Synchrotron radiation and pair-production
Many important results of the theory of synchrotron radiation are obtained within the framework of classical electrodynamics (see e.g. Jackson, 1975). The classical treatment of the synchrotron radiation is limited by the condition
where
is the so-called critical value of the magnetic field relevant to quantum effects.
Generally, the energy of synchrotron photons are much less than the energy of parent electrons. However, in specific astrophysical environments, e.g. in pulsar magnetospheres or in magnetised accretion disks, the synchrotron radiation could be close to the "quantum threshold" of about 
Gauss. The production of electron-positron pairs in a magnetic field by high energy Y-rays is, by definition, a quantum process. In the quantum regime these two processes are tightly coupled, and lead to an effective cascade development.
For interactions of electrons and photons with magnetic fields it is convenient to introduce interaction probabilities instead of standard total cross-sections (Anguelov and Vankov, 1999). But in the literature this parameter is still formally called a cross-section. These probabilities, normalised to the strength of the magnetic field, are shown in Fig. 3.6.
The probabilities of both synchrotron and pair production processes depend on a single
where B is the component of the magnetic field perpendicular to the vector of the particle speed. This parameter is an analog of the parameters k0 and s0 in the photon field. While the probability of synchrotron radiation at
is constant, the probability of pair production below
drops dramatically (proportional to
After achieving its maximum at
the probability of pair production decreases with
(see Erber, 1966). At large X0, the probability of synchrotron radiation has a similar behaviour, but its absolute value exceeds by a factor of 3 the probability of pair production.
The differential spectra of Y-rays due to synchrotron radiation and the electrons due to the magnetic pair-production are shown in Fig. 3.7. The cross-sections for synchrotron radiation and pair production are from Akhiezer et al. (1994).
Fig. 3.6 Total cross-sections (interaction probabilities) of the synchrotron radiation and magnetic pair production.
At
the synchrotron Y-ray spectra are very steep, but at large values,
Y-rays are characterised by a flat distribution. This implies a rather catastrophic character of interaction like in the photon gas. Note that in the photon gas at
the fraction of the parent electron energy that is transferred to Y-rays exceeds 0.5 and asymptotically approaches 1. In the magnetic field the energy transfer is smaller. At
it is approximately 0.1, and asymptotically approaches to 16/63 ~ 0.25 at extremely large xo (see Table 3.1). The energy distribution of pair-produced electrons is obviously a symmetric function.
Fig. 3.7 Differential cross-sections of synchrotron radiation (upper panel) and the magnetic pair production (bottom panel) normalised to the total cross-sections of these processes. The values of the parameter
are indicated by the curves.
Although at large xo the electron spectrum increases when
the pair production spectra in a magnetic field are flatter than in radiation fields, and correspondingly the energy of the primary Y-ray in the photon gas is transferred to the leading electron with somewhat higher efficiency (see Table 3.2).
Synchrotron radiation of protons
Generally, proton synchrotron radiation is treated as an inefficient process. However, under certain conditions the synchrotron cooling time of protons can be comparable or even shorter than other timescales that characterise the acceleration and confinement regions of ultrarelativistic protons. Moreover, in compact accelerators of ~ 1020 eV protons, very high energy synchrotron or curvature Y-radiation of protons always accompanies acceleration of the highest energy particles.The formalism of proton synchrotron radiation is quite simple and identical to the theory of electron synchrotron radiation. Nevertheless, it is worth discussing some basic features of this process relevant to the magnetic-field dominated environment in which the protons are accelerated at the theoretically highest possible rate.
The comprehensively developed theory of electron synchrotron radiation (see e.g. Ginzburg and Syrovatskii, 1965) can be readily applied to the proton-synchrotron radiation by re-scaling the Larmor frequency 
by the factor
For the same energy of electrons and protons,
the energy loss rate of protons
times slower than the energy loss rate of electrons. Also, the characteristic frequency of the synchrotron radiation 
emitted by a proton is
times smaller than the characteristic frequency of synchrotron photons emitted by an electron of the same energy. The synchrotron cooling time of the proton,
and the characteristic energy of the synchrotron photon
are then
and
where
Hereafter it is assumed that the magnetic field is distributed isotropically, i.e.
