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X-GTF-n-334PREPRINT

. A«il .33 S JY.J^^ w ^

SYNCHROTRON RADIATION FROMELECTRONS IN HELICAL %.".#':tBITS

KUNITOMO SAXURA1

(ACC 51 NUMBER)(THRU)

S (PA ES)(CODE)

(NASA CR f1R TMx OR AD NUMBER)v_

AUGUST 1968

lco- GODDARD SPACE FLIGHT CENTEP

GREENBELT, MARYLAND

SYNCHROTRON RADIATION FROM ELECTRONSIN HELICAL ORBITS

Kunitomo Sakurai*NASA Goddard Space Flight Center

Greenbelt, Maryland

ABSTRACT

This paper considers correction factors of some of

the commonly used re.:,ults of synchrotron radiation theory

both in vacuo and in ambient plasmas. These revisions

are important when the spectral distribution of synchro-

tron radiation and the wave frequency range in which

the influence of ambient plasmas becomes serious are

considered.

*NAS-NRC Postdoctoral Resident Research Associateship

1

2

Synchrotron radiation from energetic electrons is very

important in explaining various characteristics of solar radio

type IV bursts and galactic radio emission (for example, Kundu

1965; Ginzburg and Syrovatxkii, 1964, 1965". In recent time

both the effect of helical motion of radiating electrons (Epstein

and Feldman, 1967) and the influence of ambient plasmas (Ramaty

and Lingenf elter, 1967; Ramaty, 1968) on the characteristics

of synchrotron radiation have attracted attention.

The influence on the characteristics of synchro-

tron radiation of the helical motion of radiatinging

electrons will be here estimated for both cases, (1) in vacuo

and (2) in the plasma medium.

Spectral Distribution of Synchrotron Radiation

The angular distribution (angle e) of the radiation intensity

from electrons of arbitrary energy in helical orbits with the

pitch angle +y, for the S-th harmonic component is given by

dI ss 2e2W2 02(1-02) 0sinBsinY^ 2^c - '(1-0cos cosy") [sinY^Js (s 5cos'-'ocosy

core-scosy ssinesiny+C`Tsin6'^ Js (s ^cos^co,Sf) I 2 ) (1)

1where w is the gyrof requency which is defined by eHo /moc and

co is the velocity,of electrons, (e.g., Takakura, 1964; Kundu,

1965; Be-kefi, 1966) , and Ho is the strength of external magnetic

f ield .

J11

3

The total intensity of radiation of the s-th harmonic is

obtained., by integrating with respect to the solid angle, as

follows :dI

I s f (-S ) dn

2se 2W 2 1- 0*2 2c

B*_s (1- 0 *2 ) f J2s (2s g)d ^]

O

where d(')-2rrsin 8d 9 and S * = SsinT/ 1-s Cos Y.

In the above equation, we define an apparent Lorentz factor

y* in order to proceed with the discussion of the case for

T-r/2, which has been already dealt with by many authors (for

example, Schott, 1912; Sehwinger, 1949; Francia, 1959; Landau

and Lifshitz, 1961) . This apparent Lorentz factor is defined

as

'Y*^ 1 Y '1-^ (3)

^/1-^ p * 2

where 2, is the true Lorentz factor. If we assume

y»1, i.e. S Qe 1, y* is reduced to y sin 7 according to equation

(3) . It thus follows that -y*»1 because the pitch angle T is

taken as constant in the above discussion as far as the calculation

of the spectral distribution is concerned.

(2)

4

For Y*»1(.'. *=1) and s»1, the first and the second terms

its the parenthesis on the right-hand side of equation (2) are

expressed as

it ( 2s O*)" .— I ^. K2/3(2s/3-y*3) (4)'TT Y*

and

s (1-0 *2),r, *J2s (2 sF )df0

12^. * a K1/3(r^)dr,, (g)

by making use of the approximate formulae in Watson's book

(1945) , where n,-2s/3y* 3 . By using the equality

ae

2K2/3 (a) - ,^'aK1/3 ( ,n) dn- J a K5/3 (r) dr (6)

the right-hand side of equation (2) can be revritten for the

case y»1 4s follows:

2 w 2 0eI s ew H _.s ^'

K5/3 ( TI ) d o,J3 rrc ^y* a

e 2 W 2 s mf aK5/3 (r) dr (7), 37c 1'

.sin ,y

Here we define the characteristic harmonic number s byr

3/2.y*(-3/2,^y3si ,i3*), analogous to the discussion for the case*-rr/2 as considered by Schwinger ( 1949) . When this number s

c

n,

5

is substituted into equation (7), the following equation in

obtained:, 13-

e 2 2

10

wH18/6 K5/3

(71 ) dr2r^,c sin* s

C C

If the relation 18 -, W H /2TTy I f is taken into account, the

spectral distribution with respect to the wave frequency is

obtained as follows:

2,15rre 2

1 fa

T_ r K (9)

f C - f H sin* c

0 f /f c

5/3 (r) dr

where f---w/2r, 1 H__ W H /2v and fc W C /27T. This result is the same

as bits earlier been obtained by many authors except for a

factor (sing,)_ 1. In fact, it is known that, as the pitch angle

of radiating electrons becomes smaller, the peak intensity

radiated into the direction of the instantaneous velocity vector

of these electrons becomes higher (Sakurai and Ogawa, 1968).

