## 2. Outline of the modelIn our model the nebula is regarded as a sphere of radius , 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 Both new particles and new magnetic flux are injected into the nebula within a spherical layer at a distance 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: , where
is a
constant, , and with a constant and
representing the particles' spectral
and spatial density at time 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, . is related to the injected particles' spectrum through the particles' number conservation law: with Apart from knowledge of the injected spectrum, in order to
calculate from this equation,
knowledge of the particles' age ()
and initial energy () as functions of
their present position ( We consider the energy evolution of each particle due to adiabatic and synchrotron losses: Once the magnetic field and the fluid bulk velocity are known at each time, Eq. (4) can be integrated to determine . The determination of the time dependence of the magnetic field
strength Integration of Eq. (5), where is a constant, yields the magnetic field strength at each time (see, for instance, PS). Knowing , and further assuming that the magnetic field is largely azimuthal, , we can use the flux freezing condition, as an equation for the fluid bulk velocity , which we assume to be radial: . Integrating Eq. (6) with the boundary condition that the fluid velocity field matches the expansion velocity of the nebula at its outer edge (), we find: The velocity field starts at at a
fraction of the speed of light of order
(,
see below) and then it decreases roughly as
to match the nebular expansion
velocity at Eq. (7) can also be written as which yields, in its integral form: This last equation simply states that the magnetic flux injected
into the nebula during the time interval
, with
being the time at which a particle
that at time Presently what is left to determine, before being able to calculate as a function of the injected spectrum, is the evolution of the particles' energy from initial to present value. Introducing the expression found for into Eq. (4), we obtain: We integrate Eq. (10) after replacing 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 : with The meaning of the energy is
apparent: it represents the maximum possible energy for particles that
at time Inserting our findings into Eq. (3) we are finally able to relate 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: For simplicity, we assume all the energy cuts to be time independent and the ratio between and to remain constant. Since we have assumed that a constant fraction of the pulsar's energy outflow goes into feeding the magnetic field, the remaining will be used for accelerating particles, and then the constraint on the total slowing-down power converted into particles, implies Finally we have for the spectral and spatial distribution of particles across the nebula: with where the energies ,
and
are the evolved minimum, maximum and
intrinsic break energy at radius © European Southern Observatory (ESO) 2000 Online publication: July 13, 2000 |