Igor V. Moskalenko

# Direct measurements of cosmic rays
and their possible interpretations

###### Abstract

The last two decades have brought spectacular advances in astrophysics of
cosmic rays (CRs) and space- and ground-based astronomy. Launches of missions
that employ forefront detector technologies enabled measurements with large
effective areas, wide fields of view, and precision that we recently could not
even dream of. Meanwhile, interpretation of the individual slices of
information about the internal working of the Milky Way provided by such
experiments poses challenges to the traditional astrophysical models. New
mysteries arise in the composition and spectra of CR species at low and high
energies, in the energy range where we thought the main features were already
understood fairly well. This accumulation of unsolved puzzles highlights the
peculiarity of the current epoch and means that major breakthroughs are still
ahead. In my talk, I review the current state of direct measurements of CRs
and discuss their possible interpretations. Unfortunately, many important
ideas and publications are not discussed here due to the space limitations.

##  1 Introduction

It is appropriate to start with the definition of the term “direct
measurements.” The latter implies direct contact between CR particles and an
instrument, which can only be done at the top of the atmosphere or in space.
Apparently, the design, weight, and exposure of a scientific payload depend on
the current technology and change over time, and so does the definition of the
“direct measurements.” A plot of the all-particle CR spectrum, originally made
by Simon Swordy in 2001 [1], and its well-known variations placed direct
measurements below $\sim$1 TeV, reflecting the technology of the late 20th
century, although there are several exceptions, such as, e.g., Sokol
spacecraft [2, 3] and JACEE experiment [4]. Now, two decades later, we
routinely make direct precise measurements approaching the energy of the knee
at $\sim$3 PeV, thanks to the talents of experimentalists and recent
technological advances. I enthusiastically predict that in about two decades,
by 2050, direct measurements will advance the next three orders of magnitude
in energy and reach 1-10 EeV range. Today, this can be considered a very
challenging goal, but not impossible. Meanwhile, several breakthrough
experiments have been proposed, e.g., AMS-100 [5], HERD [6], HERO [7], ALADInO
[8], which significantly extend capabilities of the current instrumentation.
For example, AMS-100 is designed to have a geometrical acceptance of $\sim$100
m2 sr.

CR research is currently experiencing a golden age. Thanks to the outstanding
progress in the energy coverage and precision of direct measurements, we are
witnessing a constant stream of discoveries of new features in the spectra of
CR species and anomalous behavior of elemental and isotopic ratios in the
energy range from MV to 100s TV, and even more new experiments are preparing
to launch (e.g., GAPS, HERD, TIGERISS, HERO, HELIX, COSI). A natural
consequence is an _infinite_ number of publications with interpretations of
new observed features. Unfortunately, this means that many important ideas and
publications are not mentioned here due to space limitations.

For most of the 20th century, the two popular models describing the observed
spectra and composition of CRs were the leaky-box and the Galactic diffusion
model with halo [9]. The leaky-box model is the simplest and some people still
use it. This model considers the Galaxy as a volume uniformly filled with gas,
sources, and CRs with a small leakage – hence the name “leaky-box.” Tuned to
local measurements, it can correctly reproduce the fluxes of stable nuclei at
a single point in the Galaxy. The diffusion model is a realistic model in
which gas and sources are distributed in the Galactic disk. CRs fill a large
volume around the disk called the halo, and can escape into the intergalactic
space through its outer “boundaries.” The spatial distributions of CR species,
gas, background radiation, and magnetic field are non-uniform and generally
consistent with observations of diffuse Galactic thermal and non-thermal
emissions. This model is widely used today.

At the turn of the 21st century, the general success of the diffusion model
led Vitaly Ginzburg to the conclusion [10]: “In respect of CR with $E_{\rm
CR}$$<$$10^{15}$$-$$10^{16}$ eV, there generally remain some vague points, but
in the whole the picture is _clear_ enough…” This is reminiscent of the
popular view at the turn of the 20th century (ca. $\sim$1900), formulated by
Lord Kelvin (William Thomson): “There is nothing new to be discovered in
physics now. All that remains is more and more precise measurement…”
Interestingly, increasingly precise measurements is exactly what led us to the
current situation! Today we have such excellent data that the whole picture
becomes rather _unclear_. This creates exciting opportunities for theorists
and experimentalists and promises new breakthroughs!

A word of caution though. A large amount of new material and attempts to
comprehend it often lead to the development of overly complicated models,
which add many parameters in an attempt to achieve consistency with the data.
Such models are featuring two halos with different diffusion coefficients,
anisotropic diffusion, local sources with different injection spectra, sources
in spiral arms, slow diffusion zones, etc., sometimes combined within a single
model. Although this allows models to reproduce some of the observed features,
it does not necessary lead to a better understanding of the underlying
processes. Here we must adhere to time-tested wisdom, such as that formulated
by William of Ockham (known as Occam’s razor): _Numquam ponenda est pluralitas
sine necessitate_ (“Plurality must never be posited without necessity”) or
follow the advise attributed to Albert Einstein: “Everything should be as
simple as it can be but not simpler.”

##  2 Low-energy features

Low-energy measurements provide unique information about isotopic spectra and
composition of CRs. However, such measurements are subject to heliospheric (or
solar) modulation, which was difficult to handle due to the lack of
measurements outside the heliosphere. This problem was largely resolved with
Voyager 1, 2 entering the very local interstellar medium (ISM), V1 in 2012 and
V2 in 2018, and beaming the elemental spectra of CR species from ISM space
[11]. This unimaginable breakthrough provides a solid ground for studies of
low-energy particles.

Low-energy features, often called excesses, are observed in the spectra of
some elements when compared to the spectra of their neighbors, there are also
mismatches between the data taken by different instruments operating in
different energy ranges, even after their correction for solar modulation,
and/or significant deviations from model predictions. Let us start with iron
and its radioactive isotope, 60Fe ($\beta^{-}$ decay, half-life of 2.6 Myr
[12]), which may provide some clues to the origin of low-energy features in
the spectra of other species.

