On possible stationary equilivrium of dusty self-gravitating space plasmas
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Цитувати: | On possible stationary equilivrium of dusty self-gravitating space plasmas / V. Yaroshenko, A. Debosscher, M. Goossens, G. Morfill // Вопросы атомной науки и техники. — 2002. — № 4. — С. 115-117. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-803202015-04-15T03:02:25Z On possible stationary equilivrium of dusty self-gravitating space plasmas Yaroshenko, V. Debosscher, A. Goossens, M. Morfill, G. Space plasma 2002 Article On possible stationary equilivrium of dusty self-gravitating space plasmas / V. Yaroshenko, A. Debosscher, M. Goossens, G. Morfill // Вопросы атомной науки и техники. — 2002. — № 4. — С. 115-117. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.27.Lw; 95.30.Qd http://dspace.nbuv.gov.ua/handle/123456789/80320 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Space plasma Space plasma |
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Space plasma Space plasma Yaroshenko, V. Debosscher, A. Goossens, M. Morfill, G. On possible stationary equilivrium of dusty self-gravitating space plasmas Вопросы атомной науки и техники |
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Yaroshenko, V. Debosscher, A. Goossens, M. Morfill, G. |
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Yaroshenko, V. Debosscher, A. Goossens, M. Morfill, G. |
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Yaroshenko, V. |
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On possible stationary equilivrium of dusty self-gravitating space plasmas |
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On possible stationary equilivrium of dusty self-gravitating space plasmas |
| title_full |
On possible stationary equilivrium of dusty self-gravitating space plasmas |
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On possible stationary equilivrium of dusty self-gravitating space plasmas |
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On possible stationary equilivrium of dusty self-gravitating space plasmas |
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on possible stationary equilivrium of dusty self-gravitating space plasmas |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2002 |
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Space plasma |
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http://dspace.nbuv.gov.ua/handle/123456789/80320 |
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On possible stationary equilivrium of dusty self-gravitating space plasmas / V. Yaroshenko, A. Debosscher, M. Goossens, G. Morfill // Вопросы атомной науки и техники. — 2002. — № 4. — С. 115-117. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT yaroshenkov onpossiblestationaryequilivriumofdustyselfgravitatingspaceplasmas AT debosschera onpossiblestationaryequilivriumofdustyselfgravitatingspaceplasmas AT goossensm onpossiblestationaryequilivriumofdustyselfgravitatingspaceplasmas AT morfillg onpossiblestationaryequilivriumofdustyselfgravitatingspaceplasmas |
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2025-07-06T04:17:13Z |
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2025-07-06T04:17:13Z |
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1836869677253394432 |
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SPACE PLASMA
ON POSSIBLE STATIONARY EQUILIVRIUM OF DUSTY SELF-
GRAVITATING SPACE PLASMAS
V. Yaroshenko(1),(3), A. Debosscher(2),
M. Goossens(2) and G. Morfill(1)
(1)Max-Planck-Institut für Extraterrestrische Physik, D-85740, Garching, Germany
(2)Centre for Plasma Astrophysics, K.U. Leuven, Heverlee, 3001, Belgium
(3)Institute of Radio Astronomy of the National Academy of Science of Ukraine
PACS: 52.27.Lw; 95.30.Qd
INTRODUCTION
Cosmic dust is a common component of many
astrophysical objects. Prime examples in the solar system
are noctilucent clouds, cometary tails and comae,
circumsolar and planetary dust rings. Other occurrences
are in the asteroid belt and interstellar dust clouds. The
dust grains in space come in sizes ranging from
macromolecules to micron-sized grains and even to rock
fragments and asteroids. When dust particles are
immersed in plasmas and radiative environments, they
inevitably become charged (usually negatively because of
the greater mobility of electrons) and thus contribute to
the plasma collective effects as a separate species. These
particles have much larger masses than those of the
plasma ions, but much smaller charge-to-mass ratios. The
combination of charged dust and plasma is referred to as a
dusty plasma.
For certain dusty plasmas containing rather heavy charged
grains, it is assumed that the gravitational intergrain
interactions could become important and the name a self-
gravitating plasma is more appropriate. When self-
gravitating effects are incorporated, the dusty plasma
contains a highly diverse range of collective modes,
unstable behavior, and linear and nonlinear waves [1].
Although self-gravitating dusty plasmas have received
wide attention in recent years, for most of the considered
collective phenomena, a homogeneous equilibrium state
is assumed to exist. What is implied by a homogeneous
equilibrium? In ordinary plasmas the model is an infinite
system with vanishing electric fields in the zeroth order.
