RELATIVISTIC WITH

EJECTION A STRONG

FROM

COMPACT

MAGNETIC

STARS

FIELD

I. G. MITROFANOV and A. I. TSYGAN A. F. Ioffe Physico-Technical Institute, Academy of Sciences of the U.S.S.R., Leningrad, U.S.S.R. (Received 1 September, 1981) Abstract. The electron cyclotron resonance leads to a large enhancement of radiative force and may result in the ejection from magnetized compact stars (Papers I and II). On this ground, the acceleration of charged particles by radiation in a strong magnetic field is considered. Different regimes of ejection, the dependence on intensity, spectrum, angular distribution and polarization of accelerating radiation, and the influence of the opacity of ejecting plasma are analyzed. The energy of ejected plasma is shown to increase up to relativistic values, in many cases the gamma-factor appears to be >>1. A possible connection of relativistic ejection with the origin of gamma-ray bursts and other astrophysical consequences are discussed. 1. Introduction

M a n y observable properties of neutron stars and degenerate dwarfs are known to be determined b y the strong magnetic fields of these objects. The discovery of spectral features of X-ray pulsars (Tr/imper, et al., 1978; W h e a t o n et al., 1979; and P r a v d o et al., 1979) and g a m m a - r a y bursts (Mazets et al., 1980) allows us to estimate magnetic fields of corresponding neutron stars as 2-7 x 1012 G. Similar lines in the optical spectra of AM Herculis indicate that magnetic fields of degenerate dwarfs in these close binaries are about 3 x 107-3 x 108G (e.g., Mitrofanov, 1979; L a t h a m et al., 1981). The electron cyclotron f r e q u e n c y OB = eB/meC appears to be comparable with the typical f r e q u e n c y tOo of radiation of these objects. For neutron stars the cyclotron energy htOB--~ l l . 6 B l z k e V falls into the X-ray range; for degenerate dwarfs with B ~> 10SG wavelength of cyclotron resonance AB--~1 0 7 0 0 B ~ I A corresponds to optical band. In the case tO0-tOB the electron cyclotron resonance leads to the significant e n h a n c e m e n t of a radiative force acting on electrons in c o m p a r i s o n with the field-free case (Mitrofanov and Pavlov, 1981a, b, hereafter referred to as P a p e r s I and II). This e n h a n c e m e n t results in a decrease of the critical flux of radiation for which the radiative force acting on electrons b e c o m e s equal to the gravitational attraction of protons. As a result, the critical luminosity Lcr of magnetized c o m p a c t stars appears m u c h below the Eddington limit LEd~--1.25 X 1038(Mst/M| e.g., for X-ray pulsar Her X-1 L c ~ - 3 • 10 -3 LEd. Mitrofanov and P a v l o v (Papers I and II) concluded that the radiation of magnetized c o m p a c t stars is often generated in a supercritical regime, w h e n the emitting plasma has to be ejected along the magnetic lines. The acceleration of charged particles by radiation in a strong magnetic field is considered below. A general expression for radiative force is presented in Section 2. Astrophysics and Space Science 84 (1982) 35-51. 0004--640X/82/0841-0035502.55. Copyright 9 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

36

I. G. M I T R O F A N O V

A N D A. I. T S Y G A N

In Section 3, the different regimes of acceleration are considered. Maximum gamma-factors are evaluated for different physical conditions in Sections 4 to 6. In Section 7, the main conclusions are presented and the possible connection of relativistic ejection with the origin of gamma-ray bursts and other astrophysical consequences are discussed.

2. Radiative Force on a Relativistic Electron in a Strong Magnetic Field

L e t us assume that a charged particle moves with velocity V along the homogeneous magnetic field B. The axes OZ and OZ' of fixed and co-moving systems of coordinates are directed along B. In a co-moving system, a particle with a scattering cross-section O-sc(OY,n') is acted by the radiative force

f,= f d,o'f

dn' 9n' 9 Crsc(~o',n)W'(~o', n'),

(1)

where W'(~o', n') is the spectral density of flux of radiation with frequency ~o' propagating along a direction n' inside a unit solid angle. Transforming (1) to the fixed system of coordinates, one obtains the general expression for this force

f= ~f dwfdn[n-Vy2(1-/3x)l(1-/3x)osc(O)'(w,n),n'(n))W(w,n), (2) where / 3 = V / c , 7 = ( 1 - / 3 2 ) -m, ~o'=~o7(1-/3x), x = n z = c o s O and x ' = n ~ = cos O'. In the field-free case the cross-section ~rsc of the electron is equal to the Thomson value o-r, and (2) is simplified to the formula obtained by Tsygan (1980). In the magnetic field, O-sc depends on the frequency of scattering radiation and has a strong cyclotron resonance at coB. Due to this resonance, the radiative force is significantly enhanced, provided 0.01 ~o0<~ ~oB<~ 10 ~o0 (Papers I and II). In this case for evaluation of the force, the resonance part of the cross-section Crres should be taken into account only (see Pavlov et al., 1980; and Paper II) l

