Ad~. $pacc $~s. Vol.3, No.10-12, pp.279-285, 1984 0273-1177/84 $0.00 + .50

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PHYSICAL PROCESSES IN THE S T R O N G M A G N E T I C FIELDS OF A C C R E T I N G N E U T R O N STARS

P. Mdsz~ros*

Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, U.S.A.

ABSTRACT

The key radiative processes are discussed, numerical methods of magnetized radiative transfer are presented, and these are considered in their application to X-ray pulsar models.

KEYWORDS

Strong magnetic fields; Neutron Stars; Cross Sections; X-rays; Radiative Transfer

INTRODUCT I0N

We shall review here some results on physical processes in very strong magnetic fields (B ~ 1012 Gauss), which are useful in understanding accreting magnetized neutron stars. The material presented here concentrates mainly on the work of groups with whom the author has

had active interaction. By consulting also the reviews by Drs. Gnedin and Pavlov in this same Symposium, the reader should obtain a fairly complete picture of this field. We begin with a brief discussion of the propagation normal modes in the presence of a quantizing magnetic field, and then apply these results to calculations of the main radiative cross sections. We then discuss some methods of solving the radiative transfer equations, in particular numerical ones. Applications of these to particular models of X-ray pulsar emis- sion regions are then considered, and current problems are outlined.

PROPAGATION NORmaL MODES

The plasmas one has to deal with have temperatures wlich are in the X-ray range, typical of neutron star accretion problems, and the (electron) cyclotron ground harmonic is at

~H = 11.6 BI2 KeV. (i)

Cyclotron lines have been experimentally found in X-ray pulsars (TrNmper et al 1978, White et al. 1983). One can readily see that in such plasmas, the polarization of the photons is important for calculating the radiative opacities. For example, a photon of type 2 (ordi- nary) with electric field parallel to ~NS will find it easy to shake the atmospheric electrons, since the magnetic field does not offer a resistance; scattering therefore occurs almost at the same strength as if no B were present. However, a photon of type 1 (extra- ordinary), with electric vector perpendicular to B, will try shaking the electron across the B-field, but in this direction the electron impulse is quantized in orbits of finite energy (i). For ~ £ ~H a stiff resistance is encountered, and scattering occurs at considerably diminished strength. Fo[ propagation nearly parallel to the field, there is a difference between photons 2, wit h E rotating in opposite sense to the rotation of an electron, and be- tween photons i, with E rotating in the same sense as the electron. ~lile under inclusion of spin and vacuum effects one finds that both waves are resonant near ~II, it is clear that the extraordinary wave I, rotating the same way as the electron, is the more strongly reso- nan t.

The effect of the plasma on the propagation of X-rays is, in the most common regime, ade- quately described by taking into consideration the electron component and the virtual elec- tron-positron component. The first of these is familiar from classical magnetoionie theory, while the second is a quantum electrodynamical consequence of the very strong magnetic field (similar to the virtual pairs induced near a very strong Coulomb field). A small wavelike

*Smithsonian Visiting Scientist~ on leave from Max-Planck-Institut fur Physik and Astrophysik

I~E; partly supported by NASA NA~q-246

279

280 P. M6sz~ros

perturbation E rue -i(wt-k'r), introduced into Haxwell's equations, leads to a wave equation in Fourier Space

(~ + ~X + ~) " Ek : 0 (2)

: ++ N2( ÷ ~ ÷-~ Here the tensors A = 1 - 1 - kk) and k = s - 1 are t'he same as in lower field plasmas (~: dielectric tensor, N: complex index of refraction, k: wave vector). The typically pulsar-like Q.E.D. effects, caused by virtual pairs, appear in the tensor

,~ = - 2~ + N24£0kxbkxb + 76bb. Ilere

,~ = (~./45~)(B/Bcr)2 ~ 5 10-5(B/Bcr )2 (3)

measures the effect of the vacuum (virtual e+e -) polarizability, for fields less than the critical Bcr = m^2c3/e~ = 4.414 1013 Gauss (at which ~ ~H = m^c2)" The noteworthy thing is

