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Plasma polarization electric field derived from radio sounding of solar wind acceleration region with spacecraft signals
Journal Pre-proofs Plasma polarization electric field derived from radio sounding of solar wind acceleration region with spacecraft signals Yuri V. Pisanko, Oleg I. Yakovlev PII: DOI: Reference:
S0273-1177(19)30746-X https://doi.org/10.1016/j.asr.2019.10.012 JASR 14493
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Advances in Space Research
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18 July 2019 20 September 2019 7 October 2019
Please cite this article as: Pisanko, Y.V., Yakovlev, O.I., Plasma polarization electric field derived from radio sounding of solar wind acceleration region with spacecraft signals, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.10.012
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1 Plasma polarization electric field derived from radio sounding of solar wind acceleration region with spacecraft signals. Yuri V. Pisanko1,2 1Institute
of Applied Geophysics, Rostokinskaya st.9, Moscow 129128, Russia; [email protected] 2Moscow Institute of Physics and Technology (National Research University), Institutskiy lane 9, Dolgoprudny, Moscow Region 141701, Russia; Oleg I. Yakovlev3 3Institute
of Radio Engineering and Electronics of Russian Academy of Sciences, Vvedenskogo sq.1, Fryazino, Moscow Region 141190, Russia; [email protected] Abstract. Presented is the analytical approximation of averaged solar wind velocity radial dependence in the solar wind acceleration region at heliolatitudes below 60o under low and moderate solar activity. This empirical approximation is based on the data of radio sounding of the solar corona with radio signals from various spacecraft. Deduced is an equation connecting the solar wind velocity radial dependence and the radial dependence of solar wind plasma polarization electric field intensity. This allows constructing a semi-empirical radial dependence of plasma polarization electric field corresponding to the empirical radial dependence of solar wind velocity. Main properties of the semi-empirical dependence, which is based on radio sounding data, are described. Key words: solar wind; exobase; plasma polarization. 1. Introduction Studies of the propagation of monochromatic highly stable radio waves in the solar corona were carried out during the missions to Mars and Venus. Variations of radio wave frequency, phase and delay as functions of the distance between the ray path and the Sun were first detected during radio contacts with American spacecraft “Mariner-4” and “Pioneer-6” (Hollweg and Harrington, 1968; Goldstein, 1969) and Russian ones “Mars-2”, “Mars-7” and “Venera-10” (Yakovlev et al, 1977; Kolosov et al., 1978). Analysis of these preliminary data indicated that the radio sounding of the solar corona with spacecraft signals is an effective method of the solar wind acceleration region investigation. Since 1970 almost all planetary space missions included the solar corona radio sounding when moving in ecliptic plane spacecraft set over the Sun and then rose from behind the Sun. In such experiments deep space radio communication ground stations were used as complex radio-physical installations precisely registering variations of frequency, phase, amplitude and delay of radio signals resulted due to the coronal plasma influence on radio wave propagation. These data serve to calculate electron concentration, velocity and parameters of turbulence in the solar wind acceleration region. In this paper we analyze experimental results of solar wind velocity heliocentric distance dependencies obtained by means of the radio sounding of the solar corona with radio signals from various spacecraft. Then we propose a reliable empirical approximation for the “solar wind acceleration curve” – the solar wind velocity heliocentric distance dependence, applicable to heliocentric distance interval between 3 and 20 solar radii and for heliolatitudes below 60o. When analyzing experimental data one should take into account that the solar wind is essentially non-stationary phenomenon: solar wind velocity heliocentric distance dependencies obtained in
2 various time periods differ, in general, from each other because of possible strong deviations from spherical symmetry during coronal mass ejections. Therefore we consider experimental data obtained only in periods of low and moderate solar activity for the construction of the analytical approximation of the “solar wind acceleration curve”. Methods and peculiarities of obtaining the solar wind velocity heliocentric distance dependencies from radio sounding data are discussed in the reviews (Bird and Edenhofer, 1990; Armand et al., 1987; Armand et al., 2010; Yakovlev and Pisanko, 2018) so here we give only references of corresponding original papers. The purpose of the present paper is to calculate the plasma polarization electric field heliocentric distance dependence in the solar corona, which corresponds to the empirical approximation of the “solar wind acceleration curve”, derived from solar corona radio sounding experiments. We would like to demonstrate that besides electron concentration, velocity and parameters of turbulence it is possible also to derive the information about plasma polarization electric field from the radio sounding of solar wind acceleration region with spacecraft signals. 2. The empirical heliocentric distance dependence of solar wind velocity To derive the plasma polarization electric field from the radio sounding of the solar wind acceleration region with spacecraft signals we need the empirical heliocentric distance dependence of solar wind velocity. Two modifications of radio sounding techniques are used to determine the solar wind velocity heliocentric distance dependence. The first exploits the peculiarities of frequency fluctuations registered at two ground stations spaced on the large distance. Radial moving with the velocity V solar wind plasma inhomogeneities create similar frequency fluctuations observed on two ground stations with the time delay t. One calculates the cross-correlation function of frequency fluctuations registered at two ground stations and determines the time delay t corresponding to the maximum of the cross-correlation function. The solar wind velocity V = (r1-r2)/t is calculated from known difference between distances r1 and r2, corresponding to ray paths for two ground stations and known t value. The second is based on the relationship between the velocity of transportation of inhomogeneous medium through the radio link and the peculiarity of the temporal spectrum of radio amplitude fluctuations recorded at a single ground station. Spectra allow determining the characteristic frequency of amplitude fluctuations, which is proportional to the solar wind velocity. Let us note that the determination of the solar wind velocity from spectra of radio wave amplitude fluctuations is the subject of confusing influence of plasma waves so that the solar wind velocity derived this way may differ from true bulk solar wind velocity. When determining the solar wind velocity heliocentric distance dependence one assumes that the main contribution to radio signal fluctuations occurs at the proximate point to the Sun along the ray path. Under minimum to moderate solar activity a few radio sounding experiments were conducted to determine the solar wind velocity heliocentric distance dependence in the solar wind acceleration region: in April 1976 with “Venera-10” spacecraft (Kolosov et al., 1978; Yakovlev et al., 1980; Efimov et al., 1981; Kolosov et al., 1982), in March-September 1984 with two spacecraft “Venera-15” and “Venera-16” (Yakovlev et al., 1987; Rubtsov et al., 1987; Yakovlev et al., 1988; Yakovlev et al., 1989). Results of many particular determinations of solar wind velocities and primary experimental data processing have been described in detail in mentioned references. Let us note that velocity values obtained at the same heliocentric distance but at different time periods may differ from each other sometimes strongly. The difference is conditioned on both velocity variations and measurement errors. To determine averaged heliocentric distance dependence of solar wind velocity used were all the experimental values presented in tables and graphs in these publications. To decrease the dispersion individual experimental velocity values were averaged over heliocentric distance intervals of the two solar
3 radii length. Each such an interval contained from 6 to 12 velocity measurements. Obtained this way the averaged heliocentric distance dependence of solar wind velocity is presented by rectangles in Fig.1. It is clear from Fig.1 that the main solar wind acceleration takes place between 5 and 20 solar radii.
