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Introduction to PLASMA ASTROPHYSICS (Selected 10 lectures)
Boris V. Somov Astronomical Institute and Faculty of Physics Moscow State University

Pushino-na-Oke, 2008


Contents
Ab out These Lectures 1 Particles and Fields: Exact Self-Consistent Description 1.1 Liouville's theorem . . . . . . . . . . . . . . . . . . . 1.1.1 Continuity in phase space . . . . . . . . . 1.1.2 The character of particle interactions 1.1.3 The Lorentz force, gravity . . . . . . . . . 1.1.4 Collisional friction . . . . . . . . . . . . . . . 1.1.5 The exact distribution function . . . . 1.2 Charged particles in the electromagnetic field . . . . . . . . . 1.2.1 General formulation of the problem . 1.2.2 The continuity equation for electric charge . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Initial equations and initial conditions 1.2.4 Astrophysical plasma applications . . 1.3 Gravitational systems . . . . . . . . . . . . . . . . . . . . . . 1.4 Practice: Exercises and Answers . . . . . . . . . . . . . . . . 2 Statistical Description of Interacting Particle Systems 2.1 The averaging of Liouville's equation . . . . . . . . . . . . . 2.1.1 Averaging over phase space . . . . . . . 2.1.2 Two statistical p ostulates . . . . . . . . 2.1.3 A statistical mechanism of mixing . 2.1.4 Derivation of a general kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A collisional integral and correlation functions . . . . . . . . 2.2.1 Binary interactions . . . . . . . . . . . . . . 2.2.2 Binary correlation . . . . . . . . . . . . . . 1 5 9 9 9 12 15 16 18 22 22 24 25 26 28 29 33 33 33 35 37 40 43 43 46

. . . . . . . .


2 2.2.3 2.3 2.4

CONTENTS

. Equations for correlation functions . . . . . . . . . . . . . . . Practice: Exercises and Answers . . . . . . . . . . . . . . . .

The collisional integral and binary correlation . . . . . . . . . . . . . . . . . . . . .

48 51 54

3 Weakly-Coupled Systems with Binary Collisions 55 3.1 Approximations for binary collisions . . . . . . . . . . . . . . 55 3.1.1 The small parameter of kinetic theory 55 3.1.2 The Vlasov kinetic equation . . . . . . . 58 3.1.3 The Landau collisional integral . . . . . 59 3.1.4 The Fokker-Planck equation . . . . . . . 61 3.2 Correlations and Debye-Huckel shielding . . . . . . . . . . . . 64 Å 3.2.1 The Maxwellian distribution function 64 3.2.2 The averaged force and electric neutrality . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.3 Pair correlations and the Debye-Huckel Å radius . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Gravitational systems . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Comments on numerical simulations . . . . . . . . . . . . . . 71 3.5 Practice: Exercises and Answers . . . . . . . . . . . . . . . . 73 4 Macroscopic Description of Astrophysical Plasma 4.1 Summary of microscopic description . . . . . . . . . . . . . . 4.2 Definition of macroscopic quantities . . . . . . . . . . . . . . 4.3 Macroscopic transfer equations . . . . . . . . . . . . . . . . . 4.3.1 Equation for the zeroth moment . . . . 4.3.2 The momentum conservation law . . . 4.4 The energy conservation law . . . . . . . . . . . 4.4.1 The second moment equation . . . . . . 4.4.2 The case of thermo dynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 The general case of anisotropic plasma 4.5 General properties of transfer equations . . . . . . . . . . . . 4.5.1 Divergent and hydro dynamic forms . 4.5.2 Status of the conservation laws . . . . . 4.6 Equation of state and transfer coefficients . . . . . . . . . . . 4.7 Gravitational systems . . . . . . . . . . . . . . . . . . . . . . 77 77 78 80 80 82 86 86 89 90 92 92 94 95 98


CONTENTS 5 The 5.1 5.2 5.3 5.4 Generalized Ohm's Law in Plasma The classic Ohm's law . . . . . . . . . . . . . . . . . . . . . Derivation of basic equations . . . . . . . . . . . . . . . . . The general solution . . . . . . . . . . . . . . . . . . . . . . The conductivity of magnetized plasma . . . . . . . . . . . 5.4.1 Two limiting cases . . . . . . . . . . . . . . 5.4.2 The physical interpretation . . . . . . . Currents and charges in plasma . . . . . . . . . . . . . . . . 5.5.1 Collisional and collisionless plasmas 5.5.2 Volume charge and quasi-neutrality Practice: Exercises and Answers . . . . . . . . . . . . . . .

3 101 101 102 106 107 107 108 111 111 115 117 119 119 119 120 123 126 126 128 131 133 134 135 135 139 141 143 146

5.5

5.6

. . . . . . . . . . . . . . . . . . . . . . . . . .

6 Single-Fluid Mo dels for Astrophysical Plasma 6.1 Derivation of the single-fluid equations . . . . . . . . . . . . 6.1.1 The continuity equation . . . . . . . . . . 6.1.2 The momentum conservation law . . 6.1.3 The energy conservation law . . . . . . 6.2 Basic assumptions and the MHD equations . . . . . . . . . 6.2.1 Old simplifying assumptions . . . . . . 6.2.2 New simplifying assumptions . . . . . 6.2.3 Non-relativistic MHD . . . . . . . . . . . 6.2.4 Energy conservation . . . . . . . . . . . . . 6.2.5 Relativistic magnetohydro dynamics 6.3 Magnetic flux conservation. Ideal MHD . . . . . . . . . . . 6.3.1 Integral and differential forms of the law . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 The ideal MHD . . . . . . . . . . . . . . . . 6.3.3 The `frozen field' theorem . . . . . . . . 6.4 Magnetic reconnection . . . . . . . . . . . . . . . . . . . . . 6.5 Practice: Exercises and Answers . . . . . . . . . . . . . . . 7 MHD in Astrophysics 7.1 The main approximations in ideal MHD . . . . . . . . . . . 7.1.1 Dimensionless equations . . . . . . . . . 7.1.2 Weak magnetic fields in astrophysical plasma . . . . . . . . . . . . . . . . . . . . . 7.1.3 Strong magnetic fields in plasma . . 7.2 Accretion disks of stars . . . . . . . . . . . . . . . . . . . . 7.2.1 Angular momentum transfer . . . . . .

149 . 149 . 149 . . . . 152 153 157 157


4

CONTENTS 7.2.2 Accretion in cataclysmic variables 7.2.3 Accretion disks near black holes . . 7.2.4 Flares in accretion disk coronae . . Astrophysical jets . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Jets near black holes . . . . . . . . . . . 7.3.2 Relativistic jets from disk coronae Practice: Exercises and Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 160 161 162 162 165 166 169 169 172 173 175 181 183 183 188 191

7.3

7.4

8 Plasma Flows in a Strong Magnetic Field 8.1 The general formulation of a problem . . . . . . . . . . . . 8.2 The formalism of 2D problems . . . . . . . . . . . . . . . . 8.2.1 The first typ e of problems . . . . . . . . 8.2.2 The second typ e of MHD problems 8.3 The existence of continuous flows . . . . . . . . . . . . . . . 8.4 Flows in a time-dependent dipole field . . . . . . . . . . . . 8.4.1 Plane magnetic dip ole fields . . . . . . 8.4.2 Axial-symmetric dip ole fields . . . . . 8.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Ab out These Lectures
If you want to learn the most fundamental things about plasma astrophysics with the least amount of time and effort - and who doesn't? - this text is for you. The textbook is addressed to students without a background in plasma physics. It grew from the lectures given at the Moscow Institute of Physics and Technics (the `fiz-tekh') since 1977. A similar full-year course was offered to the students of the Astronomical Division in the Faculty of Physics at the Moscow State University over the years after 1990. The idea of the book is not typical for the ma jority of textbooks. It was suggested by S.I. Syrovatskii that the consecutive consideration of physical principles, starting from the most general ones, and of simplifying assumptions gives us a simpler description of plasma under cosmic conditions. On the basis of such an approach the student interested in modern astrophysics, its current practice, will find the answers to two key questions: 1. What approximation is the b est one (the simplest but sufficient) for description of a phenomenon in astrophysical plasma?
5


6

About These Lectures

2. How can I build an adequate mo del for the phenomenon, for example, a flare in the corona of an accretion disk? Practice is really imp ortant for the theory of astrophysical plasma. Related exercises (supplemented to each chapter) serve to better understanding of plasma astrophysics. As for the applications, preference evidently is given to physical processes in the solar plasma. Why? - Because of the possibility of the all-round observational test of theoretical models. For instance, flares on the Sun, in contrast to those on other stars, can b e seen in their development. We can obtain a sequence of images during the flare's evolution, not only in the optical and radio ranges but also in the EUV, soft and hard X-ray, gamma-ray ranges. It is assumed that the students have mastered a course of general physics and have some initial knowledge of theoretical physics. For beginning students, who may not know in which subfields of astrophysics they wish to specialize, it is better to cover a lot of fundamental theories thoroughly than to dig deeply into any particular astrophysical sub ject or ob ject, even a very interesting one, for example black holes. Astrophysicists of the future will need to ols that allow them to explore in many different directions. Moreover astronomy of the future will be, more than hitherto, precise science similar to mathematics and physics.

see http://www.springer/com/ http://adsabs.harvard.edu/


About These Lectures

7

The first volume of the book covers the basic principles and main practical to ols required for work in plasma astrophysics.
Figure 1:

The second volume "Plasma Astrophysics, Part II, Reconnection and Flares" represents the basic physics of the magnetic reconnection effect and the flares of electromagnetic origin in the solar system, relativistic ob jects, accretion disks, their coronae. Never say: "It is easy to show...".


8

About These Lectures


Chapter 1

Particles and Fields: Exact Self-Consistent Description
There exist two ways to describe exactly the behaviour of a system of charged particles in electromagnetic and gravitational fields.

1.1
1.1.1

Liouville's theorem Continuity in phase space

Let us consider a system of N interacting particle. Without much justification, let us introduce the distribution function f = f (r, v, t) for particles as follows. We consider the six-dimensional (6D) space called phase space X = { r, v} shown in Fig. 1.1. The number of particles present in a small volume dX = d 3 r d 3 v at a point X at a moment of time t is defined to be dN (X, t) = f (X, t) dX.
9

(1.1)

(1.2)


10

Chapter 1. Particles and Fields

v

X dv
3

dX

X

dr 0
Figure 1.1:

3

r

The 6D phase space X . A small volume dX at a

point X .
Accordingly, the total numb er of the particles at this moment is N (t) = f (X, t) dX f (r, v, t) d 3 r d 3 v . (1.3)

If, for definiteness, we use the Cartesian coordinates, then X = { x, y , z , vx , vy , vz } is a point of the phase space (Fig. 1.2) and X = { vx , vy , vz , vx , vy , vz } is the velo city of this point in the phase space. Suppose the coordinates and velocities of the particles are changing continuously - `from point to point', i.e. the particles move smo othly at all times. So the distribution function f (X, t) is differentiable. (1.4)


1.1. Liouville's Theorem

11

Moreover we assume that this motion of the particles in phase space can be expressed by the continuity equation: f + divX f X = 0 t or f + divr f v + divv f v = 0 . t
v . X

(1.5)

S X J

dS U

0
Figure 1.2:

r

The 6D phase space X . The volume U is enclosed by the surface S .
Equation (1.5) expresses the conservation law for the particles, since the integration of (1.5) over a volume U enclosed by the surface S in Fig. 1.2 gives t f dX +
U U

divX f X dX =

by virtue of the Ostrogradskii-Gauss theorem


12

Chapter 1. Particles and Fields

= Here

N (t) t

+
U S

N (t) f X dS = t

+
U S

J ž dS = 0.

(1.6)

J = fX

(1.7)

is the particle flux density in the phase space. Thus a change of the particle number in a given volume U of the phase space X is defined by the particle flux through the boundary surface S . The reason is clear. There are no sources or sinks for the particles inside the volume. Otherwise the source and sink terms must be added to the righthand side of Equation (1.5).

1.1.2

The character of particle interactions

Let us rewrite Equation (1.5) in another form in order to understand the meaning of divergent terms. The first of them is divr f v = f divr v + (v ž
r

) f = 0 + (v ž

r

)f ,

since r and v are independent variables in the phase space X . The second divergent term is divv f v = f divv v + v ž
v

f.

So far no assumption has been made as to the character of particle interactions.


1.1. Liouville's Theorem

13

It is worth doing here. Let us restrict our consideration to the interactions with

divv v = 0 , then Equation (1.5) takes the following form f +vž t or
r

(1.8)

f+

F ž m

v

f =0 (1.9)

f +X t

X

f = 0,

where X= vx , v y , v z , Fx Fy Fz , , mmm . (1.10)

So we `trace' the phase tra jectories of particles when they move under action of a force field F(r, v, t). Thus we have found Liouville's theorem in the following formulation: f +vž t
r

f+

F ž m

v

f = 0.

(1.11)

Liouville's theorem: The distribution function remains constant on the particle phase trajectories if condition (1.8) is satisfied. We call Equation (1.11) the Liouville equation. The first term in Equation (1.11), the partial time derivative f / t , characterizes a change of the distribution function f (t, X ) at a given point X in the phase space with time t. Define also the Liouville op erator


14

Chapter 1. Particles and Fields

D +vž +X Dt t X t

r

+

F ž m

v

.

(1.12)

This operator is just the total time derivative following a particle motion in the phase space X . By using definition (1.12), we rewrite Liouville's theorem as follows: Df = 0. Dt

(1.13)

What factors do lead to the changes in the distribution function? Let dX be a small volume in the phase space X .

v

(a) v J
r

v J
v

(b)

dX J
r

F

dX

J

v

0
Figure 1.3:

r

0

r

Action of the two different terms of the Liouville operator in the 6D space X .
The second term in (1.11), v ž r f , means that the particles come into and go out of the volume element dX because their velocities are not zero (Fig. 1.3a).


1.1. Liouville's Theorem

15

So this term describes a simple kinematic effect. If the distribution function f has a gradient over r, then a number of particles inside the volume dX changes because they move with velocity v. The third term, (F/m) ž v f , means that the particles escape from the volume element dX or come into it due to their acceleration or deceleration under action of the force field F (Fig. 1.3b).

1.1.3

The Lorentz force, gravity

In order the Liouville theorem to be valid, the force field F has to satisfy condition (1.8). We rewrite it as follows: v 1 F = =0 v m v or F = 0, v In particular, this condition holds if the component F of the force vector F does not depend upon the velocity component v . This is a sufficient condition, of course. The classical Lorentz force F = e E + obviously has that property. 1 ( v × B ) c (1.15) = 1, 2, 3 . (1.14)


16

Chapter 1. Particles and Fields

The gravitational force in the classical approximation is entirely independent of velocity. Other forces are considered, depending on a situation, e.g., the force resulting from the emission of radiation (the radiation reaction) and/or absorption of radiation by astrophysical plasma. These forces when they are important must be considered with account of their relative significance, conservative or dissipative character, and other physical properties taken. 1.1.4

Collisional friction

As a contrary example we consider the collisional drag force which acts on a particle moving with velocity v in plasma: F = -kv, where the constant k > 0. In this case the right-hand side of Liouville's equation is not zero: -f divv v = -f div because v = v
v

(1.16)

3k F = f, m m

= 3.

Instead of Liouville's equation we have Df 3k = f > 0. Dt m (1.17)

Thus the distribution function (i.e. the particle density) does not remain constant on particle tra jectories but increases with time. Along the phase tra jectories, it increases exp onentially:


1.1. Liouville's Theorem

17

f (t, r, v) f (0, r, v) exp

3k t. m

(1.18)

The physical sense of this phenomenon is obvious. The friction force decelerates the particles. They go down in Fig. 1.4 and are concentrated in the vicinity of the axis v = 0.
v F

0

r

Particle density increases in the phase space as a result of action of the friction force F.
Figure 1.4:


18

Chapter 1. Particles and Fields

1.1.5

The exact distribution function

Let us consider another property of the Liouville theorem. We introduce the N -particle distribution function of the form ^ f (t, r, v) =
N

(r - ri (t)) (v - vi (t)) .
i=1

(1.19)

The delta function of the vector-argument is defined as usually:
3

(r - ri (t)) =
=1

= (1.20)

i i = rx - rx (t) ry - ry (t) rz - rzi (t) .

We shall call function (1.19) the exact distribution function. It is illustrated by Fig. 1.5.
< f X

The one-dimensional analogy of the exact distribution function.
Figure 1.5:

Let us substitute the exact distribution function in the Liouville equation. Action:


1.1.5. Exact Distribution Function

19

+vž t

r

+

F ž m

v

==>

^ f

==>

= 0.

The resulting three terms are ^ f = t +
i

(-1) (r - ri (t)) r (v - vi (t)) + i
i

i (-1) (r - ri (t)) (v - vi (t)) v ,

(1.21)

vž =
i

r

^ f ^ f v = r (1.22)

v (r - ri (t)) (v - vi (t)) ,

F ž m =
i

v

^ ^ F f = f m v (1.23)

F (r - ri (t)) (v - vi (t)) . mi

Here the index = 1, 2, 3 or (x, y , z ). The prime denotes the derivative with respect to the argument of a function. The overdot denotes differentiation with respect to time t. Summation over the rep eated index (contraction) is implied: r = x rx + y ry + z rzi . i i i The sum of terms (1.21)-(1.23) equals zero. Let us rewrite it as follows


20

Chapter 1. Particles and Fields

0=
i

-r + v i - v + i
i

i

(r - ri (t)) (v - vi (t)) +

+

F mi

(r - ri (t)) (v - vi (t)) .

This can occur just then that all the co efficients of different combinations of delta functions with their derivatives equal zero as well. Therefore we find
i d r i = v (t) , dt i d v 1 = F (ri (t), vi (t)) . dt mi

(1.24)

Thus the Liouville equation for an exact distribution function is equivalent to the Newton set of equations for a particle motion, both describing a purely dynamic behavior of the particles.

It is natural since this distribution function is exact. No statistical averaging has b een done so far. Statistics will appear later on when, instead of the exact description of a system, we begin to use some mean characteristics such as temperature, density etc. The statistical description is valid for systems containing a large number of particles. We have shown that finding a solution of the Liouville equation for an exact distribution function


1.1.5. Exact Distribution Function

21

^ Df =0 Dt is the same as the integration of the motion equations.

(1.25)


22

Chapter 1. Particles and Fields

However for systems of a large numb er of interacting particles, it is much more advantageous to deal with the single Liouville equation for the exact distribution function which describes the entire system.

1.2
1.2.1

Charged particles in the electromagnetic field General formulation of the problem

Let us recall the basic physics notations and establish a common basis. Maxwell's equations for the electric field E and magnetic field B are well known to have the form: rot B = 4 1 E j+ , c c t 1 B , c t (1.26) (1.27) (1.28) (1.29)

rot E = -

div B = 0 , div E = 4 q .

The fields are completely determined by electric charges and electric currents. Note that Maxwell's equations imply: ž the continuity equation for electric charge (see Exercise 1.5) ž the conservation law for electromagnetic field energy (Exercise 1.6).


1.2. Initial Equations
t

23

e1 t

ei

Å B Å ÅÅ ÅÅ ri (t) ÅÅ 0 qÅ t

t

t v (t) q i t te
N

Figure 1.6:

A system of N charged particles.

Let there be N particles with charges e1 , e2 , . . . ei , . . . eN , coordinates ri (t) and velocities vi (t), see Fig. 1.6. By definition, the electric charge density q (r, t) =
N

ei (r - ri (t))
i=1

(1.30)

and the density of electric current
N

j (r, t) =
i=1

ei vi (t) (r - ri (t)) .

(1.31)

The coordinates and velocities of particles can be found by integrating the equations of motion - the Newton equations: ri = vi (t) , vi = 1 e mi
i

(1.32) E (ri (t)) + 1 vi × B (ri (t)) . c (1.33)

Let us count the number of unknown quantities: the vectors B, E, ri , and vi . We obtain: 3 + 3 + 3N + 3N = 6 (N + 1). The numb er of equations = 8 + 6N = 6 (N + 1) + 2.


24

Chapter 1. Particles and Fields

Therefore two equations seem to b e unnecessary. Why is this so?

1.2.2

The continuity equation for electric charge

At first let us make sure that the definitions (1.30) and (1.31) conform to the conservation law for electric charge. Differentiating (1.30) with respect to time gives q =- t ei r . i
i

(1.34)

Here the index = 1, 2, 3. The prime denotes the derivative with respect to the argument of the delta function. The overdot denotes differentiation with respect to time t. For the electric current density (1.31) we have the divergence div j = j = r
i ei v . i

(1.35)

Comparing (1.34) with (1.35) we see that q + div j = 0 . t Therefore the definitions for equation.
q

(1.36)

and j conform to the continuity

As we shall see it in Exercise 1.5, conservation of electric charge follows also directly from the Maxwell equations. The difference is that above we have not used scalar Equation (1.29).


1.2. Initial Equations

25

1.2.3

Initial equations and initial conditions

Operating with the divergence on Equation (1.26) Action: div ==> rot B = 4 1 E j+ , c c t

and using the continuity Equation (1.36), Action: div j = - we obtain 0= Thus, we find that ( div E - 4 q ) = 0 . t (1.37) q . t 1 div E . c t

4 q - c t

+

Hence Equation (1.29) will be valid at any moment of time, provided it is true at the initial moment. Let us operate with the divergence on Equation (1.27): Action: div ==> rot E = - 1 B , c t (1.38)

div B = 0 . t

Equation (1.28) implies the absence of magnetic charges or, which is the same, the solenoidal character of the magnetic field. Conclusion. Equations (1.28) and (1.29) play the role of initial conditions for the time-dependent equations


26

Chapter 1. Particles and Fields

B = - c rot E t and E = + c rot B - 4 j . t

(1.39)

(1.40)

Thus, in order to describe the gas consisting of N charged particles, we consider the time-dependent problem of N bodies with a given interaction law. The electromagnetic part of interaction is described by Maxwell's equations, the time-independent scalar equations playing the role of initial conditions for the timedependent problem. Therefore the set consisting of eight Maxwell's equations and 6N Newton's equations is neither over- nor under-determined. It is closed with respect to the time-dependent problem, i.e. it consists of 6 (N + 1) equations for 6 (N + 1) variables, once the initial and boundary conditions are given.

