A&A 402, 971-983 (2003)
R. Taillet1,2 - D. Maurin1,3
1 - Laboratoire de Physique Théorique LAPTH, 74941 Annecy-le-Vieux, France
2 - Université de Savoie, 73011 Chambéry, France
3 - Institut d'Astrophysique de Paris, 98bis Bld Arago, 75014 Paris, France
Received 6 December 2002 / Accepted 25 February 2003
The propagation of Galactic cosmic ray nuclei having energies between 100 MeV/nuc and several PeV/nuc is strongly believed to be of diffusive nature. The particles emitted by a source located in the disk do not pervade the whole Galaxy, but are rather confined to a smaller region whose spatial extension is related to the height of the diffusive halo, the Galactic wind and the spallation rate. Following the pioneering work of Jones (1978), this paper presents a general study on the spatial origin of cosmic rays, with a particular attention to the role of spallations and Galactic wind. This question is different, and to a certain extent disconnected, from that of the origin of cosmic rays. We find the regions of the disk from which a given fraction of cosmic rays detected in the solar neighborhood were emitted (f-surfaces). After a general study, we apply the results to a realistic source distribution, with the propagation parameters obtained in our previous systematic analysis of the observed secondary-to-primary ratios (Maurin et al. 2002a). The shape and size of these f-surfaces depend on the species as well as on the values of the propagation parameters. For some of the models preferred by our previous analysis (i.e. large diffusion slope ), these f-surfaces are small and in some extreme cases only a fraction of a percent of the whole Galactic sources actually contribute to the solar neighborhood cosmic ray flux. Moreover, a very small number of sources may be responsible for more than 15% of the flux detected in the solar neighborhood. This may point towards the necessity to go beyond the approximations of both homogeneity and stationarity. Finally, the observed primary composition is dominated by sources within a few kpc.
Key words: ISM: cosmic rays
The propagation of charged cosmic ray nuclei, in the energy range going from a few 100 MeV/nuc and a few PeV/nuc, is strongly affected by the Galactic magnetic field. It is a diffusive process, so that the cosmic rays emitted by a single source spread out in time, pervade the whole Galaxy, and can escape the Galaxy when reaching its boundaries. Those coming from a source located far from the Sun have a larger probability of escaping than reaching the solar neighborhood. It is the opposite for nearby sources, so that the cosmic ray fluxes in the solar neighborhood are more sensitive to the properties of the local sources (as opposed to the remote sources). Other effects like spallations and Galactic wind further limit the distance cosmic rays travel before being detected. Some consequences of the Galactic wind were studied in Jones (1978) where convective escape was compared to escape through the top and bottom boundaries of the Galaxy.
The goal of this paper is to go one step beyond by providing a general study on the spatial origin of cosmic rays, i.e. to answer the question "from which region of the Galaxy were emitted the cosmic rays detected in the solar neighborhood?''. This question is different, and to a certain extent disconnected, from that of the origin of cosmic rays ("What are the astrophysical objects which are responsible for the acceleration of cosmic rays?'') which is still much debated. We believe that it is nevertheless an interesting question, for several reasons. First, we find that the answer may cast some doubt on the validity of the stationary model, upon which most studies on cosmic rays are based. Second, it gives some clues about the spatial range beyond which the cosmic ray studies are blind to the sources. Finally, this study may be of interest to optimize the propagation codes based on Monte-Carlo methods, by focusing the numerical effort on the sources that really contribute to the detected flux.
The reader who does not want to go through the pedagogical progression can go directly from the general presentation of the method in Sect. 2 to its application in realistic cases in Sect. 7. For the others, the effect of escape is studied in Sect. 3 and that of spallations and Galactic wind is studied in Sect. 5. Then, Sect. 6 studies the effect of a realistic source distribution. Finally, the fully realistic case is considered in Sect. 7. The results and the perspectives are discussed in the last section. For convenience, we will use the word f-surfaces to describe the surfaces in the thin disk within which the sources form the fraction f of cosmic rays detected at the observer location.
A stationary point source emits particles that diffuse in a given volume. At the boundaries of this volume, the particles are free to escape and the density drops to zero. After a sufficiently long time, the stationary regime is eventually reached and the density profile is established inside the diffusive volume. If several sources are present (or even a continuous distribution of sources), their contributions add linearly at each point.