with 
The average energy of synchrotron photons produced by a particle of energy E is equal to
(Ginzburg and Syrovatskii, 1965). Correspondingly, the characteristic time of radiation of a synchrotron photon of energy e by a proton in a magnetic field B is 
For comparison, the time needed for radiation of a synchrotron Y-ray photon by an electron is shorter by a factor of
The spectral distribution of synchrotron radiation is given by
where
is the modified
Bessel function of 5/3 order. The function F(x) can be presented in a simple analytical form
(e.g. Melrose, 1980). Numerical calculations show that with
this approximation provides very good, less than 1 per cent error, accuracy in the region of the maximum at 
and still reasonable (less than several per cent error) accuracy in the broad dynamical region
In Fig. 3.8a four different examples of possible proton spectra are presented. Curve 1 corresponds to the most "standard" assumption for the spectrum of accelerated particles – power-law with an exponential cutoff at energy
Curve 2 corresponds to a less
realistic, truncated power-law spectrum, i.e.
and
While the cutoff energy E0 in the spectrum of accelerated particles could be estimated quite confidently from the balance between the particle acceleration and energy loss rates, the shape of the resulting spectrum in the cutoff region depends on several circumstances – the specific mechanisms of acceleration and energy dissipation, the diffusion coefficient, etc. For example, it has been argued that in the shock acceleration scheme one may expect not only spectral cutoffs, but perhaps also pile-ups preceding the cutoffs (Melrose and Crouch 1997; Protheroe and Stanev 1999; Drury et al., 1999). Two such spectra are shown in Fig. 3.8a. Curve 3 represents the extreme class of spectra containing a sharp (with an amplitude of factor of 10) spike at the very edge of the spectrum. The Curve 4 corresponds to a smoother spectrum with a modest pile-up (or "bump") and super-exponential (but not abrupt) cutoff. The corresponding spectral energy distributions
of synchrotron radiation are shown in Fig. 3.8b. In the high energy range,
is defined by Eq.(3.30) as
the radiation spectrum from the proton distribution with sharp pile-up and abrupt cut-off is quite similar to the synchrotron spectrum from mono-energetic protons,
Fig. 3.8 (a) Possible spectra of accelerated protons (left panel), and (b) the corresponding Spectral Energy Distributions of their synchrotron radiation (right panel). At energies
all proton spectra have power-law behaviour with
but in the "cutoff" region around Eo they have very different shapes. Curve 1 corresponds to the proton spectrum described by a power-law with exponential cutoff; curve 2 corresponds to the truncated proton spectrum; curve 3 corresponds to the proton spectrum with a sharp pile-up and an abrupt cutoff at Eo; curve 4 corresponds to the proton spectrum with a smooth pile-up and a super-exponential cutoff. For comparison, in the right panel the spectrum of the synchrotron radiation of mono-energetic protons, 
is also shown (curve 5).
All synchrotron spectra shown in Fig. 3.8b exhibit, despite their essentially different shapes, spectral cutoffs at approximately x ~ 1, if one defines the cutoff as the energy at which the differential spectrum drops to 1/e of its extrapolated (from low energies) power-law value. Therefore the energy
could be treated as an appropriate parameter representing the synchrotron cutoff for a quite broad class of proton distributions. In the case of mono-energetic protons, the cutoff energy coincides exactly with e0. This is true also for the power-law proton spectrum with exponential cutoff for which the SED of the synchrotron radiation has a shape close to
The maximum energy of the synchrotron radiation depends on the spectrum of accelerated protons. The high energy cutoff in the spectrum of protons is determined by the balance between the particle acceleration and cooling times. It is convenient to present the acceleration time of particles tacc in the following general form
where
The so-called gyro-factor
characterizes the energy-dependent rate of acceleration. For almost all proposed models n remains a rather uncertain model parameter. On the other hand, any postulation of acceleration of EHE protons in compact objects like in small scale AGN jets or in transient objects like GRBs, requires n to be close to 1.
If the energy losses of protons are dominated by synchrotron radiation, the maximum energy of accelerated particles is determined by the condition 
The relevant cutoff in the electron spectrum appears at a lower energy,
Substituting Eq.(3.34) into Eq.(3.30) we find that the position of the cutoff in the synchrotron spectrum is determined by only two fundamental physical constants, the mass of the emitting particle and the fine-structure constant
Thus, for n =1 the self-regulated cutoffs in the spectra of synchrotron radiation by electrons and protons appear at energies
and 
respectively. If Y-rays are produced in a relativistic ally moving source with Doppler factor
the electron and proton spectral cutoffs are shifted towards the GeV and TeV domains, respectively.
In relatively small magnetic fields, synchrotron radiation is the only channel of proton interactions. However, at sufficiently large values of the product
protons start to produce secondary particles – electron- positron pairs, pions, etc. The energy threshold of production of a particle of mass mx is estimated as
(e.g. Ozernoy et al., 1973). For example, in pulsars with magnetic fields as strong as 1012 G pair production starts at proton energies of about 100 GeV. The production of n-mesons requires « 5 orders of magnitude larger energies.