###  2.1 60Fe as a tracer of supernova activity in the solar neighborhood

Evidences of the past SN activity in the local ISM are abundant [13, 14, 15].
There is no general agreement on the exact number of SN events and their exact
timing, but it seems quite clear that several events may have occurred at
distances of up to 100 pc in the last $\sim$10 Myr. The most recent SN events
in the solar vicinity occured 1.5–3.2 Myr and 6.5–8.7 Myr ago [13, 14]. The
measured signal spread of $\sim$1.5 Myr implies a series of SN explosions.
Besides, the Local Bubble is a low-density region about $\sim$200 pc around
the sun filled with hot H ii gas, itself formed by a series of SN explosions
[16, 17]. Studies suggest 14–20 SNe within a moving group, the surviving
members of which are now in the Scorpius-Centaurus stellar association [17,
15].

An excess of radioactive 60Fe found in deep ocean core samples of FeMn crust
[18, 19, 14], in the Pacific Ocean sediments [20], in lunar regolith samples
[21, 22, 23], and in the Antarctic snow [24] indicates that it may have been
deposited by SN explosions in the solar neighborhood. Fifteen 60Fe events and
only one 61Co event were observed by ACE-CRIS [25] in CRs, while there were
about equal numbers of 58Fe and 59Co events. This implies that the 60Fe events
are real and not the spillover of a more abundant neighboring isotope.
Meanwhile, only an upper limit was established for 59Ni ($\tau_{1/2}$$\sim$76
kyr) [26], suggesting $\gtrsim$100 kyr time delay between the ejecta and the
next SN.

The low-energy feature in the _iron spectrum_ [27], perhaps also associated
with SN activity in the solar neighborhood, was revealed for the first time
when comparing data from Voyager 1 [11] and ACE-CRIS with AMS-02 [28]. It is
most clearly visible as a bump in the Fe/He, Fe/O, and Fe/Si ratios at 1–3 GV,
while a similar feature in the He/O and Si/O ratios is absent. The large
fragmentation cross section and fast ionization losses of iron hint at a local
origin of the excess. The calculations [27] use the Monte Carlo code HelMod
[29] based on the Parker equation and developed to describe the CR transport
through the heliosphere from interstellar space to the Earth.

Interestingly, the Ni/Fe ratio reported by CALET [30] (see also a highlight
CALET talk by Shoji Torii [31]) is constant between 10 and 200 GeV/n,
indicating the same origin of elements of the iron group. Precise measurements
of the sub-Fe/Fe = (Sc+Ti+V)/Fe ratio can shed light on the origin of the iron
group and provide further details about CR sources in our local neighborhood.

###  2.2 Aluminum excess

The excess in the Al spectrum in the narrow rigidity range of 3–10 GV
($\sim$0.8–4 GeV/n) becomes clearly visible if we compare the Al/Si ratio
measured by AMS-02 [32] with the model predictions [33], while a similar
feature in the Na/Si ratio is absent. There are four possible physical reasons
for the discrepancy between data and model calculations [33]: (i) incorrect
spectrum of 28Si, the main progenitor of secondary 26,27Al, (ii) errors in the
total Al inelastic cross sections, (iii) errors in the production cross
sections of 26,27Al isotopes, and (iv) additional local component of primary
Al.

Reason (i) can be rejected because all model calculation are based on
available data. The Si spectrum is tuned to data from Voyager 1, ACE-CRIS, and
AMS-02 [34]. Importantly, AMS-02 data is available above 2 GV, the same
rigidity range for all CR species. Contributions of other CR species are very
minor and cannot be the cause of the observed excess. (ii) Significant errors
in the total inelastic cross section of Al can be excluded as the primary
cause of the excess, taking into account the accelerator data. The total
inelastic cross section of 27Al is measured below $\sim$1 GeV/n ($<$3 GV) [35]
and in the rigidity range 10–19 GV in the inverse kinematics [36]. The
parameterizations of the total inelastic cross sections used in the model
calculations are tuned to the available data. One can also notice the absence
of similar excesses in the spectra of neighboring nuclei, such as Ne, Na, Mg,
Si [34, 33]. (iii) Almost 100% of secondary 27Al is produced through
fragmentation of 28Si with minor contributions from 29Si, 32S, and 56Fe [37].
Unfortunately, the isotopic production cross section ($p+^{28}$Si$\to^{27}$Al)
is the major source of uncertainty. Only a couple of data points are available
for this reaction, at 1.4 GV and 2.3 GV in the inverse kinematics, which
constrain 27Al production in the lower range of rigidities where the excess is
observed. Fortunately, measurements of 28Si fragmentation 28Si$+p$$\to$Al
performed by a Siegen group [38] indicate that the cross section remains flat
between 1 and 14.5 GeV/n corresponding to the rigidity range from 3.5–31 GV
(27Al), which covers the excess range and extends to significantly higher
rigidities. (iv) The observed low-energy excess in the Al spectrum is most
likely due to the stable isotope 27Al. However, because it is stable, its
yield and Galactic distribution are difficult to measure. Meanwhile, an
abundant literature exists on observations of the distribution of the diffuse
$\gamma$-ray 1.809 MeV line emission from the decay of the radioactive 26Al
isotope and its origin.

Observations of the diffuse Galactic 1.809 MeV emission line by COMPTEL [39]
and INTEGRAL [40] have shown that 26Al nucleosynthesis is ongoing in the
present Galaxy. Potential sources include AGB stars, novae, core-collapse SNe,
and Wolf-Rayet stellar winds [41, 42]. There are also reports of the discovery
of a close SNR (G275.5+18.4) with an angular diameter of $24^{\circ}$ in the
constellation Antlia Pneumatica [43]. The SNR distance is estimated as 60–340
pc, but is most likely $\sim$200 pc [44]. A marginally significant feature is
detected in the 1.8 MeV $\gamma$-ray emission line within the Antlia SNR.