The question we wish to answer is the following:
a cosmic ray being detected at the position
of an observer
(in practice, this will be the position of the Sun, and we refer
to this position as the solar neighborhood), what is the
If the sources are distributed according to
the probability that a cosmic ray detected at
from a surface
is given by
The region in which diffusion occurs is limited by surfaces (hereafter the boundaries) beyond which diffusion becomes inefficient at trapping the particles, so that they can freely escape at a velocity close to c. The density outside the diffusive volume is very small, and it is very reasonable to suppose that the boundaries are absorbers, i.e. they impose a null density (N=0).
It is well-known that the shape and location of the boundaries play a crucial role for diffusive propagation. This section shows that the cosmic rays emitted from standard sources in the disk are not sensitive to the radial extension of the Galaxy, but only to its top and bottom edge. To this aim, it is sufficient to concentrate on pure diffusion and to neglect spallations, the Galactic wind and reacceleration. Indeed this is a conservative case as these effects can only make the diffusion process even less sensitive to the presence of the boundaries (see below). Moreover, we consider the case of a homogeneous source distribution located in the disk , which also leads to a conservative result if compared to a realistic radial distribution of sources.
We first consider the pure diffusion
equation with a Dirac source term
the solution for a point source in this particular
geometry is given in Appendix A.
The probability density that a particle reaching the observer was
emitted from a point located at a distance
from the center is
thus given by (with
The influence of the
boundaries, in the case
of an infinite disk (
is now considered.
In this limit, the sum over Bessel functions can be replaced by an
integral and we obtain (see Appendix B.3)
Figure 4 shows the probability
density computed above as a function of
for unbounded space,
for the cylindrical geometry with several halo
sizes L, i.e. Eq. (7), and for the two limiting cases
|Figure 4: Cosmic ray probability density as a function of (distance of the source in the disk), for several values of L and for a disk of radius R=20 kpc. Big stars are for unbounded model, dotted line is for a spherical boundary at radius R, small stars are for top and bottom boundaries, and solid lines are for cylindrical boundaries.|
|R=20 kpc||R=20 kpc||R=20 kpc|
|-||6.2 kpc||-||14.1 kpc||-||18.2 kpc|
|L = 20 kpc||12.6 kpc||6.1 kpc||39 kpc||14 kpc||68 kpc||18.2 kpc|
|L = 5 kpc||3.1 kpc||2.95 kpc||9.5 kpc||8.6 kpc||17 kpc||14.6 kpc|
|L = 1 kpc||0.63 kpc||0.63 kpc||1.9 kpc||1.9 kpc||3.4 kpc||3.4 kpc|
An important consequence is that as long as the observer and the sources are not too close to the side boundary, the density only depends on the relative distance to the source in the disk, so that it may be assumed, for numerical convenience, that the observer is either at the center of a finite disk, or in an infinite disk. In all the paper, i.e. for standard sources in the disk, we will consider the limit , i.e. we use the integral representation described in Appendix B.3.
We find, in the case
(see Appendix B.3), and for a
homogeneous distribution of sources,
|R=20 kpc||R=20 kpc||R=20 kpc|
|-||8.6 kpc||-||15.3 kpc||-||18.5 kpc|
|L = 5 kpc||5.5 kpc||5.3 kpc||12.5 kpc||12 kpc||25 kpc||17.2 kpc|
|L = 1 kpc||1.1 kpc||1.1 kpc||2.5 kpc||2.5 kpc||4.4 kpc||4.4 kpc|
To sum up, Eq. (11) is valid as long as
the propagation of the unstable species can be then
considered as local, with a typical scale
This is no longer the case if the lifetime
which is the case at high energy because of the relativistic factor
even if the proper lifetime
-probability is straightforwardly derived.
As on these typical scales,
the source distribution can safely taken to be constant, the distance
is expressed as
Cosmic ray sources also emit electrons and positrons.
In contrast with the nuclei, these particles are light, so that they
are subject to much stronger energy losses, due to synchrotron radiation
and inverse Compton.
This results in an effective lifetime given by (e.g. Aharonian et al. 1995)
The results given in the previous section on radioactive species can
be applied to this case, with a scale length
The important conclusions at this point are that i) most of the stable primary cosmic rays that reach the solar neighborhood were emitted from disk sources located within a distance of the order of L, such that the boundary can reasonably be discarded ii) the secondary species composition is determined by sources located farther away than those determining the primary composition; iii) radioactive species may come from very close if their lifetime is so short that , high energy electrons and positrons definitely do.
