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Simplified spectrum and power formulae for resonant inverse Compton scattering in a strong magnetic field
6 August 2001
Physics Letters A 286 (2001) 309–313 www.elsevier.com/locate/pla
Simplified spectrum and power formulae for resonant inverse Compton scattering in a strong magnetic field Jun-han You ∗ , Dang-bo Liu, Ya-di Xu, Lei Chen Institute for Space and Astrophysics, Physics Department, Shanghai Jiao-Tong University, Shanghai 200030, PR China Received 3 April 2001; accepted 12 June 2001 Communicated by P.R. Holland
Abstract The resonant inverse Compton scattering (RICS) of a relativistic electron in an intense magnetic field is an important radiation mechanism in hard X-ray and γ -ray astrophysics. So far the available formulae describing RICS radiation are quite complicate in mathematics and not easy to understand in physics. In this Letter, we present the markedly simplified, analytical formulae for both the spectral and the total power of the RICS process. We will show that the RICS radiation has good monochromaticity which concentrates in hard X-ray and γ -ray wavebands, and has extremely high efficiency when compared with the coexistent, nonresonant inverse Compton scattering, if the “accommodation condition”, derived in this Letter, is satisfied. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The magnetic inverse Compton scattering of a relativistic electron in a very strong magnetic field (e.g., B ∼ 108 –1012 G s) of the neutron star or strange star is divided into two parts, the resonant and the nonresonant [1–11]. The nonresonant part of scattering is basically same as the ordinary field-free inverse Compton scattering due to the approximately same crosssection, σsnres ≈ σT , when the incident frequency νi of the scattered photons in the co-moving frame (the electron-rest frame (ERF)), S , is markedly higher than the Landau frequency, νi νB ≡ eB/2πmc. If νi is much lower than νB , νi νB , the scattering vanishes because σs → 0. In this Letter, we do not distinguish the terminology “nonresonant inverse Compton scat-
* Corresponding author.
E-mail address: [email protected] (J.-h. You).
tering” from the “inverse Compton scattering”, and simply note as “ICS”. The second part, which we call as “resonant inverse Compton scattering” (RICS), occurring in the vicinity of νB , νi ≈ νB , is a very special kind of scattering due to its remarkable properties. The RICS arises from the cyclotron-resonant absorption and subsequent instantaneous re-emission of the incident photon by the relativistic electron. We show in this Letter that the resonance nature makes the RICS potentially a very important radiation mechanism in high frequency band, more powerful than ever thought. The rapid progress in observations of X-ray and γ ray astronomy in recent years, e.g., the γ -ray pulsars, the γ -ray bursts, etc., encourages us to re-examine and improve the related theories and formulae system in previous literature. So far, no simple formulae describing the RICS radiation have been available. Therefore, in this Letter, we try to give markedly simplified analytical formulae for both the spectral and
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 1 7 - 0
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total powers of a single relativistic electron which have clear physical meaning and very convenient for the purpose of astrophysical application. In fact, detailed theoretical analysis of the magnetic Compton scattering of the soft photons by the relativistic electrons beamed along the direction of the strong magnetic field has been well-done by Dermer [6,7], and the related formulae have been markedly simplified than previous one [4]. But in his papers, the resonant and nonresonant scattering are taken into consideration together by use of the complicated QED formula of the scattering cross-section. Thus the resultant formulae are complicate in mathematics, which are inconvenient to use in astrophysical practice, even cause a misunderstanding of the basic physical meaning of RICS. For example, the delta function approximation of the resonance cross-section (Eqs. (A3) and (A4) in [6]) give σres ≈ σeff δ(ε − εB ) and σeff ≈ 323σT , where ε is a dimensionless energy of the incident photon in unit of the static energy of electron me c2 , ε ≡ hν/me c2 , εB ≡ hνB /me c2 . Such an approximate expression Eq. (A4) is easy to cause the impression that resonance scattering cross-section is only 323 times higher than the ordinary inverse Compton scattering even near the peak of the