Klein-Nishina steps in the energy spectrum of galactic cosmic ray electrons

The full Klein-Nishina cross section for the inverse Compton scattering interactions of electrons implies a significant reduction of the electron energy loss rate compared to the Thomson limit when the electron energy exceeds the critical Klein-Nishina energy E_K = gamma_K * m_e c^2 = 0.27 m_e^2 c^2/(k_BT), where T denotes the temperature of the photon graybody distribution. As a consequence the total radiative energy loss rate of single electrons exhibits sudden drops in the overall energy loss rate ~ gamma ^2 - dependence when the electron energy reaches the critical Klein-Nishina energy. The strength of the drop is proportional to the energy density of the photon radiation field. The diffuse galactic optical photon fields from stars of spectral type B and G-K lead to critical Klein-Nishina energies of 40 and 161 GeV, respectively. Associated with the drop in the loss rate are sudden increases (Klein-Nishina steps) in the equilibrium spectrum of cosmic ray electrons. Because the radiative loss rate of electrons is the main ingredient in any transport model of high-energy cosmic ray electrons, Klein-Nishina steps will modify any calculated electron equilibrium spectrum irrespective of the electron sources and spatial transport mode. To delineate most clearly the consequences of the Klein-Nishina drops in the radiative loss rate, we chose as illustrative example the simplest realistic model for cosmic ray electron dynamics in the Galaxy, consisting of the competition of radiative losses and secondary production by inelastic hadron-hadron collisions. We demonstrate that the spectral structure in the FERMI and H.E.S.S. data is well described and even the excess measured by ATIC might be explained by Klein-Nishina steps.


Introduction
Recent measurements of the energy spectrum of local galactic cosmic ray electrons at energies above a few hundred GeV by the ATIC instrument 1 have reported a significant excess in the all-electron intensity that agrees at lower energies with the measurements of the PAMELA satellite experiment 2 , which also has observed a dramatic rise in the positron fraction starting at 10 GeV and extending up to 300 GeV. The significant ATIC excess has not been confirmed by the electron spectrum determinations with the FERMI satellite 3 and the H.E.S.S. air Cherenkov 4 experiments, although these measurements indicate some spectral structure deviating from a pure power law behaviour in the ATIC energy range. These observations have motivated a large number of interpretations, from possible signatures of dark matter annihilation (e.g. 5 ) to nearby astrophysical electron sources.
Here we explain the ATIC excess by a classical effect which sofar has not been discussed in this context: during their galactic propagation positrons and electrons with energies above 10 GeV are subject to synchrotron radiation losses in the galactic magnetic field of about 3µG and inverse Compton radiation losses in galactic target photon fields listed in Table 1, including the universal microwave background radiation field, infrared photons and optical stellar photons. The diffuse galactic optical photons can be characterized by the superposition of two graybody distributions (see the discussion in section 2.3 of 6 ): 1) photons from stars of spectral type G-K with energy density W G = 0.3 eV cm −3 and temperature T G = 5000 K, cor-⋆ e-mail: rsch@tp4.ruhr-uni-bochum.de ⋆⋆ e-mail: jr@tp4.ruhr-uni-bochum.de responding to a mean photon energy < ǫ > G = 2.7k B T G = 2.327 · 10 −4 T G = 1.16 eV; 2) photons from stars of spectral type B with energy density W B = 0.09 eV cm −3 and temperature T B = 20000 K, corresponding to a mean photon energy < ǫ > B = 2.7k B T G = 2.327 · 10 −4 T B = 4.65 eV.
For electron Lorentz factors γ much smaller than the critical Klein-Nishina Lorentz factor γ K = 0.27m e c 2 /k B T = 1.58 · 10 9 /T (K) (see Eq. (3) below), the inverse Compton scattering cross section of a single electron can be well approximated by the Thomson cross section resulting in the standard energy loss rate of single electronsγ = −4cσ T Wγ 2 /(3m e c 2 ), where σ T = 6.65 · 10 −25 cm 2 denotes the Thomson cross section and c the speed of light. However, for Lorentz factors γ ≥ γ KN the full Klein-Nishina cross section has to be used 7,8,9,10 resulting in a significant reduction of the inverse Compton loss rate. For the two graybody optical photon distributions the respective critical Klein-Nishina Lorentz factors are γ KN,G = 3.2 · 10 5 , corresponding to an electron energy of E KN,G = 161 GeV, and γ KN,B = 7.9 · 10 4 , corresponding to an electron energy of E KN,G = 40 GeV. We will demonstrate that this Klein-Nishina reduction of the inverse Compton energy loss rate leads to Klein-Nishina steps in the cosmic ray electron equilibrium spectrum which describes the observed FERMI and H.E.S.S. data well. In Sect. 2 we determine the galactic synchrotron and inverse Compton energy loss rates in the full Klein-Nishina case. For the illustrative example of a purely secondary origin of galactic electrons we show in Sect. 3 the resulting Klein-Nishina steps in comparison with the recent electron spectrum observations.