###  2.3 Lithium excess

The standard propagation model [45] tuned to the B/C ratio data by Voyager 1
[11], ACE-CRIS [46], and AMS-2 [47], demonstrates good agreement with
measurements of CR species in a wide energy range, implying that Li spectrum
should also be well reproduced by the same model. However, a comparison of the
model calculations of secondary Li with data exhibits a significant excess
over the model predictions above a few GV [45]. This may be an indication of
errors in the production cross sections or an unexpected primary Li component.

From a compilation of the majority of Li production cross sections [48], one
can see that the main production channels are the fragmentations of 12C and
16O measured in several different experiments. Although they are not measured
perfectly, each contributes 12%-14%, and thus a 20% error in one of them would
correspond to only 2%-3% of total Li production. Other production channels
contribute at a level of 1%-2% or less. It is not impossible, but rather
unlikely that the cross section errors are all biased in the same direction,
resulting in the observed 20% excess.

The obvious “solution” is to renormalize the LiBeB production cross sections
to match the CR data and/or reduce the diffusion coefficient to boost Li
production, while ensuring the calculated B/C ratio is still consistent with
the experimental data [49, 50]. The scientific terms for this approach are
scale factors and nuisance parameters. While these may seem cool and produce
the desired results, hastily removing the excess risks throwing the baby out
with the bathwater.

Another cross section hypothesis suggests that the contribution of Fe
fragmentation is calculated incorrectly, and with new updated cross sections,
the Li data can be reproduced well enough [51]. Meanwhile, a back-of-the-
envelope estimate shows that the contribution of Fe to Li production is less
than 5% at 22 GV (at maximum excess).

Indeed, the _local ISM fluxes_ of main progenitors taken at $E_{\rm
kin}$=10.44 GeV/n ($\approx$22 GV) are: $F_{\rm Fe}$$\approx$$4.476\times
10^{-3}$, $F_{\rm Si}$$\approx$$6.723\times 10^{-3}$, $F_{\rm
Mg}$$\approx$$7.904\times 10^{-3}$, $F_{\rm Ne}$$\approx$$6.456\times
10^{-3}$, $F_{\rm O}$$\approx$$4.131\times 10^{-2}$, $F_{\rm
N}$$\approx$$9.089\times 10^{-3}$, $F_{\rm C}$$\approx$$3.979\times 10^{-2}$,
$F_{\rm B}$$\approx$$7.709\times 10^{-3}$ in units of m-2 s-1 sr-1 [GeV/n]-1,
where the fluxes of B–Si are taken from [34], and Fe – from [27]. The
heliospheric transport in [34, 27] is calculated using the HelMod code, but
the solar modulation at $\approx$22 GV is weak anyway. At 10 GeV/n, the cross
sections of the reactions $A+p$$\to$Li $+X$, where $A=$ B, C, N, O, Ne, Mg,
Si, Fe, and Li=6He+6Li+7Li, are about the same in the range of 23–32 mb, see
Figs. 2, 3 in [51]. Assuming they are _all the same_ , we can simply use
fluxes $F_{A}$ shown above. This gives for the iron contribution $F_{\rm
Fe}/\Sigma_{A}F_{A}$$\approx$$0.036$ (3.6%), where
$\Sigma_{A}F_{A}$$\approx$$0.123$. Even if the Fe$\to$Li cross section is 50%
larger (unlikely), the Fe contribution increases to $\approx$$0.053$ (5.3%).
These numbers are calculated without taking into account the contribution of
all other species, so the Fe contribution is even smaller in reality. Note
that the effect of possible cross section errors in reactions with He target
is minor [51]. Although the contribution of Fe increases with rigidity due to
its hard spectrum, this estimate shows that incorrect cross sections cannot be
the main cause of the excess.

An exciting possibility is that the primary Li may come from nova explosions
[45]. Indeed, the $\alpha$-capture reaction of 7Be production
3He$(\alpha,\gamma)^{7}$Be in stars was proposed a while ago [52, 53]. A
subsequent decay of 7Be (half-life of 53.22 days) yields 7Li isotope. 7Be
should be transported into cooler layers where it can decay to 7Li, the so-
called Cameron-Fowler mechanism in AGB stars. The production of 7Li in the
same reactions in novae was discussed in [54, 55, 56]. Observation of blue-
shifted absorption lines of partly ionized 7Be in the spectrum of a classical
nova V339 Del [57] about 40-50 days after the explosion is the first evidence
that the mechanism proposed in 1970s is working indeed [58]. Consequent
observations of several other novae (V1369 Cen, V5668 Sgr, V2944 Oph, V407
Lupi, V838 Her) also reveal the presence of 7Be lines in their spectra.
Meanwhile, there could be other sources of 6,7Li. Low-energy measurements by
PAMELA [59] show that 7Li/6Li$\approx$1.1 below 1 GeV/n. The _latest_
preliminary AMS-02 analysis [60] indicates 7Li/6Li$\approx$1 below 10 GeV/n,
but this still may change when the analysis is completed. Precise measurements
of the isotopic ratio can shed light on the origin of Li in CRs.

##  3 Inventory of Galactic cosmic ray sources

The main sources of Galactic CRs remain SNe and SNR with their total kinetic
energy of the ejecta in the range of $10^{51}$ erg. The Green catalog
currently lists 294 SNR. Meanwhile, the isotopic and spectral anomalies
observed recently force us to look at other sources, especially local, which
can also contribute to the observed CRs. These are primarily Wolf-Rayet stars
(currently 354 are known) and O-stars (20,000 observed), which over their
fairly short lifetimes provide, respectively, $10^{51}$ erg and $10^{50}$ erg
in high-velocity winds reaching $(2-4)\times 10^{3}$ km/s, pulsars
($\sim$1,500 observed) with their total rotating power reaching $4\times
10^{49}$ erg (Crab), and novae providing $10^{45}$ erg (estimated frequency
30–50/year, $\approx$350 observed). For comparison, countless stellar flares
can provide up to $10^{36}$ erg each, and can also add to low-energy CRs.

