A&A 482, 793802 (2008)
DOI: 10.1051/00046361:20079043
Magnetohydrodynamic model of equatorial plasma torus
in planetary nebulae
K. H. Tsui
Instituto de Física  Universidade Federal Fluminense Campus da Praia Vermelha, Av. General Milton Tavares de Souza s/n Gragoatá, 24.210346, Niterói, Rio de Janeiro, Brasil
Received 9 November 2007 / Accepted 29 January 2008
Abstract
Aims. Some basic structures in planetary nebulae are modeled as selforganized magnetohydrodynamic (MHD) plasma configurations with radial flow.
Methods. These configurations are described by time selfsimilar dynamics, where space and time dependences of each physical variable are in separable form. Axisymmetric toroidal MHD plasma configuration is solved under the gravitational field of a central star of mass M.
Results. With an azimuthal magnetic field, this selfsimilar MHD model provides an equatorial structure in the form of an axisymmetric torus with nested and closed toroidal magnetic field lines. In the absence of an azimuthal magnetic field, this formulation models the basic features of bipolar planetary nebulae. The evolution function, which accounts for the time evolution of the system, has a bounded and an unbounded evolution track governed respectively by a negative and positive energy density constant H.
Key words: magnetohydrodynamics (MHD)  methods: analytical  stars: AGB and postAGB  ISM: planetary nebulae: general
Planetary nebulae are believed to be intermediatemass stars
that are expelling their outer layers, at slow velocity,
during the asymptotic giant branch (AGB) phase.
As the temperature of the star increases, the ejected material
becomes tenuous but moves with increasing speed.
The hydrodynamic interaction of these slow and fast winds,
at the intersection of their trajectories, generates shock waves
that create the characteristic morphologies of planetary nelulae
(Dyson & de Vries 1972; Kwok et al. 1978).
The outwardshock compresses and heats the slow dense wind
that is ahead, and the inwardshock decelerates and heats the fast wind behind.
This interactingwind model reproduces the spherical features
of planetary nebulae. In contrast, the elliptical and bipolar features may be created
by a dense toroidal cloud in the equatorial plane
(Kahn & West 1985; Mellema et al. 1991). Although toroidal clouds are imaged by highresolution instruments, how such structures formed during the AGB phase is still a debated issue. Besides spherical, elliptical, and bipolar features,
highresolution images have revealed pointsymmetric features
(Miranda & Solf 1992; Lopez et al. 1993; Balick et al. 1993).
These pointsymmetric features, plus the detection of significant
magnetic fields inside central stars (Jordan et al. 2006),
appear to call for a magnetohydrodynamic (MHD) approach
to the study of planetary nebulae
(Pascoli 1993; Chevalier & Luo 1994; GarciaSegura 1997;
Bogovalov & Tsinganos 1999; Matt et al. 2000; Gardiner & Frank 2001).
In this paper, we present such an MHD analysis,
which is able to recreate axisymmetric structures
such as an equatorial plasma torus and a bipolar nebula.
We consider an MHD plasma that simulates a radialflow explosion
in spherical coordinates, and we solve for solutions that are
selfsimilar in time (Sommerfeld 1950; Landau & Lifshitz 1978).
Selfsimilar MHD analysis was pioneered by Low (1982a,b, 1984a,b),
in astrophysical stellarenvelope applications.
The association of selfsimilar solutions to eruptive phenomena
can be best demonstrated by the generation of selfsimilar shocks
in powerful atmospheric detonations. The shocks are the results
of selforganization, through dissipative processes, following chaos.
The shock is built up as the gas expands outward, and
it takes its fully developed form beyond a certain radius.
These ordered structures could have a scale invariance
that lowers the dimensionality of the timedependent fluid system,
and could exist between one spatial coordinate and time.
This invariance is identified as the selfsimilar parameter,
and similarity is temporal.
Time evolution would be a separable factor, and, as a result,
the time and space dependencies of selfsimilar
variables are represented in separable form.
For steadystate systems, the scale invariance could be between
one spatial coordinate and another,
and the similarity is said to be spatial.
Selfsimilarity is often regarded as a method to derive
