# A Minimal Model for Understanding Secondary Cosmic Rays

Ramanath Cowsik  Dawson Huth

###### Abstract

We take a phenomenological approach in a minimal model to understand the
spectral intensity of secondary cosmic-ray particles like positrons,
antiprotons, Lithium, Beryllium and Boron. Our analysis shows that cosmic rays
at $\sim$ GeV energies pass through a significant amount of matter in regions
surrounding the sources. This grammage decreases with increasing cosmic-ray
energy and becomes negligible beyond $\sim 100$ GeV. During the subsequent
propagation in the interstellar medium cosmic rays of all energies up to $\sim
10^{5}$ GeV/$n$ pass through about 1-2 g cm-2 of matter before leaking into
the intergalactic medium. It is in the interstellar medium that the bulk of
the positrons and antiprotons are generated. Also cosmic-ray nuclei like C, N,
and O at all energies generate additional amounts of Li, Be and B nuclei with
a spectrum similar to those of C, O etc. The implications of these findings of
the minimal model to the observations of gamma rays and also the importance of
spatial and temporal discreteness of cosmic-ray sources for modeling cosmic-
ray propagation are briefly pointed out.

###### keywords:

††journal: Advances in Space Research

[inst1]organization=Washington University in Saint Louis, Department of
Physics & McDonnell Center for the Space Sciences,addressline=One Brookings
Drive, city=Saint Louis, postcode=63130, state=Missouri, country=United States
of America

##  1 Introduction

In recent years a large number of theoretical models have been described to
understand the origins, propagation and spectral intensities of the observed
cosmic rays ([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]). The
main aim of the present paper is to present as briefly as possible a minimal
model to understand the intensities of secondary cosmic rays. Here we focus on
direct inferences that can be made from the observed intensities of various
cosmic-ray components, and attempt at a parameter-free analysis as far as
possible. In order to guide our phenomenological approach we begin with a
broad review of the observations:

(i) Gamma-ray emission from the Galaxy: Observations in the gamma-ray band
with dedicated instruments such as SAS-II, COS-B, Egret, CGRO, Fermi, Veritas,
HESS, etc. have revealed the presence of a large number of high-energy gamma-
ray sources in the Galaxy ([17, 18, 19, 20, 21, 22, 23, 24, 25, 16, 26]).
Besides a diffuse Galactic background in gamma-rays these discoveries have not
quite fulfilled the early expectations that this is a sure way of identifying
the cosmic ray sources ([27]). However, these observations show that cosmic
rays subsequent to their acceleration interact with matter, magnetic and
radiation fields in the source regions. The recent work of Blasi and his
collaborators show that magnetohydrodynamic analysis indicates that such a
shell or bubble formation of energetic charged particles before subsequent
leakage into the interstellar medium is expected on theoretical grounds
([16]). Eichler and Nath have pointed out that the number density of these
gamma-ray sources noted above is sufficiently large so that the line of sight
of the gamma-ray telescopes is likely to intersect several of these sources in
the disc of the Galaxy ([28]). We will return to these theoretical ideas in
the discussion section. Observations have also revealed the existence of a
diffuse background of energetic gamma-rays associated with our Galaxy. The
intensity of this diffuse background is such that cosmic rays at spectral
intensities observed near the Earth are adequate to generate this background
while interacting with the interstellar medium and radiation fields ([29, 30,
31, 32, 33]).

The gamma-ray observations, briefly noted above, have identified two regions
where the primary cosmic rays subsequent to their acceleration suffer
interactions with the ambient matter and with magnetic and radiation fields:
1) in the immediate vicinity of the sources and 2) in the general interstellar
medium. These observations imply that other products of the cosmic-ray
interactions generated in these two regions such as antiprotons, positrons and
spallogenic Li, Be, B, etc. would also be observable in cosmic rays. As the
ratio of the energy of parent particles like protons to that of $\bar{p}$ and
$e^{+}$ is very large, the contribution to the fluxes of $\bar{p}$ and $e^{+}$
will be suppressed if the high energy particles diffuse away from the source
regions more rapidly at higher energies. Even though the Eichler-Nath study
makes it difficult to assign precise ratios of the relative contributions of
these sources, two distinct sources of secondary cosmic rays have been clearly
identified by these observations. It appears that the contributions of these
two sources to the observed intensities are of the same order of magnitude.
One of the objectives of the present study is to provide an estimate of their
relative contributions.

Radiative losses modify significantly the spectral intensities of the
electronic components of cosmic rays during their propagation subsequent to
acceleration or production as secondaries in cosmic-ray interactions.
Accordingly in the assessment of their spectra and of nuclei that suffer
radioactive decay we need to keep in mind the various categories of possible
sources with characteristics that are four combinations of their temporally
and spatially discrete or continuous nature, each of which leave their
distinctive signatures on the observed spectra.

(ii) Muons - the penetrating component of cosmic rays: We have borrowed this
phrase from Bhabha ([34]) a pioneer in cosmic-ray studies. The muons generated
in the atmosphere through cosmic-ray interactions and arise mainly through the
decay of pions. Their study dates back to the earliest days of cosmic-ray
physics ([35, 36, 37, 38]). Because of their high mass and long lifetime their
spectrum at high energies suffers very little modification and has provided
important information on the spectral intensities of the nucleonic component
of cosmic rays. Their intensity and spectrum is well-measured with magnetic
spectrometers and hodoscopes located at sea level and also deep underground
and under water ([39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56]). Scaling of high energy interactions and straight-forward
calculation of pion decay in an exponential atmosphere yields the result (for
example [57, 58, 59, 60])

$\begin{split}\frac{dN_{\mu}(E)}{dE_{mu}d\omega}=AE^{-\beta}\left\\{\left(\frac{E}{\epsilon_{\pi}\eta_{\pi\mu}}+1\right)^{-1}+\dots\right\\}\\\
\dots\mathrm{cm}^{-2}\mathrm{s}^{-1}\mathrm{Sr}^{-1}\mathrm{GeV}^{-1}.\end{split}$
(1.1)

Here $\epsilon_{\pi}=\frac{h_{o}m_{\pi}}{c\tau_{\pi}}$, $h_{o}$ is the scale
height of the Earth’s atmosphere, $\eta_{\pi\mu}$ is the weighted fraction of
the energy carried away by the muons in the decay of the pion, the ellipsis
inside the flower bracket represents a similar term for the contribution of
kaons ($\sim 5$%) and ellipsis outside the bracket is to account for the decay
of the muons (important at $\sim$ 1 GeV here). $A$ is a constant, depending on
the normalization to the cosmic-ray fluxes and various parameters like
production cross-sections, interaction mean free path etc. and $\beta$ is the
spectral index of the spectral intensities of nucleons in primary cosmic rays.
In the 1960’s $\beta$ was chosen to be 2.67 and in recent decades $\beta=2.7$
has been the choice. The muon fluxes and the expression given in Equation 1.1
have been used widely in the study of neutrino physics (for example [61, 62,
63, 64, 65, 66, 67, 68]) and for normalizing the theoretical fluxes of
neutrinos. Here we use the muon spectrum as a guide in the context of
analyzing the spectral intensities of secondary cosmic rays like $e^{+},\
\bar{p}$, Li, Be, B etc, especially for the calculation of the spectrum of
$e^{+}$, which is produced by the decay of the muons.

(iii) Constancy and isotropy of the cosmic-ray flux: Cosmic rays interacting
with C, N and O nuclei in the Earth’s atmosphere generate 10Be nuclei, which
suffer $\beta$-decay with a half-life of $1.387\times 10^{6}$ years (lifetime
$\sim 2.0\times 10^{6}$ years). These nuclei get leached by rain and get
incorporated in the sea sediments. The study of the concentrations of 10Be
nuclei with depth in the sediments establishes the constancy of the cosmic-ray
fluxes to within $\sim 10\%$ on the average over the past $\sim$10 million
years. Similarly the concentrations of various cosmogenic radioactive nuclides
in primitive meteorites, including 10Be and 40K, establishes the approximate
constancy over much longer periods of $\sim$1 billion years ([69]). The recent
observation of 60Fe in cosmic rays (T${}_{\frac{1}{2}}=2.62\times 10^{6}$
years or $\tau=3.8\times 10^{6}$ years) at the $\sim 7.5\times 10^{-5}$ level
compared with 56Fe do indicate the possibility of a small but significant
contribution to the local cosmic ray flux by nearby supernovae at distances of
$\sim 300-500$ pc from which they can propagate without significant decay. The
low level of 60Fe and the other data discussed here show there has not been a
major contribution from a single supernova event to the cosmic ray flux at the
Earth over the past tens of millions of years ([70]).

Galactic cosmic rays diffuse into the solar system and this inward diffusion
is counteracted by outward convection in the magnetized solar wind. However,
diffusion becomes progressively more dominant at high energies and the cosmic-
ray flux at energies greater than $\sim 300-1000$ GeV can penetrate through
the solar magnetic fields and reach the Earth retaining their intrinsic
anisotropy. Observations compiled by ([7]) indicate that the anisotropy in the
fluxes is less than $\sim 10^{-3}$ at energies beyond $10^{3}$ GeV at least up
to $\sim 3\times 10^{5}$ GeV. These features show that there are no transients
in the cosmic-ray flux and multiple sources contribute to the cosmic-ray flux.

(iv) Observations of spectral intensities of cosmic rays: Detailed
measurements of the composition and spectra of various components of cosmic
rays have been going on for a long time with progressively better instruments
([71, 72, 73, 74, 75, 76, 77, 78, 79, 80]). In this brief paper we mainly
focus on the data provided by the Alpha Magnetic Spectrometer and CALET
because of their broad energy range and nearly complete coverage of all the
charged components up to reasonably high Z values. Also, we thought it will be
useful to analyze the data on various components from only the AMS and CALET
instruments as the efficiency of observation will not be significantly
different between neighboring charge groups and geometric factors will be
similar.

