Radiation from accreting magnetized neutron stars

R A D I A T I O N F R O M A C C R E T I N G M A G N E T I Z E D N E U T R O N

S T A R S

P. MI~SZARO S*

Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, U.S.A.

(Received 21 May, 1983)

Abstract. We review some recent developments in our understanding of accreting magnetized neutron stars. A brief summary of the observations is given, on which current phenomenological models are based. The main part of this paper is a discussion of recent work by several groups on the radiative transfer problem in a strong magnetic field and its application to models of the structure and properties of self-consistent neutron star polar cap emission regions. The assumptions and uncertainties involved are discussed, recent progress is evaluated, and current and future problems are indicated.

1. Introduction

Accreting magnetized neutron stars, often referred to as X-ray pulsars, are one of the most concrete, and yet ill-understood, of the exotic objects in high energy astrophysics. While black holes have as yet been only tentatively identified with some particular classes of observed galactic and extragalactic objects, no one doubts that magnetized neutron stars are to be identified with either radio pulsars (e.g., Hewish et al., 1968) or X-ray pulsars (e.g., Giacconi et aL, 1971). Another family of observed objects, including X-ray bursters and some galactic bulge sources, has been identified with neutron stars where the magnetic field does not play a dominant role (e.g., Lewin and Joss, 1982). It is, however, the presence of a strong magnetic field, of order B .,~ 1012 G, which provides for a wide variety of complicated and fascinating observational material, that gives us much of the information we have on the internal structure and environment of neutron stars. Of these, the radio pulsars, while fascinating in themselves, are rather harder to deal with theoretically, because the emission mechanisms are certainly non- thermal, and possibly involve coherent processes. This is due to the fact that radio pulsars blow off particles from the surface in a wind, which is tenuous enough to allow the acceleration of relativistic particles, and the production and propagation of radio radiation, whose wavelength is large compared to interparticle distances. The physics involved in accreting X-ray pulsars, on the other hand, is relatively more amenable to analysis. This is because of the higher densities involved in the inflowing matter, usually accreted from a binary companion star, which leads to larger emission region densities, and to a quasi-thermalization of the particle distributions, involving characteristic temperatures of order of tens of keV. This in turn implies typical X-ray photon wave- lengths, smaller than interparticle distances, and the absence of coherent effects. In this

* Smithsonian Visiting Scientist, partially supported through NASA Grant NAGW-246, on leave from Max-Planck-Institut ffir Physik und Astrophysik MPA, Garching.

Space Science Reviews 38 (1984) 325-351. 0038-6308/84/0384-0325504.05. �9 1984 by D. Reidel Publishing Company.

326 P. M g s z A R o s

review we shall deal with the accreting X-ray pulsars, mainly from a theoretical point of view.

x 25

8 zo

2. The Observational Material

We shall describe here briefly some of the key observational results that are used in deriving the currently accepted models. More extensive information on the observations is available in excellent reviews by, e.g., Rappaport and Joss (1981), and White et al.

(1983). X-ray pulsars were first recognized as a separate class after the launch of the X-ray

satellite Uhuru (Giacconi et al., 1971; Tananbaum et al., 1972). Most &the luminosity of these objects has been detected in the band of a few to several tens of keV, and it arrives in regular pulses (see Figure 1). The pulse periods P of individual objects

i o

Fig. 1.

I t ,A

~ 2oo 240 280 32o ~~ 440 4~0

ti

BINS

Photon counts as a function of time accumulated in 0.096 s bins from Cen X-3 during a 100 s pass (from Schreier et al., 1972).

(typically > s) are remarkably well defined, although less so than in radio pulsars, typically to 6 or 7 figure accuracy. There is a slow change in time of the period, typically - P I P ~ 10-2-10-5, and an interesting relationship exists between b, P, and the luminosity L x of individual objects, shown in Figure 2 (e.g., Rappai~ort and Joss, 1977; Ghosh and Lamb, 1979). The fact that X-ray pulsars are members of a binary system, with a nondegenerate stellar companion that feeds gas to the neutron star, was discovered shortly after the first observations (Schreier et al., 1972). While the period P is clearly to be identified with the spin period of the neutron star, the simplest explanation for P is related to the luminosity, and to the exchange of angular momentum with the companion, via the acereted gas. The fact that the X-ray pulsar is in a binary orbit is evident from the regular Doppler shifting of its pulse period around the orbit (Figure 3 a) and in some cases even by the occultation or eclipse behind its normal star companion

R A D I A T I O N F R O M A C C R E T I N G M A G N E T I Z E D N E U T R O N STARS 327

,(3_ i

v

o

o

0 -

-2

SMC X-I

I 1

M=I.3 M e

A 0 5 5 5 + 2 6

4U 0552+50

J

�9 GX 5 0 1 - 2

I 4U 0900-4C

Cen X-5

4-U 0115 + 63

~ ~.~,,~,~-~;~,~: ~.5 . -

I - 6 .............. r X2.'! ......... I I

0 I 2 3

log Io (PL 317 ]

Fig. 2. Values offi as a function of(PL 3/7) for several X-ray pulsars. This relationship has been attributed to the interaction of an accretion disk with the dipole neutron star magnetic field (from Ghosh and Lamb,

1979).

v

250

B

~o ~ * * o ~ / \ / -,

C ,~ , , ~

~ �9 , , i ' " ' "~ !

~ �9 ~176 ~176

0

I

I I I t l MAy S MAY

�9 ~

,." _ . . . . ' : o oo ~ ~

MAY 7 MAY 8

Fig. 3. Variation of the pulse period P (here labeled T) as a function of time, due to Doppler shifting as the neutron star orbits its normal star companion. (b) X-ray intensity as a function of time, showing eclipse

by the normal star binary companion (from Schreier et al., 1972).

328 v. M~.SZARO S

(Figure 3b). The study of the details of the orbital motion (e.g., B ahcall, 1978; Rappaport and Joss, 1981) allows the determination of one of the crucial parameters of a neutron star, its mass (Figure 4). This turns out to be, until now, remarkably close to the Chandrasekhar limit ~ 1.4M o . These determinations of P, ]', and orbital parameters give information about the internal properties of neutron stars (e.g., total mass, core- crust structure, etc.) and also about more distant processes (e.g., mass exchange, interaction of magnetosphere and accretion flow, etc.).

I I

I 4 U 0 9 0 0 - 4 0 '

SMC X - t

0 2

: Cen X - 3

H e r X - t

PSR 1915+16

i 4 U I 5 5 8 - 5 2

I i 3 4 5

N e u t r o n - S t a r M a s s (M o)

Fig. 4. Neutron star masses as determined from orbital dynamics. The shaded region is the likely mass range predicted by conventional particle and many-body physics theories. From Rappaport and Joss

(1981).

