Radiophysics and Quantum Electronics, Vol. 40, NoB. 1-2, 1997

GAMMA-RAY FROM

SPECTRAL

PHOTON

FEATURES

SPLITTING

OF NEUTRON

IN A STRONG

STARS

MAGNETIC

ORIGINATING

FIELD

E. V . D e r i s h e v , V . V. K o c h a r o v s k y , a n d VI. V . K o c h a r o v s k y

UDC 524.354

The spectral evolution o/the soft (~ 1 Me V) gamma-ray radiation of neutron stars in a strong magnetic field (B ~ 6-1012 (7) is analyzed. It is shown that the radiation transfer equation for the photon splitting cascade has a one-parameter set of self similar solutions whose integral expansion is an efficient method for study of the general solution. An arbitrary initial spectrum converges quickly to a self similar solution provided that most o/the radiation energy is concentrated in the hard spectral range. We consider the possible observational consequences o/the photon splitting, including the polarization and softening o/the output spectrum as well as the occurrence of a spectral break and condensation of all hard-energy radiation near that break.

1. I N T R O D U C T I O N

At present, the observed X-ray and gamma-ray radiation of neutron stars is related, as a rule, to the existence of a hot electron-positron plasma under extreme physical conditions over the surface of those stars. The formation of the spectrum of this radiation as it escapes from the neutron star magnetosphere is of more general, not only astrophysical, interest, since in many cases this process is accompanied by exotic quantum-electrodynamical effects which are generally insignificant in the astrophysics of other stars or in laboratory physics. These effects are manifested most clearly for photons with energies comparable to the ener~D, of an electron at rest, i.e., about 50 keV or greater. In this paper, we consider an effect which has not been observed yet - - the splitting of a photon in a strong magnetic field, 7 "-* 7 + "Y [1-3]. We show that independently of the magnetosphere model and the specific mechanism of radiation, the photon splitting can be responsible for the formation of a peculiar spectrum in the absence of plasma effects. The possible identification of this spectrum with the observed radiation spectrum of neutron stars in the range of 50 k e V - 1 MeV could be indirect evidence that the photon splitting exists in vacuum. Specifically, we mean the observations of X-ray and gamma-ray pulsars and the so-called SGB. (soft gamma-ray repeaters); see, e.e., [4-6]. Of course, splitting is possible in vacuum because in the presence of a magnetic field a vacuum is a nonlinear birefringence medium which owes its extraordinary properties to the virtual electron-positron pairs. In such a medium, the 7-quanta, besides the usual two-photon processes of production and annihilation of electron-positron pairs, are also subjected to one-photon processes, namely, photon splitting (due to the mentioned nonlinearity of the Maxwell equations in a strong magnetic field) and one-photon pair production 7 -* e + + e - (due to transmission of excess transverse m o m e n t u m to the magnetic field). A characteristic scale in such processes is the critical field Be, ~ 4.4. l0 is G in which the energy of a cyclotron radiation quantum is equal to the rest energy of electrons. The existence of such strong magnetic fields at the surfaces of neutron stars is generally recognized at present. In some cases, the measurements of the deceleration velocity of pulsars due to the radiation of a rotating magnetic dipole give magnetic fields over 1014 G. The presence of stars with fields B ~, 1012-10 is G is proved by identifying the observed spectral features with the cyclotron lines.

Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenil, Radiofizika, Vol. 40, Nos. 1-2, pp. 146-160, January-February, 1997. Original article submitted December 27, 1996. 0033-8443/9T/4001-0093518.00 ~1997 Plenum Publishing Corporation

93

The observed spectrum of hard X-ray and gamma-ray radiation is formed by a competition between the above three processes of the destruction of high-energy photons (7 --* T + "~, 7 -~ e+ § e-, and 7 + 7 -~ e+ + e-), which, as a rule, are supplemented by the comptonization process (7 + e+ ~ -'* ~/+ e• ." All these processes generally lead to the softening of the energy of output photons as compared with the initialradiation, and the typical cross-sectionsof pair production in the allowed range of energies exceeds by many times the cross-section of the photon splitting. The particular role of the energy softening is connected with the nonthreshold (in energy) nature and actual irreversibilityof that process. At the same time, the pair production imposes a lower limit on the energy of the photons participating in this process and is accompanied by the reverse process of annihilation in actual practice. The share of photons with energy h~ exceeding the rest energy of the electron-positron pair 2rnec2 m a y eventually become so small that a determining role in the further evolution of the radiation spectrum will be played by the relatively weak photon splittingaccompanied, possibly,by the comptonization process. It is exactly this situation,related to energies of from I to 0.05 MeV, that is of interest to us in this paper. The radiation intensity of the electron-positron plasma directly in this range of energies can be relativelysmall and, therefore,insignificantas compared to the hard radiation intensity with energy 1 M e V transformed to the sub-MeV interval.If the magnetic fieldof a neutron star exceeds (3-5) .1012 G, then the leading role in this transformation is played by the absorption of quanta due to the one-photon production of pairs and their subsequent annihilation,while the photon splittingin not significantin this case. 2. S I N G L E S P L I T T I N G

Consider in more detail the single event of a photon decay [7]. According to Farry's theorem, the contributions of the diagrams containing closed loops with an even number of external ends are counterbalanced exactly due to symmetry of the QED with respect to the charge conjugation. The four-peak diagram for splitting also gives a zero amplitude in the coUinear approximation which is fulfilled well in the magnetosphere of a typical neutron star. Thus, the photon splitting process can be described by six-peak diagrams in the least nonvanlshing order. For simplicity's sake, we will consider only the case eB ~ 0.4B~ and e < 2, e - h~/mec 2, where the photon decay is described by simple analytical formulas and the one-photon pair production is excluded (the case eB ~ B e was condidered, for example, in [3, 8]). We emphasize that taking account of the birefringence and dispersion is customary, despite the smAll-ess of these effects. It is well known that natural waves in a magnetized vacuum have linear polarization and refractiveindices [9, 7] 7 a B2 nil = I + 90 7r B--~sin2O

and

2 a B2 nj. = 1 + 45 ~rB--~rsin2 |

(1)

where a ~ 1/137 is the fine-structureconstant, | is the angle between the vectors/~ and k, and the indices J. and l]are related, correspondingly, to the waves with the electricfieldvector perpendicular and parallel to the plane in which the vectors B and k lie.Formulas (1) are valid for eB ~ Bcr sin| For estimates, we will take sin | = 1 without assuming the particular role of the photons propagating at small angles to the magnetic field. Since the photons do not have electric charge, the energy and momentum must be equal in the initial and final states, i.e., ~ = ~' +~", ~= ~+~', (2) where the wave vectors ~, ~' and frequencies~', w" are related to the produced photons. It is seen from (I) that nll > nj. and, therefore, the decays [i--*J.+ .i. and i[--*.i.+ H are forbidden by the conservation laws (2). O n account of the dispersion,the refractiveindices appear to be growing functions of velocity,so that the processes J.--,.l_+ • and []--,[]+ I[are also forbidden, i.e., photons with [[-polarizationdo not split at all [1, 7]. "The roleof the standard processesof photon productiondue to bremsstzahlung,synchrotron,or annihilationradiationof the electron-posltronplasma is assumed to be known and isnot discussedhere. 94

Of the two remaining processes, only the decay • + H has a noticeable amplitude in the weak dispersion approximation, and its total cross-section is given by the formula [1, 2]

o3

)6

Ca)

where ~c = h/mec is the Compton wavelength of the electron and r = ha~/mec2 is the normalized energy of the initial photon. In a strong field B ~ Bcr expression (3) contains an additional factor M(B) [2], which is completely inessential to this reasoning and will be omitted for simplicity. By the order of magnitude, the decay .L--..L + II is a(B/Bcr) 2 times weaker. The differential cross-section of splitting of a _k-photon into II-photons with energies f~ and (g - f~) is equal to [1, 2]

O'sp(E);

6r(61

= O'sp(e ).

