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Home , Bremsstrahlung, Electron, Proton

A&A 406, 31–35 (2003) DOI: 10.1051/0004-6361:20030782 & c ESO 2003 Astrophysics

Proton-

E. Haug

Institut f¨ur Astronomie und Astrophysik, Universit¨at T¨ubingen, 72076 T¨ubingen, Auf der Morgenstelle 10, Germany

Received 16 May 2003 / Accepted 22 May 2003

Abstract. The collision of energetic with free is accompanied by the emission of bremsstrahlung. If the target electrons are approximately at rest, this process is designated electron- bremsstrahlung or suprathermal proton bremsstrahlung. The kinematics and the fully relativistic cross section of proton-electron bremsstrahlung in Born approximation is given. The -ray spectrum produced by protons with a power-law spectrum is calculated for thin and thick targets.

Key words. mechanisms: nonthermal – X-rays: general

1. Introduction 1972). The calculation by means of the Weizs¨acker-Williams method (Jones 1971) which is inherently simpler, yields poor The galactic and solar cosmic radiation consists largely of en- results at the high- tails of the PEB spectrum (Haug ergetic (suprathermal) protons. When a beam of these pro- 1972). tons is incident on a , appreciable X- and gamma ra- diation is produced in collisions with ambient electrons which In view of the renewed interest in the X- and gamma-ray are approximately at rest. This process is much the same as production through PEB (Dogiel et al. 1998; Pohl 1998; Valinia the normal electron-proton bremsstrahlung except that now the & Marshall 1998; Baring et al. 2000) it is worthwile to provide center of of the proton-electron system is virtu- a fully relativistic cross-section formula where nearly all of the ally that of the energetic proton. Therefore it was designated angular integrations are performed analytically thus allowing to suprathermal proton bremsstrahlung (Brown 1970; Boldt & calculate the X-ray spectra for arbitrary energy distributions of Serlemitsos 1969), inverse bremsstrahlung1, or proton-electron the incident protons with substantially reduced computational bremsstrahlung (PEB, Heristchi 1986). The PEB process was expense. considered to be a potential production mechanism for the diffuse γ-ray background (Boldt & Serlemitsos 1969; Brown 1970; Pohl 1998) and for solar flare hard X-rays (Boldt & 2. Kinematics of the process Serlemitsos 1969; Emslie & Brown 1985; Heristchi 1986). In the nonrelativistic case (proton velocity v c) the The variables of the SPB process are displayed in Fig. 1. In bremsstrahlung produced by a proton of E has the following the proton energy  (including rest energy) is ex- 2 the same spectrum as that of an electron of kinetic energy pressed in units of mpc ≈ 938 MeV and the momentum p  T = (me/mp)E in collisions with a stationary proton (me and mp in units of mpc, whereas the electron energy 1 and the pho- 2 are the rest of electron and proton, respectively). Since, ton energy k are given in units of mec , the electron momen- however, the accuracy of the nonrelativistic Bethe-Heitler cross tum p1 and the momentum k in units of m ec. Energy and section for bremsstrahlung falls off rapidly at higher momentum of the outgoing are designated by primed (Haug 1997) the corresponding PEB cross section is of small quantities. In order to calculate the maximum , value. At relativistic energies, the derivation of the PEB cross kmax, the finite rest of the proton has to be taken into ac- section causes more trouble. The usual bremsstrahlung cross count. Otherwise kmax could be greater than the proton kinetic 2 section differential in both the energy and angles of the emitted energy E for E  mpc , in contradiction to energy conser- photon has to be transformed to the electron rest frame and to vation (Heristchi 1986). In an arbitrary frame of reference the be integrated over the emission angle (Brown 1970; Haug and momentum is most conveniently expressed in terms of the four-momenta which are denoted by e-mail: [email protected] underlined quantities. Taking into account the different energy 1 The term “inverse bremsstrahlung” is inappropriate since it may units for protons, electrons, and the relation reads be confused with the -electrodynamical inverse of conven- tional bremsstrahlung, the photon absorption by a free electron in the   field of a nucleus. mp p + me p − mek = mp p + me p . (1) 1 1

