\documentclass{article}

\begin{document}
\title{Magneto-acoustic Effects in Stellar Winds}
\author{ Danail Vassilev\\
	 \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}

 
\date{ }
\maketitle
\begin{abstract}

 Here we treat the outflowing material from the stars as a highly
 conductive plasma
in order to derive relevant equations governing the wind behaviour.
An estimation , based on above approach, has been made to determine the mass loss rate 
of a red giant star.
	 \\[1em]
\end{abstract}


\pagebreak
%\section{Abstract}\label{S:intro}
%Magnetoacoustic waves play an important role in stellar wind theories.
% Here we treat the outflowing material from the stars as a highly
% conductive plasma
%in order to derive relevant equations governing the wind behaviour.
%
%
%
%

\section{Introduction}

Stars emit not only radiation, but also particles.
 The emission of particles is called the stellar wind. 
The two most important parrameters regarding a stellar wind that
 can be derived from observations are mass loss rate $dM/dt$, 
which is the amount of mass lost by the star per unit time, and the 
terminal velosity $v_{term}$, which is the velosity of the stellar wind
 at a large distance from the star. By convention the mass loss rate 
$dM/dt$ is expressed in units of sollar masses per year.
\begin{equation}
M_\odot{yr}^{-1}=6.303\times{10}^{25}gs^{-1}
\end{equation}
A star with $dM/dt=10^{-6}M_\odot{yr}^{-1}$ which is not
 unusual value loses an amount equal to the total 
mass of the earth in three years. The terminal velosity $v_{term}$ 
of a stellar wind ranges typically from 10 km/s
 for a cool  supergiant star to  3000 km/s for a luminous hot star. If we
assume that the winds are stationary and spherically symmetric then 
the equation of continuity yields:
\begin{equation}\label{E:cong1}
dM/dt=4{\pi}{r^2}{\rho(r)}{v(r)}
\end{equation}
In the above equation $r$ is the distance from the center of the star,
 $\rho$ is the density and $v$ is the velocity. 

The values of $dM/dt$ and $v_{term}$ are important because:

(1)$dM/dt$ describes how much material is lost by the star per unit
 time. This is important for the evolution of the stars because stars 
with high mass loss rates will evolve differently from those with low 
mass loss rates.

(2)The gas that escapes from the star into space carries kinetic 
energy into the interstellar medium. The amount of kinetic energy that a
 stellar wind deposits into the interstellar medium per unit time is 
\begin{equation}\label{E:cong2}
E=\frac{1}{2}{v_{term}^2}\frac{dM}{dt}\ 
\end{equation} 
In order to study the effect of a stellar wind on its surrounding interstellar
 material E has to be known.

(3)Different stellar models predict different mass loss rates and different 
terminal velosities for a star. By comparing the observed values with the
 predictions we can learn which mechanism is responsible for the mass loss from 
a star.

Another important thing is that the stellar wind is treated usually as
 a high conductive plasma which has significant consequences.

\section{Alfven Model}

In hot stars the stellar winds are driven from the surface by the
 radiation pressure of the star.  However 
in order to make the picture more realistic we have to concider 
the effect of the magnetic field. Moreover some stars(red giants) have 
wind speeds of less than about 30km/s, which are well below the escape
speed from the surface of the star:
\begin{equation}\label{E:cong3}
v_{esc}=\sqrt\frac{2GM_\ast}{r_\ast}   
\end{equation}

In order to give an explanation to this paradox Hartmann and Mac Gregor 
 (1980) proposed that the winds are driven by Alfven waves, and the
 terminal speed is reduced by wave damping [1].