##  4 Protons/Helium ratio

The monotonic decrease of the H/He ratio was first noticed in the PAMELA data
when plotted vs. rigidity [61] and has been confirmed by other experiments
covering the impressive rigidity range from $\sim$100 MV to 50 TV, such as
Voyager 1 [11], AMS-02 [62] (Fig. 79), and CALET [63], while DAMPE [64]
presents only their latest proton and He spectra, but not their ratio.
Meanwhile, we note that a flatter spectrum of He vs. H was already spotted by
many earlier experiments, such as Sokol [2, 65], JACEE [4], ATIC [66], and
CREAM [67]. However, theory told us that the spectral indices in rigidity
should not depend on the specific properties of the primary particles. Thus,
many researchers then have attributed this difference to systematic effects,
and I myself am no exception.

Apparently, not only He behaves differently from protons. Accurate
measurements by AMS-02 [62] show that other, mainly primary species, such as
C, O, Si, have spectral indices above $\sim$60 GV very similar to He. This
raises a question, whether the spectrum of primary CR species depends on the
$A/Z$ ratio, which is equal to 1 for protons, and 2 for most abundant isotopes
of He, C, O, Si.

###  4.1 Hypothesis of the spatial distribution of elements

The main idea is that SN explodes into the pre-SN wind, which is composed of
lighter elements when the star is young, but becomes increasingly enriched in
heavier elements in its later stages [68, 69]. The young SN shell then
accelerates heavier elements when it is young, and lighter elements when it
fates. This would make the spectra of heavier species flatter as they are
accelerated by a stronger shock, while the spectra of lighter elements are
produced at later stages. SN can also explode into the medium enriched with
heavier elements from previous SN explosions.

An argument against this hypothesis is that the spectra of He, C, and O have
the same index, while the spectra of Ne, Mg, and most importantly Si are
somewhat steeper [62]. This implies that the spatial distribution of C and O
in the pre-SN wind should match the distribution of He, while heavier Ne, Mg,
and Si should be accelerated by a weaker shock at a later time.

###  4.2 Hypothesis of two components in the H spectrum

This is an empirical hypothesis [62], which suggests that the observed CR
proton spectrum is a combination of spectra, which come from two distinctly
different types of proton sources. One of them is a regular source that
accelerates all particles and injects them into the ISM with spectra similar
to He. Another source is enriched with hydrogen (or depleted in heavier
species) and injects protons with a steeper spectrum (by $\approx$0.3 in
index).

Earlier, a similar idea [70, 71] was proposed to reproduce the observed
positron excess. The harder spectrum (younger) sources should be surrounded by
gas producing more secondary species including positrons, and this could
explain their flatter spectrum.

An argument against this hypothesis is that it requires sources that are
unique for protons. It is not clear what kind of sources these are, and what
makes them so unique.

###  4.3 Hypothesis of different acceleration efficiency

The ideas proposed in [72] and [73] are somewhat different, but the authors
came to similar conclusions: most particles in the shock are protons
($A/Z$$=$$1$), which generate Alfvén waves and became frozen into the
generated turbulence. The nuclei with $A/Z$$>$$1$ or $A/Q$$>$$1$ (Q is the
charge of a partly ionized atom) are not in synch with Alfvén waves generated
by protons, and are more efficiently injected into the shock and then
accelerated.

The hypothesis predicts that the injection efficiency of heavier species
increases relatively to protons with increase of the shock Mach number and
$A/Q$ value. The same applies to all species $A/Q$$>$$1$, but the efficiency
should come to saturation for sufficiently high $A/Q$ values.

##  5 Silicon & fluorine puzzles

An increase in accuracy of CR data reveals unexpected puzzles in the O-Si
groups, implying that CR acceleration and transport processes are still far
from being fully understood. Let us look at the Si/O ratio in the local ISM
[27], i.e. take their local interstellar spectra [34], which eliminates the
solar modulation. The most abundant isotopes, 16O and 28Si, are primaries with
$A/Z$$=$$2$. The larger fragmentation cross section and faster ionization
energy losses of Si nuclei result in the rise of the Si/O ratio with rigidity
below $\sim$10 GV. In the absence of energy losses in the middle range
($\sim$10–300 GV), the Si/O ratio is const, while at higher rigidities, the
ratio decreases for no apparent reason. Note that the He/O ratio is flat above
60 GV.

It is also interesting to examine the differences in two secondary/primary
ratios, B/O and F/Si. If we look at the rigidities $>$10 GV, where the solar
modulation is small, the ratios diverge in the entire range up to 1 TV with
the F/Si ratio being flatter by $\approx$0.052 in the index (Fig. 3 in [74]),
albeit with large error bars in the F/Si ratio. This usually implies a
difference in the interstellar propagation, with the indices of the effective
diffusion coefficients probed by these ratios differing by that same number.
Then, a weak monotonic increase can be expected for the F/B ratio. However,
the increase is observed only up to 100–200 GV, while in the range $\sim$100
GV–1 TV the ratio becomes constant (Fig. 1 in [74]), again with large error
bars. The break at $\approx$200 GV, if confirmed, perhaps has the same origin
as the break observed in the spectra of all CR species and discussed in the
next section. Striking is that the increase in the F/B ratio is observed in
the range 10–200 GV, where Si/O$\approx$const. The latter defies the
hypothesis of the difference in propagation, as in this case, the Si/O ratio
cannot be constant, assuming that the injection spectra of O and Si are the
same.

