DISTRIBUTION

FUNCTION IN A STRONG E.V. SUVOROV

OF RELATIVISTIC MAGNETIC

ELECTRONS

FIELD

and Y U . V. C H U G U N O V

Radiophysical Research Inst#ute, GorkL U.S.S.R.

(Received 20 March, 1973) Abstract. The motion and radiation of relativistic particles with radiation reaction in a strong magnetic field has been considered. The kinetic equation determining the relaxation of the distribution function with radiation reaction has been investigated. The universal one-dimensional distribution function is found to which any isotropic ultrarelativistic distribution in a strong magnetic field is relaxed. It is of power type e-3 for ultrarelativistic energies e >>mc 2. Estimations are made which indicate that under the pulsar conditions the one-dimensional electron distribution, function is likely formed due to radiation losses while for ions the one-dimensionalization is associated with the conservation of the adiabatic invariant. The problem of taking into account the influence of radiation damping on the distribution function of relativistic electrons in a strong magnetic field is of particular interest because strong magnetic fields ( H to 1012 Oe) and ultrarelativistic particles (with the energies e to 104 m c 2) exist likely in the pulsar magnetosphere where the powerful electromagnetic radiation generates. Under these circumstances the radiation reaction proportional to the squares of the magnetic field intensity and particle energy will be rather effective and can play an important part in formation of the electron distribution function. One of the important aspects of the problem is the distribution function 'onedimensionalization' due to radiation damping. In some papers (see, for example, Sen Gupta, 1970; Kaplan and Tsytovich, 1972) it is stated that a particle through radiation looses in the main, the transverse (with respect to the external magnetic field) component of its momentum. Due to this, the electron distribution function being ultrarelativistic becomes practically one-dimensional, otherwise without any dispersion over the transverse momenta. In the paper by Sen G u p t a (1970) it is concluded from the longitudinal velocity conservation that the radiation draws all of its energy from the transverse component of the motion. In the monograph by Kaplan and Tsytovich (1972) the one-dimensional distribution function formed through the radiation losses is taken in the form f,,,~ ( e + e , ) -~ where the characteristic energy e, is considerably larger than the rest energy e, >>m c 2. This point of view, however, requires essential correction. It is well known (see Landau and Lifshits, 1967; Ginzburg and Syrovatsky, 1963) that at large pitch-angles (0 >>mc2/e) the energy of an ultrarelativistic particle decreases approximately twice at a time of t 1 ~ t o mc2/~ (where t = 3 m 3 c S / 2 e 4 H z, H is the magnetic field intensity, e is the charge of an electron, e/me 2 is the ratio of the electron energy to its rest energy). Therefore one can speak about the ultrarelativistic distribution function which forms due to radiation losses (of the t y p e f ~ (8 + e , ) - r, e, >>m e z) considering the relaxation of the initial isotropic distribution only up to a time of t >~ti. However for this time Astrophysics and Space Science 23 (1973) 189-199. All Rights Reserved Copyright 9 1973 by D. Reidel Publishing Company, Dordrecht-Holland

190

E. V. SU'VOROV AND YU. V. CHUGUNOV

the 'one-dimensionalization' cannot occur because a particle remains ultrarelativistic and consequently the force o f radiation d a m p i n g is directed against the electron velocity ( L a n d a u and Lifshits, 1967) and cannot change its pitch-angle. This means, in particular, that the emitted energy is drawn b o t h f r o m transverse and longitudinal m o t i o n o f a particle*. The latter circumstance was pointed out by Shen (1970). O n the other h a n d at t--+ oo the transverse m o m e n t u m c o m p o n e n t t h r o u g h the radiation losses falls to zero and the energy reaches the finite value e~ = m c Z / x / ( 1 - f l ) ~ (where flz is the ratio o f projection o f the velocity o f a particle on the magnetic field to the velocity o f light)**. The m o t i o n o f a particle becomes one-dimensional, however, at a time o f tN to >>tl. This time is also typical o f formation o f the one-dimensional distribution function. Therefore the m o t i o n o f an ultrarelativistic particle with the large initial pitch-angle 00 >>mcZ/eo has two characteristic stages. F o r small periods o f time t 1 < t ~ to the energy o f a particle is essentially changed but not its pitch-angle. F o r large periods o f time t>~ t o, the energy o f a particle approaches its finite value and the pitch-angle begins to decrease, i.e. the m o t i o n becomes one-dimensional. To investigate the process o f 'one-dimensionalization' it needs to consider the relaxation of the initial isotropic (or quasi-isotropic) distribution function at the time t ~>t 0. As the solution o f the kinetic equation by the characteristic m e t h o d depends on the integrals o f m o t i o n o f a separate particle, we first consider the m o t i o n o f a single particle with radiation reaction in the external megnatic field and then examine the relaxation o f the distribution function. The results obtained below for the distribution function relaxation are related to a sufficiently rarefied plasma. Only in this case we can neglect the collective effects associated with the influence o f self-consistent fields arising at various instabilities, as well as the influence o f the m e d i u m on the spontaneous radiation f r o m charged particles. 1. The force o f radiation reaction in the uniform magnetic field H directed along the axis z is equal to ( L a n d a u and Lifshits, 1967)* f,=3m2c--~ v