specific types of timedependent or steadystate solutions.
In this paper, we consider selfsimilarity to be a property
of selforganized states, at least some of which are derived from turbulence.
It is essential to note that the appearance of selforganized
states, in fluids and magnetofluids, owes itself to the existence
of multiple quadratic invariants in the absence of dissipations
(Hasegawa 1985; Biskamp 1993). In the case of MHD fluids, the invariants are the total energy density (magnetic, plasma, and thermal), magnetic helicity,
and in the case of incompressible fluids, crosshelicity.
In the presence of dissipation, these invariants could undergo
constrained changes, for example, from an MHD configuration,
permitted by the MHD equations, to another topologically accessible
configuration of lower energy. It is important to point out
that these configurations are isolated in ideal cases.
The continuous topological transformation from one to
another configuration is only possible through dissipative paths
such as magnetic reconnections. Mathematically speaking, this evolution amounts
to the application of the variational principle,
on the total energy density, under the constraint of
global magnetic helicity conservation (Hasegawa 1985).
The exact processes of dissipation,
in taking the system to selforganization,
need not be specified under the variational description.
For this reason, MHD systems have the tendency to develop
selforganized states with structural stability.
Since the selfsimilar method solves for the end configurations,
directly from the governing equations,
there is no need to specify the initial conditions.
The plasma would find its way through magnetic reconnection,
and other dissipative means,
to reach one of the possible end configurations.
The only condition is that the initial configuration
is dissipatively transformable to the final configuration.
If the initial conditions are close to those of the end
configuration, the time to reach the end configuration is brief.
Naturally, there are many end configurations, and only one of them
will be selected by dissipative reorganization. For this reason,
we cannot predict which selfsimilar state the system will reach.
We can only match a system to a given selfsimilar solution
that bears some resemblance.
One of the fundamental obstacles to visualizing selfsimilar
solutions is an attempt to associate a given initial
configuration with a specific selfsimilar end solution.
It is possible to predict the final state of ideal MHD systems, in particular, by iterating equations of motion for an initial configuration. In the presence of dissipations, it can, however, be difficult to predict the final state, because of intermediate, chaotic
events that occur while states are being reorganized.
The initial configuration could undergo topological changes,
through dissipative processes, such as magnetic reconnections,
subject to the constraint of quadratic invariants.
To describe the selforganized states arising from chaos,
we have to override the forward time iteration process
and attempt to identify the end configurations directly
from the governing equations of the system.
Although these end configurations have originated from
dissipative reorganization through chaos,
it is important to emphasize that these end configurations
are determined by the ideal governing equations.
Ideal MHD cannot, however, determine the dissipative path
taken to reach one configuration from another.
As a consequence, we are unable to recover the initial
conditions by following the selfsimilar configurations
backward in time. The initial conditions are lost from memory.
Recently, time selfsimilar analysis was applied to represent
extragalactic jets as a polarlaunched ejection event
(Tsui & Serbeto 2007), which differs from the classical accretionejection space
selfsimilar steady state MHD transport model (Blandford & Payne 1982).
A selfsimilar description has been applied to different
problems in physics, such as atmospheric ball lightnings
(Tsui 2006; Tsui et al. 2006), and interplanetary magnetic ropes
(Osherovich et al. 1993, 1995; Tsui & Tavares 2005).
Here, we follow the methods of time selfsimilar MHD analysis
to model plasma torus, by developing solutions that are launched
onto the equatorial plane.
The basic MHD equations, in Eulerian fluid description, are given by

