EPJ Web of Conferences SuGAR2015–Searching for the Origin of the Galactic
Cosmic Radiation

11institutetext: Space Science Division, US Naval Research Laboratory,
Washington, DC USA

# Impact of Fermi-LAT and AMS-02 results on cosmic-ray astrophysics

Charles D. Dermer 11<EMAIL_ADDRESS>
###### Abstract

This article reviews a few topics relevant to Galactic cosmic-ray
astrophysics, focusing on the recent AMS-02 data release and Fermi Large Area
Telescope data on the diffuse Galactic $\gamma$-ray emissivity. Calculations
are made of the diffuse cosmic-ray induced $p+p\rightarrow\pi^{0}\rightarrow
2\gamma$ spectra, normalized to the AMS-02 cosmic-ray proton spectrum at
$\approx 10-100$ GV, with and without a hardening in the cosmic-ray proton
spectrum at rigidities $R\gtrsim 300$ GV. A single power-law momentum “shock"
spectrum for the local interstellar medium cosmic-ray proton spectrum cannot
be ruled out from the $\gamma$-ray emissivity data alone without considering
the additional contribution of electron bremsstrahlung. Metallicity
corrections are discussed, and a maximal range of nuclear enhancement factors
from 1.52 to 1.92 is estimated. Origins of the 300 GV cosmic-ray proton and
$\alpha$-particle hardening are discussed.

##  1 Introduction

On the way to writing this proceedings article for the 2nd SUGAR (Searching
for the sources of galactic cosmic rays) Workshop, held in the Département de
Physique nucléaire et corpusculaire at Université Genève on 21 – 23 January
2015, the Alpha Magnetic Spectrometer Collaboration held a significant data
release on its “AMS days at CERN," which took place 15 – 17 April 2015 AMS02 .
Since new data is bound to make old theory more interesting, I’ve redirected
my thoughts towards this latest high-quality cosmic-ray data.

The major results presented at the AMS days were the $\bar{p}/p$ ratio from
kinetic energies $T_{p}\lesssim 1$ GeV to $T_{p}\approx 350$ GeV, and the
cosmic-ray proton and $\alpha$ particle spectra for rigidities 10 GV $\lesssim
R\lesssim 1500$ GV (p) and 10 GV $\lesssim R\lesssim 2500$ GV ($\alpha$).
Combining these results with the previous AMS-02 results on the positron
fraction 2013PhRvL.110n1102A  and the cosmic-ray electron and positron fluxes
2014PhRvL.113l1102A , in addition of course to the vast wealth of cosmic-ray
data Strong , gives an unprecedentedly large and accurate data set for
studying cosmic-ray physics.

In the most recent AMS-02 data AMS02 , the $\bar{p}/p$ ratio is reported to be
roughly constant for $10$ GeV $\lesssim T_{p}$ $\lesssim 350$ GeV with a value
of $\approx 2\times 10^{-4}$. The measured $\bar{p}/p$ ratio is claimed to be
a factor of $2$ – $3\times$ larger than predicted by secondary cosmic-ray
production models constrained by, for example, the positron fraction
e+/(e++e-) and the B/C ratio 2009PhRvL.102g1301D . Given the associated rising
positron fraction 2013PhRvL.110n1102A  at $\gtrsim 10$ GeV, which is often
explained as due to additional sources like pulsars, and not from enhanced
cosmic-ray induced production, one may ask whether there is a consistent
explanation for the overabundant $\bar{p}$ and the enhanced positron fraction.

AMS-02 confirms, though at a higher value of break rigidity, the hardening in
the cosmic-ray proton and $\alpha$ spectrum AMS02BoverC  reported from PAMELA
results PAMELA . The hardening appears in the AMS-02 p and $\alpha$ spectra at
$\approx 300$ – 400 GV, compared to $\approx 240$ GV for cosmic-ray p and
$\alpha$ spectra in PAMELA data 2014PhR…544..323A . Nevertheless, as was
remarked more than once at this workshop, there must be a hardening between
$\approx 100$ GeV/nuc and the knee, and it is great that AMS-02 has provided
more precise details (which improves upon the preliminary spectra released
during the 2013 ICRC AMS02 ).

