THE RADIATION ACCRETED

OF A HOT MAGNETIZED

BY DEGENERATE MAGNETIC

STARS WITH

PLASMA A STRONG

FIELD

G. G. PAVLOV, I. G. MITROFANOV and Yu. A. S H I B A N O V A. F, Ioffe Institute of Physics and Technology, Academy of Science of the U.S.S.R., Leningrad, U.S.S.R.

(Received 31 March, 1980) Abstract, The absorption coefficients for extraordinary and ordinary electromagnetic modes are

found for a tenuous hot magnetized plasma, taking into account the collisions between plasma particles and the scattering of photons. An approach is suggested which generalizes 'collisionless' and 'cold-plasma' approximations, The simple formulae obtained are valid both near, and at a distance from, the cyclotron harmonics. In particular, the ordinary mode is shown to have resonance at the cyclotron frequency. The number of noticeable reasonances of absorption coefficient at cyclotron harmonics is estimated for both modes. Using the coefficients obtained, the intensity, Stokes parameters and polarization of radiation of a homogeneous plasma slab are calculated for conditions which may be realized in the heated regions of accreted plasma in an AM Herculis-type system. The large difference between the absorption coefficient of extra-ordinary and ordinary modes near the cyclotron harmonics may result in the emission of the broad polarized continuum together with the narrow cyclotron lines. The polarization of these lines has a complicated spectral dependence. The results obtained are shown to be useful for explaining the main properties of AM Hercufistype objects. 1. I n t r o d u c t i o n

It is widely a c c e p t e d that m a n y galactic X - r a y sources are accreting n e u t r o n stars with strong m a g n e t i c fields B = 10H-1013 gauss. G n e d i n and S u n a y e v (1974) have suggested the o b s e r v a t i o n of c y c l o t r o n lines as a m e t h o d f o r m e a s u r i n g these fields. F u r t h e r m o r e , r e c e n t l y d i s c o v e r e d A M H e r c u l i s - t y p e o b j e c t s have b e e n f o u n d to emit polarized optical light and X - r a y s due to a c c r e t i o n b y d e g e n e r a t e d w a r f s with fields B = 107-109gauss (Tapia, 1977). For both classes of o b j e c t s the e x i s t e n c e o f such fields has actually b e e n confirmed b y the discoveries of lines which m a y be interpreted as c y c l o t r o n r e s o n a n c e s in X - r a y s p e c t r a o f Her X-I and 4 U 0115 + 63 (Triimper et al., 1978; W e a t o n , 1978) and in optical s p e c t r a o f A M H e r c u l i s - t y p e stars V V Pup, A M H e r and M V L y r ( M i t r o f a n o v , 1979, 1980; V o y k h a n s k a y a and M i t r o f a n o v , 1980; W i c k r a m a s i n g h e and V i s v a n a t h a n , 1979). F o r an u n e q u i v o c a l interpretation o f the o b s e r v a t i o n a l data it is n e c e s s a r y to u n d e r t a k e a theoretical investigation o f the e m i s s i o n f r o m m a g n e t i z e d p l a s m a u n d e r a p p r o p r i a t e conditions. T h e usually a d o p t e d estimations of e l e c t r o n c o n c e n t r a t i o n n and t e m p e r a t u r e T of a radiating p l a s m a (n = 1015-1018cm -3 and T = 105-108 K f o r s y s t e m s with d e g e n e r a t e d w a r f s ; n = 102~ c m -3 and T = 107-109K f o r s y s t e m s with n e u t r o n stars) s h o w that the non-relativistic (/3 Astrophysics and Space Science 73 (1980) 63-82. 0004--640X/80/0731-0063503.00 Copyright 9 1980 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

64

G . G . P A V L O V ET AL.

(2kT]mc2)l/2~ 1) approach and the tenuous-plasma approximation ([eik- 6ikl< 1, where eik is a dielectric tensor) are valid. In a magnetized tenuous plasma it is convenient to describe the radiative transfer in terms of the two normal modes (extraordinary mode EM and ordinary mode OM) with different polarizations and absorption coefficients/xj (where j = 1 for EM and j = 2 for OM). The qualitative picture of EM and OM absorption is given in Section 2 for a moderately hot plasma, the significance of electron-ion collisions and photonelectron scattering being emphasized. In Section 3 a method is suggested for numerical calculation of the absorption coefficients /xj using the classical approach to the electron-radiation interactions (i.e., hws, hoJ ~ kT, where o98 = eB[mc is the cyclotron frequency and w is the frequency of radiation) and neglecting the polarization of an electron-positron vacuum. For AM Herculistype objects these conditions are always fulfilled; for neutron stars, sufficiently small B and large n and T are necessary for their validity. The quantum effects, including the vacuum polarization, have been considered in other papers (e.g., Pavlov and Shivanov, 1979; Pavlov et al., 1979; Pavlov et al., 1980). The method suggested is valid in a wide range of plasma parameters for frequencies near and far from the cyclotron resonances o)= s00s (s = 1,2 . . . . ). The approximate analytical formulae for P-i are also obtained in this section. The resonant absorption of OM near the cyclotron frequency wB is considered in detail because, as far as we know, this question has not yet been accurately investigated for a tenuous plasma. The maximum numbers of noticeable cyclotron resonances of p.,(w) are also estimated for different parameters. In Section 4 - u s i n g the calculated absorption coefficients-the spectrum, polarization and angular distribution of radiation from a homogeneous plasma slab are investigated. The conditions for the emission of narrow cyclotron lines together with a strongly polarized continuum are found. Finally, in Section 5 the results obtained are briefly discussed and a comparison is made with the previous results of other authors. Some observational data concerning AM Herculis-type objects are analysed.

