Serb. Astron. J. � 192 (2016), 27 - 34 UDC 523.62–726DOI: 10.2298/SAJ160220004M Original scientific paper

ORBIT LIMITED THEORY IN THE SOLAR WIND - κ DISTRIBUTIONS

M. M. Martinovic1,2,3

1LESIA, Observatoire de Paris5, place Jules Janssen, 92195 Meudon, France

2Department of Astronomy, Faculty of Mathematics, University of BelgradeStudentski trg 16, 11000 Belgrade, Serbia

3IHIS Techno-experts d.o.o. - Research and Development CenterBatajnicki drum 23, 11080 Belgrade, Serbia

E–mail: mihailo.martinovic@obspm.fr

(Received: February 20, 2016; Accepted: April 15, 2016)

SUMMARY: When a solid object is immersed into ionized gas it gets broughtto a certain value of electrostatic potential and surrounded by a space charge regioncalled ‘plasma sheath’. Through this region, particles are attracted or repelled fromthe surface of the charge collecting object. For collisionless plasma, this process isdescribed by the so-called orbit limited theory, which explains how the collection ofparticles is determined by the collector geometry and plasma velocity distributionfunction (VDF). In this article, we provide explicit orbit-limited currents expres-sions for generalized Lorentzian (κ) distributions. This work is useful to describethe charging processes of objects in non-collisional plasmas like the solar wind,where the electrons VDF is often observed to exhibit quasi power-law populationsof suprathermal particles. It is found that these ‘suprathermals’ considerably in-crease the charge collection. Since the surface charging process that determines thevalue of electrostatic potential is also affected by the plasma VDF, calculation ofthe collector potential in the solar wind is described along with some quantitativepredictions.

Key words. plasmas – Sun: solar wind

1. INTRODUCTION

When an object is immersed into the plasmait causes electrostatic disturbances in the surround-ing area. The reason of these disturbances that donot exist in neutral gases relies on collection by theobject’s (afterward also called collector) surface ofcharged particles from the plasma. Since the lightplasma electrons are faster than the ions, they willmore frequently collide with the surface of the col-

lector. This way the surface gets charged and a‘plasma sheath’ is formed around the object (Lang-muir 1923). The sheath size is of the order of theplasma Debye length LD =

√ε0kbT/qn (where n, T

and q are density, kinetic temperature and charge ofparticles, while ε0 and kb stand for dielectric permit-tivity of vacuum and the Boltzmann constant) andit contains a net amount of charge in order to bal-ance the potential of the collector surface. If thereare no additional processes responsible for the elec-

27

M. M. MARTINOVIC

tron emission (photoelectric effect, secondary emis-sion, sputtering...) then the potential is expectedto be negative. On the other hand, photoelectronsejected from the illuminated surface are dominant inthe sheath around objects in the solar wind, so thepotential becomes positive in order to prevent loss oftoo many electrons.

In the pioneer work on this topic Mott-Smithand Langmuir (1926) first established a theory of or-bit limited motion of particles, calculating the incom-ing particle flux on the collector surface for multi-ple geometries and plasma velocity distribution func-tions (VDFs). This work has been extended by manyauthors. For laboratory plasmas the theory has alot of limitations, mostly due to very short Debyelengths (Allen 1992, Annaratone et al. 1992). On theother hand, in probe design (Laframboise and Parker1973) and space plasmas it has found a great numberof applications, especially in determining spacecraftfloating potentials (see e.g. Scudder et al. 2000, Kel-logg et al. 2009), while the quantitative descriptionof suprathermal particles is important for interpreta-tion of electric field fluctuations measurements (Mar-tinovic et al. 2016) and charging of dust particles(Meyer-Vernet 1982, Rosenberg and Mendis 1992,Mishra et al. 2013) in the solar wind.

None of the VDFs which have been consideredby previous authors took accurately into account theeffect of suprathermal particles in the surface charg-ing process. However, non-maxwellian distributionsexhibiting large suprathermal tails are ubiquitous inseveral non-collisional plasma environments and, inparticular, in the solar wind (Maksimovic et al. 2005,Stverak et al. 2009).