Synchrotron and inverse Compton energy loss rates
The synchrotron energy loss rate of a single electron in a largescale random magnetic field of constant strength B is 11 where U B = B 2 /8π = 0.22b 2 3 eV cm −3 if we scale the galactic magnetic field strength as B = 3b 3 µG.
In the Appendix we approximately calculate the inverse Compton energy loss rate of a single electron in one graybody photon field as where the critical Klein-Nishina Lorentz factor is given by For small electron Lorentz factors γ ≪ γ K the general inverse Compton energy loss rate (2) reduces to the Thomson limit whereas for large electron Lorentz factors γ ≫ γ K we obtain the energy-independent extreme Klein Nishina limit The total radiative (synchrotron and inverse Compton) energy loss rate of a single electron is given by the sum of rate (1) and rates (2) for the four diffuse galactic radiation fields listed in Table 1, yielding In Figure 1 we show the resulting radiative energy loss rate for the local galactic magnetic field and photon energy densities for relativistic electrons with energies between 1 and 10 7 GeV. One clearly notices the four sudden drops whenever the electron energy reaches each of the critical Klein-Nishina energies. The strength of the drop is proportional to the energy density of the photon field. For electron Lorentz factor below the smallest critical Klein-Nishina Lorentz factor all four graybody photon fields plus the magnetic field energy density contribute to the loss rate. Once the electron Lorentz factor has exceeded the critical Klein-Nishina Lorentz factor of a particular photon field, this photon field no longer contributes to the radiative loss rate due to the much reduced inverse Compton loss rate in the Klein-Nishina limit. At Lorentz factors above the maximum critical Lorentz factor from the microwave background photons γ k,4 = 5.9 · 10 8 only synchrotron losses contribute to the radiative loss rate. log 10 |γ| R /γ 2 log 10 (E e /GeV)