2, On the Absorption of Radiation in a Hot Magnetized Plasma

In a completely ionized plasma without a magnetic field the absorption of radiation is caused by the free-free transitions in electron-ion collisions ('true' absorption) and by the scattering on electrons. The absorption coefficient may be represented in the form ~llB:0 = , oC~.O ~ c2 + n~T =

O)2(1Jc + /Jr) Cb) 2

'

(1)

where 4

niZ2e 4 -

(2)

THE RADIATION OF A HOT MAGNETIZED PLASMA

65

is the effective frequency of electron-ion collisions, L ~ In (4kT/1.78hto) is the Coulomb logarithm, ur(to)= 2to2e2/3mc 3 is the damping frequency (the radiative width), o-r = (87r[3)(e2/mc2) 2 is the Thomson cross-section and top = (4rrne2/m) 1/2 is the plasma frequency. In a magnetic field the absorption of radiation depends on its polarization and, consequently, is different for EM and OM. The orientation of the EM polarization ellipse almost coincides with the projection of an electron Larmor orbit onto the plane perpendicular to the wave vector, electrons and EM field rotating in the same directions. As a result, the cyclotron resonance of EM absorption (the fundamental resonance) arises at the frequency toB of Larmor rotation. On the other hand, the polarization ellipse of OM is nearly orthogonal to that of EM, and the OM field rotates contrariwise. Therefore, the influence of the magnetic field on OM absorption is much smaller. The width of the fundamental resonance of absorption is determined both by changes of transversal energy of an electron after absorption of a photon with frequency toB and by longitudinal thermal motion. The transversal energy of an excited electron is changed mainly by collisions with plasma particles or by the spontaneous emission of a photon. The last process dominates if the characteristic time of cyclotron losses tB appears to be much smaller that the mean time between collisions. In this case collisions do not change the electron state between the absorption and emission of a photon, and spontaneous emission may be considered as a final part of the scattering process. The value of tB is equal to the inverse damping frequency at to = toB i.e., tB =vB ~ vr(toB). The mean time between electron collisions with plasma particles may be roughly estimated as v~-~, at least for a hydrogen plasma with negligible turbulence. Thus, if uc >> uB, the transversal energy is changed mainly by collisions, and the absorption of a photon may be considered as a 'true' absorption. On the other hand, if uc ~ uB, 'pure' scattering takes place. If a plasma is sufficiently cold (i.e., if v~ + vB >>vo, where vo = to/3[cos al is the Doppler width, 13 = (2kT/mc2) 1/2 and a is the angle between the wave vector, q and the magnetic field), the cyclotron resonance is broadened by the changes of electron transversal energy and has approximately a Lorentz profile of width -(v~ + us) and height -toEp/c(v~ + vB). In this case the dependence of absorption and polarization of both modes on temperature is caused only by the temperature dependence on the collision frequency. Hence, for a wide spectral range which also includes the vicinity of the cyclotron resonance, the so-called 'cold plasma approximation' (see, e.g., Ginzburg, 1964) appears to be valid. Therefore, in a 'cold' magnetized plasma for to >~ toA the appearance of the fundamental cyclotron resonance in EM absorption is the main distinction from the field-free case. In the opposite case of a moderately hot plasma (v~ + v~ <<-VD) the thermal motion of electrons becomes essential and the situation appears to be more complicated.

66

G . G . PAVLOV ET AL.

(i) The longitudinal thermal motion results in Doppler broadening of the cyclotron resonance in EM absorption. It adopts a Voigt type profile which is the convolution of the Lorentz and Doppler profiles. In the extreme case, vc + vB ~ vD, the Voigt profile consists of a Doppler 'core' with width -vD and height -wZflcvD and of Lorentz 'wings' which are almost unaffected by thermal motion. (ii) Contrary to the cold plasma case, the dipole approximation becomes insufficient for the description of radiative processes in a hot plasma. Taking into account higher multipolar terms, we can find the resonances at high cyclotron harmonics w = so~B (s = 2, 3 . . . . ) additionally to the fundamental resonance (s = 1). The harmonics result from the Doppler shift of frequency of the incident wave provided the rotational velocity Vl_l_B of electron has a component V• sin a along the wave vector q. The intensities of high resonances are proportional to ( V • 2s for both modes. Their spectral widths vo ~ stud3 Icos al result from the dispersion of longitudinal velocities of electrons provided up ~> vc + yr. With increasing temperature, the resonances overlap, being greater for harmonics with the larger numbers. The non-overlapped resonances are noticeable until they prevail over the high-frequency Lorentzian wing of the fundamental r e s o n a n c e - t h a t is, until the condition f12s-3> (re + vr)lcon for a -- 1 is fulfilled. (iii) Similarly to the resonances at high harmonics, we should assume that thermal motion also results in the resonance absorption of OM around coB.As far as we know, this question has not been considered in detail for a tenuous plasma. We show that this resonance does exist at a height of -to2vlJlwc for ~--l.

