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tokens: that (except for initially small pitch angle cases) sin θ o ≫ 1/γ o or γ ∥ /γ o ≪ 1 and, from equation ( 17 ), we can write This implies that for τ = γ ∥ /η ≈ 1/(ηsin θ o ) we have γ ≈ 1.3/sin θ o . Thus, even if an electron had started with an almost infinite energy, it loses most of its kinetic energy in a time interval of the order of 1/η, reducing to perhaps a mildly relativistic status (for not too small an initial pitch angle). For instance, let us consider γ o = 10 3 and θ o = π/4; then γ ∥ ≈ 1/sin θ o = √2. Then from equation ( 17 ) or equation ( 19 ) we obtain, for τ = 1/η, γ = 2.3. In another example, taking γ o = 10 4 and θ o = π/3, for τ = 1/η we obtain γ ∥ ≈ 1/sin θ o = 2/√3 and γ = 1.7. Thus 1/η represents the characteristic decay time of synchrotron electrons over which they turn from ultrarelativistic into mildly relativistic ones. On a shorter time-scale, where γ is still highly relativistic, we can write Then we have Equation ( 21 ) is the same relation as in equation ( 3 ), which tells us that there is an upper energy cut-off for highly relativistic electrons in a synchrotron source, . Also, while τ 1/2 = 1/(ηsin 2 θ o γ o ) gives the half-lifetime of the synchrotron electrons (Kardashev 1962 ; Pacholczyk 1970 ; Rybicki Lightman 1979 ), τ ∼ 1/η gives the decay time over which radiating synchrotron electrons turn from ultrarelativistic into mildly relativistic ones. This follows from equation ( 17 ), which gives an exact formula true for all γ, including mildly relativistic electrons, and includes the effects of the pitch-angle changes implicitly. An explicit expression for the pitch-angle changes is derived in the next section. 4 PITCH ANGLE CHANGES Both β and θ in β ⊥ = β sin θ are functions of time. Therefore we can rewrite equation ( 13 ) as Also, since , we have Eliminating from equations ( 22 ) and ( 23 ), we obtain (also see Petrosian 1985 ) This is the relation for the rate of change of the pitch angle of a charge undergoing synchrotron radiative losses. The negative sign implies that the pitch angle decreases with time and aligns with the magnetic field. The rate of alignment is very slow for low pitch angles (θ ≈ 0) as well as for high pitch angles (θ ≈ π/2) and the highest rate of change of pitch angle is for θ = π/4. With the help of equation ( 16 ), we can integrate equation ( 24 ) to obtain In equation ( 25 ), θ θ o , because the pitch angle always reduces with time. There are many notable points. If θ o = π/2, then θ = π/2 also, because if the pitch angle is π/2 then the radiating electron always moves in a circular path in the plane perpendicular to the magnetic field. Also, if θ o = 0 then θ = 0 too, as there is no more reduction in pitch angle. For any 0 θ o π/2, θ → 0 as τ → ∞. For large γ o values, which can be used to estimate the change in pitch angle with time. For example, for, say, θ o = π/3 and θ = π/6, cosh (ητsin θ o ) = 3, which gives τ ≈ 2/η for this change in the pitch angle. Thus there are