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 
where
is the mass density,
is the bulk velocity,
is the current density,
is the magnetic field,
p is the plasma pressure, is the free space permeability,
is the polytropic index, and M is the central mass,
which provides the gravitational field.
We consider a radially expanding plasma, and look for selfsimilar
solutions in time, where time evolution is described by the
dimensionless evolution function y(t). For this purpose, it is most convenient to think of Lagrangian fluid description, and consider the position vector of a given
laminar flow fluid element .
Under selfsimilarity, the radial profile is time invariant
in terms of the radial label
,
which has the dimension of r. Physically, is the Lagrangian
radial position of a fixed fluid element.
With a finite plasma, the domain of
is bounded by mass
conservation
As for the plasma velocity, we consider selfsimilar structures
deriving from a spherically symmetric radial velocity that
can be written as

(7) 
Our selfsimilar parameter ,
defined through the Lagrangian fluid label, explicitly represents
the fluid velocity by the time evolution function y(t). This evolution function will be solved selfconsistently with respect to the spatial structures of the plasma.
We emphasize that selfsimilarity, as a method, can be
applied in different ways other than the one we use here.
For example, Lou and his collaborators have treated an
aggregating fluid under its selfgravitational field with a
similarity variable x=r(t)/at, where a is the sound speed,
for isothermal fluid, ,
(Lou & Shen 2004; Bian & Lou 2005)
and for a polytropic gas, ,
(Lou & Wang 2006; Lou & Gao 2006)
to study relevant astrophysical phenomena.
Extensions to a magnetofluid have been considered by Yu & Lou (2005) and by Lou & Wang (2007). Because of the linear dependence on time, this similarity
variable x refers to a reference frame moving at speed a,
which is different from the radial plasma flow velocity v.
Furthermore, different from our similarity variable ,
x here is not the Lagrangian label of a given fluid element.
For this reason, the convective derivative remains explicit
in the x representation. As a result, the similarity variable of Lou amounts to finding
the plasma structures in an adequate moving frame in the Eulerian x fluid description.
This resembles the analytic technique of going to a moving frame
to look for stationary profile solutions for nonlinear phenomena
such as nonlinear Alfven waves, solitons, etc.
Because of this fundamentally different definition of
selfsimilarity, the nature of the phenomena intended to
describe is different. Our Lagrangian label selfsimilarity parameter is aimed to find
spatial plasma configurations, and to determine the radial plasma
flow velocity, consistent to the spatial configurations,
through the evolution function.
Since we are considering an isotropic radial plasma flow,
a natural solution would be a hydrodynamic onedimensional
expanding plasma, with radially dependent mass density and plasma pressure p, and with
and .
Nevertheless, this onedimensional hydrodynamic solution
is highly unlikely because magnetic field fluctuations
can be generated from current density fluctuations,
even in the absence of a preexisting magnetic field.
With the magnetic fields, which are basically a two or
threedimensional structure, coupling to the plasma
will generate likewise two or threedimensional
and p.
To examine the possible spatial structures of MHD plasmas,
we consider a twodimensional case with azimuthal symmetry in .
In this case, the magnetic field, through the vector potential ,
can be expressed as
This enables the magnetic field, the current density, and the electric field, to be expressed
as two scalar functions P and Q respectively

(8c) 
The independent variables are transformed from
space to
space. We determine the explicit
dependence of y on each one of the physical variables
with this radial velocity using functional analysis.
First, making use of Eq. (7), Eq. (1) renders
To reach the second equality, we note that the first bracket,
in the first equality, corresponds to the total time derivative
of an Eulerian fluid element, which amounts to the time derivative
of a Lagrangian fluid element. As for the second bracket, it can be reduced by using
and
.
Solving this equation for y scaling, by separating the time part,
gives

(9b) 
As for Eq. (6), with
where
is a constant that carries the physical dimension so that F is
a dimensionless function, it follows

(10a) 

(10b) 
As for Eq. (3), using the representation of Eq. (8a), the magnetic
functions P and Q are

(11a) 

(11b) 

(12b) 
We have reduced the general set of timedependent ideal MHD
equations, Eqs. (1)(6), to a set of selfsimilar equations with
appropriate time scalings, Eqs. (7)(12), and will seek solutions
of Eq. (2) and the time evolution function y(t).
The general ideal MHD set of equations has nonlinear terms
of the convective type
in Eqs. (1) and (2),
and interaction type
in Eq. (2)
and
in Eq. (3). By using the fluid label description, the
convective terms are absorbed into the Lagrangian time derivative
representation. The
interaction term is also absorbed,
through an adequate representation of the magnetic field,
by scalar functions P and Q. The structure of the nonlinear terms,
absorbed into the Lagrangian fluid label formulation,
will appear in the profile of the system.
The remaining task is to solve Eq. (2) for the selfsimilar structure, under the presence of the
interaction term.
We note that the equation of
and
the equation of P are of the same form, and conclude that F=F(P)is a functional of P only, or
.
The fact that two equations have the same form does not
automatically imply that the solution of one equation
is a functional of the solution to the other.
We can make this assertion only when considering selfsimilar solutions.
We proceed to write the
dependences in terms of
in
and ,
to obtain

(13a) 
p 
= 



= 

(13b) 
Furthermore, since
is a function
of P only, the
and dependences should be in a
separable form in both
The dimensions of mass density and pressure appear explicitly
in
and p_{0} respectively. In this form,
,
,
,
and
are dimensionless functions.
By taking
with the physical
dimension, we have

(14a) 

(14b) 
such that
should be a functional of
only.
Making use of Eq. (4) to eliminate the current density in Eq. (2), we derive the momentum equation, which has three components. The component, which contains only the magnetic force,
gives



(15) 

(16) 
The function P plus the functional form of Q, therefore,
determines the magnetic fields. As for the component,
with
,
it reads