resonance region, which leads to a serious underestimation of the importance of RICS mechanism. In fact, our calculation shows that, if B ∼ 1012 G s, the peak of resonance cross-section at line-center νB is so high as σres (νB ) ≈ 108 σT , and the average value in the resonance region is σres ≈ 104 σT . Our treatment is different with theirs in that we deal with the coexistent RICS and ICS processes separately by considering the fact that the RICS operates only at harmonic frequencies νi = νB , 2νB , . . . , whereas the ICS responds to a continuous spectrum of incident photons νi = νB . Furthermore, we use a simple classical quantum cross-section of scattering Eq. (1) which is exactly as the QED formula in the vicinity of νB , as we pointed out [1]. These lead to a great simplification of the formulation of RICS. In addition, we derived the “accommodation condition” under which the RICS process becomes predominant over the coexistent conventional ICS. In order to understand the origin and the characteristics of RICS process, it seems necessary to restate some aspects of the RICS physics, based on the classical quantum theory for clarity and simplicity. We start from the description of the kinetic behavior of
a relativistic electron in a strong magnetic field. It is obvious that the relativistic electron cannot keep the relativistic velocity in the directions perpendicular to the magnetic strength B due to its extremely short synchrotron lifetime when the magnetic field is very strong. For example, if B ∼ 1012 G s and the Lorentz factor γ ∼ 103 , the synchrotron lifetime of the electron is only τsy ∼ 109 B −2 γ −1 ∼ 10−18 s. Therefore the perpendicular component of the electron velocity will drop quickly down to v⊥ c. Only the relativistic motion along the magnetic field line can be retained for a long time. Thus the fast electron will move in a tightened helical orbit along the field line with v⊥ c and v c. Such a special configuration of motion determines that RICS is in fact the cyclotron-resonant scattering. The resonance nature can be better understood in a co-moving frame of reference S , fixed at the center of the circular orbit of the electron around the magnetic field line; we called this frame the electronrest frame (ERF) [1]. Therefore in S system the ve c, relocity components become v = 0 and v⊥ spectively. Thus in S one sees a nonrelativistic electron moving in a circular orbit with corresponding energy level (n + 1/2)hνB (n = 0, 1, 2, . . .), where νB = eB/2πm0 c is the Landau frequency. The most probable emission and/or absorption occurs at base frequency νB due to the most important transitions occur between the neighboring levels n n + 1, especially 0 1. Therefore in the S frame, absorption occurs as long as the frequency of the incident photon equals the base frequency, νi = νB , and the corresponding absorption transition is 0 → 1. The emission transition 1 → 0 follows immediately due to an extremely high probability of spontaneous transition, e.g., a10 = 1015 s−1 for B ∼ 1012 G s [1]. Therefore the combined process of a resonant absorption transition 0 → 1 and the subsequent re-emission 1 → 0 is equivalent to a “resonant scattering” of an incident photon with base frequency νi = νB . Returning to the laboratory system S, we get a cyclotron-resonant RICS of a fast electron. The resonance behavior leads to a series of special properties and advantages of the RICS radiation mechanism [3–8]: (1) Very high radiation efficiency. From Eq. (1) of the Letter we show that the total cross-section of resonance scattering at the line center νi = νB is σ s (νB ) ∼ 108 σT , for B ∼ 1012 G s, where σT is the
J.-h. You et al. / Physics Letters A 286 (2001) 309–313
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(0–2γ νB ). However, as shown in Fig. 1, the calculated RICS spectrum of a single electron still retains a good monochromaticity. The RICS radiation power of a fast electron with energy γ , passing through a low frequency soft photon field, is calculated via Lorentz transformation of reference frames from the co-moving frame S to the laboratory frame S [13]. For shortness, in the following we omit the complicate derivations and only present the resultant formulae which are related to the application of RICS in hard X-ray and γ -ray astrophysics.
Fig. 1. The radiation spectrum of the RICS of a single electron with energy γ shows a good monochromaticity, which is quite different from the ordinary inverse Compton scattering. Note the sharp peak at the maximum frequency 2γ νB and the weak low-frequency tail extending to ν → 0.
