- Spatial origin of Galactic cosmic rays in diffusion models
- 1 Introduction
- 2 Description of the method
- 3 The escape through the diffusive volume boundaries
- 4 Secondary and radioactive species
- 5 The effects of spallation and convection
- 6 Realistic source distribution
- 7 Application to the propagation parameters deduced from the observed B/C ratio
- 8 Summary, conclusions and perspectives
- References
- Appendix a: Online Material

A&A 402, 971-983 (2003)

DOI: 10.1051/0004-6361:20030318

**R. Taillet ^{1,2} - D. Maurin^{1,3}**

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

**Abstract**

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.

2 Description of the method

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
probability density

(1) |

that this cosmic ray was emitted from a source located at the position ? Such a question falls among classical problems of statistics. A rigorous theoretical frame is provided by the Bayes approach that summarizes the proper use of conditional probabilities. A cruder but sufficient (and equivalent) treatment is given by the frequency interpretation. The probability written above is simply given by

(2) |

where is the number of paths reaching and is the density of paths going from to . We finally notice that the latter number determines the density of cosmic rays that reach the position , when a source is placed at . We can thus write

where the density is the solution of the propagation equation for a point source located at . The normalization factor in this relation is obtained by imposing that actually is a probability density, i.e. is normalized to unity. We refer to the contours on which the probability density is constant as

If the sources are distributed according to
,
the probability that a cosmic ray detected at
was emitted
from a surface
is given by

This probability contains all the physical information about the spatial origin of cosmic rays. We define the

3 The escape through the diffusive volume boundaries

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

In unbounded space, the solution is given by . The influence of the boundaries is estimated by solving this equation in three situations: first we consider only a side boundary, then only a top plus bottom boundary, and finally all the boundaries.

3.1 Boundaries influence

Our Galaxy can be represented as a cylindrical box with radial extension

With
,
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
)

normalization being obtained by imposing . The -probability is given by

This probability is independent of the value of the diffusion coefficient

The escape from the side boundary (located at

3.1.2 Top and bottom boundaries

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)

which allows to compute the -probability as before, which is a function of only. These integrals are somewhat intricate to compute numerically, due to the very slow convergence. In this particular case, the accuracy of the numerical calculation can be checked for , as a detailed study of the function (8) shows that in this limit

It is also noticeable that the quantity

gives the fraction of cosmic rays emitted from a distance that has escaped the diffusive halo before reaching us.

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
corresponding to
or
.

We also show, in Table 1, the radii of the

% | % | % | ||||

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.

4.1 Progenitors of stable secondaries

We find, in the case
(see Appendix B.3), and for a
homogeneous distribution of sources,

The resulting integrated probabilities are shown in Table 2. The source of the primary that will give the secondaries observed at a given point is located farther away than the sources of the primary we detect (compare Tables 2 and 1). This may be of importance if for instance the source composition or the source intensity varies with position: in the ubiquitous secondary-to-primary ratio, the numerator is sensitive to sources located on a greater range than the denominator. Moreover, these secondaries set the size of an effective "local'' zone outside of which the particles reaching the solar neighborhood have never been. The local observations tell nothing about the propagation conditions outside of this zone. One could object that this conclusion is mainly based on the

% | % | % | ||||

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 |

4.2 Radioactive secondaries

where . This expression can be transformed using the identity (Lebedev 1972)

The approximation in the last step is valid if the exponential term decreases with fast enough (i.e. is large so that the upper limit can be set to 1 in the integral). We then recognize in (10) the Fourier-Bessel transform of , so that finally the normalized probability reads

where the following typical length has been introduced

Indeed, this result can be derived much more straightforwardly starting from the stationary equation (with a source at the origin) in unbounded space. This is also in full agreement with the expression given in Appendix B (see also Sect. 4.1) of Donato et al. (2002), where we found the same expression starting from the propagator of the non-stationary diffusion equation in unbounded space.