Let us look at the fluorine puzzle from a different perspective. A comparison
of standard propagation calculations tuned to the B/C or B/O ratio with the
measured F/Si ratio shows a deficit in secondary fluorine, which increases
with the rigidity up to the break at 200 GV [75]. However, it becomes
consistent with the AMS-02 data [74] and thus with the B/O ratio above 200 GV,
albeit with large error bars. This is a serious issue, which cannot be cured
by simple renormalization of the cross sections – the latter are flat above
$\sim$1–2 GeV/n. For example, if we assume that the fluorine production cross
sections are off by $\approx$10% and renormalize the calculated fluorine
spectrum down by the same factor, we obtain excesses from $\approx$3–10 GV and
at $\sim$100 GV.

The described rigidity-dependent discrepancies imply different origins of the
Si group and the CNO group, or an as yet unclear difference in their
propagation, or perhaps a non-negligible primary F component. The F anomaly is
also confirmed in other studies, e.g., [76]. Precise measurements of the
15P/16S and sub-Fe/Fe ratios should be able to clarify the issue or add more
puzzles.

##  6 200 GV & 10 TV breaks or the TV bump

The $\sim$200 GV breaks in the spectra of protons and He are clearly visible
in the data collected by ATIC-2 [66], CREAM [67], and even earlier
experiments, but initially looked like a calibration problem between lower and
higher energy ($\lessgtr$200 GeV) experiments. The break becomes widely
accepted after the PAMELA publication [61], and a confirmation by Fermi-LAT
[77], which used observations of the CR-induced $\gamma$-ray emission from the
Earth’s limb. The latter actually confirmed the flatter proton spectrum above
$\sim$200 GeV, in agreement with PAMELA measurements.

The break is most clearly visible in the latest data by AMS-02 [62], the
instrument that is best suited for this energy range. A comparison of the fits
made in the range from 30-50 GV to 200 GV shows an interesting picture, where
Fe has the hardest spectrum followed by He, O, C, and then Si, S, Ne, Mg. The
steeper spectra are observed in H, Al, N, Na, F, B, Be, and the steepest is Li
(partly tertiary). The fluorine spectrum is flatter than the spectrum of
boron, as already discussed above, and may indicate a different origin or
presence of the primary component.

Yet another break, at 10 TV, was also observed by the ATIC [66] and CREAM [78]
teams, but the former made no claim, while the latter stated that more data is
needed. The first decisive evidence was provided by the NUCLEON team [79],
which observed the break in the spectra of protons, He, and light elements at
the same rigidity. This break is now confirmed by several instruments: DAMPE
[80], CALET [81], and ISS-CREAM [82]. A striking increase in anisotropy in the
same energy range from $\sim$0.2–100 TeV [83] indicates that these two breaks
form a single bump structure from $\sim$0.2–100 TV rather than being two
independent features; note that for protons dominating in CRs, their kinetic
energy is approximately equal to the rigidity at high energies.

###  6.1 Anisotropy map

The anisotropy map of the entire sky at 10 TeV, which corresponds to the bump
maximum, was produced using the combined data of HAWC and IceCube experiments
[83]. It includes the large-scale and small-scale anisotropy features, and
provides information that is crucial for understanding the bump origin. The
large-scale map features a very sharp jump in the relative CR intensity across
the magnetic equator – a hint at the proximity of the source. The dominant
spot in the residual small-scale anisotropy map (Region A, Figs. 5, 11 [83])
points to the direction to the source, which coincides with the Galactic
_anti-center_ , the direction of the local B-field, and is about $45^{\circ}$
off the “tail” of the heliosphere. This is in a remarkable contradiction with
the conventional understanding that the phase of dipole anisotropy should
point to the direction of the Galactic center, where the majority of CR
sources are located and the CR number density is the highest.

##  7 Models of the TeV bump

On March 3, 2011, PAMELA reported about a break in the spectra of H and He at
about the same rigidity 230–240 GV [61]. No one knew yet that there was
another break at higher rigidity. Therefore, the early papers proposed
interpretation of this first break. The first paper [84], submitted on August
4, 2011, proposed four different scenarios of the origin of the break:
interstellar propagation, the injection spectrum (or spectra from two
distinctly different source populations), and a local source at low or high
energies. The analysis showed that the propagation scenario in which the break
is associated with a change in the rigidity dependence of the diffusion
coefficient is preferable. These scenarios and their variations are still
discussed in the literature. Physical interpretation for the propagation
scenario [85] was proposed 10 months later (submitted on May 30, 2012): “…the
diffusive propagation is no longer determined by the self-generated
turbulence, but rather by the cascading of externally generated turbulence
(for instance due to supernova bubbles) from large spatial scales to smaller
scales where CRs can resonate."

The propagation scenario naturally explains the same break rigidity for all
species and reproduces the observed difference between the spectra of primary
and secondary species in subsequent AMS-02 publications. Essentially, the
values of the spectral breaks in the spectra of primary (C, O) and secondary
(B) species are connected as $\Delta\delta_{\rm
sec}$$\approx$$2\Delta\delta_{\rm pri}$, where $\Delta\delta$ is the
difference in the spectral power-law indices below and above the break.

Meanwhile, it is difficult to imagine a compelling physical reason for the
repeated change in the diffusion coefficient, now at 10 TV, and so another
explanation is needed.

###  7.1 A local SNR surrounded by gas clouds

There are many models discussing the origin of the observed spectral break at
200 GV [86], but only a few of them are trying to reproduce the entire TV bump
with two breaks at 200 GV and 10 TV. One of the most popular is the model
speculating on the idea of the local SNR (e.g., Geminga SNR at $\sim$300 pc),
surrounded by gas cloud(s) where the secondary species are produced. It is
claimed that this model reproduces all the observed features in the spectra of
CR nuclei, $e^{\pm}$, $\bar{p}$, and dipole anisotropy. Various versions of
this model are discussed in [71, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97,
98, 99] and elsewhere.