VDFs observed by particle analyzers in the so-lar wind frequently exhibit a power-law behaviour atlarge energies and are therefore conveniently mod-eled by κ distributions (Maksimovic et al. 1997).Commonly used models separate three fractions ofparticles - the core, halo and strahl. The bi-Maxwellian core contains thermal particles, whilehalo is described by the ‘bi-kappa’ function, bothbeing defined by parallel and perpendicular temper-atures with respect to the magnetic field direction.The strahl is defined as a beam of particles movingoutward from the Sun and is also described by the bi-kappa distribution, shifted by certain mean velocityfrom the first two components (Stverak et al. 2009).This kind of the VDF persists in the solar wind asthe plasma moves away from the Sun (Zouganelis etal. 2004).

Presence of these fast non-thermal particlesmodifies both the value of the surface potential andthe amount of particles collected by the charged sur-face of a certain potential. The orbit limited theoryquantitatively describes both of these effects, and isgiven in Section 2 for the case of κ distributions. Thetheory presented below can be observed as a gener-alization of the method given by Laframboise andParker 1973 to non-thermal plasma. Some applica-tions to the solar wind are discussed in Section 3.

2. ORBIT LIMITED THEORYFOR κ DISTRIBUTIONS

2.1. Definition and characteristics ofκ distributions

This kind of the VDF is actually a general-ized Lorentzian (Scudder 1992) power law distribu-tion with a higher percentage of suprathermal parti-cles compared to a classic Maxwellian and is definedas

f(v) =Γ(κ+ 1

)(πκ)3/2v3

0κΓ(κ− 1/2

) 1(1 + v2

κv20κ

)κ+1 , (1)

where Γ(x) denotes the gamma function and v0κ isthe thermal speed related to the kinetic temperatureT as

v0κ =

(2κ− 3κ

kbT

m

)0.5

, (2)

with m being the particle mass. The VDF is de-fined for κ > 1.5 and for κ → ∞ it reduces to theMaxwellian distribution. This property is importantfor the solar wind since the value of the κ indexcan be used as a measure of ‘nonthermality’ of theplasma (low values of κ assume significant portionof suprathermal particles while for κ ≥ 10 the VDFis so close to the Maxwellian that it can be consid-ered thermal). Further on, it is convenient to definethe average velocity as the first moment of the 3Ddistribution (Chateau and Meyer-Vernet 1991)

< v >κ= 2√κ

π

Γ(κ− 1

)Γ(κ− 1/2

)v0κ , (3)

which converges to the Maxwellian value of < v >=√8kbT/πm for very large values of the κ index.

2.2. Basics of the model

Since the solar wind plasma we observe hereis not Maxwellian and has extremely long mean freepaths (∼ 1AU) it means that collisions can be ne-glected. We will assume that the radius of the col-lector rc is much smaller than the particles’ meanfree path and that every particle has its own trajec-tory that comes from the ‘infinity’ towards the col-lector. Also, for cylindrical and spherical collectors,the plasma local Debye length (and the size of theplasma sheath) is required to be very large comparedto the radius of the collector. Since LD ≈ 5 − 10mat distances 0.3 − 1 AU from the Sun, both of theseconditions are satisfied in the solar wind.

We observe a particle moving with the veloc-ity �v in the electrostatic potential φ(�r) and note vr

and vt to be radial and tangential components ofthe particle velocity vector with respect to the col-lector surface. It can be shown directly from the

28

ORBIT-LIMITED THEORY - κ DISTRIBUTIONS

laws of conservation of angular momentum and en-ergy (Mott-Smith and Langmuir 1926) that a parti-cle, in order to be able to reach the collector, needsto satisfy following conditions

v2r + v2

t +2qφm

> 0, vr > 0 . (4)

Fig. 1. Regions in velocity space used for in-tegration of particles for attracting (left) and re-pelling (right) collector potential. Substitution β =√

2|qφ|/m is made for clarity.

Solutions of Eq. (4) define two hyperbolas with semi-axes ±√2|qφ|/m. Results for both qφ < 0 (attract-ing potential) and qφ > 0 (repelling potential) aregiven in Fig. 1. Flux of particles that reach the sur-face are calculated by integrating the VDF over theshaded surfaces. This method is analogues to oneused by Laframboise and Parker (1973).

2.3. Flux corrections

Now we deal with corrections to the flux of in-coming particles per unit surface area of the chargedcollector. If the collector which is immersed into theplasma is not charged (has zero potential) then theflux of incoming particles will be

I0 =n < v >

4, (5)

which is a well known result of classical thermody-namics, not depending on the VDF. On the otherhand, this result will be modified for a finite value ofthe potential. As seen below, this change depends onboth the collector geometry and VDF of particles.