Rather than repeat the points made in my presentation, which is available at
the conference
website111https://indico.cern.ch/event/332221/session/5/contribution/38 and
reviews some of the work found in my previous conference papers
2013arXiv1303.6482D ; 2013arXiv1307.0497D , it seems more useful to examine a
few of the implications of the AMS-02 data in view of the goals of this
meeting.

First we look at the cosmic-ray proton and $\alpha$-particle spectra from low
energies to high energies, which represents the minimum local interstellar
medium cosmic-ray proton ($\alpha$) (LISMCRp ($\alpha$)) spectrum from which
to calculate a minimum $\gamma$-ray emissivity.

Second, we make some calculations of galactic emissivity from the cosmic-ray
proton data, using a simple nuclear enhancement factor $k_{nuc}$ to correct
for metals. Even with $k_{nuc}=2$, a power-law momentum spectrum is allowed,
so a contribution of bremsstrahlung $\gamma$ rays is required to infer a low-
energy break in the LISMCRp spectrum from the Fermi-LAT $\gamma$-ray data. A
brief discussion of metallicity ($Z>2,A>4$) corrections, including cosmic-ray
and ISM metals, is given. We consider whether the breaks in the spectra of
cosmic-ray protons and ions are a consequence of propagation effects, multiple
injection sources with different spectral indices, or due to hardening
produced by nuclear collisions which preferentially depletes the low-energy
particles that traverse larger grammages.

##  2 Cosmic rays and $\gamma$ rays

###  2.1 Cosmic-ray p and $\alpha$ spectra

Figure 1: Data, as labeled, are from ATIC Pa06 , PAMELA PAMELA , AMS-02 AMS02
, and Voyager 1 Voyager1 , expressed in the form $R^{2.7}\times$ particle
number intensity. Smooth curves show a power-law proton flux in momentum with
index $s=2.8$ in the same representation. Upper and lower panels show the
fluxes on a linear and logarithmic scale, respectively.

In Fig. 1, we construct a plot of the cosmic-ray proton intensity (or flux)
from 4 sets of data (see Strong ) measured at or near Earth and at $\sim 122$
AU (Voyager 1). The highest energy data are from the ATIC (Advanced Thin
Ionization Calorimeter)-2 balloon campaign during 2002 – 2003 Pa06 . Also
shown are PAMELA (Payload for AntiMatter Exploration and Light-nuclei
Astrophysics) data PAMELA , the recently reported AMS-02 data AMS02 , and
Voyager 1 data Voyager1 . One thing about which we can be quite certain is
that due to Solar modulation of the cosmic ray, the LISMCRp spectrum will be
above the level of the data. Though it could be that the Voyager 1 has sampled
the “true" LISMCRp spectrum, it is also possible that during the billions of
years that the Solar wind has been generated, a much more complicated and
extended magnetic structure around the Sun has formed. In any case, the data
should provide a solid lower limit to the LISMCRp spectrum. Another thing is
that AMS-02 and PAMELA differ in normalization by $\lesssim 10$%, and more
like $\sim 5$% at $10\lesssim$R(GV)$\lesssim 100$, where modulation effects
are believed to be small. These differences are less than uncertainties in
nuclear cross sections 2012PhRvD..86d3004K .

