The study of diffuse galactic Y-ray emission, i.e. the radiation components produced in interactions of electronic and nucleonic components of CRs with the interstellar gas and photon fields, provides information about the density and energy spectra of CRs in different parts of the Galactic Disk, and thus provides a key insight into the character of propagation of CRs in the Galaxy on kpc scales. A proper understanding of these processes is a necessary condition for accurate estimates of the production rates of rela-tivistic electrons and protons/nuclei in our Galaxy. Presently, our knowledge of GCRs is based on conclusions derived from the concept that interprets the secondary CR nuclei and anti-particles (positrons, antiprotons) as result of interactions of primary CRs with interstellar gas throughout the entire Galaxy. It is not obvious, however, that the locally observed CRs can be taken as undisputed representatives of the whole galactic population of relativistic particles (see Sec. 4.2.4). Therefore, the diffuse galactic Y-radiation is believed to be the most informative and model-independent channel telling us about the energy and spatial distributions of CRs in the Galactic Disk. The synchrotron radiation of the ISM at radio and possibly also at X-ray wavelengths provides additional and complementary information, but it concerns only the electronic component of CRs in two extreme energy bands below 1 GeV and above 100 TeV, respectively.
The identification and separation of the truly diffuse Y-ray emission is not an easy task because of a non-negligible contamination due to contributions from weak but numerous unresolved discrete sources. Before the launch of the Compton Gamma Ray Observatory the observations of the diffuse galactic Y-ray background were limited to the energy range between 100 MeV and few GeV explored by the SAS-2 and COS-B Y-ray satellites. The results of these important missions revealed noticeable correlations between the high energy Y-ray fluxes and the column density of the interstellar hydrogen, and thus demonstrated the existence of a truly diffuse galactic Y-radiation (for a review see Bloemen, 1989).
The highly successful observations conducted in the 1990s by detectors aboard the Compton Gamma Ray Observatory resulted in good quality data from the Galactic Disk over five decades of energy, from 1 MeV to 10 GeV (see Hunter et al., 1997b, and references therein), and initiated new theoretical studies of diffuse high-energy Y-rays (e.g. Bertsch et al., 1993; Gralewicz et al., 1997, Porter and Protheroe, 1997; Pohl and Espos-ito, 1998; Moskalenko and Strong, 1998; Strong et al., 2000; Atoyan, 2000, etc.). Because of several competing processes, the problem of identification of specific radiation mechanisms is rather complicated and confused. Nevertheless the existing data do allow definite conclusions concerning the relative contributions of different processes in each specific Y-ray energy band. The discussion of the problem in the following sections will be limited to the inner parts of the Galactic Disk within
and
which is not only the best experimentally studied region, but is also of prime interest because the CR sources are believed to be concentrated in that region of the Galaxy.
CR spectra in the inner Galaxy
In this section a simple model (Atoyan, 2000) of propagation of relativistic particles in the Galactic Disk will be described, which will be used for calculations of the resulting Y-radiation from the disk at galactic latitudes
Although the halo of galactic CRs may extend up to heights of a few kpc (e.g. Bloemen et al., 1993), one needs to know the mean spectrum of CRs only in a region close to the galactic plane. Below this region is approximated as a disk with a half-thickness
and radius
The diffusion equation for the energy distribution
of relativistic particles can be written in a general form as (Ginzburg and Sy-rovatskii, 1964)
where u = u(r) is the fluid velocity of the gas containing relativistic particles, and
describestheir total energy losses, including the adiabatic energy loss term
(e.g. Owens and Jokipii, 1977; Lerche and Schlickeiser, 1980). D = D(r, E) is the spatial diffusion coefficient, and ] is a functional representing acceleration terms of relativistic particles.
The integration of Eq.(4.17) over the volume
results in a convenient equation for the total energy distribution function of particles 
in the Galactic Disk at
where Tesc is the "diffusive + convective" escape time of particles:
The parameter Tconv describes the convective escape of particles from the disk through its surface due to the galactic wind driven by the pressure of CRs and the thermal gas (Jokipii, 1976; Berezinsky et al., 1990; Breitschwerdt et al., 1991; Ptuskin et al., 1997),
where u is the wind velocity on the surface of the Galactic Disk, which could reach ~ 50km/s (Zirakashvili et al. 1996) at the height h =1 kpc, and the (generally energy-dependent) parameter
is the ratio of the mean density of particles in the disk to their density at the disk surface (Atoyan, 2000). Correspondingly, the mean convective escape time of CRs from the Galactic Disk is estimated to be
The diffusive escape time is very often presented in a convenient power-law form
where t10 corresponds to the particle escape time at energy E =10 GeV. Because the diffusive escape time cannot be less than the light travel time, 
the energy dependence of Tesc above a certain energy disappears. On the other hand, since Tdif increases with decreasing E, Tesc given by Eq.(4.19) becomes energy-independent below some E* defined from the condition Tdf (E*) = Tconv. Neglecting at these energies Tmin in Eq.(4.21), from Eq.(4.19) it then follows that the overall escape time can be presented in the form
The implications of the Leaky-Box type Eq.(4.18) and the limits of its validity are discussed by Atoyan (2000). Here we briefly mention that the power-law index S in Eq.(4.21) for the diffusive escape time in the Leaky-Box type Eq.(4.18) doesnot coincide with the power-law index Si of the diffusion coefficient
in the general Eq.(4.17),but is effectively confined within the limits
Generally, CR propagation requires a more detailed treatment of diffusion and convection effects (see Strong et al. (2000) for numerical 3D-simulations of CR diffusion, and Breitschwerdt et al. (2002) for a self-consistent analytical approach to the problem), as well as certain assumptions about the distribution of CR sources in the Galactic Disk (e.g. Pohl and Esposito, 1998). Also, the disk-halo transition at distances ~ 1 kpc is an important part of the picture of CR transport by the galactic wind, and should be carefully taken into account in detailed calculations of the diffuse Y-radiation, in particular at high galactic latitudes.