~rres(~O', X') = ~O'r (1 + (X')2 + P~(1 - (X') 2) + 2 P ' x ' ) X X

f0=

e x p ( - ~2)(O1 t)2 i 2 + (v3 z' dE ( w ' - tos + vD~)

(3)

where vb=oYlx'l(2kTelmec2) m is the Doppler width of resonance, v'r= 2e2(w')2/3mec 3 is the damping frequency (the radiative width), and integration over ~ = (mev~]2kTe) x/2 corresponds to the averaging over thermal velocities of the electrons. If V'r>>Vb, the thermal broadening is not essential and the resonance has a L o r e n t z profile with a width - v'r. In the opposite case it has a Voigt profile which is the convolution of the Lorentz and Doppler ones. The

RELATIVISTIC EJECTION FROM COMPACT STARS

37

degree of linear polarization is accepted to be negative P~ < 0 , if the plane of preferred oscillations of the electric-field of radiation is perpendicular to the magnetic field. The degree of circular polarization is assumed to be positive P ~> 0, if the electric-field rotates around the wave-vector in the direction of the Larmor motion of an electron. A Lorentz transformation of coordinates does not change the values of ]PI[ and IPc] (Landau and Lifshitz, 1967). It is clear that the invariability of IPc[ is followed by the invariability of Pc, but the phase (the position angle) of linear polarization is usually changed. However, it was assumed that in the co-moving system of the coordinates position angle X is determined by the direction of the magnetic field, X = ~-/2 and 0 corresponding to P~ < 0 and P~ > 0, respectively. In this case, the Lorentz transformation with V along B does not change X and, therefore, Pt = P}. The resonant condition in the co-moving system w ' = ~oB is followed by the similar condition ~,B

(4)

- 3,(1 -/3x)

in the fixed system of coordinates. Assuming that infalling radiation has a sufficiently broad and smooth spectrum, i.e., that the width of resonance Vres is much smaller than the width of the spectrum, one may use the a-function approximation for a narrow resonance in Expression (3) and obtain 7/"

OJ2

f=eO'T'b'r(OgB)~ (1-x2)

dn[n-

L

{ x - ~ ~2

~ ~/2(1 -- ~X)l(1 "+ ,1 -/3x / +

+,~v {x-~55

wB

In the general case, the direction of f does not coincide with the directions of the magnetic field B or the wave-vector n (Paper II). However, the radiative acceleration is caused only by the projection fll on the magnetic lines, and only this component will be considered below. If the angular distribution of infalling radiation is symmetrical around direction B, then

(1 - x z)

_~_

This force does not depend on the profile and broadening of the resonance. Expression (6) is valid also in the case when the main process of interaction between plasma and radiation is the inverse bremsstrahlung (Paper II). The formulae (3) (5) and (6) permit us to examine the acceleration of magnetized plasma by radiation with an arbitrary spectrum, angular distribution and polarization.

38

I.G. MITROFANOV AND A. I. TSYGAN

3. On the Acceleration of Plasma by Radiation in a Strong Magnetic Field

To consider the acceleration of electron-proton plasma along magnetic lines one may use the equation of motion d

~-PI] = III('Y) --/gravll

(7)

of the particle with the electron's cross-section and the proton's mass, because the exchange of momentum between electrons and protons by collisions and the electric field is sufficiently effective. Here Pll = mpc/37 and fgr~v is the gravity attraction of the proton. For the electron-position plasma mp and fH(~/) should be substituted by me and l(fte+)('y)wfle-)(q/))---= fll('Y, Pc = 0). This force does not depend on the circular polarization of infalling radiation, because of the opposite directions of Larmor precession of the electron and positron. So, the expression for this force follows from Expressions (5) and (6) with Pc = 0. Let us assume that the radiative flux is supercritical i.e., that fit(Y) ,> fgrav- It follows from (6) that fll(7) = f/+)+ ft -), where fl +) and f/-) are the accelerating and decelerating forces corresponding to infalling photons with x >/3 and x >ft -). One may say that a regime of free acceleration (FA) takes place. With an increasing/3 fl +) decreases and fl -~ increases, until they become equal. At this moment in the co-moving system of coordinates, the angular distribution of scattered radiation is quasi-isotropic: the total flux of the accelerating photons (from the lower hemisphere) is equal to the total flux of the decelerating photons (from the upper hemisphere) (Figure 1). This restriction of free acceleration was discussed first by Tsygan (1980). It does not depend on the power of radiative flux but is determined only by the angular distribution of photons and the velocity of electrons. Therefore, it may be called the 'kinematical restriction' (KR). The angular distribution of photons scattered by the resonance part of the cross-section depends on the spectrum of radiation (see condition (4)). Accelerating and decelerating photons have frequencies yo~B< o~ <~23,o~B and ~oB/y~O) fluxes of polarized radiation. The value of 3'~ increases provided that Pe > 0. Above the hot emitting spot on the surface of a compact star, the angular

