Alfvn waves propagation in homogeneous and dusty astrophysical plasmas

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Alfvén waves propagation in homogeneous and dusty
astrophysical plasmas

• M. C. de Juli, D. Falceta-Gonçalves and
• V. Jatenco-Pereira
• Instituto de Astronomia, Geofísica e Ciências
  Atmosféricas IAG/USP

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Index

• 1. Introduction
• 2. The dusty plasma physics
• 2.1 Dusty plasma versus electron-ion plasma
• 2.2 Electric charge and dust charging process
• a) Regular variations
• b) Stochastic variations (charge fluctuations)
• 3. Astrophysical Applications
• 3.1 Stellar winds
• 3.2 Star formation regions
• 4. Summary

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1. Introduction

• Although the universe is 99 plasma, there are
  few astrophysical problems where plasma physics
  solutions have been suggested.
• Astrophysical plasma coexist with dust particles
  in many situations.
• These particles are charged either negatively or
  positively depending on their surrounding plasma
  environments.
• This system of such charged dust, electrons, and
  ions forms a so-called dusty plasma.

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2. The dusty plasma physics

• - Dusty plasma
• It is a fully or partially ionized plasma.
• A low temperature plasma containing disperse
  particles of solid material, dielectric or
  conductor.
• - Dusty particles or dust grains
• These particles are highly massive (md 106
  1018 mp), and highly charged (q 103 104 e).
• Their electric charge depends of grains size,
  their composition and conditions of surrounding
  plasma.
• The electric charge signal will be determined
  through competition among different charging
  processes.

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2.1 Dusty plasma versus electron-ion plasma

• The dust particles properties determine the
  peculiar behavior of the dusty plasmas in compare
  with an electron-ion plasma.
• Some of the differences between a dusty plasma
  and an electron-ion plasma, are
• a) The charge-mass ratio of dust particles is
  very small.
• So, the dust plasma frequency and the dust
  cyclotron
• frequency, given respectively by

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are very smaller than these frequencies for
electrons and ions. In these expressions nd0
is the dust particle density in the equilibrium,
Zd is electric charge number in the dust
particles, md is their mass, B0 is external
magnetic field strength, and c is the light
velocity in the vacuum.
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• New modes of propagation in a dusty plasma arise
  from the fact that frequencies associated with
  dust particles are smaller than electrons and
  ions one.
• Dust modes
• These modes have ultra-low frequencies (? ? ?d)
  and are associated with the dust particles
  inertia.
• Charged dust grains have a collective behavior
  and take part in the wave dynamic.
• Examples
• dust acoustic waves (DAW) and
• the electrostatic dust-cyclotron waves (EDCW).

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b) Electron and ion number in a dusty plasma is
not equal. This fact occurs because the dust
particles are charged. They get their electric
charge by electron and ion capture, electron
emission, and others charging processes. The
new quasi-neutrality condition of the plasma
where is dust charge in
the equilibrium and
for positive and
negative charged dust,
respectively.
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• The presence of dust particles can modify the
  propagation of waves modes.
• - Ion modes
• Modes of an electron-ion plasma modified via
  quasi-neutrality condition.
• These modes have low-frequencies ( ? ? ?i ) and
  are associated with the ion inertia.
• Examples
• dust-ion-acoustic waves (DIAW) and
• electrostatic dust-ion-cyclotron waves (EDICW).

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• c) The gravitational force has an important role
  in the dust particle dynamics because these
  particles have a large mass.
• Example modifications in the Jeans criterion of
  instability.
• d) The electric charge of the dust particle is
  not constant.
• The electric charge of dust grains is determined
  by electric potential of surrounding plasma
  environments and if a wave modify this potential,
  electric charge of dust grain will be affected.
• Since the dust charge varies in time, in general,
  it will be necessary to use a new variable, q,
  electric charge of grains, in order to describe
  the plasma fd fd ( r, p,t,q) .

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• e) The Debye shielding in a dusty plasma is
  different of one in an electron-ion plasma.
• In a dusty plasma, the electric charge shielding
  of dust
• particles, by the others plasma particles, in not
  exponential.
• This fact occurs due the existence of charging
  processes of dust particles.
• In these processes electrons and ions flow
  towards the grains, that results in an imperfect
  shielding of the dust particles.

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• f) The grain size is not uniform.
• There is a size distribution modeled, in
  general, by a power law in the dust particle
  radius.
• That implies in a continuous range of the
  charge-mass ratio of dust particles.
• Consequently the frequencies associated to dust,
  ?d and ?pd, assume different values in a
  particular band.