(17) 
We remark that the first three terms of this equation represent the
nonlinear forcefree field equation with
.
In the particular case of a linear functional
,
this equation describes the linear forcefree fields,
which can be verified easily from Eqs. (8) with
.
Such a forcefree magnetic configuration is the marker of
selforganized plasmas, and is the result of the variational
principle that minimizes the energy density of the magnetic field,
under the constraint of global magnetic helicity conservation.
This justifies the use of selfsimilarity method to describe
selforganized configurations.
The last term of Eq. (17) is the plasma pressure term.
For an isotropic pressure independent of ,
this last
term would be null and such plasma pressure
has no effect on the magnetic field which remains forcefree.
For an axisymmetric pressure
,
the above equation would be independent of the time evolution
function y should
,
and it would describe
a plasma pressure balanced axisymmetric magnetic field.
Such a pressurebalanced selforganized configuration could be
derived using the variational principle,
by minimizing the energy density that contains
not only magnetic energy but also plasma thermal energy
under the same constraint.
The solution of this equation of P, with different representations
of
and
,
is the core
problem of selfsimilar description.
As for the r component of the momentum equation, it reads
The term
on the left side refers to the total radial
derivative of the plasma pressure, which includes both an explicit and implicit dependence in P.
The right side of the radial equation provides an expression
for the magnetic force. Making use of the meridian component, Eq. (17), the right side
is equal to the implicit part of the radial pressure gradient,
which cancels the same term on the left side.
This leaves only the explicit radial pressure gradient







(18b) 
In terms of selfsimilar parameters, the radial equation reads

(19) 
With
as the separation constant,
as a
dimensionless constant, and
,
we derive

(20b) 

(20c) 
The first of these three equations provides the mass density
profile of the radial label. The second one specifies the
dependence between mass density and plasma pressure.
The third one defines the time history of the evolution function.
After describing our selfsimilar formulation, we solve Eqs. (14), (17), and (20), for axisymmetric plasma configurations. We begin with comparing Eqs. (14b) and (20b), and conclude that

(21) 
The solution to Eq. (17), for the magnetic field function ,
depends on the functionals
and
.
This is where different types of selfsimilar configurations
can be produced. Although Low pioneered the use of selfsimilar MHD in astrophysics,
he did not explore the full potential of the method.
We remark that selfsimilar solutions are highly dependent on
the functional representations of
and
.
Under a given geometry, there are different selfsimilar solutions,
with different characteristics to represent different selforganized plasmas,
depending on how the magnetic fields and plasma parameters are modeled.
To study configurations on and about the equatorial plane, we consider

(22a) 

(22b) 
For example, if we assume a quadratic function to have
we derive a polar ejection configuration that tends
to collimate magnetic fields and plasma density, onto the polar axis,
as plasma pressure builds up (Tsui & Serbeto 2007).
Using these functional representations in Eq. (17), we derive



(23) 
To solve this equation, we separate the variables by writing
.
Assuming
and n(n+1) to be the separation constant, we have

(24a) 

(24b) 
The first equation provides the wellknown solution,
in terms of the Legendre polynomial P_{n}(x),

(25) 
We note that the second equation is not in a separable form.
Nevertheless, should we choose n=1 with
P_{1}(x)=x,
such that
,
where we assume that C_{0}=1,
Eq. (24b) is separable. The general solution is given by a homogeneous solution
and a particular solution

(26) 
where
with n=1 is the spherical Bessel function regular at .
The coefficient A_{0} of the homogeneous solution carries the
amplitude and dimension to reflect the magnetic field through Eq. (8a).
Since plasma pressure and mass density are expressed in terms of
P via
and
,
we could normalize the entire selfsimilar solution,
with respect to the magnetic field, by taking A_{0}=1.
With
given by Eq. (20a), denoting ,
and
,
the particular solution is described by
We note that the coefficient A, which has two factors, is dimensionless.
The first factor
is related
to the plasma ,
which is the ratio of plasma pressure to magnetic pressure.
The second dimensionless factor with GM is basically the ratio of
the gravitational potential energy density to the plasma pressure.
The right side of this equation can be expanded into binomial terms,
and can be solved using power series, by taking one term at a time.
The particular solution is, therefore, a superposition
of five subsolutions R_{1(12)}, R_{1(9)}, R_{1(6)},
R_{1(3)}, and R_{1(0)}, where the bracketed number in the subscript denotes the power
of z in the binomial term, on the right side.
We denote the coefficients
A^{(12)}=A,
,
,
,
.
We solve each binomial term of the particular solution
using a power series of the form
Using standard powerseries techniques, we derive solutions of the form
R_{1(12)}(z) = a_{2}z^{2}+a_{4}z^{4}+a_{6}z^{6}+a_{8}z^{8}
+a_{10}z^{10},

(27a) 
where
a_{10}=A^{(12)},
,
,
,
,
and
,
R_{1(9)}(z)
= a_{9}z^{9}+a_{11}z^{11}+a_{13}z^{13}+a_{15}z^{15}
+ .....,