Thompson cross-section, which implies a very high RICS efficiency. (2) Very high scattering frequency. In S the scattering frequency is given by the resonance condition νs = νi = νB . Returning to the observer frame S, the Doppler formula gives νs = γ νs (1 + β cos ψs ) = γ νB (1 + cos ψs ) ∼ γ νB (taking β 1), i.e., the typical RICS frequency is on the order of γ νB . In the case when B ∼ 1012 G s, hνB ∼ 10 keV, the scattering photon would then have an energy hν ∼ γ hνB , or 100 keV–10 MeV if γ ∼ 101–3. The RICS mechanism is hence very important for hard X-ray and γ -ray astronomy, e.g., in the studies of γ -ray pulsars and γ -ray bursts, etc. [12]. (3) Very good beaming of RICS radiation. The photons produced by the RICS radiation propagate along the magnetic field lines within an extremely narrow angular cone with θ ∼ 1/γ . This beaming property greatly suppresses the strong absorption of the RICS γ -ray photons by the magnetic and/or γ –γ annihilations, which would otherwise attenuate heavily the γ ray radiation with hν 1 MeV. (4) Good monochromaticity of the RICS spectra of a single fast electron, concentrating most of radiation energy in the high frequency band. In S system, the scattering frequency is exactly monochromatic, given by the resonance condition νs ≡ νi = νB . In observer frame S, it becomes νs = γ νB (1 + cos ψs ), due to the Doppler effect. Thus νs distributes over a wide range
2. The differential scattering cross-section The simplified scattering cross-section in S frame, derived from the classical quantum theory [1,2], is 3 σ s (νi , ψi , ψs ) = r0 c 1 + cos2 ψi 1 + cos2 ψs 32 × φ(νi − νB ), (1) where ψs is the scattering angle in S , r0 is the classical radius of an electron, νB = eB/2πm0 c = 2.8 × 106B Hz, and the Lorentz profile φ(νi − νB ) = (Γlu / 4π 2 )[(νi − νB )2 + (Γlu /4π)2 ]−1 , with Γlu ≡ Γl + Γu being the quantum damping constant of the upper (u) and the lower (l) levels.
3. The resonance condition The resonance condition in S is νi = νB . Transforming to the laboratory frames S, S → S, from the Doppler formulae we get νB . νi (2) γ (1 − cos ψi ) Eq. (2) is the expression of the resonance condition νi = νB in the laboratory frame S, which indicates that the frequency of the incident photons to be resonantly absorbed cannot be taken arbitrarily, but is restricted by the given angle ψi , νi = νi (ψi ). 4. The approximate resonance condition — accommodation condition The fractional number of photons with small incident angles ψi 0 is very small in a general, isotropic
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radiation field. Furthermore, Eq. (2) shows that ψi 0 corresponds to a high frequency νi . Since the number of photons with very high νi is nominally very small in a low-frequency radiation field, therefore, neglecting the part of small incident angle scattering ψi 0, (2) becomes approximately γ hνi hνB .
(2 )
The approximate resonance condition (2 ) also provides a useful semi-quantitative criterion to estimate the efficiency of RICS of a relativistic electron passing through a low-frequency radiation field. The RICS radiation is significant only when the averaged electron energy γ¯ times the averaged photon energy of the lowfrequency radiation field (h¯ν ) is comparable to hνB . In fact, in this case, the coexistent, ordinary inverse Compton scattering is negligibly small and needs not be taken into account. Therefore, (2 ) is regarded as an “accommodation condition” for the energy spectrum of electrons in a given low-frequency radiation field and a given magnetic field B.
where x ≡ ν/2γ νB = ν/νmax is a dimensionless scattering frequency in unit of the maximum scattering frequency 2γ νB , and the function f (x) is 3 2 f (x) = 2x − 2x + x, if 0 x 1, (4) 0, if x > 1. The function f (x), depicted in Fig. 1, specifies the spectral shape of RICS of a single electron, which can be regarded as a dimensionless spectral power of RICS. Fig. 1 shows that the RICS spectrum still retains a good monochromaticity — a sharp peak near the maximum frequency 2γ νB with a weak low-frequency tail extending to ν 0 — despite the very wide frequency band. In some semi-quantitative discussions, we therefore adopt a quasi-monochromatic approximation by assuming an approximate one-to-one correspondence between the electron energy γ and the emission frequency ν, i.e., an electron with energy γ produces a monochromatic line radiation with frequency 2γ νB , γ ↔ 2γ νB .