Fig. 1 is a plot of the LISMCRp flux multiplied by $R^{2.7}$, where $R\equiv
P/Z$ is the particle rigidity, $P=\sqrt{E^{2}-m^{2}}$ is its momentum, $E$ is
its total energy, and $T=E-m$ is the particle kinetic energy. For protons,
then,

$R^{2.7}\,{dN\over dAdtd\Omega dR}={R^{3.7}\over
ET^{2}_{p}(R)}\,\left[T_{p}^{2}\,{dN\over dAdtd\Omega dT_{p}}\right]\;.$ (1)

The upper and lower panels show the measured cosmic-ray proton flux on a
linear and logarithmic scale, respectively. There are several points to be
made. First is that the flux, as shown in this representation multiplied by
$R^{2.7}$, deviates barely by a factor of 2 and is remarkably close to a power
law from $10\lesssim R$ (GV) $\lesssim 10^{4}$. Second is that three
experiments—ATIC, AMS-02, and PAMELA—now converge on a break rigidity for
protons of $R\approx 300$ GV, with the low-energy ($10\lesssim R$ (GV)
$\lesssim 300$) slope equal to $\approx 2.8$, and the high-energy
($300\lesssim R$ (GV) $\lesssim 10^{4}$) slope $\approx 2.6$, based primarily
on the ATIC data (the high-energy extension of the AMS-02 data suggests that
the hardening is less, perhaps to a slope of $\approx 2.7$).

###  2.2 The cosmic-ray spectrum in the local interstellar medium

What, then, is the LISMCRp spectrum and how is it made? The prevailing theory
is that Galactic cosmic rays are accelerated by processes taking place in SNR
shocks as the expanding supernova ejecta overtakes and sweeps up material in
the surrounding medium 2013A&ARv..21…70B . This theory has been taught so much
at school that to even question the SNR origin of Galactic Cosmic Rays at this
stage is treated as heresy.222Rather than burnt at the stake, one’s proposal
isn’t funded. In any event, this theory has received strong support from
Fermi-LAT and AGILE observations of the $\pi^{0}\rightarrow 2\gamma$ feature
in IC 443 and W44 2013Sci…339..807A , as well as in the spectrum of a 3rd
Fermi-LAT SNR, W51C, reported at this workshop by Dr. Jogler.

Although the $\pi^{0}\rightarrow 2\gamma$ decay hardening in the low-energy
$\gamma$-ray spectrum below several hundred MeV has now been reported,
confirming innumerable predictions and calculations going back to Ginzburg &
Syravotskii and Hayakawa, some open questions remain. For example, the SNR
$\gamma$-ray spectral indices can be soft (i.e., steeper than $2.5$), and
unusually low high-energy cutoffs in the $\gamma$-ray spectra are inferred
from Fermi-LAT and ground-based $\gamma$-ray detectors. For example, W51C and
W44 have cutoffs below 10 GeV, whereas Cas A has a break near 1 TeV and RX
J1713.7-2942 has a break at $\approx 10$ TeV. If the $\gamma$-ray emission is
due to cosmic rays accelerated to the knee ($\approx 3$ PeV), $\gamma$ rays at
a factor of 10 less energy ($\approx 300$ TeV), are expected, and no photons
near that high are detected. (The highest energy photons, at $\approx 100$
TeV, are observed in the steeply declining part of the RX J1713.7-3946 SNR
spectrum 2007A&A…464..235A .)

Putting these concerns to the side for the moment, we marvel instead that the
cosmic-ray proton spectrum is so close to a power law, as expected in first-
order test-particle Fermi acceration theory, as reviewed in be87 . One of the
central results of this theory is that the distribution function of
transmitted particles $f(p)\propto p^{-3r/(r-1)}$, so that $dN/dp\propto
p^{-A_{tp}}$, where the test-particle spectral index

$A_{tp}={2+r\over r-1}\;,$ (2)

Here $r$ is the compression ratio which, for strong shocks in a medium with
adiabatic index of $5/3$, appropriate to a nonrelativistic monoatomic gas,
cannot exceed a value of $r=4$, implying a test-particle index of $A_{tp}\cong
2$ for a strong nonrelativistic shock.

