope.
Conclusion
— The energy released into cosmic rays during the flares on dwarf stars is enough to explain the density of cosmic ray energy observed in our Galaxy, both in Galactic disc and in halo.
— When upper chromosphere and low corona are heated the detachment of electrons from ions takes place and the large-scale electric fields are produced. These fields accelerate charged particles up to high energies. The strongest electric fields arise from solar flares when temperature reaches 107-108 K.
References
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2. Hershberg P. E. Solar Type Activity of Main Sequence Stars. — Odessa: Astro-print, 2002. — 688 p. — In Russian.
3. Physics of Space. — Small Enciclopedia. — 1986. — in Russian.
4. Kopysov Y. S., Stozhkov Y. I. About Cosmic Ray Sources with Energies up to (1013-1014)eV // Proceedings of the 29*h ICRC. — Vol. 3. — Pune, India: August 3-10, 2005. — Pp. 141-144.
5. Podgorny I. M., Podgorny A. I. Acceleration of Particles in Current Sheet // Izvestiya Russian Academy of Sciences, ser. Phys. — Vol. 61, No 6. — 1997. — Pp. 1067-1069. — In Russian.
6. Electron, Photon, and Ion Beams from Relativistic Interaction of Petawatt Laser Pulses with Solid Target / S. P. Hatchet, C. G. Brown, T. E. Cowan et al // Phys. Plasmas. — Vol. 7, No 5. — 2000. — Pp. 2076-2082.
7. Intense High-Energy Proton Beams from Petawatt Laser Irradiation of Solids / R. A. Snavely, M. H. Key, S. P. Hatchet et al // Phys. Rev. Lett. — Vol. 85, No 14. — 2000. — Pp. 2945-2948.
8. Bochkarev S. G., Bychenkov V. Y., Tikhonchuk V. T. Investigation of Ion Acceleration at Flying Apart of Laser Plasma Using Boltzman-Vlasov-Poisson Model // Plasma Physics. — Vol. 32, No 3. — 2006. — Pp. 230-247. — In Russian.
9. Feldman Y., Asbridge J., Beim S., Gosling J. Solar Plasma and Magnetic Fields. — In the book "Solar Output and its Variation"/ Editor O. R. White. — Colorado Associated University Press. — Boulder, Colorado. — 1977.
10. de Boer W., Sander C., Zhukov V. et al. EGRET Excess of Diffusive Galactic Gamma Rays as a Tracer of Dark Matter. — In arXiv: astro-ph/0508617. — V. 1, 29 Aug. 2005.
11. Picozza P. on behalf of PAMELA collaboration, Experiment PAMELA // Proceedings of the 20th ECRS. — Lisbon, Portugal: 2006. — in press.
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УДК 539.12
Электрогазодинамическая модель ускорения заряженных частиц во вспышках карликовых звёзд Главной последовательности
Ю.С. Копысов *, Ю. И. Стожков +
* Институт Ядерных Исследований РАН Россия, 117312, Москва, Проспект 60-летия Октября, 7а ^ Физический Институт РАН им. П.Н. Лебедева Россия, 119991, Москва, Ленинский проспект, 53
Во вспышках карликовых звёзд Главной последовательности происходит генерация заряженных частиц. В нашей Галактике полное число таких звездных вспышек составляет около 1011. Мощности и частоты звездных вспышек вполне достаточно для обеспечения спектра космических лучей до энергии частиц 1015 вУ. Рассматривается электрогазодинамическая модель ускорения заряженных частиц во вспышках карликовых звёзд. Физический механизм включает в себя ускорение частиц электрическим полем, создаваемым плазменной ударной волной, порождённой звездной вспышкой.
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