Most of the information we glean from X-ray pulsars, however, is transmitted by radiation which must originate right near the surface. This is obvious from the high luminosities of many pulsars, L x < 1038 erg s - 1, which suggests efficient conversion of the gravitational energy of the accreted matter 2~/, close to the neutron star radius R N S ~ l06 cm. This is also implied by the high effective temperature kTef>~ 10 keV, which suggests a small emitting region lying on the surface (in fact, the polar cap region). The most useful information about the radiation emission region comes from accurate spectroscopic observations and pulse shape determinations. The spectrum gives information about the frequency distribution, and the pulse shape about the directionafity distribution of the emitted radiation. The continuum spectrum is generally a power law from a few to a few tens ofkeV, and drops off above that, though often not exponentially (e.g., White et al., 1983). A number of such spectra are shown in Figure 5. Spectral features are seen in some objects, such as lines due to Fe around 7 keV.

RADIATION FROM ACCRETING MAGNETIZED NEUTRON STARS 329

Fig. 5.

o o

Z 0

i

0

.-R:

I0 ~

I0-'

Io -~

10 .4

io ~

I0 -~

10 .4

10. s

Io"

iO -~

t0. ~

I0 "~

t0. s

HER X-I 1.245 3"/'.4 ........ j ....... '^iv"

N ~ Io -~

= iO'*

, = ~, ,~, , I . . . . . . . . I0"; i v

5 I0 50 I00 ENERGY (keY)

4U0900-40 283 s 56.4

- o

--f.-

I I I I ] I I I ] I I I I 1 I I


X PER 835 s 33.6 i i i I I l l l I I I I l l l l l


g o

e l I

Z 0

o ~

I0

i0 -i

10-:

io-'

I0-'

10 -I

io -I

10-21

10-31

10 -41

10. 51

io-el

CEN X-3 4.84 s 37.9 i i J i i , , , I J , i r t l r t

4- i i i i ~ 1 1 1 i 1 , 1 1


4UI223-62 700 s 36.4 J r t l J l J I i i ~ I I ~ l l

- I ' -

.+

J i i i , , H I ,

5 10 5O 100 ENERGY (keV)

4Ull45-61 292s 35.0 i i i l l l l l i I I I l l l l l

t ~

5 I0 50 I00 ENERGY (keV)

Spectra of several X-ray pulsars. The number in the top middle is the period P, and on the top right is the logarithm of L x in e r g s - i (from White et al . , 1983).

The most striking spectral features associated with X-ray pulsars are cyclotron lines (e.g., Tramper etal. , 1978; Voges etal. , 1982). This is shown in Figure 6, where an emission feature in Her X-1 around 54 keV, or equivalently an absorption feature at 38 keV, corresponds to a magnetic field ofB ~ 5 x 10 ~2, or B ~ 3.5 x 1012 G, assuming this is the ground harmonic. The possibility of a higher harmonic is also evident in the figure. Another object that shows clear evidence for cyclotron lines is 4U0115 + 63 (Wheaton et al., 1979; White et al., 1983).

330 P. MI~SZAROS

Fig. 6.

LI_I

t . j OJ t / )

t_l

t,/3 Z ,9_o C3 "1- 13_

10-2

I0-3

i0-~

10-5

10-6 10 0

, ' �9 i .... i , , . . .... T . . . , ....

HER X-I

050-8 AUGUST. 1975

$ THiS WORK I MAY 3.1976

, , I , , I H , , i , , p l , , ,

101 10 2 10 3 PHOTON ENERGY IN KEV

The spectrum of Her X- 1, showing the features between 40 and 60 keV due to the cyclotron process, leading to B -~ 4 x 1012 G (from Trtimper etaL, 1978).

The pulse shapes, which convey information about the directionality of the radiation, are usually single peaked at higher energies, and often multiple peaked at lower energies, cf. Figure 7. There is a fair amount of variation from object to object. Sometimes the pulses are asymmetric about the maximum, which, under the assumption that the pulse shape is determined in the emission region, would indicate an asymmetric emission region, or a non-dipole magnetic field. The pulse shapes, in addition, may be influenced by absorption near the magnetospheric boundary, e.g., Basko and Sunyaev (1976); McCray and Lamb (1976).

3. Basic Model and Uncertain Therein

While the basic model used to interpret pulsating X-ray sources has remained essentially unchanged since its inception (Pringle and Rees, 1972; Davidson and Ostriker, 1973; Lamb et al., 1973), it is useful to underline again the ambiguities and uncertainties involved, many of which are at the focus of recent work. In the usual picture, matter is assumed to be lost from the companion star either via a stellar wind or via slow leakage through the inner Lagrangian point (see Figure 8). At some distance, loosely labeled the Alfv6n radius, it is stopped by the magnetic pressure exerted by the accreting neutron star's magnetic field, and then it somehow threads the magnetic lines (see Figure 9).

R A D I A T I O N F R O M A C C R E T I N G M A G N E T I Z E D N E U T R O N S T A R S 331

N I--

N

g z

4UI538-52 529"s 37.0 2.0 , " ' ', ,

0

I

01, , , " ' i

I-3 keY

0 1 1 I l I I

0.0 0.5 1.0 0.5

X PER 835s 33.6 2.0~, 7-2'5 keV

I I I i

3-10 keY

co 1-3 keY {

0.0 0.5 1.0 05

X-RAY HER X-I 1.24s 37.0

o

1.5

0 I ' I I

O. 8 - 1.3 leV

0 / , t , ,

0.5-0.8 keY

0.0 0.5 I.O 0.5

PULSAR LIGHT CURVES 4UI626-67 7.6s 37.1

2"0 / ' ' 14-30' k,eV '

1.0} ~ H ~ f L ~ ~ ~ ' ~

3-14 keY

L0 ~ - J ~ - ~ - .. . . . .

0 0.7-1.9 key

1.0 - " ~ ' ~ " ~ n ~ ~ -

0 ' ' t.J0 ' 0.0 0.5 0.5

4UI I45-61 292s 35.0 2.0~,

7-25 keY

,ol 01 . . . . .

3-I0 keY

0/+ ' I-3' keV ' I

0.0 0.5 1.0 0.5

2.0

r.o

4U 1258-62 272 s 35 8 i t i I 7

7-25 keV

J 3-10 keV

I 0 _ ~ ~ j S

0 I I J

1.5-3 keV

' ' ' 0'5 000 05 I0

ARBITRARY PHASE

Fig. 7, Pulse shapes at different energy bands for several X-ray pulsars. The period in seconds is in the top middle and the logarithm of the 0.5-60 keV luminosity on the top right (from White et al., 1983).

How this occurs exactly is one of the key factors determining the structure of the emitting region, since from this point on, matter probably is channeled down more or less rigidly along the lines to the surface, where it will do most of its radiating. The transverse structure of the emitting region (that is, in the 0, ~ variables, at r ~ RNs) may thus depend crucially on the details of the latching on to field lines at R a "~ G M N s p ( R A ) 8 ~ T / B 2 ( R A ) (e.g., Lamb etal., 1973; Basko and Sunyaev, 1976;

332 P. MI~SZAROS

_ ~iR~che Lobe (o)

\\ \ Shock front

~ M o g n e t o s p h e r e

~-'\ N.