RELATIVISTIC EJECTION FROM COMPACT STARS

f

Z

7

deoele\\

\

',

39

//ration

'\

/

/

/

] z'

]

/

(a)

Fig. 1. The difference of the angular distribution of incident photons in fixed (a) and co-moving (b) systems of coordinants.

distribution of radiation narrows with rising height. Thus, the value of yK also has to grow up with height, and the energy of the ejected plasma increases in accordance with the relation 3' = 3'K(z). In this case, acceleration occurs in the regxme of KR. On the other hand, with increasing 3/ and rising height, the radiative force ft*)(3') decreases. So, the regime of KR has to finish when the radiative force becomes so small that its work along some path Az does not provide the necessary growth of the gamma-factor A3, = A3"K(Z). In this case, the regime of energetical restriction (ER) comes in. The growth of 3' gradually slows down and it approaches the final value 3' = 7E, limited by ER. It is clear that due to the enhancement of the radiative force, 3'E significantly increases in the presence of a strong magnetic field. Moreover, leaving out an account of the magnetic field one may come to the erroneous conclusion that the ejection from a compact star does not take place at all. Therefore, a strong magnetic field may cause the ejection and significantly influences this process.

4. Acceleration above the Infinite Emitting Plane

L e t us assume that radiation is emitted by the infinite plane which is perpendicular to the magnetic lines. Consider the black-body radiation (h~o0= kTr) with a polarization which is independent of the f r e q u e n c y and angular distribution described by the law - ( l + q x ) , q = 0 - 2 . In this case fll(3', q ) =

40

I. G. M I T R O F A N O V A N D A. I. T S Y G A N

f11(7 = 1, q = O) x 1(% q), where

8

1(% q) = 3(q + 2)72

f/

dx

(x-/3) F, (1-/3x)3L + vVz- j

+ P~

(1-x

+

x-/3 + 2ec(-~-_-~-~) J(exp b -1)(exp(,y(lb/3x.)- l)-'(l+ qx), (8) 135 , hc

m e c2

~--2 X 103frTrslb3(e b - 1) -1,

(9)

fT = O-TO'Tr/C,4e is the Stephan-Boltzmann constant and b = ho)B/kTr. Expression (9) for fll(Y = 1, q = 0) has been obtained in Paper I. The flux and the angular distribution of radiation do not depend on the height z above the infinite plane. If yK < yE, the final energy of the accelerated plasma is limited by KR. At the end of the regime of FA, the gamma-factor approaches 7n = const. For 2y~oB~ kTr all scatterkng photons correspond to the Rayleigh-Jeans part of the Planck spectrum. In this limit, the integral 1(7, q) may be found analytically to be given by

1(% q ) = 3(q 82)72[((1+/3)(/3+ q ) + ~-(1-/32)1n(1-/3))x x (3 + / 3 2

_

Pl(3 -/32) _ 4Pc/3) +

+ (~(1 +/3)2(2 -/3)(/3 + q) - q(1 +/3)(1 -/32)) • x(3P/-3+2/3ec)+(~(/3+q)ln(1-/3)+q)x x (1 +/32 _ Pl(1 -/32) _ 2Pc/3) - q ( l +/3)2(2 -/3)(1 -/32) x (1 - P,) + l ( l +/3)3(/3 _~ q)(3 - 3/3 +/32)(1 - P,)]. ..1

(101

3

The values of Yn following from the equation 1(% q) -- 0 are presented in Table I for different q, Pl and Pc. The magnitude yn is larger for larger q, i.e., for a more narrow angular distribution. With increasing positive circular polarization yn also increases, approaching infinity for Pc coming to 1. Of course, very large values of yK have no physical sense because either ER stops the acceleration, or deceleration due to the neglected nonresonance part of the cross-section becoming significant, or the accepted low-frequency limit 2yKtoB~ kTr becoming wrong. However, for electron-proton plasma values, 7K --- 10 seems to be possible. As it was mentioned, for electron-positron plasma, the degree of circular polarization in the expressions for radiative force should be accepted as being

q = 2

q = 0

Parameter q

P~ = - ( 1 -

p~)~lz

1.09

1.69

co

Pc = (1

P~ = 0

1,05

Pc = - (1 - p~)1/2

p2)t/2

1.42

oo

0

1.15

1.67

34.19

1.09

1.40

24.46

+0.3

1.06

1.72

292.92

1.03

1.45

212.05

-0.3

1.24

1.65

3.71

1.14

1.39

2.73

+0.6

D e g r e e of l i n e a r p o l a r i z a t i o n PI

Pc = 0

P , = (1 - p ~ ) m

c i r c u l a r polarization P~

D e g r e e of

-

TABLE I

1.11

1.75

8.87

1.06

1.48

6.50

-0.6

1.33

1.64

2.44

1.19

1.38

1.88

+0.8

1.23

1.77

4.02

1.14

1.50

3.07

-0.8

1.45

1.63

1,90

1.27

1.37

1.54

+0.95

1,45

1.80

2.46

1.29

1.52

1.98

-0.95

1.63

1.37

+ 1

T h e d e p e n d e n c e of 7I< o n p o l a r i z a t i o n a n d a n g u l a r d i s t r i b u t i o n of a c c e l e r a t i n g r a d i a t i o n (infinite s o u r c e w i t h R a y l e i g h - J e a n s s p e c t r u m )