2 2 that the vacuum ~6-dependent) terms dominate, in X-ray pulsars, the usual (W/WH) , (W/Wp) terms due to the real electrons. The far reaching consequences of this fact for X-ray pulsar opacities and spectra was noted independently by Gnedin, Pavlov and Shibanov (1978) and li~sz~ros and Ventura (1978). This can be seen from the complex eigenvalues N i of the wave equation (2)

2 NI, 2 = 1 + S + 46 sin2~ - D cos@[b-+(l+b2)I/2], (4)

and from the eigenvectors ~i' which are the polarizations eigenmodes, in the cartesian system with kl i z,

~i -~ (i Ki, l, 0). (5)

Here K i ~ (i E~ /E~,). = b ± (i + b2) I/2 is the ellipticity of mode i (i = i, 2) and everythin~ 7 i .

is seen to be a funetzon of the simple parameter

b = -(sin20/2 cosO)D -I (P-S+36), (6)

where S, P and D in (4) and (6) are elements of the susceptibility tensor of a hot plasma,

X = ~ - 1 = S (7) P

and usually a llaxwell-Boltzmann distribution in Pz is assumed (Kirk 1980, Pavlov et al 1980, Herold et al. 1981). Interesting effects are revealed by an analysis of the frequency and angle dependence of the normal modes, which are caused by the virtual e+e - pairs. Since both N and b are complex• double-valued functions of ~ and ~ (the angle between k and B), there occur branch points. For kT e < ~ ~H' with most electrons in the ground (n = O) Landau level, there are two such points (e.g. Pavlov and Shibanov 1979, Ventura, Nagel and M6sz~ros 1979, Soffel et al 1983). One is very near ~ ~ ~ll and G ~ sin -± [.943 pl/2(kTe/~ wil) -I/4 (B/lOBcr)-9/4] ~ small, and the other is at the "vacuum" frequency w = w v and 8 ~ w/2, where

~pe(3d)-i/2 w v = ~ 3(ne/1022)i/2(B/10 Bcr )-I KeV (8)

While almost everywhere else the normal modes are well defined and quasi-orthogonal ([e~ " $21 ~ 0), near these two critical (branch) points a "mode collapse" occurs, with

[~[ " e2] # O. This occurs where the real part of b becomes zero, which is brought about by a mutual cancellation of the effects induced by the real and virtual components,

Real (b) ~ (sin2~/2 cos 8)ul/2(l+~3v-lu-l(l-u)) = 0. (9)

Here the common factor is due to the real particles, and the second term in the brackets is the vacuum correction, u = (WH/W)2 and v = (w e/W)2. For w v ~ w £ WH, the second term (~) dominates, and when it becomes -i such as atF~ , Re (b) ~ O. There the modes, as evidenced

• v > by (5), become circularly polarized, the sense of rotation being opposite for w < w v. This, in turn, is reflected in the cross sections (c.f. below).

RADIATIVE OPACITIES

a) Thomson: This cross section is most easily obtained via the optical theorem from the com- plex r~fr~ctive index, d~ = 2~(neC)-i Imag(Ni). Using the transverse polarizations @t = (I - kk). ~ and the polarizability tensor ~pl = -v-i ~' defining I = k x b, one can obtain from the wave equation

Physical Processes in Strong Magnetic Fields 281

N i = i - v(eti , ~pl.ei) + 46(eti, ~i.e i) + 76(eti, bb.e i) (i0)

To first order in 6, the optical theorem gives for the (coherent) scattering of mode i at ~,@

2 il 2 2 °i : °T 2 I% + 7~[ e + I% (11)

(~+c~it) ( ~' -~0 H )

Here the polarization eigenmodes are written in rotating coordinates, with e+ = e x ± i e y" The factor multiplying tell 2 introduces a resonant behaviour, but notice that the levi 2 quantify itself is also resonant near <OH, when the 6-corrections are included in e , e , e , as well as thermal effects. (M6sz&ros and Ventura 1978, 1979, Gnedin, Pavlov and ~[~ibanov z 1978, Kirk and H6sziros 1980). The angular behaviour is shown in Figure i.

6 ° I0 o 2% ° 3'0 o &%o 5'0 ° 6% ° 7~0 o' 8'0 ~ O

Figure 1

Scattering cross section oj~ from polarization j into i, and o.] = I$(7"i at two frequencies, with vacuum polarization. B = 0.i Bcr , n e = 1021 cm -3. (from H4sz~os and Ventura 1979).