 Now we shall treat the wind 
as a highly conductive fluid in order to study the propagation of
Alfven waves. According to the Ohm's law we have:
\begin{equation}\label{E:cong4}
\vec{j}=\sigma(\vec{E}+\frac{\vec{v}}{c}\times\vec{B})
\end{equation}
or
\begin{equation}\label{E:cong5}
\vec{E}=\eta\vec{j}-\frac{1}{c}\vec{v}\times\vec{B}
\end{equation}
Combining (6) with the Maxwell equations we can find 
an equation for the temporal behaviour of $\vec{B}$:
\begin{equation}\label{E:cong6}
\nabla\times\vec{E}=-\frac{1}{c}\frac{\partial \vec{B}}{\partial t}
\end{equation}
\begin{equation}\label{E:cong7}
\frac{\partial \vec{B}}{\partial t}=-\nabla\times{c}\eta\vec{j}+
\nabla\times(\vec{v}\times\vec{B})
\end{equation}
\begin{equation}\label{E:cong8}
\frac{\partial \vec{B}}{\partial t}=\nabla\times(\vec{v}\times\vec{B})
-\frac{c}{4\pi}\nabla\times(\eta\nabla\times\vec{B})
\end{equation}
\begin{equation}\label{E:cong9}
\frac{\partial \vec{B}}{\partial t}=\nabla\times(\vec{v}\times\vec{B})
-\frac{c}{4\pi}\nabla\eta\times(\nabla\times\vec{B})+\frac{c\eta}{4\pi}
\nabla^2\vec{B}
\end{equation}
The second term on the RHS represents the effect of the spatial
 variation of resistivity on the currents generated
 by magnetic field. For simplicity we will now assume that the
 resistivity is uniform so that term disappear. 
\begin{equation}\label{E:cong10}
\frac{\partial \vec{B}}{\partial t}=\nabla\times(\vec{v}\times\vec{B})+
\frac{c\eta}{4\pi}
\nabla^2\vec{B}
\end{equation}
The first term on the RHS in the last equation is called 
the convective term and will be seen to represent the tendency of 
magnetic field to be 'frozen' in the fluid. The second term is
 diffusive and represents the resistive leakage of magnetic field 
lines across the conductive fluid. 

When the conductivity is infinite, $\eta$ tends to zero and therefore
\begin{equation}\label{E:cong11}
\frac{\partial \vec{B}}{\partial t}=\nabla\times(\vec{v}\times\vec{B})
\end{equation}
Using $\nabla.\vec{B}=0$ we get 
\begin{equation}\label{E:cong12}
\frac{d\vec{B}}{dt}=(\vec{B}.\nabla)\vec{v}-(\nabla.\vec{v})\vec{B}
\end{equation}
From continuity equation we get:
\begin{equation}\label{E:cong13}
\frac{d\vec{B}}{dt}=(\vec{B}.\nabla)\vec{v}+\frac{\vec{B}}{\rho}
\frac{d\vec{B}}{d\rho}
\end{equation}
\begin{equation}\label{E:cong14}
\frac{d}{dt}\frac{\vec{B}}{\rho}=(\frac{\vec{B}}{\rho}.\nabla)\vec{v}
\end{equation}
Let us now concider the evolution of a small vector $\delta\vec{x}$ 
connecting two neighbouring points in the fluid, as the fluid moves with
the prescribed velocity field. The point initially at position
 $\vec{x}$ at time $t$ will be displaced to the position $\vec{x}+v(\vec{x})\Delta{t}$
at time $t+\Delta{t}$. The point initially at  $\vec{x}+\delta\vec{x}$
at time $t$ will be displaced to the position  $\vec{x}+\delta\vec{x}+\vec{v}
(\vec{x}+\delta\vec{x})\Delta{t}$ at time $t+\Delta{t}$.Hence we find that
\begin{equation}\label{E:cong15}
\frac{d}{dt}(\delta\vec{x})=(\delta\vec{x}.\nabla)\vec{v}
\end{equation}
We see that $\frac{\vec{B}}{\rho}$ and $\delta\vec{x}$ satisfy the same equations for all time.
Hence if initially $\delta\vec{x}=\rho^{-1}\vec{B}$ this same 
relation will  hold for all time. This justifies the concept of the frozen field. Whenever 
the plasma starts to move, the magnetic field will be 'frozen' 
and follow the motion of plasma [2,3].   