We show in Figure 1 the spectra of $p,\ \bar{p},\ \mathrm{and}\ e^{+}$ and in
Figure 2 the spectra of Li, Be, B, C, O, (Li + Be + B) and (C + O) all
measured by AMS ([80]). We fit these spectra with simple power laws in the
rigidity region 4 GV to 500 GV, away from rigidity domains where the spectra
are not significantly affected by solar modulation or by proximity to maximum
detectable momenta. We have listed in Table 1 the spectral indices of these
components of cosmic rays. It is seen from Table 1 that the spectra of $p,\
\bar{p},\ \mathrm{and}\ e^{+}$ have very similar spectral indices (within $\pm
0.02$ of their mean value) but the spectral indices of Li, Be, and B are
significantly steeper than the spectra of C and O, (steeper by $\sim 0.3$ in
the index). This important difference between the characteristics of the two
groups with respect to the spectrum of their progenitors is addressed by the
minimal model.

Figure 1: AMS-02 spectra of $p,\ \bar{p},\ \mathrm{and}\ e^{+}$. Note that the
spectral index of these three components are very close to one another,
indicating that $\bar{p}$ and $e^{+}$ are products of high energy interactions
of the abundant primary proton component of cosmic rays Figure 2: The spectra
of the primary C and O nuclei and those of the spallogenic Li, Be and B nuclei
are shown (including the (C+O) and(Li+Be+B) spectra). Note that the spectra of
Li, Be and B are significantly steeper - a feature which is addressed in the
main body of this paper.

Particle | $q_{o}$ | $\beta$ |  $b$ (GeV Myrs)1 |  $\tau$ (Myrs) | Fit range (GV or GeV)
---|---|---|---|---|---
$e^{+}$ | $8.49\pm 0.11$ | $2.81\pm 0.01$ | $1.6\times 10^{-3}$ |  $\sim$1 | 4 - 500
$e^{-}$ | $272.65\pm 4.24$ | $3.09\pm 0.01$ | $1.6\times 10^{-3}$ |  $\sim$1 | 8 - 500
$p$ | $(1.62\pm 0.01)\times 10^{4}$ | $2.772\pm 0.003$ | - | - | 20 - 500
$\bar{p}$ | $3.29\pm 0.16$ | $2.77\pm 0.02$ | - | - | 20 - 500
He | $(2.27\pm 0.02)\times 10^{-3}$ | $2.659\pm 0.003$ | - | - | 20 - 500
Li | $29.58\pm 0.77$ | $2.95\pm 0.01$ | - | - | 20 - 500
Be | $13.21\pm 0.22$ | $2.93\pm 0.01$ | - | - | 20 - 500
B | $41.86\pm 0.84$ | $2.96\pm 0.01$ | - | - | 20 - 500
C | $63.70\pm 0.70$ | $2.624\pm 0.003$ | - | - | 20 - 500
O | $65.72\pm 1.09$ | $2.622\pm 0.004$ | - | - | 30 - 500

Table 1: Table of best fit parameters for various cosmic-ray particles. All
spectra were fit with a power law except for the $e^{+}$ and $e^{-}$ spectra
which were fit using the model that includes radiative losses shown in
Equation 2.

##  2 The analysis of Data and the minimal model

###  2.1 Brief description of the minimal model

Cosmic rays are generated in a large number of sources distributed in the disc
of the Galaxy and subsequent to acceleration they are contained in a region
surrounding the sources forming a bubble or a shell not unlike the one
described by the theoretical studies of ([16]). The escape of the cosmic rays
from this bubble is dependent on energy, becoming more rapid at high energies.
The subsequent propagation in the interstellar medium leading to the leakage
of cosmic rays from the Galaxy is essentially independent of their energy up
to $\sim 10^{5}$ GeV (More detailed description is to be found in ([81, 82],
[8] and [9]). Beyond this energy the leakage rate increases with energy and
the spectrum of cosmic rays is modeled as rigidity dependent leakage of cosmic
rays injected into the Galaxy by the remnants of Type-I and Type-II supernovae
by ([83])

###  2.2 Spectral intensities of positrons

We first address the spectral index of the positrons, as this component
exemplifies nearly all the features of the minimal model. We noted that the
spectral intensity of positrons in the interval $4-500$ GeV matches very well
with that of the dominant proton component:

  1. 1.

Scaling of the pion production cross sections in high energy interactions and
the subsequent $\pi^{+}\rightarrow\mu^{+}\rightarrow e^{+}$ decay ensures that
the production spectrum of positrons $q_{e^{+}}(E)$ has the same spectral
slope as that of the protons observed in primary cosmic rays.

  2. 2.

This similarity between proton and positron spectra signifies the parent-
daughter relationship between the two components.

  3. 3.

The observed positron spectrum $f_{e^{+}}(E)$ is just the product
$q_{e^{+}}(E)\tau(E)$, where $\tau(E)$ is the effective residence time of the
positrons in the Galaxy. Because of the scaling of the cross-section for
higher energy interactions $q_{e^{+}}(E)$ has the same energy dependence as
their parents the protons. Thus the similarity of the two spectra further
implies that $\tau(E)$ is independent of energy, $E$.

  4. 4.

The observed intensity level of $f_{e^{+}}(E)$ corresponds to an effective
grammage, $\lambda\approx 2$ g cm-2 for cosmic rays in the interstellar medium
([6, 7, 8]). [84] have suggested a similar grammage of $\sim 1-2$ g cm-2 in
their analysis of the ionization rate of the molecular clouds in the
interstellar medium by cosmic rays.

  5. 5.

The radiative energy losses suffered by the positrons during their residence
in the interstellar medium for a duration $\tau$, should modify their
spectrum. There is indeed a noticeable and progressive steepening of the
spectrum of positrons beyond $\sim$ 300 GeV corresponding to the energy losses
suffered in a magnetic field of $\sim 5\ \mu$G and the cosmic microwave
background at 2.7 K. At these high energies the Klein-Nishina cross sections
are relevant for the scattering of positrons by starlight and such scatterings
do not contribute significantly to the energy loss ([8]). In the analysis of
the positron spectrum we may note that the source of positrons being the
interactions of primary cosmic rays with the interstellar medium is both
spatially and temporally continuous.

The resulting equilibrium spectrum of positrons takes the form

$\displaystyle f_{e^{+}}(E)$
$\displaystyle=\int_{0}^{1/bE}q_{o}E^{-\beta}(1-bEt)^{\beta-2}e^{-t/\tau}dt$
(2.1) $\displaystyle\approx q_{o}\tau E^{-\beta}$ $\displaystyle\mathrm{\ for\
}E\ll 1/b\tau$ (2.2)
$\displaystyle\approx\frac{q_{o}}{b(\beta-1)}E^{-(\beta+1)}$
$\displaystyle\mathrm{\ for\ }E\gg 1/b\tau$ (2.3)

Here, $q_{o}E^{-\beta}$ is the input source spectrum of positrons per unit
time into the interstellar medium, $b$ the radiative energy loss rate of
positrons $\sim 1.6\times 10^{-3}$ Myr-1 GeV-1 ($bE^{2}$ determines the actual
energy loss rate) and $\tau$ is the leakage lifetime of cosmic rays taken to
be independent of energy. We display in Figure 3. $f_{e^{+}}(E)$ for various
values of the lifetime $\tau$. From a perusal of the figure we may conclude
that an energy independent lifetime $\tau\sim 1-2$ Myr fits the data,
including the slight steepening seen at energies beyond $\sim$ 500 GeV. At the
highest energies beyond $\sim 1000$ GeV the positron spectrum generated in the
interstellar medium will reach a larger spectral index, one higher than the
nucleonic component that generated it, that is $\sim\beta=-3.7$ for $E\ >$
1000 GeV. Around $10^{4}$ GeV the position spectrum may steepen again
reflecting the steepening of the primary spectrum beyond $\sim 10^{5}$
GeV/$n$.

Figure 3: AMS-02 positron spectra with various choices of $\tau$ in the model
with radiative losses, all other parameters are taken from Table 1. Even
though the spectral intensities at low energies will increase linearly with
the assumed lifetime $\tau$, all the theoretical spectra are normalized at
$\sim 10$ GeV

It may be noted that the amount of grammage needed to generate the intensities
of the positrons is about 2 g cm-2, which is equal to the grammage that will
be traversed by cosmic rays moving in the interstellar medium for a period of
1-2 Myr estimated from the spectral shape. The amount of matter traversed by
the cosmic rays in the interstellar medium can not exceed $\sim 2$ g/cm-2,
otherwise it will lead to spectral intensities of positions exceeding the
observations.

The observation of sources of GeV gamma rays, as noted before, show that
cosmic rays interact with matter in the vicinity of the sources before they
diffuse into the interstellar medium. What is the level of contribution of
these interactions to the positron flux?

According to the kinematics of pion production $\sim 10\%$ of the energy of
the incoming nucleon gets transferred to the leading pion, which decays
transferring about $\sim 75\%$ of its energy to the muon, which in turn
transfers $\sim 33\%$ of its energy to the positron. The net effect is that
the weighted average energy of the proton which contributes to the production
of a positron is $\sim 25$ times that of the positron, for energies beyond
$\sim 5$ GeV. Thus the energy per nucleon needed even for the production of
positrons of a few GeV is around $60-80$ GeV/$n$. In the minimal model (see
for e.g. [8, 9]) at these energies the nucleonic component diffuses away from
the vicinity of the sources without much interaction with the matter in the
ambient medium. Thus the production kinematics ensures that positrons of
$E\gtrsim$ 5 GeV get negligible contribution from the interactions of cosmic
rays in matter surrounding the sources. We will revisit some of these
distinctive features induced by kinematics later in the discussion of
antiprotons and spallation products.