These conclusions are expected to be stronger when spallations, Galactic wind and a realistic source distribution are taken into account. All these effects will limit even more the range that the particles can travel before reaching the solar neighborhood.
The diffusion of cosmic rays may be disturbed by the presence of a
convective wind of magnitude ,
directed outwards from the disk.
For numerical convenience, a constant wind has been considered,
although other possibilities (especially a linear dependence) are
probably more justified on
theoretical grounds (see discussion in Maurin et al. 2002a).
The effect is to blow the particles away from the disk, so that
those detected in the solar neighborhood come from closer sources
(compared to the
With an infinite halo, the probability density in the disk is
|Figure 5: Deviation from the pure density profile, , due to the various effects studied here: escape from the boundaries, spallations and Galactic wind. In this latter case, the choice has been made to show the similar behavior at large . The case of a radioactive species has also been shown. It should be noticed, however, that in most interesting cases, the scale length is much smaller than the others, so that in this case the propagation is dominated by radioactive decay and spallations and Galactic wind can be safely discarded.|
It is interesting to note that the effect of
is similar (though
not rigorously identical) to the effect of L (see Fig. 5).
As a matter of fact, this was noticed by Jones (1978) who studied
the propagation properties in a dynamical halo and provided a very simple
picture (along with a rigorous derivation) of the effect of the wind.
Consider a particle initially located at a distance z from the disk.
It takes a time
to diffuse back in the disk.
In the meantime, convection sweeps the particle in a distance
Both processes are in competition and
the particle will not reach the disk if
This define an effective
This is our parameter
a factor 2.
|Figure 6: Integrated probability that a particle detected at the origin was emitted inside the ring of radius , in the three situations considered. The solid dark line is obtained when only the leakage through the boundaries is considered, in which case the radii scale as with . The dotted, respectively dashed, line is obtained when only the spallations, respectively only the convective wind, are considered. The solid grey line indicates the probability that the primary progenitor of a secondary detected in the solar neighborhood was emitted from within a given distance.|
The Galactic disk contains interstellar gas mostly made of hydrogen.
When cosmic rays cross the disk, they can interact with this gas.
This interaction may result in a nuclear reaction (spallation),
leading to the destruction of the incoming particle and to the
creation of a different outgoing particle (secondary).
We present two approaches to the problem of diffusion in presence of
a spallative disk. When the halo is infinite in extent, the solution
may be obtained by using the interpretation of diffusion in terms of
This will be treated in Appendix C.
In the general case, the Bessel developments can be used as before.
Starting from Eq. (A.5), the expression for the
probability density is readily obtained.
is noteworthy, as the resulting
expression isolates the influence of spallations:
For small values of
the convergence of the
previous integral is
slow, and other forms obtained by integration by parts, as
developed in the Appendix B.3, might be preferred.
However, in this particular case, the identity
|, 1 GeV/nuc||10.2||1.45||0.59|
|, 100 GeV/nuc||115||16.4||6.7|
When all the effects above are considered, Eq. (A.5) gives
Several properties (energy losses, amount of reacceleration,
secondary-to-primary ratio) of the cosmic ray flux detected in the solar
neighborhood are determined by the number of times a given cosmic
ray has crossed the disk since it was created.
The distribution of disk-crossings is computed in Appendix C in the case of an
infinite diffusive volume and in the absence of Galactic wind.
In the most general situation, the mean number of crossings
(though not the entire distribution of crossing numbers)
can be computed as follows.
Each time a particle crosses the disk, it has a probability
of being destroyed by a spallation.
The number N(r) of surviving particles can thus be
obtained from N0(r), the number of particles diffusing without
|Figure 7: Grammage crossed as a function of the origin, for some of the models discussed in the text and for a typical value of K=0.03 kpc2 Myr-1.|
As a cross-check, it can be noticed that in this approach, the mean grammage
The diffusion coefficient actually depends on energy.
A commonly used form (see Maurin et al. 2002b for a discussion) is
For the sake of definiteness, we will consider from now on that
the cosmic ray sources for stable primaries are located in the disk and
that their radial distribution
follows that of the pulsars and supernovae
remnants, given by
The results are not much affected by taking an angular dependence into account. Considering for example the spiral arms modelling of Vallée (2002), Fig. 8 shows that the extension of the f-surfaces is almost not affected by these small scale structures. In the rest of this paper, the purely radial distribution (19) is assumed.