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2.2 Electric charge and dust charging processes

• A dust particle, in a plasma, is charged by
  different processes. This electric charge in not
  constant.
• The dust charge variation is an important
  characteristic that differ a dusty plasma from an
  electron-ion plasma.
• We can divide the charge variation of the dust
  particles in two different cases

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• a) Regular variations
• - The charge variation of dust particles is
  associated with spatial and temporal variations
  in the environment of the plasma parameters, like
  temperature and density of the electrons and
  ions, electric currents, etc.
• Spatial variation gradients effects of the
  plasma parameters (plasma inhomogeneities)
• Temporal variations charge variations
  associated with plasma oscillations.
• - The electric charge q, that a grain has in a
  particular time instant, is determined by equation

where I(r,q,t) is the total charging current the
reach the grain surface.
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The total charging current that reach the grain
surface is given by

• - The current Iext is associated to one or more
• of the following processes
• photoemission by incidence of ultraviolet
  radiation
• secondary electron emission by electron or ion
  impact
• thermionic emission
• radioactivity, etc.

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• - The currents I?(r,q,t) with ? e,i, are
  constituted by electrons and ions from plasma.
• The dust grains get their electric charge
  through inelastic collisions with these
  particles.
• Since, electrons have a higher mobility than
  ions, the grains in plasma will acquire a
  negative charge.
• In general, a summation of the processes
  included in Iext and I?(r,q,t) will determine
  the effective steady-state charge of the dust
  particle.
• In our work only I?(r,q,t) currents are included
  in the charging process.

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b) Stochastic variations (Charge fluctuations)

• In this case, electric charge variation of dust
  particles arise from stochastic nature of
  charging processes and from discrete character of
  electric charge.
• In equilibrium situation, the dust particle
  captures, in average, an equal electron and ion
  number by time unit, which implies in a constant
  mean charge.
• However, the capture of an electron by dust
  particle is not immediately following by the
  capture of a positive ion, that results in an
  instantaneous fluctuation of electric charge of
  dust particles.
• The stochastic nature of charging process of the
  dust particles must only be considered in the
  case of dust particles with a size about one
  nanometer.

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3. Astrophysical Applications

• We discuss the effects of the dust particles on
  the propagation and absorption of the Alfvén
  waves in
• 3.1) Stellar Winds
• 3.2) Star formation regions

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3.1 Stellar Winds

• - Alfvén waves in stellar winds
• Since the early observations of MHD waves in the
  solar wind various authors have suggested that
  Alfvén waves could be important to transfer
  momentum to the wind.
• The ? of, B, with the distance of the star, r,
  is smaller than the ? of the gas density, ?, ?
  the Alfvén velocity, increase with distance (vA
  B/(4? ?)1/2).
• The energy flux, per area unit, transported by
  the wave, ?M, is
• ?M ? ? vA ? (1/2) ?0 ??v2?
  vA,
• where ? is the energy density of the waves.
  This energy flux is constant, when there is no
  damping, or decreases due to damping, such that ?
  decreases with r.

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• Since the pressure associated with the wave is ?
  to ? then this pressure decreases with r. The
  result is a pressure gradient that accelerates
  the gas.

- Previous damping mechanisms studied

• The non-linear damping occur when two opposite
  modes interact, generating acoustic waves that
  accelerates the plasma.

• The resonant damping will occur at the surface
  of the tube, because of the gradient of density
  between the two mediums.

• Turbulent damping similar to the Kolmogorov
  turbulence.

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- The no-charged dust grain influence in an
Alfvén wave driven late-type winds

• P-Cygni profile for CaII K line is observed in
  late-type stars, indicating the presence of
  massive and cool winds.
• The observed wind terminal velocities are, in
  general, lower than the surface escape velocity (
  ).
• Authors have proposed several mechanisms for the
  wind acceleration, from radiation pressure to
  (MHD) waves. The most promising involves the
  damping of Alfvén waves.
• These works have been developed using a pure
  plasma wind, but observations confirms the
  presence of grains in these regions. The
  nucleation region is close to the sonic point of
  the wind.
• Since the presence of grains is important, the
  effect of grains presence on Alfvén waves damping
  must be evaluated.

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- The Model

• The model used is based on that presented by
  Jatenco-Pereira Opher (1989), where an Alfvén
  wave flux is responsible for accelerating the
  wind.