2 The differential scattering cross section is obtained in the form doji/d~} = r lej(~), ~pl.ei(k) l 2, which gives (Ventura 1979) o

[ d(cos @) = 81~ [e+ ~le!l + ------ w 2 , 'i '' " ] -- ~ie_ lle j ]_ + lez~le~! ] .

(~-~H) (12)

In these expressions, a damping term has been used in the dielectric tensor, which is the radiative damping 7tad = (2/3)(e2~/meC3).

b) Bremsstrahlung: For ~ << kTe, one can obtain the (inverse) bremsstrahlung absorption cross section by just using an appropriate collisional damping Ycoll and the optical theorem. IIowever for ~ ~ > kT e a quantum mechanical treatment is required, e.o.o Virtamo and Jauho (1975) Pavlov and Panov (1976). Using second order perturbation theory, one seeks the transition probabilitv Wfi = (2T/~) l<fITl i>12 6(E~-Ei) , where <f|Tli> = rE{<flVlr><rlHli>(E~-Er)-i + <f[HIr><rTV i> (Ei-Er)-l}, and V '= Ze2~r, H = (e/2c)(V~ + A+V). If one considers only transi- tions beginning and ending in the ground n = 0 state, and intermediate states n = O, n = i, four graphs contribute to this, one each involving e+, e and two involving e z. The resultin~ absorption cross section is (Nagel 1980)

2 2 [ ~ , i 2 ~ ei]2 i 2 ] (13)

K i = K 0 -- ie+ g~ + -- i g+ + iezl gl (~+~H)2 (~-~H)2 - ~ ,

2 2 3 2 2 -i -3 -1/2 where K 0 = 4~ Z ~ ~ c aim e L0 (vmkT/2) , and g±, g|| are magnetic Gaunt factors. This expression again reveals a similar angular and frequeNdy dependence as in the scattering case, and similar ~, 6 dependences. Angle averaged values of Ki, compared to angle averaged values of the scattering cross section are shown in Fig. 2.

282 P. M6sz&ros

i

E ~ 0 . o ~

L L W 0 10 -4

Z 0

10 .6

o

10 -e

I0 -::

.,ptasmo. v(]cuum

. . . . . . . ~ . . . . . . . . _°7__, . . . . . . . . / , . , '~< " , 01 / /

/ / / " x ~ 2

. 0 : o . 1 t to t o o FREQUENCY ( k e V )

Figure 2

Scattering (BI~ ~ ' ~'2 = 2~) and bremsstrahlung absorption (kl, k?) coefficients after angl~ averaging. 0~i B~r, n e $ 1022 cm -3, T = i0 keV (from Ventura e~ al. 1979).

c) Compton: When incoherent scattering (w' # w) is considered, the transition probability sought is--W~ = (2~/~)6(E~-Ef){~<flHl[r><rlHlli>(Ei-Er)-I + <fIH21i>}, where H 1 = (e/2c) (~ + ~), H2 (e /2meC2~2. Consider initial and final states n = 0, intermediate states 0, 1 and keeping frequency change terms to first order (in the nonrelativistic approximation) one obtains

d2o 3 ~' me dwdr 8~ ~T -~-~k~ f(po) l<e,l~(Po)le>121,,, (14)

(Nage11981b; c.f. also Kirk and ll6sz~ros 1980, and Gnedin and Sunyaev 1974). Here cAk = w'cosQ' - w cos8 and Po = me(Aw/Ak) + ~£K/2, f being the distribution function and (~-i)+ _ z has components -WH(W'+~ll-Pk'+k'2/2)-i , ~H(w-wu-pk-k2/2) -I and (p+k/2)(p+k-k'/2)

2~0 -i , , , , ,2 -I ~ " Z ' (w-pk-~ /~) - (p-k /2)(p-k +k/2)(w -pk +k /2) . The expresszon (i0 agazn has a structur~ of the form (13) or (12), with an expansion in %+e$*e+, etc. A plot of the angular dependence of d2o/dwd~ is shown in Fig. 3.