1.2.4

Astrophysical plasma applications

The set of equations described above can be treated analytically in just three cases: 1. N = 1 , the motion of a charged particle in a given electromagnetic field, e.g., drift motions and adiabatic invariants, waveparticle interaction, particle acceleration in astrophysical plasma. 2. N = 2 , Coulomb collisions of two charged particles, i.e. binary collisions.


1.2. Initial Equations

27

This is important for the kinetic description of physical processes, e.g., the kinetic effects under propagation of accelerated particles in plasma, collisional heating of plasma by a b eam of fast electrons or/and ions. 3. N , a very large numb er of particles. This case is the frequently considered one in plasma astrophysics, because it allows us to introduce macroscopic descriptions of plasma, the widely-used magnetohydrodynamic (MHD) approximation. Intermediate case: Numerical integration of Equations (1.26)-(1.33) in the case of large but finite N , like N 3 × 106 , is possible by using modern computers. The computations called particle simulations are increasingly useful for understanding many properties of astrophysical plasma and for demonstration of them. One important example of a simulation is magnetic reconnection in a collisionless plasma. This process often leads to fast energy conversion from field energy to particle energy, flares in astrophysical plasma (see Part II).


28

Chapter 1. Particles and Fields

Generalizations: The set of equations described can be generalized to include consideration of neutral particles. This is necessary, for instance, in the study of the generalized Ohm's law which is applied in the investigation of physical processes in weakly-ionized plasmas, e.g., in the solar photosphere and prominences. Dusty and self-gravitational plasmas in space are interesting in view of the diverse and often surprising facts about planetary rings and comet environments, interstellar dark space.

1.3

Gravitational systems

Gravity plays a central role in the dynamics of many astrophysical systems - from stars to the Universe as a whole. A gravitational force acts on the particles as follows: mi vi = -mi Here the gravitational potential
N

.

(1.41)

(t, r) = -
n=1

G mn , | rn (t) - r |

n = i,

(1.42)

G is the gravitational constant. We shall return to this sub ject many times, e.g., while studying the virial theorem. This theorem is widely used in astrophysics. Though the potential (1.42) lo oks similar to the Coulomb potential of charged particles, physical properties of gravitational systems differ so much from properties of astrophysical plasma.


1.4. Practice: Exercises and Answers

29

We shall see this fundamental difference in what follows.

1.4

Practice: Exercises and Answers

Exercise 1.1. Show that any distribution function that is a function of the constants of motion - the invariants of motion - satisfies Liouville's equation. Answer. A general solution of the equations of motion (1.24) depends on 6N constants Ci where i = 1, 2, ... 6N . If the distribution function is a function of these constants of the motion f = f ( C1 , ... Ci , ... C
6N

),

(1.43)

we rewrite the left-hand side of Equation (1.13) as Df = Dt
6N i=1

DC Dt

i

f C

.
i

(1.44)

Because Ci are constants of the motion, DCi /Dt = 0. Therefore the right-hand side of Equation (1.44) is also zero. Q.e.d. This is the so-called Jeans theorem. Exercise 1.2. Rewrite the Liouville theorem by using the Hamilton equations. Answer. Rewrite the Newton set of equations (1.24) in the Hamilton form: q = H , P H P = - , q = 1, 2, 3 . (1.45)


30

Chapter 1. Particles and Fields

Here H (P, q ) is the Hamiltonian of a system, q and P are the generalized coordinates and momenta, respectively. Let us substitute the variables r and v in the Liouville equation by the generalized variables q and P: f + t
P

Hž

q

f-

q

Hž

P

f = 0.

(1.46)

Recall that the Poisson brackets for arbitrary quantities A and B are defined to be
3

[A, B ] =
=1

A B A B - q P P q

.

(1.47)

Applying (1.47) to (1.46), we find the final form of the Liouville theorem f + [f , H ] = 0. t Note that for a system in equilibrium [f , H ] = 0. (1.49)

(1.48)

Exercise 1.3. Discuss what to do with the Liouville theorem, if it is impossible to disregard quantum indeterminacy and assume that the classical description of a system is justified. Consider the case of dense fluids inside stars, for example, white dwarfs. Comment. Inside a white dwarf star the temperature T 105 K, but the density is very high: n 1028 - 1030 cm-3 . The electrons cannot be regarded as classical particles.


1.4. Practice: Exercises and Answers

31

We have to consider them as a quantum system with a Fermi-Dirac distribution. Exercise 1.4. Recall the Liouville theorem in a course of mechanics - the phase volume of a system is independent of t. Show that this formulation is equivalent to Equation (1.13). Exercise 1.5. Show that Maxwell's equations imply the continuity equation for electric charge. Answer. Operating with the divergence on Equation (1.26), Action: div we have 0= Substituting (1.29) Comment: (1.29) : div E = 4 q , ==> rot B = 4 1 E j+ , c c t

4 1 div j + div E . c c t

in this equation gives us the continuity equation for the electric charge q + div j = 0 . t (1.50)

Exercise 1.6. Starting from Maxwell's equations, derive the energy conservation law for an electromagnetic field. Answer. Multiply Equation (1.26) by the electric field vector E and add it to Equation (1.27) multiplied by the magnetic field vector B. The result is


32

Chapter 1. Particles and Fields

W = - j E - div G . t Here W= E2 + B 8
2

(1.51) (1.52)

is the energy of electromagnetic field in a unit volume of space; G= c [E × B] 4 (1.53)

is the flux of electromagnetic field energy through a unit surface in space, i.e. the Poynting vector. The first term on the right-hand side of Equation (1.51) is the power of work done by the electric field on all the charged particles in the unit volume of space. In the simplest approximation evE = d E, dt (1.54)

where E is the particle kinetic energy. Hence instead of Equation (1.51) we write the following form of the energy conservation law: t E 2 + B 2 v + 8 2
2

+ div

c [E × B] = 0. 4

(1.55)


Chapter 2

Statistical Description of Interacting Particle Systems
In a system which consists of many interacting particles, the statistical mechanism of `mixing' in phase space works and makes the system's behavior on average more simple.

2.1
2.1.1

The averaging of Liouville's equation Averaging over phase space

As was shown above, the exact state of a system consisting of N interacting particles can be given by the exact distribution function in the 6D phase space X = { r, v}. This function is the sum of -functions in N points of the phase space: ^ f (r, v, t) =
N

(r - ri (t)) (v - vi (t)) .
i=1

(2.1)

We use Liouville's equation to describe the change of the system state: ^ f +vž t
r

^ Fž f+ m
33

v

^ f = 0.

(2.2)


34

Chapter 2. Statistical Description

Once the exact initial state of all the particles is known, it can be represented by N p oints in the phase space (Fig. 2.1). The motion of these points is described by Liouville's equation.
v 1 2 N X

r

Figure 2.1:

Particle tra jectories in the 6D phase space X .

In fact we usually know only some average characteristics of the system's state, such as the temperature, density, etc. Moreover the behavior of each single particle is in general of no interest. For this reason, instead of the exact distribution function, let us introduce the distribution function averaged over a small volume X of phase space at a moment of time t: ^ f (r, v, t)
X

=

1 X

^ f (X, t) dX .
X

(2.3)

The mean number of particles that present at a moment of time t in an element of volume X is ^ f (r, v, t)
X

ž X =
X

^ f (r, v, t) dX .

(2.4)

Obviously the distribution function averaged over phase volume differs from the exact one (Fig. 2.2).


2.1. Averaging of Liouville's Equation
(a) < f X < f >X < (b) X

35

The 1D analogy of the distribution function in phase space X : (a) the exact distribution function (2.1), (b) the averaged function (2.3).
Figure 2.2:

2.1.2

Two statistical p ostulates

Let us average the exact distribution function (2.1) over a small time interval t centered at a moment of time t: ^ f (r, v, t)
t

=

1 t

^ f (r, v, t) dt .
t

(2.5)

Here t is small in comparison with the characteristic time of the system's evolution: t
ev

.

(2.6)

We assume that the following two statistical p ostulates are applicable to the system considered. The first p ostulate:


36

Chapter 2. Statistical Description

^ ^ The mean values f X and f t exist for sufficiently small X and t and are indep endent of the averaging scales X and t. Clearly the first postulate implies that the number of particles should be large. For a small number of particles the mean value depends upon the averaging scale: if, e.g., N = 1 then the exact distribution function (2.1) is simply a -function, and the average over the variable X is ^ f
X

= 1/X .

For illustration, the case (X ) 1 > X is shown in Fig. 2.3.
< < f >X < < f >X X < f >X <
1

f

Figure 2.3:

^ Averaging of the exact distribution function f which is equal to a -function.
The second p ostulate is ^ f (X, t)
X

^ = f (X, t)

The averaging of the distribution function over phase space is equivalent to the averaging over time.

< ( X )
1

X

t

= f (X, t) .

(2.7)


2.1. Averaging of Liouville's Equation

37

While speaking of the small X and t, we assume that they are not to o small: X must contain a reasonably large number of particles while t must be large in comparison with the duration of drastic changes of the exact distribution function, such as the duration of the particle collisions: t c . (2.8)

It is in this case that the statistical mechanism of particle `mixing' in phase space is at work and the averaging of the exact distribution function over the time t is equivalent to the averaging over the phase volume X .

2.1.3

A statistical mechanism of mixing

Let us try to understand qualitatively how the mixing mechanism works in phase space. We start from the dynamical description of the N -particle system in 6N -dimensional phase space in which = { ri , vi } , i = 1, 2, . . . N , (2.9)

a point is determined (t = 0 in Fig. 2.4) by the initial conditions of all the particles. The motion of this point is described by Liouville's equation. The point moves along a complicated dynamical tra jectory because the interactions in a many-particle system are extremely intricate and complicated. The dynamical tra jectory has a remarkable prop erty.


38
vi

Chapter 2. Statistical Description



t =0 r

10

23

i

Figure 2.4:

The dynamical tra jectory of a system of N particles in the 6N -D phase space .

Imagine a glass vessel containing a gas consisting of a large number N of particles. The state of this gas at any moment of time is depicted by a single point in the phase space . Let us imagine another vessel which is identical to the first one, with one exception. At any moment of time t, the gas state in the second vessel is different from that in the first one. These states are depicted by two different p oints in the space . For example, at t = 0, they are points 1 and 2 in Fig. 2.5. With the passage of time, the gas states in both vessels change, whereas the two points in the space draw two different dynamical tra jectories (Fig. 2.5). These tra jectories do not intersect. If they had intersected at just one point, then the state of the first gas, determined by 6N numbers (ri , vi ), would have coincided with the state of the second gas. These numbers could be taken as the initial conditions which, in turn, would have uniquely determined the motion.


2.1. Averaging of Liouville's Equation
vi t =0 1 2 r
i

39
1 2

The dynamical tra jectories of two systems never cross each other.
Figure 2.5:

The two tra jectories would have merged into one. For the same reason the tra jectory of a system cannot intersect itself. Thus we come to the conclusion that

only one dynamical tra jectory of a many particle system passes through each point of the phase space .

Since the tra jectories differ in initial conditions, we can introduce an infinite ensemble of systems (glass vessels) corresponding to the different initial conditions. In a finite time the ensemble of dynamical tra jectories will closely fill the phase space , without intersections. By averaging over the ensemble we can answer the question: what is the probability that, at a moment of time t, the system will be found in an element = ri vi of the phase space : ^ dw = f (ri , vi )


d .

(2.10)


40

Chapter 2. Statistical Description


^ Here f (ri , vi )

is a function of all the coordinates and velocities.

It plays the role of the probability distribution density in the phase space and is called the statistical distribution function or simply the distribution function.

It is obvious that the same probability density can be obtained in another way - through the averaging over time. The dynamical tra jectory of a system, given a sufficient large time t, will closely cover the space . Since the tra jectory is very intricate, it will rep eatedly pass through the phase space element . Let (t) be the time during which the system locates in . For a sufficiently large t, which is formally restricted by the characteristic time of evolution of the system as a whole, the ratio (t) /t tends to the limit lim ( t ) dw ^ = = f (ri , vi , t) t d
t

t

.

(2.11)

By virtue of the role of the probability density, it is clear that the statistical averaging over the ensemble (2.10) is equivalent to the averaging over time (2.11) as well as to the definition (2.5).

2.1.4

Derivation of a general kinetic equation

Now we have everything what we need to average the exact Liouville equation ^ f +vž t
r

^ Fž f+ m

v

^ f = 0.


2.1. Averaging of Liouville's Equation

41

Since the equation contains the derivatives with respect to time t and phase-space coordinates (r, v), the procedure of averaging is defined as follows: f (X, t) = 1 X t ^ f (X, t) dX dt .
X t

(2.12)

Averaging the first term of the Liouville equation gives 1 X t ^ f 1 dX dt = t t = 1 t


X t

t

1 t X

^ f dX dt =
X

t

f f dt = . t t

(2.13)

In the last equality the use is made of the fact that, by virtue of the second postulate, the averaging of a smooth averaged function does not change it. Let us average the second term in Equation (2.2): 1 X t v
X t

^ f dX dt = r




1 = X =

X

1 v r t v

^ f dt dX =
t

1 X

X

f f dX = v . r r

(2.14)

Here the index = 1, 2, 3. To average the term containing the force F, let us represent it as a sum of a mean force F and the force due to the difference of the real force field from the mean (smooth) one:


42

Chapter 2. Statistical Description

F= F +F .

(2.15)

Substituting (2.15) in the third term in Equation (2.2) and averaging it, we have 1 X t ^ F f dX dt = m v


X t

F 1 = m X

X

1 v t

^ f dt dX +
t

+

1 X t

X t

^ F f dX dt = m v ^ F f dX dt . m v

=

F f 1 + m v X t

(2.16)

X t

Gathering all three terms together, we write the averaged Liouville equation in the form ^ f t

f +vž t

r

f+

F ž m

v

f=

,
c

(2.17)

where ^ f t 1 X t ^ F f dX dt . m v (2.18)

=-
c

X t


2.2. Collisional Integral

43

Equation (2.17) and its right-hand side (2.18) are called the kinetic equation and the collisional integral, respectively. Thus we have found the most general form of the kinetic equation with a collisional integral, which cannot be directly used in plasma astrophysics, without making some additional simplifying assumptions. The main of them is the binary character of collisions.

2.2
2.2.1

A collisional integral and correlation functions Binary interactions

The statistical mechanism of mixing in phase space makes particles have no individuality. However, we have to distinguish different kinds of particles, e.g., electrons and protons, because their b ehaviors differ. ^ Let fk (r, v, t) be the exact distribution function of particles of the kind k ^ fk (r, v, t) =
Nk

(r - rki (t)) (v - vki (t)) ,
i=1

(2.19)

the index i denoting the ith particle of kind k , Nk being the number of particles of kind k . The Liouville equation for the particles of kind k takes a view ^ fk +vž t
r

^ Fk ^ fk + ž mk

v

^ fk = 0 ,

(2.20)

mk is the mass of a particle of kind k . The force acting on a particle of kind k at a point (r, v) of the phase ^ space X at a moment of time t, Fk, (r, v, t), is the sum of forces acting on this particle from all other particles (Fig. 2.6):


44

Chapter 2. Statistical Description

^ F

Nl k,

(r, v, t) =
l i=1

^ F

(i) kl,

(r, v, rli (t), vli (t)) .

(2.21)

v (t)
li

e

li

Fkl

(i)

e

k

r (t)
li

z

r

x

y

An action of a particle e li located at the point r li on a particle of kind k at a point r at a moment of time t.
Figure 2.6:

^ So the total force Fk, (r, v, t) depends upon the instant positions and velocities of all the particles. By using the exact distribution function, we rewrite formula (2.21) as follows: ^ F (r, v, t) =
lX 1

k,

F

kl,

^ (X, X1 ) fl (X1 , t) dX1 .

(2.22)

Here we assume that an interaction law Fkl, (X, X1 ) is explicitly independent of time t; ^ fl (X, t) is the exact distribution function of particles of kind l, the variable of integration is designated as X1 = { r1 , v1 } and dX1 = d 3 r1 d 3 v1 .


2.2. Collisional Integral

45

Formula (2.22) takes into account that the forces considered are binary ones, i.e. they can be represented as a sum of interactions between two particles.

Making use of the representation (2.22), let us average the force term in the Liouville equation, as this has been done in formula (2.16). We have 1 X t = 1 X t 1^ F mk
k,

(r, v, t)

X t

^ fk dX dt = v

X t

lX 1

1 F mk

kl,

^ (X, X1 ) fl (X1 , t) ×

×

^ fk (X, t) dX dX1 dt = v 1 F mk

=

1 X

kl,

(X, X1 ) ×


X

lX

1



1 × v t

^ ^ fk (X, t) fl (X1 , t) dt dX dX1 .
t

(2.23)

Here we have taken into account that the exact distribution func^ tion fl (X1 , t) is independent of the velocity v, which is a part of the variable X = { r, v } related to the particles of the kind k . Formula (2.23) contains the pair pro ducts of exact distribution functions of different particle kinds, as is natural for the case of binary interactions.


46

Chapter 2. Statistical Description

2.2.2

Binary correlation

^ Let us represent the exact distribution function fk as ^ fk (X, t) = fk (X, t) + k (X, t) , ^ (2.24)

where fk (X, t) is the statistically averaged distribution function, k (X, t) is the deviation of the exact distribution function from the ^ averaged one. It is obvious that, according to (2.24), ^ k (X, t) = fk (X, t) - fk (X, t) ; ^ hence k (X, t) = 0 . ^ (2.25)

Let us consider the integrals of pair pro ducts in the averaged force term (2.23). In view of definition (2.24), they can be rewritten as 1 t ^ ^ fk (X, t) fl (X1 , t) dt =
t

= fk (X, t) fl (X1 , t) + fkl (X, X1 , t) , where fkl (X, X1 , t) = 1 t k (X, t) l (X1 , t) dt . ^ ^
t

(2.26)

(2.27)

The function fkl is referred to as the correlation function or, more exactly, the binary correlation function. The physical meaning of the correlation function is clear from (2.26).


2.2. Collisional Integral

47

The left-hand side of (2.26) means the probability to find a particle of kind k at a point X of the phase space at a moment of time t under condition that a particle of kind l places at a point X1 at the same time. By definition this is a conditional probability. In the right-hand side of (2.26) the distribution function fk (X, t) characterizes the probability that a particle of kind k stays at a point X at a moment of time t. The function fl (X1 , t) plays the analogous role for the particles of kind l.

If the particles of kind k did not interact with those of kind l, then their distributions would be independent, i.e. probability densities would simply multiply:

^ ^ fk (X, t) fl (X1 , t) = fk (X, t) fl (X1 , t) . So in the right-hand side of (2.26) there should be fkl (X, X1 , t) = 0 . There would be no correlation in the particle distribution.

(2.28)

(2.29)

We consider a system of interacting particles. With the proviso that the parameter characterizing the binary interaction, e.g., Coulomb collision considered below, i e2 l mv 2
2

,

(2.30)

is small under conditions in a wide range, the correlation function must be relatively small.


48

Chapter 2. Statistical Description

If the interaction is weak, the second term in the right-hand side of (2.26) must be small in comparison with the first one.

This fundamental property allows us to construct a theory of plasma in many cases of astrophysical interest.

2.2.3

The collisional integral and binary correlation

Now let us substitute (2.26) in formula (2.23) for the averaged force term: ^ 1 1^ fk F k, (X, t) dX dt = X t mk v
X t

=

1 X

X

lX 1

1 F mk

kl,

(X, X1 )

[ fk (X, t) fl (X1 , t) + v

+ fkl (X, X1 , t) ] dX dX1 = since fk (X, t) is a smooth function, its derivative over v can be brought out of the averaging procedure: = fk (X, t) × v

1

×



1 X

X

lX 1

1 F mk

kl,

(X, X1 ) fl (X1 , t) dX dX



+

+

1 X

X

lX

1

1 F mk

kl,

(X, X1 )

fkl (X, X1 , t) dX dX1 = v


2.2. Collisional Integral

49

=

1 F mk

k,

(X, t)

fk (X, t) + v f k l (X , X1 , t ) dX1 . v (2.31)

+
lX 1

1 F mk

kl,

(X , X1 )

Here we have taken into account that the averaging of smooth functions does not change them, and the following definition of the averaged force is used: F
k,

(X, t) =

1 X

F
X lX
1

kl,

(X, X1 ) fl (X1 , t) dX dX1 =

=
lX 1

F

kl,

(X, X1 ) fl (X1 , t) dX1 .

(2.32)

This definition coincides with the previous definition (2.16) of the averaged force, since all the deviations of the real force force Fk are taken care of in the ^ real distribution functions fk and and fl . ^ Fk from the mean (smooth) deviations k and l of the ^ ^ ^ from their mean values fk fl

Thus the collisional integral is represented in the form ^ fk t 1 F mk fkl (X, X1 , t) dX1 . v

=-
c lX 1

kl,

(X, X1 )

(2.33)

Let us recall that for the Lorentz force as well as for the gravitational one the condition F v
kl,

(X, X1 ) = 0

(2.34)


50

Chapter 2. Statistical Description

is satisfied. So, we obtain from formula (2.33) the following expression ^ fk t v 1 F mk

=-
c



kl,

(X, X1 ) fkl (X, X1 , t) dX1 .

(2.35)

lX 1

Hence the collisional integral can be written in the divergent form in the velo city space v : ^ fk t =-
c

J v

k,

, (2.36)

where the flux of particles of kind k in the velocity space is 1 F mk

J

k,

(X, t) =
lX 1

kl,

(X, X1 ) fkl (X, X1 , t) dX1 .

(2.37)

Therefore the averaged Liouville equation or the kinetic equation for particles of kind k fk (X, t) fk (X, t) F k, (X, t) fk (X, t) + v + = t r mk v v 1 F mk

=-



kl,

(X, X1 ) fkl (X, X1 , t) dX

1

(2.38)

lX 1

contains the unknown function fkl . Hence the kinetic Equation (2.38) for distribution function fk is not closed. We have to find the equation for the correlation function fkl .