6. The resonance scattering efficiency A(γ , B) 5. RICS radiation spectrum of a single relativistic electron Before we calculate the spectral power of RICS of an electron with energy γ , it is helpful to give a qualitative analysis for the spectrum of RICS. In the S system, the main scattering frequency is given by the resonance condition, νs ≡ ν = νi = νB , which shows a strict monochromaticity of scattering in the S system. However, returning to the laboratory system, we get the scattering frequency in the S system (again taking β 1), νs ≡ ν = γ ν (1 + β cos ψs ) γ νB (1 + cos ψs ). Due to the approximate isotropy of scattering in S (see Eq. (1)), the range for the scattering angle ψs is (0, π). Therefore, the scattering frequency should spread in a wide range (0, 2γ νB ), with a large highfrequency cutoff, 2γ νB . However, our calculation shows that the quantitative RICS spectrum still keeps a very good monochromaticity. The calculated RICS spectrum of a fast electron with energy γ in a strong magnetic field B is given by dpRICS = (3πr0 c)A(γ , νB )γf (x), dx
(3)
The quantity A in Eq. (3) denotes an integral π I (νi , ψi )(1 − cos ψi ) sin ψi dψi ≡ A(γ , B),
2π 0
(5) where I (νi , ψi ) denotes the intensity of a given monochromatic incident beam with frequency νi and incident angle ψi in the low-frequency field. Note that the intensity I is always a function of ψi , even when the radiation field of soft photons is isotropic, for which the dependence of the intensity I on ψi appears not explicit, I in fact is still related to ψi through the function νi (ψi ) (see Eq. (2)), i.e., I (νi ) = I (νi (ψi )), and the form of I (νi ) is determined by the given radiation field. Thus for the π isotropic lowfrequency field, Eq. (5) becomes 2π 0 I (νi (ψi ))(1 − cos ψi ) sin ψi dψi ≡ A(γ , B). The quantity A, a function of γ and B, can be regarded as a measure of the “resonance scattering efficiency”, in the sense that it includes all the “eligible” soft photons incident from all directions which are capable of being resonantly absorbed (scattered) by the electron with energy γ . It
J.-h. You et al. / Physics Letters A 286 (2001) 309–313
can be seen that, the stronger the intensity I (νi (ψi )) at the resonance frequency νi = νi (ψi ), the higher the “resonance scattering efficiency” A, hence the RICS radiation power (see Eq. (3)).
313
Acknowledgement This research is supported by the National Natural Science Foundation of China under Grant No. 19573005, the Climbing Program of State Committee of Science and Technology of China.
7. The total RICS power The total power P RICS of a fast electron is thus 1 P
RICS
=
dpRICS dx = (πr0 c)A(γ , νB )γ . dx
References
(6)
0
Thus P RICS is proportional to the product of the “efficiency” A and energy γ of the fast electron. We emphasize that, in an intense magnetic field B, the conventional ICS process still coexists with the RICS process. As discussed above, RICS process occurs only when the incident frequency νi = νB in the electron-rest frame S . If the resonance condition is not satisfied, i.e., if νi = νB , we then have the ordinary ICS. In general, for a given system of fast electrons in a low-frequency radiation field, as long as the “accommodation condition” (2 ) prevails, the conventional ICS is negligibly weak compared with the coexistent RICS.