>hi 10124

r,," I 0 - 2 5

(/)

% (D

~ 1 0 - 2 6 Z 0 F-

~ ! 0 -27

(Z) O9 0 C~ 0 l 0 - 2 O

' ' ' ' ' ' ' 1 2 ° ° ' ' ' ' ' 1

J o t, oo 1

L J 20 40 60 eO

PHOTON ENERGY (KEV)

Figure 3

Differential scattering cross section d2~21/dwd~ (2 ~ i), T e = i0 keV, B = 0.i Bcr , Bin = 45 °, ~in = 40 keV, and Bou t = 20, 70, 120 and 170 ° , as a function of ~ Wou t. (from Nage11981b).

Physical Processes in Strong Magnetic Fields 283

GENERAL REMARKS ON CROSS SECTIONS

The three main radiative processes in X-ray pulsars are those listed above, if kT < ~ ~H" In this case the usual cyclotron process (radiative absorption at ~H followed by collisional deexcitation) is far outweighed by radiative excitation followed by radiative deexcitation ( resonant scattering). Thus the real source of resonant photons is bremsstrahlung (at the resonance). All cross sections are strongly dependent on frequency, angle and polarization, due to the magnetic field. In addition to the usual resonance effects at the multiples of the cyclotron frequency, similar sharp frequency variations are o~served near the critical branch points (Re(b) = 0), e.g. at ~v ~ 3(ne/i022)i/2 (B/10 Bcr )-± keV. This is seen for instance in Fig. 2, and the corresponding spectral consequences (features in the polarized spectrum) have been noted both by Ventura et al 1979 and Pavlov and Shibanov 1979.

RADIATIVE TRANSFER

The general equation of transfer in a magnetized medium involves the polarization matrix ~B' whose components are made up of the four Stokes parameters. This set of four equations in four incognitae can be+considerable÷ simplified if one transforms the basis system ex, ey to the normal mode basis el, e 2. If one is in the regime where the modes are orthogonal

• e2) ~ 0, and if there is strong faraday depolarization (many rotations of the wave ( ~ ÷ electric vector between scatterings) then the system simplifies to two equations for the two normal intensities (Gnedin and Pavlov 1974)

(~ • ~ ) I . = - k . Z . + S . . + E. ( 1 5 ) j j ] j l 3

Here k4 is the total (scattering and absorption) opacity, S~ is the inscattering term from j j± other polarizations and directions into the beam, and Ej is thermal emistivity.

In general, one is interested both in the frequency and direction dependence of the escaping radiation, which is a difficult problem (e.g. Yahel 1980). Analytical solutions are possible only for the most simplified cases. In the case of coherent scattering, two-stream or dif- fusion approximations have been made, which average over the angle dependence and give an approximate spectrum, provided a simple atmosphere is adopted, e.g. homogeneous, or exponen- tial (Nagel 1980, t,~sz~ros et al 1980, Kaminker et al 1982). Directionality effects can also be found for homogeneous, semiinfinite media by numerical means (Silant'ev 1980), or approxi- mately by analytic methods (Kanno 1980, Nagel 1981a, Xaminker et al 1982, Pavlov et al 1983). For media of finite optical depth, taking into account incoherent scattering and inhomogeneou structures, numerical methods become necessary. This can be done, e.g., taking discrete ordinates, for 2 polarizations, N O angles and N frequencies, leading to N = Z • N~ • N~ equations in N variables. One can start either from the integral equations of transfer (M~sz~ros and Bonazzola 1981) or from the differential equations (Feautrier scheme, Nagel 1981 a,b). The first one was solved by iterations, which can be done for finite slabs (e.g. T T ~ 10-15). An example is shown in Fig. 4.

i ~o ~o so 60 90 ANGLE ANGLE

Figure 4

Beam shape for polarization i, 2 (PI, P2), energy indicated in keV, full line with vacuum polarization (dashed without) for a slab of rT = 7, n e = 1023 cm -3, T = i0 keV, B = 0.i Bcr (from li~sz~ros and Bonazzola 1981).