2.3. Correlation Functions

51

2.3

Equations for correlation functions

To derive the equations for correlation functions, it is not necessary to introduce any new postulates or develop new formalisms. All the necessary equations and averaging procedures are at hand. Looking at definition fkl (X, X1 , t) = where ^ k (X, t) = fk (X, t) - fk (X, t) , ^ we see that we need an equation which will describe the deviation of ^ distribution function from its mean value, i.e. the function k = fk - fk . ^ In order to derive such equation, we simply have to subtract the averaged Liouville equation fk (X, t) fk ( X , t ) + v + ... = ... t r from the exact Liouville equation (2.2) ^ fk +vž t The result is ^ k (X, t) ^ k (X, t) ^ Fk + v + t r m = Here v
, k r

1 t

k (X, t) l (X1 , t) dt , ^ ^
t

^ Fk ^ fk + ž mk

v

^ fk = 0 .

^ fk Fk - v m

, k

fk = v (2.39)

lX 1

1 F mk

kl,

(X, X1 ) fkl (X, X1 ) dX1 .


52

Chapter 2. Statistical Description

^ F

k,

(X, t) =
lX 1

F

kl,

^ (X, X1 ) fl (X1 , t) dX

1

(2.40)

is the exact force (2.22) acting on a particle of the kind k , and F
k,

(X, t) =
lX
1

F

kl,

(X, X1 ) fl (X1 , t) dX1

(2.41)

is the statistically averaged force. Considering that we need the equation for the pair correlation function fkl (X1 , X2 , t) = let us take two equations: one for k (X1 , t) ^ k (X1 , t) ^ +v t and another for l (X2 , t) ^ l (X2 , t) ^ +v t
2, 1,

k (X1 , t) l (X2 , t) , ^ ^

k (X1 , t) ^ + ... = 0 r1,

(2.42)

l (X2 , t) ^ + ... = 0. r2,

(2.43)

Now we add the equations resulting from (2.42) multiplied by l ^ and (2.43) multiplied by k . ^ We obtain l ^ or k ^ l ^ + k ^ +v t t
1,

k ^ l + . . . = 0 ^ r1,


2.3. Correlation Functions

53

(k l ) ^^ +v t

1,

(k l ) ^^ +v r1,

2,

(k l ) ^^ + ... = 0. r2,

(2.44)

On averaging Equation (2.44) we have the equation for the pair correlation function: fkl (X1 , X2 , t) + t +v + +
1,

fkl (X1 , X2 , t) +v r1,

2,

fkl (X1 , X2 , t) + r2,
l,

F

k,

(X1 , t) fkl (X1 , X2 , t) F + mk v 1, 1 F mk 1 F ml
kn,

(X2 , t) fkl (X1 , X2 , t) + ml v 2,

fk (X1 , t) v 1,

(X1 , X3 ) fnl (X3 , X2 , t) dX3 + (X3 , X1 , t) dX3 =

n

X

3

fl (X2 , t) + v 2, =- - Here v 1,

ln,

(X2 , X3 ) f

nk

n

X

3

n

X3

1 F mk

kn,

(X1 , X3 ) f
k ln

k ln

(X1 , X2 , X3 , t) dX3 -

v 2,

n

X

3

1 F ml

ln,

(X2 , X3 ) f

(X1 , X2 , X3 , t) dX3 . (2.45)

fkln (X1 , X2 , X3 , t) =

1 t

k (X1 , t) l (X2 , t) n (X3 , t) dt ^ ^ ^
t

(2.46)

is the function of triple correlations. Thus Equation (2.45) for the pair correlation function contains the unknown function of triple correlations.


54

Chapter 2. Statistical Description

In general, the chain of equations for correlation functions is unclosed: the equation for the correlation function of sth order contains the function of the order (s + 1).

2.4

Practice: Exercises and Answers

Exercise 2.1. By analogy with formula (2.26), show that ^ ^ ^ fk (X1 , t) fl (X2 , t) fn (X3 , t) = = fk (X1 , t) fl (X2 , t) fn (X3 , t) + + fk (X1 , t) fln (X2 , X3 , t) + fl (X2 , t) f + fn (X3 , t) fkl (X1 , X2 , t) + f
k ln kn

(2.47)

(X1 , X3 , t) +

(X1 , X2 , X3 , t) .

Exercise 2.2. Discuss a similarity and difference between the kinetic theory presented in this Chapter and the famous BBGKY hierarchy theory developed by Bogoliubov, Born and Green, Kirkwood, and Yvon. Hint. Show that essential to both derivations is the weak-coupling assumption, according to which

grazing encounters, involving small fractional energy and momentum exchange between colliding particles, dominate the evolution of the velocity distribution function. The weak-coupling assumption provides justification of the widely appreciated practice which leads to a very significant simplification of the original collisional integral.


Chapter 3

Weakly-Coupled Systems with Binary Collisions
In a system of many interacting particles, the weak-coupling assumption allows us to introduce a well controlled approximation to consider the chain of the equations for correlation functions. This leads to a significant simplification of the collisional integral in astrophysical plasma but not in self-gravitating systems.

3.1
3.1.1

Approximations for binary collisions The small parameter of kinetic theory

The infinite chain of equations for the correlation functions does not contain more information in itself than the Liouville equation for the exact distribution function. Actually, the statistical smoothing allows to lose `useless information' - the information about the exact motion of particles. The value of the chain is that it allows a direct introduction of new physical assumptions which make it possible to break the chain off at some term (Fig. 3.1) and to estimate the resulting error.
55


56

Chapter 3. Weakly-Coupled Systems

We call this procedure a well controlled approximation because it looks, in a sense, similar to the Taylor expansion series.

LT f
k

<>

KE
X

BC f
kl

... f
kln

f

k

How to break the infinite chain of the equations for correlation functions? LT is the Liouville theorem for an ^ exact distribution function fk . K E and B C are the kinetic Equation for fk etc.
Figure 3.1:

There is no universal way of breaking the chain off. It is intimately related, in particular, to the physical state of a plasma. Different states (as well as different aims) require different approximations. The physical state of a plasma can be characterized, at least partially, by the ratio of the mean energy of two particle interaction to their mean kinetic energy i e2 l mv 2
2

,

If mean kinetic energy can be reasonably characterized by some effective temp erature T , then i e2 (kB T )-1 . l (3.1)

As a mean distance between the particles we take l n
-1/3

.


3.1. Binary Collisions

57

Hence i = n1/ e2 × kB T
3

(3.2)

is termed the interaction parameter. It is small for a sufficiently hot and rarefied plasma. In many astrophysical plasmas, e.g., in the solar corona, the interaction parameter is very small. So the thermal kinetic energy of plasma particles is much larger than their interaction energy. The particles are almost free or moving on definite tra jectories in the external fields if the later are present. We call this case the approximation of weak Coulomb interaction. While constructing a kinetic theory, it is natural to use the p erturbation pro cedure with respect to the small parameter i . This means that the distribution function fk must be taken to be of order unity, the pair correlation function fkl of order i , the triple correlation function fkln of order i2 , etc.

We shall see in what follows that this principle has a deep physical sense in kinetic theory. Such plasmas are said to be `weakly coupled'. An opposite case, when the interaction parameter takes values larger than unity, is dense, relatively cold plasmas, for example in the interiors of white dwarf stars. These plasmas are `strongly coupled'.


58

Chapter 3. Weakly-Coupled Systems

3.1.2

The Vlasov kinetic equation

In the zeroth order with respect to the small parameter i , we obtain the Vlasov equation with the self-consistent electromagnetic field: fk (X, t) fk (X, t) + + v t r ek 1 fk (X, t) + E + v×B = 0. mk c v

(3.3)

Here E and B are the statistically averaged electric and magnetic fields obeying Maxwell's equations: 1 B , c t

curl E = -

div E = 4 ( 0 + q ) , (3.4)

curl B =

1 E 4 0 + ( j + jq ) , c t c

div B = 0 .

0 and j 0 are the external charges and currents; they describe the external fields, e.g., the uniform magnetic field B0 . q and j q are the statistically smo othed charge and current due to the plasma particles:

q (r, t) =
k

e

k v

fk (r, v, t) d 3 v , v fk (r, v, t) d 3 v .
v

(3.5) (3.6)

j q (r, t) =
k

ek

Therefore the electric and magnetic fields are also statistically smo othed. If we are considering processes which occur on a time scale much shorter than the time of collisions,


3.1. Binary Collisions

59



ev

c ,

(3.7)

we use a description which includes the averaged electric and magnetic fields but neglects the microfields resp onsible for binary collisions. This means that F = 0, therefore the collisional integral is also equal to zero. The Vlasov equation together with the definitions (3.5) and (3.6), and with Maxwell's Equations (3.4) is a nonlinear integro-differential equation. It serves as a classic basis for the theory of oscillations and waves in a plasma with the small parameter i . The Vlasov equation is also a proper basis for theory of waveparticle interactions in astrophysical plasma and collisionless shock waves, collisionless reconnecting current layers.

3.1.3

The Landau collisional integral

Using the perturbation procedure with respect to the small parameter i in the first order, and neglecting the close Coulomb collisions, we find the kinetic equation with the collisional integral given by Landau ^ fk t =-
c

J v

k,

,

(3.8)

Here the flux of particles of kind k in the velocity space is J
k,

=

2 ek ln mk

e
l

2 l vl

fk

fl ml v

-f
l,

l

fk mk v

×
k,

×

(u2 - u u ) 3 d vl . u3

(3.9)


60

Chapter 3. Weakly-Coupled Systems

u = v - vl is the relative velocity, d 3 vl corresponds to the integration over the whole velocity space of `field' particles l. ln is the Coulomb logarithm which takes into account divergence of the Coulomb-collision cross-section. The kinetic equation with the Landau integral is a nonlinear integrodifferential equation for the distribution function fk (r, v, t). Two approaches correspond to different limiting cases. The Landau integral takes into account the part of the particle interaction which determines dissipation while the Vlasov equation allows for the averaged field, and is thus reversible. For example, in the Vlasov theory, the question of the role of collisions in the neighbourhood of resonances remains open. The famous paper by Landau (1946) was devoted to this problem. Landau used the reversible Vlasov equation as the basis to study the dynamics of a small perturbation of the Maxwell distribution function, f (1) (r, v, t). In order to solve the linearized Vlasov equation, he made use of the Laplace transformation, and defined the rule to avoid a pole at =k v in the divergent integral by the replacement + i0. This technique for avoiding singularities may be formally replaced by a different procedure. Namely it is possible to add a small dissipative term - f (1) (r, v, t) to the right-hand side of the linearized Vlasov equation. In this way, the Fourier transform of the kinetic equation involves the complex frequency = + i ,


3.1. Binary Collisions

61

leading with 0 to the same expression for the Landau damping. Note, however, that the Landau damping is not by collisions but by a transfer of wave field energy into oscillations of resonant particles.

The Landau method is really a beautiful example of complex analysis leading to an important new physical result. The second approach reduces the reversible Vlasov equation to an irreversible one. Although the dissipation is assumed to be negligibly small, one cannot take the limit 0 directly in the master equations: this can be done only in the final formulae. This method of introducing a collisional damping is natural. It shows that even very rare collisions play the principal role in the physics of collisionless plasma.

3.1.4

The Fokker-Planck equation

The smallness of the interaction parameter signifies that, in the collisional integral, the sufficiently distant Coulomb collisions are taken care of as the interactions with a small momentum and energy transfer. For this reason, it comes as no surprise that the Landau integral can be considered as a particular case of a different approach which is the Fokker-Planck equation. Let us consider a distribution function indep endent of space so that f = f (v, t).


62

Chapter 3. Weakly-Coupled Systems

The Fokker-Planck equation describes the distribution function evolution due to nonstop overlapping weak collisions resulting in particle diffusion in velocity space: f = t ^ f t =-
c

2 [ a f ] + [b v v v



f ].

(3.10)

This equation coincides with the diffusion equation for some admixture with concentration f , e.g., Brownian particles in a gas, on which sto chastic forces are exerted by the molecules of the gas. The coefficient b plays the role of the diffusion coefficient and is expressed in terms of the averaged velocity change v in an elementary act - a collision: b = The other coefficient is a = v


1 v v 2



.

(3.11)

.

(3.12)

It is known as the coefficient of dynamic friction. A Brownian particle moving with velocity v through the gas experiences a drag opposing the motion (Fig. 1.4). In order to find the mean values appearing in the Fokker-Planck equation, we have to make clear the physical and mathematical sense of expressions (3.11) and (3.12). The mean values of velocity changes are in fact statistically averaged and determined by the forces acting between a test particle and scatterers (field particles or waves).

For test particles interacting with the thermal electrons and ions in a plasma, such calculations give us the Landau integral.


3.1. Binary Collisions

63

Thus one did not anticipate any ma jor problems in rewriting the Landau integral in the Fokker-Planck form. The kinetic equation found in this way will allow us to study the Coulomb interaction of accelerated particle beams with astrophysical plasma. Collisional friction slows down the particles of the beam and moves them toward the zero velocity in the velocity space (Fig. 3.2). Diffusion disp erses the distribution of beam particles in the velocity space.

f ( v || )

t=0

t >0

A beam of fast particles in plasma. We illustrate only the effects of Coulomb collisions.
Figure 3.2:

0

v

||

During the motion of a beam of fast particles in a plasma a reverse current of thermal electrons is generated, which tends to compensate the electric current of fast particles - the direct current. The electric field driving the reverse current makes a great impact on the particle beam kinetics. That is why, in order to solve the problem of accelerated particle propagation in, for example, the solar atmosphere, we inevitably have to apply a combined approach. This takes into account both the electric field influence on the accelerated particles (as in the Vlasov equation) and their scattering from the thermal particles of a plasma.


64

Chapter 3. Weakly-Coupled Systems

3.2

Correlations and Debye-Huckel shielding Å

We are going to understand the most fundamental property of the binary correlation function. With this aim, we shall solve the second equation in the chain illustrated by Fig.

KE ? f
k

BC f
kl

... f
kln

?

Here B C is the Equation (2.45) for the correlation function f kl . To determine and to solve this equation we have to know two functions: the distribution function fk from the first link in the chain and the triple correlation function fkln from the third link. 3.2.1

The Maxwellian distribution function

Let us consider the stationary ( / t = 0) solution to the equations for correlation functions, assuming the interaction parameter i to be small and using the successive approximations in the following form. First, we set fk l = 0 in the kinetic equation. Second, we assume that the triple correlation function fkln = 0


3.2. Debye-Huckel Shielding Å

65

in Equation (2.45) for the correlation function fkl etc. The plasma is supposed to be stationary, uniform and in the thermodynamic equilibrium state, i.e. the velocity distribution is assumed to be a Maxwellian function fk (X ) = fk (v 2 ) = ck exp - mk v 2 . 2kB Tk (3.13)

The constant ck is determined by the normalizing condition and equals ck = n
k

mk 2 kB T

3/2

.
k

It is obvious that the Maxwellian function satisfies the kinetic equation under assumptions made above if the averaged force is equal to zero: F
k,

(X, t) = F

k,

(X ) = 0 .

(3.14)

Since we shall need the same assumption in the next Section, we shall justify it there.

3.2.2

The averaged force and electric neutrality

Let us substitute the Maxwellian function in the kinetic equation, neglecting all the interactions except the Coulomb ones. We obtain the following expression for the averaged force: F (X1 ) =
lX
2

k,

F

kl,

(X1 , X2 ) fl (X2 ) dX2 =
2

since plasma is uniform, fl does not depend of r =
l r2

F

kl,

(r1 , r2 ) d 3 r2 ž
v2

fl (v2 ) d 3 v2 =


66

Chapter 3. Weakly-Coupled Systems

=-
r2 l

r1

,

ek el | r1 - r2 |

d 3 r 2 ž nl = nl el .
l

=-
r2

r1,

ek | r1 - r2 |

d 3 r2 ž

(3.15)

Therefore F
k,

= 0,

(3.16)

if the plasma is assumed to be electrically neutral:

nl el = 0 .
l

(3.17)

Balanced charges of ions and electrons determine the name plasma according Langmuir (1928). So the averaged (statistically smoothed) force (2.32) is equal to zero in the electrically neutral plasma but is not equal to zero in a system of charged particles of the same charge sign: positive or negative, it does not matter. Such a system tends to expand. There is no neutrality in gravitational systems like stellar clusters. The large-scale gravitational field makes an overall thermo dynamic equilibrium impossible. Moreover, on the contrary to plasma, they tend to contract and collapse.


3.2. Debye-Huckel Shielding Å

67

3.2.3

Pair correlations and the Debye-Huckel radius Å

As a first approximation, on putting the triple correlation function f
k ln

= 0,

we obtain from Equation (2.45), in view of condition (3.16), the following equation for the binary correlation function fkl : fk l +v r1, =-
n X3

v

1,

2,

fk l = r2,
kn,

1 F mk

(X1 , X3 ) fnl (X3 , X2 ) (X3 , X1 ) fl v 2,

fk + v 1, (3.18)

+

1 F ml

ln,

(X2 , X3 ) f

nk

dX3 .

Let us consider the particles of two kinds: electrons and ions, assuming the ions to be motionless and homogeneously distributed. Then the ions do not take part in any kinetic processes. Hence i 0 ^ for ions; and the correlation functions associated with i equal zero ^ too: f ii = 0 , fei = 0 etc. (3.19)

Among the pair correlation functions, only one has a non-zero magnitude fee (X1 , X2 ) = f (X1 , X2 ) . (3.20)

Taking into account (3.19), (3.20), and (3.13), rewrite Equation (3.18) as follows


68

Chapter 3. Weakly-Coupled Systems

v

1

f f + v2 = r1 r2 = 1 kB T [ v1 ž F (X1 , X3 ) f (X3 , X2 ) fe (v1 ) +
X
3

+ v2 ž F (X2 , X3 ) f (X1 , X3 ) fe (v2 ) ] dX3 .

(3.21)

Since v1 and v2 are arbitrary and refer to the same kind of particles (electrons), (3.21) takes the form f 1 = r1 kB T F (X1 , X3 ) f (X3 , X2 ) fe (v1 ) dX3 .
X
3

(3.22)

Taking into account the Coulomb force in the same approximation as (3.16) and assuming the correlation to exist only b etween the positions of the particles in space (rather than between velocities), we integrate both sides of (3.22) over d 3 v1 d 3 v2 . The result is g (r1 , r2 ) ne2 =- r1 kB T Here the function g (r1 , r2 ) =
v1 v2

r1 r3

1 g (r2 , r3 ) d 3 r3 . | r1 - r3 |

(3.23)

f (X1 , X2 ) d 3 v1 d 3 v2 .

(3.24)

We integrate Equation (3.23) over r1 and designate the function
2 g (r1 , r2 ) = g (r12 ) ,

where r12 = | r1 - r2 | .


3.2. Debye-Huckel Shielding Å

69

So we obtain the equation
2 g (r12 ) = -

ne2 kB T

r

3

2 g (r23 ) 3 d r3 . r13

Its solution is g (r) = where

c0 r exp - r rDH kB T 4 ne
1/2 2

,

(3.25)

rDH =

(3.26) is the Debye-Huckel radius or, more exactly, the electron DebyeÅ Huckel radius. Å The constant of integration c0 = - (Exercise 3.8). Substituting (3.27) in solution (3.25) gives the sought-after pair correlation function g (r) = - This formula shows that the Debye-Huckel radius is a characteristic length of the pair Å correlations in a fully-ionized equilibrium plasma. As one might have anticipated, the binary correlation function reproduces the shape of the actual potential of interaction, i.e. the shielded Coulomb potential: r 1 e2 exp - kB T r rDH . (3.28) 1 2 4 rDH n (3.27)


70

Chapter 3. Weakly-Coupled Systems

g (r) (r)

1 r exp - r rDH

. (3.29)

Astrophysical plasmas exhibit collective phenomena arising out of mutual interactions of many particles. Since the radius rDH is a characteristic length of pair correlations, 3 the number n rDH gives us a measure of the number of particles which can interact simultaneously. The inverse of this number is the so-called plasma parameter p = n r
3
DH

-1

.

(3.30)

This is a small quantity as well as it can be expressed in terms of the interaction parameter i (Exercise 3.1). In many astrophysical applications, the plasma parameter is really small. Thus, the number of particles inside the Debye-Huckel sphere is very Å large (Exercise 3.2). So the collective phenomena can be really important in astrophysical plasma in many places where it is weakly coupled.

3.3

Gravitational systems

A fundamental difference between the astrophysical plasmas and the gravitational systems lies in the nature of the gravitational force: there is no shielding to vitiate this long-range 1/r2 force. The collisional integral formally equals infinity.


3.4. Numerical Simulations

71

The conventional wisdom of such systems asserts that they can be described by the collisionless kinetic equation, the gravitational analog of the Vlasov equation. Comment: ==> 0 !!!

On the basis of what we have seen above, the collisionless approach in gravitational systems, i.e. the entire neglect of particle pair correlations, constitutes an uncontrolled approximation. Unlike the plasma, we cannot derive the next order correction to the collisionless equation in the perturbation expansion. We may hope to circumvent this difficulty by first identifying the mean field force F , acting at any given point in space and then treating fluctuations F away from the mean field force. However this is difficult to implement concretely because of the apparent absence of a clean separation of scales.

3.4

Comments on numerical simulations

The astrophysical plasma processes are typically investigated in well developed and distinct approaches. One approach, described by the Vlasov equation, is the collisionless limit used when collective effects dominate. In cases where the plasma dynamics is determined by collisional processes and where the self-consistent fields can be neglected, the Fokker-Planck approach is used. At the same time, it is known that


72

Chapter 3. Weakly-Coupled Systems

both collective effects and Coulomb collisions can play an essential role in a great variety of astrophysical phenomena.

Besides, collisions play the principal role in the physics of collisionless plasma. Taking collisions into account may lead not only to quantitative but also qualitative changes in the plasma behavior. Even in the collisionless limit, the kinetic equation is difficult for numerical simulations, and the `macroparticle' method is widely used algorithms. Instead of direct numerical solution of the kinetic equation, a set of ordinary differential equations for every macroparticle is solved. These equations are the characteristics of the Vlasov equation. In the case of a collisional plasma, the position of a macroparticle satisfies the usual equation of the collisionless case r dr = v(t) . dt (3.31)

However the momentum equation is modified owing to the Coulomb collisions. They are described by the Fokker-Planck operator (3.10) which introduces a friction and diffusion in velocity space. Thus it is necessary to find the effective collisional force Fc which acts on the macroparticles: v dv 1 = ( FL + Fc ) . dt m (3.32)

The collisional force can be introduced phenomenologically but a more mathematically correct approach can be constructed using the stochastic equivalence of the Fokker-Planck and Langevin equations. Sto chastic differential equations are regarded as an alternative


3.5. Practice: Exercises and Answers

73

to the description of astrophysical plasma in terms of distribution function. The Langevin approach allows one to overcome difficulties related to the Fokker-Planck equation and to simulate actual plasma processes, taking account of both collective effects and Coulomb collisions. Generally, if we construct a method for the simulation of complex processes in astrophysical plasma, we have to satisfy the following obvious but conflicting conditions. First, the method should be adequate for the task in hand. For a number of problems the simplified mo dels of the collisional integral can provide a correct description and ensure a desired accuracy. Second, the method should be computationally efficient. The algorithm should not be extremely time-consuming. In practice, some compromise between accuracy and complexity of the method should be achieved. A `recip e': the choice of a particular model of the collisional integral is determined by the importance and particular features of the collisional processes in a given astrophysical problem.