Lobe N.

sk

(b)

Fig. 8. Schematic diagram of the two mass exchange possibilities: (a) due to a wind from the companion star, or (b) due to Roche lobe overflow, through the inner Lagrangian point (from B6rner, 1980).

& B f

ALFVEN SURFACE \ I / \

, . ~ _ J DISK

Fig. 9. Interaction of an accretion disk with the magnetosphere. The matter is channeled down the field lines to the polar cap regions, where it produces the X-rays.

Ghosh and Lamb, 1979; Anzer and BOrner, 1980; Arons and Lea, 1980). It remains unclear whether the 'threading' region is small or not, by comparison with RA; this could determine whether the accretion column is a thin-walled hollow funnel, or whether it is a more or less completely filled cylinder. It is also unclear to what extent large scale instabilities are present, such as the exchange instability or the Kruskal-Schwarzschild


B

(o)

B

(c)

(b)

B

(d)

Fig. 10. Possible transverse geometries of the emission region at the polar cap. (a) Filled funnel. (b) Hollow sectional funnel. (c) Pancaked. (d) Sphaghettis.

instability. If they are inhibited, the hollow and filled funnel picture may be qualitatively correct, but departures from cylindrical symmetry could arise from an uneven contact between accretion disk and magnetosphere (Basko and Sunyaev, 1976; McCray and Lamb, 1976), so that only segments of a funnel are present, or the cross section may be non-spherical. If instabilities are predominant, blobs of matter may penetrate into the magnetosphere and land unevenly over the surface, over regions of different surface field strength (Elsner and Lamb, 1976). Alternatively, instabilities may arise even in the flow near the polar cap, breaking it up into pancakes or spaghettis (Hameury et al., 1980). The geometry and the 'filling factor' of the emitting region is therefore highly uncertain (see Figure 10). The vertical (z, or radial) structure of the emitting region is also problematic. Aside from large scale instabilities, the main uncertainty is whether plasma effects may induce a collisionless shock above the surface or not (the collision mean free path being always > AR ~ RNs ). The plasma physics of such shocks is controversial even in the absence of a strong longitudinal B-field (e.g., Zeldovich and Shakura, 1969; Alme and Wilson, 1973; McKee and Hollenbach, 1981), and is virtually unexplored in such a strong B ~ 1012 G field. It is, therefore, almost as justified at the moment to assume its existence than to assume its absence, and calculations have been performed on both assumptions (Langer and Rappaport, 1982; M6sz~ros et al., 1982; cf. w 5). In the shock case, the standoff distance is a fraction of the stellar radius, so that the emitting postshock region is a cylinder sticking out above the surface, and there is a large, if not predominant, component of emission from the sides, in a fan beam

334 v. M~SZAROS

pattern. In the absence of a shock, for L x < 1037 erg s - 1, the deceleration occurs via multiple Coulomb encounters or nuclear collisions with atmospheric particles, occurring in the denser part of the atmosphere. The deceleration path length is ,~ 30-60 g cm - 2, which for a typical atmosphere represents a Az ~ R. The atmosphere is a plane parallel section of the polar cap, which does not significantly stick out, emitting upwards in a pencil beam (see Figure 11).

B B

s

p-

tt (o) (b)

Fig. 11. Possible longitudinal geometries of the emission region. (a) Pillbox shape, arising from radiation or collisionless shock deceleration; gives a fan beam pattern, radiation being emitted sideways. (b) Plane parallel atmosphere, which does not stick out above the surface and arises if Coulomb and nuclear particle

encounters produce the deceleration. This produces a pencil beam, radiation escaping upwards.

Finally, radiation pressure is an added complication, when L x > 1037 erg s - 1, which

can both decelerate matter in a diffuse quasi-shock structure (B asko and Sunyaev, 1976; Wang and Frank, 1981) and possibly also influence the transverse 0, q~ structure of the flow. Since the scattering and absorption cross sections are strongly angle and frequency dependent in the large B-field, the pressure tensor is anisotropic and dependent on the flow velocity, because of Doppler shifts. The resonant structure of the cross sections should play an important role here.

4. Physical Processes and Polarizability in a Strongly Magnetized Plasma

The key fact in X-ray pulsar atmospheres is that the electron levels are quantized in the transverse direction, and at X-ray temperatures only the lowest few levels are populated, the fundamental of the cyclotron frequency being

heB hco~- - 11.6 B12 keY. (1)

meC


The energy of an electron is

E = (n + i + cr)hCO H + p~ /2me

= (17 + �89 + G)mec2(B/Bc) + p 2 / 2 m e . ( 2 )

Here B c = 4.414 x 1013 G is the critical magnetic field where hco H --- mcc 2, n is the principal quantum number (0, 1, 2 . . . . ), a is spin quantum number (+ �89 and Pz is unquantized. In the non-relativistic limit, the levels are equidistant (Canuto and Ventura, 1977), and, neglecting spin, one obtains the level structure of a classical oscillator. This makes it possible to derive many useful results either from a microphysical or from a classical point of view. The most important consequence of the transverse constraint on the electrons is that electromagnetic waves propagating in such a plasma have now well-defined polarization normal modes. In a coordinate system with B along z, and e +_ = (�89 (e x + iey), these are written as

e, = (e + , e _ , e x ) i . (3)

For a cold electron plasma, co >> cope, co >> (me~rap)COil, these components are

(e+_)i = 2 - 1/2(1 + K~)- 1/2 e-V-i,[Ki cos 0 + 1] ( 4 )

(ex) i = (1 + K 2 ) - 1/2 Ki sin 0 v

with i = 1 : extraordinary, i = 2: ordinary, cos 0 = K ' B, and q~the cylindrical coordinate. The ellipticity parameter K,.(CO, 0) is the ratio of axes of the polarization ellipse,

K i = (Ex/Ey) i = bo[1 + ( - 1)e(1 + b0-2)1/2], (5)

which is determined by the parameter bo,

b o ~ u 1/2 2 - 1(1 - v)- 1 sin2 0/cos 0, (6)

with

and

u = (CO/,/CO)2,

v = ( % / ~ o ) 2 ,

1 / 3 - 4 r~1/2 COp = (4~rnee2/me) 1/2 = 3.7 x . . . . 20 keV. (7)

On the zy plane, both modes describe 0rthogonal ellipses, the sign of K determining the sense of rotation, andK1 = - K 2 -1 for IbJ "> 1, it is almost linear. For Ibl ~ ( / K I N 1), almost circular (see Figure 12). The normal modes reflect the structure of the dielectric tensor, in a macroscopic sense (e.g., Ginzburg, 1970) or of the dimensionless forward scattering matrix, in a microscopic sense (e.g., Herold e ta l . , 1981).

