Formation of the 255-Kev Annihilation Line Near the Magnetic Poles of Pulsars V
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JETP Letters, Vol. 79, No. 6, 2004, pp. 241–244. Translated from Pis’ma v Zhurnal Éksperimental’noÏ i TeoreticheskoÏ Fiziki, Vol. 79, No. 6, 2004, pp. 299–302. Original Russian Text Copyright © 2004 by Kontorovich, Flanchik. Formation of the 255-keV Annihilation Line near the Magnetic Poles of Pulsars V. M. Kontorovich1, 2 and A. B. Flanchik2 1 Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkov, 61002 Ukraine 2 Kharkov State University, Kharkov, 61077 Ukraine Received December 22, 2003; in ﬁnal form, February 17, 2004 The possibility of existance is discussed for pulsars emitting soft γ radiation near their magnetic poles upon the annihilation of ultrarelativistic positrons from the magnetosphere and electrons from the surface of a star. With an increase in the energy of incident positrons, the photon energy of this backward radiation tends to a constant 2 value mec /2 = 255 keV. This radiation is shown to be directed opposite to the positron ﬂux direction. © 2004 MAIK “Nauka/Interperiodica”. PACS numbers: 97.60.Gb; 97.10.Ld; 98.70.Rz 1. A pulsar is a source of electromagnetic radiation and, with allowance for the general-relativity effects, over a wide range from radio waves to hard γ radiation the coefﬁcient A is given by the formula [5, 6] 8 γ up to a photon energy of 10 MeV. As is known, rays 3ΩB ω can be emitted due to both ultrarelativistic electrons A = -------------0 4---- cosαε+ cosφ sinα . (2) 2 cR Ω 0 m and positrons moving in a strong magnetic ﬁeld and various collision processes between charged particles. Here, h < H is the altitude over the pulsar surface, H is In the former and latter cases, this radiation is called the gap altitude determined by the condition for gener- bend radiation and collisional bremsstrahlung, respec- ating the secondary component of e–e+ plasma, Ω is the tively. In this work, we analyze the possibility of gener- angular velocity of the pulsar, R is the pulsar radius, B0 ating soft γ radiation caused by the annihilation line ε Ω is the magnetic ﬁeld at the pulsar surface, 0 = R/c formed due to the annihilation of ultrarelativistic is the pulsar geometric factor, α is the angle between positrons from the magnetosphere and electrons from 3 the magnetic and rotation axes, ω = ΩR /r3, and φ is the pulsar surface. Since the electron velocities in the g m pulsar surface are much lower than the speed of light, the azimuth angle measured from the direction of the northern magnetic semiaxis of the pulsar. The mecha- we consider electrons to be at rest. As is known [1], nism considered for generating γ radiation can be real- charged particles in the pulsar magnetosphere move ized only if E|| < 0. According to Eq. (2), there are mag- along magnetic lines of force, because the motion in the netic lines of force on which the inequality E|| < 0 is sat- plane perpendicular to the magnetic ﬁeld rapidly isﬁed. Such lines were previously called preferable. becomes impossible due to synchrotron radiation. Since the second term in Eq. (2) is small, preferable Positrons can be accelerated in the pulsar magneto- lines are most magnetic lines of force on which sphere by an electric ﬁeld. However, this acceleration is positrons moving along them are accelerated by ﬁeld not possible over the entire magnetosphere, because, in (1) toward the pulsar surface. plasma-ﬁlled regions rotating with the pulsar, the con- The mechanism of forming the 255-keV annihila- dition E · B = 0 for electric E and magnetic B ﬁelds is tion line is shown in the ﬁgure. When moving along the fulﬁeld [2]. Positrons are accelerated toward the pulsar magnetic lines of force, ultrarelativistic positrons from the magnetosphere are incident on the pulsar surface, surface in a gap, i.e., in the region over the magnetic γ pole of the pulsar, where E · B ≠ 0 and an electric ﬁeld and radiation arises due to their annihilation with electrons. Radiation in the direction of positron motion, directed along the magnetic ﬁeld exists. In this work, i.e., forward radiation, carries almost the entire positron the electric ﬁeld of the pulsar is treated in the Arons energy to the pulsar interior, and a soft backward γ radi- model [3, 4]. In this model, the longitudinal electric ation with an energy of 255 keV is emitted opposite to ﬁeld in the gap is represented in the form the positron beam. The spectrum and angular distribu- tion of this radiation will be discussed below. 