The previous sections present a complete description of the origin of cosmic rays in a stationary diffusion model (energy losses and gains are discarded). To each process by which a cosmic ray may disappear before it reaches the solar neighborhood is associated a parameter: L (escape through the top and bottom boundaries), (convection), (destructive spallation). The relative importance of these parameters may be measured by the two quantities and . One can distinguish three regimes which determine the diffusion properties of the system: i) the escape through the boundaries dominates for and ; ii) convection dominates for and ; iii) spallations dominate for and .
We now use the sets of diffusion parameters consistent with the B/C data given in Maurin et al. (2002a) (hereafter MTD02) to evaluate realistic values for these quantities.
The left panel displays for three rigidities: 1 GV, 10 GV and 100 GV. Up to several tens of GV, convection is in competition with escape; afterwards escape dominates. The noticeable fact is that models corresponding to are escape-dominated, whereas convection dominates only for large at low energy. It appears that all other parameters being constant, is fairly independent of L(indicating a similar relative importance of convection and escape for the models reproducing the B/C ratio, see MTD02). However, the spatial origin does depend on L and and not only on their ratio.
The right panel of Fig. 9 plots for GV and 1 GV for various nuclei. Protons are the most abundant species in cosmic rays. Boron and CNO family are important because they allow to constrain the value of the propagation parameters, e.g. through the B/C ratio. Last, the Fe group provides another test of the secondary production via the sub-Fe/Fe ratio. The evolution of for these species is conform to what is very well known from earlier leaky box inspired studies: for heavier nuclei, spallation dominates over escape and for this reason, the induced secondary production is particularly sensitive to the low end of the grammage distribution.
To summarize, the left panel shows the evolution from convection-domination to escape-domination as a function of and , the effect of the wind being negligible above 100 GeV whatever ( ). The right panel gives the evolution from spallation-domination to escape-domination as a function of , and the species under consideration. The effect of spallation is more important for heavy than for light nuclei, but this difference is too small to produce an evolution of the average logarithmic mass for high energy (TeV) cosmic rays (Maurin et al. 2003a).
From the previous discussion, it appears that spallations and Galactic wind play a role at low energy. The results will be shown for the particular value 1 GeV/nuc which is interesting for various astrophysical problems. First, once modulated, it corresponds to about the very lowest energy at which experimental set-ups have measured Galactic cosmic rays. Second, the low energy domain is the most favorable window to observe (resp. ) from exotic sources (see companion paper Maurin & Taillet 2003), as the background corresponding to secondaries (resp. ) is reduced. Last, these energies correspond to that of the enduring problem of the diffuse GeV -ray radial distribution. This was first quoted by Stecker & Jones (1977) and further investigated by Jones (1979) taking into account the effect of a Galactic wind.
|Figure 9: Left panel: as a function of the diffusion spectral index for different rigidities ; from top to bottom, GV, GV and GV. The parameter , as well as , is not very sensitive to the halo size L. Right panel: as a function of for GV (upper curves) and GV (lower curves) for four species: p ( mb), ( mb), B-CNO ( mb) and Fe ( mb). For the latter species we plotted the same three L values as in left panel. The behavior for other species is similar so that we only plotted the case L=6 kpc.|
From the sets of diffusion parameters that fit the B/C ratio,
the values of the parameters
are computed (see Table 5) for
the four nuclei shown in Fig. 9 and for three
|Figure 10: (50-90-99)%-surfaces (protons and Fe nuclei are considered), in a typical diffusion model with L=6 kpc and .|
As regards the last effect, it can first be seen from Fig. 13 that the heavier species come from a shorter distance (because the spallations are more important). The secondary species can be treated simply by using a source function obtained by multiplying the primary density by the gas density. It would be straightforward to apply the previous techniques to a realistic gas distribution (taking into account, in addition to the fairly flat HI distribution, that of molecular H2 and ionized HII which are more strongly peaked in the inner parts, see e.g. Strong & Moskalenko 1998 for a summary and references) and to infer the contours inside which the secondaries are created. The corresponding f-surfaces are not shown here, as they would be quite similar to those of the primaries (see left panel). What we do display in the right panel are the f-surfaces of the primaries that lead to given secondaries, as these progenitors determine the secondary-to-primary ratios (see Sect. 4.1).