If the injection spectrum of cosmic-ray protons or ions is a power-law in
momentum or rigidity, then what is the form of the intensity
$j_{sh}(T_{p})=dN/dAdtd\Omega dT_{p}$ that enters into the calculations of
$\gamma$-ray emissivity? Most simply, rigidity-dependent escape steepens the
injection spectrum by the index $\delta$, so that the steady-state number
spectrum $dN/dp\propto p^{-A_{tp}-\delta}\propto p^{-s}$, where
$s=A_{tp}+\delta$. The intensity

$j_{sh}(T_{p})\propto\beta(T_{p})\,{dN\over
dT_{p}}\propto\beta(T_{p})\,|{dp\over dT_{p}}|\left({dN\over dp}\right)\propto
p^{-s}\;$ (3)

2012PhRvL.109i1101D . In Fig. 1, a power-law momentum spectrum, normalized to
the AMS-02 data at $10$ GV $\lesssim R\lesssim 200$ GV with index $s=2.8$ is
shown. It clearly disagrees with the Voyager 1 data, though as noted, this
might not be a fatal concern if the Voyager 1 data does not yet sample the
LISMCR spectrum.

In any case, however, one would have to be extremely naive to assume that a
power-law momentum spectrum should describe the LISMCRp intensity. Besides
diffusive escape from the target-rich Galactic disk, convective Galactic winds
from the superpositions of the hot gases of O/B stars and supernovae will
cause a systematic expulsion of cosmic rays to the halo of the Galaxy. MHD
turbulence in the Galactic magnetic field could cause systematic acceleration
of cosmic rays during transport smp07 . Although breaks may be expected,
demonstrating such a break in the cosmic-ray proton spectrum has proven to be
difficult, for reasons of (1) limitations on our knowledge of the secondary
pion distributions in p-p collisions, and (2) metallicity corrections on
secondary pion production 2012PhRvL.109i1101D ; 2012PhRvL.108e1105N .

A crucial issue in the scenario that strong SNR shocks accelerate the Galactic
cosmic rays should be noted. Compression ratios $r\rightarrow 4$ imply
injection indices of cosmic rays near $2$. In theory the bulk of the cosmic-
ray population is swept downstream and is confined to the remnant until late
in the SNR’s radiative evolution when the shock is not so strong, at which
time the cosmic rays diffusively escape. The “real" injection index of cosmic
rays into interstellar space may be somewhat softer, $\approx 2.1$ – $2.2$. On
the other hand, the highest energy particles will have escaped near the Sedov
time. For these injection indices, the rigidity-dependent steepening would
have to be by $\delta\approx 0.6$ – 0.7. This can be compared with analyses
finding $\delta\approx 0.44\pm 0.03$ and a source injection index of $2.34$
2015arXiv150403134G  using the form $D(R)=D_{0}\beta(R/R_{0})^{\delta}$ for
the diffusion coefficient. PAMELA analysis finds $\delta\cong 0.4$
2014ApJ…791…93A .

The secondary to primary ratios of cosmic-ray nuclei can also be modeled by an
average escape grammage $\Lambda(R)$ through which cosmic-ray particles with
rigidity $R$ pass smp07 . Analysis of cosmic-ray ion composition often uses
the empirical function $\Lambda(R)=\beta\Lambda_{0}(R/R_{0})^{-a}$ for
$R>R_{0}$ and $\Lambda(R)=\beta_{0}\Lambda_{0}$ for $R<R_{0}$, where
$\beta=p/E=ZR/\sqrt{Z^{2}R^{2}-m^{2}}$. When fit to the B/C ratio, a typical
parameter set fitting the data has $\Lambda_{0}=11.8$ g/cm-2 and $a=0.54$, and
$R_{0}=4.9$ GV/c, with a rigidity index of $2.35$. The value of $a$
essentially coincides with the value of $\delta$ in diffusion theory.