In such a model [95], the local SNR accelerates primary species ($e^{-}$ and
nuclei). Secondary species (LiBeB, $\bar{p}$, $e^{\pm}$) are produced by
accelerated nuclei in the gas cloud(s) surrounding the shell. The nuclear
species coming from the SNR have a _convex_ spectral shape, when scaled with
$E_{\rm kin}^{2.6}$, formed by the cutoff of the injection spectrum
$\propto$$R^{-\nu}e^{-R/R_{c}}$, with $R_{c}$$=$15 TV, while the low-energy
decrease is formed due to a time delay in the propagation of particles from
the SNR that have not yet reached us. The SNR age, distance, and the diffusion
coefficient are tuned in such a way that these bumps fit in between 200 GV and
100 TV. To match the observed CR spectra and make room for the SNR component,
the spectra of Galactic CR species must have a _concave_ shape. The latter is
done by adjusting the parameters of the two-halo scenario [100]. The
steepening in the nuclear spectra, which becomes visible at 5 TV, is tuned to
reproduce the observed steepening in the $e^{+}$ spectrum at $\approx$300 GV,
assuming that all observed excess positrons are produced in the proposed
scenario (5 TV/300 GV$\approx$17 is the mean fraction of the kinetic energy of
the primary proton transferred to the secondary pion per collision [101,
102]). Primary electrons lose energy due to the inverse Compton scattering and
synchrotron emission to make the observed break at 1 TV (see Sect. 8). The
suggested source is Geminga SNR with an age 330 kyr and a distance of 330 pc.

There are some issues with the local SNR model. First, it requires fine
tuning. To fit the observed data, it is necessary to make a dip in the spectra
of Galactic species and a bump in the corresponding local SNR components at
the same energy simultaneously. The number of free parameters in this model,
not counting the normalizations of the Galactic CR species, is about 50. These
include 8 transport parameters + 6 spectral parameters + 28 (may be somewhat
less) individual normalizations of SNR components for each species + 7
parameters for primary electrons + 1 gas cloud grammage.

Second, a simple estimate of the diffusion length shows that it cannot
reproduce the observed sharp jump in the relative CR intensity across the
magnetic equator [83]. Gyroradius of 10 TV particles in the interstellar 3
$\mu$G magnetic field is $\sim$0.003 pc. For a source at $\sim$330 pc, this
gives $\sim$$10^{5}$ mean free paths – there is no way to produce the sharp
jump in the anisotropy map at such a distance. Even if the mean free path is
$\sim$1 pc, it is still $\sim$330 mean free paths. In fact, all models of the
TV bump with relatively distant sources have the same problem.

This leads us to the conclusion that the “source” should be much closer to the
solar system, and it should be a different type of source.

###  7.2 Reacceleration bump

The large number of free parameters discussed above can be avoided if, instead
of a source surrounded by gas clouds, we assume that pre-existing CRs are
_reaccelerated_ in a local shock [103, 104, 105]. If the shock is located at a
few particle’s mean free paths from the observer or connected to the observer
by the magnetic field line, it can also explain the observed sharp jump in the
relative CR intensity across the magnetic equator. Such a model requires only
a moderate reacceleration below rigidity $\sim$50 TV, a shock with the Mach
number of $\sim$1.5 should suffice. Reaccelerated particles below $\sim$0.2 TV
are convected with the ISM flow and do not reach us, thus creating the bump.
This single universal process acts on all CR species in the rigidity range
below 100 TV. The position of the sun relative to the shock defines the bump
parameters, which can change over time.

The model does not specify the location of the shock. The passing stars’ bow
shock, shock from the old SNR, or any local shock with a small Mach number
$\sim$1.5 can do the job. The distance-shock-size relation gives an estimate
of the distance: $\zeta_{obs}{\rm(pc)}$$\sim$$100\sqrt{L_{\perp}{\rm(pc)}}$;
for sufficiently large bow shocks, $L_{\perp}$$=$$10^{-3}$–$10^{-2}$ pc, the
distance is $\zeta_{obs}$$=$3–10 pc.

The _only three_ unknown bump-specific fitting parameters can be obtained from
a fit to the proton spectrum, best measured among CR species. Spectra of all
other CR species can be calculated using a simple analytical formula, where
the input parameters are their normalizations and spectral power-law indices
derived from their local interstellar spectra (LIS) below the bump. The
steeper the spectrum of ambient particles, the stronger the effect of
reacceleration, and larger the bump. The LIS for all species H–Ni are provided
in [34], and updated Fe LIS – in [27].

Interestingly that the increased CR intensity feature in the small-scale
residual anisotropy map aligns well with the direction of the local magnetic
field [83] and may indicate the direction to the shock. One of the favorite
candidates is the $\epsilon$ Eri star [105], with its configuration of the bow
and termination shocks, projected just 6.7∘ off of the direction of the local
magnetic field. $\epsilon$ Eri is a solar like star, K2 dwarf (5 000 K) with
the mass 0.82$M_{\odot}$ and radius 0.74$R_{\odot}$. It is located at the
distance of 3.2 pc and has the speed 20 km/s, a bit small, but has an
astonishing mass loss rate: $\dot{m}$$\sim$30–1500$\dot{M}_{\odot}$. It also
has a huge astrosphere – 8000 au, 47′ as seen from Earth (larger than the
angular size of the Moon!).

Other candidate stars are $\epsilon$ Indi, a triplet star K4.5V
(0.77$M_{\odot}$)+T1.5 (0.072$M_{\odot}$)+T6 (0.067$M_{\odot}$) at a distance
of 3.6 pc and moving with the speed of 40.4 km/s (radial), or Scholz’s Star, a
duplet M9.5 (0.095$M_{\odot}$)+T5.5 (0.063$M_{\odot}$) at a distance of 6.8
pc, moving with the speed of 82.4 km/s (radial). There could be other types of
shocks in the solar neighborhood, see discussion in [103, 105].