A large role is played by the virtual electron-positron pairs (vacuum polarization) which exist in large numbers for magnetic fields not far below the critical value B c = 4.414 x 1013 G (Adler, 1971; Novick et aL, 1977). These virtual pairs contribute

336 P. M]ESZAROS

dominantly to the polarization properties of the medium, through the dielectric tensor, and modify the values of the ellipticity parameter K~, in the limit of B ~ Be, so that bo has to be replaced by (M6sz#os and Ventura, 1978, 1979; Gnedin et aL, 1978)

b = bo[1 + 35(1 - u) / (uv)] , (8)

= (45/r ) - 1 ( e 2 / h c ) ( B / B c ) 2 .

The correction factor, for B >~ 1012 G, n e <~ 1 0 2 4 c m - 3 , and co ~ I019 S - 1 by far exceeds the contribution of the (real) plasma electrons. This tends to make the modes more linearly polarized (Figure 12b), since it increases b for frequencies between the 'vacuum

I , , I I l

0.5

c~

-0.5

co,/co 2 < < 1 0.5

f 3 / . . - - q ~ . . . - . 4 ~ ~ _..L-

0 ~ 20 ~ 40 ~ 60 ~ 80 ~

>_0

0.5

-1 -1 0 ~ 20 ~ &0 ~ 60 ~ 80 ~

Fig. 12. T h e e ~ i p t i c i t y p a r a m e t e r ~ ( ~ ) = K l = ~ K ~ t ~ c h a r a ~ t e r i z i n g t h e p r ~ p a g a t i ~ n n ~ r m a ~ m ~ d e s ~ f ( a ) the cold plasma at various frequencies, without taking into account vacuum polarization; (b) the same but with vacuum polarization, at ~o = 0.54~ n and various values of the electron density, parametrized with %/o~ n. As n e decreases, the modes are increasingly dominated by the vacuum, eventually becoming

predominantly plane polarized (from M6sz~tros and Ventura, 1979).

b

R A D I A T I O N F R O M A C C R E T I N G M A G N E T I Z E D N E U T R O N S T A R S 337

feature frequency'

ha) v ~- 3n~/22(B/4 x 10 I2)-I keV, (9)

and a frequency very close to co, . Below and above these frequencies the real plasma dominates (Ventura et al., 1979; Pavlov and Shibanov, 1979). At the critical points co I ~ co v and co 2 ~ co,, a switchover of the modes occurs. The nature of this has been further investigated by Kirk (1980), Pavlov et al. (1980), Herold et al. (198l), and Soffel et al. (1983).

The importance of the polarizability of the medium extends far beyond the possibility of measuring polarization. All the radiative opacities depend on the normal modes, since the cross sections are given by expressions of the type

d a n (e' IMI e), (10)

where M is an operator and e are states of the E M wave field. This implies that the vacuum polarization, which dominates the transverse part of the dielectric tensor, greatly affects the radiative processes (M6szfiros and Ventura, 1978; Gnedin e ta l . ,

1978), via the e in Equation (10). The total and differential (coherent) scattering cross sections have been calculated as

explicit functions of co, 0, and q~ (Canuto et al., 1972; Hamada and Kanno, 1974; B6rner and M6szfiros, 1978), but are most transparent in the rotating coordinates of Equation (3) (Ventura, 1979), in which their dependence on the normal modes is explicitly written out:

{( 'eJ+' 2 leJ--I 2 } = aT 1 + ul/2) 2 + (1 - ul/2) 2 "}- lez]2 '

(11)

�9 "' e j . e y,_ 2 dao(O, O ' ) _ r2 ~ e ~ + eJ+ + _ _ + # * e l dO' I 1 + u 1/2 1 - - / , / I / 2 - z - z "

Here ax = Thomson cross section, r o = electron radius, j = 1, 2, and u = (co,/c~) 2. A plot of these is shown in Figure 13. The bremsstrahlung cross section acquires a very similar form (Nagel, 1980; cf. also Virtamo and Jauho, 1975):

( ~ ) ( Z e 2 ) ( G c ~ 3 ) ( 1 - - e - h ~ f fv j = aT \ h V t h / \ 603 /

[{ ] + 7ud2)2fg• + [ed[2gLi . X (1 lea+ 12 d#-j2 ) (12) q_ /,/1/2)2 (1

where g . and gjj are Gaunt factors. These expressions apply also in the case of strong vacuum contributions, by inserting the appropriate vacuum modes ej (M6sz/tros and Ventura, 1979). A plot of the angle averaged, vacuum modified scattering and bremsstrahlung cross sections, showing also the 'vacuum feature' of Equation (9) and the cyclotron resonance is shown in Figure 14.

338 P. Ms

Fig. 13.

1

/ /

21

10 -1

. . . . . , , \

21 "" 10 4 / t

- /

.... ~ / ~ =t/2

--------- ~Icoff =1/5

o o ibo :~oo 3'0o I &0 o 5'0 ~

\ \

\ J \ J

, J I ~ [ ; I

6'0 o 70 o 80 ~

Partial scattering cross sections au, from mode i into j, as well as the total a,. = ~ a u, for two frequencies as a function of angle (from M6sz~tros and Ventura, 1979).

I

i

E 3O.Ol h I t .

o w 10-4 (..1

Z O

F- 10-6 13.. r O (/)

10 -~

10-1c

,~0td ptQsrnn

.__;:x__ .:.-12 \ o,/.i--

" ,

, a . . . . . . . . L . . . . . . . . . i . . . . . . . . i . . . . . . . . J

.01 0 . I I I0 I00 FREQUENCY (keY)

(o)

1

:E uo.ot

O~ l 0 - 4

g ~ 10-s

m 10-e

10-1c

,,.i . . . . . . . . i . . . . . . . . i . . . . . . . . i . . . . . . , , 1

|

ptosmo +vocuum

" ' - . o2 A ~ . . . . . . . -x . . . . . . . . . . , - - . . . . . / , I \

"-, 1 ol// N,, /

.01 0 . 1 1 t0 100 FREQUENCY ( k e Y )

(b)

Fig. 14. The angle averaged coherent scattering (a) and absorption (x) opacities as a function of frequency, for e&r = 55 keY. (a) Without vacuum. (b) With vacuum, for an electron density of n e = 1022 cm - 3. Notice

vacuum feature at hco v ~ 3nJ/22( 3 x 1012) - 1 keY (from Ventura et at . , 1979).