Fig.1. The heliocentric distance dependence of solar wind velocity. The heliocentric distance (abscissa axis) r is measured in solar radii, while the solar wind velocity (ordinate axis) V – in km s-1. Averaged (over heliocentric distance intervals of the two solar radii length each) experimental velocity values are shown by rectangles. The rms error bar of each average value is shown by vertical line segment. The approximation is shown by solid line. The averaged heliocentric distance dependence of solar wind velocity is quiet well described by the approximation (Yakovlev, Yakovlev, 2019), which we rewrite in the form: V (r )
V0 . 1 e e r
(1)
Here e=2.7183, the parameter values are as follows V0=325 km s-1, =0.2. Eq. (1) approximates the solar wind acceleration curve between 3 and 20 solar radii. The heliocentric distance r in Eq. (1) is measured in solar radii, while the solar wind velocity V – in km s-1. The approximation is shown by solid line in Fig.1. It corresponds to averaged experimental data within the limits of 20%. 3. A semi-empirical heliocentric distance dependence of plasma polarization electric field Let us now demonstrate how to reconstruct the plasma polarization electric field heliocentric distance dependence in the solar wind acceleration region from known empirical
4 heliocentric distance dependence of solar wind velocity derived from radio sounding of the region with spacecraft signals. Electron and proton thermal velocities Ve=(2kT/me)1/2, Vp=(2kT/mp)1/2 (where T is the coronal temperature, mp is the proton mass, me is the electron mass, k is the Boltzmann constant) under various coronal temperatures are displayed in table 1 while escape velocities at various heliocentric distances are displayed in table 2. It is clear from the tables that electron thermal velocities exceed escape velocities under coronal temperatures while proton thermal velocities are smaller then escape velocities in the corona. T(oK) 106 2.0 106 3.0 106
Table 1. Thermal velocities of electrons and protons under various coronal temperatures. Ve (km s-1) Vp (km s-1) 5504 128 7784 181 9534 222
r (in solar radii) 1 3 5
Table 2. Escape velocities at various heliocentric distances. Ves (km s-1) 618 357 276
This means that the solar gravitation confines coronal protons but can not confine coronal electrons. The Coulomb attraction between protons and electrons leads to the creation of potential electric field of coronal plasma polarization, which prevents the free escape of electrons from the solar gravitational well, but obliges them to tow protons into the interplanetary space. In the case of static corona the plasma polarization electric field is known as the PannekoekRosseland field (Pannekoek, 1922; Rosseland, 1924). The first attempt to calculate theoretically dynamical characteristics of evaporating corona under the existence of polarization electric field was made by Pikel’ner (1950). After the discovery of the solar wind phenomenon at the close of the nineteen fifties the theory places high emphasis on self-consistent kinetic calculations of solar wind characteristics including the polarization electric field (Jokers, 1970; Lemaire and Scherer, 1971a,b; Maksimovic et al., 1997; Zonganelis et al., 2004 and references herein). Among various kinetic approaches to the description of the solar wind the simplest is exospheric one where binary collisions between solar wind particles are neglected above a certain heliocentric distance called “exobase” (see, for example, Zonganelis et al., 2004) which is situated no further then a few solar radii, i.e. somewhere at the base of the solar wind acceleration region. Given from radio sounding experimental results empirical acceleration curve of the solar wind opens evident opportunity to calculate a semi-empirical heliocentric distance dependence of plasma polarization electric field, which corresponds to the empirical acceleration curve. At the same time there is no need to define concretely the model of solar wind electron component. It is quite enough to consider only solar wind protons, which one may assume collisionless above the exobase, and which are under the action of the two forces – the gravitational force from the Sun and the electric force from solar wind electrons. The spherical symmetry of the problem is defined by limitations of the empirical acceleration curve construction procedure. The motion equation of collisionless solar wind protons
m p nV
m p nM dV G qE dr r2
along with the Coulomb law (the Gauss theorem)
(2)
5
1 d 2 q r E 2 0 r dr
(3)
under the constancy of the solar wind mass flow nVr 2 K const
(4)
lead (by substitution of Eq. (3) and Eq. (4) to Eq. (2)) to an equation:
mp K
m p MK d 2 dV r E . G 0E 2 dr dr Vr
(5)
Here mp is the proton mass, n – the concentration of protons, V – the solar wind velocity, r – the heliocentric distance, G – the gravitational constant, M – the solar mass, q – the electric charge density, E – the polarization electric field intensity, 0 – the permittivity of vacuum, K – the solar wind mass flow. The value of the solar wind mass flow K here and everywhere below is for distinctness taken from (Feldman et al, 1977). Let us choose V0=325 km s-1 as the unit of velocity, the heliocentric distance to exobase rex as the unit of distance, the electric field intensity E0 at the exobase as the unit of polarization electric field intensity, and introduce dimensionless variables. If we denote by “dl” dimensionless distance, velocity and electric field intensity, and substitute the relationships r rex rdl , V V0Vdl , E E 0 E dl into Eq.(5), then after algebraic transformations we obtain:
GMm p K 1 m p KV0 dVdl d rdl2 E dl E dl 2 2 3 2 2 drdl V0 rex 0 E 0 V0 rdl 0 E 0 rex drdl
(6)
After omitting the label “dl” and introducing dimensionless parameters
N1
N2
2m p KV0
0 E 02 rex2 2m p KGM
0 E 02V0 rex3
Eq. (5) in dimensionless variables is as follows:
N1r 2
N dV d 2 2 2 r E . dr V dr
(7)
The dimensionless parameters are, correspondingly, the ratio of solar wind kinetic energy density to polarization electric field energy density at the exobase level (N1), and the ratio of solar wind potential gravitation energy density at the exobase level to polarization electric field energy density at the exobase level (N2). Let us specify the empirical heliocentric distance dependence of solar wind velocity according to Eq. (1), which in dimensionless variables is as follows: V
1 , 1 e e r
(8)
6 where =rex/R. Here R is the solar radius. The substitution of Eq. (8) to Eq. (7) gives:
d 2 r E dr
2
r 2 e e r
N1
1 e
e r 2
N 2 1 e e r .
(9)
Integrating Eq. (9) from the exobase to the current heliocentric distance gives: 2
r 2 e e r
r
r E
2
1 N1 1
1 e
e r 2
r
dr N 2 1 e e r dr .
(10)
1
After the integration one obtains: r
1 e
e r
1
dr r 1 1 e
r 2 e e r
r
1 e
e r 2
1
dr
e2
3
e
e e r .