For a sufficiently hot plasma the absorption coefficients within the Doppler cores of cyclotron resonances depend only slightly on quantities u~ and yr. Moreover, they may be approximately derived when collisions and photon scattering are formally neglected. In this case, the so-called 'collisionless' approximation is valid for the analysis of cyclotron absorption (Zheleznyakov, 1970). There seems to exist an analogy between resonance absorption by a magnetized electron and by an atom. For vo >>vr + vr in both cases the profiles of resonances have almost Doppler shapes that are independent of v~ and v~. Nevertheless, resonant cyclotron absorption results from 'true' absorption (which dominates, if vc >> vr) and 'pure' scattering (which dominates, if vc ~ v~). A comparison between v~ and vr is also necessary to check on the validity of the local thermodynamical equilibrium (LTE) approximation. If the frequency vth of thermalizing collisions of electrons with plasma particles is larger than vB, electrons appear to be thermalized before any act of spontaneous emission takes place. This thermalization seems to occur mainly due to electron-electron collisions and Vth is accepted to be about Vee- v,. Therefore, the source function of the transfer equation is the Planck function only if v~ >> v~. For the plasma under consideration the validity of LTE should be checked against the labora-

THE RADIATION OF A HOT MAGNETIZED PLASMA

67

tory case where the condition u~ >> uB is almost always fulfilled. Note also that if the radiation energy W >~ BZ/8~r, the Compton processes can violate L T E as well as the cyclotron cooling does. Our purpose is to determine the absorption coefficients and to investigate the properties of radiation from a hot magnetized plasma for a wide range of parameters. We are interested in the frequencies both near and far from the cyclotron resonances. In this case the electron collisions with plasma particles and scattering of photons should be taken into account in addition to the thermal motion of electrons.

3. Absorption Coefficients and Polarization of Normal Modes

For the analysis of normal modes it is convenient to proceed from the following expression for the dielectric tensor e~k of a hot magnetized tenuous plasma: i ~ / ~ v e_ x

eyy=exx

i~/~ v ~[cosollZxe-X

sZls

Z(Is'-ls)Ws;,

ex~ = e~ = v to tan a e_ x ~, sI, (1 + iX/'-~ zsW~); we "7' X ey~ = - e~y = iv w-w-tan a e x ~ (I's - / D ( 1 + i~/-~ zsWD; O)B

ez~ = 1 -~

2/)

[3 1cos OL[

s

e -x ~ I,z,(1 + i~/-~ zsWs).

The used coordinate system has a z-axis along B and an x-axis in the plane containing B and the wave vector q. For a tenuous plasma, Iql~ w[c. The 2 2 notation in (3) is: v =oJffoJ, X = /~ 20) 2 sin2~/2w~, Is = I s ( x ) is a modified BesseI function, I'~ = dIs/dx, z, = x, + iys, xs = (w - SwB)/UD, y, = (Uc + ur)/u~ the summation is carried out over all integer s. The plasma dispersion function W~ = W ( z D = H , + iF, = iTr -1 f2~ dt(zs - 0 -1 exp ( - t 2) may be expressed (see, e.g., Shafranov, 1967) in terms of the probability integral of a complex argument. In the spectroscopy Hs = H ( x , y,) is known as the Voigt function. Contrary to the well-known equations for collisionless plasma (see, e.g., Zheleznyakov, 1977), Equations (3) take into account the collisional absorption and photon scattering by including an imaginary term iy, in the argument of Ws. Strictly speaking, these processes should be taken into account before averaging with the electron distribution function. Nevertheless, the components eik obtained in this way appear to coincide with (3) for ]oJ - swBI >>max (up, u~ + Ur).

68

G . G. P A V L O V

ET AL.

For [cO-scOs[~> u~ + u, Therefore, in a hot plasma Equations (3) may be considered as a quite accurate interpolation over a wide spectral range. It should be noted that in a magnetic field the f r e q u e n c y of collisions equals v. for s = -+ 1 and uiT for s = 0 because of the different cross-sections of f r e e - f r e e transitions for radiation with longitudinal 0 and transversal (_1_) polarizations. The expressions for vii,• may be obtained from (2) substituting Lll,• for L (Pavlov and Panov, 1976). The difference between u~ and vii,• is essential in the quantizing magnetic field (fioJB ~> k T ) , but it is negligible in the classical case for oJ ~> coB. Using Equations (3), two points should be taken into account. First, in deriving these equations the quadratic Doppler effect was neglected. It is true only if [cos a[->fl or f l z c O ~ u~+ u,.. Second, in fundamental resonance the tenuous-plasma approximation is valid if v ~ max (fl [cos a], (u~ + Ur)/OJB), but for IcO- cOBI~> max (vo, u~ + v~) the weaker condition, v ~ 1, is sufficient. For a tenuous plasma the refractive indices Kj. and the absorption coefficients i,zi = (2,:o/c)kj of the normal modes may be found (Shafranov, 1967) by use of the components e;k nj = ~i + ikj = n~ + (n~ + n2c)l/z,

] = 1, 2,

(4)

where n1

=

Kt + ikl

=

1

+ ~i ( e y y +

exx

C O S 2 o~ - -

exz sin 2a + ez: sin z a - 2),

nL = KL + ikL = ](eyy -- exx COS2 a + ex: sin 2a - e~z sin z a),

(5)

i

nc = Kc + ikc = ~ (exy cos a + ey~ sin a).

Equations (1)-(5) appear to be sufficient for the numerical calculation of k i and Kj for given cO, uo, uc, Ur. Some results of these calculations are presented in Figures 1 and 2. For the analytical investigations of the normal modes it is convenient (Pavlov and Shibanov, 1979) to introduce the real parameters q and p, refined by q + ip =- n~ nc

sin 2 a 2cos a iX/-~ W1 - z=] + 2zo I + 2 co fl cos a(1 + iM~-~ zl W1) cOB tO

iV/-'~ W1 + z:l - - - fl cos a tan 2 a (1 + i V ~ zl WO O) B

(6) The equality in (6) is obtained from (3) if we neglect the terms of the order of flz with respect to unity and use the asymptotic formula W~ -.~ i [ V ~ z~ for s = 0, - 1.