Article published by EDP and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20030782 32 E. Haug: Proton-electron bremsstrahlung

The last inequality implies that the photon energy never ex- ceeds the kinetic energy of the proton, as it should be. Neglecting, however, me in the denominator of (8), one obtains

kmax(,1) ≈ ( − 1)( + p), (9)

which is only valid for  mp/(2me) ≈ 918, or E 860 GeV. Solving Eq. (7) for  and p to the minimum energy and momentum, respectively, of the proton required to produce photons of energy k. In the following we will restrict to the case that the target electron is at rest in the laboratory system, i.e., 1 = 1, p1 = 0. Then we have

2 2 2 1 + k + 2(me/mp)k + (me/mp) k min = , (10) [1 + (me/mp)k](1 − k) + kR

Fig. 1. Momenta and angles in the PEB process. 1 + (me/mp)[2 − k + (me/mp)k] pmin = k , (11) k[1 + (me/mp)k] + (1 − k) 2 2 2 Squaring this equation yields, utilizing p =  − p = 1 where and k2 = 0, 2 2 R = 2k(1 + me/mp) + (me/mp) k . (12) 2 + 2 + − − 2 mp me 2memp (pp1) (pk) 2me(p1k) Formula (10) is needed for the calculation of the photon spectra  = 2 + 2 +   from protons with an energy distribution f ( ) (see Sect. 4). mp me 2memp(p p1) (2) For  mp/2me, Eqs. (10) and (11) reduce to or √ k k + 2k  ≈ 1 + √ , p ≈ √ · (13)   = − − , min min mp(p p1) mp(pp1) mp(pk) me(p1k) (3) 1 + 2k 1 + 2k where (ab) = a b − a · b denotes the invariant product of the 0 0 3. Cross sections four-vectors a = (a0, a) and b = (b0, b). Since We consider the collision of an energetic proton with a station- (p p ) =  − p · p ≥  − p p ≥ 1, (4) 1 1 1 1 1 ary electron (p1 = 0). Designating the variables in the proton we have rest system by an asterisk, the invariance of the four-product (pk) yields m (pk) + m (p k) = k m ( − p cos θ) + m ( − p cos ϑ) ∗ p e 1 p e 1 1 k = k( − p cos θ). (14) ff ≤ mp (pp1) − 1 The invariance of the doubly di erential bremsstrahlung cross section implies = mp 1 − pp1 cos θ1 − 1 , (5) 2σ 2σ∗ d Ω= d ∗ Ω∗, Ω dk d ∗ Ω∗ dk d (15) yielding dk d dk d where dΩ=sin θ dθ dϕ is the element of solid angle of the  − θ − ≤ θ, θ ,ϑ = 1 pp1 cos 1 1 , emitted photon. k kmax( 1 )  − θ + /  − ϑ (6) p cos (me mp)( 1 p1 cos ) Using the relativistic transformation formula for the photon θ where θ and ϑ is the photon emission angle relative to the in- emission angle , cident proton and electron momentum, respectively, and θ 1 is  cos θ − p cos θ∗ = , (16) the angle between the incident proton and electron (Fig. 1). For  − p cos θ fixed values of the energies  and 1 it is easily seen that the ab- solute maximum of the photon energy is reached for θ = ϑ = 0 one gets and θ = π, resulting in 2σ 2σ∗ 1 d = 1 d · Ω  − θ ∗ Ω∗ (17)  + pp − 1 dk d p cos dk d k (, ) = 1 1 · (7) max 1  − + /  − ∗ ∗ ∗ p (me mp)( 1 p1) The cross section d2σ /(dk dΩ ) is the common  = , = bremsstrahlung cross section (neglecting proton recoil) If the incident electron is at rest ( 1 1 p1 0) this reduces to ∗ = , ∗ = ∗ − ∗, ∗ θ∗ where the quantities 1 2 1 k k , and cos  − 1 are expressed by the corresponding quantities in the rest kmax(,1) = < (mp/me)( − 1). (8)  − p + me/mp system of the electron by means of Eqs. (14) and (16). Using E. Haug: Proton-electron bremsstrahlung 33 the fully relativistic bremsstrahlung cross section in Born Applying the cross section (18), nearly all of the integrations approximation as given by Sauter (1934) it has the form 2 in (19) can be performed analytically. According to (20) one 2σ∗ α 2 ∗ has to distinguish between two cases: d r0 p2  −  − = ( − p cos θ) 1 ≤ ≤ 1 dk∗dΩ∗ 2π p2k a) k :  + p + me/mp  − p + me/mp  1 2 2 2 ×  + − 2(2 + 1) cos θ dσ αr2 5  − p 4 p  − p cos θ 1 = 0 W ( − p) − − + dk p2k 1 k3 k2 k2 2 12 4 3 + −  θ + p −  θ − k(p cos ) ∗ k(p cos ) p  1 1  + p 1 2 p 2 + 2k − p2 + p + + 1/p2 − − k( − p) 2 p2k 3 2 p2 3 3 2 ∗ ∗ ∗2 + + ∗ 2p ln( + p ) p p2 2k p2 k − 2 2 + +  + p ∗ ln 2 ( ) p ( − p cos θ) ∗ ∗ ∗ ∗ 6p 2 p p 2 + 2k p 2 + 2k − p 2 2 2 2 22 + 3 22 82 1 1 1 2 +L ( − p) − + + + − k 1 3 2 − × 1 − 2k + k( − p cos θ)2 + p( cos θ − p) 12k 3k 3k 6k 2 2k 1 p∗2 + 2k  − 2 L2( p) 3 2 2 ∗ − +  +  +  − + −   − θ 3 p 2 p ( p )k ( p) 1 p(p p2) k( p cos ) p + ∗ ln p k( − p cos θ) 3 1 2 −k 22 + + 3/p2 + k ( − p)2 ×  2 θ  − θ / 2 + −  2 2 2 sin k( p cos )(3 p 1) 1 2 2 2 2 2 2 2 3 2 + 2 + 4( k/p )( + 2) + ( + p )k ( − p) + 2 − + / 2 +  −  − θ (2 1)(1 k p ) 2 k( p cos ) 3 k 2k − (2 + 2)( − p)4 + L ( − p) −(k/p2)( − p cos θ)(5 + pk cos θ) , (18) p2 2k − 1 3 2 x0 + − L3(x) − L1(x) , where α ≈ 1/137 is the fine-structure constant, r0 is the classi- 2 (1 k) dx (21) −p W2(x) x cal electron radius, and p∗ = ∗2 − 1 is the momentum of the 2 2 where final electron in the rest frame of the proton. ff 2 2 2 It is instructive to compare the di erential cross W1(x) = p − 2kx + k x , section (18) with that of the normal electron-proton 2 2 2 bremsstrahlung where the electron has the same velocity, i.e., W2(x) = p − 2kx + 2k + k x , (22) ∗ =  1 . One notes the following: at low proton energies the 2 p − kx + pW1 behaviour of the two cross sections is similar. If the energies L1(x) = ln( − kx + W1), L2(x) = ln , (23) become higher, this changes drastically. The electron-proton kx W1 + W2 cross section has a sharp maximum at small angles originating L3(x) = ln √ , (24) from the denominator k∗(∗ − p∗ cos θ∗). Passing on to the rest 2k 1 1  − frame of the proton, this expression transforms to k so that the 1 me x0 =  − p cos θ0 = − · (25) proton-electron cross section becomes more isotropic, except k mp ≈ for k kmax. In any case protons can emit photons of higher In deriving the expression (21) the approximations energy than electrons. For instance, protons of kinetic energy E = 1 GeV (v ≈ 0.875c) produce photons of maximum energy W1(x0) = L1(x0) = L2(x0) = L3(x0) = 0 (26) ν ≈ . h max 2 105 MeV, whereas electrons with the same velocity were made. The exact value would be, e.g., (E1 ≈ 545 keV) generate photons with hν

