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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



q

f-

q



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