The first studies of the diffuse galactic Y-radiation by the SAS-2 and COS B satellite missions revealed a relatively small, by a factor of < 2, spatial gradient of CRs in the Galactic Disk (see e.g. Bloemen, 1989). EGRET measurements generally confirm this conclusion (Hunter et al., 1997a). A small variation of the CR density on large galactic scales can be naturally explained by effective mixture of contributions from individual sources during the CR propagation in the Galactic Disk on timescales
Meanwhile, the gradient of cosmic rays, both in the absolute flux and spectral shape, can be much stronger on smaller (sub 100-pc) spatial scales, especially in the vicinity of young CR accelerators. Consequently, we may expect significantly higher Y-ray emissivity in these regions relative to the average Y-ray production rate in the Galactic Disk.
Below, the emissivity of n0-decay Y-rays in the ~
region of a proton accelerator will be discussed for both impulsive (i.e. burst-like) and continuous scenarios of injection of relativistic protons into the ISM. Though in this section the sources of cosmic rays are not specified, it is assumed that the total CR energy per "average" particle accelerator does not exceed
which is the typical amount of energy believed to be produced in the form of relativistic particles during their diffusive acceleration by SNR shocks (e.g. Drury, 1983). On the other hand, approximately this amount of nonthermal energy per supernova is needed to explain the current fluxes of galactic cosmic rays (e.g. Ginzburg and Syrovatskii, 1964). Pulsars comprise the second important class of potential suppliers of galactic cosmic rays. An upper limit on the total energy release in cosmic rays by a pulsar may be obtained assuming that during
years a pulsar effectively accelerates particles with a rate Lp close to the rotational energy loss rate
Assuming now that the accelerated particles during their propagation in the ISM reach a radius R(t) at time t, the mean energy density of particles in the occupied region is estimated as
Thus, in the regions up to several tens of parsecs around CR accelerators with
the fluxes of relativistic particles at certain stages, depending on the time history of injection and the character of their propagation in the ISM, may exceed the average level of the "sea" of GCRs,
Correspondingly, at these stages we should expect higher Y-ray fluxes. Moreover, in the case of energy dependent diffusive propagation of particles, the spectral shape of the expected Y-ray flux may differ significantly from the spectrum of Y-rays produced by the "sea" of galactic cosmic rays. This circumstance, coupled with the possible location of high density regions near the particle accelerators, would result in enhanced Y-radiation. Also, the hard proton spectra which appear at some stages of their propagation significantly increase the probability for detection of TeV Y-rays (Atoyan, 1996a), and thus allow us to probe CR sources to very high energies. The star formation regions, which are believed to be potential settings of acceleration of CRs by supernova shocks, strong stellar winds, pulsars, etc., are of special interest (Montmerle, 1979; Casse and Paul, 1980; Paul 2001).
Proton fluxes in the ISM near the accelerator
Let assume that relativistic particles accelerated by a single source escape the source and enter the ISM. The energy spectrum of particles at a given time and distance from the source depends on (1) the time history and the injection spectrum, (2) the energy loss rate, and (3) the character of propagation of cosmic rays. In the standard diffusion approximation, the spherically symmetric propagation of cosmic rays produced by an impulsive source is described by Eq.(A.1); the general solution for an arbitrary energy source of relativistic particles injected into the ISM Q(E, R, t), energy loss rate P(E), and the diffusion coefficient D(E), is given by Eq.(A.2). The energy losses of the protons in the gas are due to ionization and nuclear interactions,
In the hydrogen medium, at kinetic energies of protons above 1 GeV the nuclear energy losses dominate over ionization losses.
Impulsive source
In the case of a power-law injection spectrum of an impulsive source, i.e.
and power-law diffusion coefficient,
the general solution reduces to
where
is the diffusion radius. It corresponds to the radius of the sphere up to which particles of energy E effectively propagate during the time t after their injection into the interstellar medium.
Generally, the spectrum of cosmic rays at the given time t and distance R from the accelerator may noticeably differ from the source (acceleration) spectrum. While the energy-independent diffusion leads to variation of the cosmic ray flux only in time, and does not change the form of the primary spectrum, the energy-dependent diffusion results in significant spectral changes. The modification of the particle spectrum is defined mainly by the parameter
For timescales less than the energy loss time,
the effective diffusion radius is reduced to the form
respectively. The energy-dependent diffusion coefficient D(E) with power-law index
is assumed. (a) and (b) are for
and
respectively. The hatched curve shows the fluxes of cosmic ray protons observed near the Earth.
Fig. 4.4 The temporal evolution of the energy spectrum of relativistic protons in the vicinity of an impulsive accelerator during their energy-dependent propagation. The power-law source spectrum with exponent r = 2.2 and total energy
are assumed. The differential fluxes of protons at different distances R and times t are shown. The curves plotted by fancy (1), solid (2), dashed (3), and dot-dashed (4) lines correspond to the age of the source
For a given energy, the maximum cosmic ray flux at a fixed distance from the source is reached when
i.e. at
At
the particles have not yet reached the point R, while at
the cosmic ray flux decreases due to the spherical expansion as
At sufficiently high energies, for which the time of the maximum flux has already passed, the modification factor at a distance R from an impulsive source is proportional to
Therefore the particles are distributed in a power-law form with
At lower energies, for which the maximum flux has not yet been reached, the primary spectrum is exponentially suppressed. Note that for D(E) = const the differential flux of cosmic rays just repeats the shape of the injection spectrum with a time-dependent amplitude proportional to g(t).