284 P. M6szSros

However this same method can be treacherous when extended to semiinfinite media, as pointed

out by Pavlov et al (1983). (c.f. also ~iihalas 1978: the number of iterations required be- comes too large). The Feautrier method, on the other hand, appears useful both for finite and semiinfinite media. An example is shown in Fig. 5. Until now, calculations have been

~ , 1 0 a3

klA

l o a O

LU 09

l O 27 eJ i y-

(D

× t O 24

U_

Z

O I O_

I,T ........ J ........ r

1 0 2 1

lOie .~ . . . . . . . . . . . . . . . . . . . I I 0 i 0 0 PHOTON ENERGY (KEV)

Figure 5

Total s~ectra including comptonization from slabs of ~T = 10-2, i0 -I, I, i0, 102 , 103 ,

p = i0 -L cm -3, T = i0 keV, B = 0.i B . Topmost is a Wien curve (from Nagel 1981b). cr

made either for one frequency (coherent scattering) and different angles (Fig. 4) or one

angle and many frequencies (Fig. 5), although the general case is only a matter of increased computing time.

The beam shapes obtained numerically reveal that near 8 = 0 ° (along the field) the intensity

is not always a maximum, but especially for ~ ~ < ~ii/4 may turn into a minimum (depending on the thermalization length, etc), a feature already expected from early work (Basko and

Sunyaev 1975). This behaviour can now be studied over a wide range of general atmospheric structures. Thus, double pulses are more deeply cleft in finite width slabs than in semi- infinite atmospheres, density gradients can be included, etc. If one knows something about the viewing angle (between observer and field directions), one can estimate the field strength from the frequency dependence of the pulse shape. However, an inclusion of incoherent scat- tering in the beam shape calculation is an urgent future task.

As far as cyclotron line shapes, one can see that with increasing scattering (or total) depth of the atmosphere, the line changes from emission to extinction (caused by scattering out of the core into the neighbouring continuum). I1ere also the inclusion of more angles is a future task, as well as a satisfactory treatment of the nonorthogonality problems near ~H" The in- creased effect of radiation pressure associated with the resonant scattering cross section is found to represent an increase (over the corresponding nonmagnetic case) of a factor of 10-20, much less than expected from the cross section increase alone, since the photon density at the resonance is also reduced by outscattering (Nagel 1982), for X-ray pulsar conditions. (Gamma-ray bursters may present a different case).

I l O D E L S O F X - R A Y P U L S A R S

Depending on their luminosity, current models for accreting X-ray pulsar emission regions (or "atmospheres") fall into one of three categories. For L x > few 1036 erg s -I (the exact value is disputed, depending on magnetic effects in Prad) one expects a radiation-induced deceleration of the matter, in a broadened shock structure (Davidson 1973, Basko and Sunyaev 1976, Wang and Frank 1981). Recently, analytic solutions for particular subregimes have been proposed (Kirk 1983). The biggest problem is to couple the magnetic angle and frequency dependence of the cross sections with the hydrodynamic equations, an enormous task. Recent progress of this has been obtained by using particular polarization and angle/frequency averages of the cross sections which include magnetic effects approximately (Riffert 1983). Time dependent calculations with two-dimensional hydrodynamic structure are in progress now (Arons, Klein and Lea, 1983). This should yield interesting results on the general beaming and variability at luminosities similar to, e.g., Her X-I or Cen X-3.

Physical Processes in Strong Magnetic Fields 285

For L ~ 1036 erg s -1, radiation pressure is less important, and deceleration might occur x possibly via a collisionless (thin) shock, due to collective effects (e.g. Langer and Rappaport 1982). The dynamics has been studied for this case including magnetic microphysics assuming the atmosphere to be optically thin (the shock would stand well above the surface, producing a cylinder-like emission region similar to the previous case). Braun and Yahel (1983) have considered the effects of radiation pressure on this type of shock model. Both in this, and in the previous case, detailed spectral predictions or pulse shapes must await a detailed radiative transfer calculation.

The alternative possibility for L x < 1036 erg s -I is that a collisionless shock does not occur, in which case deceleration proceeds by Coulomb encounters or nuclear collisions (e.g. Basko and Sunyaev 1974). The inhomogeneous temperature and density structure of such atmo- spheres, in pressure and energy balance with the infalling protons, have been calculated by li~szgros et al (1983) and Harding et al (1983), using polarized magnetic radiative transfer. While this work is still preliminary, having used coherent scattering with some allowances for comptonization, it indicates an atmosphere structure somewhat similar to the non-magnetic case of Zeldovich and Shakura (1969). The line cooling seems to act as a thermostat, bottom temperatures stabilizing at values such as to ensure significant line contributions, and the continuum spectra show a tendency to increase in hardness with increasing field strength. The next task here will be to include incoherent scattering in a fuller manner, to evaluate line profiles and pulse-phase spectroscopic effects near eH" One may thus hope in the next few years to achieve models which can be discriminated when compared to some of the recent compilations of observed properties (e.g. White et al 1983).