_

_

_

_

-1

1.80

1.53

4~

> (/3

>

9

9

9

,<

42

I.G. MITROFANOV AND A. I, TSYGAN

equal to zero. So, for this case, values of 2/K are presented in the lines of Table I which correspond to Pc = 0. If the spectrum is less sloping than the Rayleigh-Jeans one corresponding to co2, the balance between accelerating (x >/3, to > 2/toB) and decelerating (x < /3, to < 2/toB) photons occurs at smaller 2/K (see the line ~ = 0 in Table III). For unpolarized radiation with a fiat spectrum, the magnitude 2/K equals to --1.2 (for q = 0) which coincides with the field-free case. On the other hand, for a more steep spectrum - such as one corresponding to to" for n > 2 - the balance occurs for larger 2/rc's. These values may be found from equation I(2/, q, n ) = 0 , where integral 1(% q, n) is similar to (8) and depends on the spectral index n. For q = 0, this equation is rather simple: i.e., 42/n-3 [ l(2/,q=O, Pz=Pc=O,n)=-~ -

1+/32 3 n @ l2 -n-Z~-_ 2 (1 - (1 -/3)"-2) + x

• (1 +/3)(1 - (1 -/3)n-l) - 3(1 +/3)2(1 - (1 -/3)')

_

+

+ l + n ( 1 +/3)3(1 - (1 -/3)"+')].

(11)

Its solutions 2/K(n) are presented in Table II. With increasing n t> 2, 2/K(n) grows up and formally becomes infinite for n = 3. For n t> 3 KR is absent. Of course, the growth of 2/K for n t>3 would be limited either by ER or due to other physical restrictions (nonresonance scattering or a high-frequency drop of the spectrum). Thus, for the spectrum -to"(exp(-to/to0), n ~> 3, the predominant increase of accelerating flux with increasing 2/stops for 2 / - 2/~,x- toO/2toB. The spectrum of radiation generated by the common action of inverse Bremsstrahlung and Compton scattering is thought to have the broad spectral region with Wien's law -to3 exp(-to/to0) (Illarionov and Sunyaev, 1972). The ratio of too to the low-frequency boundary of this region too/to, ~ y l / 6 = (kTe/mecZ)l/6.r~3may be sufficiently large (~'r is the optical thickness of emitting plasma for Thomson scattering). If coB-to, plasma above this source may be accelerated up to ")/max ~ Y 116.

For the switched-on source with a large optical thickness rr >> 1, parameter y has the sense of non-dimensional time, and describes the evolution of spectrum T A B L E II The dependence of yK on the spectral index of accelerating radiation (infinite plane source, spectrum - w " , zero polarization, q = 0) Spectral indexn

2.0

2.2

2.4

2.6

2.8

~K

1.4

1.5

1.6

1.8

2.3

3.0

RELATIVISTIC EJECTION FROM COMPACT STARS

43

(Illarionov and Sunyaev, 1974). For y ~< 1, radiation has the flat spectrum; if 10 ~< y ~ 10 4 Wien's spectrum is formed for frequencies to > to, = kTdhy ~/6,and if y ~ 10 4, the spectrum approaches the black-body one. Above this nonstationary source during the intermediate interval of time corresponding to 10 ~ y ~ 104, plasma may be accelerated up to maximum energies 7m~-y~/6~<5 (provided toR ~ ~o,). Before and after these time intervals, 3' is limited by KR and should be much smaller. 5. Acceleration above the Local Emitting Region

The existence of a boundary of an emitting region with radius R becomes essential when the height z of an accelerated particle above this region becomes comparable with R. For the sake of simplicity consider the acceleration above the center of circular spot emitting black-body radiation. As before, the magnetic field is accepted to be perpendicular to the spot's surface. In this case, the radiative force may be evaluated using (8) and (9) provided the lower limit of integral (9) is substituted by x0(~) = ~(1+ ~2)-~t2, where ~ = z/R is the nondimensional height. For any given 3' = 3'K(~) the increase of x0(~) due to increase A~ of ~ leads to a decrease of decelerating flux (corresponding to/3 > x >~x0(O) and does not affect the accelerating one. As a result, the acceleration of the outflowing plasma up to larger 3' has to occur in accordance with law 3' -- 3"K(~). One may obtain a rough estimation of 3'K(~). For a given height ~, the regime of FA occurs until/3 < x0(~) (f/-) = 0), and is replaced by a regime of KR when /3 >~x0(~). For ~ ~> 1 equality/3 = x0(~) corresponds to 3' = ~ which appears to be asymptotic for dependence 3'K(~)Plasma moving away from the spot is influenced by decreasing radiative flux. The growth of 3' in accordance with law 3'~(~) has to cease when the work of the radiative force does not provide the necessary increase of energy: i.e., when