2.3.1. Attracting potential (qφ < 0)

If the potential is attractive (has the opposite sign of particles’ charge) then it will increase theincoming particle flux. The exact correction of particle flux for Maxwellian VDF has been calculated byLaframboise and Parker (1973) for three different cases - plain, cylindrical and spherical geometry. Here,we extend those results for κ distributions, showing that a higher portion of suprathermals will make theplasma more sensitive to the potential of the collector.

Plain geometry produces one-dimensional potential well. We will assume that the object surface isinfinite in the yz-frame and potential φ is dispersed over the x-axis. For Maxwellian distribution we have

I = nπ−3/2v−30 eη

∫ ∞√−2qφ

m

dvx

∫ ∞

−∞dvy

∫ ∞

−∞dvzvxe

−v2x/v2

0e−v2y/v2

0e−v2z/v2

0 = I0 , (6)

with v0 =√

2kbT/m and η = −qφ/kbT . Analogously, for κ distributions we have

Iκ = nΓ(κ+ 1)

Γ(κ− 1/2)π3/2κ3/2v30κ

∫ ∞√−2qφ

m

dvx

∫ ∞

−∞dvy

∫ ∞

−∞dvz

vx(1 +

v2x+v2

y+v2z+ 2qφ

m

κv20κ

)κ+1 . (7)

From here we make substitutions x = vxκ−1/2v−1

0κ , y = vyκ−1/2v−1

0κ and z = vzκ−1/2v−1

0κ . Putting these intoEq. (7) we obtain

Iκ = nΓ(κ+ 1)

Γ(κ− 1/2)κ1/2v0κ

π3/2

∫ ∞

√ηκ

dx

∫ ∞

−∞dy

∫ ∞

−∞dz

x(1 + x2 + y2 + z2 − ηκ

)κ+1 , (8)

where

ηκ = − 22κ− 3

qφ

kbT. (9)

The integral in Eq. (8) is equal to π/(2κ(κ− 1)) and is not dependent on φ. Replacing Eq. (3) into Eq. (7)we obtain

Iκ = I0κ =n < v >κ

4(10)

concluding that there is no change to the particle flux with potential in the plane geometry.

29

M. M. MARTINOVIC

Cylindrical geometry. In this geometry it is convenient to use a cylindrical coordinate system withz-axis parallel to the central axis of the collector and angle θ as the angle between the velocity of a particleand the line which is orthogonal to the z-axis. We start from Eq. (6) in Laframboise and Parker (1973) forthe flux per unit surface

I = nπ−3/2v−30 eη

∫ ∞√−2qφ

m

dvr

∫ π/2

−π/2

dθ

∫ ∞

−∞dvze

−v2r/v2

0e−v2z/v2

0v2r cos θ , (11)

for Maxwellian distribution, which gives the well-known result

I = I0[2π−1/2η1/2 + eη erfc (η1/2)] (12)

where erfc (x) is the complementary error function. Now we use the same approach for the κ distributionby rewriting Eq. (11)

Iκ = nΓ(κ+ 1)

Γ(κ− 1/2)π3/2κ3/2v30κ

∫ ∞√−2qφ

m

dvr

∫ π/2

−π/2

dθ

∫ ∞

−∞dvz

v2 cos θ(1 + v2

r+ 2qφm +v2

z

κv20κ

)κ+1 . (13)

From here we make substitutions x = vrκ−1/2v−1

0κ and y = vzκ−1/2v−1

0κ and obtain

Iκ = I0κ4Γ(κ+ 1/2)π1/2Γ(κ− 1)

∫ ∞

√ηκ

x2dx

(1 + x2 − ηκ)κ+1/2. (14)

For the zero potential we have ∫ ∞

0

x2dx

(1 + x2)κ+1/2=π1/2Γ(κ− 1)4Γ(κ+ 1/2)

, (15)

and, consequently, we have a trivial result Iκ(φ = 0) = I0κ.In the general case (the non-zero potential), Eq. (14) can be represented in terms of regularized

hypergeometric function 2F1 as

Iκ = −2I0κπ−1/2Γ(κ+ 1/2)ηκ−1

κ 2F1

(κ− 1, κ+ 1/2, κ, 1− ηκ

). (16)

For different integer values of κ the general solution is given in Table 1 and Fig. 2. It is clearly visiblethat increase of the flux is not negligible if the Maxwellian is replaced with the κ VDF.

Table 1. Expressions for correction of the particleflux due to attracting potential of a cylindrical col-lector for multiple values of κ index.