In the neighborhood of the resonance coz_ / and its multiples, the hot vacuum plus plasma modes should be used (Kirk and M6szfiros, 1980). Incoherence and recoil effects are very important near e)~ in the scattering cross section, typical shifts for k T e < hco H being Aco ~ ogi_i(he)H/mec 2) cos 0 per non-resonant scattering (Wasserman and Salpeter, 1980). The ratio of non-resonant to resonant scattering (the former having a ~ aT ~ rg) is roughly, for X-ray photons,

aT ro roO) 10_4 (13) aRe s it c

and several non-resonant (incoherent) scatterings suffice to shift a line photon out into the neighboring continuum (Nagel, 1981b; Langer and Rappaport, 1982). 'Cyclotron', or resonance photons are chiefly produced by e-p collisions followed by radiative deexcitation, e-e collisions accounting for a smaller fraction. The former is the bremsstrahlung process at the resonance, for which Nagel and Ventura (1982) have carried out calculations using a two-level system and two polarizations, including vacuum; these have indicated an interesting resonant structure peaked at co H and at c%, [ 1 + cos O(hcoza/meC2)l/2]. Calculations of the bremsstrahlung emissivity have also been reported by Lieu (1983). Cade~ and Javornik (1981) report bremsstrahlung calculations in a white dwarf field using LTE levels. Melrose (1981) discussed a scattering- type line photon transfer equation.

The Coulomb particle collision rates are strongly dependent on B (Ventura, 1973). Calculations by Basko and Sunyaev (1975) and Pavlov and Yakovlev (1976) for the diffusion and stopping of test protons in a magnetized electron gas indicated energy loss rates smaller than in the nonmagnetic case by approximately a factor i/sin20 (pitch angle), leading to stopping lengths >~ 100 g cm -2, which makes the nuclear inelastic collision length of ~ 55 g c m - 2 a more likely mechanism for stopping protons. Quantum mechanical calculations of proton-electron and electron-electron collisions were made by Bussard (1980) and Langer (1980), who used helical wavefunctions for both particles. Kirk and Galloway (1982) performed a semiclassical calculation of the proton deceleration, using helicoidal proton paths and evaluating the medium response via the longitudinal part of the dielectric tensor. This involved a numerical calculation of the Fokker-Planck coefficients for a distribution of protons, with the electrons in the n = 0 level. This treatment gives shorter stopping lengths (30-40 g cm -2) than previous estimates. Departures from the commonly adopted Maxwellian longitudinal distribution for the electrons along the field were investigated by Langer et al. (1980), who found indications of a deficiency of electrons in the tail above hco/~. The transverse population of excited Landau levels has also been investigated by Nagel and Ventura (1982), the level structure being dominated by radiative processes.

The line-level structure of atoms, in particular, heavy elements (e.g., Fe), in a strong magnetic field has been extensively by Ruder et al. (1981) and Wunner and Ruder (1980), and references therein. The relativistic quantum aspects of the plasma properties (e.g., dielectric tensor in a magnetic field) have been discussed by Kirk (1980) and

340 p. MI~SZ,~ROS

Herold et aL (1981). A relativistic quantum treatment of the Compton scattering cross sections in a magnetic field was given by Herold (1979). It is important to underline that in the non-relativistic limit with lowest order quantum corrections, the quantum treat- ments reproduce the expressions derived from a semiclassical plasma treatment. The Positronium level structure and annihilation rate in a strong field has been investigated by Wunner et al. (1981), while the one-photon and two-photon e + -e - direct annihilation rates were discussed by Wunner (1979) and Daugherty and Bussard (1980). Pair production in a strong B has been discussed by Daugherty and Harding (1983).

One dark area where almost no work has been done is the investigation of plasma instabilities and collective effects, which could have an important effect on the particle distributions, the fast proton deceleration problem, and the formation or not of coUision- less shocks. It would be desirable also to have a more exhaustive comparison of the work of Pavlov and Yakovlev (1976), Langer (1980), and Kirk and Galloway (1982), since these have used different methods. In general, the further extension of the particle kinetic and particle-photon interaction processes to the relativistic quantum regime seems highly desirable, both for hard X-ray pulsar and 7-ray burster applications.

5. Radiative Transfer

Even for simple, homogeneous atmospheres, the direction, frequency, and polarization dependence of the cross sections makes the transfer problem difficult. The most immediate way to attack such a complicated problem, is, in principle, a Monte Carlo approach. Yahel (1979, 1980a, b) and Pravdo and Bussard (1981) have performed such calculations using a cylindrical atmosphere of given temperature and density illuminated from the base by a given input spectrum. The results confirmed suspicions that the direction and width of the beaming would be energy dependent, and gave indication of phase-dependent spectral effects. The difficulty with these calculations is that an inordinate amount of time seems to be required to get even a single case, and the physics, even if there, is not easily visualizable.

A less time-consuming and more transparent approach is to solve the integro-differential equations of radiation transfer

O.VI(O, co) = - ~:(o, ~) I (~ , co) + ~(f2, ~)Bo~ +

f do)' dO' d2~ +

J dO dco (O', co', O, co)I(O', co'), (14)

where ~: is the sum of scattering plus absorption, e is the absorption modified for stimulated emission, B~, is the source function (Planck function, for a Maxwell-Boltz- mann distribution), and da is the differential scattering cross section. A major simplification is introduced by the fact that, expressed in the normal modes, the polarization matrix is diagonal (Gnedin and Pavlov, 1974), due to the fact that between scatterings the electric vector of a wave performs many rotations and phase information is lost (Faraday depolarization). It follows that instead of four equations for four Stokes


parameters, only two equations for the normal modes need to be solved (although near the resonance other complications arise, see below). Further simplifications are possible, depending on what information one is after.

A phase-averaged spectrum is most easily obtained if one performs some kind of angular average over the equations, as is done in the diffusion approximation or in the discrete ordinate method. Under some conditions (if z 2 ( k T e / m e C 2 ) ~ 1, where z = Thomson optical depth) it is also possible to neglect incoherent scattering (Comp- tonization) at least for the continuum, which then simplifies the frequency integral in the scattering term of Equation (14).

The introduction of slab, or cylinder symmetry, and use of the two-stream approximation, or the diffusion approximation (M6szS.ros e ta l . , 1980; Nag,l, 1980; Kaminker et aL, 1981) leads to four first-order equations for I, + , or equivalently, two second-order equations for u, = (�89 (I; + + / i - ), the photon energy density being 4nu,/c,

,,,r d,,, + - K,u, + ~. Suu j + A,K; = 0, (15) L dr 2 \ dr./_J j

where A = (~(0) ) , S = ( a (0 ) ) , K = (•(0)) = A + S and ~=(0 ,1 ) , D = = ( (cosZ0/~(0)) , (sinZ0/2~:(0))) for slab-cylinder geometry, and K is the value in equilibrium. The boundary conditions (with origin in the middle) are (dui/dr)r=o,

u,(R) + 2D,(dui/dr)[ r = 0, and the flux related quantity F; = (�89 - I i - ) is

F; = D, duj.. (16) dr

For these simple geometries one can get analytic solutions for the photon density and emergent spectrum (e.g., Figure 15). Simple random walk arguments help to understand the results qualitatively (see e.g., M6szhros et al., 1980; Nagel, 1982b). In the case of weak coupling (S o. ~. Ai) , both polarizations behave approximately independently, and the radiation of polarization i is at its LTE value up to a distance