A3'K(O ~>~(Tm~cl2)R 1(+)(% q, Xo)a~,

(12)

where I(+)is the part of integral (8) corresponding to accelerating photons. This inequality may be presented in the form A~I(+)(T, q, x0) ~ 3 x 104b3(e b -- 1)-1 TrsRs. 3

(13)

Therefore, an acceleration in the regime of KR is limited by ER, provided that equation (13) is satisfied. The values of 7K(~) were calculated for q = 2, b = 0.01 and 1.0 and different (Table llI and Figure 2). As is expected, 7K(O grows up with increasing ~ and, for ~ ~> 1, approaches the mentioned asymptote. For a fixed ~ larger 3'K cor-

44

I. G. M I T R O F A N O V A N D A. I. T S Y G A N

TABLE III The dependence of `/K and F = 2,/2(1 - 13~x0) on ~ (thermal-like spectrum, zero polarization, q = 2) yK

0.0 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6.0 8.0 10.0

F

b =0.01

b = 0.1

b = 1.0

b = 0.01

b = 0.1

b = 1.0

1.7 1.8 2.00 2.2 2.5 2.8 4,6 8,5 12,4 16.4 20.4

1.6 1.8 1.9 2.2 2.4 2.7 4.3 7.6 10.7 13.4 16.2

1.4 1.5 1.6 1.8 1.9 2.1 3.1 5.1 6.1 8.1 10.1

5.6 5.5 5.4 5.4 5.4 5.3 5.3 5.3 5.2 5.2 5.1

5.3 5.2 5.1 5.1 5.0 5.0 4.8 4.4 4.1 3.8 3.6

3.9 3.8 3.7 3.5 3.4 3.3 2.9 2.5 2.0 2.0 2.0

2O

10

/

91

[ .

I

I

.5

I

[

I

I

10

1

Fig. 2. The increase of ,/n with height above a local emitting region. The dashed lines indicate the limits corresponding to ER for T = l0 s and R = 105 cm. The dash-dotted line indicates the asymptotic ~,=~.

responds

to a smaller

corresponds

b. H o w e v e r ,

t o b - 0.1, b e c a u s e

photon

decreases.

crease

of

Parameter

a frequency

i n c r e a s i n g ~.

of

the maximum

f i n a l y, d e t e r m i n e d

by ER,

f o r b ~ 1 a n d b >~ 1 t h e f l u x o f t h e a c c e l e r a t i n g F = 2y2(1-13xx0),

scattered

photons

describing (see

Section

the

maximum

7) f a l l s

in-

off with

R E L A T I V I S T I C E J E C T I O N F R O M C O M P A C T STARS

45

6. Acceleration of Plasma with Large Cyclotron Opacity D u e to the finite width of c y c l o t r o n r e s o n a n c e , a p h o t o n with ~o, x related b y (4), m a y be s c a t t e r e d by electrons with g a m m a - f a c t o r s which m a y differ in A~, = ,/~ Vres 1 -- r C0B ~2--~-1 "

(14)

T a k i n g into a c c o u n t the fact that y of the e j e c t e d p l a s m a increases with the height ~, one m a y relate A , / w i t h the optical thickness of c o r r e s p o n d i n g layer A~ for resonance photons yr ~oB 1 - ~x 1 - 13x TB = O-resNA~ = l"r 2VV(~) ~ Vr(We) ~----~ -- TB(1) ~ - - - ~ ] ,

(15)

w h e r e N is the p l a s m a c o n c e n t r a t i o n , Vy(r is the dimensionless gradient d3~(r and cr = o'TNR~ is the total optical thickness of p l a s m a for T h o m s o n scattering. T h e ratio WB/Vr(WB) = 104B i I is so large that the c y c l o t r o n o p a c i t y of a g e o m e t r i c a l l y - t h i n l a y e r A~ m a y be noticeable, ~'B ~> 1, in spite of the small total T h o m s o n thickness ~'r ~ 1 of the ejected plasma. This case will be c o n s i d e r e d below. U s i n g r e s o n a n c e condition (4), one m a y e x a m i n e the d e p e n d e n c e of 3J of electrons on f r e q u e n c i e s w of scattering radiation with a fixed direction of p r o p a g a t i o n x. P h o t o n s with ~o > wm~ =- w B ( 1 - x 2 ) -~/2 could not be s c a t t e r e d (Figure 3). T h e f r e q u e n c y w~• c o r r e s p o n d s to the m a x i m u m of d e p e n d e n c e w(y)

decelerat ~ i n g

Fig. 3. The relation between y and ~o which is necessary for resonance scattering of a photon with fixed x (a). The interaction of a layer with given 3' and photons with different ~o and x (b).