Processes very similar to the production of positrons also generate electrons,
except in this case the $\pi^{-}$ decay chain is involved. As the net isospin
of the nuclear active component is positive more positrons are produced than
electrons. Thus the positron fraction, $e^{+}/(e^{+}+e^{-})$ at production is
$\sim 0.6$ ([6, 7]). This has interesting consequences for the expectation of
the positron fraction at high energies. We note here that this source of
positrons and electrons by cosmic-ray interactions in the interstellar medium
is both spatially and temporally continuous.

###  2.3 The electronic component of cosmic rays

Observations with instruments such as CALET do not distinguish $e^{+}$ from
$e^{-}$ but yield the sum of the intensities of these two particles, which is
called the electronic component. To understand this component it is useful to
subtract from it the secondary electronic component generated in the
interstellar medium by the nuclear component of cosmic rays - the spatially
and temporally continuous source ([6]). The CALET data extends the earlier
data beyond 1500 GeV up to $\sim 4000$ GeV and confirms with better statistics
the steepening of the electronic component beyond $\sim 800$ GeV. The
subtraction of the secondary electronic component generated in the
interstellar medium will tend to sharpen the steepening leading to an
essential cut-off in the primary electronic component beyond a few TeV. (As an
aside, we note that a very early analysis with a doubly continuous source
model had indicated a lifetime $\sim 1-2$ Myr for the electronic component
([85])) Two important points need to be made here:

  1. 1.

The secondary electronic component generated in the interstellar medium is
spatially continuous (to the extent that the ISM is smooth) and temporally
continuous (to the extent that cosmic ray intensity in the ISM is constant).
Accordingly, its spectrum will steepen to a form $\sim E^{-3.7}$ and will
extend up to $\sim 10^{4}$ GeV. Beyond this it would steepen reflecting the
steepening of the cosmic ray spectrum beyond $\sim 10^{6}$ GeV.

A direct consequence of this persistence of the secondary component is that at
very high energies the positron fraction will tend to $\sim 0.6$ reflecting
its production by the nuclear component in collisions with the interstellar
matter.

  2. 2.

The primary electronic component which is produced in the sources will take a
finite time to diffuse from the sources to the Earth where it is detected.
Earlier analysis that assumes that even though the sources are spatially
discrete, they are temporally continuous ([86, 8, 87]) indicate a cut-off in
the primary electronic component of the form

$f_{n}(E)\sim\mathrm{exp}\left[\frac{-br_{n}^{2}EE_{x}}{4\kappa(E_{x}-E)}\right]$
(2.4)

Here $r_{n}$ is the distance to the $n$th nearest source, $E_{x}$ the maximum
energy up to which the primary electronic component is accelerated, $b\sim
1.6\times 10^{-3}$ GeV-1 Myr-1 is a parameter which characterises the
radiation energy losses in the form $\dot{E}=-bE^{2}$, and $\kappa$ is the
value of the diffusion constant. For large values of $E_{x}$ the cut-off in
the spectrum occurs at

$\begin{split}E_{c}\approx\frac{4\kappa}{br_{n}^{2}}\sim 800\mathrm{\
GeV}&\mathrm{\ for\ }\kappa\sim 10^{28}\mathrm{cm}^{2}\mathrm{\ s}^{-1}\\\
&\mathrm{\ and\ }r_{n}\sim 300\mathrm{pc}\end{split}$ (2.5)

If the source is transient and generated a burst $\tau$ years ago at a
distance $r_{n}$ then it would take a finite amount of time for the particles
to diffuse the intervening distance in significant numbers.

$\tau_{n}\sim\frac{r_{n}^{2}}{6\kappa}$ (2.6)

and the spectrum would be cut off at $E_{c}\approx 1/b\tau$ for large $E_{x}$.
Furthermore, there will be considerable dilution of the flux during the
further time evolution subsequent to the arrival of the diffusion front ([6]).

###  2.4 The antiproton component

It is remarkable that the antiproton component, at energies $E\gtrsim 10$ GeV
has the same spectral slope as the positrons and also the protons (that are
the dominant particles in the primary cosmic-ray spectrum). This strongly
endorses a joint origin for $e^{+}$ and $\bar{p}$, namely, as the secondary
particles generated in the collisions of primary cosmic rays in the
interstellar medium. The kinematics of the production of antiprotons are
different from those of the positrons. The threshold for $\bar{p}$ production
in $p-p$ collisions is $\sim 7m_{p}\sim 6.6$ GeV and the antiproton emerges
even at threshold with a total energy of $\sim 1.7$ GeV. Accordingly cosmic-
ray interactions in matter surrounding the sources contribute a small but
significant flux at energies below $\sim 10$ GeV.

Antiprotons in cosmic rays suffer negligible amounts of annihilation in the
interstellar medium and no losses of energy through radiation. Accordingly
their spectrum is not modified during their lifetime in the Galaxy. Beyond
$\sim$ 5-10 GeV their intensities are consistent with their residence in the
Galaxy for $\sim$ 1-2 Myr.

###  2.5 Spectral intensities of L and M nuclei

The grammage of $\sim 2$ g cm-2 in the interstellar medium estimated to
generate the observed positron and antiproton intensities will concomitantly
lead to the production of Li, Be and B nuclei through the spallation of C, O
and other nuclei. Marrocchesi et al. of the CALET collaboration ([88]) compare
the flux of B measured by them over a 5 year period with those by AMS and
others. They confirm the progressive flattening of the Boron spectrum beyond
$\sim 200$ GeV/$n$ and extend it to $\sim 1500$ GeV/$n$; in this energy domain
we note the spectrum fits $\sim E^{-2.7}$ dependence, while at energies below
$\sim 100$ GeV the spectrum is steeper, $\sim E^{-3.1}$.

The flat portion of the Boron spectrum could be attributed to the production
of these nuclei in the interstellar medium and storage for a time $\sim 2$
Myr. Extending the $E^{-2.7}$ spectrum to lower energies, we may note that it
contributes to $\sim 25\%$ of the observed Boron spectrum at $\sim 10$
GeV/$n$. The difference of $\sim 75\%$ is an estimate of the contribution of
the spallation in the regions surrounding the sources. The figure 9. of ([88])
displays the observed B/C ratio up to 1500 GeV/nucleon, and illustrates this
point very well. If we draw a horizontal line through the point at $\sim 200$
GeV/nucleon, it reproduces the flux ratio at high energies and contributes
about $30\%$ of the ratio at 10 GeV/nucleon. The difference drops steeply with
increasing energy and becomes negligible beyond $\sim 100$ GeV/nucleon. In the
minimal model this difference represents the energy-dependent grammage
traversed by the cosmic rays in the regions surrounding the sources. The
evidence for such a clear separation of the two components in the spectra of
secondary nuclei is not as clear in the spectra of Lithium and Beryllium, but
the AMS and CALET data do not preclude such a separation.

###  2.6 The injection spectrum

In the minimal model cosmic rays are injected with the same spectral
characteristics as the observed spectra of protons and other nuclei $\sim$
E-2.7. The spectra of gamma rays and non-thermal radio waves from young
supernova remnants indicate the presence of energetic particles with flatter
spectra $\sim$ E-2 – E-2.5 in apparent contradiction with the requirements.
This issue has not been resolved definitively; however there are many
possibilities: Völk and collaborators have argued that the spectra of young
supernova remnants need not necessarily represent the spectrum of particles
that are finally injected into the interstellar medium ([89]). The analysis of
the evolution and structure of supernova remnants ([90]) clearly indicates the
softening of the spectrum with age with synchrotron spectrum reaching an index
of $\sim-0.75$ in $\sim 10^{4}$ years and suggests the possibility that by
about $10^{5}$ years when the energetic particles completely diffuse and merge
into the interstellar medium it is possible that the spectral softening will
reach the required value of $–0.85$. These particles responsible for the
synchrotron emission are generally attributed to those accelerated in the
reverse shock of the remnant. The acceleration in the forward shock could have
different characteristics; softening of the shock profile due to the presence
of neutrals, back reaction of the accelerated particles or their loss from the
system due to leakage will lead to the production of a steeper spectrum. There
may also be other possibilities for generating the requisite injection
spectrum. At present we need to treat it as a requirement of the model – the
injected spectrum of the cosmic rays ought to have the same spectral index as
the observed spectrum of primary nucleonic components up to $\sim 10^{5}$ GeV.

###  2.7 The residence time of cosmic rays in the Galaxy

In principle there are three possibilities for achieving energy-independent
leakage of cosmic rays from the Galaxy: (1) The loss of cosmic rays from the
galaxy is dominated by convection or advection; ([91]) invoke this mechanism
to fit the observations of gamma-rays in the central regions of the Galaxy up
to a radial distance of $\sim 5$ kpc. (2) Parker instability occurs in the
galactic plane and causes magnetic flux tubes to emerge out of the Galactic
plane inflated by cosmic-ray pressure. As this happens the interstellar gas
slides back to the disk forming clouds and the cosmic rays flow out of the
galaxy ([92, 93, 94, 95]). (3) The turbulence spectrum has a slope of $-2$, so
that the quasilinear theory predicts a diffusion constant independent of
energy; keeping in mind the findings of Reichherzer et al. this requires that
the fluctuation fields have a very small amplitude.

###  2.8 The anisotropy of cosmic rays

The diffusive current of cosmic rays generates anisotropy in the local
interstellar medium of magnitude $\delta\sim 3\kappa\Delta n/cn$, where $n$ is
the density of cosmic rays. The lifetime of cosmic rays gives us an estimate
for the value of the diffusion coefficient $\tau\sim s^{2}/4\kappa$, $s$ being
the scale length of the containment volume of cosmic rays. Combining these two
the anisotropy is seen to be $\sim 3\kappa\Delta n/cn\sim 3s/4c\tau$,
independent of energy in the minimal model, small and consistent with the
observational upper bounds. It should be noted that at the Earth the
anisotropies are noticeable only at energies beyond about $300\ –\ 1000$ GeV,
because at lower energies the diffusive current of cosmic rays into the solar
system is counteracted by the outward directed convection current in the solar
wind. At high energies diffusion dominates and the interstellar anisotropies
become measurable at the Earth.