2,, ,-., (D,/A,) u2 (17)

of the edge (thermalization length). Outside of that ue starts dropping steeply as photons diffuse out, and at distances less than D, from the edge, photons escape freely (u, ~ constant), similarly to the one-polarization situation (e.g., Felten and Rees, 1972). For strong coupling, (S~ >> A,) radiation of both polarization is at LTE until a distance

A t ~'~ [ ( O 1 q- D2)/(A I + A2)] I/2 (18)

(coupled thermalization length), then both drop together until a distance

2 m ... [DID2/SI2(DI + D2)] I/2 (19)

(mixing length), where the mode of longer mean free path (i, say) becomes optically thin (u~ becomes constant, free flight), while the shorter mean free path mode j continues to drop in density until it is within Dj of the edge, where it levels off. Since S and A depend

342 P. M~SZAROS

1026

>

1024

v 1022

F-

Z I , I

I-- ~ 102 o

u i . . . . . . . . i . . . . . . . . i . , . . . . . . E

cow ptos~

/ 1 /

1 /

. . . . . . i . . . . . . . . 1 . . . . . . . . i

~ : 1 " ' l 1o lOO FREOUENCY (keV)

10 z6 F"

"~ 1024 [

v 1022 l ~ .

z hi

102~ I

. . . . . . . . i . . . . . . . . i . . . . . . . . i

/ ; 7 ', /

I / /

/

. . . . . . . . . . .,'r0 . . . . . . . . . ....

�9 I 1 I0 I00 FREQUENCY (keY)

(o) (b)

Fig. 15. Spectra using the opacities (a) and (b) of Figure 14, showing the cyclotron and vacuum features obtained for a homogeneous medium of n e = 1022 cm -3, L = 105 cm, T = 10 keV, if comptonization is neglected. The two outcoming polarization are indicated, the top curve being a blackbody (from Ventura

et al . , 1979).

1

10 -1

10 -2

X i0-3

- . J

I J_

10 -4 z c) I.-- o I0 -5

(3- 10-6

10-7

10 -B

i . . . . i . . . .

i i i i I i i i i

0 ~ 30 ~ 60 ~ 90 ~

DIRECTION

Fig. 16. Directionality of the outcoming radiation as a function of angle respect to magnetic field, from plasma slabs of various optical depths. Plasma parameters: density 10-3 g cm-3 , temperature 10 keV, cyclotron frequency at 50 keV. Photon frequency 2.5 keV. The Thomson optical depths ofthe slabs are 10 2,

10 1, . . . , 104. Each curve gives the totale flux (from Nagel, 1981a).


differently on density mad temperature, and both depend on frequency, different regimes depend on these parameters, but the same slab can be in different regimes at different frequencies. This is why it is crucial to treat two polarizations. This is particularly so in the resonance region, where the coupling S o . between modes is directly responsible for producing a marked departure from the continuum value (setting S U = 0 would give an almost smooth spectrum).

A more elaborate approach is required to get the directionality (beaming) of the radiation. Essentially, the two-stream approximation has to be extended to a multistream approach, say to n angles (0i, ... On), for every one of which there is a n / i + , I - , and two polarization. One can do this either by seting up a simultaneous system of 2n equations like (15) (Feautrier method, used by Nagel, 198 la) or by setting up a system of two coupled angular integral equations for the source functions Bi (M6sz~os and Bonazzola, 1981). An example of the outcoming beam of radiation as a function of angle is shown in Figure 16. Somewhat similar work with a different emphasis has been done by Silant'ev (1981) and Kaminker e t al. (1982). An approximate analysis of Kanno (1980) already indicated some of the qualitative behavior expected. This consists in solving for the photon density u ( z ) with, say, a simple two-stream or diffusion approach first, and then to assume that at each depth the intensity scattered in I2 is N ( c / 4 n ) u ( z ) " a ( f~ ) . Multiplying by the escape probability e - ~(a), and integrating over z (Nagel, 1981a, gives a two-polarization version of this) leads to

I ( 0 ) ~ const + [cos 0/x(0)] (20)

in optically thick situations, because the radiation emerges from a layer of effective depth d ~ cos 0/re(0). For optically thin situations, the intensity is proportional to the emissivity (i.e., to ~c(0)) and to the projected area, so

1(0) ,,~ ~(0)/cos 0. (21)

This qualitative behavior, with finer details due to absorption, mode switching, etc., is seen in the detailed numerical calculations. Even in semi-infinite media, for o9 sufficiently below ogn (usually og/ogn < 10) the opacity near 0 = 0 is so reduced that a central hollow appears in the beam, so that for low og/ogn, beams are double or even triple, but simple for higher og/ogH. Double peaks based on a reduced opacity near 0 = 0 had already been proposed by Basko and Sunyaev (1975). For slabs of finite width, calculations show that the central hollow appears at frequencies closer to co n because the slab becomes thin at frequencies less removed from o9,~/than in a semi-infinite medium, while if one introduces outside illumination, this tends to fill in the hollow (M6sz~ros and Bonazzola, 1981).

A treatment of incoherent scattering effects becomes of particular importance near con, because of the sharp frequency and angle dependence of the scattering cross section there. For electrons of a given longitudinal momentump, and a photon with initial co, 0 scattered into 0', the change in frequency is

Ao9 = A k ( p / m e ) - h ( A k ) Z / Z m e , (22)

344 P. MI~SZAROS

where Am = co' - co, Ak = h(co' cos 0' - co cos 0). Thus, to treat comptonization one needs to consider a range of angles and frequencies. One approach consists in neglecting spatial diffusion and considering only photon diffusion in frequency (Bonazzola et aL,

1979) or in both frequency and direction (Wasserman and Salpeter, 1981). Diffusion in space, angle, and frequency can be attacked either by the diffusion equation or integral equation method by introducing a discretized frequency dimension, e.g., in Equations (15) one would have 2 x n x rn equations, if we consider 01, ... 0 n angles and (.D1, . . . O) m frequencies. This also tends to become time consuming, although a calculation by Nagel (1981b) with only two angles (60 ~ 120 ~ already gives a qualitative idea of the possible resonance line structure, which is complementary to that indicated by Wasserman and Salpeter (1981), and also to related work by Bussard and Lamb (1982). This suggests that, for low densities and low optical depths, the line will appear in 'emission' (uniform temperature and density are assumed), but for larger densities/optical depths, photons will be scattered out of the core (-,~ coz_/) and diffuse to the wings (see Figure 9). For k T e ~ h o ~ , the red wing will be predominantly populated, due to the finite Compton recoil, but a small blue wing continues to remain.

There are several difficulties with the cyclotron resonance region, which await further work. Even at the level where one neglects spatial diffusion effects, it appears very desirable to seek an extension of the coupled electron population-photon distribution kinetic equations to include several levels (n > 0, 1) at the same time as departures from a longitudinal Maxwellian distribution. The desirability stems from the observation that there may be many systems where, unlike in Her X-l, the field value may not be high by comparison to k T e (e.g., Wheaton etal., 1979; Pravdo etaL, 1979; White etal.,

1983). Another difficulty which appears if one treats the spatial diffusion, aside from the need for many angles, is the fact that at the resonance, the inclusion of vacuum polarization causes the normal modes to be slightly non-orthogonal and actual departure depending on field strength and density (M~sz~os and Ventura, 1979; Soffel et al.,

1983). In this regime the four equations for four Stokes parameters would not simplify to two equations, in essence because a 'normal' mode does not remain unchanged; it has a finite coupling to the other mode, and the full set of four equations needs to be considered.