46

I.G.

MITROFANOV

A N D A. I. T S Y G A N

which occurs at Ymax= ( 1 - X2)-m, i.e., at /3max= X. Photons with frequencies toB ~< to < tom,x m a y be scattered b y electrons with 3, f r o m two different intervals [ % -+ A3,] and [3'- -- AT], where y_+= toB(1 - x2)-1(1 _+x(1 -

x2)) ~/2

Photons with to < toB m a y be scattered only b y electrons with 7 from [T+ + AT]. The layer of plasma with a given V is accelerated b y photons with x >/3 and is decelerated b y photons with x x~ =/3(1+(1-/32)1/2) -1 follows f r o m to >t toB, see Figure 3). If the lower layers have noticeable opacity ~-s ~> 1, the flux of photons with x~ <~ x
toB -[- Pres

~(1 -/3x) <~ to ~< y(1 -/3x)"

(17)

If rB > 1 the flux of photons with to f r o m the resonance interval (17), which m a y reach the layer of electrons with % depends on to as

(

r~C(to, x) = W(to, x) exp -~'B

V--~e~

(18)

/

where Ato = (tos + Vres)Y-l(1 - / 3 x ) - 1 - to if toby/-1(1 - / 3 x ) -1 increases with an increasing % or hto = to - (toB -- Vres)3~-l(1 --/3X) -1 if toB~/-I(1 --/3X) -1 decreases with an increasing % So, only photons with frequencies from intervals 3to = Vresl"Bly-l(1-/3X) -1 near the 'red' or 'blue' wings of resonance reach this layer without attenuation (Figure 3). Substituting fires exp( - htol3to) b y 3-function, one obtains from (2) and (3) the expression for the radiative force for the source of black-body radiation fll(~/, q, x0, ~'B) = fll(~ = 1, q = 0)1(% q, x0, l-s), where

8 ff -/3)Ix-/31 I('y, q, xo,'rs)= 3(q + 2),rB(1)y2 ___odx (x (1 _/3x)4 ( x - / 3 ~2

(1 - x 2)

x x -/3

(19) and where E(x,/3) describes the attenuation of incident flux of photons with x~ ~< x
RELATIVISTIC EJECTION FROM COMPACT STARS

47

resonance condition (4). A layer of plasma scatters photons from any one 'state' (~o, x) to all others. It is a screen if the total flux scattered to directions x >/3 is smaller than the total flux scattered from these directions. These fluxes are equal if KR takes place. Indeed, in the dipole approximation, the probabilities of scattering to the upper and lower hemisphere are equal in the co-moving system of coordinates. On the other hand, for electron with 1/= 3,K total fluxes of photons infalling from both these semispheres are also equal. So, this electron does not reduce the mean flux of photons with x >/3. On the other hand, layers with ~/< y~: (i.e., in the regime of FA) should be a good screen for accelerating photons. Due to this effect, ejected plasma is freely accelerated until accelerating photons with x >/3 dominate decelerating ones with x0 ~> 1 is well approximated by ~ ( ~ ) = 2~2+ 1 resulting from equality x0(~) = x~ (decelerating photons are absent for x0(~) > x~). Of course, the final energy of accelerated plasma is also limited by the ER (12), but in this case the radiative force is slightly decreased l/I-B(1). Consider a freely-accelerated layer of plasma which at some height ff~ attains the gamma-factor 3'1 = ~:(~1) limited by KR. Due to the growth of ?K(~) with if, its further acceleration is possible, but all layers above ff~ are no longer the screen for this layer. Only decelerating photons with max(x0, x~)~
q = 2) f~

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

F

b =0.01

b=0.1

b=l.0

b =0.01

b=0.1

b=0.1

1.9 2.1 2.5 2.9 3.5 4.3 5.3 6.5 7.8 9.3 10.9

1.8 2.0 2.3 2.8 3.3 4.0 4.9 5.9 7.1 8.4 9.8

1.5 1.6 1.8 1.9 2.5 3.1 3.9 4.9 6.1 7.5 9.0

7.2 7.4 7.9 8.7 10.0 11.5 13.7 16.4 19.2 22.6 26.0

6.8 6.6 7.0 8.2 8.8 10.1 11.9 13.7 16.1 18.7 21.2

4.5 4.4 4.4 4.6 5.3 6.3 7.8 9.8 12.2 15.0 18.0

48

I.G.