To sum up, Eq. (11) is valid as long as
and
:
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
is large,
which is the case at high energy because of the relativistic factor
,
even if the proper lifetime
is short.
The
-probability is straightforwardly derived.
As on these typical scales,
the source distribution can safely taken to be constant, the distance
is expressed as

It means that the sources that contribute to the fraction

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

Formulae (11) and (13) can be used with . This effect is discussed by Aharonian et al. (1995) to show that a nearby source may be necessary to explain the high energy electron flux observed on the solar neighborhood.

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.

5 The effects of spallation and convection

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
no-wind case).
With an infinite halo, the probability density in the disk is
given by

where the characteristic radius has been defined. The expression in Eq. (14) is a function of only. The deviation from a pure law, as well as deviations due to escape, radioactive decay and spallation (see next section), is shown in Fig. 5.

The -probability is given by

(15) |

Some values are indicated in Table 4 and plotted in Fig. 6.

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
halo size
.
This is our parameter
up to
a factor 2.

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
random walks.
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.
The limit
is noteworthy, as the resulting
expression isolates the influence of spallations:

= |

where the quantity has been defined. Would there be no spallation, the behavior would be recovered. The term has the effect to kill the contributions of in the integral, with . It leads to a decrease of the integral on scales . Some typical values, for (see Sect. 5.5) with and are given below at 1 GeV/nuc and 100 GeV/nuc. The heavy species are more sensitive to spallations, so that they come from a shorter distance. This could in principle affect the mean atomic weight of cosmic rays if the composition of the sources is not homogeneous (see e.g. Maurin et al. 2003a). See Sect. 7.1) for the results with realistic propagation parameters.

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

yields the more useful form

This expression is in full agreement with Eq. (C.3) obtained with the random walk approach (see Appendix C). For large values of , the convergence can be checked by comparing the results to the asymptotic development

Finally, the - probability can be computed as before

Some values are indicated in Table 4.

p |
O | Fe | |

44 | 309 | 760 | |

, 1 GeV/nuc | 10.2 | 1.45 | 0.59 |

, 100 GeV/nuc | 115 | 16.4 | 6.7 |

The -probability is displayed in Fig. 6 as a function of . The effect of the Galactic wind is very similar to that of the top and bottom boundaries, whereas the effect of spallations is quite different. In the latter case, the cutoff in the density is a power law in and decreases much more slowly than the exponential cutoff due to the wind or to escape. As a result, the 99%-surfaces are much larger than the 90%-surfaces. This can also be seen in the first three lines of Table 4.

When all the effects above are considered, Eq. (A.5) gives

where , et . The smallest of these three numbers indicates the dominant effect. Various -probabilities are shown in Table 4.

L(kpc) |
(km s^{-1}) |
(mb) | (kpc) | (kpc) | (kpc) |

0 | 0 | 6.3 | 19 | 34 | |

0 |
4.9/V_{10} |
18.6/V_{10} |
41/V_{10} |
||

0 | |||||

5 | 0 | 0 | 3.1 | 9.5 | 17 |

5 | 10 | 50 | 2.05 | 6.8 | 13.1 |

For a radioactive species, the spallations and the Galactic wind have a negligible effect on propagation as long as (see Sect. 4.2) is smaller than

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 *N*_{0}(*r*), the number of particles diffusing without
spallations, as

so that the number of crossing is readily obtained from the densities with and without spallations as

Notice that this expression applied to Eq. (16) leads to Eq. (C.2) when , , and when

The evolution of the grammage with the distance of the source is displayed in Fig. 7. The effect of escape, spallations and Galactic wind is shown.

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 kpc^{2} Myr^{-1}. |

As a cross-check, it can be noticed that in this approach, the mean grammage

yields the right order of magnitude for the usual grammage derived from leaky box analysis (9 g cm

5.5 The energy dependence

The diffusion coefficient actually depends on energy.
A commonly used form (see Maurin et al. 2002b for a discussion) is

where stands for the rigidity, and . The previous results were given for , typical for a proton with an energy of 1 GeV. This implies that the parameters , are larger at higher energy. They eventually become larger than

6 Realistic source distribution

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

with kpc, , for Case & Bhattacharya (1998). This distribution is now closer to the distribution adopted by Strong & Moskalenko (1998) ( and ), a flatter distribution designed to reproduced radial -ray observations (see Fig. 16). This distribution can be inserted in Eq. (4), which is then used to compute the

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.