[1] You Jun-han, Chen Jun-feng, Deng Jing-song, Phys. Lett. A 232 (1997) 367. [2] Chen Jun-feng, Deng Jing-song, Xu Yi, You Jun-han, Sci. China A 16 (1997) 379. [3] Xia Xiao-yang, Qiao Guo-jun, Wu Xin-ji, Astron. Astrophys. 93 (1985) 152. [4] H. Herold, Phys. Rev. D 19 (1997) 2868. [5] J.K. Daugherty, A.K. Harding, Astrophys. J. 309 (1986) 362. [6] C.D. Dermer, Astrophys. J. 347 (1989) L13. [7] C.D. Dermer, Astrophys. J. 360 (1990) 197. [8] A.K. Harding, J.K. Daugherty, Astrophys. J. 374 (1991) 687. [9] C.D. Dermer, S.J. Sturner, Astrophys. J. 420 (1994) L75. [10] S.J. Sturner, C.D. Dermer, Astrophys. J. 420 (1994) L79. [11] S.J. Sturner, C.D. Dermer, F.C. Michel, Astrophys. J. 445 (1995) 736. [12] Huang Yong-feng, Dai Zi-gao, Lu Tan, Chin. Phys. Lett. 16 (1999) 775. [13] W. Tucker (Ed.), Radiation Processes in Astrophysics, MIT Press, Cambridge, 1979.
Physics Letters A 286 (2001) 309–313 www.elsevier.com/locate/pla
Simplified spectrum and power formulae for resonant inverse Compton scattering in a strong magnetic field Jun-han You ∗ , Dang-bo Liu, Ya-di Xu, Lei Chen Institute for Space and Astrophysics, Physics Department, Shanghai Jiao-Tong University, Shanghai 200030, PR China Received 3 April 2001; accepted 12 June 2001 Communicated by P.R. Holland
Abstract The resonant inverse Compton scattering (RICS) of a relativistic electron in an intense magnetic field is an important radiation mechanism in hard X-ray and γ -ray astrophysics. So far the available formulae describing RICS radiation are quite complicate in mathematics and not easy to understand in physics. In this Letter, we present the markedly simplified, analytical formulae for both the spectral and the total power of the RICS process. We will show that the RICS radiation has good monochromaticity which concentrates in hard X-ray and γ -ray wavebands, and has extremely high efficiency when compared with the coexistent, nonresonant inverse Compton scattering, if the “accommodation condition”, derived in this Letter, is satisfied. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The magnetic inverse Compton scattering of a relativistic electron in a very strong magnetic field (e.g., B ∼ 108 –1012 G s) of the neutron star or strange star is divided into two parts, the resonant and the nonresonant [1–11]. The nonresonant part of scattering is basically same as the ordinary field-free inverse Compton scattering due to the approximately same crosssection, σsnres ≈ σT , when the incident frequency νi of the scattered photons in the co-moving frame (the electron-rest frame (ERF)), S , is markedly higher than the Landau frequency, νi νB ≡ eB/2πmc. If νi is much lower than νB , νi νB , the scattering vanishes because σs → 0. In this Letter, we do not distinguish the terminology “nonresonant inverse Compton scat-
* Corresponding author.
E-mail address: [email protected] (J.-h. You).
tering” from the “inverse Compton scattering”, and simply note as “ICS”. The second part, which we call as “resonant inverse Compton scattering” (RICS), occurring in the vicinity of νB , νi ≈ νB , is a very special kind of scattering due to its remarkable properties. The RICS arises from the cyclotron-resonant absorption and subsequent instantaneous re-emission of the incident photon by the relativistic electron. We show in this Letter that the resonance nature makes the RICS potentially a very important radiation mechanism in high frequency band, more powerful than ever thought. The rapid progress in observations of X-ray and γ ray astronomy in recent years, e.g., the γ -ray pulsars, the γ -ray bursts, etc., encourages us to re-examine and improve the related theories and formulae system in previous literature. So far, no simple formulae describing the RICS radiation have been available. Therefore, in this Letter, we try to give markedly simplified analytical formulae for both the spectral and
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 1 7 - 0
310
J.-h. You et al. / Physics Letters A 286 (2001) 309–313
total powers of a single relativistic electron which have clear physical meaning and very convenient for the purpose of astrophysical application. In fact, detailed theoretical analysis of the magnetic Compton scattering of the soft photons by the relativistic electrons beamed along the direction of the strong magnetic field has been well-done by Dermer [6,7], and the related formulae have been markedly simplified than previous one [4]. But in his papers, the resonant and nonresonant scattering are taken into consideration together by use of the complicated QED formula of the scattering cross-section. Thus the resultant formulae are complicate in mathematics, which are inconvenient to use in astrophysical practice, even cause a misunderstanding of the basic physical