Where are the Pevatrons that Form the Knee in the Spectrum of the Cosmic Ray
Nucleon Component around 4 PeV?

© 2023 A. A. Lagutin${}^{\boldsymbol{1)}^{*}}$ and N. V.
Volkov${}^{\boldsymbol{1)}^{**}}$††1)Altai State University, Radiophysics and
Theoretical Physics Department†† ∗E-mail<EMAIL_ADDRESS>∗∗E-mail:
<EMAIL_ADDRESS>
> Abstract—The paper discusses an approach that made it possible to estimate
> the distance to the nearest pevatrons, which form a knee in the spectrum of
> the cosmic ray nucleon component of about $4$ PeV. It is based on the
> spectra of nucleons and electrons obtained by the authors in the framework
> of the superdiffusion model of nonclassical cosmic rays diffusion, which
> have a knee, on the assumption that nucleons and electrons are accelerated
> by the same type sources and their propagation in an inhomogeneous turbulent
> galactic medium is characterized by the same diffusion coefficient, and also
> on the knee in the spectrum of the electronic component in the region of
> $0.9$ TeV, established in the DAMPE experiment.
>
> It is shown that pevatrons, which form a knee in the spectrum of the cosmic
> ray nucleon component of about $4$ PeV, are located at distances of the
> order of $0.75$ kpc from the Earth.

##  1 Introduction

Despite more than 100 years of research, the spatial distribution of the main
cosmic rays (CRs) sources and the mechanisms of particle acceleration in them
have not been finally established. To solve the problem of searching of
galactic CR sources the energies of the order of $10^{15}$ eV is a key place
because at these energies the CR spectrum has a break (so-called ‘‘knee’’).
Today it is generally accepted that CRs with energies around the knee are
mainly of galactic origin, and their sources are called pevatrons.

The search of galactic pevatrons is currently one of the priority task solved
jointly by all ground-based and orbital astrophysical observatories operating
in the very high energy region (see for example reviews HAWC:2020 ;
Cristofari:2021 ; Fermi:2022 ; LHAASO:2023 ). One of the most important
results achieved is the detection of gamma rays with energies above 100 TeV,
clearly indicating that there is an effective acceleration of CR particles up
to energies of the order of $10^{15}$ eV. For many years, the scientific
community has been dominated by the hypothesis that the main sources capable
of accelerating CRs to such energies are supernova remnants (SNRs). Despite
this, today there are no reliable experimental data confirming the fact that
supernovae accelerate CR nuclei to energies of $\sim 4$ PeV, i.e., to the knee
energy in the CRs spectrum Caprioli:2011 ; Blasi:2019 ; Evoli:2021 ; Vieu:2023
; Cardillo:2023 . The detection of ultrahigh-energy gamma-rays from regions
not associated with SNRs indicates that there are other astrophysical objects
that can claim the role of pevatrons (see materials of HONEST Workshops
HONEST:2022 ).

Main goal of this paper is to discuss an approach that made it possible to
estimate the distance to the nearest pevatrons, which form a knee in the
spectrum of the CRs nuclear component about $4$ PeV.

##  2 Proposed approach

The key elements of the proposed approach are based on the following results
and assumptions.

  * •

Our approach is based on the spectra of nuclei and electrons obtained by the
authors in the framework of the superdiffusion model of nonclassical CRs
diffusion, which have a knee. The main provisions of the nonclassical
diffusion model and the results of its application for the interpretation of
the spectra of the leptonic and nuclear components of CRs, as well as the
spectrum and mass composition in the ultrahigh energy region, are given in our
previous papers Lagutin:2001 ; Lagutin:2003 ; Lagutin:2004a ; Lagutin:2004b ;
Lagutin:2009 ; Lagutin:2021a ; Lagutin:2021b ; Lagutin:2023 .

  * •

We assume that nuclei and high energy electrons and positrons are accelerated
by the same type sources and their propagation in an inhomogeneous (fractal-
like) turbulent galactic medium is characterized by the same diffusion
coefficient.

  * •

We use the fact that there is a knee in the high-energy cosmic-ray electrons
plus positrons spectrum in the region $\sim 0.9$ TeV. Early indications of the
presence of this knee were obtained by ground-based Cherenkov detectors of the
H.E.S.S. collaboration HESS:2008 . Recent results of direct measurements by
DAMPE DAMPE:2017  and CALET CALET:2017  space observatories confirmed the
presence of this knee in the total spectrum of electrons and positrons.