κ Iκ/I0κ

2 1−(2η)3/2

1−2η

3 9+61/2η3/2(2η−5)(3−2η)2

4 −500+101/2η3/2(175+12η(η−7))4(2η−5)3

5 19208+141/2η3/2(−5145+2η(1323+10η(2η−27)))8(7−2η)4

Fig. 2. Particle flux correction factor in cylindri-cal geometry due to attracting potential for differentvalues of κ index. Black line is the result given byMott-Smith and Langmuir (1926), and Laframboiseand Parker (1973) for Maxwellian electrons. See theelectronic edition of the Journal for a color versionof this figure.

30

ORBIT-LIMITED THEORY - κ DISTRIBUTIONS

Spherical geometry. In this case we use spherical coordinates with potential φ in the x-directionand vx = v cos θ. We obtain for Maxwellian distribution

I = nπ−3/2v−30 eη

∫ ∞√−2qφ

m

dv

∫ π/2

0

dθ

∫ 2π

0

dψe−v2/v20v cos θv2 sin θ , (17)

resulting with I = I0(1 + η). On the other hand, the same method for κ distribution gives

Iκ = nΓ(κ+ 1)

Γ(κ− 1/2)π3/2κ3/2v30κ

∫ ∞√−2qφ

m

dv

∫ π/2

0

dθ

∫ 2π

0

dψv3 sin θ cos θ(

1 + v2+ 2qφm

κv20κ

)κ+1 . (18)

Substituting x = vκ−1/2v−10κ we obtain

Iκ = I0κ2κ(κ− 1)∫ ∞

√ηκ

x3dx

(1 + x2 − ηκ)κ+1, (19)

which is solved to give

Iκ = I0κ

(1 +

2κ− 22κ− 3

η

). (20)

The solution is given in Fig. 3, clearly noting that spherical collectors are more sensitive to the variation ofthe VDF than cylindrical ones.

Fig. 3. Particle flux correction factor in spher-ical geometry due to attracting potential for differ-ent values of the κ index. Black line is the result ofMott-Smith and Langmuir (1926), and Laframboiseand Parker (1973) for Maxwellian electrons. See theelectronic edition of the Journal for a color versionof this figure.

2.3.2. Repelling potential (qφ > 0)

All the above calculations assume positive ηand ηκ, meaning that the potential is attractive. Ifthis is not the case then we deal with the repellingpotential which will effectively reduce the incomingparticle flux. To calculate this flux, we make useof the same condition given in Eq. (4), integratingalong the surface given in Fig. 1 (right). This ef-fectively means that, when ηκ is negative, the lowerlimit of integration in Eqs. (8), (14) and (19) can’tgo below zero. All three of these equations give thesame result for the repelling potential

Iκ = I0κ(1 − ηκ)1−κ . (21)

For very large values of κ, Eq. (21) convergesto the well-known Maxwellian limit eη. The result isgiven in Fig. 4.

Fig. 4. Particle flux correction factor due torepelling potential for different values of the κ in-dex. Black line is the result given by Mott-Smithand Langmuir (1926), and Laframboise and Parker(1973) for Maxwellian electrons. See the electronicedition of the Journal for a color version of this fig-ure.

2.4. Validity of theory for non-idealcollector shapes

If the collector is not an ideal cylinder orsphere than it is necessary to acknowledge limits forusage of the theory given above. As noted in Sec-tion 2.2, the ‘orbit limited criterion’ for the results

31

M. M. MARTINOVIC

above to be valid is that all particles arrive to thecollector from ‘infinity’, so none of the particles canoriginate from the collector surface. If we assumethe opposite - that a particle does originate from thesurface of the collector and then returns to it, thansuch a particle needs to pass through an equipoten-tial point where it will have zero kinetic energy E(and the maximum distance from the surface) beforeit starts to ‘descend’ back. Consequently, the crite-rion is E = 0 for spherical surfaces and E⊥ = 0 forcylindrical ones.

Since these analyses are observing only a sin-gle particle without any collective effects, than theresults given by Laframboise and Parker (1973) (seeSection 3 of that paper) are valid for κ distributionsas well as for Maxwellian one. This is the reason whythe detailed analysis will not be repeated here, butonly the results.