##  3 Discussion

It is perhaps useful to begin the discussion as suggested by the referees with
a brief review of alternate views regarding the same aspects of Galactic
cosmic rays as those discussed in this paper. Before discussing the models, we
note that the AMS - Collaboration favor the view that their measurements of
the positron fluxes and the positron fraction imply that the positrons arise
from the annihilation of Galactic dark matter particles or from pulsars ([96,
97, 98, 99, 100]). The alternate view generally assumes diffusive transport of
cosmic rays leading to their loss from the Galaxy into the intergalactic
medium with a rate that increases with the rigidity or energy of the
particles, thereby reducing their lifetime with increasing energy (hereafter
referred to as the ’energy-dependent leakage model’). By an appropriate choice
of this energy dependence the declining ratio of the intensities of secondary
cosmic-ray nuclei like Boron to the intensities of their parents like Carbon
can be reproduced. In order that the spectra of primary cosmic rays like
protons, helium and carbon match the observations, the sources of cosmic rays
are assumed to inject a relatively flat spectrum into the interstellar medium
to compensate for the enhanced leakage from the Galaxy at high energies. This
alternate view was advanced nearly 50 years ago soon after the “nested leaky-
box model” was proposed by ([81, 82]), which is the minimal model presented in
this paper. In an early review of the status of the field ([101]) remarked in
conclusion that the measurement of the positron spectrum will dictate the
definite choice between the two models. Today we have good measurements of the
spectrum of positrons, as well as those of antiprotons, which are both similar
to that of the protons and as we have shown are consistent with the
predictions of the nested leaky box model.

The energy-dependent lifetime model evolved with the inclusion of many details
including the possible distribution of sources and parameters pertaining to
the Galaxy, and also convection and other transport mechanisms, all coded into
the software called “GALPROP”. This provided considerable flexibility to vary
the large number of parameters easily and explore the consequences. These
rapid developments in the field were reviewed by ([1]).

Subsequently new sets of extensive observations of cosmic rays, especially
space-borne detectors like PAMELA, AMS and CALET stimulated a large body of
elaborate work with the participation of a great many scientists investigating
even more complex sets of scenarios with computer simulations and new software
packages.

The motivation for the energy-dependent lifetime models were in part provided
by a wide variety of observations of the interstellar magnetic fields: These
observations, mainly using radio-astronomical techniques indicate that the
smooth magnetic field of the Galaxy is superimposed by random magnetic fields
of about the same or even greater strength ([102, 103]) and in the
interplanetary space ([104, 105]).The power-law index of the wavenumber
spectrum, $\alpha$ varies over a wide range from about $-2$ to $-5/3$
(Kolmogorov) or even down $-3/2$ (Kraichnan). The quasi-linear theory of
particle transport predicts a diffusion constant $\kappa$ that increases with
rigidity as R2-α and the assumption of the Kolmogorov value $\alpha$ = 5/3
which immediately helps to understand the falling (B/C) ratio in cosmic rays.
The relatively flat spectra predicted by the highly efficient acceleration of
cosmic rays in high Mach number shocks, and the observations of the spectra of
gamma rays from several supernova remnants. If the injection spectra of cosmic
rays correspond to these theoretical expectations and observations, then the
energy-dependent lifetime would generate a steeper spectrum in the steady
state in conformity with the observations. This is just a very brief summary
of a field which incorporates many subtle aspects of physics and astrophysics
reviewed by ([101, 1]) and by ([15]) and also referred to in context in this
paper ([74, 7, 2, 3, 12, 14, 106]). In such a scenario the spectra of
positrons and antiprotons tend to be noticeably steeper than that of their
parent particles, the protons and the other nuclei in cosmic rays.
Accordingly, several papers have discussed alternate sources for these
particles, sources such as annihilating Galactic dark matter and positrons
diffusing out of pulsar magnetospheres ([96, 97, 98, 99]). The review by JB
Tjus and L Merten provides a good overview with a more extensive reference
list. However, these models do not provide a natural explanation for the
observation that the spectra of both positrons and antiprotons have the same
spectral index and their intensities are both compatible with a traversal of
$1-2$ g cm-2 of interstellar matter by the primary cosmic rays.

Two papers that have appeared recently prompt us to reconsider this picture:
In the first paper by ([16] shows that cosmic rays streaming out of the
sources generate various modes and that these modes scatter cosmic rays very
efficiently reducing their diffusion. Accordingly, cosmic rays are confined to
a bubble or a shell surrounding the sources for sufficient time to
significantly affect the ratios of secondary to primary cosmic rays like B/C.
In summary they state “Our results … provide a physical explanation of the
numerous claims of suppressed cosmic-ray diffusion around Galactic sources…”.
Thus the analysis of cosmic-ray phenomena within the framework of the energy-
dependent leaky-box model will require reconsideration. The second paper by
([91]) note that even though Kolmogorov spectrum is an idealization and the
interstellar turbulence is driven on many scales from scales of $\sim$100 pc
of super bubbles down to the kinetic scales of cosmic rays themselves.
However, they adopt a Kolomogorov spectrum in their explicit simulations of
the transport based on Taylor-Green-Kubo formalism. One of the results of
their analysis is that the diffusion constant and its dependence on particle
rigidity depends not only on the exponent $\alpha$ of the power spectrum, but
also on the ratio of the strength of the random magnetic fields $b$ to the
smooth fields $B$ in the interstellar medium. They note that the index
$\delta$ depicting the increase of the diffusion constant with rigidity does
not quite reach the value $\delta=2-\alpha$, ($\delta$ = 1/3 for Kolmogorov),
predicted by the quasilinear theory until the ratio $b/B\leq 0.05$, and
increases with increasing value of the ratio. On the other hand, the radio
astronomical observations yield a ratio $b/B\sim 1$. This again points to the
need to reexamine the energy-dependent leakage models.

We are not aware of any recent paper that has critically reviewed these recent
developments, nor has a consensus view emerged regarding the parameters of the
energy-dependent leakage models. These energy-dependent lifetime models are
able to fit the observed positron spectrum assuming that that this arises from
leakage of positrons from the magnetospheres of pulsars. However, they have
not paid equal attention to explain the spectrum of antiprotons.

The extensive data collected with the AMS and CALET instruments provide good
coverage of the spectra with good statistics up to $\sim 5\times 10^{4}$ GeV
for protons and up to $\sim 1.5\times 10^{3}$ GeV/$n$ for the C and O nuclei.
These primary cosmic-ray nuclei are most relevant for the generation of the
secondary cosmic rays discussed in this paper. The data indicate breaks in the
spectra with the spectral indices changing by $\sim 0.1$ up to $\sim 0.4$ in a
narrow interval $\Delta E$ at various energies $E$. A key feature of these
breaks is that they occur over a very narrow range of energies $\sim 0.5E$ to
$2E$

There have been suggestions that because of the multiplicity and discrete
nature of the sources such breaks are to be expected when spectra of slightly
different spectral indices from a few sources overlap on top of one another.
However, such a superposition, say of two power laws, would only result in a
slow flattening over an energy band $\sim 0.2E$ to $5E$

With regard to the alternate possibility that the changes could be associated
with the acceleration process, it may be noted that for even in the first
order diffusive Fermi acceleration the energy gain is by a factor
$(1+\frac{\nu_{s}}{c})\approx 1+2\times 10^{-2}$ at each crossing of the
shocks, and changes as sharp as indicated by the data are not expected.

Protons dominate the nuclear component of cosmic rays, which contain about 99%
of the energy density of cosmic rays, $\sim 10^{-12}$ erg cm-3. The only
Galactic source that has been identified that contains adequate power and can
in principle generate this component is the supernova. The debris of the
explosion carries typically $10^{50}-10^{51}$ erg and higher values in the
explosion of very massive stars, the rate of such events being about one in
50-100 years. The efficiency of transfer of kinetic energy of the debris to
relativistic particles, through acceleration in shocks for example, has to be
very high $\gtrsim 10\%$ in order to replenish the cosmic rays which steadily
leak out of the Galaxy by diffusion.

The presence of various isotopes of Fe, Co, Ni and r-process nuclei in cosmic
rays clearly point to a supernova connection. For example 59Co is produced by
the electron capture of 59Ni, which in turn is the decay product of 59Cu.
These nuclei are generated in the very final stages of evolution in the cores
of massive stars, lasting less than a few days just before the explosion. At
the same time, the observation that 59Ni has decayed by electron capture to
generate the flux of 59Co in cosmic rays indicates that supernovae should
accelerate the products of nucleosynthesis after a period of about 105 years,
the half-life of 59Ni. On the other hand the observation 60Fe nuclei in cosmic
rays indicate that the cosmic rays that are observed at the Earth went through
the acceleration process and then diffused to the Earth within one or two
lifetimes for the decay of 60Fe, about 3.7 Myr. The observations of these
isotopes therefore confines the acceleration process in the interval between
$10^{5}-4\times 10^{6}$ ergs after the supernova explosion ([107, 108, 109,
70]).

In our present discussion we have not included these and other constraints
posed by the observations of composition, anisotropy, etc., and focused on the
implications of the observations of high energy cosmic rays by CALET and AMS.
In particular we have studied the implication of the positron and antiprotons
being the secondary particles generated by the primary nuclear component of
cosmic rays.