69

T H E RADIATION O F A HOT M A G N E T I Z E D P L A S M A

For Izll >> 1 - i.e., [o~ - w~l >> vo or vc + vr >>vo - we obtain sin 2 a ~oR 2cosol w

q

P=-

q

vc + Vr oJ

(7)

In the opposite case [ZmJ~< 1 - i.e., IoJ - We[ <~ vD for Vc + vr ~ vo, sin2a . { 1 + /3 [ ( l + c o s 2 a ) x ~ q = 2cos a cos ce - (1 + 2 cos

x/-g

,

(8)

sin 2a 13 ( l + 2 c o s 2a) H1 2cos a cos a ~ ( H 1 z+ F~)'

P =

where H~ ~ exp (-x~), 1::1~- F(Xl, 0). It follows from (7)-(9) that IPl < moreover, pZ ~ 1 + q2. Using the last condition, we obtain from (4) 2 kc + qkL P~_kc kj- = k~ + (1 + q2)1t2 -7-2(1 q )3/2"

(9)

[ql

and,

(10)

The upper and lower signs correspond hereafter to EM and OM, respectively. The third term in the right-hand side of (10) is, at least,/3 -2 times smaller than the first or the second. If we omit the third term and evaluate the quantities kx, kL and kc retaining only the dominant parts in each term of the series (3) (these parts are proportional to/32~-3 for s/> I), then =8Jl_(o~+~OB) 2 l+cos2a--+

s n( l +o-q 2 )coso t 1/2

+2v~+v~ 1~

2)1/2 sinZa+/3~

•

q sin2 a + 2 c o s a ) ] (1 + q2) m .

l+cos2a-

N" (/3~osina. (11)

In this expression the formula (7) should be used for q with adopted accuracy. Neglecting in (11) the harmonics s ~>2 and using the asymptotic form of H~ for Iz~l >> 1, we obtain the known (e.g., Pavlov and Shibanov, 1978) formulae for collisional and/or Thomson absorption in the cold-plasma approximation. On the other hand, in the limit vc, v , ~ 0 , Equation (I1) reduces to the collisionlessapproximation formula for the cyclotron absorption at harmonics of wB (Zheleznyakov, 1970). The EM absorption is described by (11) with sufficient accuracy. However, for OM this equation is erroneous around ~OB.*The expression in the last parenthesis of (I1) vanishes at o~ = ~o~ since the dominant parts (~ vfl -~ and ~ v) of the first

*This fact has not been mentioned in previous papers (e.g., Zheleznyakov, 1970, 1977).

70

G.G, PAVLOV ET AL.

and second terms in (10) cancel each other. Hence, for a correct evaluation of k2 we should evaluate kr, kL and kc with an accuracy ~ v/3. The third term of (10) is just of the same order, and is also essential in this case. Moreover, the more accurate expressions (8) and (9) should be used for q and p. As a result, for ]60 -- cOBI~ PD and VD >>V~ + Vr we obtain k2 - - ~

1 [(pc "+"/Tr)f-D PC-[- Pr s i n 2 ot -- (.02 1-~-COS20g [(CO -}- COB)2 COS2 a - + - ~ - ~ - ~ /3 sin 4 a(1 + 2 c o s 2 O r) 2 . H1 ] 4~/-~ [cos a[(1 + c o s 2 a) 3 H ~ F~ "

(12)

For ]c0 -coB] >> VD the coefficient k2 may be evaluated by use of Eq. (11). The last term of Equation (12) represents the resonant absorption of OM at co ~ wB. The height of the OM resonance differs from that of the EM resonance by the factor -/32. This results from the opposite characters of the polarizations of OM and EM at co ~-cob (see Section 2). For the EM polarization the electric dipole absorption is permitted, whereas the OM absorption is associated with the higher multipolarity transitions. The OM resonance at co = ,oB is comparable with that of EM at co = 2co~ (see Figures 1 and 2). It disappears if sin 4 c~ ~ (Pc + Vr)/VD. The last term of (12) appears to coincide with the expression for k2 derived by Akhiezer et al. (1975) and Zheleznyakov (1977) for a moderately dense plasma. The condition v >>/3 [cos ~[ (but v ~ 1) which they used is inverse to that of our paper. This coincidence is caused by the fact that for OM even moderately dense plasma could be considered as tenuous because of the exact cancellation of the dominant terms in (10). It is seen from (11) that even for a hot plasma (VD>> Vr + Vc) the cold-plasma approximation is still valid in the wings of cyclotron resonances, [w - stoBI >>VD provided their Doppler cores do not overlap (s/3 [cos a / ~ 1). This approximation is also valid near the centres of harmonics s ~>2 for EM and s/> 1 for OM, provided the angle a is small enough (a < a~s)) or the number of harmonics exceed some particular value s*. We can estimate a~s) and s* from (11) and (12). For instance, the first resonance of OM disappears for a < a ~ l)~[(vc + vr)//3~os] ~/4. The numbers are presented in Table I for different /3 and y-~ Vc/O~s in the case vr ~ vc. As a rule, s* = s * - 1. The table illustrates, in particular, that for sufficiently large/3 the number s* appears to be independent of 7. For instance, it is the case for/3 = 0.141 ( k T = 5.1 keV) provided y ~ 10-4. It means that for such temperatures the Doppler cores of resonances with s > s* are completely overlapped. For smaller /3 and/or larger 7 the resonances with s > s * are almost imperceptible on the background of 'collisional' (highfrequency) wing of the first resonance. Of course, in this case s* depends on 7 (cf. Table I). As mentioned above, the normal modes are elliptically polarized. For [Pl ~ [q[ (see Equations (7)--(9)) the degrees of circular (P~)) and linear (P~)) polarization

THE RADIATION OF A HOT MAGNETIZED

PLASMA

71

TABLE I The number of noticeable resonances of the EM absorption coefficient for c~= 60~ and different fl and 3' = uc(oJB)/ws

/3

10-1 3x 10-z 10 -2

10-2

10-'*

10-6

10-s

4

6

6

6

3 2

4 3

5 4

6 5

of the jth m o d e m a y be written in the form P~) = -+ (1 + q2)-1/2 sign (cos o0, P I j) = w-Iql (1 +

q2)-1/2.