If we have a cylindrical collector than it willnot be possible for any of the particles that satisfyconditions from Eq. (4) to pass the surface twice, sowe conclude that the ‘orbit limited criterion’ is validfor cylinders for any kind of VDF. Another impor-tant point is that, since near the infinite cylinder wealways have a zero electric field ( �E → 0), we can ap-ply the theory not just for circular cylinders but forany kind of a convex cylinder. On the other hand,for spheroids there are limitations for the ratio ofthe major to minor axis. For a prolate spheroid, themaximum ‘allowed’ ratio is 1.653 and for the oblate‘Earth-like’ one, the maximum ratio is 2.537. If thespheroid is not completely convex (there are flat sur-faces on it), the theory is not applicable. This state-ment has been confirmed for geometries of severalspacecraft (see e.g. Kellogg et al. 2009).

3. APPLICATIONS INTHE SOLAR WIND

The velocity distribution for both protons andelectrons has been theoretically derived by many au-thors and measured with particle analyzers duringvarious missions. These results show that the κ in-dex varies with the heliographic distance and so-lar wind speed. The closest distance covered withspacecraft that give reliable VDF measurements isup to ∼ 0.3AU. By analyzing the data, it has beenproven (see e.g. Stverak et al. 2009) that, beside theMaxwellian core that carries ∼ 90% of all particles,the κ index decreases with distance from the Sun forboth the halo and strahl components. Moreover, itis also evident that κ is lower for the fast solar wind.

Measured values give κ ≈ 10 at 0.3AU de-creasing with ∼ r0.65±0.15 for the slow wind (for boththe halo and strahl), κ ≈ 7 at 0.3AU decreasing with∼ r0.45±0.1 (halo) and κ ≈ 14 at 0.3AU decreasingwith ∼ r0.9±0.1 (strahl) for the fast wind (Maksi-movic et al. 2005, Stverak et al. 2009). Approxi-mate values (with ∼ 20% measurement errors) aregiven in Fig. 5.

Fig. 5. Evolution of the κ index for non-thermalparts of VDFs measured by HELIOS between 0.3 and1 AU. Results are given in details in the work byStverak et al. 2009. Uncertainties of the results areless than 25%.

During the missions that cruise towards theSun, spacecraft need to make the journey from 1AUwhere they experience the surroundings of the solarwind described by VDF of κ ≈ 4 − 5 to conditionswhere the κ index is greater than 10. Beside this,the temperature and density of the solar wind sig-nificantly change along the way, which gives growthto the random flux of particles given by Eq. (5).Since the random flux of protons is approximately√mp/me ≈ 43 times smaller than the one of the elec-

trons, it can be neglected when we consider chargingin the solar wind.

On the other hand, the illuminated surface ofthe collector will emit electrons due to the photoef-fect. The flux of the photoelectrons jph depends onlyon the sunlight intensity (decreasing with the squareof the distance from the Sun) and properties of thesurface material. For a given cover material and fre-quency of incident radiation, the rate at which pho-toelectrons are ejected from the surface is directlyproportional to the intensity of the incident light.

Since the current of secondary electrons canbe neglected in the solar wind (Escoubet et al. 1997)we have the equilibrium condition that the currentproduced by the photoelectrons escaping from thecollector potential well and the current of electronscollected from the surrounding plasma I0κ are mu-tually balanced. If the distribution of emitted pho-toelectrons is Maxwellian, we have for the currentbalance equation:

S⊥jphe− eφ

kbTph = SIκ (22)

where S⊥ and S are sunlit and total surface of thecollector, charge q is replaced by −e and Iκ is givenby Eqs. (10), (16) and (20), depending on the geom-etry. With Tph we mark the photoelectron temper-ature. This parameter has been measured by manyauthors and is found to be ∼ 3 eV (Henri et al. 2011)for spacecraft covers and ∼ 2 − 2.7 eV for standard

32

ORBIT-LIMITED THEORY - κ DISTRIBUTIONS

BeCu antennas (Scudder et al. 2000, Pedersen et al.2008, Kellogg et al. 2009). Assuming these parame-ters, the potential φ of the collector can be found asthe solution of Eq. (22). Since the values of Iκ are byan order of magnitude below the values of jph in thesolar wind, it is clear that potential of the antennaand spacecraft surface will be positive - attractingelectrons. A negative potential can appear only inshadowed areas.

Affection of the strahl. By the name strahlis presumed an anisotropic beam-like fraction of theVDF. Although participation of the strahl in the to-tal particle population is less than 5% and decreaseswith distance from the Sun, it carries a significantfraction of the heat flux since the beam is facedaway from the Sun along the magnetic field direc-tion (Maksimovic et al. 2005). The strahl is welldescribed by the κ distribution shifted by the meanvelocity vbeam. Consequently, in the reference framemoving with the velocity vbeam, it can be describedby a standard κ VDF with the mean velocity< v >s0

given by Eq. (3). On the other hand, in the space-craft reference frame (if we consider the magneticfield to be radial in the first approximation) the par-ticle random flux on the sunlit surface originatingfrom the strahl is given as I0κs = ns < v >s /4where ns is the number density of particles in thebeam and < v >s=< v >s0 +vbeam. The contri-bution of this component is additive to the isotropicpart.