The situation is essentially different when self-
gravitational forces are included in the analysis. Since
gravitation, contrary to the electromagnetic forces cannot
be shielded, but is always attractive, gravitational collapse
of extended regions with distributed masses is inevitable
(Jeans instability), unless counteracted upon by pressure
or other effects. In fact, there is no way to make the
gravitational potential disappear as the mass densities do
not satisfy any condition analogous to charge
quasineutrality. This implies that a truly homogeneous
equilibrium is impossible, and even more so an infinite
one. Hence, it is physically relevant to investigate basic
stationary nonuniform states in self-gravitating plasmas,
and clarify the problem how heavy dust species influence
the plasma equilibrium state, especially keeping in mind
that the physics of self-gravitating dusty plasmas is
becoming increasingly relevant in determination of the
macroscopic behavior of extended systems in
astrophysical scenarios.
Attempts to investigate a fundamental problem of
stationary equilibria for dusty self-gravitating plasmas
were recently made by several groups independently and
almost simultaneously [2,3] . Physically, both approaches
are considered the stationary state invoking the charge
quasineuntallity of dusty plasmas. Although this condition
is somewhat restrictive and limits the possible number of
self-consistent equilibria, nonlinear equilibrium structures
of hot quasineutral self-gravitating plasmas without and
with uniform rotation of the dust were considered by
Tsintsadze et al. [3]. The consequences of small
perturbations were also studied. On the other hand, the
equilibria by avoiding the usual Jeans swindle were
considered for one-dimensional, cylindrical as well as
spherical symmetrical cases [2] The stationary state was
governed by a nonlinear differential equation for the
inhomogeneous and stationary plasma flow, which has a
singularity at the dust-acoustic velocity. The singularity is
a manifestation of the equilibrium gravitational potential,
which is inhomogeneous.
In the present paper we give a fully consistent analysis of
the stationary state of a self-gravitating dusty plasma,
rejecting the Jeans swindle assumption, as has been done
by Rao [2]. But contrary to the paper [2], we consider the
closed set of exact equations, including the Poisson
equation for the electrostatic potential instead of using the
charge quasineuntallity condition. In Cartesian geometry
a mathematically correct nonlocal treatment can be
performed in one dimension only, because there is no
additional direction, along which the medium could be
assumed strictly uniform. Moreover, in this model the
computations can be done explicitly, for the stationary
state. The considered equilibrium is obtained for a plasma
with zero plasma flows.
GENERAL FORMALISM
We consider an infinite, unmagnetized and hot self-
gravitating plasma. Then the plasma species are governed
by the standard set of fluid equations
,0
2
=
∂
∂
+
∂
∂
+
∂
∂+
∂
∂
+
∂
∂
x
n
n
v
xxm
q
x
vv
t
v TJE α
α
α
α
αα
α
α ψψ
( ) 0=
∂
∂+
∂
∂
αα
α vn
xt
n
. (1)
Two Poisson equations relate the electrostatic Eψ and the
gravitational Jψ potentials to the particle densities αn
∑−=∇ ααπψ nqE 42 , (2)
Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 115-117 115
∑=∇ ααπψ nmGJ 42 . (3)
Here αq and αm denote the particle charges and masses,
αv and ααα mTvT = the fluid and thermal velocities,
for the particles of species α , G is the gravitational
constant.
We are now looking for a stationary equilibrium, letting
0=∂∂ t . Then the continuity equations (2) give the
conservation of the particle fluxes constJvn == ααα . At
the same time, the momentum equations (2) lead to the
conservation of the total energy, consisting of the kinetic,
electrostatic, thermal and gravitational components
0ln
2 0
2
2
=+++
α
α
α
α
αα ψψ
n
nv
m
qv
TJE , (4)
where αJ and α0n are integration constants.