2 proton injection rate (protons per cm , s, and MeV) by f (E0) and the target area by A, the photon production rate is given by ∞ N(hν) = A Z(hν) f (E0)dE0 Emin ∞ −1 E0 dσ dE = v . Ane dE0 f (E0) dE ν (E) (37) Emin Emin d(h ) dt Switching to the energy units  and k, this has the form ∞  −1 mp 0 pd dσ N(k) = A nec d0 f (0) d m    dt dk e min(k) min(k) − m ∞ pd 1 dσ ∞ = p    . A nec d  d 0 f ( 0) (38) me min(k) dt dk  If we choose again a power law for the proton injection rate, −δ f (0) = I0(0 − 1) , (39) the second integral of (38) can be easily solved yielding − m ∞ pd 1 dσ = p  N(k) I0A nec d  me min(k) dt dk ∞ −δ × d(0 − 1)(0 − 1) −1 √ − I Am n c ∞ 2 − 1 d 1 = 0 p e δ −  ( 1)me min(k) dt Fig. 2. PEB photon spectra from protons with a power-law spectrum, dσ spectral index δ = 2.6, for thin (upper curve) and thick targets. The × ( − 1)−(δ−1) d. (40) dk proton fluxes are arbitrary. Considering protons impinging on a target with γ ≈ . = 4 number density nH, the energy loss rate is given by index takes the value 2 23 at k 10 . That is, even though the spectral indexes approach each other in the thick and thin-  2 d = − π 2 mec Λ , target cases, the thick-target PEB spectra are flatter in the whole 4 r0nH p (41) dt mpv energy range. v = / where (p )c is the proton velocity. The Coulomb loga- References rithm Λp is different for plasmas consisting of neutral or - ized hydrogen (Emslie 1978). Baring, M. G., Jones, F. C., & Ellison, D. C. 2000, ApJ, 528, 776 The thick-target photon spectrum plotted in Fig. 2 was cal- Boldt, E., & Serlemitsos, P. 1969, ApJ, 157, 557 Brown, J. C. 1971, Sol. Phys., 18, 489 culated for a completely ionized hydrogen target with electron Brown, R. L. 1970, Lett. Nuovo Cimento, 4, 941 number density n using the energy loss rate (Lang 1980) e Dogiel, V. A., Ichimura, A., Inoue, H., & Masai, K. 1998, PASJ, 50, 2 2 2 2 567 d mec m v W v = −2πr2n ln e + 1 − 2 , (42) Drell, S. D. 1952, Phys. Rev., 87, 753 0 e v πα3 2 dt mp 2 cne c Elwert, G. 1939, Ann. Phys. (Leipzig), 34, 178 where Elwert, G., & Haug, E. 1969, Phys. Rev., 183, 90 Emslie, A. G. 1978, ApJ, 224, 241 2 2p 2 Emslie, A. G., & Brown, J. C. 1985, ApJ, 295, 648 W = mec (43) 2(me/mp) + 1 Haug, E. 1972, Astrophys. Lett., 11, 225 Haug, E. 1997, A&A, 326, 417 is the maximum kinetic energy transferred to free electrons. To Heristchi, D. 1986, ApJ, 311, 474 facilitate the comparison with the thin-target curve the spectral Jones, F. C. 1971, ApJ, 169, 503 index was again chosen δ = 2.6 (the magnitude of the photon Lang, K. R. 1980, Astrophysical Formulae, 2nd edition (Berlin: fluxes is arbitrary). At very low energies the photon spectrum Springer-Verlag) can be approximated by a power law with γ ≈ δ−1 = 1.6, again Pohl, M. 1998, A&A, 339, 587 in agreement with the nonrelativistic result of Brown (1971). Sauter, F. 1934, Ann. Phys. (Leipzig), 20, 404 With increasing k it steepens continuously and the spectral Valinia, A., & Marshall, F. E. 1998, ApJ, 505, 134