In the above equation, the characteristic frequency f C is

I

given by

f3 f 2 sin 2 3 eH o y 2 sin 2c'!F -4-- -H Y Tr woe

which is differept from the result by Epstein and Feldman

(10)

(1967) by a factor sin *. The reason is due mainly to the

vagueness in their definition of gyrofrequency. Whenever we

observe the Doppler-shifted gyrofrequency, we must always

define this frequency by

6

eHo sin* ./1-$

wA (l1)cmo 1-6cos*come

since only this component Ho A.(-HosJ , n*) is effective in the

gyration of an electron. If the form H0 „Lis used, equat ion ( l0)

I

reduces to

eH:ic - ^ Y2 sin +V

0(Ila)

In conclusion, equation (9) must always be used in considering

the spectral characteristics of synchrotron radiation from

relativistic electrons, in which case the frequency f m , where

the emission intensity I f attains the maximum, is given by

fm 'Z 0.29 f (12)

where f c also contains the factor sin 2 y. Therefore, this

frequency always becomes smaller, than * the case for *-rr/2.

This discrepancy becomes serious when the pitch angle * approaches

zero,

The factor sin* does not appear in the polarization

equation defined by Ginzburg and Syrovas: ii (1965) . It thus

follows that this factor only affects the spectral distribution

of the emission power.

Synchrotron Radiation in Plasmas

The characteristics of synchrotron radiation are somewhat

modified in the presence of ambient plasmas as has been considered

7

by Ginzburg and Syrovatskii (1865) and Ramaty and bingenf elter

(1867). In dealing with the synchrotron radiation from energetic

electrons in helical orbits in ambient plasmas, it must, how-

ever, be remarked that the power radiated par unit frequency

interval is altered as follows;

2./JTTe 2 f H 1 1 fC1-,(1 ^4(

)Y a cl cl

where the refractive index-A( 2 - 1-f p2 /f 2 , the plasma frequency2

f p 2(-- a ne ; ne , electron number density) and the characteristicTrmo

frequency in this case is given by

f^ 3eHo ^2

sin2 +VC1 +(1- 2 )^v2 i',^ (1.4)G 1 TTmoc A

In equation (13), the :influence of ambient plasmas is

neglected when

( I_4) Y2 << 1 . (15)

Then, from this equation, it follows that

2 4 neec

Ho sin

Since the essential, part of synchrotron radiation is confined

to the frequency range f .,, f c , the criterion

I

4n ec

3110sin ^

8

must be satisfied for the influence of ambient plasmas to be

neglected.

Consequently, since the factor (sin 2 *) -1 is larger than

unity unless * - n/2, the critical frequently f o becomem larger

by this factor in comparison with the case whore the pitch

angle * of radiating electrons is * - r/2. , It is, therefore,

impossible to neglect it in estimating the Lifluence of ambient

plasmas on the characteristics of synchrotron radiation from

energetic electrons in helical orbits. The criterion given

by equation (17) differs by a factor (sin 2 0 -1 from that given

by Ramaty and Lingenfelter (1967).

I

0

oftI "Flo

9

REFERENCES

Bekefi, G., Radiative Processes in Plasmas, John Wiley, New

York (1966) .

Epstein, R. I. and Feldman, P. A., Synchrotron radiation from

electrons in helical orbits, Ap. J., 150, L109-114(1967).

Francia, C. T. di, Introduction to the theory of synchrotron

radiation, Radioastronomxa Solare, 414-423, ed. by

Righini. , G., Societa Italia di Fisica (1959) .

Ginzburg, V. L. and Syrovatrkii, S. I., The Origin, of Cosmic

Rays, Macmillan, New York (1984).

Ginzburg, V. L. and Syrovatskii, S. I . , Cosmic magneto--

bremsetrahlunc, Ann. Rev. Astr,..i. Astrophys., 3, 297-350,

(1965).

Kundu, M. R., Solar Radio Astronomy, John Wiley, New York

(1965).

Landau, L. D. and Lifshitz, E. M. , Th , : Classical. Theory of Fields,

Pergamon, Oxford (1961).

Ramaty, R., InXluence of the ionized medium on synchrotron emission

of intermediate electrons, J. Geophys. Res., 73, 3573-_.,..,.»

3582 (1968).

Ramaty, , R. and Lingenfelter, , R. E. 0 The influence of the ionized

medium on synchrotron emission spectra in the solar corona,

J. Geophys. Res., 72, 879-$83 (1967).

10

Sakurai, K. and Ogawa, T., Some characteristics of gyro-

synchrotron radiation from energetic electrons and the

evolution of solar .radio type IV bursts, Uchusen- Ken.-kyu,

12 1 503-531 (1966).

Schott, G. A., Electromagnetic Radiation, Cambridge Univ.,ti

Cambridge (1912).

Schwinger, J., On the classical radiation of accelerated

electrons, Phys. Rev., 75, 1912-1.925 (1949).

Takakura, T., Synchrotron radiation from intermediate energy

electrons in helical orbits and solar radio bursts at

microwave frequencies, Pub. Astron. Soc. Japan, 12,

352-375 (1960).

Watson, G. N., A Treatise on the Theory of Bessel Functions,

Cambridge University, Cambridge (1945) .

I