3.5

Practice: Exercises and Answers

Exercise 3.1. Show that the interaction parameter i is related to the plasma parameter p as follows: i = 1 4
2/3 p

.

(3.33)


74

Chapter 3. Weakly-Coupled Systems

Exercise 3.2. How many particles are inside the Debye-Huckel sphere Å in plasma of the solar corona? Answer. For an electron-proton plasma with T 2 × 106 K and n 2 × 108 cm-3 , the Debye-Huckel radius Å rDH = kT 8 e2 n
1/2

4.9

T n

1/2

0.5 cm .

(3.34)

The number of particles inside the Debye-Huckel sphere Å N
DH

=n

4 r 3

3
DH

108 .

(3.35)

Hence the typical value of plasma parameter in the corona is really small: p 10-8 . The interaction parameter is also small: i 10-6 . Exercise 3.3. Estimate the interaction parameter (3.2) in the interior of white dwarf stars (see also Exercise 1.3). Exercise 3.4. Let w = w (v, v) be the probability that a test particle changes its velocity v to v + v in the time interval t. The velocity distribution at the time t can be written as f (v, t) = Show that the Fokker-Planck equation follows from the Taylor series expansion of the function f (v, t) given by formula (3.36). f (v - v, t - t) w (v - v, v) d 3 v . (3.36)


3.5. Practice: Exercises and Answers

75

Exercise 3.5. Express the collisional integral in terms of the differential cross-sections of interaction between particles. Exercise 3.6. Show that the Fokker-Planck collisional model can be derived from the Boltzmann collisional integral under the assumption that the change in the velocity of a particle due to a collision is rather small. Exercise 3.7. The Landau integral is generally thought to approximate the Boltzmann integral for the 1/r potential to a `dominant order', i.e. to within terms of order 1/ln, where ln is the Coulomb logarithm. However this is not the whole truth. Show that the Landau integral approximates the Boltzmann integral to the dominant order only in parts of the velocity space.

Exercise 3.8. Find the constant of integration c 0 in formula (3.25). Exercise 3.9. Write and discuss the gravitational analog of the Vlasov equation. Answer. The basic assumption is that the gravitational N -body system can be described in terms of a statistically smooth distribution function f (X, t). The Vlasov equation manifests that this function will stream freely in the self-consistent gravitational potential (r, t) associated with f (X, t), so that f (X, t) f (X, t) f (X, t) + v - = 0. t r r v Here (3.37)


76

Chapter 3. Weakly-Coupled Systems

= - 4 G (r, t) and (r, t) = f (r, v, t) d 3 v .

(3.38)

(3.39)

Note that, in the context of the mean field theory, a distribution of particles over their masses has no effect. Applying for example to the system of stars in a galaxy, Equation (3.37) implies that the net gravitational force acting on a star is determined by the large-scale structure of the galaxy rather than by whether the star happens to lie close to some other star.

The force acting on any star does not vary rapidly, and each star is supposed to accelerate smoothly through the force field generated by the galaxy as a whole. In fact, gravitational encounters are not screened, they can be thought of as leading to an additional collisional term on the right side of the equation - a collisional integral. However very little is known mathematically about such possibility. Exercise 3.10. Discuss a gravitational analog of the Landau integral in the following form ^ f t =
c

v

v

2 | v - v | ž - vv v v

× (3.40)

× [ f (r, v, t) f (r, v , t) ] d 3 v . Here is a constant determined by the effective collision rate.


Chapter 4

Macroscopic Description of Astrophysical Plasma
In this Chapter we treat individual kinds of particles as continuous media, mutually penetrating charged gases which interact between themselves and with an electromagnetic field. This approach gives us the multi-fluid mo del which is useful to consider many properties of astrophysical plasmas, e.g., the solar wind.

4.1

Summary of microscopic description

The kinetic equation gives us a microscopic (though averaged in a statistical sense) description of plasma. Let us consider the transition to a less comprehensive macroscopic description. We start from the kinetic equation for particles of kind k fk (X, t) fk (X, t) + v + t r + F
k,

(X, t) fk (X, t) = mk v
77

^ fk t

.
c

(4.1)


78

Chapter 4. Macroscopic Description of Plasma

Here the statistically averaged force is F
k,

(X, t) =
lX
1

F

kl,

(X, X1 ) fl (X1 , t) dX1

(4.2)

and the collisional integral ^ fk t =-
c

J v

k,

(X, t) ,

(4.3)

where the flux of particles of kind k 1 F mk

J

k,

(X, t) =
lX
1

kl,

(X, X1 ) f kl (X, X1 , t) dX1

(4.4)

in the 6D phase space X = { r, v}.

4.2

Definition of macroscopic quantities

Before the deduction of equations for the macroscopic quantities or macroscopic transfer equations, let us define the following moments of the distribution function. (a) The zeroth moment (without multiplying the distribution function fk by the velocity v) fk (r, v, t) d 3 v = nk (r, t)
v

(4.5)

is the number of particles of kind k in a unit volume. It is related to the mass density of particles of kind k k (r, t) = mk nk (r, t) . The plasma mass density is accordingly


4.2. Definition of Macroscopic Quantities

79

(r, t) =
k

mk nk (r, t) .

(4.6)

(b) The first moment of the distribution function, i.e. the integral of the product of the velocity v to the first power and the distribution function fk , v fk (r, v, t) d 3 v = nk u
v k,

(4.7)

is the particle flux, i.e. product of the number density by their mean velocity u
k,

(r, t) =

1 nk

v fk (r, v, t) d 3 v .
v

(4.8)

Consequently, the mean momentum of particles of kind k in a unit volume is expressed as follows mk nk u
k,

=m

k v

v fk (r, v, t) d 3 v .

(4.9)

(c) The second moment of the distribution function is defined to be
(k)

(r, t) = mk
v

v v fk (r, v, t) d 3 v =
k, uk,

= mk nk u Here we have introduced

+p

(k)

.

(4.10)

v = v - u

k,

which is the deviation of the particle velocity from its mean value (4.8), so that v v = 0 ;


80

Chapter 4. Macroscopic Description of Plasma

and p = mk
v (k )

v v fk (r, v, t) d 3 v ,

(4.11)

is the pressure tensor. is the tensor of momentum flux for particles of kind k . Its component is the th component of the momentum transported by the particles of kind k , in a unit time, across the unit area perpendicular to the axis r . Once we know the distribution function fk (r, v, t), we can derive all macroscopic quantities related to these particles. So, higher moments of the distribution function will be introduced as needed.
(k) (k)

4.3

Macroscopic transfer equations

Note that the deduction of macroscopic equations is just derivation of the equations for the distribution function moments. 4.3.1

Equation for the zeroth moment

Let us calculate the zeroth moment of the kinetic equation: fk 3 dv+ t v
v

v

fk 3 d v+ r ^ fk t d 3v .
c

+
v

Fk, fk 3 dv= mk v

(4.12)

v

We interchange the order of integration over velocities and the differentiation with respect to time t in the first term and with respect to coordinates r in the second one.


4.3. Macroscopic Transfer Equations

81

Under the second integral v fk v = (v fk ) - fk = (v fk ) - 0 , r r r r

since r and v are indep endent variables in the phase space X . Taking into account that the distribution function quickly approaches zero as v , the integral of the third term is taken by parts and equals zero (Exercise 4.1). The integral of the right-hand side of (4.12) describes the change in the number of particles of kind k as a result of collisions with particles of other kinds. If the processes of transformation, during which the particle kind can be changed (such as ionization, recombination, charge exchange etc., see Exercise 4.2), are not allowed for, then the last integral is zero as well: ^ fk t d 3v = 0 .
c

(4.13)

v

Thus, by integration of (4.12), the following equation is found nk + nk u t r
k,

= 0. (4.14)

This is the continuity equation expressing the conservation of particles of kind k or (i.e. the same, of course) conservation of their mass: k + k u t r
k,

= 0.

(4.15)

Equation (4.14) for the zeroth moment nk depends on the unknown first moment uk, . This is illustrated by Fig. 4.1.


82

Chapter 4. Macroscopic Description of Plasma

f

k

f <>
X

k

f

kl

f

kln

LT

KE <>

BC
v

...

m m
Figure 4.1:

0

n n u

k

1

k

K E is the kinetic equation, m0 is the equation for the zeroth moment of the distribution function fk .
4.3.2

The momentum conservation law

Now let us calculate the first moment of the kinetic equation multiplied by the mass mk : m
k v

fk v d 3 v + t

+m

k v

v v



fk 3 d v+ r

v F
v

k,

fk 3 d v= v

=m

k v

v



^ fk t

d 3v .
c

(4.16)

With allowance made for the definitions (4.7) and (4.10), we obtain the momentum conservation law (mk nk u t
k,

)+

r



mk nk u

k, uk,

+ p -

(k)


4.3. Macroscopic Transfer Equations

83

-F
(k)

k,

(r, t)

v

=

F

(c) k,

(r, t)

v

.

(4.17)

Here p is the pressure tensor (4.11), i.e. a part of the unknown second moment (4.10). The mean force acting on the particles of kind k in a unit volume is F
k,

(r, t)

v

=
v

F

k,

(r, v, t) fk (r, v, t) d 3 v .

(4.18)

In the case of the Lorentz force, the mean force F or F
k, k,

(r, t)

v

= nk ek E +

1 ( uk × B ) c

(r, t)

v

= q E + k

1 ( j q × B ) . k c

(4.19)

q Here k and j q are the mean densities of electric charge and current, k produced by the particles of kind k .

Note that the mean electromagnetic force couples all the charged components of astrophysical plasma together because the electric and magnetic fields, E and B, act on all charged comp onents and, at the same time, all charged components contribute to the electric and magnetic fields according to Maxwell's equations. The right-hand side of Equation (4.17) contains the mean force resulting from collisions, the mean collisional force F
(c) k,

(r, t)

v

=m

k v

v



^ fk t

d 3v .
c

(4.20)


84

Chapter 4. Macroscopic Description of Plasma

Substituting (4.3) in definition (4.20) and integrating gives us the most general formula for the mean collisional force F =
l=k v v1 r1 (c) k,

(r, t)

v

=m

k v

J

k,

(r, v, t) d 3 v =

(4.21)

F

kl,

(r, v, r1 , v1 ) fkl (r, v, r1 , v1 , t) d 3 r1 d 3 v1 d 3 v .

Note that for the particles of the same kind, the elastic collisions cannot change the total particle momentum per unit volume. That is why l = k in the sum (4.21). Formula (4.21) contains the unknown binary correlation function fkl . The last should be found from the correlation function Equation (2.45) indicated as the second link B C in Fig. 4.2. Thus the equation for the first moment of the distribution function is as much unclosed as the initial kinetic equation. Therefore the equation for the first moment is unclosed in two directions. If each of kinds of particles is in thermodynamic equilibrium, then the mean collisional force can be expressed in terms of the mean momentum loss during the collisions of a particle of kind k with the particles of other kinds: mk nk (uk, - ul, ) . kl

F

(c) k,

(r, t)

v

=-
l=k

(4.22)


4.3. Macroscopic Transfer Equations

85

f

k

f

kl

f

kln

KE <>

BC
v

...

Figure 4.2:

m0 = n n m1 = u

k

k

f ...

kl

...

the first link two

m0 , m1 are equations for the two moments. The m1 is unclosed in directions.

- Here kl 1 = kl is the mean frequency of collisions between the particles of kinds k and l.

If u

k,

>u

l,

then the mean collisional force is negative:

the fast particles of kind k slow down by collisions with the slowly moving particles of other kinds. The force is zero, once the particles of all kinds have identical mean velocities. Therefore the mean collisional force, as well as the mean electromagnetic force, tends to make astrophysical plasma be a single hydrodynamic medium.


86

Chapter 4. Macroscopic Description of Plasma

4.4
4.4.1

The energy conservation law The second moment equation

The second moment of a distribution function fk is the tensor of momentum flux density
(k)

(r, t) = mk
v

v v fk (r, v, t) d 3 v =
k, uk,

= mk nk u

+p

(k)

.

In order to find an equation for this tensor, we should multiply the kinetic equation fk fk Fk + v + t r m
, k

fk = v

^ fk t

c

by the factor mk v v and integrate over velocity space v. In this way, we could arrive to a matrix equation in partial derivatives. If we take the trace of this equation we obtain the partial differential scalar equation for energy density of the particles. This is the correct self-consistent way which is the basis of the moment metho d. For our aims, a simpler direct procedure is sufficient and correct. In order to derive the energy conservation law, we multiply Equation (4.1) by the particle's kinetic energy
2 mk v /2

and integrate over velocities, taking into account that v = u and
k,

+ v ,

v

v

= 0,


4.4. The energy conservation law

87

2 v = u

2 k,

+ ( v ) + 2 u

2

k,

v .

A straightforward integration yields t r
2 k u k + k 2 k

+

+



k u

k,

2 uk (k ) + k + p u 2

k,

+q

k,

=

q = k ( E ž uk ) + Fk ž u

(c)

k

+Q

(c) k

(r, t) + Lk (r, t) .

(r )

(4.23)

Here

1 mk k (r, t) = nk = mk 2nk

v

mk (v )2 fk (r, v, t) d 3 v = 2
2

(v ) fk (r, v, t) d 3 v
v

(4.24)

is the mean kinetic energy of chaotic (non-directed) motion per single particle of kind k . Thus the first term on the left-hand side of (4.23) represents the time derivative of the energy of the particles of kink k in a unit volume, which is the sum of kinetic energy of a regular motion with the mean velocity uk and the so-called internal energy. As every tensor, the pressure tensor can be written as p = pk
(k )

+ .

(k )

(4.25)

On rearrangement, we obtain the following general equation t
2 k u k + k 2 k

+


88

Chapter 4. Macroscopic Description of Plasma

+

r



k u

k,

2 uk + wk + 2

(k)

u

k,

+q

k,

=

q = k ( E ž uk ) + Fk ž u

(c)

k

+Q

(c) k

(r, t) + Lk (r, t) .

(r)

(4.26)

Here wk = k + is the heat function per unit mass. Therefore the second term on the left-hand side contains the energy flux k u
k, 2 uk +w 2 k

pk k

(4.27)

,

which can be called the `advective' flux of kinetic energy. Let us mention the known astrophysical application of this term. The advective co oling of ions heated by viscosity might dominate the cooling by the electron-ion collisions, e.g., in a high-temperature plasma flow near a rotating black hole. In an advection-dominated accretion flow (ADAF), the heat generated via viscosity is transferred inward the black hole rather than radiated away locally like in a standard accretion disk model. However, discussing the ADAF as a solution for the important astrophysical problem should be treated with reasonable cautions. Looking at Equations (4.23) for electrons and ions separately, we see that too many assumptions have to be made to arrive to the ADAF approximation.


4.4. The energy conservation law

89

For example, this is not realistic to assume that plasma electrons are heated only due to collisions with ions and, for this reason, the electrons are much cooler than the ions. The suggestions underlying the ADAF model ignore several effects including reconnection and dissipation of magnetic fields (regular and random) in astrophysical plasma. This makes a physical basis of the ADAF model uncertain.

4.4.2

The case of thermo dynamic equilibrium

In order to clarify the definitions given above, let us, for a while, come back to the general principles. If the particles of the k th kind are in the thermo dynamic equilibrium, then fk is the Maxwellian function with the temperature Tk : f
(0) k

(r, v) = nk (r)

mk 2 kB Tk (r)
2

3/2

×

× exp -

mk | v - uk (r) | 2 kB Tk (r)

.

(4.28)

In this case, according to (4.24), the mean kinetic energy of chaotic motion per single particle of kind k mk k = 3 k Tk . 2B (4.29)

The pressure tensor (4.11) is isotropic: p = pk , where the scalar pk = nk kB Tk is the gas pressure of the particles of kind k . (4.31)
(k)

(4.30)


90

Chapter 4. Macroscopic Description of Plasma

This is also the equation of state for the ideal gas. Thus we have found that the pressure tensor is diagonal. This implies the absence of viscosity for the ideal gas: = 0 .
(k )

(4.32)

The heat function per unit mass or, more exactly, the specific enthalpy is wk = k + pk 5 k B Tk = . k 2 mk (4.33)

It was a particular case of the thermodynamic equilibrium.

4.4.3

The general case of anisotropic plasma

In general, we do not expect that the particles of kind k have reached thermodynamic equilibrium. Nevertheless we often use the mean kinetic energy (4.24) to define the effective kinetic temp erature Tk according to definition (4.29). A kinetic temperature is just a measure for the spread of the particle distribution in velocity space. The kinetic temperatures of different components in astrophysical plasma may differ from each other. Moreover, in an anisotropic plasma, the kinetic temperatures parallel and p erp endicular to the magnetic field are different. Without supposing thermodynamic equilibrium, in an anisotropic plasma, the part associated with the deviation of the distribution function from the isotropic one is distinguished in the pressure tensor: p
(k)

- pk



=

(k)

.

(4.34)


4.4. The energy conservation law

91

Here

(k)

is called the viscous stress tensor.

Recall that we did not derive an equation for this tensor. The term uk, in equation (4.23) represents the flux of energy released by the viscous force in the particles of kind k . The last term on the left-hand side of the energy equation, the vector q
k, (k )

=
v

mk (v )2 v fk (r, v, t) d 3 v 2

(4.35)

is the heat flux density due to the particles of kind k . Formula (4.35) shows that a third-order-moment term appears in the second order moment of the kinetic equation. The right-hand side of the energy conservation law (4.23) contains the following four terms: (a) The first term
q k ( E ž uk ) = nk ek E u k,

(4.36)

is the work done by the Lorentz force (without the magnetic field, of course) in unit time on unit volume. (b) The second term F
(c) k

ž uk = u

k, v

mk v



^ fk t

d 3v
c

(4.37)

is the work done by the collisional force of friction of the particles of kind k with all other particles in unit time on unit volume. The work of friction force results from the mean momentum change of particles of kind k (moving with the mean velocity uk ) owing to collisions with all other particles.


92

Chapter 4. Macroscopic Description of Plasma

This work equals zero if uk = 0. (c) The third term Qk (r, t) =
v (c)

mk (v )2 2

^ fk t

d 3v
c

(4.38)

is the rate of thermal energy release (heating or cooling) in a gas of the particles of kind k due to collisions with other particles. Recall that the collisional integral depends on the binary correlation function fkl . (d) The last term Lk (r, t) takes into account that a plasma component k can gain energy by absorbing radiations of different kinds and can lose the energy by emitting radiations.
(r)

4.5
4.5.1

General prop erties of transfer equations Divergent and hydro dynamic forms

Equations (4.14), (4.17), and (4.23) are referred to as the equations of particle, momentum and energy transfer. They are written in the `divergent' form. This essentially states the conservation laws and turns out to be convenient in numerical work, to construct the conservative schemes for computations. Sometimes, other forms are more convenient. For instance, the equation of momentum transfer or simply the equation of motion can be brought into the frequently used form:
k

uk t

,

+u

k,

uk, r
v

=-
(c) k,

p r
v

(k)

+ (4.39)

+F

k,

(r, t)

+F

(r, t)

.


4.5. Properties of the Transfer Equations

93

The so-called substantial derivative appears on the left-hand side of this equation: d (k) = +u dt t = + uk ž r t

k,

r

. (4.40)

This substantial or advective derivative - the total time derivative following a fluid element of kind k - is typical of hydrodynamictyp e equations, to which the equation of motion (4.39) belongs. In the frame, in which the fluid element is not moving, the mean velocity uk = 0 but the time partial derivative / t does not vanish of course. The total time derivative with respect to the mean velocity uk of the particles of kind k is different for each kind k . In the one-fluid MHD theory, we shall introduce the substantial derivative with respect to the average velocity of the plasma as a whole. For the case of the Lorentz force, the equation of motion of the particles of kind k can be rewritten as follows: d
k (k)

u dt

k,

=-

p r

(k)

q + k E +

1q ( j × B ) + ck (4.41)

+F

(c) k,

(r, t)

v

.

Here the right-hand side represents the forces acting on the fluid element of kind k , in particular, the last term is the mean collisional force. The left-hand side of (4.41) is the change of the momentum of this fluid element.


94

Chapter 4. Macroscopic Description of Plasma

4.5.2

Status of the conservation laws

As we saw above, when we treat a plasma as several continuous media (the mutually penetrating charged gases), for each of them,

the main three average properties (density, velocity, and a quantity like temperature) are governed by the basic conservation laws for mass, momentum, and energy in the media.

These conservation equations contain more unknowns than the number of equations. The transfer equations for local macroscopic quantities are as much unclosed as the initial kinetic equation (see K E in Fig. 4.3).
f f <>
X

k

k

f

kl

f

kln

LT

KE <>

BC
v

...

m0 = n n m1 = u m2 =

k

k

f f

kl

... ...

k

kl

K E and B C are the kinetic equation and the equation for the correlation function. m0 , m1 etc. are the chain of the equation for the moments.
Figure 4.3:

For example, formula for the mean collisional force contains the unknown correlation function fkl . The last should be found from the correlation function Equation (the second link B C in Fig. 4.3).


4.6. Equation of State and Transfer Coefficients

95

The terms (4.37) and (4.38) in the energy conservation equation also depend on the unknown function fkl . It is also important that the transfer equations are unclosed in `orthogonal' direction: Equation for the zeroth moment (the link m0 in Fig. 4.3), density nk , depends on the unknown first moment, the mean velocity uk , and so on. This pro cess of generating equations for the higher moments could be extended indefinitely depending solely on how many primary variables (nk , uk , k , ...) we are prepared to introduce. Anyway we know now that the conservation laws for mass, momentum, and energy in the components of astrophysical plasma represent the first three links in the chain of equations for the distribution function moments. It certainly would not be easy (if possible) to arrive to this fundamental conclusion and would b e difficult to derive the conservation laws in the form of the transfer Equations (4.14), (4.17), and (4.23) in the way which is typical for the ma jority of textbooks: from simple specific knowledge to more general ones.

