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5. SYNCHROTRON RADIATION 5.1 Charge Motions in a Static Magnetic

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5. SYNCHROTRON RADIATION 5.1 Charge Motions in a Static Magnetic 5. SYNCHROTRON RADIATION1 5.1 Charge motions in a static magnetic field Charged particles moving inside a static magnetic field continuously accelerate due to the Lorentz force and continuously emit radiation. In the case of non-relativistic motions, the emitted radiation is named cyclotron radiation and is due to the spiraling motion of particles and consequent acceleration. The radiation frequency is just that of the gyration qB/mc, where q and m are the electric charge and mass of the particle. This is a line emission, if we neglect broadening factors. The nature of the emission changes radically if the particle becomes relativistic, and the emission is then named synchrotron emission. The motion of a generic charge is ruled by the Lorentz dynamical equation of motion [5.0] and the energy conservation equation 1 This Section particularly benefited, in addition to the Rybicki & Lightman book, by lecture notes on High Energy Astrophysics by A. Cavaliere. 5.1 (the second term has been neglected because of the scalar product and because an assumption about the lack of significant large scale electric fields - that would be immediately erased by charge motions - was made). Then we use the second eq. into the former and get (see the figure below for an illustration of the relevant quantities) [5.1] The solution is then an helicoidal motion (see graph below), with the velocity along the direction of the B field keeping constant, and a uniform circular motion around that direction with rotational pulsation given by 5.2 qB ω = . [5.2] B γ mc 5.2 Total synchrotron emitted power The total emitted power by the particle is easily studied in the instantaneous reference frame of the electron (K'). Here the particle will not remain at rest for long time, but just for a short time interval and the electron motion in that frame will in any case be non-relativistic during this time. So we can use here the Larmor formula for the emitted power. Alternatively, one might use the full relativistic eq. (1.21), but this would be a quite more complicated derivation (see Sect. 5.7 below). Because energy and time transform in the same way from K' to the system K of the observer (and of the B field), we have dW= γγ dW'; dt= t ' dW dW ' =PP =' = dt dt ' 2 2q d 2 PP='' = a 3c3 We need to consider how the vector acceleration transforms from one system (K) to the other (K'). It can be seen (e.g. Rybicki & Lightman) that the following relativistic rules apply if the particle is at rest in K', where parallel and orthogonal refer to the direction of the instantaneous velocity vector 32 aa';'= γγ a⊥⊥= a . From this and [5.2] we have qB ar=vv =ωω2 = ⊥⊥γ mc BB ⊥ 2 22 22q4 qB 2 2 22 2 P= γ v = rcβγ B [5.3] 33c3γ 2 mc 22 ⊥ 0 ⊥ 2 e2 v r = ; β2=⊥ = βa 22sin 0 mc22⊥ c 5.3 r0 is the classical electron radius, qe= the electron charge. a is the pitch angle, that is the angle between the velocity and B field vectors, as shown in the figure: For an isotropic distribution of charge velocities, we average over α as: ββ222 βa22= sin dΩ= , such that ⊥ 43π ∫ 2 2 2 22 2 P= rc0 βγ B 3 2 or, if we use the Thomson cross section σπT = 83r0 and the energy density of the magnetic field UB 4 22 2 P= σ c βγ Uwith UB= / 8π [5.4] 3 TB B For the cyclotron emission we have instead 4v2 PU= σ . 3 TBc 5.3 Aberration, beaming, angular distribution of radiation Recall that in the lab reference system, in which the electron is seen moving with relativistic velocity, the emission is strongly enhanced (beamed) in the direction of the velocity. According to the relativistic aberration law, we have 5.4 where α is in the lab frame (K) and α' in the electron frame (K'). This implies that , except when . We can see this by a power series development If this condition does not verify, then (see below figure for an illustration of effect) [5.5] Case of Case of 5.5 5.4 The transition from the cyclotron to the synchrotron spectrum Let us consider here the evolution of a synchrotron spectrum at increasing electron velocity and energy. This makes a transition from a cyclotron line emission to the synchrotron continuum one that is illustrated by the following figures. 