--v

~c5c2 ( 1 - f l ~ ) + ~ c o s 0

,

(1)

* Despite the conservation of the longitudinal velocity v~, the longitudinal momentum changes due to a decrease in the mass of a particle. ** Note that ff the initial angle is 0 >>mc2/8, a particle through the radiation losses a considerable fraction of its energy (e0 >>e~) Therefore, the quasi-isotropic initial distribution function cannot transform into the one-dimensional distribution such as fi N (e + e,)-v falling weakly up to ultrarelativistic energies e. >>mc 2. * The expression for the radiation force (1) is obtained by the perturbation method on the assumption that in the frame of reference associated with a particle, it is smaller than the external force. However, in the laboratory frame of reference this condition is weaker and in particular does not exclude that the radiation force may be of the order of magnitude and even more than the external force F ( f i . ~ Fe/mc ~ when the particle moves perpendicularly to the external force). The latter circumstance shows the relativistic mass anisotropy roll~ m j_ and that for large energies e >>me 2 the force of radiation damping is anti-parallel to the velocity while the external force may have the component orthogonal to it.

RELATIVISTIC ELECTRONS IN A STRONG MAGNETIC FIELD

191

where e, m, v, e are the charge, the rest mass, the velocity and the energy of a particle, respectively, c is the velocity of light in vacuum, fiz=V/e cos& In the synchrotron limit when (e/mc2) 2 (1-82)>> 1 the radiation force is directed against the velocity and the pitch-angle 0 does not vary. The transverse to the velocity component of the radiation force which decreases the pitch-angle is of importance for the dipole approximation (e/mc2) 2 (1 _ f i z ) ~ 1. Taking into account that fi~ = const, the motion of a particle influenced by the radiation force (1) is obtained by transfer from the frame of reference in which it moves transversely to the magnetic field*. The energy of an electron moving transversely to H is changed according to the law (Landau and Lifshits, 1967)

e'(t') = cth ( t ' + const) me 2 \to '

(2)

where t o = 3m3eS/2e4H z and the constant is determined by the energy at t =0. Allowing for the transformation laws into the laboratory frame of reference for the electron energy and time e' = e x / 1 - - ~ , t'=tx/i-flz~ we obtain e(t)=~oocth(~oX/1-/~2+const ),

(3)

where In the ultrarelativistic limit at t ~ t o / ~ expression (3) for energy is essentially simplified and one can find the particle trajectory. In this limit the angular velocity of an electron in the external field is linearly varied with the time ~ = o)nmcZ/e (t) ~- oh1 x x [(t/to)(1-fl~) + const] (On = eH/mc is the gyrofrequency of an electron). Therefore, in the plane perpendicular to the magnetic field, a particle moves along Cornu spiral

x _- c,/2t0

, f cos e 2 de, it/

0

,/---f

y ~- c 2to/co~

sine 2 d e , O

with the characteristic scale** c~/(2to/cgn) with the velocity v• the axis z it moves with the velocity v~-c cosO.

c sin O while along

* T h e fact that flz = const is also obtained by transfer into a n e w f r a m e of reference. Indeed, in the coordinate system where fix = 0 a particle m o v e s perpendicularly to the m a g n e t i c field a n d the r a d i a t i o n force projection o n the m a g n e t i c field is equal to zero. So, in this f r a m e o f reference the velocity c o m p o n e n t r e m a i n s equal to zero at all m o m e n t s o f time, in a n y other f r a m e o f reference m o v i n g with the c o n s t a n t velocity a l o n g z, vz = const. ** By the characteristic scale o n the plane I H we m e a n the distance f r o m the point where the particle h a d the infinite energy to the stop point.