(4)

Hereafter, f], as e, denotes the photon energy normalized to the rest energy of an electron. Expressions (3) and (4) are valid under the condition of smallness of the eight-peak diagrams, i.e., under the condition r ~ B~. The small difference in the refractive indices nj. and nlj leads to a slightly incomplete coMnearity between the wave vectors of the initial and newly produced photons. However, the coMnear appro~dmation is quite reasonable in this case. For example, for B = B~, sin | = 1, and equal energies of the produced photons, their wave vectors make an angle of only about 1 ~ with the initial direction. Strictly speaking, if one of the produced photons has energy much smaller than the energy of the initial electron, then the colllnear approximation is disturbed so strongly that a parametric decay into an energetic photon propagating in the same direction and a counterpropagating optical photon with energy 12 ~ e (see, e.g., [10]) becomes possible. However, this effect is not significant in the considered evolution of the X-ray and soft g~mm~-ray spectra. In the absence of a mechanism changing the polarization of radiation, the photons will undergo splitting only one time. In the initial (nonpolarized) radiation, only high-energy photons with transverse polarization can be decomposed with an excess of softer longitudinally polarized quanta. The latter can be subject to splitting only to the extent that this is allowed by the geometry of curved magnetic field lines. Only in the dipole field of a neutron star will the angle between the polarization vector and the field line direction change along the line of sight; therefore, multiple decays of photons are conceivable. However, in a regular magnetic field (with the scale of line direction variation comparable to or greater than the scale of field intensity variation) the probability of such a "cascade= splitting decreases abruptly with increasing number of steps in the cascade, so that the single splitting approY~mation is quite su~cient in many cases. Under such conditions, the single splitting of photons has at least three observed consequences. Firstly, it leads to the general softening of the spectrum. Secondly, it smooths out all spectral signatures, since each line Ucasts a shadow" in the region of low energies in the form of a bell-shaped spectrum (4). Thirdly, the polarization selectivity of splitting leads to the fact that the "/-ray radiation appears polarized in the plane of the magnetic field and the beam. The degree of polarization reaches 100~ for the hardest part of the spectrum and dlmlnlshes to zero in the smaller energy direction if the initial radiation is nonpolarized. In the spectra of young pulsars with a strong noncoaxial magnetic field, we must observe, during one pulse of radiation, a peculiar "pulsation" from hardest when the angle between the magnetic field axis and the line of sight is smallest to softest when this angle is maximal. Up to this time, we have spoken exclusively of the photon splitting while its main rivals, the one- and two-photon pair production processes, were neglected. In the case of one-photon pair production, this is reasonable for the subthreshold values of the 7-ray quanta: < 2rn, c2/sin |

(5)

It foUows from expression (3) that the optical depth to splitting for the near-threshold photons propagating across the lines of force exceeds unity. In this case, the surface value of the field B ~ 6 9 1012 G. The 95

asymptotic formula for the cross-section of the one-photon pair production ~pp can be found in [11]; the ratio ~pp/r depends solely on e sin | and B; for the magnetic fields of interest to us, B ~ 6 9 101: G, this ratio is much greater than unity for a considerable excess of e sin | over the threshold value 2. Hence, in the region where the one-photon pair production is possible this process dominates in the absorption of energetic photons and leads to breaking of the spectrum if hw > 2m~cZ/sin | The case is different for the two-photon process whose rate is determined not only by the crosssection [12], 0"2pp :

~0" T

\mec2/

- 1

(6)

(~T ~ 6"10 -2s cm 2 is the Thomson cross-section a n d / ~ are the photon energies in the inertia center system), but also by the density of the approaching photons (with energy ~ > mec2). For a typical neutron star, this fact limits the luminosity in the range/k~ > 0.5 MeV:

L < Lpp ~ 1OZSerg/s.