(27b) 
where
,
,
,
,
,
R_{1(6)}(z)
= a_{2}z^{2}+a_{4}z^{4},

(27c) 
where
a_{4}=A^{(6)},
,
and
2)a_{2} =0,
R_{1(3)}(z) = a_{3}z^{3}+a_{5}z^{5}+a_{7}z^{7}+a_{9}z^{9}
+ .....,

(27d) 
where
,
,
,
,
.
As for
R_{1(0)}(z), this solution has to be solved with series
of negative powers, which describes the divergent nature of the
gravitational field
at ,
although this limit is irrelevant here. We derive

(27e) 
where a_{0}=0,
a_{2}=A^{(0)},
,
,
,
.
We note that the homogeneous solution corresponds to forcefree
magnetic field configurations. Due to the plasma pressure in Eq. (17),
there are particular solutions that maintain the pressure balance.
After solving for the spatial structure of ,
the magnetic fields are given by
B_{r} 
= 



= 

(28a) 
The selfsimilar homogeneous solution
,
provided by Eq. (26), enables oscillations in
to be modeled,
and vanishes at the zeros of the spherical Bessel function.
To understand the magnetic structures,
we first set aside the particular solution
.
At the first zero of
,
we have
,
,
and the only nonvanishing
field is
according to Eqs. (28).
The meridian selfsimilar solution ,
given by Eq. (25), oscillates in x.
Together they describe the magnetic fields of Eqs. (28).
Within this region of ,
the topological center,
defined by
and
,
has B_{r}=0,
,
and the only nonvanishing
field is
.
This is the magnetic axis.
The field lines about the magnetic axis are, therefore, given by
By axisymmetry, the third group is decoupled from the first two groups.
For the field lines on an
plane,
we consider the first equality between B_{r} and
,
which gives

(29a) 
The nested field lines are given by the contours of
on the
plane.
At the topological center, the magnetic axis, we have
at its maximum value,
and
at its maximum, such that
takes its maximum value.
Since

(29b) 
by Eq. (28c), the line integral of
on the magnetic axis
is a maximum. We, therefore, have a sequence of plasma tori,
due to the periodic nature of the spherical Bessel function.
The presence of the particular solution
modifies
this structure. Due to the divergent nature of
,
as we will see in the next section, only the first torus prevails.
As for the temporal part, the evolution function in Eq. (20c) is described by

(30a) 

(30b) 
Here, H is an integration constant that is independent of time.
To understand the meaning of ,
we note that,
using Eq. (20c), plasma acceleration in Lagrangian coordinates is

(31a) 
A positive
corresponds to an outward, decelerating flow.
The deceleration becomes smaller as y, or as r, becomes larger,
and
equals the intensity of the deceleration,
for a given radial label .
To derive an expression for H, we multiply Eq. (30a) by
,
and use Eq. (31a) to obtain
The first term on the right side is the kinetic energy density of
the fluid element, and the second term is its work on expansion,
due to the explicit part of the pressure gradient.
We note that the magnetic energy does not appear in this expression
because it is cancelled by the work due to the implicit pressure
gradient, as discussed in Eq. (18a).
It is clear that H measures the total energy of the fluid element.
We consider the explicit radial dependent part of the mass
density, in Eq. (20a). Since
is positive,
should also be positive. Differentiating with respect to
indicates that
has a minimum at

(32) 
Because we are considering a finite plasma, the range of the radial
label
is finite. We assume that
is enclosed in the interval
.
For
,
increases
because of the
term, where
is defined by Eq. (20c)
and measures the inertia of the plasma.
A positive
corresponds to a decelerating plasma.
Beyond the external bound is interstellar space,
where plasma mass density decreases abruptly.
In the presence of the gravitational term, ,
in Eq. (20a),
the mass density
rises again for
.
This gravitational term is singular at .
However this singularity is not included,
because the internal bound of
is at the surface of the star.