Klein-Nishina steps in the electron equilibrium spectrum
In this section we calculate the equilibrium spectrum of galactic cosmic ray electrons above 10 GeV taking into account the modified radiative loss rate (6), as well as non-thermal bremsstrahlung, adiabative deceleration losses in a possible galactic wind with the velocity v gw and Coulomb and ionization losses 15 . Because the radiative loss rate of electrons is the main ingredient in any transport model of high-energy cosmic ray electrons, Klein-Nishina steps will modify any calculated electron equilibrium spectrum irrespective of the electron sources and spatial transport mode. To delineate most clearly the consequences of the Klein-Nishina drops in the radiative loss rate, we chose as illustrative example the simplest realistic model for cosmic ray electron dynamics in the Galaxy. Extensions to more sophisticated models of cosmic ray electron dynamics (influence of localized point sources, spatial diffusion, convection and distributed reacceleration), where the consequences of the modified inverse Compton losses also occur, are the subject of future work. At electron energies above 10 GeV the electron's radiative loss time τ R = γ/|γ R | ∝ γ −1 is so short that the Galaxy behaves as a thick target or fractional calorimeter 14,15 for the electrons. The equilibrium energy spectrum of cosmic ray electrons N(γ) then results from the balance of electron production, expressed as injection spectrum Q(γ), and radiative energy losses from the solution of the balance equation implying Moreover, we assume here that all electrons are secondaries resulting from inealastic hadron-hadron collisions of primary cosmic ray hadrons with interstellar gas atoms and molecules during their confinement in the Galaxy. It is well established 16,17 that secondary production accounts for the major part of the observed galactic cosmic ray electrons at relativistic energies. The locally measured hadron spectrum 18 at energies below 4.4 · 10 15 GeV is a power law ∝ γ −s h , with spectral index s = 2.74. Using the hadron-hadron cross section templates 19 the resulting electron injection spectrum at energies above 10 GeV, Q(γ) = Q 0 γ −s , then follows a power law with the hadron spectral index s. With this injection spectrum and the radiative energy loss rate (6) the equilibrium spectrum (8) becomes which is shown in Figure 3 in comparison with the observed energy spectrum of galactic cosmic ray electrons. It can be seen that the spectral shape of the H.E.S.S. and FERMI data is well fitted. In Figure 2

Summary and conclusions
The full Klein-Nishina cross section for the inverse Compton scattering interactions of electrons implies a significant reduction of the electron energy loss rate compared to the Thomson limit when the electron energy exceeds the critical Klein-Nishina energy E K = γ K m e c 2 = 0.27m 2 e c 2 /(k B T ), where T denotes the temperature of the photon graybody distribution. As a consequence the total radiative energy loss rate of single electrons exhibits sudden drops in the overallγ ∝ γ 2 -dependence when the electron energy reaches the critical Klein-Nishina energy. The strength of the drop is proportional to the energy density of the photon radiation field. The diffuse galactic optical photon fields from stars of spectral type B and G-K lead to critical Klein-Nishina energies of 40 and 161 GeV, respectively. Associated with the drop in the loss rate are sudden increases (Klein-Nishina steps) in the equilibrium spectrum of cosmic ray electrons (see Figure 2). Because the radiative loss rate of electrons is the main ingredient in any transport model of high-energy cosmic ray electrons, Klein-Nishina steps will modify any calculated electron equilibrium spectrum irrespective of the electron sources and spatial transport mode. To delineate most clearly the consequences of the Klein-Nishina drops in the radiative loss rate, we chose as illustrative example the simplest realistic model for cosmic ray electron dynamics in the Galaxy, consisting of the competion of radiative losses and secondary production by inelastic hadron-hadron collisions. We demonstrate that the spectral structure in the FERMI and H.E.S.S. data is well described and even the excess measured by ATIC might be explained by Klein-Nishina steps.
After completing this work we noticed the recent preprint by Stawarz, Petrosian and Blandford (2009) who also explain the recently measured galactic electron spectrum by the Klein-Nishina suppression of the inverse Compton energy loss of relativistic electrons in an optical photon field with an energy density of 3 eV cm −3 .
The inverse Compton power of a single electron in a general target photon field n(ǫ) is (Ch. 4.2 in 6 ) where ǫ s denotes the scattered photon energy. The differential Klein-Nishina cross section 8 is given by By integrating over all kinematically allowed scattered photon energies we find for the inverse Compton energy loss rate of a single electron where ǫ s,max = Γγmc 2 /(Γ + 1) corresponds to q = 1. Using q as integration variable instead of ǫ s results in For the graybody photon distribution the inverse Compton energy loss rate then is Jones 7 already noted that the double integral I(γ, T ) cannot be solved exactly, so that approximations (see e.g. 12 ) are required. It has been noted 13 that the integral J(Γ) is reasonably well approximated by J(Γ) = The series 1 leads to For A < 1 we approximate the integral (A.11) by We combine the two expansions (A.14) and (A.15) to