4.6

Equation of state and transfer co efficients

The first three transfer equations for a plasma component k would be closed with respect to the three unknown variables k , uk , and k , (k) if it were possible to express the other unknown quantities pk , , ( qk) , etc. in terms of these three variables. Thus, we have to know the equation of state and the so-called transfer co efficients.


96

Chapter 4. Macroscopic Description of Plasma

How can we find them? Formally, we should write equations for higher moments of the distribution function. However these equations will not be closed either. So, how shall we proceed? According to the general principles of statistical physics, by virtue of collisions in a closed system of particles, any distribution function tends to assume the Maxwellian form. The Maxwellian distribution is the kinetic equation solution for a stationary homogeneous gas in the absence of any mean force in the thermal equilibrium, i.e. for a gas in thermo dynamic equilibrium. Then spatial gradients and derivatives with respect to time are zero. In fact they are always nonzero. For this reason, the assumption of full thermodynamic equilibrium is replaced with the lo cal thermodynamic equilibrium (LTE). Moreover if the gradients and derivatives are small, then the real distribution function differs little from the local Maxwellian one, the difference being prop ortional to the small gradients or derivatives. If we are interested in a process occurring in a time t, which is much greater than the collision time , and at a distance L, which is much larger than the mean free path , t , L , (4.42)

then the distribution function fk (r, v, t) is a sum of the lo cal Maxwellian distribution


4.6. Equation of State and Transfer Coefficients

97

f

(0) k

(r, v, t) = nk (r, t)

mk 2 kB Tk (r, t)
2

3/2

×

mk | v - uk (r, t) | × exp - 2 kB Tk (r, t) and some small additional term f Therefore fk (r, v, t) = fk (r, v, t) + f Since the function fk (0) (0) fk / t and fk / r .
(0) (0) (1) k (1) k

(4.43)

(r, v, t).

(r, v, t) .

(4.44)

depends on t and r, we find the derivatives

By using these derivatives, we substitute function (4.44) in the kinetic equation and linearly approximate the collisional integral by using one or another of the models introduced in Chapter 3; see also Exercise 4.5. Then we seek the additional term f
(1) k

in the linear approximation.

For example, in the case of the heat flux q , the flux q is chosen to be proportional to the temperature gradient. Thus, in a fully ionized plasma without magnetic field, the heat flux in the electron component of plasma qe = - where e 1.84 × 10-5 T ln
5/2 e e

Te ,

(4.45)

(4.46)

is the coefficient of electron thermal conductivity. In the presence of strong magnetic field in astrophysical plasma, all the transfer coefficients become highly anisotropic.


98

Chapter 4. Macroscopic Description of Plasma

Since the Maxwellian function and its derivatives are determined by the parameters nk , uk , and Tk , the transfer co efficients are expressed in terms of the same quantities and magnetic field B , of course. This procedure makes it possible to close the set of transfer equations for astrophysical plasma under the conditions (4.42).

The first three moment equations were extensively used in astrophysics, for example, in investigations of the solar wind. They led to a significant understanding of phenomena such as escape, acceleration and co oling. However, as more detailed observations become available, it appeared that the collisionally dominated models are not adequate for most physical states of the solar wind. A higher order, closed set of equations for the six moments have been derived for multi-fluid, mo derately non-Maxwellian plasma of the solar wind. On this basis, the generalized expression for heat flux relates the flux to the temperature gradients, relative streaming velocity, thermal anisotropy, temperature differences of the components.

4.7

Gravitational systems

There is a big difference between astrophysical plasmas and astrophysical gravitational systems (Sect. 3.3). The gravitational attraction cannot b e screened. A large-scale gravitational field always exists over a system because the neutrality condition (3.17) cannot be satisfied.


4.7. Gravitational Systems

99

The large-scale gravitational field makes an overall thermodynamic equilibrium impossible. Therefore those results of plasma astrophysics which explicitly depend upon the plasma being in thermodynamic equilibrium do not hold for gravitational systems. For systems, like the stars in a galaxy, we may hope that the observed distribution function reflects something about the initial conditions rather than just the relaxation mechanism. So galaxies may be providing us with clues on how they were formed.

If we consider processes on a spatial scale which is large enough to contain a large number of stars then one of the main requirements of the continuum mechanics is justified. Anyway, several aspects of the structure of a galaxy can be understood in hydro dynamic approximation. More often than never, hydrodynamics provides a first level description of an astrophysical phenomenon governed predominantly by the gravitational force. For example, the early stages of star formation during which an interstellar cloud of low density collapses under the action of its own gravity can be modeled in the hydrodynamic approximation. However, when we want to explain the difference between the angular momentum of the cloud and that of the born star, we have to include the effect of a magnetic field.


100

Chapter 4. Macroscopic Description of Plasma


Chapter 5

The Generalized Ohm's Law in Plasma
The multi-fluid models of astrophysical plasma allow us to derive the generalized Ohm's law and to consider different physical approximations, including the collisional and collisionless plasma models.

5.1

The classic Ohm's law
j = E ,

The usual Ohm's law, relates the current j to the electric field E in a solid conductor in rest. As we know, the electric field in every equation of motion determines acceleration of particles rather than their velocity. That is why, generally, such a simple relation as the classic Ohm's law does not exist. Moreover, while considering astrophysical plasmas, it is necessary to take into account the presence of a magnetic field and the motion of a plasma as a whole or as a medium consisting of several moving components, their compressibility.
101


102

Chapter 5. Generalized Ohm's Law

Recall the way of deriving the usual Ohm's law. The current is determined by the relative motion of electrons and ions. Let us assume that the ions do not move. An equilibrium is set up between the electric field action and electronson-ions friction: 0 = - e ne E + me ne ei ( 0 - u resulting in Ohm's law j = - e ne u Here = is the electric conductivity. In order to deduce the generalized Ohm's law for a plasma with magnetic field, we have to consider at least two equations of motion - for the electron and ion components.
e, e,

),

=+

e2 ne E = E . me ei

(5.1)

e2 ne me ei

(5.2)

5.2

Derivation of basic equations

Let us write the momentum equations for electrons and ions: me (ne u t
e,

1 )=- - ene E + ( ue × B ) r c + me ne ei (u
(i) i,

(e)

+


-u

e,

), 1 ( ui × B ) c

(5.3)

mi (ni ui, ) = - + Zi en t r

i

E+

+



5.2. Derivation of Basic Equations

103

+ me ne ei (u Here the tensor of momentum flux

e,

- ui, ) .

(5.4)

(r, t) = me ne u and (r, t) = mi ni u
(i)

(e)

e,

u

e,

+ p
(i)

(e)

(5.5) (5.6)

i,

u

i,

+ p .

The last term in (5.3) represents the mean momentum transferred, because of collisions, between electrons and ions. It is equal, with opposite sign, to the last term in Equation (5.4). We assume that there are just two kinds of particles, their total momentum remaining constant under the action of elastic collisions. Now let us suppose that the ions are protons (Zi = 1), and electrical neutrality occurs: ni = ne = n . Let us multiply (5.3) by -e/me and add it to (5.4) multiplied by e/mi . The result is [ en (u t 1 1 +e n + me m
2 i,

-u

e,

)] =

e F mi

i,

-

e F me

e,

+

i

e2 n E + c
i,

ue ×B me )+ me (u mi F

+


ui ×B mi
e,

-


- ei en (u Here F
e,

-u
(e)

e,

i,

-u

).

(5.7)

=- r

and

i,

= . r

(i)

(5.8)


104

Chapter 5. Generalized Ohm's Law

Let us introduce the velocity of the centre-of-mass system u= Since m
i

mi u i + me u e . mi + me me ue ui . mi (5.9)

me , u = ui +

On treating Equation (5.7), we neglect the small terms of the order of the ratio me /mi . We obtain the equation for the current j = en (u i - u e ) in the system of coordinates (5.9). This equation is j e2 n e 1 = (j × B)- E + (u × B) - t me c me c - ei j + e e Fi - Fe . mi me (5.10)

The prime designates the current in the system of moving plasma, i.e. in the rest-frame of the plasma. Let E u denote the electric field in this frame of reference, i.e. Eu = E + 1 u × B. c (5.11)

Now we divide Equation (5.10) by ei and represent it in the form j= (e) e2 n Eu - B j × n - me ei ei e e Fi - Fe . mi me (5.12)

-

1 1 j + ei t ei


5.2. Derivation of Basic Equations

105

Here n = B/B and
(e) B =

eB me c

is the electron gyro-frequency. Thus we have derived a differential equation for the current j . The third and the fourth terms on the right do not depend of magnetic field. Let us replace them by some effective electric field E where =
eff

=-

e 1 j + ei t ei

1 1 Fi - Fe , mi me

(5.13)

e2 n me ei

(5.14)

is the plasma conductivity in the absence of magnetic field. Combine the fields (5.11) and (5.13), E = Eu + E in order to rewrite (5.12) in the form
(e) B j = E - j × n. ei eff

,

(5.15)

(5.16)

We shall consider (5.16) as an algebraic equation in j , neglecting the j / t dependence of the field (5.13). Note, however, that the term j / t is by no means small in the problem of the particle acceleration by a strong electric field in astrophysical plasma.


106

Chapter 5. Generalized Ohm's Law

Collisionless reconnection is the phenomenon in which particle inertia of the current replaces classical resistivity in allowing fast reconnection to occur.

5.3

The general solution

Let us find the solution to Equation (5.16) as a sum j = E + E + H n × E . Substituting (5.17) in (5.16) gives == e2 n , me ei 1 1+
B

(5.17)

(5.18)

=

(e)
B



2 ei

,

(5.19)

(e)

H =

ei
2

1 + B ei

(e)

.

(5.20)

Formula (5.17) is called the generalized Ohm's law. A magnetic field in a plasma not only changes the magnitude of the conductivity, but the form of Ohm's law as well: the electric field and the resulting current are not parallel, since = . Thus the conductivity of a plasma in a magnetic field is anisotropic. Moreover the current component j magnetic and electric fields.
H

is perpendicular to b oth the

This component is the so-called Hall current (Fig. 5.1).


5.4. Conductivity of Magnetized Plasma

107

B6 E
T T U

E

The direct (j and j ) and Hall's (j H ) currents in a plasma with electric (E ) and magnetic (B) fields.
Figure 5.1:

j nT
Q







j
E

E

j

E

H

5.4
5.4.1

The conductivity of magnetized plasma Two limiting cases


The magnetic-field influence on the conductivity conductivity H is determined by the parameter
(e)
B

and on the Hall

ei .

This is the turning angle of an electron on the Larmor circle in the intercollisional time. Let us consider two limiting cases. (a) The turning angle be small:
(e)
B

ei

1.

(5.21)

Obviously this corresponds to the weak magnetic field or dense co ol plasma, so that the electric current is scarcely affected by the magnetic field: = , H (e) B ei 1. (5.22)

Thus the usual Ohm's law with isotropic conductivity holds.


108

Chapter 5. Generalized Ohm's Law

(b) The opposite case, when the electrons spiral freely between rare collisions of electrons with ions:
(e)
B

ei

1,

(5.23)

corresponds to the strong magnetic field and hot rarefied plasma. This plasma is termed the magnetized one. It is frequently encountered under astrophysical conditions. In this case =
(e)
B



ei

(e) H B ei .

2

(5.24)

Hence in a magnetized plasma, for example in the solar corona In other words, the impact of the magnetic field on the direct current is especially strong for the component resulting from the electric field E . The current in the E direction is considerably weaker than it would be in the absence of a magnetic field. Why? 5.4.2 H . (5.25)

The physical interpretation

The physical mechanism of the perpendicular current j is illustrated by Fig. 5.2. The primary effect of the electric field E in the presence of the magnetic field B is not the current in the direction E , but rather the electric drift in the direction perpendicular to both B and E .


5.4. Conductivity of Magnetized Plasma
+ p E j vd

109

B

e

1 3 2

vd

u e

Initiation of the current in the direction of the perpendicular field E as the result of rare collisions (1, 2, 3, ...).
Figure 5.2:

The electric drift velocity is independent of the particle's mass and charge. The electric drift of electrons and ions generates the motion of the plasma as a whole with the velocity v = vd = c E×B . B2 (5.26)

This would be the case if there were no collisions at all. Collisions, even the rare ones, disturb the Larmor motion, leading to a displacement of the ions (not shown in Fig. 5.2) along the field E , and the electrons in the opposite direction (Fig. 5.2). The small electric current j appears in the direction E . To ensure the current across the magnetic field, the electric field is necessary, i.e. the electric field component perpendicular to both the current j and the field B.


110

Chapter 5. Generalized Ohm's Law

The Hall electric field balances the Lorentz force acting on the carriers of the perpendicular electric current in a rarely collisional plasma due to the presence of a magnetic field, i.e. the force F ( j ) = en u c
i

×B-

en u c

e

×B=

=

1 e n (u c

i

-u

e

)×B

(5.27)

Hence the magnitude of the Hall electric field is EH = 1 j × B. en c (5.28)

The Hall electric field in astrophysical plasma is frequently set up automatically, as a consequence of small charge separation within the limits of quasi-neutrality. In a fully-ionized rarely-collisional plasma, the tendency for a particle to spiral round the magnetic field lines insures the great reduction in the transversal conductivity. However, since the dissipation of the energy of the electric current into Joule heat, jE, is due solely to collisions between particles (if the particle acceleration can be neglected), the reduced conductivity does not lead to increased dissipation. On the other hand, the Hall electric field and Hall current can significantly modify conditions of magnetic reconnection. Compared with ordinary resistive MHD, the Hall MHD reconnection is distinguished by qualitatively different magnetic field distributions, electron and ion signatures in reconnecting current layers.


5.5. Currents and Charges in Plasma

111

Although the Hall effect itself is nondissipative, jH E = 0 , (5.29)

it can lead to dissipation through a turbulent "Hall cascade", magnetic energy cascading from large to small scales, where it dissipates by ohmic decay. The Hall effect can dominate ohmic decay of currents in the crust of neutron stars and therefore can determine evolution of their magnetic field. In an initial poloidal dip ole field, the toroidal currents "twist" the field. The resulting poloidal currents then generate a quadrup ole poloidal field.

5.5
5.5.1

Currents and charges in plasma Collisional and collisionless plasmas

Let us point out another property of the generalized Ohm's law. Under laboratory conditions, as a rule, one cannot neglect the gradient forces. On the contrary, these forces usually play no part in astrophysical plasma. We shall often ignore them. However this simplification may be not justified in reconnecting current layers, shock waves and other discontinuities. Let us also restrict our consideration to very slow (say hydrodynamic) motions of plasma. These motions are supposed to be so slow that the following three conditions are fulfilled. (A) It is supposed that


112

Chapter 5. Generalized Ohm's Law

=

1



ei

or

ei

1,

(5.30)

where is a characteristic time of the plasma motions. Thus departures of actual distribution functions for electrons and ions from the Maxwellian distribution are small. This allows us to handle the transfer phenomena in linear approximation. Moreover, if a single-fluid model makes a sense, the electrons and ions could have comparable temperatures, ideally, the same one T which is the temperature of the plasma as a whole: Te = Tp = T . (B) We neglect the electron inertia in comparison with that of the ions and make use of (5.9). This condition is usually written in the form Thus the plasma motions have to be so slow that their frequency is smaller than the lowest gyro-frequency of the particles. Recall that the gyro-frequency of ions (C) The third condition
(i)
B



(i)
B

=

eB . mi c

(5.31)



(e)
B

.


5.5. Currents and Charges in Plasma

113



(e)
B

ei

1.

(5.32)

Hence we can use the isotropic conductivity . The generalized Ohm's law assumes the form which is specific to the ordinary magnetohydro dynamics (MHD): j = E+ 1 u×B . c (5.33)

The MHD approximation is the sub ject of the next chapter. Numerous applications of MHD to astrophysical plasma should be discussed in the remainder of the lectures.

In the opposite case, when the parameter
(e)
B

ei

1,

charged particles revolve around magnetic field lines, and a typical particle may spend a considerable time in a region of a size of the order of the gyroradius. Hence, if the length scale of a phenomenon is much larger than the gyroradius, we may expect the hydro dynamic-typ e models to work. It appears that, even when the parameter
(e)
B

ei ,

(like in the solar corona) and collisions are negligible, the 2D quasihydrodynamic description of plasma, the Chew-Goldb erger-Low(CGL) approximation is quite useful. This is because a strong magnetic field makes a plasma, even a collisionless one, more `interconnected', more hydrodynamic in the directions perpendicular to the magnetic field.


114

Chapter 5. Generalized Ohm's Law

As for the motion of particles along the magnetic field, some important kinetic features still are significant. Chew et al.: "A strictly hydrodynamic approach to the problem is appropriate only when some special circumstance suppresses the effects of pressure transp ort along the magnetic lines". There is ample experimental evidence that strong magnetic fields do make astrophysical plasmas behave like hydrodynamic charged fluids. This does not mean, of course, that there are no pure kinetic phenomena in such plasmas. There are many of them indeed. The most interesting of them is magnetic reconnection in the solar corona and solar wind.


5.5. Currents and Charges in Plasma

115

5.5.2

Volume charge and quasi-neutrality

While deriving the generalized Ohm's law, the exact charge neutrality of plasma was assumed: Zi ni = ne ,
i

i.e. the absolute absence of the volume charge in plasma: q = 0 . However there is no need for such a strong restriction. It is sufficient to require quasi-neutrality, i.e. Zi n i - n
i e

n- e

1

1.

So the volume charge density has to be small in comparison to the plasma density. Once the volume charge density q = e
i

Zi ni - ne = 0 ,

yet another term must be taken into account in the Ohm's law: jq = q u . u This is the so-called convective current. It must be added to the conductive current (5.17). The volume charge, the associated electric force q E and the convective current q u are of great importance in electrodynamics of relativistic ob jects such as black holes and pulsars. (5.34)


116

Chapter 5. Generalized Ohm's Law

Charge-separated plasmas originate in magnetospheres of pulsars and rotating black holes, e.g., a sup er-massive black hole in active galactic nuclei (AGN). A strong electric field appears along the magnetic field lines. The parallel electric field accelerates migratory electrons and/or positrons to ultra-relativistic energies.

Volume charge can be evaluated in the following manner. From Maxwell's equation div E = 4 we estimate q E . 4 L (5.35)
q

On the other hand, the equation of plasma motion yields ene E so that E kB T . eL (5.36) p n e kB T , L L

On substituting (5.36) in (5.35), we find q kT 1 1 1 B =2 ene eL 4 L ene L or r2 q DH . ene L2 kB T 4 e2 n

e

(5.37)


5.6. Practice: Exercises and Answers

117

Since the usual concept of plasma implies that the Debye radius rDH L, (5.38)

the volume charge density is small in comparison with the plasma density. When we consider phenomena with a length scale L much larger than the Debye radius rDH and a time scale much larger than the inverse the plasma frequency, the charge separation can b e neglected.

5.6

Practice: Exercises and Answers

Exercise 5.1 Consider a plasma system with given distributions of magnetic and velocity fields. Is it possible to use Equation (5.12) in order to estimate the growth rate of electric current and, as a consequence, of magnetic field in such a system, e.g., a protostar? Exercise 5.2 Evaluate the characteristic value of the parallel conductivity in the solar corona. Answer. It follows from formula (5.18) that 1016 - 1017 , s-1 . (5.39)

(e) Exercise 5.3 Estimate the parameter B ei in the corona above a sunspot. Answer. Just above a large sunspot the field strength can be as high as B 3000 G . With ep 0.1 s, we obtain



(e)
B

ei 1010 rad

1.


118

Chapter 5. Generalized Ohm's Law

So, for anisotropic conductivity in the solar corona, the approximate formulae (5.24) can be well used.


Chapter 6

Single-Fluid Mo dels for Astrophysical Plasma
Single-fluid models are the simplest but sufficient approximation to describe many large-scale low-frequency phenomena in astrophysical plasma: motions driven by strong magnetic fields, accretion disks, and relativistic jets.

6.1
6.1.1

Derivation of the single-fluid equations The continuity equation

In order to consider a plasma as a single medium, we have to sum each of the three transfer equations over all kinds of particles. Let us start from the continuity equation nk + nk u t r
k,

= 0.

With allowance for the definition of the plasma mass density , we have + div t k u
k k

= 0.

(6.1)

119


120

Chapter 6. Single-Fluid Models

The mean velocities of motion for all kinds of particles are supposed to be equal to the plasma hydro dynamic velocity: u1 (r, t) = u2 (r, t) = ž ž ž = u (r, t) , as a result of action of the mean collisional force. However this is not a general case. In general, the mean velocities are not the same, but a frame of reference can be chosen in which u =
k

(6.2)

k uk .

(6.3)

Then from (6.1) and (6.3) we obtain the usual continuity equation + div u = 0 . t We shall consider both cases.

(6.4)

6.1.2

The momentum conservation law

In the same way, we handle the momentum equation k d
(k)

u dt

k,

=-

(k) 1q q p + k E + ( jk × B ) + r c +F
(c) k,

(r, t)

v

.

On summing over all kinds of particles, we obtain the equation 1 d u =- p + q E + ( j × B ) + dt r c +
k

F

(c) k,

(r, t)

v

.

(6.5)


6.1. Single-Fluid Equations

121

Here the volume charge is q =
k

nk ek =

1 div E , 4

(6.6)

and the electric current is j=
k

nk ek uk =

c 1 E rot B - . 4 4 t

(6.7)

The electric and magnetic fields, E and B, are averaged fields associated with the total electric charge density q and the total current j. They satisfy the macroscopic Maxwell equations. Since elastic collisions do not change the total momentum, F
k (c) k,

(r, t)

v

= 0.