(13)

The positive P~) c o r r e s p o n d to a clockwise rotation of the electric field of the w a v e around the w a v e vector. The negative linear polarization of EM means that the preferred direction of the oscillations of the electric vector lies in the plane perpendicular to the magnetic field. F r o m (6), (7) and (13) it is clear that for o: = 0, 180 ~ both modes are circularly polarized, whereas for o~ = 90 ~ they are linearly polarized in the mutually orthogonal planes. For intermediate a, small non-orthogonality of the polarization ellipses takes place. The degree of the non-orthogonality (Gnedin and Pavlov, 1973) is equal to 2p(1 + q2)-1 for p 2 ~ 1 + q2. Comparing this expression with the third term in (10) we conclude that the first resonance of OM absorption results partly f r o m the non-orthogonality of the normal modes. In Figure 1 the profiles of the absorption coefficients tzi for T = 6 x los K and Vc(~B) = 5 X 10.70.}B >~ Vr(OJ) are c o m p a r e d with those obtained in the collisionless (vc ~ 0 ) and cold-plasma (13 ~ 0 ) approximations. Large errors are illustrated for these approximations and applied for the adopted parameters. In Figure 2 the d e p e n d e n c e of/zi-profiles on o~ is presented for T = 1.2 x l0 s K and Vr(~O)= 2 X 10-StOB-> Vc(W). For a = 0 the only spectral feature is the fundamental r e s o n a n c e of EM; for a > 0 the first resonance of OM and the higher r e s o n a n c e s of both m o d e s arise. With increasing a, the height of the r e s o n a n c e s increase whereas their widths decrease. The noticeable a s y m m e t r y of resonances is caused by vo = oJB/3[cos a[ and increases with oJ within the Doppler cores. 4. T h e E m i s s i o n from a H o m o g e n e o u s Plasma Slab

In order to determine the conditions in the emission region of the accreted p l a s m a we should solve the kinetic equations for the distribution function of

72

G.G.

I I

~

I

[

~

~

PAVLOV

~

ET AL.

I

9

9

t'q

~•

o~-~

I

.~

f

t

I

t

f i

.~- i~

e-

9

I:~

THE RADIATION OF A HOT MAGNETIZED

73

PLASMA

~J~

X

m

9

/

/

l ;l

iI

i

/ iI

r

X

I

/ )

'

I I I

t I

/ f

!

t f

f

ii I

/ /

I

/

//

/

/

/

!

b~

it

.cl

/

J J

..SS j

/

,,t / I

l

/ /

9I,

o

,,,,

I

T

-,

k~ O

j

v

I

!

l

~q

74

G . G . P A V L O V ET AL.

protons and electrons together with the radiative transfer equation. However, for the case of L T E the last one may be considered independently of the others if the electron temperature is known. Using results obtained in the previous section, and assuming the validity of L T E and true-absorption approximations (see Section 2) we easily find the properties of radiation from a hot magnetized plasma. L e t the emitting region have the form of a slab of thickness I with uniform electron temperature and concentration. The magnetic field is considered to be directed at an arbitrary angle 0 relative to the normal n to the face of the slab. Then the intensities of the normal modes emerging from the slab are /*=

Bo,(T) 1 - e x p ( - r j ) ) ,

2

(14)

(

where B~,(T)=(hw3/4cc3c2)[exp(hw/kT)-l] -1 is the Planck function, r i = /~il sec 0 and 0 is the angle between n and wave vector q. All Stokes parameters may be expressed in terms o f / / a s I = I1 + I2; V =

Pro(I, -

I2), Q = PI')(I~ - I2), u = o,

(15)

where Pt)) and P t u are given by Equation (13), Q and U are defined in coordinate system with the z-axis along q and the x-axis in the B-q plane. For given f r e q u e n c y the values I, Q, V depend on two angles 0 and O and on the dimensionless parameters /3 = 1.8 x 10-2 T~/2,

3' --- vc(,os)/wn = 3.1 x 10-8 nt6T6312B81 x [5.1 + In (T61Bs)]

and t =- rffw = we, O = 0) = 2.9 • 103 n16T6~/2Bs12 (1 + cos 2 0)lcos 01-1 . The last estimation is correct if y ~/3 [cos 0]. In this case the optical thickness t of the slab for EM at w---wn is associated with the often used parameter A = w ~ l / w s c (Bekefi, 1%5) by means of t = A~/-~(1 + c o s 20)/2fl Icos 01. In Figures 3-6 the values of /3, y and t appropriate for the emission regions of accreting magnetized degenerate dwarfs are chosen. The spectra of the intensity I are shown in Figure 3 for 0 = 0, O = 60 ~ and different t. Each spectrum consists of a smooth continuum and pronounced cyclotron emission lines. The number of lines depends on t but cannot exceed s*, being equal to 4 in the case under consideration (see Table I). The shapes of the lines also depend on the slab thickness t. If r~ ~ 1 in the centre of some resonance, then the profile of the lines is similar to that of/~t (the curve 'a' near w = 2ws). When r~ becomes ~ 1, the E M intensity I~ tends to reach the level of B,o(T)]2, and the line becomes saturated due to self-absorption. With increasing t the level Bo,(T)/2 is also reached in the wings of EM resonance and an EM continuum is formed. At the same time the OM radiation appears noticeable at