Figure 1:
The function
R_{0}(z)=A_{0}zj_{1}(z) of the
homogeneous solution is plotted as a function of
to show the radial domains of the axisymmetric plasma
structures. 
Open with DEXTER 
To consider the function
,
the homogeneous solution
,
with A_{0}=1, is shown in Fig. 1, with the first node at
.
In the absence of plasma pressure,
would be the only
term for .
With
,
this would
correspond to the equatorial plasma torus forcefree magnetic fields.
As for the particular solution
,
we note that
R_{1(12)}(z),
R_{1(9)}(z),
R_{1(6)}(z), and
R_{1(3)}(z)diverge as z goes to infinity, whereas
R_{1(0)}(z) diverges as z goes to zero.
These two extremes are naturally excluded, because z=0 is the center of the central star,
and
requires an infinite plasma, which is unphysical.
An adequate domain could be provided by the homogeneous solution
plotted in Fig. 1, which shows an axisymmetric equatorial plasma torus with 0<z<4.5,
if the particular solution does not significantly alter this structure.
The particular binomial solutions depend on the
parameter
,
where a is a measure of the azimuthal magnetic field given by Eq. (22a).
We assume the minimum
,
such that
.
We note that the two factors (brackets) in
in the equation of
are dimensionless quantities.
The first factor
can be expressed in terms of plasma ,
which is the ratio of the plasma pressure to the magnetic pressure.
This factor is estimated to be 10^{1}.
The second factor is basically the ratio of the gravitational potential
energy density
to the plasma pressure p_{0},
plus other dimensionless multipliers to the fourth power.
We believe that the gravitational potential energy density
and the plasma pressure are approximately similar.
We could take
,
and the remaining terms
would provide the factor 10^{6}. Multiplying the two factors together gives A=10^{7}.

Figure 2:
The power series solution
R_{1(12)}(z) of the
particular solution, with
,
is plotted as a function of z to show the plasma pressure effect on the radial domains
of the axisymmetric plasma structures. 
Open with DEXTER 

Figure 3:
The power series solution
R_{1(9)}(z) of the
particular solution, with
and
,
is plotted as a function of z to show the plasma pressure effect on the radial domains
of the axisymmetric plasma structures. 
Open with DEXTER 

Figure 4:
The power series solution
R_{1(6)}(z) of the
particular solution, with
and
,
is plotted as a function of z to show the plasma pressure effect on the radial domains
of the axisymmetric plasma structures. 
Open with DEXTER 

Figure 5:
The power series solution
R_{1(3)}(z) of the
particular solution, with
and
,
is plotted as a function of z to show the plasma pressure effect on the radial domains
of the axisymmetric plasma structures. 
Open with DEXTER 

Figure 6:
The power series solution
R_{1(0)}(z) of the
particular solution, with
and
,
is plotted as a function of z to show the plasma pressure effect on the radial domains
of the axisymmetric plasma structures. 
Open with DEXTER 
Of the five parts of the particular solution R_{1},
R_{1(12)} is unaffected by the choice of
,
and is plotted in Fig. 2. Other binomial parts R_{1(9)}, R_{1(6)}, R_{1(3)},
plotted in Figs. 35 respectively,
carry mixed contributions of plasma pressure and gravitational field,
and are negligible within the domain.
The last part R_{1(0)} in Fig. 6 is purely gravitational.
The neighborhood of the center of the star, z=0, is excluded.
These binomial parts depend on the choice of
,
but would have a negligible effect on R_{0},
within the domain of interest. The homogeneous and the particular solutions
provide the complete solution for .
Together with ,
they determine the function
,
which governs the entire axisymmetric selfsimilar profile
in magnetic fields, plasma pressure, and mass density.
To consider the magnetic field profiles, we recall that the field
lines on an
plane are given by Eq. (29a), with
R(z)=R_{0}(z)+R_{1}(z).
The presence of a plasma gives a diamagnetic effect,
as shown by
R_{1(12)}(z) in Fig. 2,
carrying an opposite sign with respect to R_{0}(z) of Fig. 1.
Examining Fig. 2, we note that R_{1(12)} vanishes at z=4.5and z=5.2. The first node of R_{1(12)} happens to be at the
same location as the first node of R_{0}, also at z=4.5.
Furthermore, Figs. 36 show that R_{1(9)}, R_{1(6)},
R_{1(3)}, R_{1(0)} are all negligibly small.
Consequently, the general solution
R(z)=R_{0}(z)+R_{1}(z)=0at z=4.5. By Eqs. (28), we conclude that the radial and azimuthal
magnetic fields, B_{r}(z) and
,
vanish at this location, while the meridian magnetic field,
,
is maximum. Consequently, the domain 0<z<4.5 constitutes a region for a
magnetic torus with closed field lines. Since plasma is tied to the magnetic field lines,
it circulates along and is confined within this torus.
We note that the sign of R_{1} is negative,
which indicates the diamagnetic nature of plasma.
We note that R_{1(12)} has a maximum at the same location as R_{0}.
As a matter of fact, the functional form of R_{1(12)} in the
interval (0,4.5) is much the same as that of the forcefree R_{0}.
Consequently, the general solution R is practically forcefree,
even when plasma pressure is taken into account, in Eq. (23).

