A&A 409, 9–19 (2003) Astronomy DOI: 10.1051/0004-6361:20031060 & c ESO 2003 Astrophysics Cyclotron-synchrotron: Harmonic fitting functions in the non-relativistic and trans-relativistic regimes A. Marcowith1 and J. Malzac2,3 1 C.E.S.R., 9 Av. du Colonnel Roche, 31028 Toulouse Cedex 4, France 2 Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, UK 3 Osservatorio di Brera, via Brera, 28, 20121 Milano, Italy e-mail: [email protected] Received 14 April 2003 / Accepted 30 June 2003 Abstract. The present work investigates the calculation of absorption and emission cyclotron line profiles in the non-relativistic and trans-relativistic regimes. We provide fits for the ten first harmonics with synthetic functions down to 10−4 of the maximum flux with an accuracy of 20% at worst. The lines at a given particle energy are calculated from the integration of the Schott formula over the photon and the particle solid angles relative to the magnetic field direction. The method can easily be extended to a larger number of harmonics. We also derive spectral fits of thermal emission line plasmas at non-relativistic and trans- relativistic temperatures extending previous parameterisations. Key words. line: profiles – magnetic fields – radiation mechanisms: non-thermal, thermal 1. Introduction GHS98, Wardzi´nski & Zdziarski (2000), hereafter WZ00 and Wardzi´nski & Zdziarski (2001). The cyclotron-synchrotron radiation is produced by charged A detailed numerical investigation of the thermalisation particles spiraling in a magnetic field. The non-relativistic limit process in the presence of cyclo-synchrotron photons has (for particle speed β = v/c 1) is the cyclotron radiation. scarcely been discussed (see however GHS98) but is of great The relativistic limit (for particle speed β = v/c ∼ 1) is the importance for the spectral and temporal modeling of the com- synchrotron radiation. At intermediate speeds, the process is plex spectral energy distribution produced in compact objects. sometimes called gyro-magnetic (for β = v/c ∼ 0.5). Low temperature plasmas that may be found in gamma-ray The cyclotron-synchrotron effect is one of the most impor- bursts (in the comoving frame) and in accretion discs may tant processes in astrophysics and has been invoked in solar, produce cyclotron lines combined with synchrotron signatures. neutron star physics and in galactic or extra-galactic jets. At Such line features can be produced in neutron star magneto- non relativistic and ultra relativistic limits, the particle emis- spheres probing the magnetic field. sivity is provided by well known analytical formulae. The present work aims to propose useful fitting formu- The main contribution to the cyclotron radiation comes lae for the first ten harmonics dominating the emissivity for from low harmonics. In most astrophysical objects, this radi- particle velocities (or normalised momentum) between β = ation is either self-absorbed or absorbed by the environment. 0.01 and p = βγ = 2 (see Fig. 2 in Mahadevan et al. 1996 It can even be suppressed by plasma effects such as in the hereafter MNY96). We consider the case of an isotropic elec- Razin-Tsytovich effect. However, as stressed by different au- tron distribution in a tangled magnetic field. The method can be thors (Ghisellini & Svensson 1991, hereafter GS91; Mason extended to an arbitrary number of harmonics. At higher ener- 1992; Gliozzi et al. 1996) the cyclotron cross-section in the gies, however, the synchrotron formulae are worth being used. non- and mildly relativistic regimes is several orders of magni- The derived functions can then be easily inserted into radiative tude above the Thompson cross section. The cyclotron mech- transfer codes. anism appears then as a very efficient thermalisation process in hot plasmas surrounding compact objects. At higher en- The article is organised as follows: Sect. 2 recalls the ergies, the synchrotron photons are a supplementary source derivation of cyclo-synchrotron emissivity. Section 3 deals with for the Comptonisation process in accretion discs coronæ, see the procedures used to compute the emission and the absorp- ffi e.g. Di Matteo et al. (1997), Ghisellini et al. (1998), hereafter tion coe cient: an integration over a small frequency interval around the cyclotron resonant frequency and a direct integra- Send offprint requests to: A. Marcowith, tion over the angle between the photon and the magnetic field. e-mail: [email