Gut

1 U2 mg1 P1

including the condition of frozenness (p is a density). Integrating Eq. 4.15.18 we have

The lower limit of integration corresponds to the initial angle of entry into the region 2 behind the front in the first instant of particle contact with the front, i.e. p = p(-n/2) is the momentum before the wave passage. Eq. 4.15.21 results in that after crossing the shock front ( = n/2), the momentum p2 = p(n/2) is determined by the relation:

which is the expression for the law of magnetic moment conservation, i.e. p11H = const. Concrete properties of the medium parameter variations in the shock wave have not been used in the deduction of the Eq. 4.15.18-4.15.22. The only limitation for the field discontinuity have come to the condition of the conservation of magnetic flux per unit volume at the crossing of a discontinuity surface: u\H\ = U2H2 . Therefore the law of conservation of magnetic moment holds for any field discontinuity with this condition.

The law of magnetic moment conservation (see Eq. 4.15.22) having been deduced exactly is a result of the assumption that the front velocity u1 is infinitely small compared with a particle's velocity v. Actually the Eq. 4.15.22 holds with the accuracy to the terms of the order of ui/ v as compared to unit. The matter of problem is that if a particle makes a finite number of revolutions during when it is on the front (ui/ v is not an infinitely small quantity), the magnetic moment is not conserved according to the general theory of a violation of adiabatic invariants in the presence of forces undergoing a discontinuity of any derivative. if a particle has time only for a single crossing of the shock front ( u1 is comparable with v), the momentum increment Ap = (ui - U2)) cosp/c2 is positive for head on (-n¡ 2 <p<n¡ 2) and negative for overtaking collisions. In this case a magnetic moment is not conserved. it is evident that the less is a particle's velocity the less is the difference of its behavior on the front from the behaviors of thermal particles which are heated non-adiabatically on the shock front.

4.15.5. Particle acceleration by oblique shock waves

The acceleration of energetic particles on the front of an oblique shock wave have been analyzed by Alekseev and Kropotkin (1970). The particles with a Larmor radius rL which is large compared to a discontinuity thickness of d were considered as energetic particles. It was assumed that the motion of such particles near the plane of discontinuity (x = 0) is determined by magnetic field which is uniform in each of semi-spaces (1 at x > 0 and 2 at x < 0). The collisions for discontinuities in collisional plasma are neglected; the latter assumption was argued for the particles with a free path A >> rg >> d . Since d is of the order of the free path length of the thermal plasma particles, and the free path of a given particle is increased with its energy, the above inequality will hold for the particles the energy of which is large compared to the thermal energy. It was assumed for collisionless discontinuities that an interaction of the particles under consideration with micro-fields (forming a jump of the magnetic field) is insignificant compared to the interaction with regular macroscopic magnetic field; this appears to be true if rg >> d . Particle interaction with an oblique shock front is essentially different from the interaction with a wave in purely transverse field (the normal field component Hn = 0) considered above. Note that there always exists the coordinate system in which an electric field E = 0 (Landau and Lifshitz, M1957). In this coordinate system a particle trajectory consists from parts of spiral lines occurring by turns either in the region 1 with the magnetic field (having the components Hx1 = Hn, Hy1 = 0, Hz1 = Ht1) and in region 2

with the field H2 (its components are Ex2 = En, Ey2 = 0, Ez2 = Et2). A

trajectory is determined by the following two parameters: pitch angle 6 and the phase 9 of Larmor precession in any point of crossing the front plane (the phase 9 is counted from a normal to the force line which is directed from the space 1 to 2). Let En/Et1 = tga and En/Ez2 = tgfi be a tilt of the force lines to the front plane in the regions 1 and 2, respectively. Then 61 and 91 are related to 62 and 92 (the subscript indicates to which of the regions the variables 6 and 9 are related) by the condition of the continuity of the velocity at the crossing of a shock front cos62 = cos^cos6 + sin^sin6sinp, sin62cosp2 = sin6cos9, sin62sinp2 =-sin Y cos 6 + cos Y sin 6 sin 9, (4.15.23)

where y = a - ¡3. Furthermore, the phases of successive crossings (m and m + 1) of the front plane are connected by the relation:

for the region 1 and by a similar relation for the region 2. For simplicity we shall assume below that the angles a and ¡are small. For an increments of a pitch angle and a phase per one revolution we then obtain

A6 = 2Ysinp, Apsinp = ctg6[2na - y( - sin2p)]. (4.15.25)

The variations of parameters 6 and 9 along a particle trajectory determine a certain dependence of 6 which is governed by the equation c,g66= . 2f+s'9;+ . (4.15.26)

The solution of Eq. 4.15.26 is

where 60 is the pitch angle for the first crossing of the front. The phase 9 varies from

0 to n at the transition from the region 1 to 2 and conversely otherwise. The Eq. 4.15.27 means that the magnetic flux enclosed by a particle during one revolution is constant. The particles moving from a region of weaker magnetic field with the pitch angle eo > ecr (sin2 ecr = fija at a> fi) are reflected from the plane of the shock front with conservation of the magnetic moment