From the previous results, it appears that only a fraction of the
sources present in the disk actually contribute to the flux in the
solar neighborhood. In this paragraph we present the fraction
sources which are located inside given f-surfaces.
|Figure 11: 99%-surfaces for several species. The left panel corresponds to primary species (protons, CNO and Fe) while the right panel corresponds to the progenitors of secondary species (B and sub-Fe), for L=6 kpc and .|
|Figure 12: 99%-surfaces for several , in the case L=6 kpc. The left panel corresponds to protons while the right panel corresponds to Fe nuclei.|
|Figure 13: 99%-surfaces for several L, in the case . The left panel corresponds to protons while the right panel corresponds Fe nuclei.|
|Figure 14: Fraction (in %) of the Galactic sources contributing to the fraction f of the cosmic ray flux at the solar position, for protons and Fe nuclei, for the particular diffusion model L=6 kpc, .|
Figure 14 also shows that a very small fraction of
sources may contribute to a non negligible fraction of the fluxes.
For example, the sphere of radius pc centered on the
solar neighborhood contains only
of the sources
but for L=6 kpc and
it is responsible for about 5%
of the proton flux and 18% of the Fe flux.
The mean age of the cosmic rays is given by
7-400 Myr (see Table 7).
For a supernova rate of three per century, the total number of sources
responsible for the flux is
This tells us that in models with the largest ,
18% of the
Fe flux can be due to
only 5 sources.
The approximation of stationarity and of continuous source
distribution is likely to break down with such a small number of
sources. Conversely, for small values of (preferred by many authors), this approximation is probably better.
|L=10 kpc||196 / 392 / 332||76 / 74 / 17|
|L=2 kpc||72/ 130 / 108||24/ 22 / 7|
Donato et al. (2002) emphasized that the existence
of a local underdensity (
cm-3) around the solar
neighborhood greatly affects the
interpretation of the flux of radioactive species at low energy
(we refer the interested reader to this paper for a deeper
discussion and references on the local interstellar medium).
The most important effect of this hole is that it exponentially
decreases the flux by a factor
pc is the radius
local underdense bubble and
is given by Eq. (12)).
This can be
easily understood as there is almost no gas in this region, hence no
spallations, leading to no secondary production.
The local bubble is obviously not spherical, but this approximation is
sufficient at this level. This attenuation factor is straightforwardly
recovered starting from the probability density as given in
Sect. 4.2, if correctly normalized to unity. To this end,
the sources (here spallation of primaries on the interstellar medium) are
considered to be uniformly distributed in the disk, except in the
empty region r<a. The probability density is zero in the hole whereas
outside, it is given by
We saw in a previous section that the high energy e+ and e-behave like unstable species. Their
can be compared to
Realistic values for
in Fig. 15.
At high energy, the Lorentz factor enhances the lifetime
of radioactive nuclei, making their origin less local, whereas the
energy losses are increased for electrons and positrons, making their
origin more local (99-90-50% of 100 GeV
come from sources located in a thin disk with radius
|Figure 15: Realistic values of and for two extreme halo sizes L and diffusion slope . As all results in this section, propagation parameters fit B/C and are taken from MTD02.|
Finally, radioactive nuclei are a very important tool for cosmic ray physics. They come from a few hundreds of parsec, and their fluxes are very sensitive to the presence of a local underdense bubble, through the attenuation factor . For example, for a typical bubble of size a=100 pc and an energy 800 MeV/nuc (interstellar energy), if , whereas . With Myr, Myr and Myr, it leads to and . For 14C, the attenuation is around 1 GeV/nuc, so that this species is heavily suppressed. However, it should be present around 10-100 GeV/nuc (as at these energies), with the advantage that solar modulation is less important at these energies.
The flux of radioactive species directly characterizes the local diffusion coefficient K0 if the local environment is specified. This would in turn allow to break the degeneracy in propagation parameters that one can not avoid at present. Last, even though the surviving fraction of a radioactive does depend on the halo size L, we emphasize that it is a very indirect way to derive the propagation parameters. In the forthcoming years, new measurements of radioactive species that do not depend on L (e.g. by PAMELA and AMS) should provide a promising path to update our vision of cosmic ray propagation.
The question of the source distribution is very present
in cosmic ray physics. With the occurrence of the old problem of short
pathlengths distribution in leaky box models (see for example
Webber et al. 1998), Lezniak & Webber (1979) studied a
diffusion model with no-near source in the solar neighborhood.