6. Self-Consistent Atmospheres

The solutions described in Section 5 were found for arbitrary, although mostly reason- able, density and temperature distributions. Finding the spatial (or temporal) dependence of p and T that satisfies energy and momentum conservation, coupled with the radiative transfer, is a further complication, again best approached piecemeal.

In the most complicated situation, where L x is not far from the Eddington limit L E = 47rGMmpC/aT = 1.3 x 1038 ( M / M o ) erg s - 1, one has to take into account radiation pressure. Since many of the X-ray pulsars first discovered had Lx ~> 1037 erg s - 1, this problem was attacked first, by simplifying to the outmost the radiative transfer, in order to concentrate on the dynamics. Davidson (1973) introduced the approximation

R A D I A T I O N F R O M A C C R E T I N G M A G N E T I Z E D N E U T R O N S T A R S 3 4 5

; , i , , . . . . . , I . . . . . . . . i

,-~ 10 ~3

1030

~10 27 Ixl

1 0 2 4

z 1021

o_

10 l0 ,~ . . . . . . . . ~ . . . . . . . . 10 100

PHOTON ENERGY (KEV)

Fig. 17. Spectrum of the escaping radiation, including incoherent scattering, for slabs ofp = 10 2 g cm 3, T = 10 keV, co w = 50 keV. The curves have, from top to bottom, Thomson optical depths of rT = 10-2, 10 - l, 1, 10 I, 10 2, 10 3, and the topmost is a blackbody curve. This shows the change from cyclotron emission to 'absorption', actually due to scattering out of the core, as the optical depth increases (from Nagel,

1981).

.a

.6

.7.

|

-I

-Z

1.5 P 1.0

P

t5

I.Q

I m l l I !

l lS/L

�9 I "~ ...... i "--'~ o.s g tO

Fig. 18. Matter deceleration due to radiation pressure, for L x = 1037 erg s - ~,M = Mo , RNS = 106 cm, and polar cap solid angle of 0 o = 0.1 radians, neglecting cyclotron resonance effects. (a) Dimensionless radiation energy density U, and dimensionless velocity ~3 = v/c against radius f = r/RNs , for three values of ~ = 0/0 o - solid: 0 = 0, dashed: 0 = 0.5, dotted: 0 = 0.9. (vb) Contours of equal plasma velocity in the r, 0 plane (full lines), and normalized radiation flux vectors (arrows), for the same parameters as (a) (from Wang and

Frank, 1981).

346 v. M]~SZ~ROS

of considering only radiation pressure (cold gas), which by neglecting all magnetic effects on the cross sections becomes isotropic. The field appears only in the constraint of 1-D motion along the funnel, but the grey transfer can be done in 2-D. This approach was greatly refined by Basko and Sunyaev (1976) who solved in this approximation the hydrodynamic and transfer equations, and investigated the topology and energetics. As is to be expected, for L x > 4 x 1036, a broad 'radiation shock' or deceleration front starts forming (see Figure 1 la) which moves away further from the surface as L x increases, reaching standoff distances larger than RNs for high Lx. The flow behind is in a slow-settling regime, and the mass and radiation density continues to increase as it approaches the surface, and v --, 0. Still keeping the cold gas, and grey transfer approach, Wang and Frank (1981) introduced a measure of difference between the parallel and perpendicular scattering cross sections, and also treated approximately the case of particle collisions at the bottom of the funnel. They present extensive numerical results on the two-dimensional structure of the shock and radiation field (see Figure 18), finding the radiation pressure dominated region to occur for L x > 7 x 1036 erg s - ~, depending somewhat on the funnel topology (hollow tunnel, interrupted circumference, etc., cf. Basko and Sunyaev, 1976, and Figure 10).

The assumption of a collisionless shock decelerating the inflow was raised for nonmagnetic neutron stars by Zeldovich and Shakura (1969), Alme and Wilson (1973), and Shapiro and Salpeter (1975). The consequences of extending this assumption to a strongly magnetized neutron star have been investigated by Langer and Rappaport (1982) (see Figure 1 la). They took also the (adiabatic) postshock electron temperature different from the proton temperature, avoiding thus the large 7-ray flux ( ~ �88 of total L) expected if right behind the shock T e = Tp, e.g., Shapiro and Salpeter (1975). Behind the shock the two-fluid hydrodynamic equations were solved, using fully magnetized Coulomb friction between components and fully magnetized polarization averaged cross sections for the radiation, which was assumed to escape freely, transfer effects being neglected. The accretion rates chosen were small enough ()Q < 1016 g s - 1) that radiation pressure can be neglected, also making the transverse Thomson optical depth < 1. They estimate a cyclotron photon escape probability

ee ~ exp(- ~e,), (23)

where Pt is the thermalization probability (ratio of resonant absorption to resonant scattering, cf. Equations (11) and (12), averaged over a thermal distribution) and N is the average number of scatterings before the photons leave the column. Depending on ne, B, Te, one has P t ~ 10-8-10-lO, while for resonant scatterings ~; ~ ~2,s ,,~ 108. However, nonresonant scatterings have a lower cross section by ~10 -4 (Equation (13)). Several nonresonant scatterings would suffice to shift a photon out of the resonance to the continuum, for which the column is almost transparent (ZT < 1). Thus, N ~ (ar~s/aT) X f e w ~ 10-3-10 -4, giving P~ -~ 1. As a consequence, cyclotron photons represent most of the energy loss in this calculation, and the spectrum is a superposition of the broadened resonant emission coming from different parts of the flow. The shock standoff distance is given by the proton-electron energy exchange time,

R A D I A T I O N F R O M A C C R E T 1 N G M A G N E T I Z E D N E U T R O N STARS 347

since the proton energy has to be radiated by the electrons. This time is tp e ~ p- 1 ~ M - 1, so the lower accretion rate X-ray pulsars have larger standoff distances, over which B ~ B N s ( r / r N s ) - 3 varies significantly, resulting in a continuous- looking spectrum, of average slope in energy ~ 0 and dropping off at the cyclotron frequency of the surface field BNs. The low energy end is expected to vary strongly with M for M < 1015. For 21;/~> 10 is s -1, by contrast, a broadened hump spectrum is produced, due mostly to the surface field (see Figure 19).