The value $\delta\approx 0.4$ is near the theoretically favored value of
$\delta\approx 1/3$ for Kolmogorov turbulence, but observationally then
requires a soft injection index $\approx 2.3$ – 2.4. In any case, cosmic rays
must sample a wide range of environments, and it may be surprising that the
cosmic-ray proton spectrum is as smooth as it is. Now we consider $\gamma$-ray
constraints on deviations from a single momentum power law at the low- and
high-energy parts of the Galactic cosmic-ray proton distribution.

###  2.3 $\gamma$-ray constraints on cosmic rays

We can ask if the power-law momentum spectrum shown in Fig. 1 can be ruled out
by the $\gamma$-ray data. The expression $R^{2.7}\,dN/dtd\Omega dAdR=1.8\times
10^{4}(R^{0.9}/E)$ GV1.7/(s-sr-m2) translates into a cosmic-ray proton flux

$j_{sh}(T_{p})=1.8R^{-2.8}\;({\rm s\leavevmode\nobreak\ sr\leavevmode\nobreak\
cm}^{2}{\rm\leavevmode\nobreak\ GV})^{-1}\;.\;$ (4)

(In comparison, I 1986A&A…157..223D  used the expression
$j_{sh}(T_{p})=2.2E_{p}^{-2.75}\;({\rm s\leavevmode\nobreak\
sr\leavevmode\nobreak\ cm}^{2}{\rm\leavevmode\nobreak\ GeV})^{-1}$, with
$E_{p}$ the total energy in GeV, based on the cosmic-ray proton spectrum shown
in Simpson’s review 1983ARNPS..33..323S .)

It is a simple matter to plug the proton flux given by Eq. (4) into a code
that calculates the $\gamma$-ray emissivity per H atom due to pion-producing
reactions of relativistic protons with protons at rest, and compare with the
emissivity measured with the Fermi-LAT telescope. For this purpose, I use my
original code 1986A&A…157..223D , including the kludge between 3 and 7 GeV,
which may be the source of some structure in the $\gamma$-ray production
spectra. The improvements suggested 2013arXiv1307.0497D  to properly
characterize the different isobars according to their exclusive cross
sections, which would mitigate the effects of the kludge, have not yet been
fully implemented. We also make calculations using Kamae’s model kam06 .
Notably, a nuclear enhancement factor $k_{nuc}=2.0$ is assumed.

Figure 2: Fermi-LAT data shows early results of the Galactic $\gamma$-ray
emissivity 2009ApJ…703.1249A , and the emissivity from the 2012 analysis of
Casandjian cas12 . Calculations are for a proton momentum power-law shock
spectrum given by eq. (4) (thick), the same spectrum with a hardening by $0.2$
units at 300 GeV (thin solid), and the shock spectrum with a low-energy cutoff
at 2 GeV in the LISMCRp spectrum (thin dotted and dot-dashed); see Fig. 1. Two
$p+p\rightarrow\pi^{0}\rightarrow 2\gamma$ models are used, shown in black
1986A&A…157..223D  and blue kam06 .

Fig. 2 shows calculations compared to Fermi-LAT data. The early Fermi-LAT
emissivity study 2009ApJ…703.1249A  uses data from 2008 August 4 to 2009
January 31 ($\approx 6$ mo), whereas the 2012 study cas12  uses 3 years of
Fermi-LAT data. Besides the single momentum power-law shock spectrum, eq. (4),
we also consider a case where the shock spectrum hardens at $R_{br}=300$ GV by
0.2 units, and a case where the spectrum nose dives below 2 GeV. Even with
Pass 8 analysis and 10 years of Fermi-LAT data, it will be a big challenge to
see a statistically significant hardening above $\approx 30$ GeV. The two
emissivities—with and without the spectral hardening at $R>300$ GV in the
LISMCRp spectrum—differ by only a factor of 2 at 100 GeV photon energies.
Attempts to see the spectral hardening from Earth-limb analysis may be more
successful, but limitations of our knowledge of secondary production cross
sections can ultimately hinder analysis.