Figure 7:
The magnetic field lines using the homogeneous
solution R_{0}(z) are shown with C=+0.05, C=+0.20, C=+0.40,
C=+0.60, C=+0.80, and C=+1.00 from the outer to inner
contours respectively. The axes are labelled in
,
and
the contribution of the plasma pressure represented by R_{1}(z) is neglected here. 
Open with DEXTER 
For simplicity, we neglect the diamagnetic effect,
and take
R(z)=R_{0}(z) to highlight the poloidal field lines in Fig. 7,
which is described by first equality of the field line equation,
with solutions in Eq. (29a). The addition of
R_{1(12)}(z) would change the contours slightly,
but not the topology. Apart from these poloidal fields,
there are associated toroidal fields given by Eq. (28c).
The maxima of R and ,
and
,
give the maximum of this
toroidal field where B_{r}=0 and
respectively.
This corresponds to the magnetic axis of the plasma torus.
Together, the field lines thread out nested magnetic surfaces
enclosing the magnetic axis, one within another.
The plasma pressure and mass density of this plasma torus have
R(z)>0. With
,
this warrants
to assure positive pressure and mass density.
With
in Eq. (22b), the plasma pressure and mass density
vanish at z=4.5 since R(z)=0.
It is important to note that, for z>4.5, the particular solution R_{1}(z) dominates the homogeneous solution R_{0}(z) such that
the general solution R(z) never vanishes.
The radial component of the magnetic field is nonzero.
For this reason, the field lines are open to outer space,
and the plasma is no longer confined.
This outer open plasma structure is not shown in Fig. 7.

Figure 8:
The asymptotically bounded evolution function,
the lower curve, is plotted as a function of the normalized
time
with
,
whereas the unbounded
evolution function, the upper curve, is plotted with
. 
Open with DEXTER 
For the evolution function, it is most convenient to write Eq. (30b)
in the normalized form

(33) 
with
as the normalized time.
We assume that
is positive, and take y(0)=1to be the initial value, such that the Lagrangian label corresponds to the initial position of the Eulerian fluid element.
Equation (33) indicates an asymptotically convergent solution,
with H=H<0 negative, which is shown in the lower curve
of Fig. 8 with
.
This track represents an asymptotically stable radius
for the equatorial plasma torus. On the other hand, if the energy density H=+H>0 is positive,
there is an ever expanding evolution track with a terminal
velocity
.
This is shown in the upper curve of Fig. 8, which is inappropriate for AGB slow winds.

Figure 9:
The explicit radial dependence of pressure profile
is shown
as a function of the normalized radial label
through the thick line. The explicit radial dependence of mass density profile
is also shown through the thin line for comparisons. 
Open with DEXTER 
To better understand the amplitude and the sign of H, we multiply Eq. (30a) by
,
and integrate to derive

(34) 
The first integral on the right side is positive definite,
and corresponds to the kinetic energy of the expanding structure.
Inside the second integral, there is a plasma pressure term,
and a gravitational term. The gravitational term is always positive.
Within the plasma pressure term, we have
for
,
since
has a similar profile as
,
according to Eq. (14a). This part has a forward plasma pressure force.
For
,
we have
,
and the corresponding
plasma pressure force is backward. It is clear that the explicit part of the pressure profile
,
which is described by the explicit part of the mass density
profile
in Eq. (14a), contributes to the sign of H.
Rewriting Eq. (20a) as

(35) 
and assuming that the coefficient on the right side has a value of unity,
these two profiles are shown in Fig. 9, where we have taken
or
z=3z_{*}=6 as the upper bound in Fig. 9, as an example.
The pressure profile in the positive gradient region
helps to cancel the effects of the negative gradient,
over the region
,
in the above equation.
Assuming that the slow wind velocity is below the escape velocity,
the first integral on the right side is less than the gravitational
term of the second integral. This implies that H is negative,
due to the overall negative sign on the second integral.
The plasma torus is, therefore, asymptotically stationary.

Figure 10:
The full, explicit and implicit, radial mass density
profile
using the homogeneous solution R_{0}(z) is shown as a function of z. 
Open with DEXTER 
When deriving the full radial profile of the plasma pressure,
we caution that Fig. 9 is only the explicit part,
,
of the radial profile relevant to Eq. (34).
Another radial contribution comes from the implicit part,
.
The full radial pressure profile is a combination of the
explicit and the implicit part,
,
where
.
Since
by Eq. (14a)
and
by Eq. (20b),
we could plot the full mass density profile,
which reflects the plasma pressure profile with , as
With z_{*}=2 and considering
,
the full radial dependence
,
with the value of the coefficient in Eq. (36) taken to be unity,
is shown in Fig. 10 for the plasma torus with a very pronounced
maximum at z=3.7, and zero at z=4.5. The corresponding mass density spatial contours
of
are shown in Fig. 11.

