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the discovery of high - energy @xmath0-radiation from extragalactic compact objects has motivated many authors to consider the effects of @xmath0-ray absorption by @xmath0-@xmath0 pair production , eventually inducing pair cascades . the relevance of @xmath0-@xmath0 pair production to astrophysical systems has first been pointed out by nikishov ( 1962 ) . the first investigation of the @xmath0-@xmath0 absorption probability of high - energy photons by different soft photon fields , along with some useful approximations , can be found in gould & schrder ( 1967 ) . the energy spectrum of injected electrons and positrons due to this process has been studied by several authors ( e. g. , bonometto & rees 1971 , aharonian et al . 1983 , zdziarski & lightman 1985 , coppi & blandford 1990 ) . in most astrophysically relevant cases , simple approximations can be used for this purpose , without much loss of accuracy . these usually rely on the high - energy photon having much higher energy than the soft photons and thus dominating the energy input and determining the direction of motion of the center - of - momentum frame of the produced pairs . bonometto & rees ( 1971 ) used basically the same technique as we do , but restricted their analysis to the case @xmath1 , and did not solve the problem analytically . two recipes to calculate the full energy - dependence of the injected pairs have been published ( aharonian et al . 1983 and coppi & blandford 1990 ) , but here the reader is still left with integrations to be carried out numerically . it is the purpose of this paper to derive the full energy - spectrum of pairs , injected by @xmath0-@xmath0 pair production , exact to second order qed for the case of isotropic radiation fields . in section 2 , we give a short overview of the kinematics which are used in section 3 to calculate the pair injection spectrum . in section 4 , we compare our results to well - known approximations and specify the limitations of the various approximations . our analysis is easily generalized to non - isotropic radiation fields . the derivation presented here is widely analogous to the derivation of the pair annihilation spectrum , given by svensson ( 1982 ) . we consider an isotropic photon field @xmath2 where @xmath3 is the dimensionless photon energy in a rest frame which we call the laboratory frame . the lorentz invariant scalar product of the four - momenta @xmath4 of two photons having energies @xmath5 colliding under an angle of cosine @xmath6 in the laboratory frame is then given by    @xmath7    here , @xmath8 is the photon energy in the center - of - momentum frame . in order to allow for the possibility to create an electron - positron pair , conservation of energy implies @xmath9 , and the condition @xmath10 determines the pair - production threshold . @xmath11 is the lorentz factor of the electron / positron in the cm frame where the produced electrons move with speed @xmath12 and @xmath13 . the definition of the angle variables needed in this calculation is illustrated in fig . = 4.5 cm    the cm frame moves relative to the laboratory frame with velocity @xmath14 and lorentz factor @xmath15 . the four velocity of the laboratory frame ( @xmath16 in the cm frame ) is denoted by @xmath17 . the lorentz factors of the produced pairs in the laboratory frame are related to the cm quantities by    @xmath18    evaluating the lorentz invariant scalar product    @xmath19    in the laboratory and the cm - frame , respectively , we find    @xmath20    and    @xmath21    inserting eq . ( 4 ) into eq . ( 2 ) and using energy conservation ( @xmath9 ) fixes the angle cosine @xmath22 to    @xmath23    the differential cross section for @xmath0-@xmath0 pair production ( see eq . [ 11 ] ) depends on    @xmath24 the differential yield of produced pairs is calculated as    @xmath25 @xmath26 where @xmath27 the differential cross section has been evaluated by jauch & rohrlich ( 1959 ) :    @xmath28    where @xmath29 @xmath30 @xmath31 ^ 2 } + { 1 \over   [ 1 + \bcm x]^2 } \right ) \biggr\rbrace.