##  8 Cosmic ray electrons

Electrons in CRs are subject to severe energy losses at all energies. The
fastest losses are at low and high energies due to the ionization and the
inverse Compton and synchrotron emission, correspondingly. Therefore, in order
for very-high energy electrons to reach us, their sources must be close to us
and relatively young. Perhaps Nishimura et al. [106] were the first to suggest
that “the electron spectrum in TeV region would deviate from smooth power law
behavior due to small number of sources which are capable of contributing to
the observed flux… several bumps would be observed in the spectrum correlating
to each source…” Other early papers [107, 108] also show possible
contributions of local sources. Subsequent publications discussed the origin
of the observed spectrum and modeled the contribution of local sources above 1
TeV.

The all-electron ($e^{-}$$+$$e^{+}$) spectrum up to 1 TeV was measured by
Fermi–LAT [109, 110] (pre-Fermi–LAT measurements are summarized in [109]). It
appears to be too flat, contrary to the expectations of a steep decrease (see
Fig. 57 in [62] and Fig. 5 in [111]), and cannot be reproduced with a single
component. The sharp cutoff above $\sim$1 TeV was reported by H.E.S.S. [112,
113] and confirmed by VERITAS [114]. Subsequent measurements by PAMELA [115],
AMS-02 [116, 117, 62], CALET [118], DAMPE [119], and Fermi–LAT [120] reveal a
flattening at $\sim$60 GeV and confirmed the cutoff at 1 TeV. Interestingly,
the experiments are consistent in pairs, that is, CALET and AMS-02 are
consistent with each other, but differ by $\approx$20% from DAMPE and
Fermi–LAT, which are also consistent with each other. Above 1 TeV, H.E.S.S.,
CALET, and DAMPE measurements are consistent with each other, there is also a
slight hint at a possible feature above 3 TeV albeit with large error bars.
Slow diffusion zones observed around several pulsars [121] can increase the
energy losses of TeV electrons making observations of such features less
likely. The all-electron spectrum includes the $e^{+}$ excess (the so-called
“signal”), and may include an identical $e^{-}$ “signal” if the source is
charge-sign symmetric, such as pulsars or dark matter (DM) [62].

There are quite a number of publications discussing the contributions to the
all-electron spectrum from local sources. Such multi-component models include
the average electron spectrum from distant Galactic sources, parametrized
contributions from local catalog sources (SNRs, PWNe), and may utilize the
observed radio spectral indices of local SNRs, see examples in [122, 123].

AMS-02 data on electrons ($e^{-}$) may offer a clue to the origin of the break
at $\sim$1 TeV. Preliminary analysis [124] uses a three-component fit, which
includes low- and high-energy power-laws plus the $e^{+}$-like source term.
According to this fit, the break in the all-electron spectrum at $\sim$1 TeV
is related to the cutoff in the $e^{+}$ spectrum plus the identical $e^{-}$
component, and implies a charge-symmetric source of the excess positrons
(pulsars, DM). However, more accurate data is needed to test if the charge-
symmetry is exact (e.g., hadronic processes do not produce identical $e^{\pm}$
spectra).

##  9 Cosmic ray positrons

Unexpected behavior of the positron fraction $e^{+}/(e^{+}$$+$$e^{-})$ was
first noticed in the data from the TS93 balloon flight [125], which observed a
constant positron fraction of $0.078$$\pm$$0.016$ in the range of 5–60 GeV,
and in the HEAT experiment [126], which detected “a small positron flux of
nonstandard origin.” Earlier experiments were contaminated with protons due to
insufficient rejection capability. The results did not attract much attention
until PAMELA team reported a surprising rise in the positron fraction up to
100 GeV [127], contrary to the expectations of a monotonic decrease with
energy [128, 129]. Conventional models imply a smooth CR source distribution
and steady-state production of secondary species in the hadronic CR
interactions with ISM gas. The rise was confirmed by Fermi–LAT [130], which
used the geomagnetic field for identifying positrons, and with a higher
precision and up to $\sim$500 GeV by AMS-02 [131, 132]. The latest AMS-02 data
indicate that the $e^{+}$ flux is the sum of low-energy secondaries from CR
production plus a high-energy component from a new source with a cutoff at
$749^{+197}_{-137}$ GeV [62, 124].

Perhaps the most striking is the fact that the $e^{+}/\bar{p}$ and $\bar{p}/p$
ratios barely change from 60–525 GeV [133, 62] hinting at some connection
between the three species. Fitting a constant to the flux ratio in this range
yields $e^{+}/\bar{p}$ $=$ $2.01$ $\pm$0.03(stat.) $\pm$0.05(syst.) [134],
consistent with a constant. It is unclear whether it has a fundamental
importance or is a chance coincidence. Some authors argue that this indicates
that both species ($e^{+},\bar{p}$) are secondary and are produced in the same
process [135, 136]. However, the fit could also be performed with a moderately
rising functional dependence, because constraining are only a few lower-energy
points from 60–130 GeV.

Given the high collected positron statistics, the authors [137] point to small
but significant peaks in the positron fraction at 12 and 21 GeV, which could
be associated with a powerful explosion in the Galactic center and Fermi
Bubbles [138, 139] or with yet unknown processes.

The positron anomaly gave rise to a huge number of interpretation papers, most
of which connect the positron excess with DM. However, accurate multi-
messenger data ($\gamma,\bar{p}$, CRs) collected after the PAMELA discovery
imposes tight constraints on the most simplistic DM models (see a review talk
by Francesca Calore [140]). The astrophysical interpretations can be divided
into groups of models with primary and secondary origin of positrons. In turn,
models with primary positrons discuss their production in pulsars with
additional effects produced by pulsar bow shocks and slow diffusion zones.
Secondary production models are divided between positron production in ISM and
in SNR shocks; the latter exploit various properties of SNR shocks and various
configurations of target gas distributions. There are also models exploiting
inhomogeneity of the SNR (CR sources) distribution. Here we mention only a few
examples.