(6.8)

On substituting (6.6)-(6.8) in Equation (6.5), the latter gives the momentum conservation law d u =- p + F (E, B) . dt r

(6.9)

Here the electromagnetic force is written in terms of the electric and magnetic fields: F (E, B) = - The tensor M = 1 1 -E E - B B + (E 2 + B 2 ) 4 2 (6.11) ( E × B ) - M . t 4 c r (6.10)

is the Maxwellian tensor of stresses. The divergent form of the momentum conservation law is


122

Chapter 6. Single-Fluid Models

( E × B ) u + t 4 c

+

( r



+M



) = 0. (6.12)

The operator / t acts on two terms: u is the momentum of the plasma in a unit volume, E × B/4 c is the momentum of the electromagnetic field. The divergency operator / r acts on = p + u u , which is the momentum flux tensor =
k

(6.13)

,

(k )

(6.14)

see definition (4.10). Thus the pressure tensor p where p=
k

= p



+ p
k



,

(6.15)

is the total plasma pressure, the sum of partial pressures, and


=
k



(k)

(6.16)

is the viscous stress tensor which allows for the transport of momentum from one layer of the plasma flow to the other layers so that relative motions inside the plasma are damped out. The momentum conservation law (6.12) is applied for a wide range of conditions in plasmas like fluid relativistic flows, for example, astrophysical jets.


6.1. Single-Fluid Equations

123

The assumption that the astrophysical plasma behaves as a continuum medium is excellent in the cases in which we are often interested: the Debye length and the Larmor radii are much smaller than the plasma flow scales. On the other hand, going from the multi-fluid description to a singlefluid model is a serious damage because we loose an information not only on the small-scale dynamics of the electrons and ions but also on the high-frequency processes in plasma. The single-fluid equations describe well the low-frequency large-scale behavior of plasma in astrophysical conditions.

6.1.3

The energy conservation law

In a similar manner as above, the energy conservation law is derived. We sum Equation (4.23) over k and then substitute in the resulting equation the total electric charge (6.6) and the total electric current (6.7) expressed in terms of the electric field E and magnetic field B. The following divergent form of the energy conservation law is obtained: t u2 E2 + B + + 2 8 + r u
2

+ u + (6.17)

u2 c +w + ( E × B ) + 2 4
(c) ff



+ q ] = u F

+L

(rad)

(r, t) .

On the left-hand side of this equation, an additional term has appeared:


124

Chapter 6. Single-Fluid Models

the operator / t acts on the energy density of the electromagnetic field W= E2 + B2 . 8 (6.18)

The divergency operator / r acts on the Poynting vector, the electromagnetic energy flux G= c [E × B] . 4 (6.19)

The right-hand side of Equation (6.17) contains the total work of friction forces in unit time on unit volume u F
(c) ff

=
k

Fk, u fk t

(c)

k,

=

=
k

u

k, v

mk v



d 3v .
c

(6.20)

This work related to the relative motion of the plasma components is not zero. Recall that we consider general case (6.3). By contrast, the total heat release under elastic collisions between particles of different kinds is Q
k (c) k

(r, t) =
k v

mk (v )2 2

fk t

d 3v = 0 .
c

(6.21)

Elastic collisions in a plasma conserve both the total momentum and the total energy. If we accept condition (6.2) then the collisional heating (6.20) by friction force is also equal to zero. In this limit, there is not any term which contains the collisional integral.


6.1. Single-Fluid Equations

125

Elastic collisions have done a go o d job. Inelastic collisions are important in radiative cooling and heating. In optically thin plasma with collisional excitations of ions, the power of radiation from a unit volume of plasma is proportional to the square of plasma density n (cm-3 ):

L

(rad)

- n2 q (T ) .

(6.22)

The function q (T ) is called the radiative loss function. It depends strongly on the temperature T but weakly on the plasma density n (Fig. 6.1).

Radiative loss function vs. temperature at fixed values of plasma density: 109 cm-3 (solid curve), 1010 cm-3 (dotted curve), 1011 cm-3 (dashed curve).
Figure 6.1:


126

Chapter 6. Single-Fluid Models

6.2
6.2.1

Basic assumptions and the MHD equations Old simplifying assumptions

As we saw above, the transfer equations determines the behavior of different kinds of particles in a plasma once two conditions are complied with: (a) many collisions occur in a characteristic time of a phenomenon under consideration: c , (6.23)

(b) the mean free path c is significantly smaller than the distance L, over which macroscopic quantities change considerably: L c . (6.24)

Once these conditions are satisfied, we can close the set of transfer equations, as was discussed in Sect. 4.6. While considering the generalized Ohm's law, other three assumptions have been made. The first condition can be written in the form ei , (6.25)

where ei is the electron-ion collisional time, the longest collisional relaxation time. Thus the electrons and ions have comparable temperatures, ideally, the same temp erature T . Second, we neglect the electron inertia in comparison with that of the ions. This condition is usually written as
(i) -1
B

,

where



(i)
B

=

eB . mi c

(6.26)


6.2. MHD Equations

127

Thus the plasma motions have to be so slow that their frequency = 1/ is smaller than the lowest gyro-frequency of the particles. The third condition,
(e)
B

ei

1,

(6.27)

is necessary to write down Ohm's law in the form j= E+ 1 v × B + q v . c (6.28)

Here v is the velocity of plasma, E and B are the electric and magnetic fields in the `laboratory' system of coordinates, where we measure the velocity v. Accordingly, the field Ev = E + 1 v×B c (6.29)

is the electric field in a frame of reference related to the plasma. Complementary to the restriction (6.24) on the characteristic length L, we have to add the condition L where r
DH

r

DH

,

(6.30)

is the Debye-Huckel radius. Å

Then the volume charge q is small in comparison with the plasma density . Under the conditions listed above, we use the general hydrodynamictype equations: the conservation laws for mass (6.4), momentum (6.5) and energy (6.17). The general hydrodynamic-type equations have a much wider area of applicability in astrophysics than the equations of ordinary MHD derived below.


128

Chapter 6. Single-Fluid Models

The latter will be simpler than the equations derived above. Therefore additional simplifying assumptions are necessary. Let us introduce them. 6.2.2

New simplifying assumptions

First assumption: the conductivity is large, the electromagnetic processes being not very fast. Then, in the Maxwell's equation rot B = 4 1 E j+ , c c t

we ignore the displacement current in comparison to the conductive one. The corresponding condition is found by evaluating the currents as follows 1E c Thus 4 . 4 j c or E 4 E .

(6.31)

In the same order to the small parameter / , we neglect the convective current in comparison with the conductive current in Ohm's law. Actually, q v v div E 1 LE 1 E 4 L 4 4 E ,

once the condition (6.31) is satisfied.


6.2. MHD Equations

129

The conductivity of astrophysical plasma is often very high (Exercise 5.1). This is why condition (6.31) is satisfied up to frequencies close to optical ones. Neglecting the displacement current and the convective current, Maxwell's equations and Ohm's law result in the following relations: j= E=- c rot B , 4 (6.32) (6.33) (6.34) (6.35) (6.36)

1 c v×B+ rot B , c 4 1 div ( v × B ) , 4 c div B = 0 ,

q = -

B c2 = rot ( v × B ) + B. t 4

Once B and v are given, the current j, the electric field E, and the volume charge q are determined by formulae (6.32)--(6.34). Thus the problem is reduced to finding the interaction of two vector fields: the magnetic field B and the hydrodynamic velocity field v. As a consequence, the approach under discussion is known as magnetohydro dynamics (MHD). The corresponding equation of motion is obtained by substitution of (6.32)-(6.34) in the equation of momentum transfer (6.5). With the viscous forces as usually written in hydrodynamics, we have


130

Chapter 6. Single-Fluid Models



1 dv = - p + q E - B × rot B + dt 4 +v + + 3 div v . (6.37)

Here is the first viscosity coefficient, is the second viscosity coefficient (see Landau and Lifshitz, Fluid Mechanics ). Formulae for these coefficients and the viscous forces should be derived from the moment equation for the pressure tensor.

The second additional assumption has to be introduced now. Treating Equation (6.37), the electric force q E can be ignored in comparison to the magnetic one if v
2

c2 ,

(6.38)

that is in the non-relativistic approximation. To make certain that this is true, evaluate the electric force q E and the magnetic force 1 B2 1 | B × rot B | . 4 4 L (6.40) 1 vB vB B 2 1 v2 4 c L c 4 L c2 (6.39)

Comparing (6.39) with (6.40), we see that the electric force is a factor of v 2 /c2 short of the magnetic one. In a great number of astrophysical applications, the plasma velocities fall far short of the speed of light. The Sun is a good case in point.


6.2. MHD Equations

131

The largest velocities in coronal mass ejections (CMEs) do not exceed 3 × 108 cm/s. Thus, we neglect the electric force acting upon the volume charge in comparison with the magnetic force. However the relativistic ob jects like accretion disks near rotating black holes (Novikov and Frolov, 1989), and pulsar magnetospheres are at the other extreme. The electric force plays a crucial role in electrodynamics of relativistic ob jects. 6.2.3

Non-relativistic MHD

With the assumptions made above (2 + 3 + 2), the considerable simplifications have been obtained; and now we write the following set of equations of non-relativistic MHD: v = - , t r B = rot ( v × B ) + m B , t div B = 0 , + div v = 0 , t t B2 v 2 + + 2 8 = - div G , (6.41) (6.42) (6.43) (6.44) (6.45) (6.46)

p = p (, T ) .

The momentum of electromagnetic field does not appear on the lefthand side of (6.41).


132

Chapter 6. Single-Fluid Models

It is negligibly small in comparison to the plasma momentum v . This fact is a consequence of neglecting the displacement current. On the right-hand side of (6.41), the asterisk refers to the total momentum flux tensor , which equals


= v v + p



v - +

1 + 4 In Equation (6.42)

B2 2



- B B



.

(6.47)



m

=

c2 4

(6.48)

is the magnetic viscosity. It plays the same role as the kinematic viscosity = / in the equation of motion. The vector G is defined as the energy flux G = v v2 1 +w + [ B × ( v × B ) ] - 2 4 T. (6.49)



-

m v ( B × rot B ) - v - 4



The Poynting vector as a part in (6.49) is GP = 1 m B × (v × B) - B × rot B . 4 4 (6.50)

The energy flux due to friction is written as the contraction of the v velocity vector v and the viscous stress tensor .


6.2. MHD Equations

133

6.2.4

Energy conservation

The non-relativistic MHD equations are frequently used to model solar flares, eruptive prominences, etc. A goal of such studies is to deduce how energy of magnetic field is stored and then suddenly released to drive these phenomena. However most models use a simple energy equation, the discussion often centers on the over-simplified interpretation or just comparison of magnetic field structure in the models with corresponding features observed in emission. With new capabilities to study X-ray and EUV emission from Hino de and complementary observations from SOHO, RHESSI and other satellites, the models advance to more quantitative results. We have to develop the MHD models that include radiative losses and other dissipative processes, the energy transport by anisotropic heat conduction. The equation of state (6.46) can be rewritten in other thermodynamic variables. In order to do this, we have to make use of the thermodynamic identities d = T ds + p d 2 and dw = T ds + 1 dp .

Here s is the entropy p er unit mass. We transform the energy conservation law (6.45) from the divergent form to the hydrodynamic one: T ds m v v = (rot B)2 + + dt 4 r + div T + L
(rad)

(r, t) .

(6.51)


134

Chapter 6. Single-Fluid Models

Thus the heat abundance change dQ = T ds in a moving element of unit volume is a sum of the Joule and viscous heating, conductive heat redistribution and radiative cooling.

6.2.5

Relativistic magnetohydro dynamics

Relativistic MHD models are of considerable interest in several areas of modern astrophysics. The theory of gravitational collapse and models of sup ernova explosions are based on relativistic hydrodynamics for a star. The effects of deviations from spherical symmetry due to magnetic field require the use of relativistic MHD models. Relativistic hydrodynamics is presumably applied to the so-called quark-gluon plasma which is the primordial state of hadronic matter in the Universe. When the medium interacts electromagnetically and is highly conducting, the simplest description is in terms of relativistic MHD. From the mathematical viewpoint, the relativistic MHD was mainly treated in the framework of general relativity. This means that the MHD equations were studied in conjunction with Einstein's equations. Lichnerowicz (1967) has made a thorough and deep investigation of the initial value problem. In many applications, however, one neglects the gravitational field generated by the conducting medium in comparison with the background gravitational field as well as in many cases one simply uses sp ecial relativity.


6.3. Ideal MHD

135

Such relativistic MHD is much simpler than the full general relativistic theory. So more detailed results can be obtained (Novikov and Frolov, 1989).

6.3
6.3.1

Magnetic flux conservation. Ideal MHD Integral and differential forms of the law

Equations (6.44), (6.41), and (6.45) are the conservation laws for mass, momentum, and energy, respectively. Let us show that Equation (6.42): B = rot ( v × B ) + m B , t with
m

= 0, is the magnetic flux conservation law.

Let us consider the time derivative of the vector B flux through a surface S moving with the plasma (Fig. 6.2).
B dS z v

S L y x

v

The magnetic field B flux through the surface S moving with a plasma with velocity v.
Figure 6.2:

According to the known formula of vector analysis (see Smirnov, 1965), we have


136

Chapter 6. Single-Fluid Models

d dt

B ž dS =
S S

B + v div B + rot ( B × v ) ž d S . t

Since div B = 0, d dt B ž dS =
S S

B - rot ( v × B ) ž d S , t

or, making use of Equation (6.42), d dt B ž dS =
S m S

B ž d S . (6.52)

Thus, if we cannot neglect magnetic viscosity m , then the change rate of magnetic flux through a surface moving together with a conducting plasma is prop ortional to the magnetic viscosity. The right-hand side of (6.52) can be rewritten with the help of the Stokes theorem: d dt B ž dS = -
S m L

rot B ž d l .

Here L is the `liquid contour' bounding the surface S . By using equation j= we have d dt c rot B , 4 c j ž dl.
L

B ž dS = -
S

(6.53)


6.3. Ideal MHD

137
-1

The change rate of flux is proportional to resistivity

of the plasma.

Equation (6.53) is equivalent to the differential Equation (6.42) and presents an integral form of the magnetic flux conservation law. The magnetic flux through any surface moving with the plasma is conserved, once the electric resistivity -1 can be ignored. When is it possible to neglect resistivity of plasma? The relative role of a dissipation process can be evaluated as follows. Let us pass on to the dimensionless variables r = r , L t = t , v = v , v B = B . B0

On substituting them in (6.42) we obtain B0 B v B0 = rot ( v × B ) + t L B0 B. L2

m

Now we normalize this equation with respect to its left-hand side, i.e. B v m = rot ( v × B ) + 2 B . t L L This dimensionless equation contains two dimensionless parameters. The first one, = v , L

will be discussed later on. Here, for simplicity, we assume = 1. The second parameter,


138

Chapter 6. Single-Fluid Models

Re

m

=

L2 vL = , m m

(6.54)

is termed the magnetic Reynolds number, by analogy with the hydro dynamic Reynolds number Re = vL .

Omitting the asterisk, we write the dimensionless equation B 1 = rot ( v × B ) + B . t Re m (6.55)

The larger the magnetic Reynolds number, the smaller the role played by magnetic viscosity. So the magnetic Reynolds number is the measure of a relative imp ortance of resistivity. If Re
m

1,

we neglect the plasma resistivity and, as a consequence, magnetic field diffusion and dissipation. On the contrary, in laboratory, e.g., in devices for studying reconnection, because of a small value L2 , the magnetic Reynolds number is usually not large: Re m 1 - 3 . In this case, the resistivity has a dominant role, and dissipation is important.


6.3. Ideal MHD

139

6.3.2

The ideal MHD

Under astrophysical conditions, owing to the low resistivity of plasma and the enormously large length scales, the magnetic Reynolds number is usually huge: Re m > 1010 (e.g., Exercise 6.1). Thus, in a great number of problems, it is sufficient to consider a medium with infinite conductivity: Re
m

1.

Furthermore the usual Reynolds number can be also large (see, however, Exercise 6.2): Re 1.

Let us also assume the heat exchange to be of minor importance. This assumption is not universally true either. Sometimes the thermal conductivity is so effective that an astrophysical plasma must be considered as isothermal, rather than adiabatic. However, conventionally, while treating the `ideal medium', all dissipative coefficients as well as the thermal conductivity are set equal to zero:



m

= 0,

= = 0,

= 0.

The complete set of the ideal MHD equations has two equivalent forms. The first one is the transfer equations:


140

Chapter 6. Single-Fluid Models

v + (v ž t

)v = -

p 1 - B × rot B , 4 div B = 0 , s + (v ž t )s = 0, (6.56)

B = rot (v × B) , t + div v = 0 , t p = p (, s) .

The divergent form corresponds to the conservation laws for energy, momentum, mass and magnetic flux: t v 2 B2 + + 2 8 = - div G ,


(6.57) (6.58) (6.59) (6.60) (6.61) (6.62)

v = - t r

,

= - div v , t B = rot ( v × B ) , t div B = 0 , p = p (, s) . Here the energy flux and the momentum flux tensor are G = v and
= p

1 v2 +w + B 2 v - (B ž v) B 2 4 + v v + 1 4 B2 - B B 2


(6.63)

.

(6.64)


6.3. Ideal MHD

141

6.3.3

The `frozen field' theorem

The magnetic flux conservation law (6.60) written in the integral form d dt B ž dS = 0
S

allows us to represent the magnetic field as a set of field lines attached to the medium, as if they were `frozen into' it. For this reason, (6.60) is referred to as the `freezing-in' equation. The "frozen field" theorem can be formulated as follows.

In the ideally conducting medium, the field lines move together with the plasma. A medium motion conserves not only the magnetic flux but each of the field lines as well.

Let us imagine a thin tub e of field lines (Fig. 6.3). There is no magnetic flux through any part of the surface formed by the boundary field lines that intersect the closed curve L. Hence, the "fluid particles" that are initially in the same flux tube must remain in the flux tub e. In ideal MHD flows, magnetic field lines are therefore materialized and are unbreakable because the flux tube links the same fluid particles. As a result its top ology cannot change. Fluid particles which are not initially on a common field line cannot become linked by one later on. This general topological constraint restricts the ideal MHD motions, forbidding a lot of motions that would otherwise appear.


142

Chapter 6. Single-Fluid Models

z

FP v

FP

B y x L
Figure 6.3:

dS

The field-flux tube through the surface with a plasma with velocity v. L is the "liquid bounding the surface d S. The "fluid particles" are initially in this flux tube remain in the same

d S moves contour" (F P ) that tube.

Conversely, the fluid particle motion, whatever its complexity, may create situations where the magnetic field structure becomes itself very complex.

In general, the field intensity B is a lo cal quantity. However the magnetic field lines (even in vacuum) are integral characteristics of the field. Their analysis becomes more complicated. Nonetheless, an investigation of non-lo cal structures of magnetic fields is fairly important in plasma astrophysics. The geometry of the field lines appears in different ways in the equilibrium criteria for astrophysical plasma.


6.4. Magnetic Reconnection

143

Much depends on whether the field lines are concave or convex, on the so-called sp ecific volume of magnetic flux tubes. However even more depends on the presence of X-type points, as well as on other topological characteristics, e.g. the global magnetic helicity.

6.4

Magnetic reconnection

Reconnection of magnetic field lines is the physical process which involves a breakdown of the "frozen field" theorem. The effects of electric resistivity, normally negligible in the large, b ecome lo cally dominant with dramatic consequences in the large-scale plasma flows and magnetic field configuration. Reconnection changes top ology of magnetic field. The origin of the concept of reconnection lies in an attempt by Giovanelli (1946) to explain solar flares. Reconnection is the means by which energy stored in magnetic fields is released rapidly to produce such phenomena as solar flares and magnetospheric substorms. Furthermore, reconnection plays important roles in many areas of astrophysics. Depending on complexity of fields and conditions, reconnection can occur over an extended region in space or can be "patchy" and "unpredictable". For example, in the Earth's magnetosphere the reconnecting current layers (RCLs) are formed by the interaction between the solar wind and the geomagnetic field. Such RCLs have finite extents, and their boundary conditions often change rapidly. On the contrary,


144

Chapter 6. Single-Fluid Models

The Wind , ACE and Cluster spacecraft on 2 February 2002: The spacecraft positions are shown in units of Earth radius (RE ) and in geocentric solar ecliptic coordinates.
Figure 6.4:

in the solar wind, the magnetic field orientations on the two sides of the interplanetary current layers are usually well defined, and the boundary conditions seem to be relatively stable. Phan et al. (2006) report the 3-spacecraft observations of plasma flow associated with large-scale reconnection in the solar wind (Fig. 6.4).


6.4. Magnetic Reconnection


145

M

B j CL

A "wavy ecliptic current" layer (CL). The Sun is the center of an extensive layer.
Figure 6.5:

In the most astrophysical situations, the reconnection process is predictable and occurs in an internal scale of a phenomenon, which is responsible to the initial and boundary conditions. In the solar wind the scale of a current layer (CL) around the Sun can be very large (Fig. 6.5). The current layer (CL) separates the fields of nearly opposite directions. The average plane of the layer is approximately the plane of the equator of the Sun's average magnetic dipole (M) field. On the other hand, the high-speed solar wind that originates in coronal holes is permeated by evolved Alfv? ype fluctuations assoen-t ciated with MHD turbulence. The spacecraft Wind allows us to study reconnection in this turbulent flow.


146

Chapter 6. Single-Fluid Models

The Wind observations demonstrate that reconnection is one way in which the solar wind turbulence is dissipated and the high-speed wind is heated far from the Sun. In the solar wind, the kinetic and thermal energies of plasma exceed the magnetic energy. We neglect the magnetic force as compared to the inertia force of moving plasma and its pressure gradient. We call such a process as reconnection in a weak magnetic field. Another example of this phenomenon is the photospheric reconnection. Reconnection in a strong magnetic field is a fundamental feature of astrophysical plasmas like the solar corona. Such reconnection explains an accumulation of magnetic energy and a sudden release of this energy, a flare. This phenomenon is accompanied by fast ejections of plasma, powerful flows of heat and hard electromagnetic radiation, by acceleration of particles.

6.5

Practice: Exercises and Answers

Exercise 6.1. Estimate the magnetic Reynolds number in the solar corona. Answer. Taking characteristic values of the parallel conductivity as estimated in Exercise 5.1: = 1016 - 1017 s-1 , we obtain


6.5. Practice: Exercises and Answers

147

Re

m

=

vL 1011 - 1012 , m

(6.65)

if the length and velocity, L 104 km and v 10 km s-1 . Exercise 6.2. Show that in the solar corona, usual viscosity of plasma can be a much more important dissipative mechanism than electric resistivity. Answer. The characteristic value of kinematic viscosity = 3 × 10
15

cm2 s-1 .