(11)
arctg e arctg e e
e r
2e 1 e e e e r e e 3 ln e r e 1 e e r 1 e e 1 e e r 2 2 1 e r e 3 2 r 2 1 2e r 1 4 Li 2 1 e e r 4 Li 2 1 e e e r e 1 e 1 e
1 1 1 2 Li 2 ln 2 1 e e ln 2 1 e e r . 2 Li 2 e e r 3 1 e 1 e
(12)
Here Li2 is the dilogarithm function (Stegun, 1972). Eq. (10)-(12) allow calculating the polarization electric field heliocentric distance dependence. With known heliocentric distance dependence of plasma polarization electric field it is easy to calculate the heliocentric distance dependence of electric charge density. From Eq. (7) one obtains:
1 d 2 1 dV N 2 r E 2 N1 . 2 dr r 2V r dr 2r E
(13)
Eq. (13) along with Eq. (3) and Eq. (8) allows determining the heliocentric distance dependence of electric charge density q in the solar wind acceleration region as follows:
q
1 dV N 2 N1 . dr r 2V 2r 2 E
(14)
To express the electric charge density in SI unit system the dimensionless q should be multiplied by 0E0/rex. Let us estimate the polarization electric field intensity at the exobase. From Fig.1 one can see that the solar wind velocity in the base of the solar wind acceleration region is relatively low; for distinctness let us suppose that the polarization electric field intensity at the exobase is of the
7 order of the Pannekoek-Rosseland field intensity, i.e. is not differ dramatically from the polarization electric field intensity of the static corona. In SI unit system the latter is as follows (Pannekoek, 1922; Rosseland, 1924): E0
m p GM
2q e rex R
2
1.42856 10 6
1 (V m-1). rex2
(15)
The heliocentric distance to the exobase rex in Eq. (15) is dimensionless and is measured in solar radii while qe=1.610-19 C is an elementary electric charge. Therefore the order of magnitude of electric charge density in the solar wind acceleration region is 0E0/R ~ 10-25 C m-3. This means that the electric charge density is negligible in the solar wind acceleration region. The calculated heliocentric distance dependence of polarization electric field for the exobase situated at 3 solar radii is shown in Fig.2.
Fig.2. The heliocentric distance dependence of polarization electric field intensity if the electric field boundary value (E0) at the exobase is equal to Pannekoek-Rosseland’s and the heliocentric distance of exobase is equal to 3 solar radii. The field intensity (ordinate axis) E is expressed in V m-1, while the unit on the abscissa axis r is equal to 3 solar radii. One can see that the electric field intensity increases very rapidly with heliocentric distance, achieves its maximum (~1000 Vm-1) at approximately 3.7 solar radii and then decreases to almost zero at approximately 20 solar radii. The maximum of electric field intensity (situated not far from the inner boundary of the solar wind acceleration region) is connected with the need for solar wind protons to overcome the solar gravitation when they follow solar wind electrons on its way from the Sun keeping up given empirical acceleration curve, derived from radio sounding. The shift of the exobase closer to the Sun (to the heliocentric distance of 2.5 solar radii) does not change the general character of the electric field intensity heliocentric distance
8 dependence, but the electric field maximum intensity becomes stronger (~1400 Vm-1) and is situated closer to the Sun (approximately at 3 solar radii). The stronger maximum electric field intensity is needed to overcome the stronger solar gravitation when keeping up given empirical acceleration curve. The shift of the exobase further from the Sun (to the heliocentric distance of 3.5 solar radii) also does not change the general character of the electric field intensity heliocentric distance dependence, though the electric field maximum intensity becomes weaker (~800 Vm-1) and is situated further from the Sun (approximately at 4.3 solar radii). The weaker maximum is sufficient to overcome the weaker solar gravitation when keeping up given empirical acceleration curve. From Fig.2 it is clear that the polarization electric field vanishes at heliocentric distances beyond approximately 20 solar radii where the solar wind moves mainly due to inertia without acceleration, the solar gravitation influence is negligible, so there is no need in the substantial polarization electric field beyond the outer boundary of the solar wind acceleration region. Let us estimate the relative deposit of the induction electric field (connected to the solar rotation) to the total electric field in the solar wind acceleration region. In the framework of the Parker spherically symmetric model of the interplanetary magnetic field (Parker, 1963) the induction electric field E (directed along a heliolongitude circle) is (in SI unit system) as follows:
E r R Bs
R2 , r2
(16)
where =2.6910-6 s-1 is the angular velocity of solar equatorial regions, Bs10-4 T is the solar general magnetic field value. From Eq. (16) the induction electric field intensity is estimated as ~ 10-2 V m-1 in the solar wind acceleration region. This is substantially smaller then characteristic values of polarization electric field and justifies neglecting the induction electric field connected to the solar rotation when considering the solar wind acceleration region in middle and low heliolatitudes. Having the electric charge density heliocentric distance dependence given by Eq. (14) it is easy to calculate the total positive electric charge of the solar wind acceleration region. In spherically symmetric case presented in Fig.2 it is equal to approximately 6500 C. It is interesting to compare this value with the absolute value of the total negative electric charge of the Earth. The latter is equal to approximately 500000 C and is one hundred times as much as the absolute value of the total positive electric charge of the solar wind acceleration region. At the same time let us note that the value of 6500 C is one hundred times as much as the very crude estimate (~77 C) of the global positive electrostatic charge of the Sun given by Neslusan (2001). 4. Conclusions At present only by means of radio sounding with spacecraft signals one can obtain the experimental data based heliocentric distance dependence of solar wind velocity in the solar wind acceleration region. Basing on the analysis of radio sounding data obtained during radio contacts with various spacecraft presented is the reliable empirical approximation of the averaged solar wind velocity heliocentric distance dependence (acceleration curve), applicable to heliocentric distance interval between 3 and 20 solar radii and for heliolatitudes below 60o. The experimental velocity value dispersion at close heliocentric distances can be large (maximum error of approximation in a single measurement can achieve 40%), but averaging over 6 to 12 experimental values reduces the error to 20%. The Coulomb attraction between protons and electrons leads to the creation of potential electric field of coronal plasma polarization, which prevents the free escape of hot coronal electrons from the solar gravitational well, but obliges them to tow protons into the