##  3 The nonclassical CRs diffusion equations

For the first time, the equations of superdiffusion of cosmic rays without
taking into account energy losses and taking them into account were proposed
in our works Lagutin:2001 ; Lagutin:2004a .

For the density of particles $N(\textbf{r},t,E)$ with energy $E$ at the
location r and time $t$, generated in a fractal-like medium by Galactic
sources with a distribution density $S(\textbf{r},t,E)$, is written as

$\frac{\partial N(\textbf{r},t,E)}{\partial
t}=-D(E,\alpha)(-\Delta)^{\alpha/2}N(\textbf{r},t,E)+S(\textbf{r},t,E),$ (1)

$\frac{\partial N(\textbf{r},t,E)}{\partial
t}=-D(E,\alpha)(-\Delta)^{\alpha/2}N(\textbf{r},t,E)+\dfrac{\partial
B(E)N(\textbf{r},t,E)}{\partial E}+S(\textbf{r},t,E).$ (2)

In these equation $D(E,\alpha)=D_{0}(\alpha)E^{\delta}$ is the anomalous
diffusivity; $(-\Delta)^{\alpha/2}$ is the fractional Laplacian Samko:1993
(‘‘Riesz operator’’) (reflects a nonlocality of the diffusion process of
particles in the interstellar medium); $B(E)$ is the mean rate of continuous
energy losses of electrons and positrons.

It should be noted that when $\alpha=2$ from Eqs. (1) and (2) we obtain the
normal diffusion Ginzburg-Syrovatskii equations.

In calculations of the spectrum of electrons and positrons, the main
mechanisms of energy lossess are taken into account, i.e. ionization,
bremsstrahlung, synchrotron and inverse Compton losses. In a recent paper
Fang:2021  it was shown that in the relativistic regime (Klein-Nishina mode)
the threshold energy of inverse Compton scattering is reached even when
electrons interact with photons of visible radiation. The cross sections for
the interaction of electrons with background photons become much smaller than
the Thomson cross section traditionally used in calculations. As a result, the
rate of energy lossess of electrons during interaction with the background
electromagnetic radiation of the Galaxy decreases (the Klein-Nishina effect).
In this paper, in calculating the spectra of electrons and positrons with
energies $E>100$ GeV, the approximation expressions proposed in Fang:2021
were used to take into account the Klein-Nishina effect.

The solution of the superdiffusion Eqs. (1) and (2) was found by the Green’s
function method in both cases with energy losses and without ones. To solve
equation (2) we use the Syrovatskii functions Ginzburg:1964

$\lambda(E,E_{0})=\int\limits_{E}^{E_{0}}\frac{D(E^{\prime})}{B(E^{\prime})}dE^{\prime},\qquad\tau(E,E_{0})=\int\limits_{E}^{E_{0}}\frac{dE^{\prime}}{B(E^{\prime})}.$
(3)

5mm

Figure 1: Dependences of the Syrovatskii functions
$\lambda(E,E_{0})^{1/\alpha}$ and $\tau(E,E_{0})$ on the initial energy
$E_{0}$ of the particles for the different regimes of energy losses

The $\lambda(E,E_{0})^{1/\alpha}$ function should be understood as the average
distance over which the particle has diffused, taking into account the energy
losses from the initial value $E_{0}$ to the energy $E$. The $\tau(E,E_{0})$
function gives the ‘‘cooling time’’ from $E_{0}$ energy to $E$. Fig. 1 show
the behavior of the functions $\lambda(E,E_{0})^{1/\alpha}$ and
$\tau(E,E_{0})$ for different values of the initial energy $E_{0}$ of the
particles. The calculations were carried out for the both Klein-Nishina and
Thomson regimes of energy losses. It can be seen from the Fig. 1(a) that if
the particle energy at the point of observation is equal to $0.9$ TeV, then
for any mode of energy lossess from the initial value of $4$ PeV, the
diffusion radius of such particles is about $100-200$ pc. It can be seen from
Fig. 1(b), the ‘‘cooling time’’ of such particles varies from $10^{5}$ to
$10^{6}$ years.