We can conclude that in a perfectly conducting fluid the plasma particles behave as 
if they were tied to the magnetic field lines. So the field 
lines bahave as if they were mass-loaded elastic strings under tension. Thus by analogy 
with the transverse vibrations of elastic strings, we expect that
 whenever the conducting fluid is slightly disturbed from the equilibrium 
conditions, the magnetic field lines will
 perform transverse vibrations. The speed of propagation of these 
transverse vibrations are expected to be given 
by $v_A=(\frac{{tension}}{{density}})^{1/2}$ which is known to be the Alfven speed.

The apropriate equations needed to describe the wave motion of a
conducting fluid in the presence of magnetic field are:
\begin{equation}\label{E:cong16}
\frac{\partial\rho}{\partial{t}}+\nabla.(\rho\vec{v})=0
\end{equation}
\begin{equation}\label{E:cong17}
\rho\frac{\partial\vec{v}}{\partial{t}}+\rho(\vec{v}.\nabla)\vec{v}=-\nabla\rho-\frac{1}{4\pi}\vec{B}\times(\nabla\times\vec{B})
\end{equation}
\begin{equation}\label{E:cong18}
\frac{\partial \vec{B}}{\partial t}=\nabla\times(\vec{v}\times\vec{B})
\end{equation}
The two new equations are continuity equation (17)
and the force equation (18). We shall assume that equilibrium velocity is zero. In order to deduce dispersion relation for small amplitude 
waves let us consider small amplitude departures from the equilibrium values.
\begin{equation}\label{E:cong18}
\vec{B}(\vec{x},t)=\vec{B_0}+\vec{B_1}(\vec{x},t)
\end{equation}
\begin{equation}\label{E:cong19}
\rho(\vec{x},t)=\rho_0+\rho_1(\vec{x},t)
\end{equation}
\begin{equation}\label{E:cong20}
\vec{u}(\vec{x},t)=\vec{v_1}(\vec{x},t)
\end{equation}
\begin{equation}\label{E:cong21}
\frac{\partial^2\vec{v}}{\partial{t}^2}-{s}^2\nabla(\nabla.\vec{v_1})
+\vec{v_A}\times\nabla\times[\nabla\times(\vec{v_1}\times\vec{v_A})]
\end{equation}
where ${s}=\sqrt\frac{\partial{p}}{\partial\rho}$is the speed of sound and ${v_A}=\frac{B_0}{\sqrt{4\pi\rho_0}}$
is the Alfven speed. Assuming $\vec{v_1}(\vec{x},t)$ to be a plane wave
\begin{equation}\label {E:cong22}
\vec{v_1}(\vec{x},t)=\vec{v_1}e^{i(\vec{k}.\vec{x}-\omega{t})}
\end{equation}
equation (23) becomes:
\begin{equation}\label{E:cong23}
-\omega^2\vec{v_1}+({s}^2+\vec{v_A}^2)(\vec{k}.\vec{v_1})\vec{k}+\vec{v_A}.\vec{k}[(\vec{v_A}.\vec{k})\vec{v_1}-(\vec{v_A}.\vec{v_1})\vec{k}-(\vec{k}.\vec{v_1})\vec{v_A}]=0
\end{equation}
If $\vec{k}$ is perpendicular to $\vec{v_A}$ last term vanishes and
 we have a 
longitudinal magnetosonic wave with phase velocity $u_{long}=\sqrt{s^2+v_A^2}$.