The evolution of the differential fluxes of protons,
during their energy-dependent propagation in the interstellar medium from a single impulsive source with
is shown in Figs. 4.4. The power-law diffusion coefficient in the form of
with
is assumed, where
is the value of the diffusion coefficient at E = 10 GeV. The commonly used diffusion coefficient at 10 GeV is of about
(see e.g. Berezinsky et al., 1990).
However, much smaller values, e.g.
in particular in dense regions of the interstellar gas, cannot be excluded (Ormes et al., 1988).
Continuous source
The impulsive source of particles corresponds to a scenario in which the time interval for acceleration of the bulk of relativistic protons,
is significantly less than the age of the accelerator, t. Therefore, for relatively short timescales,
the assumption of "impulsive particle acceleration" should be considered rather as an idealised working hypothesis. In particular, for SNRs the assumption of impulsive source would be actually valid only for timescales
Though the possibility of the effective production of CRs at early stages of supernova evolution cannot be excluded, a more realistic model of diffusive shock acceleration in SNRs predicts particle acceleration in the so-called Sedov phase with a typical duration
Thus, SNRs may be treated rather as continuous accelerators. On the other hand, the case of impulsive source could be applied to the classical "continuous accelerators" like pulsars, if one assumes that the bulk of particles is produced at the early stages, e.g. before the braking of the pulsar due to the magnetic dipole radiation.
In the case of continuous acceleration of particles by a single source with time dependent evolution,
Eq.(4.1) should be convolved with the function
in the time interval
Although the acceleration of particles may have rather complicated time history, to make the interpretation of the results easier, in Figs. 4.5 are shown the spatial and temporal evolution of CRs calculated for the continuous source with a constant acceleration rate.
For the assumed luminosity of
the total energy released in relativistic protons during 105 yr by a stationary accelerator is about 3 x 1049 erg, which is only 3 times less than the total energy of accelerated protons assumed in Fig. 4.4 for an impulsive source. From comparison of Fig. 4.4 and Fig. 4.5 it is seen, however, that the character of the evolution in time of the fluxes from impulsive and continuous accelerators is qualitatively different. To understand this difference, note that for the time intervals less than the energy loss time, the energy distribution function of particles injected from a continuous source is given by the expression:
where erfc(z) is the error-function (see e.g. Atoyan et al., 1995). At a given distance R, the flux of protons with given energy E at initial stages,
, increases exponentially with time t, and when
it gradually saturates at the level
This is the maximum flux at given distance R from a continuous source, which is relevant to the situation when the decrease of the cosmic ray density due to spherical expansion of the volume occupied by the cosmic rays injected earlier is compensated for by the arrival of new particles. For a power-law diffusion coefficient, the spectrum of accelerated particles from a single continuous source observed at distance R is described by the power-law exponent
i.e. by a factor of
smaller than in the case of an impulsive accelerator (see Eq. 4.4)). For
the index
is close the spectral index of the cosmic rays observed locally near the Earth. For these parameters Eq. (4.8) is reduced to
where
From comparison of Eqs.(4.10) with the locally observed CR spectrumgiven by Eq.(4.30), we find that for a large diffusion coefficient,
the flux of protons from a continuous accelerator with
could exceed the level of the "sea" of GCRs only in the near vicinity of the source.
Fig. 4.5 The temporal evolution of the energy spectrum of relativistic protons in the vicinity of a continuous accelerator during their energy-dependent propagation.The power-law primary spectrum with the exponent 
and the source luminosity
in the protons are assumed. The
differential fluxes of protons at different distances R and times t are shown. The curves plotted by fancy (1), dot-dashed(2), dashed (3), and solid (4) lines correspond to the age of the source
respectively. The energy-dependent diffusion coefficient D(E) with power-law index
is assumed. (a) and (b) are for
respectively. The hatched curve shows the fluxes of cosmic ray protons observed near the Earth.
However, for a smaller diffusion coefficient, D28 = 0.01, much larger regions around the source, up to distances R ~ 100 pc, could be significantly enhanced by cosmic rays (see Fig. 4.5).
The case of dense gas regions
The results presented in Fig. 4.4 and Fig. 4.5 correspond to the propagation of particles in low-density medium,
where the energy losses of protons during the timescales of interest,
are negligible. The effect of energy losses of relativistic protons in a dense medium with
is demonstrated in Fig. 4.6 for (a) impulsive and (b) continuous sources, respectively.
Fig. 4.6 The temporal evolution of the proton fluxes in dense medium with
from the accelerator.The diffusion coefficient with
are assumed. The curves marked as 1, 2 and 3 correspond to the fluxes at times
respectively. (a) – impulsive accelerator with
tor with
The hatched curve indicates the level of the locally observed cosmic ray proton flux.
For time periods up to t = 104 yr the proton fluxes in the high and low density media coincide, because the cooling time of the protons tpp in a medium with
still is significantly larger than 104 yr (see Eq. 3.13). However, for
the fluxes of protons propagating in a dense medium are strongly affected by interactions with the ambient gas. In particular, in the case of an impulsive injection of CRs in the dense medium, the fluxes of protons at
are exponentially suppressed compared with the relevant fluxes in a low-density medium. For a continuous accelerator, the effect caused by energy losses of protons is less profound; it is substantially masked by the flux of recently injected particles.