(i) In the minimal model we have described here the focus has been on the
ratios of the spectral intensities of secondary cosmic rays to that of their
progenitors - the primary cosmic rays. The reason for this is that absolute
intensities especially over a wide range of energies are very difficult to
measure. This is particularly true for measurements carried out with balloon
borne and satellite borne instruments. Cosmic ray studies have given birth to
the whole field of high-energy astrophysics and although major effort has been
dedicated to their study over the past century, the challenges posed by steep
spectra and declining abundances with increasing atomic number and energy have
prevented precise measurements of the spectral intensities of various
components of cosmic rays. This is true even after the major progress achieved
recently with instruments like AMS, ATIC and PAMELA ([80, 71, 72, 73, 74, 75,
76, 79, 88]). In this context the spectral intensities of charge ratios of
muons measured at sea level provide an insight into the spectrum and
composition of the nucleonic component at high energies, especially for the
analysis of $e^{+}$ and $\bar{p}$ in cosmic rays.

(ii) The close similarity of the spectra of $p,\ \bar{p}\ \mathrm{and}\ e^{+}$
also indicates that their lifetimes are independent of energy and are nearly
the same in the Galaxy, as expected for scattering by fluctuations and
irregularities in the magnetic fields being the controlling factor in their
diffusion. Continued effort towards measurement of their spectra should be a
priority for the future.

(iii) The question proposed by 10Be/9Be is interesting, as the bulk of Be
production in the minimal model takes place in and around the source regions.
As the particles diffuse in the interstellar medium it takes a finite amount
of time for them to reach us even from the nearest of the sources. Radioactive
decay will significantly depopulate the 10Be nuclei with respect to 9Be as
discussed by ([8, 9]). It will be useful to confirm their suggestion with more
recent data.

(iv) Returning to the confinement and diffusion of cosmic rays close to their
sources, we may look at the recent theoretical study of ([106]) and ([16]) who
show that the dynamical effects of cosmic rays in and around the sources
confine them in these regions for sufficient times to interact with ambient
matter and radiation fields. These analyses have confirmed earlier kinematic
analyses and provide theoretical support for understanding gamma-ray sources
and of course cosmic ray propagation.

##  4 Conclusion

The critical review of cosmic ray data implies that cosmic rays spend
significant time in and around the sources generating gamma-rays, and
secondary nuclei like Li, Be and B up to $\sim 300$ GeV/$n$. Some production
of these secondary nuclei, and the bulk of the production of $e^{+}$ and
$\bar{p}$, especially at high energies, occur in the general interstellar
medium. The lifetime of cosmic rays in the Galaxy is nearly constant or weakly
dependent on energy up to $\sim$ PeV energies, and helps us to understand the
constancy and isotropy of the observed cosmic rays.

Thus the preponderance of evidence from observations of $\gamma$-ray astronomy
and the spectra of positrons, antiprotons and secondary nuclei indicates that
there are two regions that contribute to the secondary component in cosmic
rays: (i) The region surrounding the sources dominate at lower energies and
tapers off with increasing energy. (ii) The second region is the general
interstellar medium contributing nearly a constant amount of grammage at about
20% of the maximum at all energies.

##  5 Acknowledgements

We would like to thank our past collaborators B. Burch, M.A. Lee, and T.
Madziwa-Nussinov for their efforts in progressing our understanding of the
material in this paper. We also thank the reviewers for their carfeul reading
of the paper and productive suggestions.

## References

  * [1] A. W. Strong, I. V. Moskalenko, V. S. Ptuskin, Cosmic-ray propagation and interactions in the galaxy, Annual Review of Nuclear and Particle Science 57 (1) (2007) 285–327.  arXiv:https://doi.org/10.1146/annurev.nucl.57.090506.123011, doi:10.1146/annurev.nucl.57.090506.123011.
URL https://doi.org/10.1146/annurev.nucl.57.090506.123011

  * [2] P. Mertsch, S. Sarkar, Testing astrophysical models for the pamela positron excess with cosmic ray nuclei, Phys. Rev. Lett. 103 (2009) 081104.  doi:10.1103/PhysRevLett.103.081104.
URL https://link.aps.org/doi/10.1103/PhysRevLett.103.081104

  * [3] P. Mertsch, S. Sarkar, AMS-02 data confront acceleration of cosmic ray secondaries in nearby sources, Phys. Rev. D 90 (2014) 061301.  doi:10.1103/PhysRevD.90.061301.
URL https://link.aps.org/doi/10.1103/PhysRevD.90.061301

  * [4] P. Blasi, Origin of the positron excess in cosmic rays, Phys. Rev. Lett. 103 (2009) 051104.  doi:10.1103/PhysRevLett.103.051104.
URL https://link.aps.org/doi/10.1103/PhysRevLett.103.051104

  * [5] N. J. Shaviv, E. Nakar, T. Piran, Inhomogeneity in cosmic ray sources as the origin of the electron spectrum and the pamela anomaly, Phys. Rev. Lett. 103 (2009) 111302.  doi:10.1103/PhysRevLett.103.111302.
URL https://link.aps.org/doi/10.1103/PhysRevLett.103.111302

  * [6] R. Cowsik, B. Burch, Analysis of the spectral intensities and ratios of electrons and positrons in cosmic rays (2009).  arXiv:0906.2365.
  * [7] R. Cowsik, B. Burch, Positron fraction in cosmic rays and models of cosmic-ray propagation, Physical Review D 82 (2) (2010) 023009.
  * [8] R. Cowsik, B. Burch, T. Madziwa-Nussinov, The origin of the spectral intensities of cosmic-ray positrons, The Astrophysical Journal 786 (2) (2014) 124.  doi:10.1088/0004-637x/786/2/124.
URL https://doi.org/10.1088/0004-637x/786/2/124

  * [9] R. Cowsik, Positrons and antiprotons in galactic cosmic rays, Annual Review of Nuclear and Particle Science 66 (1) (2016) 297–319.  doi:10.1146/annurev-nucl-102115-044851.
URL https://doi.org/10.1146/annurev-nucl-102115-044851

  * [10] R. Cowsik, T. Madziwa-Nussinov, Spectral intensities of antiprotons and the nested leaky-box model for cosmic rays in the galaxy, The Astrophysical Journal 827 (2) (2016) 119.  doi:10.3847/0004-637x/827/2/119.
URL https://doi.org/10.3847/0004-637x/827/2/119

  * [11] D. Gaggero, L. Maccione, G. Di Bernardo, C. Evoli, D. Grasso, Three-dimensional model of cosmic-ray lepton propagation reproduces data from the alpha magnetic spectrometer on the international space station, Phys. Rev. Lett. 111 (2013) 021102.  doi:10.1103/PhysRevLett.111.021102.
URL https://link.aps.org/doi/10.1103/PhysRevLett.111.021102

  * [12] C. Evoli, R. Aloisio, P. Blasi, Galactic cosmic rays after the AMS-02 observations, Physical Review D 99 (10) (2019) 103023.
  * [13] C. Evoli, P. Blasi, E. Amato, R. Aloisio, Signature of energy losses on the cosmic ray electron spectrum, Physical Review Letters 125 (5) (2020) 051101.
  * [14] C. Evoli, G. Morlino, P. Blasi, R. Aloisio, AMS-02 beryllium data and its implication for cosmic ray transport, Physical Review D 101 (2) (2020) 023013\.
  * [15] J. Becker Tjus, L. Merten, Closing in on the origin of galactic cosmic rays using multimessenger information, Physics Reports 872 (2020) 1–98.
URL https://www.sciencedirect.com/science/article/pii/S0370157320301927

  * [16] B. Schroer, O. Pezzi, D. Caprioli, C. Haggerty, P. Blasi, Dynamical effects of cosmic rays on the medium surrounding their sources, The Astrophysical Journal Letters 914 (1) (2021) L13.  doi:10.3847/2041-8213/ac02cd.
URL https://doi.org/10.3847/2041-8213/ac02cd

  * [17] C. E. Fichtel, R. C. Hartman, D. A. Kniffen, D. J. Thompson, G. F. Bignami, H. Ögelman, M. E. Özel, T. Tümer, High-energy gamma-ray results from the second Small Astronomy Satellite., The Astrophysical Journal 198 (1975) 163–182.  doi:10.1086/153590.
  * [18] W. Hermsen, K. Bennett, G. F. Bignami, G. Boella, R. Buccheri, J. C. Higdon, G. Kanbach, G. G. Lichti, J. L. Masnou, H. A. Mayer-Hasselwander, J. A. Paul, L. Scarsi, B. N. Swanenburg, B. G. Taylor, R. D. Wills, COS-B Observations of Localised Gamma-Ray Emission, in: Proceedings of 15th International Cosmic Ray Conference, Vol. 11, 1977, pp. 14–19.
  * [19] D. J. Thompson, Gamma ray astrophysics: the egret results, Reports on Progress in Physics 71 (11) (2008) 116901.  doi:10.1088/0034-4885/71/11/116901.
URL http://dx.doi.org/10.1088/0034-4885/71/11/116901

  * [20] P. F. Michelson, W. B. Atwood, S. Ritz, Fermi gamma-ray space telescope: high-energy results from the first year, Reports on Progress in Physics 73 (7) (2010) 074901.  doi:10.1088/0034-4885/73/7/074901.
URL http://dx.doi.org/10.1088/0034-4885/73/7/074901

  * [21] A. Abramowski, F. Aharonian, F. Ait Benkhali, et al., Diffuse galactic gamma-ray emission with H.E.S.S., Phys. Rev. D 90 (2014) 122007.  doi:10.1103/PhysRevD.90.122007.
URL https://link.aps.org/doi/10.1103/PhysRevD.90.122007