Such treatments, however, contain a number of ad hoc assumptions, thus for the study of diffuse galactic Y-radiation the accuracy of sophisticated theoretical models should not be overrated, given the uncertainties concerning both the parameters of the interstellar medium (especially in distant, confused parts of the Galaxy) and the origin of sources of GCRs. Therefore, at this stage the simple phenomenological approach based on the approximate description of CR spectra by the Leaky-Box type equations is reasonably justified, at least for the narrow region of the galactic plane, and allows unambiguous conclusions from the existing Y-ray data.
For a time-dependent injection Q(E, t), the solution to Eq.(4.18), in terms of the spatial density functions n = N/V and q = Q/V, is
Here the variable Zt corresponds to the energy of a particle at an instant 
which has energy E at the time t, and is determined from the equation
For a quasi-stationary injection of electrons into the ISM on time-scales exceeding the escape time Tesc(E), the energy distribution of particles becomes time-independent. For both CR protons and electrons injected into the ISM we assume a stationary (continuous) source function (per unit volume) in a ‘standard’ power-law form with an index r0 and an exponential cutoff at E0. The flux of diffuse radiation with energy EY in a given direction is defined by the unit volume emissivity qY (r, EY) integrated along the line of sight:
where
is the mean emissivity, and Zj is the characteristic line-of-sight depth of the emission region.
Diffuse radiation associated with cosmic ray electrons
There are four principal processes for the production of non-thermal hard X-rays and Y-rays in the interstellar medium by CR electrons: inverse Compton (IC) scattering, bremsstrahlung, annihilation of positrons, and synchrotron radiation (provided that the electrons are accelerated up to energies exceeding 100 TeV). The first three mechanisms are discussed below; the synchrotron radiation of hard X-rays will be discussed in Sec. 4.4.4 in the context of IC radiation from the highest energy electrons.
IC gamma rays
Calculations of the diffuse IC Y-rays require knowledge of the low-frequency target photons fields. The photon fields which are important for production of IC Y-rays are the 2.7 K CMBR and the diffuse galactic radiation – the starlight and dust photons at near and far infrared frequencies, respectively. While the density of the 2.7 K CMBR is universal, with w2.7K ~ 0.25eV/cm3, the densities of diffuse galactic photon fields vary from site to site. The studies of Chi and Wolfendale (1991) show that the starlight energy density increases from the local value wNIR — 0.5eV/cm3 (Mathis et al. 1983) up to — 2.5eV/cm3 in the central 1 kpc region of the Galaxy. For the calculations below the mean value, wNIR — 1.5eV/cm3 is used. The energy density of dust emission in the galactic plane is rather uncertain; it is estimated from wFIR — 0.05 — 0.1 eV/cm3 (e.g. Mathis et al. 1983) to wFIR — 0.2 — 0.3eV/cm3 (Chi & Wolfendale 1991). For the calculations below wFIR — 0.2 eV/cm3 is used. Fortunately, large uncertainties in wFIR do not appear to have a strong impact on the calculations because at all Y-ray energies the contribution from IC up-scattering of 2.7 K CMBR photons significantly exceeds the IC fluxes produced on FIR photons (see below).
Another source of uncertainties in calculations of the diffuse IC fluxes is the lack of independent information about the high energy electrons. While radio measurements allow definite conclusions about the average electron flux below a few GeV, at higher energies the electron fluxes in the Galaxy are rather uncertain. The standard interpretation of the energy spectrum of CRs usually assumes a uniform and continuous distribution of sources in the Galaxy both in space and time. The validity of this assumption is questionable, at least for TeV electrons. Because of severe radiative losses, the source(s) of observed TeV electrons should be younger than « 105 yr, and cannot be located much beyond a few 100 pc. Therefore, the measured electron spectrum might not be applicable for calculations of Y-radiation from distant parts of the Galactic Disk.