THE

RADIATION

OF A HOT

MAGNETIZED

75

PLASMA

/ I

I/I/

//l I

12

// /

I

I

-

/ / / I I /

/ / / r

/

8

/ / /

/ / / I I

I

/

/ / / /

/ /

I

2

Fig. 3. T h e s p e c t r a o f i n t e n s i t y o f radiation (in the units co~kT/4cr3c 2) f r o m a h o m o g e n e o u s slab for /3 = 0.02 ( T = 1.2 x 106 K) 3' = 2 x 10 -6, 0 = 0, ,9 = 60 ~ and different v a l u e s o f t = ~q(oJ = oaB, O = 0) lettered b y 'a' (t = 3 x 103), 'b' (t = 3 x I06), 'c' (t = 3 x 107), 'd' (l = 7 x 108), 'e' (l = 1.4 x 109) and 'f' (t = 2 x 109). T h e d a s h e d lines r e p r e s e n t B~(T)/2 and Bo,(T).

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G. G. PAVLOV ET AL.

I

2

i

0.86"

!

!

l

4, cY

a

/,~

g,s

Fig. 4. T h e s p e c t r a of i n t e n s i t y (in the u n i t s w2okT/4~r3c 2) in the v i c i n i t y of the first r e s o n a n c e for /3 = 0.042 ( T = 5.3 • 106 K), 3, = I0 5, 0 = 0, t = l05 a n d d i f f e r e n t v a l u e s of 0 i n d i c a t e d n e a r the curves.

the resonance f r e q u e n c y and the intensity /2 increases with t until ~z ~ 1 (the curve 'a' near o ) = wB, the curve 'c' near o ) = 3o~B). For larger ~'2 the OM line saturates, resulting in an increase of the total intensity I up to the level of B,,(T). Finally, when t is so large that r2 ~> 1 between a chosen resonance and the next, this line disappears and the spectrum in the domain considered tends towards a black-body spectrum. Thus, at small t(~/3 -2) only the fundamental cyclotron resonance is noticeable in the intensity spectra. With increasing t the lines at higher harmonics increase and become stronger than the first ones which disappear due to self-absorption and give rise to the Rayleigh-Jeans continuum. For sufficiently large t ( - 101~ in our example) the disappearance of the last (fourth) line occurs and the blackbody spectrum arises up to dividing f r e q u e n c y ~o, which may be found from the equation ~-2(o9= to,)= 1. Note that o9. cannot, in general, be determined in a collisionless approximation. At o ) > to. the spectrum falls off and tends to

THE RADIATION OF A HOT MAGNETIZED PLASMA

O.5

77

o.

V 3

2 84

r

Fig. 5.

The spectra of the Stokes p a r a m e t e r V (in the units ~oZkT[4zr3cZ)and the degree of circular polarization Pc for the s a m e values of parameters as in Figure 3.

coincide with that of the optically thin slab without a magnetic field: I ( w >> co.) = Bo,( T ) l z ( B = 0)l ~ o~ uck Tl/47r2 c 3. Owing to the large difference between the heights of r e s o n a n c e s o f / z l and/zz, the inequality ~-z,~ t ~ ~-~ can take place for a wide range of t. T h e r e f o r e , around the cyclotron harmonic a relatively broad continuum, resulting f r o m the saturated E M line, m a y be o b s e r v e d together with the narrow emission line of OM (curve 'a' near wB, curve 'd' near 3oJB and 4~oB). This effect is m o s t p r o n o u n c e d at the fundamental harmonic (the c u r v e for ,9 = 80 ~ in Figure 4). The spectra of circular polarization are shown in Figure 5 for the same /3, 3' and t as in Figure 3. T h e y also consist of the cyclotron features and the continuum. If ~-i <~ 1 within some r e s o n a n c e then the 'emission line' of polarization appears due to E M emission (e.g., c u r v e 'd' near 4wB). N o t e that the circular polarization a p p e a r s to be strong e v e n for a well a w a y f r o m zero, because of the large IP~)l (see (13)). For example, Pc = 90% near ~o = 3rob for t = 3 • 103 and a = O = 60 ~ If r2 >> 1 in the r e s o n a n c e region, but ~'2 ~ 1 in both wings, then the absorption line appears b e c a u s e the polarizations of the normal modes com-

G.G.

78

T

PAVLOV ET AL.

"T"

r

Pc r

T

1-""

q7~r

o,s q2s /

I

/' /

8

/

/

I

1

I

/

I

I

I

I

/

I

/

/ I

/

/

/

/

/ // / / / /

o r

Z

tt

3"

Fig. 6. The spectra of the intensity (in the units w~kT/4rr3c 2) and the degree of circular polarization for /3 = 0.02, y = 2 x 10 6, 0 = 0, t = 3 x 107 and different values of t9 indicated near the curves.