M I T R O F A N O V A N D A. I. T S Y G A N /

/

/

/ /

/ /

/

-

:

I

/[

/,

E4 9 o ~176

ii

0

,

,

t

I

i

t

,__.J

2

Fig. 4. The increase of ~/ limited by KR due to screen-effect. The dashed lines indicate ,)K(~) and ~K(~,s --- 1). Curve (a) corresponds to acceleration from the surface of a spot, curve (b) corresponds to acceleration from the height ~ =0.6. Above the dotted line, which corresponds to y = 2~2+ 1, x0(ff)< x~ and below it x0(ff)> x~.

and b e c o m e equal to ~ ( f f , E - 1 ) when screening totally disappears. If ff > ft, (from equation ~)~(~., E - 1 ) = 2ff2,+ 1) then b e t w e e n if2 (from equation x0(~2)= x~) and ~3 (from equation x0(~3) =/31) the g a m m a - f a c t o r remains constant because forces ft+~(x >/3) and ft-~(/31 < x if3, further acceleration is possible in accordance with the law 3' = ~)K(~,E- 1). Figure 4 illustrates the growth of 3' for initial height ~ = 0 and 0.6 for chosen p a r a m e t e r s b = 0.1 and q = 2.

7. Conclusions and Discussion Significant e n h a n c e m e n t of a radiative force in a strong magnetic field may lead to the ejection f r o m c o m p a c t stars with luminosities much smaller than the Eddington limit. Outflow of plasma begins in the regime of free acceleration (FA), which finishes when kinematical restriction (KR) b e c o m e s essential. The path of free acceleration m a y be evaluated on the grounds of (7) and (8) or (19) for any chosen condition. H o w e v e r , a simple rough estimation m a y be proposed: namely, this path is m u c h smaller than the radius of the spot ~ ~ 1 if

'YK <~ 50TrsR ~13(mp[m )l/3b (e b - 1)-113 ,

(20)

RELATIVISTIC EJECTION FROM COMPACT STARS

49

where m = m e or me for electron-proton or electron-positron plasma, respectively. In the first case, it follows from (20) 3'K ~<2 for b = 0.01; 3~K~< 10 for b -- 0.1 and 3J~:~<40 for b = 1.0. For electron-positron plasma, the estimates of 7K are - 1 2 times larger. In the presence of a magnetic field, the angular distribution of scattered photons, which determines the energy limited by KR, depends on the spectrum and polarization of infalling radiation. For unpolarized radiation with a flat spectrum YK = 7K(B = 0 ) , for a spectrum with dW(o~)/do~ < 0 3~K< ~/K(B =0), and for d W ( w ) [ d w > 0 7 n > 7K(B =0). If the spectral index n 11>3, ~/n is formally equal to infinity and K R is absent. In this case acceleration is restricted by the ER, or by a downfall of spectral distribution, or by the decelerating influence of the nonresonance part of the cross-section. Polarization of radiation may also lead to an increase of 3'K with the KR formally disappearing for Pc = 1. The values of 3JK for different cases are presented in Tables I, II and III. In the transition zone between the regions of FA and KR, the maximum gamma-factors may be significantly increased by screen-effect provided ejected plasma has noticeable cyclotron opacity. (Table IV and Figure 4). The final 2/of outflowing plasma is determined by the ER. Its values may be estimated using (13) and, in some cases, appear to be as large as 75 -~ 10 + 20. The present analysis was based on the simple arbitrarily-accepted properties of accelerating radiation. H o w e v e r , they are known to be determined by the properties of emitting plasma. The investigation of this influence is a complicated physical problem and only the first steps have been taken towards its solution (see, e.g., Nagel, 1981; Kaminker et al., 1981). Were it solved, the ejection of emitting plasma would be considered quantitatively on the basis of presented results. On the other hand, an examination of ejection using the observational data about spectrum, polarization and angular distribution of radiation seems to be also inconsistent because it is this process that is likely to determine the properties of outgoing radiation. H o w e v e r , in spite of this lack of consistency, the results obtained permit us to clear up the general scheme of ejection. They allow us to understand the influence of any peculiarity of accelerating radiation on this process. For example, consider the radiation with a narrow spectral line in the neighbourhood of ~oB. If the screen-effect is essential, all decelerating photons have frequencies ~oB/7(1 -/3x0) ~< w < oJB. For ff >> 1 denominator 3'(1 -/3x0) = 1 + (8if) -4 is close to 1, and this interval approximately coincides with the region of the spectral line. Accordingly, the cyclotron absorption should lead to an increase of the gammafactor limited by KR, and, on the other hand, the emission line should result in a decrease of this factor. The initial spectrum of outgoing radiation is transformed by the scattering of photons on ejected electrons. The maximum increase of f r e q u e n c y in F = 23,2(1-/3x0) times takes place for photons which are scattered from directions with a maximum angle O0 = arc cos x0 to a direction along the velocity O = 0. For