This simple comparison does not preclude the possibility that eq. (4) could
describe the LISMCRp spectrum, at least below 300 GV. In fact, the numerical
fit using the simple shock power-law momentum spectrum, though going right
through the low energy data, must allow some emission for bremsstrahlung from
nonthermal electrons making radio synchrotron and for bremsstrahlung from the
e+ and e- made from the decay of $\pi^{\pm}$. A full study must consider
associated radio synchrotron emission 2013MNRAS.436.2127O , demodulation or
reconstruction of the LISMCR e+/e- spectrum, and the inverse Compton emissions
that are dependent on Galactic location cas15 ; str15 . A recent evaluation of
$\gamma$-ray emissivity from the AMS-02 cosmic-ray proton and $\alpha$ data,
including calculations of the associated neutrino fluxes, see
2015arXiv150400227G . Even so, emissions from leptons, because of their strong
cooling, are expected to have softer spectra and contribute relatively less to
the $\gamma$-ray emissivity at higher energies. It is not clear if the
emissivity data is systematically $\sim 10$ – 30% larger than predicted by
cosmic-ray induced emissions when using $k_{nuc}=2$, and whether this is a
reasonable value. This brings us to a reconsideration of the nuclear
enhancement factor.

###  2.4 Metallicity corrections

The calculations shown in Fig. 2 make use of a simple multiplier for the
effects of all nuclei other than protons and H. This is the nuclear
enhancement factor $k_{nuc}$, which was assigned a value of 2.0 in our
calculations. Let us make a quick calculation of the range of values of
$k_{nuc}$. The first thing to note is that in collision physics, $T_{nuc}$,
the kinetic energy per nucleon is the quantity of interest. At the same values
of $T_{nuc}$ we see that the accelerated particle distributions—the cosmic
rays—have abundances of H, He, CNO and Fe in the ratio of 1:0.07:0.0044:0.0003
2012PhRvL.109i1101D ; smp07 . From Mori 2009APh….31..341M , drawing on the
work of J.-P. Meyer Meyer , we take the composition of the target ISM material
with relative abundances of H:He:CNO:NeMgSiS:Fe in the ratio
1:0.096:1.38e-3:2.11e-4:3.25e-5.

Table 1: Relative contributions to nuclear enhancement factor $k_{nuc}$ | $\sigma\propto A_{1}A_{2}$ | $\sigma\propto A^{2/3}_{1}A_{2}^{2/3}$
---|---|---
p-H | 1 | 1
p-He + $\alpha$-H | 0.66 | 0.42
$\alpha$-He | 0.11 | 0.043
CNO | 0.11 | 0.042
NeMgSiS | 0.014 | 0.0045
Fe | 0.022 | 0.0042
Total | 1.92 | 1.52

The maximum enhancement due to nuclei can be estimated assuming that there is
no shadowing, and each nucleon participates independently in the scattering.
On the other hand, an apparent minimum value of $k_{nuc}$ can be obtained by
assuming a geometrical shadowing $\propto A^{2/3}$, as if the nucleons were
tightly packed in the nucleus.

Table 1 shows the relative contributions of various nuclei or nucleon groups
to $k_{nuc}$, under the two extremes that $\sigma\propto A_{1}A_{2}$ and
$\sigma\propto A_{1}^{2/3}A_{2}^{2/3}$. The notation is such that the row
labeled CNO includes all reactions of C, N, and O in the cosmic rays and in
the ISM involving p, He, and $\alpha$ particles. Similarly, NeMgSiS includes
all reactions of CR and ISM Ne, Mg, Si, and S in the cosmic rays and ISM that
interact with p, $\alpha$, and CNO. For this estimate, we assume the same
cosmic ray as ISM composition for NeMgSiS.