P,

pletely compensate each other. If, in the centre of the resonance ~'2~< 1 ~ r~, then only partial compensation takes place and an emission line with an absorption core is formed (curve 'c' near 3o}B). At ~o = 0.5~o8 the continuum polarization changes the sign due to intersection of the absorption coefficients tz~ and P-2 (see Figure 1 and Equations (10) and (11)). At frequencies below o}. we have a black-body spectrum of intensity and

T H E R A D I A T I O N O F A HOT M A G N E T I Z E D P L A S M A

79

the radiation is completely depolarized. The spectra of the linear polarization are not presented because they are similar to the spectra of the circular one. The spectra of I and Pc for t = 3 x 10 7 and different 0 are shown in Figure 6. For O = 0 they are smooth since the high resonances of P-~.zare absent as well as the first one of P~2; the fundamental resonance of EM does not manifest itself because of the large optical depth chosen. With increasing O the level of the continuum component of intensity increases because of growth of the optical depth along the line of sight. An increasing 0 also results in the appearance of cyclotron lines. The optical depth of some of these lines is small and they become narrower due to a decrease in the Doppler width (the second and third resonances in Figure 6). For the others with smaller numbers, the optical depth appears to be large, and they broaden with increasing 0 due to saturation (e.g., the first resonance in Figure 6, which is caused by OM only). For sufficiently large O the first resonance of OM disappears because the slab becomes optically thick for OM also. In the spectral band near the cyclotron frequency for intermediate O, the slab can emit a broad EM continuum together with a narrow OM line (Figure 4 and curve 'b' in Figure 3).

5. Discussion

Using the approximate formulas (11) and (12) we can estimate the absorption coefficients and clear up the main characteristics of the emitted radiation. The results obtained are appropriate, in particular, for a plasma accreted by a degenerate dwarf with a strong magnetic field - 108gauss. We found that, in this case, neither the cold-plasma nor the collisionless approximation was valid. Therefore, the separate consideration of cyclotron radiation and bremsstrahlung (see e.g., Fabian et al., 1976) is generally incorrect. For example, the maximum number s* of the noticeable resonances and the dividing frequency ~o, may be estimated in a collisionless approximation only if the temperature is sufficiently high and/or the density is sufficiently low (see Table I). In a general case, formulae (11) and (12) are necessary for this purpose. Thus, for the conditions adopted by Chanmugam and Wagner (1979) the number s* is 6-7 for k T = 1 keV instead of > 20 from Figure 3 of their paper. As far as we know, the OM radiation has not yet been taken into account in the analysis of emission of the objects under consideration (e.g., Masters et al., 1977; Lamb and Masters, 1979). However, even the simplest case of the homogeneous slab demonstrates that OM is important and leads to a number of new qualitative effects. Thus, it results in the generation of narrow lines together with a polarized continuum, and showing 'emission', 'absorption' or more complicated features in the spectra of polarization, etc. The omission of this mode may lead to incorrect estimations and erroneous qualitative calculations. The calculations for the simplest case of the homogeneous plasma slab seem

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to demonstrate the most essential properties of the emitted radiation, such as: (i) a Rayleigh-Jeans continuum at oJ < co, and a flat one at w ~>o2, are formed; (ii) at o) ~>to, the cyclotron resonances are noticeable provided s* > o~,/wB, (iii) at to >~o2, the radiation is polarized. These results generally confirm the proposed interpretation (see Section 1 for references) of the strong emission features and continuum polarization of AM Herculis-type objects. The variability of these features is likely to be the result of variations in the optical depth of the emitting region-e.g., due to variations of the accretion rate. Furthermore, for the equivalent widths of these lines a strong dependence on phases of rotational period may be predicted because the optical depth and the resonance profiles essentially depend on the angle between the line of sight and the magnetic field. For a definite verification of the cyclotron nature of the lines under inspection, narrow-band polarimetry should also be used. Of course, we cannot exclude the possibility that an observed line may be a cyclotron harmonic with s t> 2 or the possibility that several lines may be generated simultaneously. Nevertheless, the interpretation of the broad absorption features of VV Puppis as 6th, 7th and 8th cyclotron harmonics (Wickramasinghe and Visvanathan, 1979) seems to face serious objections. First, their calculations were performed only for EM with the wave vector perpendicular to the magnetic field. This contradicts the large periodic increase of circular polarization just at the phases when the lines were observed, since at = 90~ only linear polarization may arise. On the other hand, even if a is far from 90~ the mentioned harmonics either overlap due to temperature broadening or appear to be imperceptible on the background of the high-frequency wing of fundamental resonance (see Table I). To explain these absorption features and other observed properties, and to develop a model of AM Herculis-type objects, we should take into account heating of accreted plasma by the release of gravitational energy together with cooling by emission of both cyclotron modes. The mean temperature of protons may reach a value as high as the gravitational value, -GmpMJkRdw. The mean temperature of electrons is determined by the balance between their heating by protons and cooling by the emission of photons, provided the comptonization processes are negligible. If bremsstrahlung prevails over other cooling processes, then the distribution of electrons appears to be locally in equilibrium (LTE) with Te = Tp. On the other hand, if cyclotron cooling is more essential, LTE is still valid provided the characteristic frequency of thermalizing processes vth ~ Uee Uc is larger than vB = vr(wB). In this case Te = Tp if (melmp)uc >>us, and Te < Tp if (me/mp)u~ ~ uB. Finally, the electron distribution is anisotropic and non-equilibrium if us > u~. By use of (11) and (12), and assuming LTE, the emissivity of heated plasma may be easily evaluated from