Here Tp 2 × 106 K and np ne 2 × 108 cm-3 were taken as the typical proton temp erature and density. If the length and velocity, L 109 cm and v 106 cm s-1 , then the ordinary Reynolds number Re = Thus Re
m

vL 0.3 .

(6.66)

Re .


148

Chapter 6. Single-Fluid Models


Chapter 7

MHD in Astrophysics
MHD is appropriate for many phenomena in astrophysical plasma, that take place on a relatively large scale. The non-relativistic MHD is applied to dynamo theory, flows in the solar atmosphere, flares, coronal heating, solar and stellar winds. Relativistic MHD describes well accretion disks near relativistic ob jects, and relativistic jets.

7.1
7.1.1

The main approximations in ideal MHD Dimensionless equations

The equations of MHD constitute a set of nonlinear differential equations in partial derivatives. The order of the set is rather high, and its structure is complicated. To formulate a problem, we have to know the initial and b oundary conditions admissible by this set of equations. To do this, in turn, we have to know the typ e of equations, in the sense adopted in mathematical physics. To formulate a problem, we usually use one or another approximation, which makes it possible to point up and isolate the main effect.
149


150

Chapter 7. MHD in Astrophysics

For instance, if the magnetic Reynolds number is small, a plasma moves comparatively easily with respect to magnetic field. This is the case in laboratory and technical devices. The opposite approximation is that of large magnetic Reynolds numb ers, when the magnetic field `freezing in' takes place in plasma. This approximation is quite characteristic of the astrophysical plasma. How can we isolate the main effect in a phenomenon and correctly formulate the problem? - From the above examples, the following rule suggests itself: take the dimensional parameters of a phenomenon, combine them into dimensionless combinations, calculate their values, and use a corresponding approximation in the dimensionless equations. Such an approach is effective in hydrodynamics. Let us start with the ideal MHD equations: v + (v ž t )v = - p 1 - B × rot B , 4 (7.1) (7.2) (7.3) (7.4) (7.5) (7.6)

B = rot (v × B) , t + div v = 0 , t s + (v ž t )s = 0,

div B = 0 , p = p (, s) . Let the quantities


7.1. Main Approximations in Ideal MHD

151

L, , v , 0 , p0 , s0 , B

0

be the characteristic values of length, time, velocity, density, pressure, entropy and field strength, respectively. Rewrite Equations (7.1)-(7.6) in the dimensionless variables r = r , L t = t , ... B = B . B0

Omitting the asterisk, we obtain the equations in dimensionless variables:
2

1 v + (v ž t

)v = -

2

p1 - B × rot B ,

(7.7) (7.8) (7.9) (7.10) (7.11) (7.12)

B = rot (v × B) , t + div v = 0 , t s + (v ž t )s = 0,

div B = 0 , p = p (, s) . Here v , L 2 = v2 , VA2 2 = p0 0 VA2

=

(7.13)

are three dimensionless parameters; VA = B0 4 (7.14)
0


152

Chapter 7. MHD in Astrophysics

is the characteristic value of the Alfv? speed (see Exercise 7.1). en If the gravitational force is taken into account, Equation (7.7) contains another dimensionless parameter, g L/VA2 , where g is the gravitational acceleration. The analysis of these parameters allows us to separate the approximations which are possible in the ideal MHD. 7.1.2

Weak magnetic fields in astrophysical plasma

We begin with the assumption that 2 1 and
2

1.

(7.15)

As is seen from Equation (7.7), in the zero-order approximation relative to the small parameters -
2

and



-2

,

we neglect the magnetic force as compared to the inertia force and the pressure gradient. In subsequent approximations, the magnetic effects are treated as small corrections to the hydrodynamic ones. A lot of problems of plasma astrophysics are solved in this approximation, termed the weak magnetic field approximation. Among the simplest of them are the ones concerning the weak field's influence on hydrostatic equilibrium. An example is the problem of the influence of magnetic field on the equilibrium of a self-gravitating plasma ball (a star, the magnetoid of quasar's kernel etc.). Some other problems are in fact analogous to the mentioned ones. They are called kinematic problems, since


7.1. Main Approximations in Ideal MHD

153

they treat the influence of a given plasma flow on magnetic field; the reverse influence is considered to be negligible. Such problems are reduced to finding the magnetic field resulting from the known velocity field. An example is the magnetic field amplification and support by stationary plasma flows (magnetic dynamo). The simplest example is the magnetic field amplification by differential rotation. A leading candidate to explain the origin of large-scale magnetic fields in astrophysical plasma is the turbulent dynamo theory. 7.1.3

Strong magnetic fields in plasma

The opposite approximation - that of the strong field - reflects the specificity of MHD to a greater extent. This approximation is valid when the magnetic force Fm = - 1 B × rot B 4 (7.16)

dominates all the others (inertia force, pressure gradient, etc.). In Equation (7.7), the magnetic field is a strong one if 2 1 and
2

1,

(7.17)

i.e. the magnetic energy density greatly exceeds that of the kinetic and thermal energies: B02 8 0 v 2
2

and

B02 8

2n0 kB T0 .

From Equation (7.7) it follows that, in the zeroth order with respect to the small parameters (7.17), the magnetic field is force-free:


154

Chapter 7. MHD in Astrophysics

B × rot B = 0 . This conclusion is quite natural:

(7.18)

if the magnetic force dominates all the others, the magnetic field must balance itself in the region under consideration. Condition (7.18) means that electric currents flow parallel to magnetic field lines. If, in addition, electric currents are absent in some region, then the strong field is simply p otential: rot B = 0 , B= , = 0 . (7.19)

Let us consider the first order in the small parameters (7.17). If they are not equally significant, there are two p ossibilities. (a) We suppose, at first, that 2
2

1.

(7.20)

Then we neglect the inertia force in Equation (7.7) as compared to the pressure gradient. Decomposing the magnetic force into a magnetic tension force and a magnetic pressure gradient, Fm = - 1 1 B × rot B = (B ž 4 4 )B - B2 , 8 (7.21)

we obtain the following dimensionless equation: (B ž )B = B2 + 2p . 2 (7.22)

Owing to the gas pressure gradient, the magnetic field differs from the force-free one:


7.1. Main Approximations in Ideal MHD

155

the magnetic tension force (B ž ) B/4 must balance not only the magnetic pressure gradient but that of the gas pressure as well. Obviously the effect is proportional to the small parameter 2 . This approximation is naturally called the magnetostatics since v = 0. It works in regions of a strong magnetic field where the gas pressure gradients are large, e.g., in coronal lo ops and reconnecting current layers in the solar corona. (b) The inertia force also causes the magnetic field to deviate from the force-free one:
2

1 v + (v ž t

)v = -

1 B × rot B .

(7.23)

Here we ignored the pressure gradient as compared with the inertia force. This is the case
2



2

1.

(7.24)

The approximation corresponding (7.24) is termed the approximation of strong field and cold plasma. The main applications of this approximation are the solar atmosphere and the Earth's magnetosphere. Both ob jects are well studied from the observational viewpoint. So we can proceed with confidence from qualitative interpretation to the construction of quantitative mo dels. The presence of a strong field and a rarefied plasma is common for both phenomena.


156

Chapter 7. MHD in Astrophysics

A sufficiently strong magnetic field easily moves a comparatively rarefied plasma in many non-stationary phenomena in space. Some astrophysical applications will be discussed in the following two Sections.

In closing, let us consider the dimensionless parameter = v /L . It characterizes the relative role of the lo cal / t and transp ort (v ž ) terms in the substantial derivative d 1 = + (v ž dt t If ).

1, the flow can be considered to be stationary 2 (v ž )v = - 1 B × rot B . (7.25)

If 1, the transport term (v ž of motion takes the form 2

) can be ignored, and the equation

1 v = - B × rot B , t

(7.26)

other equations becoming linear. This case corresponds to small small p erturbations. If need be, the right-hand side of Equation (7.26) can be linearized too.


7.2. Accretion Disks

157

Generally the parameter 1, and the MHD equations in the approximation of strong field and cold plasma take the following dimensionless form:
2

dv 1 = - B × rot B , dt B = rot ( v × B ) , t + div v = 0 . t

(7.27)

(7.28)

(7.29)

In the next Chapter we shall consider some continuous flows, which are described by these equations.

7.2
7.2.1

Accretion disks of stars Angular momentum transfer

Magnetic fields are discussed as a means of angular transport in the accretion disk. Interest in the magnetic fields in binary stars steadily increased after the discovery of the nature of AM Herculis. The optical counterpart of the soft X-ray source has p olarization in the V and I spectral bands. This suggested the presence of a strong field, B 108 G, assuming the fundamental cyclotron frequency to be observed. Other similar systems were soon discovered. Evidence for strong fields was found in the X-ray binary pulsars and the intermediate p olar binaries.


158

Chapter 7. MHD in Astrophysics

MHD in binary stars is now an area of central importance in stellar astrophysics (Campbell, 1997).

SS L
1

L D

2

A binary system with an accretion disk. The tidally and rotationally distorted secondary star S S loses plasma from the unstable L1 point. The resulting plasma stream feeds an accretion disk D, centered on the primary star.
Figure 7.1:

The disk is fed by the plasma stream originated in the L1 region (Fig. 7.1) of the secondary star. In a steady state, plasma is transported through the disk at the rate it is supplied by the stream and the angular momentum is advected outwards. Such advection requires coupling between rings of rotating plasma; the ordinary viscosity is to o weak to provide this. Hence some form of anomalous viscosity must be invoked. Purely hydro dynamic turbulence does not produce sustained outward transport of angular momentum. MHD turbulence greatly enhances angular momentum transport (Balbus and Papaloizou, 1999).


7.2. Accretion Disks

159

Turbulent viscous and magnetic stresses cause radial advection of the angular momentum via the azimuthal forces.

7.2.2

Accretion in cataclysmic variables

Cataclysmic variables (CVs) are binary systems composed of a white dwarf (primary star) and a late-typ e, main-sequence companion (secondary star). The way this plasma falls towards the primary depends on the intensity of a magnetic field of the white dwarf. The strong field (B flow.
>

107 G) may entirely dominate the accretion

The magnetic field is strong enough to synchronize the white dwarf rotation (spin) with the orbital period. No disk is formed. Instead, the field channels accretion towards its p olar regions. Such synchronous systems are known as AM Herculis binaries or p olars. The intermediate (B 2 - 8 × 106 G) field primary stars harbor magnetically truncated accretion disks which extend until magnetic pressure begins to dominate. Presumably the plasma is finally accreted onto the magnetic p oles of the white dwarf. The asynchronous systems are known as DQ Herculis binaries or intermediate p olars (IPs). The accretion geometry strongly influences the emission properties at all wavelengths and its variability. The knowledge of the behavior in all energy domains can allow one to locate the different accreting regions.


160

Chapter 7. MHD in Astrophysics

The white dwarf LHS 2534 offers the first empirical data of the Zeeman effect on neutral Na, Mg, and both ionized and neutral Ca. The Na I splitting results in a field strength estimate of 1.92 × 106 G.

7.2.3 In the dence (a) (b) (c)

Accretion disks near black holes
binary stars discussed above, there is an abundance of evifor accretion disks: double-peaked emission lines, eclipses of an extended light source centered on the primary, eclipses of the secondary star by the disk.

The case of accretion disks in active galactic nuclei (AGN) is less clear. Nonetheless the disk accretion onto a sup er-massive black hole is the commonly accepted model for these ob jects. As the plasma accretes in the gravitational field of the central mass, magnetic field lines are convected inwards, amplified and finally deposited on the horizon of the black hole. As long as a magnetic field is confined by the disk, a differential rotation causes the field to wrap up tightly, becoming highly sheared and predominantly azimuthal in orientation. A dynamo in the disk may be responsible for the maintenance and amplification of the magnetic field. In the standard mo del of an accretion disk (Shakura and Sunyaev, 1973; Novikov and Thorne, 1973), the gravitational energy is locally radiated from the optically thin disk. However the exp ected p ower far exceeds the observed luminosity. There are two p ossible explanations: (a) the accretion occurs at extremely low rates, or


7.2. Accretion Disks

161

(b) the accretion occurs at low radiative efficiency. Advection results in a structure different from the standard model. The advection process physically means that the energy generated via viscous dissipation is restored as entropy of the accreting plasma flow rather than being radiated. An optically thin advection-dominated accretion flow (ADAF) seemed to be a model that can reproduce the observed spectra of black hole systems such as AGN and Galactic black hole candidates. 7.2.4

Flares in accretion disk coronae

Following the launch of several X-ray satellites, astrophysicists have tried to observe and analyze the variations of high energy flux from black hole candidates. It has appeared that there are many relationships between flares in the solar corona and `X-ray shots' in accretion disks. For example, the peak interval distribution of Cyg X-1 shows that the o ccurrence frequency of large X-ray shots is reduced. A second large shot does not occur soon after a previous large shot. This suggests the existence of energy-accumulation structures, such as non-potential magnetic fields in the solar corona. It is likely that accretion disks have a corona. Galeev et al. (1979) suggested that the corona is confined in strong magnetic loops which have buoyantly emerged from the disk. Magnetic reconnection of buoyant fields in the lower density surface regions may supply the energy source for a hot corona.


162

Chapter 7. MHD in Astrophysics

The existence of a disk corona with a strong field raises the possibility of a wind flow similar to the solar wind. This would result in angular momentum transp ort away from the disk, which could have some influence on the inflow. Another feature is the possibility of a flare energy release similar to solar flares. When a plasma in the disk corona is optically thin and has a dominant magnetic pressure, the circumstances are similar to the solar corona. Therefore it is possible to imagine some similarity between the mechanisms of solar flares and X-ray shots in accretion disk coronae. Besides the effect of heating the the disk corona, reconnection is able to accelerate particles to high energies. Some geometrical and physical properties of the flares in disk coronae can be inferred from X-ray observations of Galactic black hole candidates.

7.3
7.3.1

Astrophysical jets Jets near black holes

Jet-like phenomena, including relativistic jets, are observed on a wide range of scales in accretion disk systems. AGN show extremely energetic outflows extending beyond the outer edge of a galaxy in the form of strongly collimated jets. There is evidence that magnetic forces are involved in the driving mechanism and that the magnetic fields also provide the collimation of relativistic flows.


7.3. Astrophysical Jets

163

Rotating black holes are thought to be the prime-mover in centers of galaxies. The gravitational field of rotating black holes is more complex than that of non-rotating ones. The weak-gravity (far from the hole) low-velocity co ordinate acceleration of uncharged particle d2 r dr =g+ ×H 2 dt dt
gr

.

(7.30)

This looks like the Lorentz force with the electric field E replaced by g, the magnetic field B replaced by the vector H
gr

= rot A

gr

,

and the electric charge e replaced by the particle mass m. These analogies lie behind the words "gravitoelectric" and "gravitomagnetic" to describe the gravitational acceleration field g and to describe the "shift function" Agr (Exercise 7.6). Thus, far from the horizon, the gravitational acceleration g=- M er r2 (7.31)

is the radial Newtonian acceleration, and the gravitomagnetic field Hgr = 2 J - 3 ( J ž er ) er r3 (7.32)

is a dipole field. The role of dipole moment is played by the hole's angular momentum J= ( r × m v) dV .


164

Chapter 7. MHD in Astrophysics

The gravitomagnetic force drives an accretion disk into the hole's equatorial plane and holds it there (Fig. 7.2).

H

gr

jet J

V

D

An accretion disk D around a rotating black hole is driven into the hole's equatorial plane at small radii by a combination of gravitomagnetic forces (action of the gravitomagnetic field Hgr on orbiting plasma) and viscous forces.
Figure 7.2:

At radii where the bulk of the disk's gravitational energy is released and where the hole-disk interactions are strong, there is only one geometrically preferred direction along which a jet might emerge, which coincides with the rotation axis of the black hole. The jet might be produced by winds off the disk, in other cases by


7.3. Astrophysical Jets

165

electrodynamic acceleration of the disk, and in others by currents in the hole's magnetosphere. However whatever the mechanism, the jet presumably is locked to the hole's rotation axis. The black hole acts as a gyroscop e to keep the jet aligned. It is very difficult to torque a black hole. The fact accounts for the constancy of the observed jet directions over length scales as great as millions of light years and thus over time scales of millions of years or longer.

In the highly-conducting medium, the gravitomagnetic force couples with electromagnetic fields over Maxwell's equations. This effect has interesting consequences for the magnetic fields advected towards the black hole. It leads to a gravitomagnetic dynamo which amplifies any seed field near a rotating compact ob ject. This process builds up the dip olar magnetic structures which may be behind the bip olar outflows seen as relativistic jets.

7.3.2

Relativistic jets from disk coronae

Relativistic jets are produced perpendicular to the accretion disk plane (Fig. 7.2) around a super-massive black hole in an AGN. The sho ck of the jets on intergalactic media is considered as being able to accelerate particles up to the highest energies, say 1020 eV for cosmic rays. This hypothesis need, however, to be completed by some necessary ingradients since such powerful galaxies are rare ob jects.


166

Chapter 7. MHD in Astrophysics

The relativistic jets may be powered by acceleration of protons in a corona above an accretion disk. The acceleration arises as a consequence of the shearing motion of the magnetic field lines in the corona, that are anchored in the underlying Keplerian disk. Particle acceleration in the corona leads to the development of a pressure-driven wind. It passes through a critical point and subsequently transforms into a relativistic jet at large distances from the black hole.

7.4

Practice: Exercises and Answers

Exercise 7.1. Evaluate the Alfv? speed in the solar corona above a en large sunspot. Answer. From definition we find B VA 2.18 × 1011 , cm s n
-1

.

(7.33)

Above a sunspot B 3000 G, n 2 × 108 cm-3 . Thus (7.33) gives unacceptably high values: VA 5 × 1010 cm s This means that in a strong magnetic field and low density plasma, the Alfv? en waves propagate with velocities approaching the light speed c . So the non-relativistic formula (7.33) has to be corrected by a relativistic factor:
-1

> c.


7.4. Practice: Exercises and Answers

167

r VA el =

B 4
2

1 1 + B 2 /4 c 4 c2 .
2

,

(7.34)

which agrees with (7.14) if B

Therefore the relativistic Alfv? wave speed is always smaller than the en light speed. For values of the magnetic field and plasma density mentioned above,
r VA el 2 × 1010 cm s-1 .

Exercise 7.2. Discuss properties of the Lorentz force in terms of the Maxwellian stress tensor (6.11). Exercise 7.3. Show that the magnetic tension force is directed to the local centre of curvature. Exercise 7.4. For the conditions in the corona, used in Exercise 7.1, estimate the parameter 2 . Answer. Substitute p0 = 2n0 kB T0 in definition (7.13): For the temperature T0 2 × 106 K and magnetic field B0 3000 G 2 10-7 . Exercise 7.5. By using formula (6.63) for the energy flux in ideal MHD, find the magnetic energy influx into a reconnecting current layer. Answer. In this simplest approximation, near the layer, the magnetic field B v. In formula (6.63) the product B ž v = 0 and the energy flux density G = v B2 v2 +w + v. 2 4 (7.35)


168

Chapter 7. MHD in Astrophysics

If the approximation of a strong field is satisfied, the last term in (7.35) is dominating, and we find the Poynting vector directed into the current layer GP = B2 v. 4 (7.36)

Exercise 7.6. Consider a weakly gravitating, slowly rotating body such as the Sun, with all nonlinear gravitational effects neglected. Compute the gravitational force and gravitomagnetic force from the linearized Einstein equations (see Landau and Lifshitz, Classical Theory of Field ). Show that, for a time-independent body, these equations are identical to the Maxwell equations: rot g = 0 , rot H
gr

div g = - 4 Gm , div Hgr = 0 .

(7.37) (7.38)

= - 16 G m v ,

Here the differences are: (a) two minus signs because gravity is attractive rather than repulsive, (b) the factor 4 in the rot Hgr equation, (c) the presence of the gravitational constant G, (d) the replacement of charge density q by mass density m , and (e) the replacement of electric current j by the mass flow m v.


Chapter 8

Plasma Flows in a Strong Magnetic Field
A strong magnetic field easily moves a rarified plasma in many non-stationary phenomena in the astrophysical environment. The best studied example is the solar flares which strongly influence the interplanetary and terrestrial space.

8.1

The general formulation of a problem

As was shown above, the set of MHD equations for an ideal medium in the approximation of strong field and cold plasma is characterized 2 only by the small parameter 2 = v 2 /VA :
2

dv 1 = - B × rot B , dt B = rot (v × B) , t + div v = 0 . t

(8.1) (8.2) (8.3)

Let us represent all the unknown quantities in the form
169


170

Chapter 8. Flows in Strong Field

f (r, t) = f

(0)

(r, t) + 2 f

(1)

(r, t) + . . . .

Then we try to find the solution in three consequent steps. (a) To zeroth order with respect to 2 , the magnetic field is determined by the equation B
(0)

× rot B

(0)

= 0.

(8.4)

This must be supplemented with a boundary condition, which generally depends on time: B
(0)

(r, t) | S = f 1 (r, t) .

(8.5)

Here S is the boundary of the region G (Fig. 8.1), in which the forceS

v
G

v
B

z

||

y x

The boundary and initial conditions for the ideal MHD problems.
Figure 8.1:

free-field Equation (8.4) applies. The strong force-free field, changing in time according to the boundary condition (8.5), sets the plasma in motion.


8.1. Formulation of Problem

171

(b) Kinematics of this motion is uniquely determined by two conditions. The first one signifies the orthogonality of acceleration to the magnetic field lines B
(0)

ž

dv (0) = 0. dt

(8.6)

This equation is the scalar product of Equation (8.1) and the vector B(0) . The second condition is a consequence of the freezing-in Equation (8.2) B (0) = rot v t
(0)

×B

(0)

.

(8.7)

Equations (8.6) and (8.7) determine the velocity v(0) (r, t), if the initial condition inside the region G is given: v Here v
(0) (0)

(r, 0) | G = f 2 (r) .

(8.8)

is the velocity component along the field lines.
(0)

The velocity component across the field lines, v , is uniquely defined by the freezing-in Equation (8.7) at any moment, including the initial one. (0) Therefore we do not need the initial condition for v . (c) Since we know the velocity field v(0) (r, t), the continuity equation (0) + div t
(0) (0)

v

=0

(8.9)

allows us to find the plasma density distribution

(0)

(r, t), if we know its initial


172

Chapter 8. Flows in Strong Field



(0)

(r, 0) | G = f3 (r) .