_A model with local SNR, surrounded by gas clouds (secondary production in the
local ISM), has already been discussed above. We refer the reader to Sect.
7.1._

###  9.1 Positron production in Galactic SNR shocks

The first models [141] speculated on the idea of producing secondary species
in a SNR shock, where CRs are accelerated (originally proposed in [142]).
Therefore, the secondary $e^{\pm}$ participate in the acceleration process and
turn out to have a very flat spectrum, which is responsible, after propagation
in the Galaxy, for the observed positron excess. However, it soon becomes
clear that this process should also increase the production of other secondary
species, and thus other secondary to primary ratios ($\bar{p}/p$, B/C) should
rise too [143, 144, 145, 146], contrary to observations. Non-observation of
such rise provides significant constraints on physical conditions in the
shock.

###  9.2 Secondary positrons in the volume charge model

In this model [147], CRs accelerated in the SNR shell penetrate into the dense
gas clumps upstream, where they interact with the ISM gas and produce
secondary particles ($\bar{p},e^{\pm}$). The predominance of positively
charged particles in the shock and in the pre-cursor develops a positive
electric volume charge in the gas cloud, which preferentially expels secondary
positrons into the upstream plasma where they are accelerated by the shock.
Since the shock is modified, these positrons develop a harder spectrum than CR
electrons accelerated in other SNRs. Mixing these populations explains the
increase in the positron fraction $e^{+}/(e^{+}$$+$$e^{-})$ above 8 GeV.
Besides, there are also other sources of positrons in the ISM, such as
radioactive decay.

###  9.3 Primary positrons from pulsars

Pulsars are the primary charge-symmetric suspects as they disconnect positrons
from nuclear species and therefore remove the constraints associated with
production of other secondaries. Probably, the first mention of pulsars as
sources of CR positrons can be found in [148], and the first calculation of
secondary production and pulsar contribution to CR positrons – in [149]. A
calculation of the positron fraction using the data available at that time
(contaminated by protons above $\approx$5 GeV) was performed in [150]. This
model included secondary $e^{\pm}$, primary $e^{-}$ from SNR, and primary
$e^{\pm}$ from pulsars. Examples of modern calculations of the positron
fraction, which include $e^{\pm}$ from known sources and secondary $e^{\pm}$
can be found in [151, 152].

###  9.4 Primary positrons in the pulsar bow shock model

Pulsars with high spin-down power produce relativistic winds. Some pulsars
move relative to their surrounding ISM at supersonic speeds producing bow
shocks. Ultrarelativistic particles accelerated at the termination surface of
the pulsar wind can experience reacceleration in the converging flow system,
producing a universal spectrum similar to that of protons accelerated in the
SNR shell. This scenario naturally explains why the $e^{+}/p$ ratio remains
constant above 60 GeV. Primary positrons and electrons in this scenario have
similar spectra.

An idea of positron reacceleration in a pulsar bow shock was proposed in
[153], and further detailed in [154]. It is suggested that the 5.7 millisecond
(MSP) pulsar PSR J0437-4715 may produce the observed positrons. The pulsar
distance and velocity are $\approx$$156.79$$\pm$$0.25$ pc and $\sim$100 km/s.
It is the closest and brightest MSP in a binary system with a white dwarf
companion and an orbital period of 5.7 days. It is observed in optical, far-
ultraviolet (FUV), and X-ray bands and exhibits the greatest long-term
rotational stability of any pulsar. The model assumes that the pulsar’s
position coincides with the direction of the local magnetic field and adjusts
the parallel diffusion coefficient to match optical, FUV, and X-ray
constraints on the flux of accelerated leptons from the nebula.

##  10 Cosmic ray antiprotons and 10 GeV excess

CR antiprotons in the range 2–12 GeV were observed for the first time during
balloon flights [155, 156]. The following series of Antarctic flights by BESS
[157], N. America flights by MASS91 [158], HEAT [159], and CAPRICE98 [160]
instruments, and space experiments, PAMELA [161] and AMS-02 [62], extended the
energy range and increased accuracy of $\bar{p}$ measurements. The $\bar{p}$
spectrum is now measured up to $\sim$450 GV [62]. It features a low-energy
rise caused by the kinematics of the process [162]. Ratio $\bar{p}/p\approx
const$ from 30–450 GV [62].

Following the publication of the AMS-02 $\bar{p}$ data [133], several groups
independently noticed an excess over conventional model predictions at around
10 GeV [163, 164, 165]; all three papers are marked as published on May 9–12,
2017. Two papers [164, 165] proposed an interpretation in terms of DM, while
[163] pointed to increased systematics due to the high solar activity period
during data taking or due to the production cross section uncertaincies, see
also [166, 167, 168, 169, 170].

At present, the two hypotheses remain, (i) DM contribution, and (ii)
systematics due to the solar modulation and/or cross section uncertainties.
People like the DM hypothesis, but attempts are being made to improve on the
cross sections. Interestingly, the same proposed DM candidate
($m_{\chi}\approx 50-100$ GeV) can reproduce the 10 GeV antiproton excess,
$\gamma$-ray excess from thsce Galactic center, and the extended $\gamma$-ray
emission from the 400-kpc-across halo of the Andromeda galaxy (M31) [171, 172]
(see also references therein). Due to space limitations, I have to conclude my
review with an incomplete list of papers that discuss astrophysical
uncertainties associated with antiprotons [173, 174, 175, 176, 177, 178, 179],
and the DM interpretation of the antiproton excess [180, 181, 182, 183].

##  11 Concluding remarks – situation with production cross sections

Unfortunately, one major remaining bottleneck is the accuracy of particle and
especially isotopic production cross sections (see discussion in [48, 184]).
Every time an unexpected spectral feature is detected, there is a chance of an
error in the model predictions due to errors in the cross sections. The
elimination of such errors is easier and much cheaper than building and
successfully launching a major space mission like AMS-02, but this requires a
dedicated community effort.

_Partial support from NASA grants Nos. 80NSSC22K0477, 80NSSC22K0718,
80NSSC23K0169 is greatly acknowledged._

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