If $\vec{k}$ is paralel to $\vec{v_A}$
\begin{equation}\label{E:cong24}
(\vec{k}^2.\vec{v_A}^2-\omega^2)\vec{v_1}^2+(\frac{s^2}{{v_A}^2}-1)\vec{k}^2(\vec{v_A}.\vec{v_1})\vec{v_A}=0
\end{equation}
In this case there are two modes: a longitudinal wave with phase
 velocity equal to the sound velocity $s$ and a 
transverse wave $(\vec{v_1}.\vec{v_A}=0)$
with phase velocity equal to $v_A$.
 The magnetic field for the last mode is $\vec{B_1}=-\frac{k}{\omega}B_0\vec{v_1}$

If the conductivity of the fluid is not infinite we expect to have damping of oscillations. 
For the case of interest when we have Alfven wave parallel to the field the dispersion relation becomes:
\begin{equation}\label{E:cong25}
k^2{v_A}^2=\omega^2(1+i\frac{c^2k^2}{4\pi\sigma\omega})(1+i\frac{k^2\chi}{\rho_0\omega})
\end{equation}
where $\chi$ is the fluid viscosity
In order to simplify the result we assume the correction terms to be small. In this limit 
\begin{equation}\label{E:cong26}
k=\frac{\omega}{{v_A}}+\frac{i(\xi_m+\xi_k)\omega^2}{2{v_A}^3}
\end{equation}
where $\xi_m$ and $\xi_k$ represent the magnetic and kinetic viscosities correspondingly. The positive 
imaginary part in this expression for $k(\omega)$ implies wave damping.
The amplitude of the wave is exponentially decreasing. We see that the attenuation
increases rapidlywith frecuencyand decreases with increasing the magnetic field intensity.

An expression allowing numerical estimation of $\frac{dM}{dt}$ based 
on the above model has been given in [1].   The mass loss rate
of a typical Alfven wave driven wind is:
\begin{equation}\label{E:cong27}
\frac{dM}{dt}=9.1\times{10}^{-17}\frac{B_0^4R_\ast^{7/2}}{M_\ast^{3/2}10^{13}\rho_0}M_\odot{yr^{-1}}
\end{equation}
with $\rho_0$ in units $gcm^{-1}$.

For a red giant star $R=300R_\odot$ and $B_0=1\times{10}^{-5}Gauss$, $M_\ast=1M_\odot$. The latter yelds:
\begin{equation}\label{E:cong28}
\frac{dM}{dt}=8\times10^{-7}M_\odot{yr}^{-1}
\end{equation}
The above result matches perfectly with the observed value 
of the mass loss rate of a red giant star. Curiously to say,
this estimation is of the same order as the rate of mass loss in a 
wind driven purely by a radiation pressure gradient. But unlike the
 radiation-driven wind theories, the model that equation (29) is based
 on explains adequately the low speeds of the red giants winds.
\section{Conclusion}
The Alfven wave driving wind mechanism discussed here and the radiation
driving mechanism are the most commonly occuring classes of wind 
mechanisms across the Hertzsprung-Russell diagram. Alfven waves are
a valuable addition to the list of mass loss mechanisms, because
the waves can produce a significant mass loss even in stars which have 
neither hot coronal zones nor strong radiative fluxes. Consequently, 
Alfven waves have been considered as a possible cause for the mass loss 
in every portion of the HR diagram, where the radiative driving mechanism 
is ineffective [1]. As it was proposed by Hartmann and Macgregor, the 
winds from the red giants are driven by Alfven waves and the wave damping 
due to the finite conductivity of the plasma is responsible for the 
reduced terminal speed of the wind.

\begin{thebibliography}{99}
\bibitem{refA}Henry J.G.L.M. Lamers, Joseph P. Cassimelli,1999, 
 Introduction to Stellar Winds 
%\section{Reference}\label{S:Ref}
%1.Henry J.G.L.M. Lamers, Joseph P. Cassimelli,1999, 
% Introduction to Stellar Winds
\bibitem{refB}Peter A. Sturrock,1994,Plasma Physics
%2.Peter A. Sturrock,1994,Plasma Physics
\bibitem{refC}J. D. Jackson,1975, Classical electrodynamics
%3.J. D. Jackson,1975, Classical electrodynamics
\end{thebibliography}
\end{document}