192

E. V. SUVOROV AND YU. V. CHUGUNOV

Let us point out some peculiarities of synchrotron radiation of an ultrarelativistic particle associated with radiation reaction. It is easy to show that the local perturbations of the particle trajectory due to the radiation force are negligibly small. Therefore at each moment of time the radiation from a particle at the perturbed trajectory is the same as that from a particle with the equal energy at the unperturbed trajectory. This is the natural consequence of the perturbation method used for calculating the radiation force. However, the latter has a considerable effect on the trajectory as a whole (in the plane _I_H instead of the circle, the particle moves along the Cornu spiral). This fact leaves an impression on the particle radiation. In a fixed direction the radiation consists of separate narrow pulses. The form, frequency spectrum and radiation pattern of each pulse are such as those of the particle with the energy corresponding to the moment of radiation and moving along the unperturbed trajectory. However due to the energy losses, the pulse period decreases in time, their form being also changed. If the particle making a turn along the spiral losses a considerable fraction of its energy, the frequency spectrum and radiation pattern of two neighbouring pulses of radiation are essentially different. Despite this fact the radiation from an ensemble of particles with a given energy spectrum remains such as that when we neglect the radiation reaction*. Let us give attention to the dispersion of synchrotron radiation lines. It is known that when an electron moves, for example, transversely to the magnetic field, the synchrotron radiation spectrum consists of harmonics of relativistic gyro-frequency v~n=o~n mc2/e without allowing for the radiation reaction. With allowance for radiation losses the frequency of the electron rotation in the magnetic field and the emitted frequencies vary in time and consequently the radiation lines are broadened. It is easy to estimate at what intensity of the magnetic field and energies of relativistic particles the linewidth exceeds the distance between lines and the radiation spectrum of a single particle becomes continuous**. A change in energy of a particle for a turn is of the order of magnitude de~frc/w H that leads to broadening of the emission line of the order of dco/co~ de/s. In order the harmonics to be resolved, it needs to satisfy the condition d ~ o ~ n. At the frequencies typical of synchrotron radiation (con (e/mc2) a vJn) with taking into account expression (1) for fr, this leads to the inequality s )5

--no2

e3H ~ mZc,i.

For the electron in the magnetic field H ~ 1 0 e , the lines in the maximum of synchrotron radiation are overlapped at the energies 5> 103mc a while for the fields H N 10 l~ Oe at 5> 10 rnc 2. * To maintain the given energy spectrum, a constant injection of energetic particles into the radiation region is necessary. ** In addition the magnetic field inhomogeneity along the trajectory leads to the dispersion of radiation lines and for the radiation from an ensemble of particles, the dispersion of their energy does.

193

RELATIVISTIC ELECTRONS IN A STRONG MAGNETIC FIELD

2. Let us consider now the variation of the pitch-angle of a separate particle due to the radiation losses cos 0 (t) = x/1 _ ((tW-mc2/-~"" .

(4)

It follows from (4) that until the energy of a particle exceeds considerably its finite value e>>~ the pitch-angle is practically not changed. When the energy reaches its finite value, the pitch-angle decreases up to zero. In some limiting cases the expression for the pitch-angle is simplified. If at the initial moment a relativistic particle moves almost transversely to H (flz ~ 1), then

cos 2 0 (t) _ 82 + 2 exp ( -

t/to)"

(4a)

Expression (4a) is usable at all moments of time t > 0. When the condition l - f l z2~ l is satisfied, the approximative expression for the pitch-angle has the form of

{O2(t) ~_ (1 - fl2) 1

]2}

V cth((t/to) x/i - fl~) + eo/e~o [_(80/~)cth--((t/~to-)-x/1 =fizZ+ 1

'