9 interplanetary space. We demonstrate how to derive the information about plasma polarization electric field from the radio sounding of solar wind acceleration region with spacecraft signals. Specifically, we suppose collisionless solar wind protons and construct a semi-empirical heliocentric distance dependence of plasma polarization electric field, which corresponds to the empirical approximation of the averaged solar wind acceleration curve. This semi-empirical curve shows that one could expect the electric field intensity of the order of tens to hundreds volt per meter in the solar wind acceleration region. The electric field intensity increases very rapidly with heliocentric distance, achieves its maximum not very far from the inner boundary of the solar wind acceleration region and then decreases to almost zero at the outer boundary of the region, where the solar wind moves mainly due to inertia without acceleration and the solar gravitation influence is negligible, so there is no need in the substantial polarization electric field beyond the outer boundary of the solar wind acceleration region. The maximum is connected with the requirement for solar wind protons to overcome the solar gravitation when they follow solar wind electrons on its way from the Sun keeping up given empirical acceleration curve, derived from radio sounding. The corresponding electric charge density is negligible in the solar wind acceleration region. Acknowledgments We are grateful to the Associate Editor Dr. Stefan Ferreira and two anonymous reviewers for the attention to our paper and valuable comments and suggestions. References Armand, N.A., Efimov, A.I., Yakovlev O.I., 1987. A model of the solar wind turbulence from radio occultation experiments. Astron. Astrophys. 183(1), 135-141. Armand, N.A., Guleaev, Yu.V., Gavrik, A.L., Efimov, A.I., Matyugov, S.S., Pavelyev, A.G., Savich, N.A., Samoznaev, L.N., Smirnov, V.M., Yakovlev, O.I., 2010. Results of solar wind and planetary ionosphere research using radiophysical methods. Physics-Uspekhi (Advances in Physical Sciences). 53(5), 517-523. Bird, M. K., Edenhofer, P., 1990. Remote sensing observations of the solar corona. In: Schwenn, R., Marsch, E. (Eds.). Physics of the inner heliosphere. Springer – Verlag, Heidelberg, pp.13-97. Efimov, A.I., Yakovlev, O.I., Shtrykov, V.K., Rogalskii, V.I., Tukhonov, V.F., 1981. Space observations of frequency and phase fluctuations of radio waves scattering by the nearsolar plasma. Radio Eng. Electron. Phys. 26(2), 311-320. Feldman W.C., Asbridge J.R., Bame S.J., and Gosling J.T., 1977. Plasma and magnetic fields from the Sun, In White, D.R. (Ed.), The Solar Output and Its Variation. Colorado Associated University Press, Boulder, pp.351-382. Goldstein, R. M., 1969. Superior conjunction of Pioner-6. Science. 166(3905), 598-601. Hollweg, J.V., Harrington J., 1968. Properties of solar wind turbulence deduced from radio measurements. J. Geophys. Res. (Space Physics). 73(23), 7221-7250. Jokers K., 1970. Solar wind models based on exospheric theory. Astron. Astrophys. 6, 219-239. Kolosov, M.A., Yakovlev, O.I., Efimov, A.I., 1978. Investigation of the propagation of the decimeter radio waves at the flight of the interplanetary station Venera-10. Radio Eng. Electron. 23(9), 1829-1839. (in Russian) Kolosov, M.A., Yakovlev, O.I., Efimov, A.I., et al., 1982. Decimeter radio wave propagation in the turbulent plasma near the sun, using Venera-10 spacecraft. Radio Science. 17(3), 664-674. Lemaire, J., Scherer, M., 1971a. Kinetic models of the solar wind. J. Geophys. Res. 76, 7479-7490
10 Lemaire, J., Scherer, M., 1971b. Simple model for an ion-exosphere in an open magnetic field. Phys. Fluids, 14, 1683-1694 Maksimovic, M., Pierrard, V., Lemaire, J.F., 1997. A kinetic model of the solar wind with Kappa distribution functions in the corona. Astron. Astrophys. 324, 725-734 Neslusan, L., 2001. On the global electrostatic charge of stars. Astron. Astrophys. 372, 913-915 doi:10.1051/0004-6361:20010533 Pannekoek, A., 1922. Ionization in stellar atmospheres. Bull. Astron. Inst. Neth., 1, 107118 Parker, E.N., 1963. Interplanetary dynamical processes. John Wiley and Sons, Inc., New York. Pikel’ner, S.B., 1950. Dissipation of corona and its meaning. Dokladi Akademii Nauk SSSR. 72, 255-258 (in Russian) Rosseland, S., 1924. Electric state of a star. Mon. Not. Roy. Astron. Soc., 84(9), 720-729 Rubtsov, S.N., Yakovlev, O.I., Efimov, A.I., 1987. Концентрация, неоднородность плазмы и кинетическая энергия солнечного ветра по данным радиопросвечивания с использованием аппаратов Венера-15 и 16. Kosmicheskie Issledovaniya. 25(4), 620-625 (in Russian). Stegun, I.A., 1972. Miscellaneous functions. In: Abramowitz, M., Stegun, I.A. (Eds.). Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards, Applied Mathematics Series 55, pp.998-1010. Yakovlev, O.I., Molotov, E.P., Kruglov, Y.M., Efimov, A.I., Razmanov, V.M., 1977. Peculiarities of propagation of radio waves through the solar plasma, based on data supplied by Mars-2 and Mars-7 transmitters. Radiotekhnika i electronica. 22(2), 29-35 (in Russian). Yakovlev, O.I., Efimov, A.I., Razmanov, V.M., Shtrikov, V.K., 1980. Radio wave propagation in the turbulent solar plasma using three satellites. Acta Astronautica. 7, 235-242. Yakovlev, O.I., Efimov, A.I., Rubtsov, S.N., 1987. Dynamics and turbulence in the solar wind acceleration region from radio sounding data of “Venera-15” and “Venera-16” spacecraft. Kosmicheskie Issledovaniya. 25(2), 251-257 (in Russian). Yakovlev, O.I., Efimov, A.I., Rubtsov, S.N., 1988. Solar wind from radio occultation data using Venera-15 and Venera-16 spacecraft. Astronomicheskii Zhurnal. 65(6), 1290-1299. Yakovlev, O.I., Efimov, A.I., Yakubov, V.P., Korsak O.M., Kaftonov A.S., Erofeev A.L., Rubtsov S.N., 1989. Radio wave frequency and phase fluctuations in two spaced ground stations during radio sounding of solar plasma and solar wind velocity. Izvestiya Vuzov. Radiofisika. 32(5), 531-537 (in Russian). Yakovlev, O.I., Pisanko, Yu.V., 2018. Radio sounding of the solar wind acceleration region with spacecraft signals. Adv. Space Res., 61, 552-566. Yakovlev, O.I., Yakovlev, Y.O., 2019. Analysis of radial dependencies of the electron concentration and plasma velocity in the region of accelerated solar wind. Cosmic Research, 57(4), 237-242. Zonganelis, I., Maksimovic, M., Meyer-Vernet, N., Lamy, H., Issautier, K., 2004. A transonic collisioless model of the solar wind. Astrophys. J., 606, 542-554. Figure captures Fig.1. The heliocentric distance dependence of solar wind velocity. The heliocentric distance (abscissa axis) r is measured in solar radii, while the solar wind velocity (ordinate axis) V – in km s-1. Averaged (over heliocentric distance intervals of the two solar radii length each) experimental velocity values are shown by rectangles. The rms error bar of each average value is shown by vertical line segment. The approximation is shown by solid line. Fig.2. The heliocentric distance dependence of polarization electric field intensity if the electric field boundary value (E0) at the exobase is equal to Pannekoek-Rosseland’s and the heliocentric
11 distance of exobase is equal to 3 solar radii. The field intensity (ordinate axis) E is expressed in V m-1, while the unit on the abscissa axis r is equal to 3 solar radii.
12 Table 1. Thermal velocities of electrons and protons under various coronal temperatures. Ve (km s-1) Vp (km s-1) 6 10 5504 128 6 2.0 10 7784 181 3.0 106 9534 222 T(oK)
r (in solar radii) 1 3 5
Table 2. Escape velocities at various heliocentric distances. Ves (km s-1) 618 357 276
13
14
15 Semi-empirical radial dependence of plasma polarization electric field derived from radio sounding of solar wind acceleration region.