Note that reactions involving ions heavier that He account for up to 5% of the
emissivity, so cannot be in any sense neglected. This estimate shows that
$k_{nuc}=2.0$ exceeds the upper limit, so that the emissivity calculations in
Fig. 2 are, if anything, an overestimation. Mori 2009APh….31..341M , using
DPMJET-3 to derive cosmic-ray energy-dependent emissivities, does not give
results below 6 GeV/nuc. Recent treatments are given by 2014ApJ…789..136K
who, using different compositions, find $1.9\lesssim k_{nuc}\lesssim 2.1$, in
better agreement with the “no-shadowing" cross sections. As noted there, the
situation is more complicated than a simple value of $k_{nuc}$, which is
dependent on the photon energy and the differences between cosmic-ray proton
and ion spectra. Other treatments find corrections $\propto A^{2.2/3}$
2007NIMPB.254..187N . The low value of $k_{nuc}=1.45$ obtained in my early
study 1986A&A…157..223D  was a consequence of using a cross-section correction
closer to the $A^{2/3}$ behavior, and by using a low value for the cosmic-ray
$\alpha$ particle to proton ratio.

###  2.5 Cosmic-ray spectral hardenings

Cosmic-ray spectral hardenings at $\approx 300$ GV in the p and $\alpha$
spectra could be due to any number of physical effects. From general
principles, we can suggest either that this spectral hardening is due to (1)
the superposition of two populations with different mean injection indices;
(2) rigidity-dependent effects on escape from the Galaxy; (3) rigidity-
dependent effects at the source/accelerator; (4) acceleration during
transport; or (5) grammage effects.

Effect (5) fails because grammage makes instead a low-energy hardening when
heavier ($\gtrsim$CNO) low-energy cosmic-rays pass through significant
grammages during escape. Effect (4), which suggests turbulent reacceleration,
would make curved, not broken power-law spectra 2006ApJ…647..539B . Effects
(2) and (3) go hand-in-hand, in terms of the behavior if not in explanation.
All Fermi acceleration theories have, in the collisionless limit, a dependence
on rigidity as the fundamental quantity. A break or hardening at a given
rigidity should manifest as one of the 3 (Bernard) Peters cycle predicting
$E_{br}\propto Z$, as described by Dr. Tilav at this workshop.

Idea (1) seems to be ruled out from a number of directions. A 0.2 break from
the addition of two underlying distributions that only differ by 0.2 in index
would be extremely broad, and there would be unusual changes in composition
depending on the compositions of the 2 populations. Which returns us to (2)
and (3), for which we need an explanation why, during cascading from small to
large $k$, the wave turbulence spectrum governing diffusion would apparently
soften, so that lower energy ions that gyroresonate with these waves would
escape more readily than at high energies resulting in a hardening.

###  2.6 $\bar{p}/p$ ratio

The figure in the AMS-02 release AMS02  showing a severe disagreement of the
$\bar{p}/p$ ratio with a secondary production model prediction deserves a few
comments. First is that ratios $\bar{p}/p$ and $e^{+}/(e^{+}+e^{-})$ are far
less useful than the particle spectral intensities themselves
2014JCAP…09..051K , particularly when looking at fits to data using models
involving multiple source populations. Second is that the model
2009PhRvL.102g1301D  showing a discrepancy with the AMS-02 data, used to
constrain fits for dark matter candidates to $\bar{p}$ production, was
tailored to fit PAMELA data that becomes increasingly uncertain at high
energies.

If there are $\bar{p}/p$ enhancements compared to expectations from secondary
production, a non-Copernican explanation has us fortuitously situated some 2
Myr ago next to a SN that deposited 60Fe while enhancing cosmic-ray production
in $\bar{p}$ and e+ 2015arXiv150204158K . Fits to the $\bar{p}/p$ data with a
conventional propagation parameters ($1/3<\delta\lesssim 1/2$) in a secondary
nuclear production models are found in recent analyses 2015arXiv150404276G ;
2015arXiv150407230L . In particular, agreement with the B/C and $\bar{p}/p$
ratio with a propagation model with $\delta\approx 0.42$ is found in
2015arXiv150405175E .