\ ] ] we may express the solid angle element @xmath32 . using eq . ( 2 ) , we find    @xmath33    this enables us to carry out the @xmath22-integration in eq . ( 9 ) immediately . if we write the denominators in eq . ( 11 ) as @xmath34 with @xmath35 we find    @xmath36 @xmath37 @xmath38 ^ 2 } + { 1 \over [ a_- + b_- \cos\phi]^2 } \right ) \biggr\rbrace\ ] ] @xmath39 @xmath40 where    @xmath41    @xmath42    with    @xmath43    @xmath44    and we used the integrals    @xmath45    @xmath46    and the identity    @xmath47    which follows from eqs . ( 5 ) , ( 6 ) and ( 14 ) . now , inserting eq . ( 15 ) into eq . ( 8) yields the exact expression for the differential pair injection rate . using eq . ( 1 ) we transform the @xmath48 integration into an integration over @xmath49 . this leads us to    @xmath50 @xmath51 @xmath40 which can be calculated analytically . the integration limits follow from @xmath52 and the condition @xmath53 which yields    @xmath54    where    @xmath55 @xmath56 + 1 \pm   \sqrt { ( \gm [ e - \gm ] + 1)^2 - e^2 } \right).\ ] ] using the integrals 2.271.4 , 2.271.5 , 2.272.3 , 2.272.4 , and 2.275.9 , of gradshteyn & ryzhik ( 1980 ) , we find as final result for the differential pair yield    @xmath57 @xmath58 where for @xmath59 we have    @xmath60 @xmath61 @xmath62 and    @xmath63 @xmath64 for @xmath65 we find    @xmath66 @xmath67 now , we use the exact expression , given in eq . ( 26 ) to specify the regimes of validity and the limitations of various approximations . the first detailed computation of the pair production spectrum was presented by bonometto & rees ( 1971 ) . based on the neglect of the energy input of the soft photon , they basically follow the same procedure as described above , but do not carry out the angle - integration ( integration over @xmath8 in our formalism ) analytically . in the case @xmath1 , it is in very good agreement with the exact result , but its evaluation is even more time - consuming than using the latter . for this reason , we will not consider it in detail , but concentrate on approximations which really yield simpler expressions than the exact one . probably the simplest expression for the pair spectrum injected by @xmath0-rays interacting with a power - law of energy spectral index @xmath69 ( @xmath70 ) is based on the assumption @xmath1 and on the well - known fact that photons of energy @xmath71 interact most efficiently with photons of energy @xmath72 which motivates the approximation for the @xmath0-@xmath0 opacity of gould & schrder ( 1967 ) using a @xmath68 function approximation for the cross section , @xmath73 since in this approach pair production takes place only near the pair production threshold ( @xmath74 ) , the produced pairs have energies @xmath75 ( bonometto & rees 1971 ) . the resulting pair injection spectrum is therefore    @xmath76    ( e. g. lightman & zdziarski 1987 ) where @xmath77 ( which , of course , reduces to @xmath78 for @xmath79 ) and @xmath80 is a numerical factor , depending only on the spectral index of the soft photon distribution . this approximation yields useful results , if the power - law photon spectra extend over a sufficiently wide range ( @xmath81 ) and if for every high - energy photon of energy @xmath71 there is a soft photon of energy @xmath72 . else , the injection spectrum calculated with eq . ( 30 ) cuts off at the inverse of the respective cutoff of the soft photon spectrum , seriously underpredicting the injection of pairs of higher or lower energy , respectively , where the injection spectrum declines smoothly . nevertheless , these pairs can still carry a significant fraction of the injected power . ( 30 ) fails also to describe the injection of low - energetic pairs in case of a high lower cut - off of the @xmath0-ray spectrum even if soft photons of energy @xmath82 are present . for example , in the case of the interaction of a power - law spectrum extending from @xmath83 @xmath84 with a soft power - law spectrum extending from @xmath85  @xmath86 eq . ( 30 ) overpredicts the injection of pairs slightly above @xmath87 by an order of magnitude and cuts off below this energy . a similar problem arises at the high - energy end of the injection spectrum . an example for this fact is shown in fig . 2 . in contrast , the approximation ( 30 ) can well be used to describe the injection of pairs at all energies if both photon fields extend up to ( and down to , respectively ) @xmath88 . for more general soft photon distributions which are different from a power - law ( e. g. a thermal spectrum ) the analogous @xmath68-function approximation has first been introduced by kazanas ( 1984 ) .      