50

i.G. MITROFANOVAND A. I. TSYGAN

an optically-thin plasma this parameter is maximum near the surface of emitting spot F = 272 and decreases with increasing ~ (Table III). If the screen-effect is important, F(~:) increases with increasing ff (Table IV), for ~ -> 1 F = 4ff2. For = 2 estimated F are as large as 20-30. The transformation of a thermal spectrum by non-resonant scattering on electrons of relativistic ejected plasma with constant F leads to the origin of spectrum which, in the broad range of energies, may be well approximated by the law d N ( E ) l d E ~ E -~ e x p ( - E / E o ) , here d N ( E ) / d E is the spectral density of the number of photons and E0 --- 1.3FkT, It should be emphasized that this law appears to agree with observed spectra of gamma-bursts (Mazets et al., 1980, 1981). The typical value E 0 - 3 0 0 keV for kTr = 12 keV corresponds to F = 20. This fact confirms our supposition (Papers I, II) that hard radiation of gamma-bursts arises because of the Compton scattering of thermal photons on electrons of relativistically ejected plasma. It is known that, in a strong magnetic field, a photon with h~o > 2mec2(1 - x2)-1/2 may create an electron-positron pair (see, e.g., Klepikov, 1954). If among all photons which increase the energy due to scattering on relativistic electrons or positrons, there is one which gives birth to a pair, then above the emitting region the avalanche of a creation of electron-positron plasma should take place. This process is stopped when the optical thickness of occurring electron-positron plasma for Compton scattering approaches unity. The creation of an electronpositron avalanche is likely to accompany the generation of gamma-bursts by magnetized neutron stars, as was first suggested by Tsygan (1980). The discovery of well-pronounced lines in spectra of gamma-bursts, which may be interpreted as electron-positron annihiliation lines (Mazets et al., 1980, 1981), seems to confirm this assumption. The creation of an electron-positron avalanche in the course of relativistic ejection in a strong magnetic field will be considered in detail elsewhere.

Acknowledgement The authors thank A. D. Kaminker and G. G. Pavlov for useful discussions.

References Illarionov, A. F. and Sunyaev, R. A.: 1972, Astron. Zh. 49, 58. Illarionov, A. F. and Sunyaev, R. A.: 1974, Astron. Zh. 51, 698. Karninker, A. D., Pavlov, G. G., Silant'ev, N. A. and Shibanov, Yu. A.: 1981, A. F. Ioffe Institute preprint No. 716. Klepikov, N. P.: 1954, Zh. Eksp. Teor. Fiz. (JETP) 26, 19. Landau, L. D. and Lifshitz, E. M.: 1967, The Theory of Field, Izd. Nauka, Moscow. Latham, D. W., Liebert, J. and Steiner, J. E.: 1981, Astrophys. J. 246, 919. Mazets, E. P., Golenetskii, S. V., Aptekar, R. L., Guryan, Yu. A. and Ilyinskii, V. N.: 1980, Pisma v Astron. Zh. (Soy. Astron. Letters) 6, 706.

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Mazets, E. P., Golenetskii, S. V,, Ilyinskii, V. N., Guryan, Yu. A., Aptekar, R. L., Panov, V. N., Sokolov, I. A., Sokolova, Z. Ya., and Kharitonova, T. V.: 1981, A. F. Ioffe Institute preprint No. 719. Mitrofanov, I. G.: 1979, in M. J. Plavec et al. (eds.), 'Close Binary Stars', IAU Syrup. 88, 431. Mitrofanov, I. G. and Pavlov, G. G.: 1981a, Astron. Zh. (Soy. Astron. L) 58, 309 (Paper I). Mitrofanov, I. G. and Pavlov, G. G.: 198lb, submitted to Monthly Notices Roy. Astron. Soc. (Paper II). Nagel, W.: 1981, submitted to Astrophys. L Pavlov, G. G., Mitrofanov, I. G., and Shibanov, Yu. A.: 1980, Astrophys. Space Sci. 73, 63. Pravdo, S. H., White, N. E., Boldt, E. A., Holt, S. S., Serlemitsos, P. J., Swank, J. H., Szymkowiak, A. E., Tuohy, I., and Garmire, G.: 1979, Astrophys. J. 231, 912. Trfimper, J., Pietsch, W., Reppin, G., Voges, W., Staubert, R., and Kendziorra, E.: 1978, Astrophys. J. 219, L105. Tsygan, A. I.: 1980, A. F. Ioffe Institute preprint No. 659 (Astrophys. Space Sci. 77, 187). Wheaton, W. A., Dory, J. P., Primini, F. A., Cook, B. A., Dobzon, C. A., Goldman, A., Hecht, M., Hoffman, J. A., Howe, S. K., Sheepmaker, A., Tsiang, E. Y., Lewin, W. H. G., Matterson, J. L., Gruber, D. E., Baity, W. A., Rothschild, R., Knight, F. K., Nolan, P., and Peterson, L. E.: 1979, Nature 282, 240.