meaning of RICS. For example, the delta function approximation of the resonance cross-section (Eqs. (A3) and (A4) in [6]) give σres ≈ σeff δ(ε − εB ) and σeff ≈ 323σT , where ε is a dimensionless energy of the incident photon in unit of the static energy of electron me c2 , ε ≡ hν/me c2 , εB ≡ hνB /me c2 . Such an approximate expression Eq. (A4) is easy to cause the impression that resonance scattering cross-section is only 323 times higher than the ordinary inverse Compton scattering even near the peak of the resonance region, which leads to a serious underestimation of the importance of RICS mechanism. In fact, our calculation shows that, if B ∼ 1012 G s, the peak of resonance cross-section at line-center νB is so high as σres (νB ) ≈ 108 σT , and the average value in the resonance region is σres ≈ 104 σT . Our treatment is different with theirs in that we deal with the coexistent RICS and ICS processes separately by considering the fact that the RICS operates only at harmonic frequencies νi = νB , 2νB , . . . , whereas the ICS responds to a continuous spectrum of incident photons νi = νB . Furthermore, we use a simple classical quantum cross-section of scattering Eq. (1) which is exactly as the QED formula in the vicinity of νB , as we pointed out [1]. These lead to a great simplification of the formulation of RICS. In addition, we derived the “accommodation condition” under which the RICS process becomes predominant over the coexistent conventional ICS. In order to understand the origin and the characteristics of RICS process, it seems necessary to restate some aspects of the RICS physics, based on the classical quantum theory for clarity and simplicity. We start from the description of the kinetic behavior of
a relativistic electron in a strong magnetic field. It is obvious that the relativistic electron cannot keep the relativistic velocity in the directions perpendicular to the magnetic strength B due to its extremely short synchrotron lifetime when the magnetic field is very strong. For example, if B ∼ 1012 G s and the Lorentz factor γ ∼ 103 , the synchrotron lifetime of the electron is only τsy ∼ 109 B −2 γ −1 ∼ 10−18 s. Therefore the perpendicular component of the electron velocity will drop quickly down to v⊥ c. Only the relativistic motion along the magnetic field line can be retained for a long time. Thus the fast electron will move in a tightened helical orbit along the field line with v⊥ c and v c. Such a special configuration of motion determines that RICS is in fact the cyclotron-resonant scattering. The resonance nature can be better understood in a co-moving frame of reference S , fixed at the center of the circular orbit of the electron around the magnetic field line; we called this frame the electronrest frame (ERF) [1]. Therefore in S system the ve c, relocity components become v = 0 and v⊥ spectively. Thus in S one sees a nonrelativistic electron moving in a circular orbit with corresponding energy level (n + 1/2)hνB (n = 0, 1, 2, . . .), where νB = eB/2πm0 c is the Landau frequency. The most probable emission and/or absorption occurs at base frequency νB due to the most important transitions occur between the neighboring levels n n + 1, especially 0 1. Therefore in the S frame, absorption occurs as long as the frequency of the incident photon equals the base frequency, νi = νB , and the corresponding absorption transition is 0 → 1. The emission transition 1 → 0 follows immediately due to an extremely high probability of spontaneous transition, e.g., a10 = 1015 s−1 for B ∼ 1012 G s [1]. Therefore the combined process of a resonant absorption transition 0 → 1 and the subsequent re-emission 1 → 0 is equivalent to a “resonant scattering” of an incident photon with base frequency νi = νB . Returning to the laboratory system S, we get a cyclotron-resonant RICS of a fast electron. The resonance behavior leads to a series of special properties and advantages of the RICS radiation mechanism [3–8]: (1) Very high radiation efficiency. From Eq. (1) of the Letter we show that the total cross-section of resonance scattering at the line center νi = νB is σ s (νB ) ∼ 108 σT , for B ∼ 1012 G s, where σT is the
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311
(0–2γ νB ). However, as shown in Fig. 1, the calculated RICS spectrum of a single electron still retains a good monochromaticity. The RICS radiation power of a fast electron with energy γ , passing through a low frequency soft photon field, is calculated via Lorentz transformation of reference frames from the co-moving frame S to the laboratory frame S [13]. For shortness, in the following we omit the complicate derivations and only present the resultant formulae which are related to the application of RICS in hard X-ray and γ -ray astrophysics.