###  2.7 Secondary nuclear production models

The high-quality Fermi-LAT data demands increasingly accurate nuclear data,
which is currently lacking. Some new low-energy, $\lesssim 1$ GeV/nuc data are
presented in 2014PhRvD..90l3014K  and were discussed at this workshop by Dr.
Taylor. Another paper that would be a valuable benchmark for low-energy
secondary production calculations is 1988JPhG…14..903A . All the new data is
at $T_{nuc}\lesssim 2$ GeV/nuc. The important regime for $\gamma$-ray
astronomy is at $\approx 2$ – 10 GeV/nuc, where scaling models break down,
event generators extrapolate outside their zone of certainty, and exclusive
formulations become impossible.

Rather than providing empirical formulations that are specific, model-
dependent, and can’t be tampered with, a better direction is to have physical
models that can be modified. Detailed examination of processes in the regime
is required. Intermediate-energy nuclear astrophysics is not, unfortunately, a
growth area.

A relevant formula to conclude this contribution is the $\pi^{0}$-production
threshold kinetic energy per nucleon $T_{thr,nuc}$ of a cosmic-ray ion with
mass $M=A_{1}m_{p}$ striking an interstellar atom or ion with mass
$m=A_{0}m_{p}$. The threshold condition is that
$s=E^{2}-p^{2}=(m+M+m_{\pi^{0}})^{2}$. I obtain

$T_{thr,nuc}={m_{\pi}}\;\left({1\over A_{1}}+{1\over A_{0}}+{m_{\pi}\over
2A_{1}A_{0}m_{p}}\right)\;.$ (5)

Thus $T_{thr,nuc}=2m_{\pi}+m_{\pi}^{2}/2m_{p}\cong 280$ MeV, as is well known.
This also implies that $T_{thr,nuc}=171$ MeV/nuc for p-He and $\alpha$-p, and
$T_{thr,nuc}=69.9$ MeV/nuc for $\alpha$-He collisions. Concerns
2014PhRvD..90l3014K  about the accuracy of near-threshold $p+p\rightarrow\pi$
cross sections 1986ApJ…307…47D  are not as important as near-threshold
contributions of cosmic-ray nuclei.

##  3 Concluding remarks

These few pages should not conceal the much more detailed and exhaustive work
that must be done to satisfy cosmic-ray constraints. For this, one may consult
the detailed GALPROP treatment 2002ApJ…565..280M , which requires processes
not often studied, for example, nuclear energy losses from pion production
2015ApJ…802..114K , or semi-analytic treatments, e.g., 2012JCAP…01..010B .

The new AMS-02 confirms the hardening in the cosmic-ray proton and $\alpha$
spectrum reported with PAMELA, and the two data sets are typically within
$\sim 5$% of each other. With this accuracy, and the quality of the Fermi-LAT
emissivity data, renewed focus on metallicity corrections and secondary
production processes at the $\sim 5$% level is required.

Before explaining the shape of the LISMCRp spectrum, we must be sure that we
can determine its spectrum from as low an energy, $R\ll 1$ GV
($E_{nuc}\lesssim 100$ MeV/nuc), below which Solar energetic particles and
anomalous cosmic rays start to dominate, to as high as energy as possible. It
is generally believed that Solar modulation effects become negligible above
$R\sim 10$ GV. From this boundary condition, we should be able to track the
$\gamma$-ray spectrum back to the cosmic rays and their sources. This is work
in progress.

Acknowledgments I thank Drs. Elisa Prandini, Simona Toscano, and Andrii
Neronov for the kind invitation to visit Geneva and attend the workshop.
Comments of M. Kachelriess, S. Ostapchenko, T. Kamae, and particularly A.
Strong are much appreciated. Secondary nuclear production code used here is
available by e-mail request<EMAIL_ADDRESS>
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