Later, Webber (1993a,b) propagated -like sources
with diffusion generated by a Monte Carlo random walk for the same
purpose. Brunetti & Codino (2000) follow this line but they introduce
random walks in a more realistic environment, i.e. circular,
elliptical and spiral magnetic field configurations. In a more formal
context, Lee (1979) used a statistical treatment of means and
fluctuations (see references therein) to characterize amongst others
the possibility that nearby recent sources may dominate the flux of
Finally, it is known that the present cosmic ray models are not able
to reproduce accurately for example proton-induced -rays measurements.
To illustrate this point, we plot in Fig. 16 the
radial distribution of protons
obtained with the same diffusion parameters as used above.
None of the models shown match the data.
|Figure 16: Radial distribution of the proton flux for the models discussed in this study, compared to the source radial distribution of Case & Bhattacharya (1998) given Eq. (19). For each of the values L=2 kpc and L=10 kpc, the three values , 0.6 and 0.85 are presented, the flatter distribution corresponding to the lower . Also shown is the gamma-ray emissivity per gas atom ( COS-B Bloemen 1989), which is proportional to the proton flux, as given by COS-B (open circles, Bloemen 1989) and EGRET (triangles, Strong & Mattox 1996), along with the proton flux obtained with the Strong & Moskalenko (1998) distribution (see Sect. 6).|
We provide an answer to this question under the two important hypotheses that the source distribution is continuous and that we have reached a stationary regime: most of the cosmic rays that reach the solar neighborhood were emitted from sources located in a rather small region of the Galactic disk, centered on our position. The quantitative meaning of "rather small'' depends on the species as well as on the values of the diffusion parameters. For the generic values and L=6 kpc chosen among the preferred values fitting B/C (see Sect. 7), half of the protons come from sources nearer than 2 kpc, while half of the Fe nuclei come from sources nearer than 500 pc. Another way to present this result is to say that the fraction of the whole Galactic source distribution that actually contributes to the solar neighborhood cosmic ray flux can be rather small. For the generic model just considered, 8% (resp. 1.5%) of the sources are required to account for 90% of the proton (resp. Fe) flux. These fractions are smaller for higher and smaller L. To summarize, the observed cosmic ray primary composition may be dominated by sources within a few kpc, so that a particular care should be taken to model these source, spatially as well as temporally (Maurin et al. 2003b).
Independently of all the results, this study could be used as a check for more sophisticated Monte Carlo simulations that will certainly be developed in the future to explore inhomogeneous situations. Several other consequences deserve attention. First, the results may point towards the necessity to go beyondthe approximations of both continuity and stationarity. In particular, it could be that only a dynamical model, with an accurate spatio-temporal description of the nearby sources, provides a correct framework to understand the propagation of Galactic cosmic rays. The contribution from nearby sources would be very different in the low energy (GeV/nuc) or in the high energy regime (PeV) compared to the stationary background. Second, as discussed in Sect. 4.1, the diffusion parameters derived from the observed B/C ratio have only a local validity, and one should be careful before applying them to the whole Galaxy, since the cosmic rays are blind to most of it.
This work has benefited from the support of PICS 1076, CNRS and of the PNC (Programme National de Cosmologie). We warmly thank Eric Pilon for his expertise on asymptotic developments. We also thank the anonymous referee for his pertinent suggestions.
For a primary species, the differential density
(in energy) N(r,z) is a solution of the equation
(see for example Maurin et al. 2002b and references
For a primary point source,
and we find in the disk (z=0)
In practice, the infinite sums above are truncated to some order , chosen as a compromise between accuracy (good convergence of the series) and computer time. In the case of a point source , the profiles are singular near the source and the convergence of the series appears to be very slow. A few methods are presented to speed up this convergence.
Part of the difficulty to evaluate numerically the Bessel expansions
comes from the fact that the resulting functions are singular at the
source position. If we know a reference function
which exhibits the same
singularity and for which the Bessel coefficients
are known, it is then judicious to write the
When the disk has an infinite radius, the Bessel sum can be replaced by
an integral, and the end result is obtained from the Bessel sum by
so that in the general case
- see Eqs. (A.5) and Eq. (A.4) -,
A cosmic ray crossing the Galactic disk has a probability p to
disappear in a nuclear reaction with interstellar matter. This
probability is related to the
reaction cross section