1 2 E||()h = ---Ah()– Hh , (1) 2. The formation of the annihilation line is primarily 2 determined by the two- and three-photon electron– 0021-3640/04/7906-0241 $26.00 © 2004 MAIK “Nauka/Interperiodica” 242 KONTOROVICH, FLANCHIK From these relations, the desired energies of annihila- tion photons are expressed as ε χ ω ()ε + – p+ cos 1 = + + m ε------------------------------------------χ, + + m – p+ cos (5) ()ε ω m + + m 2 = ε------------------------------------------χ. + + m – p+ cos 3. Photon energies (5) are functions of two parame- ters: the energy of an incident positron and the emission angle χ. To determine the angular distribution of anni- hilation photons, we use the following general formula for the differential cross section for two-photon annihi- lation [8]: π 2 2 2 2 2 dσ 8 re m m m ------ = -------------------------------------- + --------------- ds tt()– 4m 2 sm– 2 um– 2 Mechanism of the formation of annihilation line: I is the gap (6) 2 2 2 2 region, where E|| ≠ 0; II is the region of open ﬁeld lines, m m 1sm– um– + -------------- + --------------- – --- --------------- + --------------- , where E|| = 0; and III is the corotation region. The annihila- 2 2 42 2 tion of a positron from the pulsar magnetosphere and a sur- sm– um– um– sm– face electron is accompanied by the backward emission of the 255-keV line. where s, t, and u are the standard kinematic invariants given by Eqs. (3) and re is the classical electron radius. Let us calculate expression (6) in the laboratory frame, positron annihilation events. In this work, three-photon where the electron is at rest. Using the expressions t = ε 2 ω annihilation affecting the width and wings of the anni- 2m(m + +) and u – m = –2m 2 and differentiating the 2 ω ε | | χ hilation line [7] is disregarded. expression s – m = –2 2( + – p+ cos ), we obtain ω ()ε χ ω χ Let us consider the two-photon annihilation kine- ds = –22d 2 + – p+ cos + 2 p+ d cos matics in the laboratory frame, where an electron is at ∂ω rest. We use the standard kinematic invariants [8] 2 = 2 ω p – --------------- ()ε – p cosχ d cosχ 2 + ∂χcos + + sp==()– k 2 ()k – p 2, – 1 2 + ()ε m + + m = – p+ ε------------------------------------------χ (7) ()2 ()2 + + m – p+ cos tp==– + p+ k1 + k2 , (3) ()εε ()χ m + + m + – p+ cos do ()2 ()2 – -------------------------------------------------------------- ------, up==– – k2 k1 – p+ , ()ε χ 2 π + + m – p+ cos where p–, p+, k1, and k2 are the 4-momenta of the elec- do = 2πχsin dχ. tron, positron, and annihilation photons, respectively. Substitution of Eq. (7) into general expression (6) gives Let us express the frequencies of annihilation photons through the angle χ between the direction of positron 2 ()ε , χ re G + motion and the photon 3-momentum k (ﬁgure). In the dσ = ------------------------- F()ε , χ do, (8) 2 2 p + laboratory frame, the electron and positron have + ε 4-momenta p– = (m, 0) and p+ = ( +, p+), respectively. where Therefore, from Eqs. (3) and the 4-momentum conser- ()ε ()ε , χ m + + m vation law p– + p+ = k1 + k2, we obtain G + = ε------------------------------------------χ + + m – p+ cos 2 2 sp==()– k ()k – p , m()εε + m ()– p cosχ , – 1 2 + – ---------------------------------------------------------------+ + + - []()ε χ 2 + + m – p+ cos m2 – 2mω = m2 – 2 p k , (4) 1 + 2 ε χ ()ε , + – p+ cos m ω ε ω ω χ ω ()ε χ F + m = -------------------------------- + ε-------------------------------χ- (9) m 1 = + 2 – p+ 2 cos = 2 + – p+ cos . m + – p+ cos JETP LETTERS Vol. 79 No. 6 2004 FORMATION OF THE 255-keV ANNIHILATION LINE 243 2m21 1 with an energy of 255 keV are emitted opposite to the + ---------ω ---- + ε-------------------------------χ- positron beam incident on the surface. Indeed, as was 2 m + – p+ cos mentioned above, a photon with momentum k2 is emit- ted at a small angle to the direction of positron motion. m41 1 2 – ------ ---- + -------------------------------- . Let us calculate the angle φ between the momenta of 2 ε χ ε ω m + – p+ cos annihilation photons. From the relation t = 2m(m + +) = 2 ω ω φ ε 2k1k2 = 2 1 2(1 – cos ), we obtain Since the denominators in expressions (9) contain + + | | χ ε | | χ mm()+ ε m – p+ cos and + – p+ cos , cross section (8) is non- φ + χ cos= 1 – ------------------------ω ω -. (15) zero only for small angles . This property is a manifes- 1 2 tation of the general property (relativistic aberration) of radiation from an ultrarelativistic particle [9]. Almost In view of limiting expressions (12)–(14), formula (15) the entire positron energy is emitted along its direction yields of motion. It follows from Eqs. (9) that there are two χ ≤ ε small-angle regions: the region χ < m/ε , where m/(ε – –1, m/ +, + + |p |cosχ) ӷ 1, and the region χ < m/ε , where cosφ = m2 2m χ2 m m (16) + + 12– ---------- – ------- – -----, ----- < χ ≤ -----. m/(ε + m – |p |cosχ) ӷ 1. ε2 χ2 ε 2 ε ε + + + + + + (i) In the region χ < m/ε , the product + φ χ –1 The limiting value of cos for 0 is equal to –1; m2 χ2 i.e., photons move in the opposite directions.