S0273-1177(19)30746-X https://doi.org/10.1016/j.asr.2019.10.012 JASR 14493
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
18 July 2019 20 September 2019 7 October 2019
Please cite this article as: Pisanko, Y.V., Yakovlev, O.I., Plasma polarization electric field derived from radio sounding of solar wind acceleration region with spacecraft signals, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.10.012
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1 Plasma polarization electric field derived from radio sounding of solar wind acceleration region with spacecraft signals. Yuri V. Pisanko1,2 1Institute
of Applied Geophysics, Rostokinskaya st.9, Moscow 129128, Russia; [email protected] 2Moscow Institute of Physics and Technology (National Research University), Institutskiy lane 9, Dolgoprudny, Moscow Region 141701, Russia; Oleg I. Yakovlev3 3Institute
of Radio Engineering and Electronics of Russian Academy of Sciences, Vvedenskogo sq.1, Fryazino, Moscow Region 141190, Russia; [email protected] Abstract. Presented is the analytical approximation of averaged solar wind velocity radial dependence in the solar wind acceleration region at heliolatitudes below 60o under low and moderate solar activity. This empirical approximation is based on the data of radio sounding of the solar corona with radio signals from various spacecraft. Deduced is an equation connecting the solar wind velocity radial dependence and the radial dependence of solar wind plasma polarization electric field intensity. This allows constructing a semi-empirical radial dependence of plasma polarization electric field corresponding to the empirical radial dependence of solar wind velocity. Main properties of the semi-empirical dependence, which is based on radio sounding data, are described. Key words: solar wind; exobase; plasma polarization. 1. Introduction Studies of the propagation of monochromatic highly stable radio waves in the solar corona were carried out during the missions to Mars and Venus. Variations of radio wave frequency, phase and delay as functions of the distance between the ray path and the Sun were first detected during radio contacts with American spacecraft “Mariner-4” and “Pioneer-6” (Hollweg and Harrington, 1968; Goldstein, 1969) and Russian ones “Mars-2”, “Mars-7” and “Venera-10” (Yakovlev et al, 1977; Kolosov et al., 1978). Analysis of these preliminary data indicated that the radio sounding of the solar corona with spacecraft signals is an effective method of the solar wind acceleration region investigation. Since 1970 almost all planetary space missions included the solar corona radio sounding when moving in ecliptic plane spacecraft set over the Sun and then rose from behind the Sun. In such experiments deep space radio communication ground stations were used as complex radio-physical installations precisely registering variations of frequency, phase, amplitude and delay of radio signals resulted due to the coronal plasma influence on radio wave propagation. These data serve to calculate electron concentration, velocity and parameters of turbulence in the solar wind acceleration region. In this paper we analyze experimental results of solar wind velocity heliocentric distance dependencies obtained by means of the radio sounding of the solar corona with radio signals from various spacecraft. Then we propose a reliable empirical approximation for the “solar wind acceleration curve” – the solar wind velocity heliocentric distance dependence, applicable to heliocentric distance interval between 3 and 20 solar radii and for heliolatitudes below 60o. When analyzing experimental data one should take into account that the solar wind is essentially non-stationary phenomenon: solar wind velocity heliocentric distance dependencies obtained in
2 various time periods differ, in general, from each other because of possible strong deviations from spherical symmetry during coronal mass ejections. Therefore we consider experimental data obtained only in periods of low and moderate solar activity for the construction of the analytical approximation of the “solar wind acceleration curve”. Methods and peculiarities of obtaining the solar wind velocity heliocentric distance dependencies from radio sounding data are discussed in the reviews (Bird and Edenhofer, 1990; Armand et al., 1987; Armand et al., 2010; Yakovlev and Pisanko, 2018) so here we give only references of corresponding original papers. The purpose of the present paper is to calculate the plasma polarization electric field heliocentric distance dependence in the solar corona, which corresponds to the empirical approximation of the “solar wind acceleration curve”, derived from solar corona radio sounding experiments. We would like to demonstrate that besides electron concentration, velocity and parameters of turbulence it is possible also to derive the information about plasma polarization electric field from the radio sounding of solar wind acceleration region with spacecraft signals. 2. The empirical heliocentric distance dependence of solar wind velocity To derive the plasma polarization electric field from the radio sounding of the solar wind acceleration region with spacecraft signals we need the empirical heliocentric distance dependence of solar wind velocity. Two modifications of radio sounding techniques are used to determine the solar wind velocity heliocentric distance dependence. The first exploits the peculiarities of frequency fluctuations registered at two ground stations spaced on the large distance. Radial moving with the velocity V solar wind plasma inhomogeneities create similar frequency fluctuations observed on two ground stations with the time delay t. One calculates the cross-correlation function of frequency fluctuations registered at two ground stations and determines the time delay t corresponding to the maximum of the cross-correlation function. The solar wind velocity V = (r1-r2)/t is calculated from known difference between distances r1 and r2, corresponding to ray paths for two ground stations and known t value. The second is based on the relationship between the velocity of transportation of inhomogeneous medium through the radio link and the peculiarity of the temporal spectrum of radio amplitude fluctuations recorded at a single ground station. Spectra allow determining the characteristic frequency of amplitude fluctuations, which is proportional to the solar wind velocity. Let us note that the determination of the solar wind velocity from spectra of radio wave amplitude fluctuations is the subject of confusing influence of plasma waves so that the solar wind velocity derived this way may differ from true bulk solar wind velocity. When determining the solar wind velocity heliocentric distance dependence one assumes that the main contribution to radio signal fluctuations occurs at the proximate point to the Sun along the ray path. Under minimum to moderate solar activity a few radio sounding experiments were conducted to determine the solar wind velocity heliocentric distance dependence in the solar wind acceleration region: in April 1976 with “Venera-10” spacecraft (Kolosov et al., 1978; Yakovlev et al., 1980; Efimov et al., 1981; Kolosov et al., 1982), in March-September 1984 with two spacecraft “Venera-15” and “Venera-16” (Yakovlev et al., 1987; Rubtsov et al., 1987; Yakovlev et al., 1988; Yakovlev et al., 1989). Results of many particular determinations of solar wind velocities and primary experimental data processing have been described in detail in mentioned references. Let us note that velocity values obtained at the same heliocentric distance but at different time periods may differ from each other sometimes strongly. The difference is conditioned on both velocity variations and measurement errors. To determine averaged heliocentric distance dependence of solar wind velocity used were all the experimental values presented in tables and graphs in these publications. To decrease the dispersion individual experimental velocity values were averaged over heliocentric distance intervals of the two solar
3 radii length. Each such an interval contained from 6 to 12 velocity measurements. Obtained this way the averaged heliocentric distance dependence of solar wind velocity is presented by rectangles in Fig.1. It is clear from Fig.1 that the main solar wind acceleration takes place between 5 and 20 solar radii.