Both the spectrum and the absolute flux of high energy electrons at TeV and higher energies may show significant variations from site to site in the Galactic Disk. In particular, one could expect a significant enhancement of the electron flux in the central region of the Galaxy due to the suspected concentration of CR sources there. Therefore one may allow deviations of the predicted electron distribution in the inner Galaxy from the observed fluxes, except perhaps for the region below a few GeV where the radio observations do provide information about the average spectrum of galactic electrons along the line of sight.
In Fig. 4.17 the average spectrum of electrons calculated for the inner part of the Galaxy is shown. The calculations are performed assuming a power-law injection spectrum with
For the assumed energy density of resulting electrons,
the calculated spectra J(E) below 1 GeV match the electron fluxes derived from radio observations. This normalisation requires an acceleration rate
Fig. 4.17 The mean flux of electrons assuming a quasi-stationary production of electrons with a power-law acceleration index
and exponential cutoff at
For the CR propagation the following escape times are assumed:
For the ISM the following parameters are assumed:
The electron injection rate is normalised so that the energy density in the resulting spectrum of electrons is
The hatched region and experimental points are the same as in Fig. 4.12. of electrons in the Galaxy per 1 km3
This implies that for the inner part of the Galactic Disk, with
8.5 kpc and a half thickness 1 kpc, the overall acceleration power should be about
Between 100 MeV and 1 GeV the dissipation of electrons is dominated by adiabatic and bremsstrahlung losses, the rates of which are proportional to energy,
and thus do not change the original (acceleration) spectrum. Therefore the spectral index of the observed synchrotron radio emission contains direct information about the acceleration spectrum of electrons in this energy region. Below 100 MeV, the electron spectrum suffers significant deformation (flattening) because of energy-independent ionization losses, while above 1 GeV the spectrum steepens because of both the escape and radiative (synchrotron and inverse Compton) losses.
Fig. 4.18 The flux of diffuse 7-rays produced by CR electrons. The open dots show the bremsstrahlung flux, and the open triangles show the overall flux of the IC radiation components due to the 2.7 K CMBR (thin dashed line), diffuse NIR/optical radiation (dot—dashed line), diffuse FIR radiation (3-dot—dashed line). The heavy dotted line shows the flux of 7-rays due to annihilation of relativistic positrons in flight. The sum of the bremsstrahlung and IC fluxes is shown by the solid line. The heavy dashed line corresponds to the overall 7-ray flux including also the annihilation radiation. The data points show the mean flux of diffuse high energy 7-rays observed by EGRET, and the hatched region shows the range of average diffuse 7-ray fluxes detected by COMPTEL from the direction of the inner Galaxy at low galactic latitudes.
The fluxes of diffuse Y-rays produced by electrons are shown in Fig. 4.18. The spectrum of IC Y-rays below the highest energy observed by EGRET, E < 30 GeV, is not very sensitive to the exact value of the cutoff energy, Eo, in the injection spectrum of electrons, provided that E0 exceeds several TeV. For the energy density of the diffuse interstellar NIR/optical radiation wNIR = 1.5 eV/cm3 is assumed, for which the IC radiation component produced on galactic starlight photons (dot-dashed line) somewhat exceeds, at energies between 10 MeV amd 30 GeV, the IC flux from the up-scattered 2.7 K CMBR (dashed line). The ‘FIR’ component of IC radiation (3-dot-dashed line), calculated for wFIR = 0.2eV/cm3, in any energy band contributes less than 25% of the total IC flux.
In its turn, the overall IC Y-ray flux can account, for the chosen infrared photon field densities, for only < 20 % of the Y-ray fluxes observed both at MeV ("COMPTEL") and GeV ("EGRET") energies (see Fig. 4.18). Formally, the IC fluxes could be increased assuming larger depths 1d of the emission region. However, the value 1d = 15 kpc assumed in Fig. 4.18 is already large, and could not realistically be increased further.
Another way to increase the flux of IC Y-rays would be to assume that the electron density in the inner Galaxy is significantly larger than we = 0.05eV/cm3. The freedom here is however restricted by radio observations. Below a few GeV the spectral index of electrons is well determined, 
where
is the photon index of the observed radio emission. Normalising the GeV electrons to the radio flux at v =10 MHz, F10mhz, we find a direct relation between the IC Y-ray and synchrotron radio fluxes:
For the radio flux at v =10 MHz,
a decrease of the magnetic field by a factor of two would lead to an increase of the electron flux shown in Fig. 4.17 by a factor of 2ar ~ 3, and correspondingly to an increase of the IC Y-ray fluxes by the same factor. Thus an increase of the IC flux would also imply very high radio fluxes, exceeding by more than one order of magnitude the flux detected from the direction of the galactic poles, unless we assume unrealistically low magnetic field, B ~ 1^G in the galactic plane. Besides, this would also automatically increase the flux of the bremsstrahlung Y-rays, resulting in an overproduction of diffuse radiation in the 10 — 30 MeV region.