The simplest case that is compatible with the basic
equations, assuming zero fluxes 00 == αα Jv . Solving
(4), one gets the Boltzmann relation for the particle
densities
+
−=
α
αα
αα
ψψ
T
mqnn JEexp0 . (5)
In order to have a reasonable model for some of the
important dusty plasmas, we represent the charged dust
by a limited number of discrete species, considering two
species 2,1=α with different masses and other
characteristics. Correspondingly, we introduce two
dimensionless potentials as
α
αα
α
ψψψ
T
mq JE+= ,
which obey two coupled nonlinear equations obtained by
substitution of (5) into the Poisson equations (2) and (3):
( ) ( )2021012
1
2
expexp ψψψ −+−=
∂
∂ bnan
x
,
( ) ( )2021012
2
2
expexp ψψψ −+−=
∂
∂ dncn
x
. (6)
Parameters cba ,, and d are completely specified by
charges, masses and temperatures of the plasma particles
−=
1
2
1
12
1
4
m
qGm
v
a
T
π
,
−=
1
21
22
1
4
m
qqGm
v
b
T
π
,
−=
2
2
2
22
2
4
m
qGm
v
c
T
π
,
−=
2
21
12
2
4
m
qqGm
v
d
T
π
. (7)
To solve the pair of equations (6), let us assume that
( ) ( ) ( )020112 ln nnxx γψψ −= , (8)
where γ is a positive constant to be determined. In view
of (5), the latter relation signifies a similar spatial
distribution of the two species
( ) ( )xnxn 12 γ= . (9)
Substituting (8) into (6) apparently gives two nonlinear
equations for the potential ( )x1ψ , known as equations
of Emden-type [4], viz.
( ) ( )1012
1
2
exp ψγψ −+=
∂
∂ nba
x
( ) ( )1012
1
2
exp ψγψ −+=
∂
∂ ndbc
x
(10)
which become identical if the adjustable parameter γ is
given by
db
ac
−
−=γ . (11)
Hereby, the pair of coupled nonlinear equations (6) is
reduced to a single equation for determining the stationary
equilibrium state
( )1
2
012
1
2
exp ψψ −=
∂
∂ ln
x
(12)
with ( )( ) 12 −−−= dbadbcl . It can be easily shown that
the solution of (12) posses the scaling and translation
invariance.
In general, the Emden equation (12) can be solved
analytically for different signs of coefficient 2l [4]. But
before discussing the qualitative properties of the
equilibrium solution admissible for (12), we first analyze
the possible values for 2l . The latter is completely
determined by the parameters of the plasma particles
through relations (7) (particle charges, masses and
temperatures) and in principle, 2l can be positive as well
as negative. The same is also true for the parameter γ ,
however this should be positive by definition. Hence, a
careful discussion of different parameter ranges is to be
considered.
EQUILIBRIUM CONSIDERATIONS
It is readily verified with the help of definitions (7) that
( )adbc − is always positive. At the same time the
difference ( )db − can be positive or negative. The only
way to have both γ and 2l positive requires ( )ac − and
( )db − to be positive simultaneously. This restricts dusty
plasma parameters such that
−<−
21
21
2
2
2
1
2
1
2
1
mm
qqG
m
qG
T
T
ν
ν
−<−
21
21
2
1
2
2
2
2
2
2
mm
qqG
m
qG
T
T
ν
ν
. (13)
For definiteness, let us assume ,21 mm < < and 01 >q
while 02 <q , bearing in mind for example a plasma
consisting of the light positive ions and heavy dust grains.
Then the first inequality in (13) can be satisfied
automatically. So the second condition in (13) should be
regarded as a main restriction on the parameters of self-
gravitating plasmas, yielding
12
2
2
2 ≥mqG . (14)
The left-hand side of (14) represents the ratio of
electrostatic and gravitational forces between the heavier
particles. Even two equal forces can still provide the
validity of (13). Therefore, the condition 02 >l actually
implies such a dusty plasma, where gravitational
interactions between heavier particles are about equally
important as electrostatic ones. An alternative possibility,
when 02 <l in the Emden equation (12) can be expected
for the limiting case where the prevailing force that
controls particle dynamics is self-gravitation. Now, the
pair of conditions ( ) 0<− ac and ( ) 0<− db has to be
satisfied simultaneously. Without going into details, it can
be shown that these inequalities can be never satisfied for
any reasonable parameters of plasma particles. This
indicates that the possibility for negative 2l should be
rejected in the present context.
Summarizing, it can be concluded that when 2l is
negative, there is no stationary state with proportional
densities as implied by (9) for a motionless dusty plasma
with dominating self-gravitating interactions. Or,
alternatively, the plasma species reach a stationary state,
with inhomogeneous particle flows akin to those
considered by Rao et al. [2].
Now we return to (12), to get the first integral admissible
for 02 >l
( ) ,exp2 2
101
2
2
1 βψψ +−−=
∂
∂ nl
x
(15)
where β enters as an arbitrary integration constant.