It is, from our present knowledge, quite possible also that the instabilities giving rise to collisionless shocks are inhibited in the strong longitudinal magnetic field. Decelera- tion would then occur in the denser part of the atmosphere, by particle encounters, either multiple Coulomb collisions between beam protons and atmospheric electrons, or nuclear p-p collisions, e.g., Basko and Sunyaev (1975b), Wang and Frank (1981), (see Figure 1 lb). The calculations of Basko and Sunyaev (1974) and Pavlov and Yakovlev (1976) indicated a Coulomb stopping length ~> 100 g cm-2, but the stopping should be limited to the lower value of 55 g cm- 2 due to the p-p inelastic nuclear collisions. The earlier radiative transfer also took a rather simplified consideration of magnetic aniso- tropies, although not of polarization and frequency dependence, A fairly complete calculation is that by M6szfiros et al. (1983) and Harding et al. (1983), who take account of the angle, frequency, polarization, and temPerature dependence of the Coulomb and radiative cross sections. The Coulomb deceleration calculation used follows Kirk and Galloway's (1982) method, leading to Coulomb stopping lengths of 30-50 g cm-2, so the effect of nuclear collisions was also included. The plane parallel atmosphere was

I0

1.0

~o.

0.0

L I 4

SOURCE SPECTRA

B~=5 xlO12 G

11

t f '111

/ \

l

1 I

I

E N E RGY/1~ w..~

Fig. 19. Dimensionless spectrum of collisionless shock accretion, neglecting radiative transfer effects, for a surface field strength B, = 5 x 1012 G, the photon energy bieng in units of the surface cyclotron frequency he). At lower accretion rates the shock stands off farther, and the spectrum is a composite of the resonance

line photons at different field values sampled below the shock (from Langer and Rappaport, 1982).

348 P. M]~SZ~.ROS

~ 1 n 3~

E i f ) Z o 1 029 o

1028 B = 5 x 1012G ~

1031

1 0 2 7 , i i , i i 1 ~ 1 i i 1111111 i

0.1 1 10 E (keV)

, , , , , , , q , , , , ~

IV1 = 10 ~sg s "1 P e s c = 1

B = 1013G / \

T

l l l l l l _ ~ l i~11

102 103

Fig. 20. Photon number spectrum for an atmosphere heated by Coulomb deceleration, M = 1M o, R = 1.19 • 106 cm, ]1~ = 1015 g c m - 1, area = 0.77 x 101~ cm 2, B = 5 x 1012 G and 1013 G. The coherent scattering approximation is used for the continuum, and cyclotron line effects are treated approximately. The arrow gives the cyclotron frequency which together with the crossbar gives an estimate of the number of line photons. The actual line profile is not computed, but should appear absorbed, core photons being

mostly redistributed into the red wing (from M6sz~tros et al., 1982).

divided into 20 to 30 slabs and a beam of protons of given initial energy was followed as it deposited energy and momentum. The radiative transfer, as far as energetics was concerned, used a two-stream scheme solved slab by slab, with a perfectly reflecting boundary condition at the bottom, taken to be where the protons stopped. The atmosphere structure in energy and momentum balance was found by iterations, using a relaxation technique, and values of )1~/< 1017g s-1 were used to avoid excessive radiation pressure. The radiative processes used were the inverse Compton cooling and bremsstrahlung, including approximately the effects of the resonant line photon production and diffusion into the continuum. The outer temperatures are typically 20-40 keV, decreasing inward in a sort of step to ,~ 7-10 keV, depending on field strength and _~/, with density increasing as a power law inwards up to 1022-1024 cm- 3. Having found the equilibrium atmosphere structure, the directional transfer scheme for finite slabs of M6sz~os and Bonazzola (1981) was used to calculate the beaming properties of the radiation, the pulse shapes as a function of viewing angle, and phase dependent spectral indices. The spectra are power laws of approximately the observed exponent (< - 1), dropping off at o9 > 09/4, for hco/4 > (kTe). The 'vacuum' feature can be seen in individual polarizations, but not in the polarization-summed spectrum, being spread out by the density gradient. The spectra are somewhat sensitive to 3~/and y (stopping length), and are most sensitive to B, higher values leading to less efficient cooling, hotter atmospheres,


60/45

~ 5 x 1 0 1 2 G

1013 G

0 1 PHASE

30/30

/vL <

PHASE

Fig. 21. Normalized pulse shapes for the pencil beam emerging from the atmosphere of Figure 12. Left panels are for viewing angles 60/45 (B to spin axis-line of sight to spin axis), and right panels for 30/30. Top panels are B = 5 x 1012 G, bottom is B = 1013 G; the photon energies are 18, 10, and 3 keV from top to bottom. This shows the splitting up of the pulses from single into multiple (from M~sz~tros et aL, 1982).

and harder spectra (see Figure 20). The line photon production acts as a damper, however, because of its sensitive exp(-hco~/kTe) dependence, so the temperature changes are not large. The pulse shapes are often single at high frequencies (~o > co~t/4) and double below that, the central hollow becoming deeper with decreasing co (see Figure 13), a feature seen in many X-ray pulsars. The hardness ratio varies with phase, this being most rapid at phase 0, which is also an observed effect.

As is evident, all the calculations of self-consistent atmospheres done until now have been made using various degrees of simplification. In the plane-parallel, pencil beam calculations of the type of M6szfiros et al. (1983), it would be desirable to introduce an inclination of the magnetic field to the normal of the surface, since that is one of the ways of obtaining asymmetric pulse shapes (Kaminker et aL, 1982). Also, the inclusion of comptonization explicitly in calculating the spectrum would be of use, but some simplification is required if a stable solution requires many iterations. In the shock calculations of the type of Langer and Rappaport (1982), a radiative transfer calculation would be of great interest, in order to get more accurate spectra and to obtain pulse shapes. These latter, in particular, when they become available, may be very instrumental in distin- guishing between shock and Coulomb models. A comparison of the Coulomb pulse shapes with observations of symmetric pulsed objects is possible, after extension of the calculations to a larger region &parameter space, especially if something is known about the field strength and viewing angle. Some comparisons using various semi-empirical geometries have been reported by Wang and Welter (1981). In the radiation shock calculations of the type of B asko and S unyaev (1976) and Wang and Frank (1981) the greatest need is for an inclusion of a non-isotropic radiation pressure, including resonant

350 P, MI~SzARos

effects with incoherent scattering and allowance for bulk Doppler shifts. A finite gas

temperature would be necessary to calculate the production of photons, especially if

cyclotron line photons dominate, as suggested by the lower luminosity calculations of

Langer and Rappapor t (1982); in this case, a non-grey transfer calculation in 2-D may

be necessary.

In conclusion, much has been done, but we are still only halfway there to our present

goal of understanding what the exact geometry is, how the spectrum arises, and what

the nature of the pulse formation is. There are certainly subclasses of X-ray pulsars,

which the present highly idealized models have not yet tackled; we are still at the stage

of determining the gross features of the most commonly applicable model, or models,

and this may be achieved within the next few years. Much further theoretical work will

be needed for that, as well as a systematization and extension of our observational

knowledge.

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Radiation from accreting magnetized neutron stars