• Similar to the solar case, the wind has
  non-radial divergence geometry on its base
  becoming radial after a distance, called
  transition radius ( rt ). The cross section of
  the flux tube, showed in figure, is given by

• The wind equation solved is given by

where u is the wind velocity, ve the escape
velocity, vt the thermal velocity, MA the Mach
number and L the damping length of the wave.
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• Havnes, et al. (1989) studied the influence of
  grains in the Alfvén wave damping.
• They noted that the waves are significantly more
  damped in regions where grains exist.
• At this work we simulated the grain presence,
  inputing an exponentially decaying damping
  length. The used damping length is given
• where, A is a damping factor, and r1 is the grain
  formation region.

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- The grain presence region

• The damping by grains must be introduced only in
  the region where grains can exist. Observations
  can not show us where the nucleation occurs, but
  theoretical models can (Gail Sedlmayr (1984,
  1986, 1987).

The figure shows the nucleation region, as
function of the effective surface temperature and
the mass loss rate for two grain type. For cool
supergiant (T?2000K) we may use
ro lt r lt 2ro.
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- Results
We present the results of a simulation of grains
presence including an non-isothermal profile
applied to a K5 supergiant star, of M16M?,
ro400R?, S5 and ?o3.36x106 erg/s/cm2.
The wind velocity profile where the dotted line
is the Jatenco-Pereira and Opher pure plasma and
isothermal wind, and the filled line is the
presented model. The grain formation region is
showed also.
At the grain formation region, the sudden damp of
the waves causes the rapid deposition of
momentum, accelerating the wind to upper
velocities when compared to JPO model.
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- Conclusions
In this work we present a model of mass loss in
late-type stars, using a flux of Alfvén waves as
an accelerating mechanism. Grains presence are
simulated.
The model was applied to an K5 star, showing that
the sudden damping of the waves causes a local
acceleration of the wind. The results were
compared to Jatenco-Pereira and Opher (1989)
model.
The comparison shows important differences
between the models. Our model results in upper
velocities. The differences are not despicable,
and more studies in the future could improve this
model.
Our model was used to simulate the Betelgeuse
(?Ori) wind also. The results are in agreement
with recent observations of this red supergiant.
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• - Present work (variable charge of the dust
  particles)
• At the moment, we are considering the Alfvén in a
  magnetized dusty plasma with variable charge on
  the dust particles, in the context of the kinetic
  theory.
• In a paper in preparation, we consider the case
  of propagation of the waves exactly parallel to
  the external magnetic field and Maxwellian
  distributions for the electrons and ions in the
  equilibrium.
• We show that the presence of dust particles with
  variable charge in the plasma produces an
  additional damping of the Alfvén waves.

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3.2 Alfvén wave pressure against dusty molecular
cloud collapse (charged dust)

• The Interstellar Medium is plenty of giant
  structures, cold and dense, which main
  constituent is quasi-neutral matter as atoms,
  molecules and dust particles.
• This structures are called as Molecular Clouds.
• They are known as the main star formation
  regions.
• Dwarf Molecular Clouds (DMCs) are 5 pc lenght
  structures, and have masses of hundreds of solar
  masses.
• Observations indicates that most DMCs can live
  more than 108 years in equilibrium.
• However they are cold, T 10 20 K, and the
  thermal pressure could not support the
  gravitational collapse.

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• The Jeans mass
• for typical parameters of DMCs is 3 solar
  masses.
• Also, the free-fall time would be
• which gives 106 years for the typical
  parameters, much lower than the 108 DMC lifetime.

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- Stability Mechanisms

• DMCs present also magnetic fields of few
  µGauss.
• Magnetic field pressure can support
  gravitational collapse, however only in the
  perpendicular direction of its field lines.
• For the parallel direction of the field there is
  no explanation yet. Some of the main cadidates
  for the support are
• rotation, internal turbulence and MHD waves.
• Among the MHD waves, Alfvén waves propagate
  through the magnetic field lines in the parallel
  direction.
• Some authors advice the need for low damping
  mechanisms acting on the Alfvén wave flux, with
  damping lenghts greater than 1pc, to guarantee
  the stability in such DMCs lenghtscales.

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- Alfvén wave damping

• There are several damping mechanisms in the
  literature. In particular, ion-neutral
  collisional damping and the non-linear damping
  are the most used in such regions.
• Typical parameters, applied to the above
  mechanisms, confirm the low damping of Alfvén
  waves in DMCs.
• However, previous authors did not considered
  that DMCs are, actually, a Dusty Magnetized
  Plasma.
• In this case, dust particles can play an
  important role on the Alfvén waves dispersion
  relation.