THE RADIATION OF A HOT MAGNETIZED PLASMA

81

Q = 2zr f f d o f f da sin a(/xl +/z2) B~(T) 2

~ nkT (2vB + 4 kT e2 ) rnc 2 hc Uc ,

(16)

where vc is effective collision frequency (2) averaged over the spectrum with the function exp(-hw/kT). It follows from (16) that cyclotron cooling (the first term) dominates the free-free cooling (the second term) if B > 112 T63/4gauss 10s ,16"1/2T61/4gauss. On the other hand, LTE is valid if B ~ 2 x 108 n16 and Te = T; if B ~ 4 x 10 6 ,~ 1/216T63/4gauss. In the right-hand sides of these rough estimations we omit square roots of the Coulomb logarithms which are of the order of unity. Thus, the conditions for prevailing of bremsstrahlung cooling and for the equality of Te and Tp are different in contrast with the assumption of Fabian et al. (1976). It seems evident that the distribution of temperature in the heated region should be inhomogeneous. In this case the emitted radiation differs from that of a homogeneous slab. Owing to the difference between the optical thicknesses for two modes, the emitted continuum may be polarized at all frequencies. Even for a very thick and heated region, the cyclotron resonances with s < s* may be observable. Furthermore, the cyclotron features may possess complicated shapes if this region is optically thin for OM and thick for EM. If in this case the temperature increases with depth, the EM absorption line should be observable with the narrow OM emission at the centre. Two absorption lines h 5700 and h 5000 ~ observed by Liebert et al. (1979) when VV Puppis was at maximum light state may be explained in this way as a superposition of broad EM absorption and narrow OM emission lines, both being centred at 5350 A. If this resonance is fundamental, then the corresponding magnetic field is about 2 x 108 gauss. A similar estimation on the basis of the h 3470 feature, observed by Liebert et aL (1978) when VV Puppis was at a state of minimum light, leads to a higher magnitude - 3 . 1 x 108 gauss (Mitrofanov, 1980). This difference may result from the fact that, in maximum state, the emitting region increases its height and becomes optically thick for EM or for both modes. The mentioned estimates of the magnetic field correspond to a difference of about 0.2 Rawarr between the heights at maximum and minimum states provided the field is dipolar. If the change of temperature with depth in the emitting region is nonmonotonous, very complicated spectral features may be expected. The specific conditions for their origin will be considered in detail elsewhere.

Acknowledgements We thank Dr J. Liebert for providing results of observations in advance of publication, and Drs V. N. Fedorenko, Yu. N. Gnedin, A. D. Kaminker and D. G. Yakovlev for useful discussions.

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References Akhiezer, A. I., Akhiezer, I. A., Polovin, R. V., Sitenko, A. G. and Stepanov, K. N.: 1975, Plasma Electrodynamics, Pergamon, Oxford. Bekefi, G.: 1%6, Radiation Processes in Plasma, Wiley & Sons. Chanmugam, G. and Wagner, R. L.: 1979, Astrophys, J. 232, 895. Ginzburg, V. L.: 1%4, The Propagation of Electromagnetic Waves in a Plasma, Pergamon Press, Oxford. Fabian, A., Pringle, J. K. and Rees, M.: 1976, Monthly Notices Roy. Astron. Soc. 173, 43. Gnedin, Yu. N. and Pavlov, G. G.: 1973, Zh. Eksper. Teor. Fiz. 65, 1806 (Soy. Phys. JETP 38, 903). Gnedin, Yu. N. and Sunayev, R. A.: 1974, Astron. Astrophys. 36, 379. Lamb, D. Q. and Masters, A. R.: 1979, Astrophys. J. 234, L117. Liebert, J., Stockman, H. S., Angel, J. R. P., Wolf, N. J., Hege, K. and Margon, B.: 1978, Astrophys. J. 225, 201. Liebert, J., Stockman, H. S., Angel, J. R. P., Margon, B.: 1979 (private communication). Masters, A. R., Pringle, J. E., Fabian, A. C. and Rees, M. G.: 1977, Monthly Notices Roy. Astron. Soc. 178, 501. Mitrofanov, I.G.: 1979, Report in 1AU Symp. 88. Mitrofanov, I. G.: 1980, Nature 283, 176. Pavlov, G. G. and Panov, A. N.: 1976, Zh. Eksper. Teor. Fiz. 71,572 (Soy. Phys. JETP 44, 300). Pavlov, G. G. and Shivanov, Yu. A.: 1978, Astron. Zh. 55, 373. Pavlov, G. G. and Shivanov~ Yu. A: 1979, Zh. Eksper. Teor. Fiz. 76, 1457. Pavl0v, G. G., Shivanov, Yu. A. and Gnedin, Yu. N.: 1979, Pis'ma Zh. Eksper. Teor. Fiz. 30, 137. Pavlov, G. G., Shivanov, Yu. A. and Yakovlev, D. G.: 1980, Astrophys. Space Sci. 73, 33. Shafranov, V. D.: 1%7, in M. A. Leontovich (ed.), Reviews of Plasma Physics, Vol. 3, Consultants Bureau, New York. Tapia, S.: 1977, Astrophys. J. 212, L125. Triimper, J., Pietsch, W., Reppin, G., Voges, W., Staubert, R. and Kendziorra, E.: 1978, Astrophys. J. 219, L105. Voykhanskaya, N. F. and Mitrofanov, I. G.: 1980, Pis'ma Astron. Zh. 6, 159. Weaton, W.: 1978, IAU Circ. 3238. Wickramasinghe, D. T. and Visvanathan, N.: 1979, Report in IAU Symp. 88. Zheleznyakov, V. V.: 1970, Radio Emission of the Sun and Planets, Pergamon Press, Oxford. Zheleznyakov, V. V.: 1977, Electromagnetic Waves in Cosmic Plasma (in Russian), Nauka, Moscow.