7.1 Evolution of and with

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.

From the sets of diffusion parameters that fit the B/C ratio,
the values of the parameters
and
are computed (see Table 5) for
the four nuclei shown in Fig. 9 and for three
values of
.

p, |
B-CNO | Sub-Fe, Fe | ||

- | - | |||

L=10 kpc |
(kpc) | /5.17/1.6 | /3.41/0.89 | /3.26/0.83 |

(kpc) | 33.5/10.2/4.0 | 4.21/1.07/0.35 | 1.46/0.37/0.12 | |

(kpc) | 6.43/4.67/2.50 | 5.23/2.92/1.11 | 4.00/2.03/0.57 | |

L=6 kpc |
(kpc) | /3.64/1.15 | /2.40/0.64 | /2.30/0.60 |

(kpc) | 24.7/7.4/2.9 | 3.10/0.80/0.26 | 1.08/0.27/0.09 | |

(kpc) | 4.93/3.5/1.88 | 3.96/2.18/0.82 | 2.99/1.63/0.54 | |

L=2 kpc |
(kpc) | /1.40/0.46 | /0.92/0.26 | /0.88/0.24 |

(kpc) | 9.7/2.9/1.2 | 1.21/0.31/0.10 | 0.42/0.10/0.03 | |

(kpc) | 2.07/1.44/0.78 | 1.63/0.87/0.33 | 1.21/0.57/0.19 |

From these values, the 50-90-99%-surfaces are derived and displayed in Fig. 10, for protons and Fe nuclei. The effect of (Fig. 12), of

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 H_{2} 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
of the
sources which are located inside given *f*-surfaces.

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. |

This fraction is presented in Fig. 14 for the particular model

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.

p |
Fe | ||

- | |||

L=10 kpc |
90% | 22.9/13.3/3.9 | 9.6/2.2/0.24 |

99% | 56.7/40.7/16.4 | 35/14.5/1.77 | |

L=6 kpc |
90% | 14.3/7.6/2.3 | 5.63/1.5/0.17 |

99% | 40.9/27/9.6 | 22.2/9.2/1.48 | |

L=2 kpc |
90% | 2.9/1.6/0.83 | 1.2/0.24/0.025 |

99% | 9.3/5.9/3.8 | 4.7/1.77/0.27 |

p |
Fe | |

- | ||

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
of the
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

(20) |

The quantity is obtained directly from the no hole case (see Sect. 4.2) by replacing by . It means that the sources that contribute to the fraction

We saw in a previous section that the high energy e^{+} and e^{-}behave like unstable species. Their
typical length
can be compared to

(21) |

The dependence in the propagation model is similar for both expressions and is contained in the last term. There is a big difference, though, as the typical distances travelled by radioactive nuclei scale as , whereas they scale as for electrons and positrons.

Realistic values for
and
are presented
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
kpc).

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. |

For 100 GeV e

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 ^{14}C, 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 *K*_{0} 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
primaries.
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). |

One is left with two alternatives: either modifying the source distribution (for example, the distribution of Strong & Moskalenko 1998 yields a better agreement), or giving up the assumption that the diffusion parameters apply to the Galaxy as a whole (Breitschwerdt et al. 2002). It is thus of importance to understand to what extent the cosmic rays detected on Earth are representative of the distribution of the sources in the whole Galaxy.

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.