(7)

For this luminosity, the optical depth to two-photon processes is smaller than unity. Simultaneously, inequality (7) limits the equilibrium density of the electron-positron plasma: ne <: 10 is cm -s. This limit means exactly that the optical depth to Thomson scattering is smaller than unity. Thus, the two-photon processes and the Thomson scattering can be neglected. In fields exceeding 1013 G, the limit (7) is not very important, since photons with energies greater than mcc2 split rapidly. 3. SELF-SIMILAR. SOLUTION As was mentioned above, the single splitting approximation applies only if the radiation can escape freely from the interaction region without undergoing noticeable polarization in the scattering by plasma partides. However, in the actual physical situation the electron density can appear greater than the boundary value ne ~ 10 Is cm -3, and the magnetosphere will become opaque for radiation. The Thomson scattering depolarizes the photons, making multiple scattering possible. (For simplicity's sake, we ignore the possibility of effective cyclotron scattering and the respective depolarization.) Generally speaking, the observed radiation spectrum will be determined by two simultaneous processes: comptonlzation and splitting of photons. In principle, comptonization has the length l/co,r%, so that for very small photon energies comptonization is faster than splitting process (whose inverse length is given by expression (3)). At the same time, for reasonably small energies e ~ 0.5 and fields B ~ 2-1013 G the opposite situation is possible in which comptonization is not as sigai~cant even if ne "~ 10 is cm -s. In this case, we can simplify the problem and write the transfer equation only with allowance for the photon splitting and negIecting the polarization dependence of the latter (Fsp and F(fl', fl) are the splitting cross-sections averaged with respect to polarization):

~F(~'~) "- --Orsp(~'~)F(~'~)"l" 2/n Q~F(fl')~(fl',fl)dfl'.

(8)

Here, we used the coUinear approximation, 0/01 means the derivative along the beam coordinate, and F(fl) is the spectral density of the quantum number flux. In reality, the photon trajectory represents a broken line, a/a93must be understood as a derivative along that trajectory, and F(fl) shows the probability density that the photon has energy f]. The number 2 before the integral term is due to the fact that the new photon with frequency fl is produced in two ways: fl ~ --, (fl) + (fl ~ - fl) and CI~ --* (fl~ - fl) + (G). Nonlinear processes such as the induced splitting and the photon convergence were neglected, because they require extremely high radiation intensities comparable with the equilibrium thermal values (with the effective temperature defined by the mean energy of the gamma-ray quanta). As is shown below, Eq. (8) has a one-parameter family of self-similar solutions which are convenient for the integral representation of general spectra. Each se]f-slmilar solution is a natural spectrum in the photon splitting problem in the sense in which a black-body spectrum is natural for the comptonlzation 96

problem in an equilibrium plasma. Study of the properties of these solutions is of particular importance in understanding the physical evolution of the photon spectra in a strong magnetic field. Introducing the coordinate-dependent frequency scale a(l) and denoting z = f~/a(l), we rewrite Eq. (8) in the form 0 az ) = -/~aSzSF(az) + 60# a5 f=o~ F(ax')z2( z' z) 2dz', (9) -~lFC where

( 13

i (BsinO

,(B, O ) = 30r2 ~3-q-~/ ~ _ Bor / ; for definiteness, we assume that the polarization is complete and instantaneous. The solution of this equation can be sought in the serf-similar form F(az) = a(t)A(~) --- A(t)A (n/a(t)), which gives

1 L[-~'f1(Z) At -- ~Z~(X)] at "- --~,5fl(X)+6ox2fz~176

x)2d2~1.

