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Astron. Astrophys. 359, 1107-1110 (2000)

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2. Outline of the model

In our model the nebula is regarded as a sphere of radius [FORMULA], expanding at a constant rate. A continuous supply of new magnetic energy and particles is provided by the central pulsar, whose energy output per unit time can be written as

[EQUATION]

Both new particles and new magnetic flux are injected into the nebula within a spherical layer at a distance [FORMULA] from the central pulsar. We shall assume that the injection region has zero extension, although very likely this is an oversimplification.

We assume that the particles' velocity distribution is isotropic and that each particle emits essentially at its characteristic frequency: [FORMULA], where [FORMULA] [FORMULA] is a constant, [FORMULA], and E and [FORMULA] are the particle's energy and pitch angle, respectively. Under these assumptions the nebular synchrotron emissivity becomes

[EQUATION]

with [FORMULA] a constant and [FORMULA] representing the particles' spectral and spatial density at time t. The integral represents the average over particles' pitch angles.

As stated before, we regard the magnetic field as spatially constant, therefore, in our model, both the spatial and spectral characteristics of the nebular synchrotron emission are directly related to the particle spatial and spectral distribution, [FORMULA].

[FORMULA] is related to the injected particles' spectrum through the particles' number conservation law:

[EQUATION]

with J being the number of newly injected particles per unit time and energy interval and [FORMULA] and [FORMULA] being, respectively, the initial energy and injection time of a given particle. The last term in Eq. (3) represents the jacobian of the transformation [FORMULA].

Apart from knowledge of the injected spectrum, in order to calculate [FORMULA] from this equation, knowledge of the particles' age ([FORMULA]) and initial energy ([FORMULA]) as functions of their present position (r) and energy (E) is needed.

We consider the energy evolution of each particle due to adiabatic and synchrotron losses:

[EQUATION]

Once the magnetic field [FORMULA] and the fluid bulk velocity [FORMULA] are known at each time, Eq. (4) can be integrated to determine [FORMULA].

The determination of the time dependence of the magnetic field strength B is straightforward under our assumptions of time constancy of the nebular expansion rate and spatial constancy of B itself. Following PS, we relate B to the nebular content of magnetic energy [FORMULA]: [FORMULA]=8 [FORMULA] [FORMULA]/ [FORMULA], with [FORMULA] representing the confining volume for the magnetic field. Then we write the time evolution of [FORMULA] due to expansion losses and the decreasing pulsar input:

[EQUATION]

Integration of Eq. (5), where [FORMULA] is a constant, yields the magnetic field strength at each time (see, for instance, PS).

Knowing [FORMULA], and further assuming that the magnetic field is largely azimuthal, [FORMULA], we can use the flux freezing condition,

[EQUATION]

as an equation for the fluid bulk velocity [FORMULA], which we assume to be radial: [FORMULA]. Integrating Eq. (6) with the boundary condition that the fluid velocity field matches the expansion velocity of the nebula at its outer edge ([FORMULA]), we find:

[EQUATION]

The velocity field starts at [FORMULA] at a fraction of the speed of light of order [FORMULA] ([FORMULA], see below) and then it decreases roughly as [FORMULA] to match the nebular expansion velocity at R.

Eq. (7) can also be written as

[EQUATION]

which yields, in its integral form:

[EQUATION]

This last equation simply states that the magnetic flux injected into the nebula during the time interval [FORMULA], with [FORMULA] being the time at which a particle that at time t is at r was born, is all contained between [FORMULA] and r. Taking [FORMULA] from Eq. (9), we have got the relation needed to connect the particles' position at each time to their age.

Presently what is left to determine, before being able to calculate [FORMULA] as a function of the injected spectrum, is the evolution of the particles' energy from initial to present value. Introducing the expression found for [FORMULA] into Eq. (4), we obtain:

[EQUATION]

We integrate Eq. (10) after replacing [FORMULA] with its time average. We take this to be 2/3, which is equivalent to state that each particle during its synchrotron lifetime experiences all the possible velocity orientations with respect to the magnetic field with the same probability. Obviously, we expect this approximation to work better the longer a particle lives, hence the smaller its initial energy is. Nevertheless we apply it to particles of all energies and finally find the expression for [FORMULA]:

[EQUATION]

with

[EQUATION]

The meaning of the energy [FORMULA] is apparent: it represents the maximum possible energy for particles that at time t reside at r. Substituting [FORMULA] with [FORMULA], Eq. (11) also defines the maximum radial distance [FORMULA] from which we expect emission to come for a given frequency.

Inserting our findings into Eq. (3) we are finally able to relate [FORMULA] to the injected particle spectrum. Concerning the latter some assumptions are necessary. Assuming that the injected electron spectrum is described by a single power law, synchrotron aging can account for just one break in the emission spectrum, while observation shows that in the case of the Crab Nebula there is another break between the optical and the X-rays; this break must be intrinsic to the injection mechanism. Then we allow for a particle energy spectrum with two different slopes in two different energy ranges, namely:

[EQUATION]

For simplicity, we assume all the energy cuts to be time independent and the ratio between [FORMULA] and [FORMULA] to remain constant. Since we have assumed that a constant fraction [FORMULA] of the pulsar's energy outflow goes into feeding the magnetic field, the remaining [FORMULA] will be used for accelerating particles, and then the constraint on the total slowing-down power converted into particles,

[EQUATION]

implies

[EQUATION]

Finally we have for the spectral and spatial distribution of particles across the nebula:

[EQUATION]

with

[EQUATION]

where the energies [FORMULA], [FORMULA] and [FORMULA] are the evolved minimum, maximum and intrinsic break energy at radius r, respectively, and can be calculated as functions of [FORMULA], [FORMULA] and [FORMULA] by means of Eq. (11).

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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