Fig.1. The heliocentric distance dependence of solar wind velocity. The heliocentric distance (abscissa axis) r is measured in solar radii, while the solar wind velocity (ordinate axis) V – in km s-1. Averaged (over heliocentric distance intervals of the two solar radii length each) experimental velocity values are shown by rectangles. The rms error bar of each average value is shown by vertical line segment. The approximation is shown by solid line. The averaged heliocentric distance dependence of solar wind velocity is quiet well described by the approximation (Yakovlev, Yakovlev, 2019), which we rewrite in the form: V (r )
V0 . 1 e e r
(1)
Here e=2.7183, the parameter values are as follows V0=325 km s-1, =0.2. Eq. (1) approximates the solar wind acceleration curve between 3 and 20 solar radii. The heliocentric distance r in Eq. (1) is measured in solar radii, while the solar wind velocity V – in km s-1. The approximation is shown by solid line in Fig.1. It corresponds to averaged experimental data within the limits of 20%. 3. A semi-empirical heliocentric distance dependence of plasma polarization electric field Let us now demonstrate how to reconstruct the plasma polarization electric field heliocentric distance dependence in the solar wind acceleration region from known empirical
4 heliocentric distance dependence of solar wind velocity derived from radio sounding of the region with spacecraft signals. Electron and proton thermal velocities Ve=(2kT/me)1/2, Vp=(2kT/mp)1/2 (where T is the coronal temperature, mp is the proton mass, me is the electron mass, k is the Boltzmann constant) under various coronal temperatures are displayed in table 1 while escape velocities at various heliocentric distances are displayed in table 2. It is clear from the tables that electron thermal velocities exceed escape velocities under coronal temperatures while proton thermal velocities are smaller then escape velocities in the corona. T(oK) 106 2.0 106 3.0 106
Table 1. Thermal velocities of electrons and protons under various coronal temperatures. Ve (km s-1) Vp (km s-1) 5504 128 7784 181 9534 222
r (in solar radii) 1 3 5
Table 2. Escape velocities at various heliocentric distances. Ves (km s-1) 618 357 276
This means that the solar gravitation confines coronal protons but can not confine coronal electrons. The Coulomb attraction between protons and electrons leads to the creation of potential electric field of coronal plasma polarization, which prevents the free escape of electrons from the solar gravitational well, but obliges them to tow protons into the interplanetary space. In the case of static corona the plasma polarization electric field is known as the PannekoekRosseland field (Pannekoek, 1922; Rosseland, 1924). The first attempt to calculate theoretically dynamical characteristics of evaporating corona under the existence of polarization electric field was made by Pikel’ner (1950). After the discovery of the solar wind phenomenon at the close of the nineteen fifties the theory places high emphasis on self-consistent kinetic calculations of solar wind characteristics including the polarization electric field (Jokers, 1970; Lemaire and Scherer, 1971a,b; Maksimovic et al., 1997; Zonganelis et al., 2004 and references herein). Among various kinetic approaches to the description of the solar wind the simplest is exospheric one where binary collisions between solar wind particles are neglected above a certain heliocentric distance called “exobase” (see, for example, Zonganelis et al., 2004) which is situated no further then a few solar radii, i.e. somewhere at the base of the solar wind acceleration region. Given from radio sounding experimental results empirical acceleration curve of the solar wind opens evident opportunity to calculate a semi-empirical heliocentric distance dependence of plasma polarization electric field, which corresponds to the empirical acceleration curve. At the same time there is no need to define concretely the model of solar wind electron component. It is quite enough to consider only solar wind protons, which one may assume collisionless above the exobase, and which are under the action of the two forces – the gravitational force from the Sun and the electric force from solar wind electrons. The spherical symmetry of the problem is defined by limitations of the empirical acceleration curve construction procedure. The motion equation of collisionless solar wind protons
m p nV
m p nM dV G qE dr r2
along with the Coulomb law (the Gauss theorem)
(2)
5
1 d 2 q r E 2 0 r dr
(3)
under the constancy of the solar wind mass flow nVr 2 K const
(4)
lead (by substitution of Eq. (3) and Eq. (4) to Eq. (2)) to an equation:
mp K
m p MK d 2 dV r E . G 0E 2 dr dr Vr
(5)
Here mp is the proton mass, n – the concentration of protons, V – the solar wind velocity, r – the heliocentric distance, G – the gravitational constant, M – the solar mass, q – the electric charge density, E – the polarization electric field intensity, 0 – the permittivity of vacuum, K – the solar wind mass flow. The value of the solar wind mass flow K here and everywhere below is for distinctness taken from (Feldman et al, 1977). Let us choose V0=325 km s-1 as the unit of velocity, the heliocentric distance to exobase rex as the unit of distance, the electric field intensity E0 at the exobase as the unit of polarization electric field intensity, and introduce dimensionless variables. If we denote by “dl” dimensionless distance, velocity and electric field intensity, and substitute the relationships r rex rdl , V V0Vdl , E E 0 E dl into Eq.(5), then after algebraic transformations we obtain:
GMm p K 1 m p KV0 dVdl d rdl2 E dl E dl 2 2 3 2 2 drdl V0 rex 0 E 0 V0 rdl 0 E 0 rex drdl
(6)
After omitting the label “dl” and introducing dimensionless parameters
N1
N2
2m p KV0
0 E 02 rex2 2m p KGM
0 E 02V0 rex3
Eq. (5) in dimensionless variables is as follows:
N1r 2
N dV d 2 2 2 r E . dr V dr
(7)
The dimensionless parameters are, correspondingly, the ratio of solar wind kinetic energy density to polarization electric field energy density at the exobase level (N1), and the ratio of solar wind potential gravitation energy density at the exobase level to polarization electric field energy density at the exobase level (N2). Let us specify the empirical heliocentric distance dependence of solar wind velocity according to Eq. (1), which in dimensionless variables is as follows: V
1 , 1 e e r
(8)
6 where =rex/R. Here R is the solar radius. The substitution of Eq. (8) to Eq. (7) gives:
d 2 r E dr
2
r 2 e e r
N1
1 e
e r 2
N 2 1 e e r .
(9)
Integrating Eq. (9) from the exobase to the current heliocentric distance gives: 2
r 2 e e r
r
r E
2
1 N1 1
1 e
e r 2
r
dr N 2 1 e e r dr .
(10)
1
After the integration one obtains: r
1 e
e r
1
dr r 1 1 e
r 2 e e r
r
1 e
e r 2
1
dr
e2
3
e
e e r .