using the full cross section for @xmath0-@xmath0 pair production as given by jauch & rohrlich ( 1959 ) instead of the @xmath68-function approximation adopted in eq . ( 30 ) does not reduce the limitations of this power - law approach significantly , but other soft photon distributions can be treated more successfully with this approximation which in the limit @xmath1 reads    @xmath89    where @xmath90 and the limits @xmath91 are given by eqs . ( 24 ) and ( 25 ) . here , we have assumed that the produced electron and positron have energy @xmath92 . this approach works equally well for power - law photon fields , but in contrast to eq . ( 30 ) , it tends to underpredict the injection of low - energetic pairs . the same is true for the interaction of @xmath0-ray photon fields with thermal soft photon fields where the high - energy tail of the injection spectrum is described very accurately ( a few  % error ) by eq . the accuracy of this approximation improves with decreasing lower cut - off of the @xmath0-ray spectrum . e. g. , the injection due to a power - law @xmath0-ray spectrum from @xmath93  @xmath84 interacting with a thermal spectrum of normalized termpature @xmath94 is described by eq . ( 31 ) with a deviation of less than 30  % from the exact result down to @xmath95 . for @xmath96 , the deviation was much less than 10  % . we show an example for the latter situation in fig . 3 .      a very useful approximation to the pair injection spectrum for all shapes of the soft photon spectrum under the condition @xmath97 has been found by aharonian et al . ( 1983 ) . they use a different representation of the pair production cross section and end up with a one - dimensional integral over @xmath98 which is equivalent to our @xmath8 integration in eq . they solve this integration analytically after simplifying the integrand and the integration limits according to the assumptions mentioned above . the resulting injection spectrum is    @xmath99 @xmath100 @xmath101 it describes the power - law tail of the pair spectrum injected by power - law @xmath0-ray photon fields perfectly and is much more accurate to the injection of low - energy pairs . interaction with a power - law soft photon field is reproduced within errors of only a few  % . even if as well the @xmath0-ray as the soft photon spectrum extend to @xmath102 , the error at @xmath95 increases only to @xmath103 10  % . problems with this approximation arise if the soft photon spectrum extends up to @xmath104 , but the @xmath0-ray spectrum has a lower cut - off @xmath105 . in this case , the injection of low - energetic pairs is seriously overpredicted by eq . ( 32 ) . for power - law soft photon fields , the integration over @xmath106 in eq . ( 32 ) can be carried out analytically , as was found by svensson ( 1987 ) . his equation ( b8 ) ( multiplying with the total absorption coefficient @xmath107 by which the total injection rate had been normalized to 1 ) is in perfect agreement with the numerical results according to eq . ( 32 ) . the interaction of power - law @xmath0-ray spectra with thermal soft photon fields is generally described within an error of a few % at all electron / positron energies if the soft photon temperature is @xmath108 , even if the @xmath0-ray spectrum extends down to @xmath109 . interestingly , even the interaction of a mildly relativistic thermal photon field ( @xmath110 ) with itself ( for which aharonian s approximation was not designed ) is reproduced reasonably well , but the result of eq . ( 32 ) differs from the exact injection rate by a roughly constant factor . when artificially introducing a factor adjusting the high - energy tails of the injection spectra , eq . ( 32 ) overpredicts the injection of low - energetic pairs by a factor of @xmath111 , but for @xmath112 there is very good agreement with the exact result . the deviation becomes more important with increasing photon temperature , and for @xmath113 the injection of cold pairs is already overpredicted by a factor of @xmath114 . 4 illustrates the accuracy of the various approximations for a compact thermal radiation of temperature @xmath113 . we find that all the statements on soft photon or @xmath0-ray power - law spectra made above are only very weakly dependent on the respective spectral index . we derived the full energy spectrum of injected pairs , produced by @xmath0-@xmath0 pair production and compared the result to the various approximations known before . we found that the simplest expressions , based on @xmath68 function approximation to the cross section can well reproduce the power - law tail of the injection spectrum , but have problems at low particle energies .
we present the first completely analytical computation of the full differential @xmath0-@xmath0 pair production rate from compact radiation fields , exact to 2nd order qed , and use this result to investigate the validity of previously known approximations .