  * [22] Y. Hanabata, H. Katagiri, J. W. Hewitt, J. Ballet, Y. Fukazawa, Y. Fukui, T. Hayakawa, M. Lemoine-Goumard, G. Pedaletti, A. Strong, et al., Detailed investigation of the gamma-ray emission in the vicinity of SNR W28 with FERMI-LAT, The Astrophysical Journal 786 (2) (2014) 145.
  * [23] A. Abeysekara, A. Albert, R. Alfaro, C. Alvarez, J. Álvarez, R. Arceo, J. Arteaga-Velázquez, D. A. Rojas, H. A. Solares, A. Barber, et al., Extended gamma-ray sources around pulsars constrain the origin of the positron flux at earth, Science 358 (6365) (2017) 911–914.
  * [24] J. Holder, Latest results from veritas: Gamma 2016, AIP Conference Proceedings 1792 (1) (2017) 020013\.  arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.4968898, doi:10.1063/1.4968898.
URL https://aip.scitation.org/doi/abs/10.1063/1.4968898

  * [25] F. Aharonian, R. Yang, E. de Oña Wilhelmi, Massive stars as major factories of galactic cosmic rays, Nature astronomy 3 (6) (2019) 561–567.
  * [26] L. Tibaldo, D. Gaggero, P. Martin, Gamma rays as probes of cosmic-ray propagation and interactions in galaxies, Universe 7 (5) (2021) 141.
  * [27] P. Morrison, Solar origin of cosmic-ray time variations, Phys. Rev. 101 (1956) 1397–1404.  doi:10.1103/PhysRev.101.1397.
URL https://link.aps.org/doi/10.1103/PhysRev.101.1397

  * [28] B. B. Nath, D. Eichler, Diffuse galactic gamma-rays from star clusters, Monthly Notices of the Royal Astronomical Society: Letters 499 (1) (2020) L1–L5.  doi:10.1093/mnrasl/slaa151.
URL http://dx.doi.org/10.1093/mnrasl/slaa151

  * [29] T. Delahaye, A. Fiasson, M. Pohl, P. Salati, The GeV-TeV galactic gamma-ray diffuse emission, Astronomy & Astrophysics 531 (2011) A37.  doi:10.1051/0004-6361/201116647.
URL http://dx.doi.org/10.1051/0004-6361/201116647

  * [30] M. Ackermann, M. Ajello, A. Allafort, L. Baldini, J. Ballet, G. Barbiellini, M. G. Baring, D. Bastieri, K. Bechtol, R. Bellazzini, et al., Detection of the characteristic pion-decay signature in supernova remnants, Science 339 (6121) (2013) 807–811.  doi:10.1126/science.1231160.
URL http://dx.doi.org/10.1126/science.1231160

  * [31] M. Ackermann, M. Ajello, A. Albert, W. B. Atwood, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, R. Bellazzini, et al., The spectrum of isotropic diffuse gamma-ray emission between 100 MeV and 820 GeV, The Astrophysical Journal 799 (1) (2015) 86.  doi:10.1088/0004-637x/799/1/86.
URL http://dx.doi.org/10.1088/0004-637X/799/1/86

  * [32] O. Kalashev, S. Troitsky, Fluxes of diffuse gamma rays and neutrinos from cosmic-ray interactions with the circumgalactic gas, Phys. Rev. D 94 (2016) 063013.  doi:10.1103/PhysRevD.94.063013.
URL https://link.aps.org/doi/10.1103/PhysRevD.94.063013

  * [33] S. Biswas, N. Gupta, Galactic gamma ray background from interactions of cosmic rays (2020).  arXiv:1907.03102.
  * [34] H. J. Bhabha, On the penetrating component of cosmic radiation, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 164 (917) (1938) 257–294.
URL http://www.jstor.org/stable/97091

  * [35] V. Hess, Über die absorption der $\gamma$-strahlen in der atmosphäre (on the absorption of gamma radiation in the atmosphere), Phys. Zeit. 12 (1911) 998–1001.
  * [36] V. Hess, Über beobachtungen der durchdringenden strahlung bei sieben freiballonfahrten (on observations of the penetrating radiation during seven free balloon flights), Phys. Zeit. 13 (1912) 1084–1091.
  * [37] V. Hess, Über den ursprung der durchdringenden strahlung (on the origin of penetrating radiation), Phys. Zeit. 14 (1913) 610–617.
  * [38] A. H. Compton, H. A. Bethe, Composition of cosmic rays, Nature 134 (3393) (1934) 734–735.  doi:10.1038/134734b0.
URL https://doi.org/10.1038/134734b0

  * [39] O. Allkofer, K. Carstensen, W. Dau, The absolute cosmic ray muon spectrum at sea level, Physics Letters B 36 (4) (1971) 425–427.
  * [40] C. Ayre, J. Baxendale, C. Hume, B. Nandi, M. Thompson, M. Whalley, Precise measurement of the vertical muon spectrum in the range 20-500 GeV/c, Journal of Physics G: Nuclear Physics 1 (5) (1975) 584–600.
  * [41] B. Rastin, An accurate measurement of the sea-level muon spectrum within the range 4 to 3000 GeV/c, Journal of Physics G: Nuclear Physics 10 (11) (1984) 1609–1628.
  * [42] M. Crouch, An improved world survey expression for cosmic ray vertical intensity vs. depth in standard rock, in: Proceedings of 20th International Cosmic Ray Conference, Vol. 6, 1987, pp. 165–168.
  * [43] Y. M. Andreyev, V. Gurentsov, I. Kogai, Muon intensity from the Baksan underground scintillation telescope, in: Proceedings of 20th International Cosmic Ray Conference, Vol. 6, 1987, pp. 200–203.
  * [44] C. Berger, M. Fröhlich, H. Mönch, R. Nisius, F. Raupach, P. Schleper, Y. Benadjal, D. Blum, C. Bourdarious, B. Dudelzak, et al., Experimental study of muon bundles observed in the Fréjus detector, Physical Review D 40 (7) (1989) 2163–2171.
  * [45] M. De Pascale, A. Morselli, P. Picozza, R. Golden, C. Grimani, B. Kimbell, S. Stephens, S. Stochaj, W. Webber, G. Basini, et al., Absolute spectrum and charge ratio of cosmic ray muons in the energy region from 0.2 GeV to 100 GeV at 600 m above sea level, Journal of Geophysical Research: Space Physics 98 (A3) (1993) 3501–3507.
  * [46] M. Aglietta, B. Alpat, E. Alyea, P. Antonioli, G. Badino, Y. Ban, G. Bari, M. Basile, A. Benelli, V. Berezinsky, et al., Neutrino-induced and atmospheric single-muon fluxes measured over five decades of intensity by lvd at gran sasso laboratory, Astroparticle Physics 3 (4) (1995) 311–320.
  * [47] M. Ambrosio, R. Antolini, G. Auriemma, R. Baker, A. Baldini, G. Barbarino, B. Barish, G. Battistoni, R. Bellotti, C. Bemporad, et al., Vertical muon intensity measured with macro at the gran sasso laboratory, Physical Review D 52 (7) (1995) 3793.
  * [48] R. Bellotti, F. Cafagna, M. Circella, G. De Cataldo, C. N. De Marzo, N. Giglietto, P. Spinelli, R. L. Golden, S. A. Stephens, S. J. Stochaj, W. R. Webber, M. P. De Pascale, A. Morselli, P. Picozza, J. F. Ormes, R. E. Streitmatter, F. Massimo Brancaccio, P. Papini, P. Piccardi, P. Spillantini, G. Basini, F. Bongiorno, M. Ricci, M. T. Brunetti, A. Codino, C. Grimani, M. Menichelli, I. Salvatori, Measurement of the negative muon spectrum between 0.3 and 40 GeV/c in the atmosphere, Phys. Rev. D 53 (1996) 35–43.  doi:10.1103/PhysRevD.53.35.
URL https://link.aps.org/doi/10.1103/PhysRevD.53.35

  * [49] J. Kremer, M. Boezio, M. e. Ambriola, G. Barbiellini, S. Bartalucci, R. Bellotti, D. Bergström, U. Bravar, F. Cafagna, P. Carlson, et al., Measurements of ground-level muons at two geomagnetic locations, Physical Review Letters 83 (21) (1999) 4241–4244.
  * [50] R. Bellotti, F. Cafagna, M. Circella, C. N. De Marzo, R. L. Golden, S. J. Stochaj, M. P. De Pascale, A. Morselli, P. Picozza, S. A. Stephens, M. Hof, W. Menn, M. Simon, J. W. Mitchell, J. F. Ormes, R. E. Streitmatter, N. Finetti, C. Grimani, P. Papini, S. Piccardi, P. Spillantini, G. Basini, M. Ricci, Balloon measurements of cosmic ray muon spectra in the atmosphere along with those of primary protons and helium nuclei over midlatitude, Phys. Rev. D 60 (1999) 052002\.  doi:10.1103/PhysRevD.60.052002.
URL https://link.aps.org/doi/10.1103/PhysRevD.60.052002

  * [51] M. Boezio, P. Carlson, T. Francke, N. Weber, M. Suffert, M. Hof, W. Menn, M. Simon, S. A. Stephens, R. Bellotti, F. Cafagna, M. Circella, C. De Marzo, N. Finetti, P. Papini, S. Piccardi, P. Spillantini, M. Ricci, M. Casolino, M. P. De Pascale, A. Morselli, P. Picozza, R. Sparvoli, G. Barbiellini, P. Schiavon, A. Vacchi, N. Zampa, C. Grimani, J. W. Mitchell, J. F. Ormes, R. E. Streitmatter, U. Bravar, R. L. Golden, S. J. Stochaj, Measurement of the flux of atmospheric muons with the CAPRICE94 apparatus, Phys. Rev. D 62 (2000) 032007.  doi:10.1103/PhysRevD.62.032007.
URL https://link.aps.org/doi/10.1103/PhysRevD.62.032007