(11)
arctg e arctg e e
e r
2e 1 e e e e r e e 3 ln e r e 1 e e r 1 e e 1 e e r 2 2 1 e r e 3 2 r 2 1 2e r 1 4 Li 2 1 e e r 4 Li 2 1 e e e r e 1 e 1 e
1 1 1 2 Li 2 ln 2 1 e e ln 2 1 e e r . 2 Li 2 e e r 3 1 e 1 e
(12)
Here Li2 is the dilogarithm function (Stegun, 1972). Eq. (10)-(12) allow calculating the polarization electric field heliocentric distance dependence. With known heliocentric distance dependence of plasma polarization electric field it is easy to calculate the heliocentric distance dependence of electric charge density. From Eq. (7) one obtains:
1 d 2 1 dV N 2 r E 2 N1 . 2 dr r 2V r dr 2r E
(13)
Eq. (13) along with Eq. (3) and Eq. (8) allows determining the heliocentric distance dependence of electric charge density q in the solar wind acceleration region as follows:
q
1 dV N 2 N1 . dr r 2V 2r 2 E
(14)
To express the electric charge density in SI unit system the dimensionless q should be multiplied by 0E0/rex. Let us estimate the polarization electric field intensity at the exobase. From Fig.1 one can see that the solar wind velocity in the base of the solar wind acceleration region is relatively low; for distinctness let us suppose that the polarization electric field intensity at the exobase is of the
7 order of the Pannekoek-Rosseland field intensity, i.e. is not differ dramatically from the polarization electric field intensity of the static corona. In SI unit system the latter is as follows (Pannekoek, 1922; Rosseland, 1924): E0
m p GM
2q e rex R
2
1.42856 10 6
1 (V m-1). rex2
(15)
The heliocentric distance to the exobase rex in Eq. (15) is dimensionless and is measured in solar radii while qe=1.610-19 C is an elementary electric charge. Therefore the order of magnitude of electric charge density in the solar wind acceleration region is 0E0/R ~ 10-25 C m-3. This means that the electric charge density is negligible in the solar wind acceleration region. The calculated heliocentric distance dependence of polarization electric field for the exobase situated at 3 solar radii is shown in Fig.2.
Fig.2. The heliocentric distance dependence of polarization electric field intensity if the electric field boundary value (E0) at the exobase is equal to Pannekoek-Rosseland’s and the heliocentric distance of exobase is equal to 3 solar radii. The field intensity (ordinate axis) E is expressed in V m-1, while the unit on the abscissa axis r is equal to 3 solar radii. One can see that the electric field intensity increases very rapidly with heliocentric distance, achieves its maximum (~1000 Vm-1) at approximately 3.7 solar radii and then decreases to almost zero at approximately 20 solar radii. The maximum of electric field intensity (situated not far from the inner boundary of the solar wind acceleration region) is connected with the need for solar wind protons to overcome the solar gravitation when they follow solar wind electrons on its way from the Sun keeping up given empirical acceleration curve, derived from radio sounding. The shift of the exobase closer to the Sun (to the heliocentric distance of 2.5 solar radii) does not change the general character of the electric field intensity heliocentric distance
8 dependence, but the electric field maximum intensity becomes stronger (~1400 Vm-1) and is situated closer to the Sun (approximately at 3 solar radii). The stronger maximum electric field intensity is needed to overcome the stronger solar gravitation when keeping up given empirical acceleration curve. The shift of the exobase further from the Sun (to the heliocentric distance of 3.5 solar radii) also does not change the general character of the electric field intensity heliocentric distance dependence, though the electric field maximum intensity becomes weaker (~800 Vm-1) and is situated further from the Sun (approximately at 4.3 solar radii). The weaker maximum is sufficient to overcome the weaker solar gravitation when keeping up given empirical acceleration curve. From Fig.2 it is clear that the polarization electric field vanishes at heliocentric distances beyond approximately 20 solar radii where the solar wind moves mainly due to inertia without acceleration, the solar gravitation influence is negligible, so there is no need in the substantial polarization electric field beyond the outer boundary of the solar wind acceleration region. Let us estimate the relative deposit of the induction electric field (connected to the solar rotation) to the total electric field in the solar wind acceleration region. In the framework of the Parker spherically symmetric model of the interplanetary magnetic field (Parker, 1963) the induction electric field E (directed along a heliolongitude circle) is (in SI unit system) as follows:
E r R Bs
R2 , r2
(16)
where =2.6910-6 s-1 is the angular velocity of solar equatorial regions, Bs10-4 T is the solar general magnetic field value. From Eq. (16) the induction electric field intensity is estimated as ~ 10-2 V m-1 in the solar wind acceleration region. This is substantially smaller then characteristic values of polarization electric field and justifies neglecting the induction electric field connected to the solar rotation when considering the solar wind acceleration region in middle and low heliolatitudes. Having the electric charge density heliocentric distance dependence given by Eq. (14) it is easy to calculate the total positive electric charge of the solar wind acceleration region. In spherically symmetric case presented in Fig.2 it is equal to approximately 6500 C. It is interesting to compare this value with the absolute value of the total negative electric charge of the Earth. The latter is equal to approximately 500000 C and is one hundred times as much as the absolute value of the total positive electric charge of the solar wind acceleration region. At the same time let us note that the value of 6500 C is one hundred times as much as the very crude estimate (~77 C) of the global positive electrostatic charge of the Sun given by Neslusan (2001). 4. Conclusions At present only by means of radio sounding with spacecraft signals one can obtain the experimental data based heliocentric distance dependence of solar wind velocity in the solar wind acceleration region. Basing on the analysis of radio sounding data obtained during radio contacts with various spacecraft presented is the reliable empirical approximation of the averaged solar wind velocity heliocentric distance dependence (acceleration curve), applicable to heliocentric distance interval between 3 and 20 solar radii and for heliolatitudes below 60o. The experimental velocity value dispersion at close heliocentric distances can be large (maximum error of approximation in a single measurement can achieve 40%), but averaging over 6 to 12 experimental values reduces the error to 20%. The Coulomb attraction between protons and electrons leads to the creation of potential electric field of coronal plasma polarization, which prevents the free escape of hot coronal electrons from the solar gravitational well, but obliges them to tow protons into the
