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

Index

1. Introduction

2. The dusty plasma physics2.1 Dusty plasma versus electron-ion plasma2.2 Electric charge and dust charging process

a) Regular variationsb) Stochastic variations (charge fluctuations)

3. Astrophysical Applications3.1 Stellar winds3.2 Star formation regions

4. Summary

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.

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.

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

m

q

m

q

d

2/12204

d

ddpd m

eZncm

BeZ

d

dd

0

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.

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).

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:

0000 dnQnq

where is dust charge in the equilibrium and for positive and negative charged dust, respectively.

dd ZeQ 0

1,1 d

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).

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) .

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 dustparticles, 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.

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.

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:

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.

),(r, tqIdt

dq

The total charging current that reach the grain surface is given by:

.),,(),(r,,

tqIItqIie

ext r

- The current Iext is associated to one or moreof the following processes:

• photoemission by incidence of ultraviolet radiation;

• secondary electron emission by electron or ion impact;

• thermionic emission;

• radioactivity, etc.

- 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.

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.

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

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.

• 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.

• 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.

2/0evu

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

• 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:

S

O

O r

rrArA

)()(

• The wind equation solved is given by:

2

A

A2

T2

vM1

M31

8

1

2

v

2

u

dr

du

u

1

2

e22

A

AT

T

2T v

4

1v

L

r

4

1v

M1

M31

4

1

dr

dv

v

r1v

r

1

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.

- The Model

• 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.

A

rrexpLL 1

rr 1

- 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:

(T2000K) we may use

ro < r < 2ro.

We present the results of a simulation of grains presence including an non-isothermal profile applied to a K5 supergiant star, of M=16M, ro=400R, S=5 and o=3.36x106 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.

- Results

- 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.

- 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.

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.

• 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.

3s

21

0

23

3J0J cρ

G

πλρM

2

1

0ff 32Gρ

3πt

• 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.

- Stability Mechanisms

• 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.

- Alfvén wave damping

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

- The role of dust (size distribution)

paCaf )(

212 uukz

da

aa

af

svvu

ma

dAd

d

iA

i

1 2

4

2max

2

2max

2

220

2

20

2

1

)(

da

a

af

svvu

ma

dAd

d

iA

i

1 2

4

2max

2

max3

220

2

03

2

)(

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).

- 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: 0 gP

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, we can determine the density profile for the stability equation.

1)ln( LvA

ikL /2

00 )()( AA

int/U

Gg 4

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.

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.

- Conclusions

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:

- 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.

- 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 << 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.