Fig. 1. The radiation spectrum of the RICS of a single electron with energy γ shows a good monochromaticity, which is quite different from the ordinary inverse Compton scattering. Note the sharp peak at the maximum frequency 2γ νB and the weak low-frequency tail extending to ν → 0.
Thompson cross-section, which implies a very high RICS efficiency. (2) Very high scattering frequency. In S the scattering frequency is given by the resonance condition νs = νi = νB . Returning to the observer frame S, the Doppler formula gives νs = γ νs (1 + β cos ψs ) = γ νB (1 + cos ψs ) ∼ γ νB (taking β 1), i.e., the typical RICS frequency is on the order of γ νB . In the case when B ∼ 1012 G s, hνB ∼ 10 keV, the scattering photon would then have an energy hν ∼ γ hνB , or 100 keV–10 MeV if γ ∼ 101–3. The RICS mechanism is hence very important for hard X-ray and γ -ray astronomy, e.g., in the studies of γ -ray pulsars and γ -ray bursts, etc. [12]. (3) Very good beaming of RICS radiation. The photons produced by the RICS radiation propagate along the magnetic field lines within an extremely narrow angular cone with θ ∼ 1/γ . This beaming property greatly suppresses the strong absorption of the RICS γ -ray photons by the magnetic and/or γ –γ annihilations, which would otherwise attenuate heavily the γ ray radiation with hν 1 MeV. (4) Good monochromaticity of the RICS spectra of a single fast electron, concentrating most of radiation energy in the high frequency band. In S system, the scattering frequency is exactly monochromatic, given by the resonance condition νs ≡ νi = νB . In observer frame S, it becomes νs = γ νB (1 + cos ψs ), due to the Doppler effect. Thus νs distributes over a wide range
2. The differential scattering cross-section The simplified scattering cross-section in S frame, derived from the classical quantum theory [1,2], is 3 σ s (νi , ψi , ψs ) = r0 c 1 + cos2 ψi 1 + cos2 ψs 32 × φ(νi − νB ), (1) where ψs is the scattering angle in S , r0 is the classical radius of an electron, νB = eB/2πm0 c = 2.8 × 106B Hz, and the Lorentz profile φ(νi − νB ) = (Γlu / 4π 2 )[(νi − νB )2 + (Γlu /4π)2 ]−1 , with Γlu ≡ Γl + Γu being the quantum damping constant of the upper (u) and the lower (l) levels.
3. The resonance condition The resonance condition in S is νi = νB . Transforming to the laboratory frames S, S → S, from the Doppler formulae we get νB . νi (2) γ (1 − cos ψi ) Eq. (2) is the expression of the resonance condition νi = νB in the laboratory frame S, which indicates that the frequency of the incident photons to be resonantly absorbed cannot be taken arbitrarily, but is restricted by the given angle ψi , νi = νi (ψi ). 4. The approximate resonance condition — accommodation condition The fractional number of photons with small incident angles ψi 0 is very small in a general, isotropic
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J.-h. You et al. / Physics Letters A 286 (2001) 309–313
radiation field. Furthermore, Eq. (2) shows that ψi 0 corresponds to a high frequency νi . Since the number of photons with very high νi is nominally very small in a low-frequency radiation field, therefore, neglecting the part of small incident angle scattering ψi 0, (2) becomes approximately γ hνi hνB .
(2 )
The approximate resonance condition (2 ) also provides a useful semi-quantitative criterion to estimate the efficiency of RICS of a relativistic electron passing through a low-frequency radiation field. The RICS radiation is significant only when the averaged electron energy γ¯ times the averaged photon energy of the lowfrequency radiation field (h¯ν ) is comparable to hνB . In fact, in this case, the coexistent, ordinary inverse Compton scattering is negligibly small and needs not be taken into account. Therefore, (2 ) is regarded as an “accommodation condition” for the energy spectrum of electrons in a given low-frequency radiation field and a given magnetic field B.