5.6 For another graphical representation we refer to Ghisellini (High Energy Astrophysics): From cyclo to synchro: if the emitting particle has a very small velocity, the observer sees a sinusoidal (in time) electric field E(t). Increasing the velocity the pattern becomes asymmetric, and the second harmonic appears. For 0 < β ≪ 1 the power in the second harmonic is a factor β2 less than the power in the first, see eq. (5.4). For relativistic particles, the pattern becomes strongly beamed, the emission is concentrated in the time ∆tA. As a consequence the Fourier transformation of E(t) must contain many harmonics, and the power is concentrated in the harmonics of frequencies ν ∼1/∆tA. Broadening of the harmonics due to several effects ensures that the spectrum in this case becomes continuous. Note that the fundamental harmonic becomes smaller at increasing γ (since νB ∝ 1/γ). The spectrum of emission of the first 20 harmonics of mildly relativistic cyclotron radiation for an electron with v = 0.4c (Bekefi, 1966). 5.7 Hard X-ray spectrum of the neutron star X0115+63 observed with Beppo-SAX, showing the first four cyclotron harmonics. (From Santangelo et al. 1999) Cyclotron lines are observed more frequently in absorption in the high-energy spectra of neutron stars, like in the above figure, where the first 4 harmonics are observed. From this, the value of the (strong) stellar magnetic field can be derived from : where the last factor accounts for the gravitational redshift . 5.5 Qualitative behavior of the synchrotron spectrum of a single pulse We have seen that the spectral emission in the case of a cyclotron is concentrated essentially in a line of frequency . All the electron orbit is seen by the observer. Instead, in the synchrotron case, because of the relativistic beaming, the radiation field seen by the observer is concentrated inside a narrow cone of width around the electron velocity vector. As a consequence, the observer will see photon emission only from a small fraction of the orbit: therefore, the emitted spectrum will 5.8 include a much wider range of photon frequencies. All this is easily understood in terms of the Fourier transformation. Let us consider the above figure (from Ribicki & Lightman) representing the portion of the orbit seen by the observer. a is the radius of the orbit. The points 1 and 2 correspond to the extremes of the sector seen: in 1 the emission cone enters the line- of-sight, in 2 it exits it. The emission cone is shown in the two points. The angle Δθ is that subtending the sector, and is given by . The distance Δs between positions 1 and 2 is then [5.6] Now the radius of the orbit can be obtained from [5.7] where is the gyration frequency of the relativistic electron from [5.2] and where α is the pitch angle (obviously, the radius a increases when α decreases, that is the velocity approaches the B direction). So, from [5.6] and [5.7] [5.8] 5.9 In the observer frame K, the transit times t1 and t2 are such that ∆=sv( tt21 − ) 2 ∆=tt21 − t = γωB sina This time interval however does not correspond to the times of arrival to the observer of the emitted photons, because these will be shorter by an amount equal to ∆sc/ , the time that radiation emitted from 1 takes to travel from 1 to 2. So the arrival times will be arr 2v1 ∆=t 1 − 3 γωaBBsin c γωasin because 2(1-v/c)∼1/γ2. In conclusion, the observed pulse will have a temporal duration 1 times the period of the electron gyration. γ 3 From consideration of the Fourier transform, we then expect the presence of characteristic photon frequencies in the observed spectrum proportionally larger by a 3 factor γ , that is 3 1132 1γ qB n= ω = γωsin a= sina= γn char 22π π B 2πγmc cycl We then define a critical frequency for the synchrotron emission as: 3 2 n= γω sina [5.9] C 2 cycl 5.10 So we expect significant emission to happen at this frequency. We will see later that indeed the peak synchrotron emission happen at this critical frequency. Let us see now some further scaling relations concerning the spectrum of a single radiation pulse by a single orbiting electron. Due to the relativistic beaming and the enormously increased emitted power in the direction of the electron velocity vector, the dependence of the electric field intensity E on the angle θ (the polar angle between the instantaneous velocity vector and the observer line-of-sight) goes under the product γθ . So also concerning the temporal dependence we have Et()∝ g[γθ () t], where t is the time in the K observer frame. This means that a given value of E will be obtained for large values of θ if γ is proportionally small, or vice-versa small θ if γ is large. Let us set the origin of times t and paths s when θ = 0 (this corresponds to when the velocity vector is along the visual).
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