  * [52] M. a. Ambrosio, R. Antolini, G. Auriemma, D. Bakari, A. Baldini, G. Barbarino, B. Barish, G. Battistoni, R. Bellotti, C. Bemporad, et al., Low energy atmospheric muon neutrinos in macro, Physics Letters B 478 (1-3) (2000) 5–13.
  * [53] C. Waltham, S. Collaboration, et al., Through-going muons in the sudbury neutrino observatory, in: Proceedings of 27th International Cosmic Ray Conference, Vol. 3, 2001, pp. 991–994.
  * [54] M. Boezio, V. Bonvicini, P. Schiavon, A. Vacchi, N. Zampa, D. Bergström, P. Carlson, T. Francke, P. Hansen, E. Mocchiutti, M. Suffert, M. Hof, J. Kremer, W. Menn, M. Simon, M. Ambriola, R. Bellotti, F. Cafagna, F. Ciacio, M. Circella, C. N. De Marzo, P. Papini, S. Piccardi, P. Spillantini, E. Vannuccini, S. Bartalucci, M. Ricci, M. Casolino, M. P. De Pascale, A. Morselli, P. Picozza, R. Sparvoli, J. W. Mitchell, J. F. Ormes, S. A. Stephens, R. E. Streitmatter, U. Bravar, S. J. Stochaj, Energy spectra of atmospheric muons measured with the caprice98 balloon experiment, Phys. Rev. D 67 (2003) 072003.  doi:10.1103/PhysRevD.67.072003.
URL https://link.aps.org/doi/10.1103/PhysRevD.67.072003

  * [55] S. Haino, T. Sanuki, K. Abe, K. Anraku, Y. Asaoka, H. Fuke, M. Imori, A. Itasaki, T. Maeno, Y. Makida, et al., Measurements of primary and atmospheric cosmic-ray spectra with the BESS-TeV spectrometer, Physics Letters B 594 (1-2) (2004) 35–46.
  * [56] P. Achard, O. Adriani, M. Aguilar-Benitez, M. Van Den Akker, J. Alcaraz, G. Alemanni, J. Allaby, A. Aloisio, M. Alviggi, H. Anderhub, et al., Measurement of the atmospheric muon spectrum from 20 to 3000 GeV, Physics Letters B 598 (1-2) (2004) 15–32.
  * [57] R. Cowsik, Y. Pal, T. N. Rengarajan, S. N. Tandon, Flux of atmospheric neutrinos of different types at sea level and cosmic ray neutrino experiments, in: Proceedings of 8th International Cosmic Ray Conference, Vol. 6, 1963, p. 211.
  * [58] R. Cowsik, Y. Pal, S. N. Tandon, R. P. Verma, 3°K blackbody radiation and leakage lifetime of cosmic-ray electrons, Phys. Rev. Lett. 17 (1966) 1298–1300.  doi:10.1103/PhysRevLett.17.1298.
URL https://link.aps.org/doi/10.1103/PhysRevLett.17.1298

  * [59] T. K. Gaisser, Primary spectrum and composition with icecube/icetop, Nuclear and Particle Physics Proceedings 279-281 (2016) 47–55.
URL https://www.sciencedirect.com/science/article/pii/S2405601416301894

  * [60] P. Zyla, R. Barnett, J. Beringer, et al., Review of Particle Physics, PTEP 2020 (8) (2020) 083C01.  doi:10.1093/ptep/ptaa104.
  * [61] R. Cowsik, Y. Pal, S. N. Tandon, Evaluation of high energy natural neutrino experiments, Proceedings of the Indian Academy of Sciences - Section A 63 (4) (1966) 217–243.  doi:10.1007/BF03049032.
URL https://doi.org/10.1007/BF03049032

  * [62] R. Cowsik, B. V. Sreekantan, A bound on violations of Lorentz invariance, Physics Letters B 449 (3) (1999) 219–222.
URL https://www.sciencedirect.com/science/article/pii/S0370269399000167

  * [63] R. Cowsik, T. Madziwa-Nussinov, S. Nussinov, U. Sarkar, Testing violations of lorentz invariance with cosmic rays, Phys. Rev. D 86 (2012) 045024.  doi:10.1103/PhysRevD.86.045024.
URL https://link.aps.org/doi/10.1103/PhysRevD.86.045024

  * [64] P. B. Dev, D. Kazanas, R. N. Mohapatra, V. Teplitz, Y. Zhang, Heavy right-handed neutrino dark matter and PeV neutrinos at icecube, Journal of Cosmology and Astroparticle Physics 2016 (08) (2016) 034.
  * [65] T. Kajita, Atmospheric neutrinos and discovery of neutrino oscillations, Proceedings of the Japan Academy. Series B, Physical and biological sciences 86 (4) (2010) 303–321, 20431258[pmid].  doi:10.2183/pjab.86.303.
URL https://pubmed.ncbi.nlm.nih.gov/20431258

  * [66] T. K. Gaisser, Atmospheric neutrinos (2019).  arXiv:1910.08851.
  * [67] N. Rius, Atmospheric neutrinos and new physics (2017).  arXiv:1705.09140.
  * [68] Y. Sui, P. B. Dev, A combined astrophysical and dark matter interpretation of the icecube hese and throughgoing muon events, Journal of Cosmology and Astroparticle Physics 2018 (07) (2018) 020.
  * [69] R. Wieler, J. Beer, I. Leya, The Galactic cosmic ray intensity over the past $10^{6}--10^{9}$ years as recorded by cosmogenic nuclides in meteorites and terrestrial samples, Space Science Reviews 176 (1) (2013) 351–363.  doi:10.1007/s11214-011-9769-9.
URL https://doi.org/10.1007/s11214-011-9769-9

  * [70] W. Binns, M. Israel, E. Christian, A. Cummings, G. A. de Nolfo, K. Lave, R. Leske, R. Mewaldt, E. Stone, T. T. von Rosenvinge, M. Wiedenbeck, Observation of the 60Fe nucleosynthesis-clock isotope in galactic cosmic rays, Science 352 (6286) (2016) 677–680.  doi:10.1126/science.aad6004.
URL https://doi.org/10.1126/science.aad6004

  * [71] H. S. Ahn, E. S. Seo, J. H. Adams, G. Bashindzhagyan, K. E. Batkov, J. Chang, M. Christl, A. R. Fazely, O. Ganel, R. M. Gunasingha, T. G. Guzik, J. Isbert, K. C. Kim, E. Kouznetsov, M. Panasyuk, A. Panov, W. K. H. Schmidt, R. Sina, N. V. Sokolskaya, J. Z. Wang, J. P. Wefel, J. Wu, V. Zatsepin, The energy spectra of protons and helium measured with the ATIC experiment, Advances in Space Research 37 (10) (2006) 1950–1954.
URL https://www.sciencedirect.com/science/article/pii/S0273117705011233

  * [72] A. D. Panov, J. H. Adams, H. S. Ahn, G. L. Bashindzhagyan, K. E. Batkov, J. Chang, M. Christl, A. R. Fazely, O. Ganel, R. M. Gunasingha, T. G. Guzik, J. Isbert, K. C. Kim, E. N. Kouznetsov, M. I. Panasyuk, W. K. H. Schmidt, E. S. Seo, N. V. Sokolskaya, J. P. Wefel, J. Wu, V. I. Zatsepin, The energy spectra of heavy nuclei measured by the ATIC experiment, Advances in Space Research 37 (10) (2006) 1944–1949.
URL https://www.sciencedirect.com/science/article/pii/S027311770500952X

  * [73] A. D. Panov, J. H. Adams, H. S. Ahn, G. L. Bashinzhagyan, J. W. Watts, J. P. Wefel, J. Wu, O. Ganel, T. G. Guzik, V. I. Zatsepin, et al., Energy spectra of abundant nuclei of primary cosmic rays from the data of ATIC-2 experiment: Final results, Bulletin of the Russian Academy of Sciences: Physics 73 (5) (2009) 564–567.  doi:10.3103/s1062873809050098.
URL http://dx.doi.org/10.3103/S1062873809050098

  * [74] J. Chang, J. H. Adams, H. S. Ahn, G. L. Bashindzhagyan, M. Christl, O. Ganel, T. G. Guzik, J. Isbert, K. C. Kim, E. N. Kuznetsov, M. I. Panasyuk, A. D. Panov, W. K. H. Schmidt, E. S. Seo, N. V. Sokolskaya, J. W. Watts, J. P. Wefel, J. Wu, V. I. Zatsepin, An excess of cosmic ray electrons at energies of 300–800 GeV, Nature 456 (7220) (2008) 362–365.  doi:10.1038/nature07477.
URL https://doi.org/10.1038/nature07477

  * [75] O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, M. Bongi, V. Bonvicini, S. Bottai, A. Bruno, et al., Measurement of boron and carbon fluxes in cosmic rays with the pamela experiment, The Astrophysical Journal 791 (2) (2014) 93.  doi:10.1088/0004-637x/791/2/93.
URL http://dx.doi.org/10.1088/0004-637X/791/2/93

  * [76] O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, et al., Ten years of PAMELA in space, Riv. Nuovo Cim. 40 (10) (2017) 473–522.  arXiv:1801.10310, doi:10.1393/ncr/i2017-10140-x.
  * [77] K. Abe, H. Fuke, S. Haino, et al., Measurements of cosmic-ray proton and helium spectra from the BESS-polar long-duration balloon flights over antarctica, The Astrophysical Journal 822 (2) (2016) 65.  doi:10.3847/0004-637x/822/2/65.
URL https://doi.org/10.3847/0004-637x/822/2/65

  * [78] R. P. Murphy, M. Sasaki, W. R. Binns, T. J. Brandt, T. Hams, M. H. Israel, A. W. Labrador, J. T. Link, R. A. Mewaldt, J. W. Mitchell, B. F. Rauch, K. Sakai, E. C. Stone, C. J. Waddington, N. E. Walsh, J. E. Ward, M. E. Wiedenbeck, Galactic cosmic ray origins and OB associations: Evidence from superTIGER observations of elements26fe through40zr, The Astrophysical Journal 831 (2) (2016) 148.  doi:10.3847/0004-637x/831/2/148.
URL https://doi.org/10.3847/0004-637x/831/2/148