Figure 11:
The mass density contours
with
are shown with C=+200, C=+300, C=+400, and C=+500 from the outer to inner contours respectively. 
Open with DEXTER 
In addition to the equatorial plasma torus, our selfsimilar MHD formulation is able to model planetary nebulae of the bipolar type. For this purpose, we consider

(37) 
to warrant
.
Following the same analysis, the radial function becomes

(38) 
The coefficients are
,
,
,
,
and
.
Although we have continued to use
as the normalized
radial label, the normalization factor a here has no
connection to
because we now assume that .
We could imagine that a=k, such that .
We notice that the particular solution consists entirely
of negative terms, apart from the final term,
whereas the homogeneous solution is positive.
With A_{0}=1, A=10^{7}, z_{*}=2,
,
the general solution shown in Fig. 12 becomes negative at
z=z_{0}=4.55.
This value of z_{0} happens to be close to the roots of
R_{0}(z) and R_{1}(z), of the equatorial plasma torus case,
where
.
This fact is purely a coincidence.
The full radial profile of the plasma density,
described by Eq. (36), explicit and implicit parts, is shown in Fig. 13.
Since plasma density is positive definite,
defines the physical domain of our selfsimilar solution.
The bipolar field contours are shown in Fig. 14.
The radial component vanishes at z_{0},
B_{r}(z_{0})=0,
which is the boundary of the expanding bipolar structure.

Figure 12:
The general solution of the radial function R(z) is shown as a function of z. 
Open with DEXTER 

Figure 13:
The full, explicit and implicit, radial mass density
profile
is shown as a function of z. 
Open with DEXTER 

Figure 14:
The magnetic field lines are shown with
C=+0.0, C=+0.1, C=+0.3, C=+0.5, and C=+0.7 from the outer to inner contours respectively. 
Open with DEXTER 
To conclude, we have represented the AGB wind by an exploding
ideal MHD plasma. The explosion is modelled by an isotropic radial
plasma velocity in spherical coordinates. Through dissipation,
this exploding plasma is believed to undergo selforganization.
The possible selforganized states may or may not be spherically
symmetric, although the velocity will be spherical symmetric.
Selforganization implies that the plasma global conservation
properties are sufficiently dominant to drive the system
through dissipation, independently of the initial conditions.
To find the possible selforganized states, we solve the ideal MHD
equations, with
,
for selfsimilar solutions in time,
where the temporal and spatial dependences of each physical
variable are in separable form.
We identify these selfsimilar solutions of ideal MHD to be
the selforganized configurations reached by dissipation.
We would like to emphasize the word ideal,
which means that this process occurs without dissipation.
Although selforganized states are physically generated
by dissipation, they are part of the ideal MHD solutions.
Dissipation provides the means by which an ideal state,
of a particular energy density, is topologically transformed
to another ideal state, of lower energy density.
Using this time selfsimilar formulation, we can derive
possible end states without time iteration,
assuming a specified initial configuration.
By circumventing intermediate stages, this selfsimilar approach,
which is assumed to be independent of selforganization
physical arguments, appears to be developing the required
solutions for the problems waiting to be solved.
Under axisymmetry, we have established two solutions for
equatorial ejection. The first solution has a finite azimuthal magnetic field,
and represents a plasma torus with closed toroidal field lines.
Beyond the plasma torus, the magnetic field lines are open
and the plasma is not confined. This plasma torus provides
the principle structure for the interactingwind model.
The second solution has a null azimuthal field,
and represents a bipolar planetary nebula.
Whether these selforganized structures continue to expand
indefinitely depends on the integration constant Hof the evolution function, which could have
an asymptotically bounded track for a stable torus,
and an unbounded track for an ever expanding bipolar nebulae.
Additional nebula features, such as filaments and jets,
could be accounted for with a fully threedimensional
selfsimilar MHD model.
We in fact believe that this description of selforganized
plasmas, using selfsimilar solutions, could represent a
fundamental process in astrophysical ejection events.
Apart from the equatorial ejection solutions of plasma torus
and bipolar structure of planetary nebulae discussed here,
the highly collimated polar ejection solutions could be
relevant to extragalactic jets, shouldwe consider these
objects to be ejection events (Tsui & Serbeto 2007).
The collimated polar ejection mechanism could also be relevant
to some asymmetric supernovae, with recoil on the neutron star,
and to quasar ejection models of active galactic nucleus
in cosmology.
Acknowledgements
The author is deeply grateful to Prof. Akira Hasegawa for the very
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