protected] These procedures are tested in Sect. 4. Section 5 provides the Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20031060 10 A. Marcowith and J. Malzac: Cyclotron-synchrotron: Harmonic fitting functions 2 fitting functions in the non-thermal case, at a given particle en- where the critical frequency is νc = 3/2 νbγ , x = ν/νc and ergy. Section 6 provides the fitting functions for the emission K5/3(t) is the modified Bessel function of order 5/3. The parti- coefficients produced in a thermal plasma. cle emissivity scales as ν1/3 for x 0.29 (0.29 is the energy of maximum emissivity) and is exponentially cut-off at high energies (Ginzburg & Syrovatskii 1969). 2. Cyclo-synchrotron emissivity The cyclo-synchrotron spectrum produced by one particle of 3. Calculation procedures mass m, charge e and velocity v = β · c embedded in a uniform magnetic field B can be derived from classical electrodynamics We developed two methods to numerically estimate the cy- (see for example Bekefi 1966, chapter 6). The emitted power clotron emissivity and absorption cross-section of one particle spectrum, the so-called Schott formula is expressed as a sum of at a given energy and frequency. Each method aims to treat the harmonics peaking at a frequency ν = n ν b/γ (n ≥ 1), where resonant term inside the Dirac function in a correct way, see γ = (1 − β2)−1/2 is the Lorentz factor of the particle. The cy- Eqs. (1) and (2). clotron frequency is νb = eB/(2π mc), where the magnetic field B is expressed in Gauss units. 3.1. Frequency integration 2 ∞ θ − β 2 dP e 2 (cos ) 2 2 2 = 2π ν J (x) + β⊥ J (x) δ(y) , (1) The first procedure consists in integrating the expression (1) dνdΩ c 2 θ n n ν ν − ∆ν/ ,ν+∆ν/ n=1 sin over in a narrow frequency range [ 2 2], with ∆ν = αν ν, α being a positive real variable. = νγβ θ/ν β = β θ β = β θ with x ⊥ sin b, cos p, ⊥ sin p and Equation (1) becomes , Jn Jn being the Bessel functions of (integer) order n and its first derivatives. 2πe2ν2 n2 2 dP θ ∼ b n The cyclotron resonance is defined by ( p) ν Ω ανc γ2 − β θ 3 d d n=n (1 cos ) ν 1 y = b − ν − β θ = . θ − β 2 n (1 cos ) 0 (2) cos 2 2 2 γ × J (x ) + β⊥ J (x ) . (6) sin θ n r n r Equation (1) is expressed in CGS units (e.g. erg s−1 Hz−1 st−1) = β θ/ −β θ as well as all equations in the text. The resonant term is xr n ⊥ sin (1 cos ). The limits n1 ν − ∆ν/ The angles are defined relative to the dominant magnetic and n2 are fixed by the resonant condition (2) taken at 2 and ν +∆ν/2. field direction: θp is the particle pitch-angle and θ is the photon angle. Integrating Eq. (1) over the photon angle and the frequency, 3.2. Direct integration averaging the pitch-angle leads to the total power emitted by one particle The second procedure eliminates the resonance through a di- rect integration over θ, the angle between the photon and the 4 B2 magnetic field. The resonance condition selects an angle for a P = σ c (γβ)2 , (3) tot 3 T 8 π given harmonic n, a given pitch-angle and β ν σ . × −25 2 n b where the Thomson cross section is T 6 65 10 cm . 1 − γν cos θr = · (7) At the non-relativistic limit, only the first harmonics con- β tribute significantly to the emissivity. For the particular pitch- angle θp = π/2, and after integration over all observer angles, Integrated over the solid angle, Eq. (1) leads to the power emitted in a given harmonic n is (Bekefi 1966) n 2 π 2ν 2 cos θ − β dP θ = 2 e r 2 + β2 2 . 8π2 e2ν2 + 2n+1 ( p) Jn (xr) ⊥ Jn (xr) (8) b (n 1)(n ) 2n dν βc sin θr η = β . (4) n=n1 n c (2n + 1)! The resonant term is now x = νγβ⊥ sinθ /ν . The sum bound- β 2 r r b The ratio of two successive harmonics scales roughly as . aries n and n define the range of n for which | cos θ |≤1in β → 1 2 r In the relativistic limit ( 1) the emissivity is dominated Eq. (7). by the harmonics of order n ∼ γ 2. The spectrum tends towards We note that for cos θp = 0, cos θr becomes undefinite and a continuum as the frequency interval between two harmonics Eq. (8) breaks down. In this case, all the power is radiated at is νb/γ. The synchrotron power radiated by one particle per unit the resonant frequencies, i.e. dP/dν(θp = π/2) = 0 except at frequency interval is given by (we still assume θp = π/2) frequencies ν = mνb/γ and at those frequencies the power per √ unit frequency is actually infinite. The total power emitted in 2 ν ν ∞ dP 3e b θ = π/ = K5/3(t)dt (5) harmonic n at pitch angle p 2 may be estimated by inte- dν c ν ν c ν grating Eq.