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- The role of dust (size distribution)
Observations indicate that the distribution of
dust sizes in space is with p 3 4,
considering different dust compositions. For ex,
for graphite dust particles the radius range of
a 10-7 10-4 cm is obtained. These dust
particles are, in general, charged with Zd 101
103 e-. If dust particles are charged, they
interact with the waves, and give rise to a
dust cyclotron resonance damping. The dispersion
relation for a dust size distribution is
calculated by Cramer, N., Verheest, F.
Vladimirov, S. (2002)
where
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The imaginary part of k gives the damping lenght
of the wave. am is the ratio of
maximum and minimum dust radii. For am 1.3
the resonance band occurs for 0.59 ??/?dmax? 1.
In the interstellar medium am 102 and we
expect resonance band can be even larger
affecting almost all spectrum of low frequency.
(a) The real part of the wave number for am 1.1
(solid line) and am 1.3 (dotted line). (b) The
imaginary part of the wave number for am 1.1
(solid line) and am 1.3 (dotted line).
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- Cloud stability

• The model used is described below
• In this scheme, we show the propagation direction
  of the Alfvén waves and the
• way as they increase the cloud
  support against gravity.
• The temporal mean of the momentum equation for a
  cloud in mechanical
• equilibrium can be written by

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• Using a WKB aproximation, the wave energy density
  gradient will be given
• by
• , where
  is the wave damping lenght.
• The Poisson equation determines the gravity by
• Using also the wave frequency spectrum
• ,
• and defining
• as the ratio of the wave density energy and
  the gas energy density,

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The wave power spectrum, damped by dust-cyclotron
resonance, for different cloud locations z. We
note that the frequency band is almost completely
damped up to z 0.01 pc. These waves cannot
reach the boundareis of DMCs ( 1 - 10 pc).
Density profile as a function of distance for
different values of the parameter ?. Dotted line
represents equilibrium without Alfvenic support,
and the dashed, dot-dashed and solid lines
represent the cases of ? 0.05, ? 0.15 and ?
0.25, respectively. The sudden damping of the
wave flux results in compact and denser cloud
cores already observed in DMCs.
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- Conclusions
Typical dwarf molecular cloud parameters as nH2
104 cm-3 and T 20 K lead to the MJ of
3 solar masses, which is ?? Mcloud (Mcloud
100 M?). Considering B 10 µGauss, the cloud
stability may be reached, however only on
the perpendicular direction. Alfvén waves
propagating along B could provide an extra
pressure in the parallel direction. On the
other hand, DMCs presents high amounts of charged
dust particles, which interact with B and
provide a dust cyclotron damping mechanism
for the waves. Using this damping mechanism and
a particle size distribution, just like the
observed in the ISM, we show that the flux is
dissipated suddenly in a region ?? 1 pc much
smaller than the cloud size of 1 10 pc. The
sudden damping of the wave flux results in
compact and denser cloud cores already
observed in DMCs, which could be explained.
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Summary

• Most of the material in the universe is in the
  plasma state and it coexists with dust particles
  in many situations.
• Dust grains become charged if they are immersed
  in a plasma.
• The system composed of charged dusts, electrons
  and ions forms a so called dusty plasma.
• We have presented the results of some works in
  which we consider the presence of dust grains in
  the plasma and their effects in the propagation
  and damping of Alfvén waves.
• We have concentrated in two astrophysical
  problems

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• - 1) Stellar winds
• In the literature, several acceleration
  mechanisms of winds have been proposed. Among
  them, one of the most promising involves the
  damping of a flux of Alfvén waves.
• Models without dust
• Direct momentum transference from waves to
  plasma particles (electrons and ions).
• Models with no charged dust
• Damping associated with the collision of
  electrons and ions with neutral particles (dust
  grains)
• Models with charged dust (variable charge of
  dust particles)
• The presence of dust particles with variable
  charge in the plasma produces an additional
  damping of the Alfvén waves. A damping associated
  with the charge variation of the dust particles.

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• - 2) Star formation regions (size distribution of
  charged dust particles)
• When wave damping is not considered, the wave
  flux can support the cloud against gravity,
  preventing its collapse, as also pointed out by
  Martin et al. (1997).
• We have presented a model in which a flux of
  waves propagating in a dwarf molecular cloud is
  damped due to resonant interaction with dust
  charged particles.
• Taking into account this wave damping, discussed
  by Cramer et al. (2002), the flux is dissipated
  suddenly (in a region ltlt 1pc), leading to the
  formation of a compact and dense core.
• In this case, the waves could not reach the
  outer layers of the cloud, and if this is so,
  they could not be used to explain the size of
  these objects, although they could still be used
  to inhibit star formation.