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Astron. Astrophys. 345, 769-777 (1999)

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Appendix A: synchrotron radiation, adiabatic expansion and continuous injection in the case of a time-variable magnetic field

We assume that adiabatic expansion and injection of particles start at time [FORMULA]. The source is continuously replenished by a constant flow of particles with a power law energy spectrum with spectral index [FORMULA] intensity B decreases according to

[EQUATION]

and the characteristic size R of the regions occupied by the particles (which should not be confused with the source size LS) grows as

[EQUATION]

The continuity equation for this model is

[EQUATION]

[FORMULA] is the differential energy spectrum of the relativistic particles at time t. The first term on the right hand side describes the effects of the synchrotron losses, [FORMULA], [FORMULA] is constant (Pacholczyk 1970). The second term accounts for adiabatic expansion ([FORMULA]). The term [FORMULA] describes the injection rate and is considered to be constant for simplicity.

The solution of Eq. (A1) is

[EQUATION]

where [FORMULA] is MAX[FORMULA] and [FORMULA] is implicitly given by

[EQUATION]

The energy [FORMULA] is

[EQUATION]

[FORMULA] is the break energy of the particle population injected at the time [FORMULA] ([FORMULA]). The solution (A2) represents the sum of all the populations injected form [FORMULA] to [FORMULA]. The integral (A2) can be written as

[EQUATION]

where [FORMULA] is equal to MAX[FORMULA],

[EQUATION]

and

[EQUATION]

In the energy spectrum two break energies are always present, [FORMULA] and [FORMULA], the break energy of the first population injected at [FORMULA]. These regions can be identified in the energy spectrum:

  • 1) [FORMULA]. In this region the spectrum is a power law:

    [EQUATION]

  • 2) [FORMULA]. The spectrum starts to gradually deviate from the low energy power law at [FORMULA] and undergoes a steepening at [FORMULA].

    [EQUATION]

  • 3) [FORMULA]. The spectrum is again a power law but with a spectral index steeper by 1.

    [EQUATION]

Since the separation between these two energies increases with time the shape of the spectrum also changes with time. But an asymptotic stationary shape still exists when [FORMULA]. In fact, in this case [FORMULA], and at every energy [FORMULA], [FORMULA]. In this way the integral

[EQUATION]

does not depend on the ratio [FORMULA] anymore.

The integral can be solved only by numerical means, but analytic asymptotic limits can still be found.

[EQUATION]

i.e. in the spectrum only the [FORMULA] break energy is present. The asymptotic solutions are two power laws. The low-energy power law spectral index is [FORMULA], the high-energy power law spectral index is [FORMULA]. Moreover, the normalization for [FORMULA] is time-independent, i.e. a perfect balance is reached between the number of particles that leave this region because the synchrotron and expansion losses and the number of particles constantly injected into the source.

A.1. CI model

The CI model reproduces the simple situation in which there is no adiabatic expansion and the magnetic field strength stays constant. This corresponds to [FORMULA] and [FORMULA] in (A3). In this case the integral can be directly solved resulting in:

[EQUATION]

where

[EQUATION]

The shape of the spectrum does not depend on time (Kardashev 1962).

A.2. CIE model

The CIE model reproduces the situation in which the volume containing the particles is adiabatically expanding at a constant rate and the magnetic field is frozen in the plasma. Since [FORMULA], the conservation of the magnetic flux requires that [FORMULA]. This corresponds to [FORMULA] and [FORMULA] in (A3). The shape of the spectrum changes with time during its rise. However an asymptotic stationary solution exists when [FORMULA]:

[EQUATION]

[EQUATION]

In this asymptotic limit the CIE spectrum shows a break energy four times greater than the break energy of the CI model. Using (A4) one finds that

[EQUATION]

this solution is the same found by Kardashev (1962). The asymptotic solutions are two power laws with the same dependence on the spectral index as in the case of the CI model. However, the energy range necessary to complete the transition between the two asymptotic power laws is wider. For [FORMULA] the break energy of the CIE model is greater than the break energy [FORMULA] of a CI model with the same [FORMULA] magnetic field. This result appears a bit surprising at first glance. In fact, due to the expansion losses and the stronger magnetic field, the break energy of the first population of particle injected at [FORMULA] in the CIE model is [FORMULA] than that of the corresponding population of the CI model. However, both the stronger mean magnetic field and the expansion losses will also decrease the energy of the first populations in a way that, at time t, they do not contribute appreciably to the integrated spectra.

A.3. CIm model

The CIm model reproduces the situation in which the volume containing the particles is adiabatically expanding and there is always equipartition between magnetic field and particle energy density. The equipartition condition [FORMULA] (u is the energy density of the relativistic particles) implies that [FORMULA]. This corresponds to [FORMULA] in (A3). An asymptotic stationary solution exists when [FORMULA]:

[EQUATION]

[EQUATION]

In this asymptotic limit the CIm spectrum shows a break energy ([FORMULA]) times greater than the break energy of the CI model. Using (A4) one finds that

[EQUATION]

For the CIm the CIE comments are also valid.

A.4. Emission spectrum

The emission spectrum is given by the convolution of the emission spectrum of the single electron [FORMULA] (with [FORMULA] constant, Pacholczyk 1970) with the energy distribution [FORMULA]:

[EQUATION]

where [FORMULA] and [FORMULA]. It is assumed that the pitch angles [FORMULA] between the electron velocity and the magnetic field direction are isotropically distributed and that the time scale for their continuous re-isotropization is much shorter than the radiative time-scale, [FORMULA] (Jaffe & Perola 1974).

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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