(8.10)

Therefore, at any moment of time, the field B (0) (r, t) is found from Equation (8.4) and the boundary condition (8.5). Thereupon the velocity v(0) (r, t) is determined from Equations (8.6) and (8.7) and the initial condition (8.8). Finally the continuity Equation (8.9) and the initial condition (8.10) give the plasma density (0) (r, t). We restrict our attention to the zeroth order relative to the parameter 2 , neglecting the field deviation from a force-free state. The question of the existence of solutions will be considered later on, using 2D problems.

8.2

The formalism of 2D problems

Being relatively simple from the mathematical viewpoint, 2D ideal MHD problems allow us to gain some general knowledge concerning the actual flows of plasma with the frozen-in strong magnetic field. The 2D problems are sometimes a close approximation of the real 3D flows and can be used to compare the theory with experiments and observations. There are two typ es of problems treating the plane flows of plasma, i.e. the flows with the velocity field v = { vx (x, y , t), vy (x, y , t), 0 } . All the quantities are dependent on variables x, y and t.


8.2. 2D Problems

173

8.2.1

The first typ e of problems

The first type incorporates the problems with a magnetic field which is everywhere parallel to the z axis: B = { 0, 0, B (x, y , t) } . The corresponding current is parallel to the (x, y ) plane: j = { jx (x, y , t), jy (x, y , t), 0 } . As an example, let us discuss the effect of a longitudinal magnetic field in a reconnecting current layer (RCL). Under real conditions, reconnection does occur not at the zeroth lines but rather at the separators. The latter differ from the zeroth lines only in that the separators contain the longitudinal field (Fig. 8.2).
y B

B

||

x

A longitudinal field B parallel to the z axis is superimposed on the 2D hyperbolic field in the plane (x, y ).
Figure 8.2:

With appearance of the longitudinal field, the force balance in the RCL is changed. The field and plasma pressure outside the RCL must balance not only the gas pressure but also that of the longitudinal field inside the RCL (Fig. 8.3)


174
y

Chapter 8. Flows in Strong Field

B

B

||

j j
x

y

x

B

||

A model of a RCL with a longitudinal component B of magnetic field.
Figure 8.3:

B=

0, 0, B (x, y , t) .

If the longitudinal field accumulated in the RCL during reconnection, the field pressure B 2 /8 would considerably limit the layer compression and the reconnection rate. However the solution of the problem of the first type with respect to B shows that another effect is of importance in the real plasma with finite conductivity. The longitudinal field compression in the RCL produces a gradient of this field and a corresponding current circulating in the transversal plane (x, y ). This current is represented schematically in Fig. 8.3. Dissipation of the circulating current leads to longitudinal field diffusion outwards from the RCL. More exactly, because of dissipation, plasma moves into the RCL relatively free with respect to the longitudinal component of magnetic field, thus limiting its accumulation in the RCL.


8.2. 2D Problems

175

8.2.2

The second typ e of MHD problems Magnetic field and its vector p otential

8.2.2 (a)

The 2D problems of the second type treat the plane flows v = { vx (x, y , t), vy (x, y , t), 0 } , associated with the plane magnetic field B = { Bx (x, y , t), By (x, y , t), 0 } . The currents corresponding to this field are parallel to the z axis j = { 0, 0, j (x, y , t) } . The vector-potential A has an its only non-zero component: A = { 0, 0, A (x, y , t) } . The magnetic field B is defined as B= A A ,- ,0 y x . (8.11)

The scalar function A (x, y , t) is often termed the vector p otential. This function is quite useful, owing to its properties. Prop erty 1. Substitute (8.11) in the differential equations describing the magnetic field lines dy dz dx = = . Bx By Bz These equations imply parallelism of the vector d l = {dx, dy , dz }


176

Chapter 8. Flows in Strong Field

to the vector B = {Bx , By , Bz }. In the case under study Bz = 0, dz = 0 , and dx dy =- A/ y A/ x or A A dx + dy = 0 . x y On integrating the last, we come to the relation

A (x, y , t) = const for t = const .

(8.12)

This is the equation for a family of magnetic field lines in the plane z = const at the moment t. Property 2. Let L be a curve in the plane (x, y ) and d l an arc element along this curve (Fig. 8.4).

dS 1 L dl 2 B

The curve L connects the points 1 and 2 situated in different field lines.
Figure 8.4:

Let us calculate the magnetic flux d through the arc element d l.


8.2. 2D Problems

177

By definition, ex e d = B ž d S = B ž (ez × d l ) = B ž 0
y

ez 1 0 =

0

dx dy = B ž { (-dy ) ex + dx ey } = = -Bx dy + By dx . On substituting (8.11) in (8.13) we find d = - A A dy - dx = - dA . y x

(8.13)

On integrating this along the curve L from point 1 to point 2 we obtain the magnetic flux = A2 - A1 . Thus the fixed value of the potential A is not only the field line `tag' determined by formula (8.12); the difference of values of the vector potential A on two field lines is equal to the magnetic flux between them. Simple rule: Plot the field lines corresponding to equidistant values of A. Prop erty 3. Let us substitute definition (8.11) in the freezing-in equation. We obtain the following equation rot dA = 0. dt


178

Chapter 8. Flows in Strong Field

Disregarding a gradient of an arbitrary function and considering the second type of 2D problems, we have dA A + (v ž dt t This equation means that the lines A (x, y , t) = const (8.15) )A = 0 . (8.14)

are Lagrangian lines: they move together with plasma. According to (8.12) they are composed of the field lines. Hence Equation (8.14) expresses the magnetic field freezing in plasma. Thus we have one of the integrals of motion A (x, y , t) = A (x0 , y0 , 0) A
0

(8.16)

at an arbitrary t. Here x0 , y0 are the coordinates of a "fluid particle" at the initial time t = 0; x, y are the coordinates of the same particle at a moment of time t or the coordinates of any other particle situated on the same field line A0 at the moment t. Property 4. Equation of motion (8.1) rewritten in terms of the vector potential A(x, y , t) is of the form 2 1 dv = - A dt A. (8.17)

In the zeroth order relative to the small parameter 2 , outside the zeroth p oints (where A = 0) and the magnetic field sources (where A = 0) we have:


8.2. 2D Problems

179

A = 0 .

(8.18)

So the vector potential is a harmonic function of variables x and y . Hence, considering the (x, y ) plane as a complex plane z = x + iy, it is convenient to relate an analytic function F to the vector potential A: F (z , t) = A (x, y , t) + i A+ (x, y , t) . (8.19)

Here A+ (x, y , t) is a conjugate harmonic function connected with A (x, y , t) by the Cauchy-Riemann condition A+ (x, y , t) = - A A dx + dy + A+ (t) = y x

=-

B ž d l + A+ (t) ,

where A+ (t) is a quantity independent of the coordinates x and y . The function F (z , t) is termed the complex p otential. The magnetic field vector dF B = Bx + i By = - i dz the asterisk denoting the complex conjugation. Now we can apply the methods of the complex variable function theory, in particular the method of conform mapping . This has been done in order to determine the structure of magnetic field:


,

(8.20)


180

Chapter 8. Flows in Strong Field

ž in vicinity of reconnecting current layer (Syrovatskii, 1971), ž in solar coronal streamers (Somov and Syrovatskii, 1972) ž in the Earth's magnetosphere (Oberz, 1973), ž the accretion disk magnetosphere (Somov et al., 2003). Markovskii and Somov (1989) generalized the Syrovatskii model by attaching four shock MHD waves at the edges of the RCL. The model reduces to the Riemann-Hilbert problem (in an analytical form on the basis of the Christoffel-Schwarz integral) in order to analyze the structure of magnetic field in vicinity of reconnection region (Bezrodnykh et al., 2007).

8.2.2 (b)

Motion of the plasma and its density

The motion kinematics due to changes in a potential field is uniquely determined by two conditions: (i) the freezing-in condition and (ii) the acceleration orthogonality with respect to the field lines dv(0) × dt A(0) = 0 . (8.21)

Equation (8.21) is a result of eliminating the unknown A(1) from two components of the vector equation dv(0) 1 = - (0) A(1) dt A(0) . (8.22)

If x(t) and y (t) are the coordinates of a fluid particle, Equations (8.21) and (8.14) are reduced to the ordinary differential equations (Somov and Syrovatskii, 1976). Once the kinematic part of the problem is solved, the tra jectories of fluid particles are known:


8.3. Continuous MHD Flows

181

x = x (x0 , y0 , t) ,

y = y (x0 , y0 , t) .

(8.23)

The fluid particle density change on moving along its tra jectory is determined by the continuity Equation (8.3), rewritten in the Lagrangian form, and is equal to (x, y , t) d U0 D (x0 , y0 ) = = . 0 (x0 , y0 ) dU D (x, y ) (8.24)

Here d U0 is the initial volume of a particle, d U is the volume of the same particle at a moment of time t; D (x0 , y0 ) x 0 y0 x0 y0 = - D (x, y ) x y y x (8.25)

is the Jacobian of the transformation that is inverse to the transformation (8.23). The 2D equations of the strong-field-cold-plasma approximation are relatively simple but useful for applications to astrophysical plasmas. In particular, they enable us to study the fast plasma flows in the solar atmosphere and to understand some aspects of the reconnection process. In spite of their numerous applications, the list of exact solutions to them is rather p o or. Still, we can enrich it significantly, relying on many astrophysical ob jects, for example in the accretion disk coronae and some mathematical ideas.

8.3

The existence of continuous flows

Thus, in the strong-field-cold-plasma approximation, the MHD equations for a plane 2D flow of ideally conducting plasma (for the secondtype problems) are reduced, in the zeroth order in the small parameter 2 , to the following closed set of equations:


182

Chapter 8. Flows in Strong Field

A = 0, dv × dt A = 0,

(8.26) (8.27) (8.28) (8.29)

dA = 0, dt + div v = 0 . t
S

v
G

v
B

||

y

x
Figure 8.5:

The boundary and initial conditions for a secondtype 2D problem.

The solution of this set is completely defined inside some region G (Fig. 8.5) once the b oundary condition is given at the boundary S A (x, y , t) | S = f1 (x, y , t) together with the initial conditions inside the region G v (x, y , 0) | G = f2 (x, y ) , (x, y , 0) | G = f3 (x, y ) . (8.31) (8.32) (8.30)


8.4. Time-dependent Magnetic Dipole

183

Here v is the velocity component along field lines. Once the potential A (x, y , t) is known, the transversal velocity component is uniquely determined by the freezing-in Equation (8.28) and is equal, at any moment including the initial one, to v (x, y , t) = (v ž A) | A A =- 2 A| t | A . A |2 (8.33)

The density (x, y , t) is found from the continuity Equation (8.29) and the initial density distribution (8.32). The next Section is devoted to an example which may have applications.

8.4
8.4.1

Flows in a time-dep endent dip ole field Plane magnetic dip ole fields

Two parallel currents, equal in magnitude but opposite in direction, engender the magnetic field which far from the currents can be described by a complex potential F (z ) = im , z m = me
i

(8.34)

and is called the plane dip ole field. 2 Il c has the meaning of the dip ole moment, I is the current magnitude, l is the distance between the currents. m= Formula (8.34) corresponds to the dipole situated at the origin of coordinates in the plane (x, y ) and directed at an angle of to the x axis. The quantity


184

Chapter 8. Flows in Strong Field

y m

x

Figure 8.6:

The field lines of a plane magnetic dipole.

Let us consider the plasma flow caused by the change with time of the strong magnetic field of the dipole = /2 and m = m(t) , m(0) = m0 . (a) Let us find the first integral of motion. According to(8.34), the complex potential F (z , t) = - m(t) x + i m(t) y . x2 + y 2 (8.35)

So, the field lines constitute a family of circles A (x, y , t) = - m(t) x = const x2 + y 2 for t = const .

They have centres on the axis x and the common point x = 0, y = 0 in Fig. 8.6. Therefore the freezing-in condition (8.16) results in a first integral of motion


8.4. Time-dependent Magnetic Dipole

185

mx m 0 x0 =2 . 2 2 x +y x0 + y0
2

(8.36)

Here x0 , y0 are the coordinates of a fluid particle at the initial time t = 0. (b) The second integral is easily found in the limit of small changes of the dipole moment m (t) and respectively small plasma displacements. Assuming the parameter = v /L to be small, Equation (7.26): 2 v 1 = - B × rot B , t

which is linear in velo city. The integration over time (with zero initial values for the velocity) allows us to reduce Equation (7.26) to the form x A = K (x, y , t) , t x y A = K (x, y , t) . t y (8.37)

Here K (x, y , t) is some function of coordinates and time. Eliminating it from two Equations (8.37), we arrive at y A = x y A . x (8.38)

Thus, not only the acceleration but also the plasma displacements are normal to the field lines. For dipole field, we obtain an ordinary differential equation dy 2xy =2 . dx x - y2 Its integral


186

Chapter 8. Flows in Strong Field

y = const x2 + y 2 describes a family of circles, orthogonal to the field lines, and presents fluid particle tra jectories. In particular, the tra jectory of a particle, situated at a point (x0 , y0 ) at the initial time t = 0, is an arc of the circle y y0 =2 2 2 x +y x0 + y0
2

(8.39)

from the point (x0 , y0 ) to the point (x, y ) on the field line (8.36) (Fig. 8.7).
y x, y
Figure 8.7:

A tra jectory of a fluid particle driven by a changing magnetic field of a plane dipole.

y

t=0

0

m

0

x

0

x

The integrals of motion (8.36) and (8.39) completely determine the plasma flow in terms of the Lagrangian coordinates x = x (x0 , y0 , t) , y = y (x0 , y0 , t) . (8.40)

This flow has a simple form: the particles are connected with the magnetic field lines and move together with them in a transversal direction. This is a result of considering small displacements under the action of the force perpendicular to the field lines.


8.4. Time-dependent Magnetic Dipole

187

(c) The plasma density change. On calculating the Jacobian for the transformation given by (8.36) and (8.39), we obtain the formula m (x, y , t) = 0 m0 +x2 y
2 2 3m2 - m0 4 m0 2 (m2 x2 + m0 y 2 )4 2 2 m2 x4 + m0 y 4 +

- 2x2 y

2

2 m0 - m2

2

.

(8.41)

On the dipole axis (x = 0) (0, y , t) m = , 0 m0 whereas in the `equatorial plane' (y = 0) m0 (x, 0, t) = 0 m
3

(8.42)

.

(8.43)

With increasing dipole moment m, the plasma density on the dipole axis grows proportionally to the moment, whereas that at the equatorial plane falls in inverse proportion to the third power of the moment. The result pertains to the small changes in the dipole moment. The exception is formula (8.42). It applies to any changes of the dipole moment. The acceleration of plasma is perpendicular to the field lines and is zero at the dipole axis. Hence, if the plasma is motionless at the initial moment, arbitrary changes of the dipole moment do not cause a plasma motion on the dipole axis (v = 0).


188

Chapter 8. Flows in Strong Field

Plasma displacements in the vicinity of the dipole axis always remain small ( 1) and the solution obtained applies. (d) In the general case of arbitrarily large dipole moment changes, the inertial effects resulting in plasma flows along the field lines are of considerable importance (Somov and Syrovatskii, 1972). The solution of the problem requires the integration of the ordinary differential equations that follows from Equation (8.21) and the freezing-in Equation (8.14). 8.4.2

Axial-symmetric dip ole fields

2D axial-symmetric problems can better suit the astrophysical applications of the second-typ e problem considered. The ideal MHD equations are written, using the approximation of a strong field and cold plasma, in spherical coordinates with due regard for axial symmetry. The role of the vector potential is fulfilled by the so-called stream function (r, , t) = r sin A (r, , t) . (8.44)

Here A is the only non-zero -component of the vector-potential A. In terms of the stream functions, the equations take the form dv = -2 K (r, , t) , dt where K (r, , t) = (Somov and Syrovatskii, 1976). d = 0, dt j (r, , t) r sin d = - div v , dt

(8.45)


8.4. Time-dependent Magnetic Dipole

189

The equations formally coincide with the corresponding Equations (8.17), (8.14) and (8.3) describing the plane flows in terms of the vector potential. As a zeroth approximation in the small parameter 2 , we may take, for example, the dip ole field. In this case the stream function is of the form
(0)

(r, , t) = m(t)

sin2 , r

(8.46)

where m(t) is a time-varying moment. Let us imagine a magnetized ball of radius R(t) with the frozen field B int (t). The dipole moment of such a ball (a star or its envelope) m(t) = 1 B 2
int

(t) R3 (t) =

1 B0 R 2

2 0

R(t) .

(8.47)

Here B0 and R0 are the values of B t = 0.

int

(t) and R(t) at the initial time

The second equality takes account of conservation of the flux Bint (t) R 2 (t) through the ball. Formula (8.47) shows that the dipole moment of the ball is proportional to its radius R(t). The solution to the problem (Somov and Syrovatskii, 1972a) shows that as the dipole moment grows the magnetic field rakes the plasma up to the dipole axis, compresses it and simultaneously accelerates it along the field lines. The density at the axis grows in proportion to the dipole moment, just as in the 2D plane case.


190

Chapter 8. Flows in Strong Field







Envelopes of nova and sup ernova stars present a wide variety of different shapes. It is common to find either flattened or stretched envelopes. Their surface brightness is maximal at the ends of the main axes of an oval image. This can sometimes be interpreted as a gaseous ring observed almost from an edge. However, if there is no luminous belt between the brightness maxima, then the remaining possibility is that single gaseous compressions - condensations - exist in the envelope. At the early stages of the expansion, they give the impression that the nova `bifurcates'. Suppose that the star's magnetic field was a dipole one before the explosion. At the moment of the explosion a massive envelope with the frozen-in field separated from the star and began to expand. The expansion results in the growth of the dipole moment. The field rakes the interstellar plasma surrounding the envelope, as well as external layers of the envelope, up in the direction of the dipole axis. The process can be divided into two stages. At the first one, the plasma is raked up by the field into the polar regions, a growth in density and pressure at the dipole axis taking place. At the second stage, the increased pressure hinders the growth of the density, thus stopping compression, but the raking-up still continues. The gas pressure gradient, arising ahead of the envelope, gives rise to the motion along the axis.


8.5. Practice

191

As a result, all the plasma is raked up into two compact condensates.

If a magnetized ball compresses, plasma flows from the poles to the equatorial plane, thus forming a dense disk or ring. This is the old problem of astrophysics concerning the compression of a gravitating cloud with a frozen-in field. Magnetic raking-up of plasma into dense disks can work in the atmospheres of collapsing stars.

8.5

Practice

Exercise 8.1. For a 3D field B, consider properties of the vectorpotential A which is determined in terms of two scalar functions and : A= + . (8.48)

Here is an arbitrary scalar function. Answer. Formula (8.48) permits B to be written as B= Hence Bž = 0 and B ž = 0. (8.50) Thus and are perpendicular to the vector B, and functions and are constant along B. The surfaces = const and = const are orthogonal to their gradients and tangent to B. Hence × . (8.49)


192

Chapter 8. Flows in Strong Field

a magnetic field line can be conveniently defined in terms of a pair of values: and . The functions and are referred to as Euler p otentials or Clebsch variables. Advantage of these variables appears in the study of field line motions. Exercise 8.2. Evaluate the typical value of the dipole moment for a neutron star. Answer. Typical neutron stars have B 1012 G. With the star radius R 10 km, it follows from formula (8.47) that m 1030 G cm3 . Some of neutron stars are the spinning super-magnetized neutron stars created by sup ernova explosions. The rotation of such stars called magnetars is slowing down so rapidly that a sup er-strong field, B 1015 G , could provide so fast braking. For a magnetar m 1033 G cm3 . Exercise 8.3. Show that, prior to a solar flare, the magnetic energy density in the corona is of about three orders of magnitude greater than any of the other types. Exercise 8.4. By using the method of conform mapping, determine the shape of a magnetic cavity, magnetosphere, created by a plane dipole inside a perfectly conducting uniform plasma with a gas pressure p0 . Answer.


8.5. Practice

193

The conditions to be satisfied along the boundary S of the magnetic cavity G are equality of magnetic and gas pressure, B2 8 = p0 = const ,
S

(8.51)

and tangency of the magnetic field, Bžn Condition (8.52) means that Re F (z ) = A (x, y ) = const , (8.53)
S

= 0.

(8.52)

where a complex potential F (z ) is an analytic function within the region G except at the point z = 0 of the dipole m. Let us assume that a conform transformation w = w(z ) maps the region G onto the circle |w| 1 in an auxiliary complex plane w = u + iv so that the point z = 0 goes into the centre of the circle (Fig 8.8). The boundary | w | = 1 is the field line S of the solution in the plane w, which we construct: F (w) = w - 1 . w (8.54)

Note that we have used only the boundary condition (8.52). The other boundary condition (8.51) allows us to find an unknown transformation w = w(z ). The field lines are shown in Fig. 8.8b. This solution can be used in the zero-order approximation to analyze properties of plasma flows near collapsing or exploding astrophysical ob jects with strong magnetic fields.


194
(a) v (b) S

Chapter 8. Flows in Strong Field
y/L 0.1
A=- 1

i

S

m

1

1

0.1

0.15

u
A= 1 A=- 2

x/L

A= 0

i

The field lines of a dipole m inside: (a) the unit circle in the plane w, (b) the cavity in a plasma of constant pressure.
Figure 8.8:

Exercise 8.5 To estimate a large-scale magnetic field in the corona of an accretion disk, we have to find the structure of the field inside an open magnetosphere created by a dipole field of a star and a regular field generated by the disk (Somov et al., 2003). Consider a 2D problem, demonstrated by Fig. 8.9, on the shape of a magnetic cavity and the shape of the accretion disk under assumption that this cavity, i.e. the magnetosphere, is surrounded by a perfectly conducting uniform plasma with a gas pressure p0 . Discuss a way to solve the problem by using the method of conform mapping.


8.5. Practice

195

p

0

Su

y z l m B x

G r Sd

A model of the star tion disk; l and r are the and Sd together with l and the domain G in the plane z
Figure 8.9:

magnetosphere with an accrecross sections of the disk. Su r constitute the boundary of .