~a 5

Here, we used the derivatives A ~ = dA/dl, a' = da/dl, the conservation condition of the total energy flux:

f n

(n)d. -

and f~ = dfl/dZ.

f

(10)

The terms A' and a t are related by

- r

(11)

The function f l ( z ) was introduced in a fashion such that it does not depend explicitly on l; thus, it follows from relationship (11) that At/A= -2aria. As a result, the variables in Eq. (10) are split:

ata [2fl(Z) Jr zfICz)] = --ZSflCz) Jr 60z l~fz ~ fl(Zt)(z t - z)Zdz t. /'~a

(12)

The choice of a particular value of the splitting constant is physically meaningless and leads only to a change in the scale of the function fl(z). For simplicity, we assume that this constant is equal to unity. Then, 1

a'

---i,a~ =

1

~

2 f l ( Z ) Jr zfI(z) = --zSfl(z) Jr 60z' !

where the quantity

~" =_ f IJdl playes

a(O)

a(O = {/1 + S~-as(o)'

I"

/ l ( z t ) ( z t --

z)Idzt,

(13a) (136)

the role of the optical depth to splitting along the photon trajectory

0

and actually is independent of the angle | under the conditions Of multiple scattering. To prove the selfsimilarity of the solution of Eq. (13b), we write this solution for the function fN(Y) = fl (Ny) (i. e., we make the substitution z --* Ny): 1

~

[2fN(~) + yf/~(y)] = -ysIM(y) + 60~'

IM(~t)(y' - y)~a~'.

It is easily seen that exactly the same equation is obtained by choice of the splitting constant 1/N s in Eq. (12) and substitution of the scale factor in Eq. (13a): a ~ Na (the law of transformation of the scale factor along the beam trajectory remains the same). Thus,

fN ( N~a~l)) = f N ( y ) - fl(NZ)= f~ ( N-~l)) .

(14)

To summarize, we state that if the initial spectrum for I = 0 is given in the form

F(fl) = A(O)fl (f//a(O)),

(15) 97

Fig. I. then its further evolution is governed by the self-slmiIar law

= .,,.(o)ro(o)]' La(/)J

fl (fZ/a(l)),

(16)

where the dependence a(1) is taken from relationship (13a). Putting it differently, the spectrum of form (15) compresses in energy and increases in amplitude, remalni~g the same in form. Moreover, it appeared that an arbitrary initial spectrum satisfying the condition of concentration of most energy in the region where the optical depth to splitting exceeds unity converges rapidly (asymptotical]y for r ~ 2) to the self-slmilax solution (16). The numerical example in Fig. 1 shows the result of the evolution of t w o different spectra, the delta-correlated function 5(w/wo - 1) and the =step" O(w/wo - 1)8(1.5 - w/wo), which axe identical as to the total energy flux. The frequency is normalized to the arbitrary value w0, and the optical depth to splitting at the frequency w/wo = 1 is equal to 2.3. Consider in more detail the form of the function .fl(z) o b t a ~ e d from Eq. (13b). The numerical solution normalized so that max(fl(z)) = 1 is presented in Fig.2. The asymptotic forms of this solution for = - , 0 and z --* co can be easily found analytically. The behavior of the solution for z -* co follows from the solutions of the same equation without the integral term: .fl(z) = Cz -2 e x p ( - z S / 5 ) ; the value C ~ 1.5 is obtained from the n~merical count. The estimation of the integral t e r ~ gives 1 2 0 z - l ~ which proves the rapid asymptotic convergence. At the zero, the function fl(z) c a n be expanded into a +oo

Laurent series: f l ( z ) =

~

C,=z'~. By direct substitution of the expansion into Eq. (13b) it can be easily

verified that the first two nonzero coefficients in the expansion axe C-2 and C~. By a numerical count we found C-2 = 0 and C2 ~ 10. This conclusion is physically apparent. The self-similax spectrum is formed by multiple superposition of the spectra of form (4) and, therefore, must have the s a m e asymptotic behavior at the zero as the differential splitting r .fl(z) or z2. We also note that t h e function f l ( z ) has a unique m a x i m u m at point = ~. 0.7 at half-peak intensity width ~ 0.9 The form of the se]t'-similax spectrum (15) is too specific to be considered a g o o d a p p r o ~ m a t i o n for 98