H = macv2 sin2 e/2H = macv2 sin2 <%/2H , (4.15.28)

where the pitch angle after coming from the front is ek = n-eo . The rest particles pass through the front changing their pitch angles according to the conservation of the magnetic moment. It is possible to determine the trajectory in the front plane by using the integrals of motion

c where Py and pz are the momentum components of a particle. In the instant when a particle crosses the plane of the shock front Py = - p sine cos (p, pz = p cose (the terms ~ a are omitted). Putting x = 0 and neglecting the rapid oscillations of a particle in the direction normal to the front, we obtain from Eq. 4.15.29 and 4.15.30:

y - yo = rg (coseo - cose), z - zo = rg (sineo cospo - sinecosp) .(4.15.31)

For definiteness we consider that a particle is positively charged. The quantity rg = pc17eHn is the Larmor radius in the field Hn ; yo and zo are the particle's coordinates at the first crossing of the front; do and po are the initial pitch-angle and phase; e and p in Eq. 4.15.31 are determined by Eq. 4.15.27. The total displacement in the yz-plane (during the time when a particle is near the front) will be

Ay = rL(coseo -cosek), Az = rL(sineo -sinek). (4.15.32) For particle reflection Eq. 4.15.32 gives

In the case of small angles aand fiEq. 4.15.25 are complicated. The variations Ae and Ap during one revolution are now not small. The pitch angle ek and pk of a particle leaving the plane of the shock front are determined by the successive solutions of the Eq. 4.15.23 and Eq. 4.15.24. In this case 6^ also depends on the initial pitch angle 6o and on the phase (po . A particle's magnetic moment is not conserved after passing through the front. However, the general character of the motion remains the same. The particles moving from the region of a weaker field will be reflected from the front if their pitch angle is more than 6cr depending on the initial phase po . The other particles will cross the front.

On the basis of the results considered, Alekseev and Kropotkin (1970) have calculated the expected particle acceleration. The matter of the problem is that in any coordinate system, except for the single one in which [uH] = 0, acceleration will take place because there is a uniform electric field parallel to the plane of discontinuity. When crossing this plane the leading center of a particle's motion moves along the electric field so that there is a corresponding change of the particle's energy (see for comparison Section 4.15.1).

Let us pass to the coordinate system K' moving relative the initial frame with the velocity u = cE/Hn along the z-axis (the z component of a magnetic field has a jump at the plane of discontinuity but the other two components are continuous). The electric field E' in the system K' is equal to zero; Hz1 = Hzt and H'z2 = Hz2 ; the magnetic field component parallel to the velocity u does not vary and is normal r 2 i i2 y/2 i , to the plane of discontinuity H 'n = I Hn - |E| I . At |E| = Hn the angle a' of force line tilt to the front plane will be small in the K' system even if the initial angle a ~ 1. A particle trajectory in the K' system has been described above. The turn transition to the initial coordinate system K makes it possible to obtain the particle trajectory in the presence of an electric field.

The change of kinetic energy AEk at the crossing of the front is given by the equation where ay is determined by Eq. 4.15.32 in the case of oblique shook wave.

4.15.6. Particle acceleration by rotational discontinuities

Alekseev and Kropotkin (1970) have also considered the trajectories of motion and acceleration of energetic particles in the vicinity of moving rotational discontinuity. Near the plane of a rotational discontinuity a magnetic field has the components:

Hx1 = Hn, Hy1 = Ht cosY/2, Hz1 = Ht sinY/2 for x > 0;

Hx2 = Hn, Hy2 = HtcosY/2, Hz2 = Ht sinY/2 for x < 0; (4.15.34b)

Here Y is the angle of the turn of the field's vector at the discontinuity. Similar to the case of an oblique shock wave, the boundary conditions (satisfied on the discontinuity surface) result in the existence of the coordinate system where E = 0 (see, for example, Landau and Lifshits, M1957). In the case of a rotational discontinuity, assuming that the tilt angle of magnetic force lines is small, we obtain the following expressions (instead of Eq. 4.15.23):

cos 6*2 = cos Ycos 61 + sin Ysin6cosp1 + «(1 - cos Y)sin6sinp1; sin62cosp2 =-sinYcos6 + cosYsin61cosp -asinYsin^sin^; sin02sinp2 =-a(1 - cos Y)cos6 + asin Y cos6 + sin^sin^. (4.15.35)

Using Eq. 4.15.35 and Eq. 4.15.24) we obtain increments of pitch angle 6and phase p during one revolution:

A 0 = 2a[(1 - cos Y)sin Y - <~sin ç(sin Y cos ç + cos Yctg 0)]; Açsinç = a{ctg0[2a(n-ç)+ (1 - cos Y)sin2ç]- 2sin Ysinç

2ctg0(cos2 ^ - sin2 Y cos2 ç)+ sin2Y cosç(1 - ctg2i

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