REFERENCES

Arons, J., Klein, R. and Lea, S. (1983) (in preparation). Basko, H.M. and Sunyaev, R.A. (1975), Astron. Astrophys. 42, 311. Basko, M.ll. and Sunyaev, R.A. (1976), M.N.R.A.S., 175, 395. Braun, A. and Yahel, R.Z. (1983) (preprint). Davidson, K., (1973) Nature Phys. Sci. 246, i. Gnedin, Yu.N. and Pavolv, G.G. (1974) Soviet Phys. JETP, 38, 903. Gnedin, Yu.N. and Sunyaev, R.A. (1976) Soviet Phys. JETP, 38, 51. Gnedin, Yu.N., Pavlov, G.G., Shibanov, Yu.A. (1978) Sov. Phys. JETP Lett. 27, 305. Harding, A.K., li~sz~ros, P., Kirk, J.G., Galloway, D.J. (1983) (preprint). llerold, ll., Ruder, H., Wunner, G., (1981) Plasma Phys., 23, 775. Kanno, S. (1980) Pub. Ast. Soc. Japan, 32, 105. Kaminker, A.D., Pavlov, G.G. and Shibanov, Yu.A. (1982) Astrophys. Sp. Sci., 86, 249. Kirk, J.G. (1980) Plasma Phys. 22, 639. Kirk, J.G. and li~sz~ros, P. (1980), Ap.J., 241, 1153. Kirk, J.G. (1983) (preprint). Langer, S.H. and Rappaport, S. (1982) Ap.J., 257, 733. Hdsz~ros, P. and Ventura, J. (1978) Phys. Hey. Lett. 41, 1544. M~sz~ros, P. and Ventura, J. (1979) Phys. Rev. D, 19, 3565. M@sz~ros, P., Nagel, W. and Ventura, J. (1980) Ap.J., 238, 1066. ~i~sz~ros, P. and Bonazzola, S. (1981) Ap.J., 251, 695. H~sz~ros, P., Harding, A.K., Kirk, J.G., Galloway, D.J. (1983) Ap.J. (Lett.), 266, L33. Hihalas, D., "Stellar Atmospheres", Freeman, San Francisco (1978). Nagel, W., (1980) Ap.J., 236, 904. Nagel, W., (1981a) Ap.J., 251, 278. Nagel, W., (1981b) Ap.J., 251, 288. Nagel, W., (1982) in "Accreting Neutron Stars", Brinkmann, W. and TrHmper, J. (eds.)

Hax-Planck-Inst. Extraterr. Physik ~E report 177. Pavlov, G.G. and Panov, A.N. (1976), Soy. Phys. JETP, 76, 1457. Pavlov, G.G., Shibanov, Yu.A., Yakovlev, D.G. (1980) Astroph~s. Sp. Sci., 73, 33. Pavlov, G.G., et al. (1983) (preprint). Riffert, H. (1983) (preprint). Silant'ev, N.A. (1981) A strophys. Sp. Sci., 82, 363. Soffel, M. et al., (1983) Astron. Astrophys. (in press). TrOmper, J. et al. (1978) Ap.J. (Lett), 219, LI05. Ventura, J. (1979) Phys. Rev D, 19, 1684. Ventura, J., Nagel, W., H~sz~ros, P., (1979), Ap.J., 233, L125. Virtamo, J., Jauho, P., (1975), Nuovo Cim., 26B, 537. Wang, Y.II. and Frank, J. (1981) Astron. Astrophys., 93, 255. ~ite, N., Swank, J.H., Holt, S.S. (1983) Ap.J., (in press). Yahel, R.Z. (1980) Astron. Astrophys., 90, 26. Zeldovich, Yu.B., Shakura, N.I. (1969) Sov. Astron., 13, 175.