(4b)

where % is the energy of a particle at t = 0. Finally, at the very small initial pitch-angles 0o "~mc2/eo the particle pitch-angle through the radiation decreases up to zero practically without changing its energy the effect of 'radiation compression' in the terminology suggested by Kaplan and Tsytovich (1972). In this case,

0 (t) ~ Ooe41-13~2-t/t~

(4c)

Expression (4c) is obtained from (4b) in the approximation 8o -coo ~ e. Therefore, the particle motion tends to be one-dimensional (0-+0) at any initial pitch-angles. The radiation compression (the decrease of the pitch-angle without changing the particle energy) takes place only in the case of very small initial pitchangles 0o ,~mc2/8o . The time-scale of 'radiation compression' is t ~ to,~me 2. For this period the particles with large initial pitch-angles 0o >>mcZ/eo relax completely. Their energy reaches the finite value and the motion becomes one-dimensionaL 3. Let us consider the relaxation of the relativistic distribution function of charged particles which may be inhomogeneous along the direction of the external magnetic field z. The kinetic equation for the momentum distribution function F (p, z, t) with taking into account the radiation reaction is of the form ~t+V~zz+~

[-vH]+f~

F

=0.

(5)

194

E . V . SUVOROV AND YU. V. CHUGUNOV

In the presence of the axial symmetry in variables p~, p• with allowance for expression (1), we obtain*

,T/eN/p2 _it_m 2r OF

OF

ot + p m e

1 {p~p2 OF

Oz - to

+

+ p; (p~ + m% ~) 0p_~ + 2 ( 2 p 2 + m z c 2) F = 0 . (6) The solution of Equation (6) takes the form nl 2 C2 ..{_

F -

p2 r

4 2

GP•

(7)

where ~b satisfies the equation

mc~/p2 +

m2r

0qb

O~ + pzmc2 0q~ c3z

1

2 8r

2

01~"]

and is the arbitrary function of three integrals of motion = ~ (Zo, Pzo, P• Pz ~o = z - . / m ~ c_z + p~ ct,

pzmc P~o = ' /N/mZc 2 + •

p•

X

e x p ( - 2`/ (mZc 2 + p2)/(macZ + p2) t/to ) + pZ,/(~m2 cZ + p2 + mc)2

exp(- 2,/(m2e2 + pi)/(m2e~

+ p2) t/to) - / , / ( , / m ~ e

~ + Pl + me) ~

2mcp•

= X P.I_O ` / m 2 c 2 _}_p2 + me

e x p ( - x/'(m2c 2 + pZx)/(m2c2 + p2-) t/to ) x

exp ( - 2`/(mZc 2 + p~)/(mZc 2 + p2) t/to ) _ p~/(x/m2c z + p2 + mc)2

(8) We first refer to the uniform problem. Let at t = 0 there be some isotropic uniform distribution, for example, the relativistic equilibrium function N e-~~ F (p, t = 0) = 4re (mc) 3 (xT/mc 2) K2 (mcZ/xr) '

(9)

* The corresponding equation for the ultrarelativistic case (PII,P• >>me) is given in the paper by Tsytovich et al. (1970).

195

RELATIVISTIC ELECTRONS IN A STRONG MAGNETIC FIELD

where xT is the temperature in the energy units, N is the particle density, K z (x) is the MacDonald function. At the initial moment almost all particles are considered to be ultrarelativistic that is xT/mc 2 >>1. We find from (7) and (8) that the evolution of the initial distribution (9) occurs in the following way

N mc 2 (mc2~ P~oP~o ~ CX/m2czq- p2 Pzo~ F(ptE, p• t) -- 4n(mc) 3 x ~ K21 \ xT ] ~GP• e x p (~ xT p~ ) { t /mZcZ+ p ~ P•

F(pii, p •

for

exp - -- - to Vm2c

for

exp - ~ m ~ c 2 + ~ j > ~ / m 2 c 2 + p ~ + m c ,

7

J

<

2+

(10)

+

where expression (8) should be used as P~o and P• It is easy to understand that in some region of the values p~, p• the distribution function is equal to zero. In fact all energies up to s = oo are presented in the initial distribution function (9). The radiation losses are such that the infinite energy is lost at the finite time (Pomeranchuk's effect). Due to this at t > 0 the particle energy cannot exceed some critical value dependent on the pitch-angle. This is shown in the distribution function (10). When considering the relaxation it is convenient to single out two time intervals: small time t ~ to when the motion of each particle occurs with the pitch-angle conservation and large time t >~t o when the pitch-angles vary. (a) At the time t ~ t o when p• Pz >>mc and the energy ~ ~-pc (at the ultrarelativistic stage of evolution), the relaxation occurs according to the simpler law which corresponds to taking into account only the first term ~ s 2 in expression (1) for the relaxation force (the synchrotron limit) F(p, 0, t ) =

IN(c)3 8n ~ 0

l { s ( 1 - s / s e t ) 4exp - x T ( l _ s / e c r

} )

for

s
for

s>ecr.