9 interplanetary space. We demonstrate how to derive the information about plasma polarization electric field from the radio sounding of solar wind acceleration region with spacecraft signals. Specifically, we suppose collisionless solar wind protons and construct a semi-empirical heliocentric distance dependence of plasma polarization electric field, which corresponds to the empirical approximation of the averaged solar wind acceleration curve. This semi-empirical curve shows that one could expect the electric field intensity of the order of tens to hundreds volt per meter in the solar wind acceleration region. The electric field intensity increases very rapidly with heliocentric distance, achieves its maximum not very far from the inner boundary of the solar wind acceleration region and then decreases to almost zero at the outer boundary of the region, where the solar wind moves mainly due to inertia without acceleration and the solar gravitation influence is negligible, so there is no need in the substantial polarization electric field beyond the outer boundary of the solar wind acceleration region. The maximum is connected with the requirement for solar wind protons to overcome the solar gravitation when they follow solar wind electrons on its way from the Sun keeping up given empirical acceleration curve, derived from radio sounding. The corresponding electric charge density is negligible in the solar wind acceleration region. Acknowledgments We are grateful to the Associate Editor Dr. Stefan Ferreira and two anonymous reviewers for the attention to our paper and valuable comments and suggestions. References Armand, N.A., Efimov, A.I., Yakovlev O.I., 1987. A model of the solar wind turbulence from radio occultation experiments. Astron. Astrophys. 183(1), 135-141. Armand, N.A., Guleaev, Yu.V., Gavrik, A.L., Efimov, A.I., Matyugov, S.S., Pavelyev, A.G., Savich, N.A., Samoznaev, L.N., Smirnov, V.M., Yakovlev, O.I., 2010. Results of solar wind and planetary ionosphere research using radiophysical methods. Physics-Uspekhi (Advances in Physical Sciences). 53(5), 517-523. Bird, M. K., Edenhofer, P., 1990. Remote sensing observations of the solar corona. In: Schwenn, R., Marsch, E. (Eds.). Physics of the inner heliosphere. Springer – Verlag, Heidelberg, pp.13-97. Efimov, A.I., Yakovlev, O.I., Shtrykov, V.K., Rogalskii, V.I., Tukhonov, V.F., 1981. Space observations of frequency and phase fluctuations of radio waves scattering by the nearsolar plasma. Radio Eng. Electron. Phys. 26(2), 311-320. Feldman W.C., Asbridge J.R., Bame S.J., and Gosling J.T., 1977. Plasma and magnetic fields from the Sun, In White, D.R. (Ed.), The Solar Output and Its Variation. Colorado Associated University Press, Boulder, pp.351-382. Goldstein, R. M., 1969. Superior conjunction of Pioner-6. Science. 166(3905), 598-601. Hollweg, J.V., Harrington J., 1968. Properties of solar wind turbulence deduced from radio measurements. J. Geophys. Res. (Space Physics). 73(23), 7221-7250. Jokers K., 1970. Solar wind models based on exospheric theory. Astron. Astrophys. 6, 219-239. Kolosov, M.A., Yakovlev, O.I., Efimov, A.I., 1978. Investigation of the propagation of the decimeter radio waves at the flight of the interplanetary station Venera-10. Radio Eng. Electron. 23(9), 1829-1839. (in Russian) Kolosov, M.A., Yakovlev, O.I., Efimov, A.I., et al., 1982. Decimeter radio wave propagation in the turbulent plasma near the sun, using Venera-10 spacecraft. Radio Science. 17(3), 664-674. Lemaire, J., Scherer, M., 1971a. Kinetic models of the solar wind. J. Geophys. Res. 76, 7479-7490
10 Lemaire, J., Scherer, M., 1971b. Simple model for an ion-exosphere in an open magnetic field. Phys. Fluids, 14, 1683-1694 Maksimovic, M., Pierrard, V., Lemaire, J.F., 1997. A kinetic model of the solar wind with Kappa distribution functions in the corona. Astron. Astrophys. 324, 725-734 Neslusan, L., 2001. On the global electrostatic charge of stars. Astron. Astrophys. 372, 913-915 doi:10.1051/0004-6361:20010533 Pannekoek, A., 1922. Ionization in stellar atmospheres. Bull. Astron. Inst. Neth., 1, 107118 Parker, E.N., 1963. Interplanetary dynamical processes. John Wiley and Sons, Inc., New York. Pikel’ner, S.B., 1950. Dissipation of corona and its meaning. Dokladi Akademii Nauk SSSR. 72, 255-258 (in Russian) Rosseland, S., 1924. Electric state of a star. Mon. Not. Roy. Astron. Soc., 84(9), 720-729 Rubtsov, S.N., Yakovlev, O.I., Efimov, A.I., 1987. Концентрация, неоднородность плазмы и кинетическая энергия солнечного ветра по данным радиопросвечивания с использованием аппаратов Венера-15 и 16. Kosmicheskie Issledovaniya. 25(4), 620-625 (in Russian). Stegun, I.A., 1972. Miscellaneous functions. In: Abramowitz, M., Stegun, I.A. (Eds.). Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards, Applied Mathematics Series 55, pp.998-1010. Yakovlev, O.I., Molotov, E.P., Kruglov, Y.M., Efimov, A.I., Razmanov, V.M., 1977. Peculiarities of propagation of radio waves through the solar plasma, based on data supplied by Mars-2 and Mars-7 transmitters. Radiotekhnika i electronica. 22(2), 29-35 (in Russian). Yakovlev, O.I., Efimov, A.I., Razmanov, V.M., Shtrikov, V.K., 1980. Radio wave propagation in the turbulent solar plasma using three satellites. Acta Astronautica. 7, 235-242. Yakovlev, O.I., Efimov, A.I., Rubtsov, S.N., 1987. Dynamics and turbulence in the solar wind acceleration region from radio sounding data of “Venera-15” and “Venera-16” spacecraft. Kosmicheskie Issledovaniya. 25(2), 251-257 (in Russian). Yakovlev, O.I., Efimov, A.I., Rubtsov, S.N., 1988. Solar wind from radio occultation data using Venera-15 and Venera-16 spacecraft. Astronomicheskii Zhurnal. 65(6), 1290-1299. Yakovlev, O.I., Efimov, A.I., Yakubov, V.P., Korsak O.M., Kaftonov A.S., Erofeev A.L., Rubtsov S.N., 1989. Radio wave frequency and phase fluctuations in two spaced ground stations during radio sounding of solar plasma and solar wind velocity. Izvestiya Vuzov. Radiofisika. 32(5), 531-537 (in Russian). Yakovlev, O.I., Pisanko, Yu.V., 2018. Radio sounding of the solar wind acceleration region with spacecraft signals. Adv. Space Res., 61, 552-566. Yakovlev, O.I., Yakovlev, Y.O., 2019. Analysis of radial dependencies of the electron concentration and plasma velocity in the region of accelerated solar wind. Cosmic Research, 57(4), 237-242. Zonganelis, I., Maksimovic, M., Meyer-Vernet, N., Lamy, H., Issautier, K., 2004. A transonic collisioless model of the solar wind. Astrophys. J., 606, 542-554. Figure captures Fig.1. The heliocentric distance dependence of solar wind velocity. The heliocentric distance (abscissa axis) r is measured in solar radii, while the solar wind velocity (ordinate axis) V – in km s-1. Averaged (over heliocentric distance intervals of the two solar radii length each) experimental velocity values are shown by rectangles. The rms error bar of each average value is shown by vertical line segment. The approximation is shown by solid line. Fig.2. The heliocentric distance dependence of polarization electric field intensity if the electric field boundary value (E0) at the exobase is equal to Pannekoek-Rosseland’s and the heliocentric
11 distance of exobase is equal to 3 solar radii. The field intensity (ordinate axis) E is expressed in V m-1, while the unit on the abscissa axis r is equal to 3 solar radii.
12 Table 1. Thermal velocities of electrons and protons under various coronal temperatures. Ve (km s-1) Vp (km s-1) 6 10 5504 128 6 2.0 10 7784 181 3.0 106 9534 222 T(oK)
r (in solar radii) 1 3 5
Table 2. Escape velocities at various heliocentric distances. Ves (km s-1) 618 357 276
13
14
15 Semi-empirical radial dependence of plasma polarization electric field derived from radio sounding of solar wind acceleration region.