3 Propagation model

   
3.1 Features of the model

The propagation of cosmic rays throughout the Galaxy is described with a two-zone effective diffusion model which has been thoroughly discussed elsewhere (Maurin et al. 2001 - hereafter Paper I, Donato et al. 2001 - hereafter Paper II). We repeat here the main features of this diffusion model for the sake of completeness but we refer the reader to the above-mentioned papers for further details and justifications.

The Milky-Way is pictured as a thin gaseous disc with radius R = 20 kpc and thickness 2 h = 200 pc (see Fig. 4) where charged nuclei are accelerated and destroyed by collisions on the interstellar gas, yielding secondary cosmic rays. The thin ridge is sandwiched between two thick confinement layers of height L, called diffusion halo.

Figure 4: Schematic view of the axi-symmetric diffusion model. Secondary antiproton sources originate from CR/ ISM interaction in the disc only; primary sources are also distributed in the dark halo which extends far beyond the diffusion halo. In the latter case, only sources embedded in the diffusion halo contribute to the signal (see Appendix B).

The five parameters of this model are K₀, $\delta$, describing the diffusion coefficient $K(E) = K_0 \beta {\cal R}^{-\delta}$, the halo half-height L, the convective velocity $V_{\rm c}$ and the Alfven velocity $V_{\rm a}$. Specific treatment related to $\bar{p}$ interactions (elastic scattering, inelastic destruction) and more details can also be found in Paper II.

Actually, a confident range for these five parameters has been obtained by the analysis of charged stable cosmic ray nuclei data (see Paper I). The selected parameters have been employed in Paper II to study the secondary antiproton flux, and are used again in this analysis (for specific considerations about the Alfvén velocity, see Sect. 5.2 of Paper II). In principle, this range could be further reduced using more precise data or considering different sorts of cosmic rays. For the particular case of $\beta$ radioactive nuclei, Donato et al. (2002) showed that with existing data no definitive and strict conclusions can so far be drawn. We thus have chosen a conservative attitude and we do not apply any cut in our initial sets of parameters (which can be seen in Figs. 7 and 8 of Paper I).

3.2 Solution of the diffusion equation for antiprotons

Antiproton cosmic rays have been detected, and most of them were probably secondaries, i.e. they were produced by nuclear reactions of a proton or He cosmic ray (CR) nucleus impinging on interstellar (ISM) hydrogen or helium atoms at rest. When energetic losses and gains are discarded, the secondary density $N^{\bar{p}}$ satisfies the relation (see Paper II for details)

 

$$h ~ \delta(z) ~ q^{\rm sec}(r,0,E) = 2 ~ h ~ \delta(z) ~ \Gamma^{\rm ine}_{\bar{p}} ~ N^{\bar{p}}(r,0,E) \left\{ V_{\rm c} \frac{\partial}{\partial z} ~ - ~ K \left( {\di... ...le \frac{\partial}{\partial r}} \right) \right) \right\} ~ N^{\bar{p}}(r,z,E) ,$$ (4)

as long as steady state holds. Due to the cylindrical geometry of the problem, it is easier to extract solutions performing Bessel expansions of all quantities over the orthogonal set of Bessel functions $J_{0}(\zeta_{i} x)$ ($\zeta_{i}$ stands for the ith zero of J₀ and $i = 1 \dots \infty$). The solution of Eq. (4) may be written as (see Eqs. (A.3) and (A.4) in Paper II)

 

$$N^{\bar{p},\;\rm sec}_{i}(z,E) \; = \; {\displaystyle \frac{2 ~ h... ...es \exp \left\{ {\displaystyle \frac{V_{\rm c} ~ \vert z\vert}{2 ~ K}} \right\}$$

$$\times\left\{ {\sinh \left\{ {\displaystyle \frac{S_{i}}{2}} ~ \l... ...} ~ / ~ {\sinh \left\{ {\displaystyle \frac{S_{i}}{2}} ~ L \right\}} \right\} ,$$ (5)

where the quantities S_i and A_i are defined as

$$S_{i} \equiv \left\{ {\displaystyle \frac{V_{\rm c}^{2}}{K^{2... ...\zeta_{i}^{2}}{R^{2}}} \right\}^{1/2} \;\;\; \mbox{and} \;\;\;$$

$$A_{i}(E) \equiv 2 ~ h ~ \Gamma^{\rm ine}_{\bar{p}} \; + \; V_... ...coth} \left\{ {\displaystyle \frac{S_{i} L}{2}} \right\} \cdot$$ (6)

We now turn to the primary production by PBHs. It is described by a source term distributed over all the dark matter halo (see Sect. 2.1) - this should not be confused with the diffusion halo - whose core has a typical size of a few kpc. At z=0 where fluxes are measured, the corresponding density is given by (see Appendix A)

$$N^{\bar{p},\rm prim}_{i}(0)= \exp\left( \frac{-V_{\rm c}L}{2K} \right) \frac{y_i(L)}{A_i\sinh(S_iL/2)}$$ (7)

where

 

$$2\int_0^L\exp\left( \frac{V_{\rm c}}{2K}(L-z')\right)$$

$$\times\sinh\left(\frac{S_i}{2}(L-z')\right)q^{\rm prim}_{i}(z'){\rm d}z'.$$ (8)

This is not the final word, as the antiproton spectrum is affected by energy losses when $\bar{p}$ interacts with the galactic interstellar matter and by energy gains when reacceleration occurs. These energy changes are described by the integro-differential equation

 

$$A_{i} ~ N^{\bar{p}}_{i} \; + \; 2 ~ h ~ \partial_{E} \left\{ b^{\... ...i} ~ - ~ K^{\; \bar{p}}_{\rm EE}(E) ~ \partial_{\rm E} N^{\bar{p}}_{i} \right\}$$

$$= 2 ~ h ~ \left\{ q_{i}^{\rm prim}(E) + q_{i}^{\rm sec}(E) + q_{i}^{\rm ter}(E) \right\}\cdot$$ (9)

We added a source term $q_{i}^{\rm ter}(E)$, leading to the so-called tertiary component. It corresponds to inelastic but non-annihilating reactions of $\bar{p}$on interstellar matter, as discussed in Paper II. The resolution of this equation proceeds as described in Appendices (A.2), (A.3) and (B) of Paper II, to which we refer for further details. The total antiproton flux is finally given by

$$N^{\bar{p},\;\rm tot}(R_{\odot},0,E) {\displaystyle \sum_{i=1}^{\infty}} ~ \left(N^{\bar{p},\;\rm sec}_{i}(0,E) +N^{\bar{p},\;\rm prim}_{i}(0,E) \right)$$ =

$$\qquad\qquad\qquad\times J_{0} \left( \zeta_{i} {\displaystyle \frac{R_{\odot}}{R}} \right)$$ (10)

where $N^{\bar{p},\;\rm sec}_{i}(0,E)$ and $N^{\bar{p},\;\rm prim}_{i}(0,E)$are given by formulæ (5), (7) and (8). We emphasize that the code (and thus numerical procedures) used in this study is exactly the same as the one we used in our previous analysis (Papers I and II), with the new primary source term described above.

As previously noticed, the dark halo extends far beyond the diffusion halo whereas its core is grossly embedded within L. We can wonder if the external sources not comprised in the diffusive halo significantly contribute to the amount of $\bar{p}$ reaching Earth. A careful analysis shows that in the situation studied here, this contribution can be safely neglected (see Appendix B).