(11) and G~ = mc2to/t sin2 0. It is seen from (11) that in time the isotropic (at first) distribution function becomes anisotropic. However this circumstance does not mean the distribution function onedimensionalization. The relaxation occurs in such a way that the value N (0)= = ~ F (p, O, t)p 2 dp = N/4zc is independent of the angle 0 as for the isotropic distribution function. This is consequence of the fact that at the ultrarelativistic relaxation stage the pitch-angle of each particle does not change (one of the characteristics of the kinetic equation is P• = const). Figure 1 shows the form of the energy spectra N (s) for different values of the pitch-angles at a certain moment of time t ~ t o. The energetic spectrum at 0 = 0 is independent of time. The energetic spectra for the angles 0 # 0 have the energy cut-off at s > s ~ and a sharp maximum near it. With increasing the time at the fixed 0, the value ecr decreases and the energetic spectrum is

196

E . V . SUVOROV AND Y U . V . CHUGUNOV -1

2 2

N($,8)&xxTN o (xT/.c)

O
(3 If

I

/89 @t

B=O

8-

o

Fig. 1. The energetic spectra N(e) for different pitchangles at a certain moment of time tl < t ~ to. The energetic spectra for the same angle 0 after a little while are shown by dashed lines. (The initial isotropic distribution function has the form of (9)).

deformed, narrowing along the abscissa axis and stretching along the ordinate axis. This is shown in Figure 1 by dashed lines. It is clear that t h o u g h the particles with large energies (8 ~ x T ) are m o v i n g at small angles to the magnetic field the main part o f particles has smaller energies ~ mc2to/(t)~> mc 2 and pitch-angles of these particles 0~1. (b) Further at t~> t o an essential part is played by the second term in radiation force (1) related with the dipole radiation. The pitch-angles of separate particles are changed that means the beginning o f the distribution function 'one-dimensionalization'. Here the relaxation is rather complicated. However it is clear that in time the distribution function tends to the one-dimensional one becoming more limited over p~. It is easy to find the one-dimensional distribution function which is asymptotically established at t > t o by integrating (11) over transverse momenta. One should bear in mind that at t ~ Go p• is m u c h smaller than mc and at the same time the value r / = exp

-- --

to

+ PV

197

RELATIVISTIC ELECTRONS IN A STRONG MAGNETIC FIELD

is much smaller than unity and is independent ofpz. Then we have m

2rime

F(p=, t-~oo)=2,~ p~F(p=,p,, t)dp~ =>~ 0

PAP'-~4,~(m~y xT • 0

,(mc2~ 0l 2 + pZ,/4m2e2)2 rla f ev/m-~e2 + pZrlZ + p~/4m2c') x K ; \ x T J -(qi22p~/4-~c2p exp~ - , xT ~12- pZz/4m2eZj" The integrand at e ~ 0 has the properties of the delta-function 3 (p,) in a sense that the result of integrating is independent of e though the interval of integration tends to zero. As a result we obtain F(P/t, Pl, t - , m )

•

[

xT

N

__1

1 + 2 c x/rn2c 2 + p~

(mc 2"] e_../.,~+pj/,,T

(~f +2

1

l ]a(pD__~ , m2c2

(12)

+ P z 2rcp•

Taking into account the high temperature of the initial distribution (xT >>mr 2) the one-dimensional distribution (12) in the momentum region Pz ~ xT/c is approximated by the formula

F(Pll, p•

N

1

a (p~)

t-+m) ~_2me (I + p~/m2c2)3/22up1"

(13)

Not only the ultrarelativistic distribution (9) but any isotropic distribution function with the mean energy e > rnc2 relaxes to the one-dimensional distribution function (13) due to the radiation reaction. The one-dimensional distribution is easy obtained from the following simple considerations. The isotropic ultrarelativistic initial distribution is the delta-distribution 6(v-e) in the velocity space. The longitudinal velocity distribution N (v~) is a 'step' function

N(v~)=2=

i o

v•

[N0

for

]v~l
[0

for

Iv=l>c.