4. CONCLUSION

Any charged collector object in the plasmawill perturb it by attracting particles with the op-posite charge and repelling particles with the chargeof the same sign. The orbit limited theory, estab-lished by Mott-Smith and Langmuir (1926), quanti-tatively describes this effect, proving that both thecollector charging and collection of surrounding par-ticles depend on the collector shape (geometry) andplasma VDF. Laframboise and Parker (1973) appliedthe theory to the plasma probe design, using a newmathematical model, applicable to space plasmas. Inthis work, the same approach is used to extend theorbit limited theory to the solar wind plasma de-scribed by the κ distributions. These results are ofimportance since the accurate determination of thespacecraft and antenna potential is needed for im-provement of in situ measurements in space missions.

By analyzing the plain, cylindrical and spher-ical geometries it is shown that the correction of theparticle flux at the surface of the collector in a spaceplasma is strongly dependent of the VDF shape.A higher percentage of suprathermals (that corre-sponds to lower values of the κ index) considerablyincreases the particle collection, with spheres beingmore sensitive than cylinders. These results are validfor any kind of convex cylinders, while spheroids havestrictly defined maximum level of elongation thatmust not be breached for the calculations to be trust-worthy.

On the other hand, the collector charging pro-cess is also defined by the same parameters. Electro-static potential of the illuminated surface is calcu-lated directly from the balance of the photoelectronand plasma current. Calculation of the surface po-tential using results presented here can be performedfor a spacecraft and its antennas more accurately incomparison with standard calculation that assumesa thermal plasma, since it enables implementationof a more realistic plasma model described by the κdistributions.

Acknowledgements – Thanks to Arnaud Zaslavsky,Milan Maksimovic and Stevo Segan for useful advicesthat improved this paper and also to the reviewer forconstructive comments and helpful suggestions on anearlier version of the manuscript. During the workon the paper the author was financially supportedby the Ministry of Education and Science of the Re-public of Serbia through the project 176002.

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TEORIJA OGRANIQENOG ORBITALNOG KRETANjAU SUNQEVOM VETRU - κ RASPODELA

M. M. Martinovic1,2,3

1LESIA, Observatoire de Paris5, place Jules Janssen, 92195 Meudon, France

2Department of Astronomy, Faculty of Mathematics, University of BelgradeStudentski trg 16, 11000 Belgrade, Serbia

3IHIS Techno-experts d.o.o. - Research and Development CenterBatajnicki drum 23, 11080 Belgrade, Serbia

E–mail: mihailo.martinovic@obspm.fr

UDK 523.62–726Originalni nauqni rad

Kada se qvrsto telo nalazi okru�enojonizovanim gasom ono biva dovedeno naodre�enu vrednost elektrostatiqkog poten-cijala i okru�eno naelektrisanom regionomzvanim ”plazmeni plaxt”. Kroz ovaj regionqestice bivaju privuqene ili odbijene odpovrxine naelektrisanog tela (kolektora). Zaplazmu u kojoj nema sudara, ovaj proces jeopisan teorijom ograniqenog orbitalnog kre-tanja, koja objaxnjava na koji naqin je inten-zitet sakupljanja qestica odre�en geometri-jom kolektora i funkcijom raspodele samihqestica. U ovom radu dati su eksplicit-ni izrazi za orbitalne struje kod funkcija

raspodele oblika generalizovanog lorenci-jana (κ raspodele). Razultati ovog rada sukorisni u opisivanju procesa naelektrisa-vanja objekata u nesudarnim plazmama kaoxto je Sunqev vetar, gde merene funkcijeraspodele sadr�e netermalne brze qestice iopisane su stepenim κ funkcijama. Prona�enoje da brze ”supratermalne” qestice znaqaj-no pojaqavaju sakupljanje qestica na kolekto-ru. Kako funkcija raspodele tako�e utiqe ina proces formiranja elektrostatiqkog poten-cijala povrxine kolektora, dat je i postupakizraqunavanja istog u Sunqevom vetru zajednosa nekim kvantitativnim predvi�anjima.

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