Fig. 2. actual radiation sources. Hence, in general, the spectr-m will represent the superposition of the functions fl(fl/a) with differentscale factors a and amplitude factors A(a): oo

0

relation to the function A(a), this equation is a ~ e d h o l m integral equation of the first kind which is reduced, by the substitutions f / ~ e", a --. e'l and A(a) ~ B(a)/a, to the contracted equation, which, in turn, is transformed formally to an algebraic one by a Fourier transform (only if this equation exists for the function F(f~) in the new variables). In practice, one has to seek the solution numerically; moreover, the problem of solving the equation of the firstkind is ill-posedand requires that specialmeasures be undertaken to reach the convergence. Fortunately, there is an important and very representativecase of the power-law spectrum F(f/) oc f~-'~ in which the solution can be easily found in explicitform. In this case, the amplitude A(a) is also a power function but with exponent smaller by unity: A(a) ~ a -n-1 . For lack of space, we give only the main results of the analysis of the power-law spectrum evolution without proofs. First of all,we note that, despite the initialform of the function A(a) in the expansion (17), the scale factor cannot exceed

= (k5

(18)

at any finitedistance from the source. If the initialspectrum is obeyed by the power law, then the resultant deformation of the "spectrum" A(a) will depend on the exponent:

a-n-1 =

A(l,a) (i_5/o, 99

For F(f/) oc f/-n and n > 7 the function /I([,a) decreases monotonically to zero as it approaches arnax in the variable a (A(a) - 0 if a > amax)- For hard (not as steep) spectra with n < 7 the function A(I, a) has an integrable power-law singularityin the vicinity of a ~ x with width smaller than that of the self-similar spectrum. In the latter case, the function F(f/, l) can be represented as the superposition of the power-law spectrum cut at the frequency ~c ~ 0.Tamax and the seLf-similarsolution (16). Such a spectrum has a singularity in the form of a flatteningor even an elevation in the energy region ~ f~c and retains its shape with change in the self-similarparameter a m a x when it evolves. From an observer's point of view, f~c is a characteristicenergy above which the radiation is disturbed. It is exactly this flattening, followed by an abrupt break in the spectrum, that must be sought, in the first place, in the hard radiation spectra of neutron stars, for example, X-ray pulsars. Recall that the function F(~) is the spectral density of the photon number flux and is related to the spectral intensity by the relationship I(w) = l~uj_F(u~). 4. I N F L U E N C E O F O P A Q U E M A G N E T O S P H E R E

The problem of applicabilityof these results,in particular, of the self-similarsolution in actual astrophysical situations, merits more detailed consideration. One refinement consists in the fact that the depolarization due to Thomson scattering assumes the diffusionalmotion of photons, so that the effective l

optical depth ~" -- f/~dI depends on the escape time of the photon te,c. The further qualitative analysis 0

requires only a rough estimate of this dependence: r 0( te,c (if the trajectoriesare very complicated). A detailed consideration of the photon diffusionproblem can be found in [13]. Following the same recipe, we represent solution (17) as