- Aharonian, F. A., Atoyan, A. M., & Völk, H. J. 1995, A&A, 294, L41 NASA ADS
- Bloemen, H. 1989, ARA&A, 27, 469 NASA ADS
- Breitschwerdt, D., Dogiel, V. A., & Völk, H. J. 2002, A&A, 385, 216 NASA ADS
- Brunetti, M. T., & Codino, A. 2000, ApJ, 528, 789 NASA ADS
- Case, G. L., & Bhattacharya, D. 1998, ApJ, 504, 761 NASA ADS
- Donato, F., Maurin, D., & Taillet, R. 2002, A&A, 381, 539 NASA ADS
- Jones, F. C. 1978, ApJ, 222, 1097 NASA ADS
- Jones, F. C. 1979, ApJ, 229, 747 NASA ADS
- Lebedev, N. N. 1972, Special functions and their applications (Dover)
- Lee, M. A. 1979, ApJ, 229, 424 NASA ADS
- Lezniak, J. A., & Webber, W. R. 1979, Ap&SS, 63, 35 NASA ADS
- Maurin, D., & Taillet, R. 2003, A&A, to appear
- Maurin, D., Donato, F., Taillet, R., & Salati, P. 2001, ApJ, 555, 585 NASA ADS
- Maurin, D., Taillet, R., & Donato, F. 2002a, A&A, 394, 1039 NASA ADS
- Maurin, D., Taillet, R., Donato, F., et al. 2002b, in Research Signposts, to appear
- Maurin, D., Cassé, M., & Vangioni-Flam, E. 2003a, Astroparticle Phys., 18, 471 NASA ADS
- Maurin, D., Salati, P., Taillet, R., Vangioni-Flam, E., & Cassé, M. 2003b, in prep.
- Stecker, F. W., & Jones, F. C. 1977, ApJ, 217, 843 NASA ADS
- Strong, A. W., & Mattox, J. R. 1996, A&A, 308, L21 NASA ADS
- Strong, A. W., & Moskalenko, I. V. 1998, ApJ, 509, 212 NASA ADS
- Vallée, J. P. 2002, ApJ, 566, 261 NASA ADS
- Webber, W. R. 1993a, ApJ, 402, 185 NASA ADS
- Webber, W. R. 1993b, ApJ, 402, 188 NASA ADS
- Webber, W. R., Kish, J. C., Rockstroh, J. M., et al. 1998, ApJ, 508, 940 NASA ADS

9 Online Material

Appendix A: General solutions of the diffusion equation in cylindrical geometry

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
therein)

with

(A.3) |

The various terms in Eq. (A.1) correspond respectively to (i) a differential operator describing convection out from the Galactic plane and isotropic diffusion

with

and

and is the Bessel transform of the source distribution (which may depend on the density of another species, in particular for secondary species).

For a primary point source,
and we find in the disk (*z*=0)

The generic solution for secondaries can be straightforwardly derived from that of primaries (e.g. Maurin et al. 2001),

We use . The distinction between and is necessary since both species have different destruction rates and rigidities.

Appendix B: Numerical evaluation of the point source solution in Bessel basis

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.

B.2 Sum representation: Comparison to a known function

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
density (
)
as

where the singularity is entirely contained in the

B.3 Integral representation for infinite radius disk

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
the substitution
and
,
so that in the general case
- see Eqs. (A.5) and Eq. (A.4) -,

with

and

The integrals of the form

where

The convergence is faster, as when . Second, using the identity (

This expression is meaningful only if

(B.2) |

The latter expression is meaningful only if

Appendix C: Alternative description of spallations: Random walk approach

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
by

where contains the dependence on the incidence angle of the particle with the Galactic plane. The propagation in the

In this expression, is the number of steps of the walk and depends on its statistical properties (for instance, for elementary steps and for

where is the variance of the elementary random step (in units of ), so that the physical time is related to by , where is the mean free path and

We are now able to compute the probability distribution of disk crossings for cosmic rays emitted from a distance

where the probability that a CR reaching distance

The above integral C.1 can be performed, yielding the final result

with . The average number of disk crossings is readily obtained:

and the associated variance tends to infinity. We can also compute the integrated probability, that more that

A particle having crossed

= | |||

= |

This can be written as

with defined in Eq. (17) and . The density of cosmic rays in the disk is then given by

The quantity seems to be related to the detailed statistical properties of the random walk under consideration, through , and . However, direct comparison with the alternative expression (16) obtained above indicates that these expressions are indeed equivalent, with .

Copyright ESO 2003