04)

Clearly, N (v~) is maintained during the relaxation through radiation since the longitudinal particle velocities do not change. At the finite stage of relaxation the transverse momentum decreases up to zero and the resulting relation between v= and the longitudinal momentum p~ is Vz= cpJx/(m% 2 +p~). Making use of this relation we find

dr=_ No 1 N(p~) = N(vz) dPz 2mc (1 + p2/m2cZ)3/2'

(15)

198

E.V. SUVOROVANDYU.V. CHUGUNOV

which is identical to the one-dimensional distribution (13). As in the case (13) expression (15) is valid for sufficiently small momenta p~ < ~/c where e is the mean energy of the initial isotropic distribution. Hence, the one-dimensional distribution (13) is universal in a sense that at the time t ~ t o any isotropic ultrarelativistic distribution is relaxed to it. This distribution may be taken as the basis for investigating the dispersion characteristics of electromagnetic waves in relativistic plasma in a strong magnetic field. Note also that the aboveconsidered integrals of motion (the characteristics of the kinetic equation) enables one to investigate some other problems inhomogeneous in space and nonstationary in time. In particular, the above-solved problem of the time relaxation of homogeneous in space distribution is transformed into the case of stationary injection into the

..{_--p2.

halfspace z > 0 by substituting t ~ ( z / c p z ) ~ / m 2 c 2 As to the application o f obtained results to the physical processes taking place near the pulsar surface we should note the following. At present the conditions of acceleration and generation of relativistic particles in the pulsar magnetosphere are unknown. However, if we assume that the relativistic particles are generated near the neutron star surface and then transfered into the external magnetospheric layers, two mechanisms will compete in forming the one-dimensional distribution. The first is discussed in the present paper, the second is connected with conservation of the adiabatic invariant mvZ,/H. For the electrons, the characteristic scale at which the one-dimensionalization occurs due to radiation losses is equal to Rr=cto~-~ 10- s (1012/HOe ) cm. The characteristic distance ofone-dimensionalization because of the particle transfer into the region of the weaker magnetic field (the conservation of adiabatic invariant) is comparable with the neutron star radius, R a ~ 10 6 c m . It follows from the comparison of these scales that with the magnetic field in the region of the relativistic particle generation more than 10 6 0 e , the radiative 'one-dimensionalization' is determinative. For smaller fields the one-dimensionalization associated with conservation of the adiabatic invariant is more effective. However, these two mechanisms lead to essentially different one-dimensional distribution functions. The relaxation through radiation is accompanied by the loss of a considerable fraction of energy of the initial isotropic distribution while the one-dimensionalization associated with the conservation of the adiabatic invariant does not change the particle energy. For ions, because of their large mass, the scale of the one-dimensionalization through radiation is essentially greater than that for the electrons. The second mechanism (the conservation of the adiabatic invariant) is equally effective both for electrons and ions. The radiative relaxation for ions is prevailling at the fields H > 1012 Oe.

Acknowledgements The authors are grateful to Dr A. A. Andronov and Prof. V. V. Zheleznyakov for the discussion of the paper.

RELATIVISTIC ELECTRONS 1N A STRONG MAGNETIC FIELD

199

References Ginzbttrg V. L. and Syrovatsky S. L: 1963, The Origin of Cosmic Rays, Izd. AS U.S.S.R., Mescow. Kaplan S. A. and Tsytovich V. N.: 1972, Plasma Astrophysics, Izd. Nauka, Moscow. Landau, L. D. and Lifshits, E. M. : 1967, The FieM Theory, Izd. Nauka, Moscow. Sen Gupta N. D.: 1970, Phys. Letters 32A, 103. Shen C. S.: 1970, Phys. Letters 33A, 322. Tsytovich V. N., Buckee J. W., and ter-Haar D. i 1970, Preprint 26/70 Oxford; Phys. Letters 32A, 471.