o o

where the function p(tcsc) is the escape time distribution of the photons. To determine how strongly the diffusioneffect will change the observed spectrum, we must rescalep(tesc)into P(f~c), using expression (18). In the calculation of re,c, we must trace the photon trajectoriesin the opposite direction through all the decay events in which it participated. The function P(f/c) normalized to unity is presented in Fig. 3 (the frequency is given in conventional units, and the point ('/c= 1 corresponds to the m a x i m u m of the function p(t~,c)). Due to the weak dependence of fie on te,c (f/co( t-~,1~ Is if T OC re,c), the width of the function p(flc) at the may;muiR appears considerably smaller than the width of f1(ft/a). Thus, it can be stated that the ~ion spread of photons changes only marginally the spectrum calculated with neglect of that spread. Taking account of the comptonization is more important. There are at least two cases where the comptonization is small compared to the photon splitting. The Erst case is a strongly nonequillbrium turbulent magnetic field in which the turbulent energy exceeds by m a n y orders the energy of the plasma which maintains the turbulence. The second case is a plasma of specific composition, in which the main contribution to the opacity is from heavy charged particles,for example, protons. However, both cases are so exotic that they can be considered only as theoreticallypossible. If the main contribution to the opacity is given not by strongly relativisticelectrons and positrons, then the splittingis only insignJAcantlystronger than the comptonization and only in a narrow window of

parameters: ~r,~2-5,

i.e., n,,,,(2-5)-10 Iscm -3,

and

B~2.10

Iso.

(21)

In any case, at the low-frequency limit f/<< f/c the form of the spectrum is determined solelyby the emission and comptonization processes, since the black-body asymptotic form F(f~) c( f/, f~ --* 0 outweighs the selfsimilar asymptotic form F(N) (x f/2. Conversely, the splittingdominates in the region f~ ~ f/c (for B = Bcr and ~ c ~ 90 keV this statement is valid at least until the densities n, ,,,1023 c m -s are reached). Thus, the characteristicfeature of a self-similarspectrum -- the abrupt breaking for N > f/,: (oc exp(-f~s/5aSmax) versus exp(-f~) for the thermal spectrum) ~ willremain observable there. In hard power-law spectra f/-'~ with n
Fig. 3. 5. CONCLUSION According to the above, the photon splitting effect in a strong magnetic field may have several observable consequences for the 7-radiation of neutron stars. In an optically thin magnetosphere with a regular magnetic field the single splitting appro~rimation applies. This results in polarization of the photons (the electric vector lies in the plane of the magnetic field and the line of sight), and this polarization reaches 100% in the hardest part of the spectrum. Determining the spectral boundary at which the polarization degree ~, 50% by observations with polarization-sensitive "/-ray detetors, we can give an independent estimate of the magnetic field strength of a neutron star (assnmln 5 that the initial radiation is not polarized): B ~ mec2/tu~so x 1013 O. Moreover, the photon splitting must lead to the general softening of the gamma-ray spectrum and blurring of spectral lines, for example, cyclotron features. In young pulsars, in which the rotation and magnetic field axes are not "setthd" yet, the "spectrum pulsation" effect must also be observed throughout each period. Obviously, the X-ray pulsar GXI+4 [6] can be considered an examph of such an object, although other explanations of its unusual brightness curve are also plausibh. : In the optically thick magnetosphere, the photon splitting competes with the comptonization; thus, the radiation transfer p r o b h m must be solved in a self-consistent manner in general. However, the qualitative features of the spectrum in an energy range of the order of or greater than the boundary value tuac (see Sec. 3) are dear even from the analysis which we performed. First of all, the splitting leads to lack of hard photons compared to the equilibrium thermal spectrum (they are transferred to the softer region of the spectrum). The flattening or even an elevation of the spectrum in the immediate vicinity of the boundary energy are unambiguous proofs of the hardness of the spectrum of the initial source whose radiation was subject to the photon splitting effect. In the absence of a low-energy background, an arbitrary given distribution of photons condenses from the region of energies greater than ~ c to a self-similar spectrum. Identification of the above features in the spectra of neutron stars can be indirect proof of the existence of the photon splitting. This proof is of great importance, since the opportunities for study of exotic processes 101

in quantum electrodynamics under laboratory conditions are very limited. The authors are grateful for the support under Contract PSS* 0992 of the European Society and under Grant No. 96-02-16045a of the Russian Foundation for Fundamental Research. REFERENCES

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102