  * [79] P. Maestro, O. Adriani, Y. Akaike, Y. Asaoka, E. Berti, G. Bigongiari, M. Bongi, P. Brogi, A. Bruno, N. Cannady, C. Checchia, M. L. Cherry, G. Collazuol, T. G. Guzik, M. Ichimura, M. H. Israel, K. Kasahara, Y. Kawakubo, J. F. Krizmanic, P. S. Marrocchesi, A. M. Messineo, J. W. Mitchell, S. Miyake, M. Mori, N. Mori, H. M. Motz, K. Munakata, L. Pacini, F. Palma, P. Papini, B. F. Rauch, S. B. Ricciarini, T. Sakamoto, M. Sasaki, F. Stolzi, A. Sulaj, T. Tamura, S. Torii, J. P. Wefel, K. Yoshida, C. A. L. E. T. Collaboration, CALET results after three years on orbit on the international space station, Physics of Atomic Nuclei 82 (6) (2019) 766–772.  doi:10.1134/S1063778819660384.
URL https://doi.org/10.1134/S1063778819660384

  * [80] M. Aguilar, L. Ali Cavasonza, G. Ambrosi, et al., The Alpha Magnetic Spectrometer (AMS) on the international space station: Part ii — results from the first seven years, Physics Reports 894 (2021) 1–116.
URL https://www.sciencedirect.com/science/article/pii/S0370157320303434

  * [81] R. Cowsik, L. W. Wilson, Is the residence time of cosmic rays in the galaxy energy-dependent?, in: Proceedings of 13th International Cosmic Ray Conference, Vol. 1, 1973, pp. 500–505.
  * [82] R. Cowsik, L. W. Wilson, The nested leaky-box model for galactic cosmic rays, in: Proceedings of 14th International Cosmic Ray Conference, Vol. 2, 1975, pp. 659–664.
  * [83] T. K. Gaisser, T. Stanev, S. Tilav, Cosmic ray energy spectrum from measurements of air showers, Frontiers of Physics 8 (6) (2013) 748–758.
  * [84] B. B. Nath, N. Gupta, P. L. Biermann, Spectrum and ionization rate of low-energy galactic cosmic rays, Monthly Notices of the Royal Astronomical Society: Letters 425 (1) (2012) L86–L90.
  * [85] R. Cowsik, Y. Pal, S. N. Tandon, R. P. Verma, 3°k blackbody radiation and leakage lifetime of cosmic-ray electrons, Phys. Rev. Lett. 17 (1966) 1298–1300.  doi:10.1103/PhysRevLett.17.1298.
URL https://link.aps.org/doi/10.1103/PhysRevLett.17.1298

  * [86] R. Cowsik, M. A. Lee, On the sources of cosmic ray electrons, The Astrophysical Journal 228 (1979) 297–301.
  * [87] J. Nishimura, T. Kobayashi, Y. Komori, K. Yoshida, Observations of high energy primary electrons and their astrophysical significance, Advances in Space Research 19 (5) (1997) 767–770.
URL https://www.sciencedirect.com/science/article/pii/S0273117796001445

  * [88] P. S. Marrocchesi, New Results from the first 5 years of CALET observations on the International Space Station, in: Proceedings of 37th International Cosmic Ray Conference, no. 395, 2021, p. 010.  doi:10.22323/1.395.0010.
  * [89] E. G. Berezhko, H. J. Völk, Theory of cosmic ray production in the supernova remnant rx j1713.7-3946, A&A 451 (3) (2006) 981–990.  doi:10.1051/0004-6361:20054595.
URL https://doi.org/10.1051/0004-6361:20054595

  * [90] R. Brose, M. Pohl, I. Sushch, Morphology of supernova remnants and their halos, A&A 654 (2021).  doi:10.1051/0004-6361/202141194.
URL https://doi.org/10.1051/0004-6361/202141194

  * [91] P. Reichherzer, L. Merten, J. Dörner, J. Becker Tjus, M. Pueschel, E. Zweibel, Regimes of cosmic-ray diffusion in galactic turbulence, SN Applied Sciences 4 (1) (2022) 1–14.
  * [92] E. Parker, The dynamical state of the interstellar gas and field, The Astrophysical Journal 145 (1966) 811–833.
  * [93] E. Parker, The dynamical state of the interstellar gas and field. iii. turbulence and enhanced diffusion, The Astrophysical Journal 149 (1967) 535–552.
  * [94] E. Parker, Galactic effects of the cosmic-ray gas, Space Science Reviews 9 (5) (1969) 651–712.
  * [95] E. N. Parker, Cosmical magnetic fields: Their origin and their activity, Oxford Classic Texts in the Ph, 2019.
  * [96] L. Accardo, M. Aguilar, D. Aisa, B. Alpat, et al., High statistics measurement of the positron fraction in primary cosmic rays of 0.5–500 GeV with the alpha magnetic spectrometer on the international space station, Phys. Rev. Lett. 113 (2014) 121101.  doi:10.1103/PhysRevLett.113.121101.
URL https://link.aps.org/doi/10.1103/PhysRevLett.113.121101

  * [97] L. Bergström, T. Bringmann, I. Cholis, D. Hooper, C. Weniger, New limits on dark matter annihilation from alpha magnetic spectrometer cosmic ray positron data, Phys. Rev. Lett. 111 (2013) 171101.  doi:10.1103/PhysRevLett.111.171101.
URL https://link.aps.org/doi/10.1103/PhysRevLett.111.171101

  * [98] T. Linden, S. Profumo, Probing the pulsar origin of the anomalous positron fraction with AMS-02 and atmospheric cherenkov telescopes, The Astrophysical Journal 772 (1) (2013) 18.
  * [99] L. Feng, R.-Z. Yang, H.-N. He, T.-K. Dong, Y.-Z. Fan, J. Chang, AMS-02 positron excess: new bounds on dark matter models and hint for primary electron spectrum hardening, Physics Letters B 728 (2014) 250–255.
  * [100] I. Cholis, T. Karwal, M. Kamionkowski, Studying the milky way pulsar population with cosmic-ray leptons, Physical Review D 98 (6) (2018) 063008.
  * [101] C. J. Cesarsky, Cosmic-ray confinement in the galaxy, Annual Review of Astronomy and Astrophysics 18 (1) (1980) 289–319.  arXiv:https://doi.org/10.1146/annurev.aa.18.090180.001445, doi:10.1146/annurev.aa.18.090180.001445.
URL https://doi.org/10.1146/annurev.aa.18.090180.001445

  * [102] J. W. Armstrong, B. J. Rickett, S. R. Spangler, Electron Density Power Spectrum in the Local Interstellar Medium, The Astrophysical Journal 443 (1995) 209–221.  doi:10.1086/175515.
  * [103] M. Iacobelli, M. Haverkorn, E. Orrú, et al., Studying galactic interstellar turbulence through fluctuations in synchrotron emission, A&A 558 (Oct 2013).  doi:10.1051/0004-6361/201322013.
URL https://doi.org/10.1051/0004-6361/201322013

  * [104] M. L. Goldstein, D. A. Roberts, W. H. Matthaeus, Magnetohydrodynamic turbulence in the solar wind, Annual Review of Astronomy and Astrophysics 33 (1) (1995) 283–325.  arXiv:https://doi.org/10.1146/annurev.aa.33.090195.001435, doi:10.1146/annurev.aa.33.090195.001435.
URL https://doi.org/10.1146/annurev.aa.33.090195.001435

  * [105] L. Gallana, F. Fraternale, M. Iovieno, et al., Voyager 2 solar plasma and magnetic field spectral analysis for intermediate data sparsity, Journal of Geophysical Research: Space Physics 121 (5) (2016) 3905–3919.  doi:https://doi.org/10.1002/2015JA021830.
URL https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1002/2015JA021830

  * [106] C. Evoli, E. Amato, P. Blasi, R. Aloisio, Galactic factories of cosmic-ray electrons and positrons, Phys. Rev. D 103 (2021) 083010\.  doi:10.1103/PhysRevD.103.083010.
URL https://link.aps.org/doi/10.1103/PhysRevD.103.083010

  * [107] M. Casse, A. Soutoul, 60fe: An unusual cosmic ray clock?, in: Proceedings of 18th International Cosmic Ray Conference, Vol. 2, 1983, pp. 327–327.
  * [108] M. Wiedenbeck, W. Binns, E. Christian, A. Cummings, B. Dougherty, P. Hink, J. Klarmann, R. Leske, M. Lijowski, R. Mewaldt, et al., Constraints on the time delay between nucleosynthesis and cosmic-ray acceleration from observations of 59ni and 59co, The Astrophysical Journal Letters 523 (1) (1999) L61–L64.
  * [109] W. R. Binns, R. G. Bose, D. L. Braun, T. J. Brandt, W. M. Daniels, P. F. Dowkontt, S. P. Fitzsimmons, D. J. Hahne, T. Hams, M. H. Israel, J. Klemic, A. W. Labrador, J. T. Link, R. A. Mewaldt, J. W. Mitchell, P. Moore, R. P. Murphy, M. A. Olevitch, B. F. Rauch, K. Sakai, F. San Sebastian, M. Sasaki, G. E. Simburger, E. C. Stone, C. J. Waddington, J. E. Ward, M. E. Wiedenbeck, The superTIGER instrument: Measurement of elemental abundances of ultra-heavy galactic cosmic rays, The Astrophysical Journal 788 (1) (2014) 18.  doi:10.1088/0004-637x/788/1/18.
URL https://doi.org/10.1088/0004-637x/788/1/18