where x ≡ ν/2γ νB = ν/νmax is a dimensionless scattering frequency in unit of the maximum scattering frequency 2γ νB , and the function f (x) is 3 2 f (x) = 2x − 2x + x, if 0 x 1, (4) 0, if x > 1. The function f (x), depicted in Fig. 1, specifies the spectral shape of RICS of a single electron, which can be regarded as a dimensionless spectral power of RICS. Fig. 1 shows that the RICS spectrum still retains a good monochromaticity — a sharp peak near the maximum frequency 2γ νB with a weak low-frequency tail extending to ν 0 — despite the very wide frequency band. In some semi-quantitative discussions, we therefore adopt a quasi-monochromatic approximation by assuming an approximate one-to-one correspondence between the electron energy γ and the emission frequency ν, i.e., an electron with energy γ produces a monochromatic line radiation with frequency 2γ νB , γ ↔ 2γ νB .
6. The resonance scattering efficiency A(γ , B) 5. RICS radiation spectrum of a single relativistic electron Before we calculate the spectral power of RICS of an electron with energy γ , it is helpful to give a qualitative analysis for the spectrum of RICS. In the S system, the main scattering frequency is given by the resonance condition, νs ≡ ν = νi = νB , which shows a strict monochromaticity of scattering in the S system. However, returning to the laboratory system, we get the scattering frequency in the S system (again taking β 1), νs ≡ ν = γ ν (1 + β cos ψs ) γ νB (1 + cos ψs ). Due to the approximate isotropy of scattering in S (see Eq. (1)), the range for the scattering angle ψs is (0, π). Therefore, the scattering frequency should spread in a wide range (0, 2γ νB ), with a large highfrequency cutoff, 2γ νB . However, our calculation shows that the quantitative RICS spectrum still keeps a very good monochromaticity. The calculated RICS spectrum of a fast electron with energy γ in a strong magnetic field B is given by dpRICS = (3πr0 c)A(γ , νB )γf (x), dx
(3)
The quantity A in Eq. (3) denotes an integral π I (νi , ψi )(1 − cos ψi ) sin ψi dψi ≡ A(γ , B),
2π 0
(5) where I (νi , ψi ) denotes the intensity of a given monochromatic incident beam with frequency νi and incident angle ψi in the low-frequency field. Note that the intensity I is always a function of ψi , even when the radiation field of soft photons is isotropic, for which the dependence of the intensity I on ψi appears not explicit, I in fact is still related to ψi through the function νi (ψi ) (see Eq. (2)), i.e., I (νi ) = I (νi (ψi )), and the form of I (νi ) is determined by the given radiation field. Thus for the π isotropic lowfrequency field, Eq. (5) becomes 2π 0 I (νi (ψi ))(1 − cos ψi ) sin ψi dψi ≡ A(γ , B). The quantity A, a function of γ and B, can be regarded as a measure of the “resonance scattering efficiency”, in the sense that it includes all the “eligible” soft photons incident from all directions which are capable of being resonantly absorbed (scattered) by the electron with energy γ . It
J.-h. You et al. / Physics Letters A 286 (2001) 309–313
can be seen that, the stronger the intensity I (νi (ψi )) at the resonance frequency νi = νi (ψi ), the higher the “resonance scattering efficiency” A, hence the RICS radiation power (see Eq. (3)).
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Acknowledgement This research is supported by the National Natural Science Foundation of China under Grant No. 19573005, the Climbing Program of State Committee of Science and Technology of China.
7. The total RICS power The total power P RICS of a fast electron is thus 1 P
RICS
=
dpRICS dx = (πr0 c)A(γ , νB )γ . dx
References
(6)
0
Thus P RICS is proportional to the product of the “efficiency” A and energy γ of the fast electron. We emphasize that, in an intense magnetic field B, the conventional ICS process still coexists with the RICS process. As discussed above, RICS process occurs only when the incident frequency νi = νB in the electron-rest frame S . If the resonance condition is not satisfied, i.e., if νi = νB , we then have the ordinary ICS. In general, for a given system of fast electrons in a low-frequency radiation field, as long as the “accommodation condition” (2 ) prevails, the conventional ICS is negligibly weak compared with the coexistent RICS.
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