Assuming that the potential 1ψ and its derivative x∂∂ 1ψ
vanish for large enough x , it follows immediately that β
prescribes the normalization constant densities α0n in (5)
through 01
22 2 nl=β and γβ /2 02
22 nl= . It is now
straightforward to obtain the general solution of the
Emden equation (12) as
= −
2
coshln 2
1
xβψ (16)
yielding the particle densities (5)
= −
2
coshln 2
0
xnn β
αα . (17)
Therefore the basic stationary state of a dusty plasma with
heavy charged particles cannot intrinsically be uniform
when self-gravitational forces between grains become
important, e.g. in the spirit of (14). Accordingly to (17),
the characteristic length of inhomogeneity 1~ −∆ β and
maximum amplitudes of the density profiles α0n are
determined by the same constant β . The latter can be
related either to the total number of particles N or to the
total charge Q or to the total mass M per unit surface in
the yz, -plane
( ) ( ) ( )γβ +=+= ∫∫
∞
∞−
∞
∞−
12
221 l
dxxndxxnN ,
( ) ( ) ( )2122211
2 qq
l
dxxnqdxxnqQ γβ +=+= ∫∫
∞
∞−
∞
∞−
, (18)
( ) ( ) ( )2122211
2 mm
l
dxxnmdxxnmM γβ +=+= ∫∫
∞
∞−
∞
∞−
.
Fixing one of these quantities determines β , and thus all
other quantities and the lengthscale ∆ .
Hence, from the standard basic equations (1)-(3) it
follows as a mathematically rigorous result, that a dusty
plasma with motionless gravitating grains has neither
global ( 0≠Q ) nor local charge neutrality (
2112 qqnn −≠=γ ) in a stationary equilibrium. This
point deserves special attention.
Recalling the definition of the parameter γ (11), and
using the adopted model in the case 21 ~ TT , we get
1
21
2
2
2
2
2
1 1
−
+
−−≅
qqq
Gm
q
qγ .
This highlights the transition from a quasineutral dusty
plasma 2112 qqnn −==γ (consisting of such light
particles, that the enormous domination of electric forces
over gravitation allows to neglect gravitational
interactions) to a charged self-gravitating plasma: as the
mass of the heavier component increases, the parameter
γ grows also. The physical consequence of this growth
of γ is the existence of a global electrostatic charge of a
self-gravitating plasma (18), which has the same sign as
the heavier charged particles. This global charge of dusty
plasmas in the equilibrium is easily understood on
physical grounds. Indeed, for the stationary equilibrium of
heavy gravitating particles it is necessary to balance their
mutual gravitational attraction by electrostatic repulsion
due to an excess of like charges.
CONCLUSIONS
In this paper we have considered the existence of self-
consistent stationary equilibrium of a dusty plasma with
heavy self-gravitating particles by avoiding the Jeans
swindle assumption. Starting with the respective basic
equations, we have shown that the possible equilibria are
governed by nonlinear coupled differential equations for
the plasma densities. The latter equations can be reduced
to a single Emden-type equation by assumption of the
similar spatial distribution for all the plasma species.
As it can be expected, the stationary equilibrium state of a
dusty plasma with heavy charged particles cannot
intrinsically be uniform when self-gravitational forces
between grains become important. The spatial scale of
inhomogeneity and maximum amplitudes of the density
profiles are determined either by the total number or the
total charge, or the total mass of plasma particles.
It is shown how that presence of heavy self-gravitating
charged grains can modify the main property of plasmas,
namely quasineutrality, at least in the context of adopted
one dimensional model of dusty plasmas.
ACKNOWLEDGMENTS
V.Y. thanks University, Leuven for the kind hospitality
during a stay in which this work was initiate ans
Alexander von Humboldt Foundation for the financial
support.
REFERENCES
[1] F. Verheest, Waves in dusty space plasmas, Kluwer,
Dordrecht (2000).
[2] Rao, N.N., Verheest, F. and Cadez, V. 2001 Stationary
equilibria of self-gravitating quasineutral plasmas. Phys.
Plasmas 2001, 8, 4740-4744
[3] Tsinsadze, T.L., Mendoça, J.T., Shukla, P.K. , Stenflo,
L.and J. Mahmoodi 2000 Regular structures in self-
gravitating plasmas. Physica Scripta, 62, 70-75.
[4] Ames, W.F., Nonlinear Ordinary Differential
Equations in Transport Processes, Academic Press, New
York and London (1968).
INTRODUCTION
EQUILIBRIUM CONSIDERATIONS
CONCLUSIONS
ACKNOWLEDGMENTS
REFERENCES
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