For point instant source
$S(\textbf{r},t,E)=S_{0}E^{-p}\delta(\textbf{r})\delta(t)$ solutions of the
superdiffusion Eqs. (1) and (2) take the form:

  * •

for the nuclear component of CR (without energy losses)

$\leavevmode\nobreak\
N(\textbf{r},t,E)=S_{0}E^{-p}(D(E,\alpha)t)^{-3/\alpha}g_{3}^{(\alpha)}(|\textbf{r}|(D(E,\alpha)t)^{-1/\alpha}).$
(4)

  * •

for the leptonic component of CR (with energy losses)

$\leavevmode\nobreak\
N(\textbf{r},t,E)=\frac{S_{0}E^{-p}}{B(E)}\lambda(E,E_{0})^{-3/\alpha}g_{3}^{(\alpha)}(|\textbf{r}|\lambda(E,E_{0})^{-1/\alpha}),$
(5)

In the expressions (4) and (5) $g_{3}^{(\alpha)}(r)$ is the probability
density of three-dimentional sphericaly-symmetrical stable distribution
Uchaikin:1999a ; Zolotarev:1999 . The key feature of this function is the
presence of a knee. In case $\alpha=2$ $g_{3}^{(\alpha)}(r)$ is the normal
distribution.

In the particular case (Tomson regime) for energy losses in the form
$B(E)=bE^{2}$ GeV/s, where $b=1.1\cdot 10^{-16}$ (GeV s)-1 we find the
solution of superdiffusion Eq. (2)

$\leavevmode\nobreak\
N(\textbf{r},t,E)=S_{0}E^{-p}(1-btE)^{p-2}\lambda(t,E)^{-3/\alpha}g_{3}^{(\alpha)}(|\textbf{r}|\lambda(t,E)^{-1/\alpha}).$
(6)

Here

$\leavevmode\nobreak\
\lambda(t,E)=D_{0}(\alpha)E^{\delta}\hat{\lambda}(t,E),\qquad\hat{\lambda}(t,E)=\dfrac{1-(1-btE)^{1-\delta}}{b(1-\delta)E}.$
(7)

The figure 2 shows the results of calculating the function
$\hat{\lambda}(t,E)$ for various values of energy $E$. It should be noted that
in wide range of parameters $E$ and $t$ we obtain $\hat{\lambda}(t,E)\equiv
t$, i.e. $\lambda(t,E)=D(E,\alpha)t$. Calculations in the Klein-Nishina mode
lead to the same conclusion.

5mm

Figure 2: Dependence of the function $\hat{\lambda}(t,E)$ on time $t$ for
different values of energy $E$

##  4 Knee in the energy spectrum

To analyze the energy dependence of the electron concentration, we write
solution (6) of the superdiffusion Eq. (2) in the form $N=N_{0}E^{-\eta}$. It
follows from this representation that

$\eta=-\dfrac{E}{N}\dfrac{\partial N}{\partial E}.$

Taking into account the property of the stable law Uchaikin:1999a
$\dfrac{dg_{3}^{(\alpha)}(r)}{dr}=-2\pi rg_{5}^{(\alpha)}(r)$, we find

$\leavevmode\nobreak\ \eta=2p-2+\dfrac{\delta-1}{\alpha}\left[3-\dfrac{2\pi
r^{2}}{\lambda(t,E)^{1/\alpha}}\dfrac{g_{5}^{(\alpha)}(|\textbf{r}|\lambda(t,E)^{-1/\alpha})}{g_{3}^{(\alpha)}(|\textbf{r}|\lambda(t,E)^{-1/\alpha})}\right]=2p-2+\dfrac{\delta-1}{\alpha}\Xi.$
(8)

In Eq. (8) $g_{5}^{(\alpha)}(r)$ is the probability density of five-
dimentional stable distribution Uchaikin:1999a .

Fig. 3 shows the energy dependence of the index $\Xi$ of the electrons and
positrons observed spectrum for various diffusion modes (parameter $\alpha$).
It can be seen that the spectrum has a knee in the superdiffusion regime
$\alpha<2$ (exponent $\Xi$ at the knee point of $0.9$ TeV is equal to zero).
At the same time, in the normal diffusion regime, the spectral index $\Xi$
increases monotonically with increasing energy. In this case the spectrum has
no knee.

5mm

Figure 3: Change in the spectral index of the observed electrons on the energy
in the case of a point instant source for the different diffusion regimes.
$r=200$ pc

It should be noted that the break in the electron spectrum obtained in the
framework of the nonclassical diffusion model, as well as comparison with
experimental data, will be considered in our next work.

##  5 Approach to estimate the distance to the nearest pevatrons

It was shown in Lagutin:2001  that the knee may be due to anomalous CR
diffusion in a turbulent (fractal type) interstellar medium, but not the
presence of maximum cosmic ray energy in sources. Recently published data of
the LHAASO Collaboration on the spectrum of diffuse gamma radiation from the
disk of the Galaxy LHAASO:2023aug , which is described by a power-law function
with an exponent $-2.99$ in the entire energy region of 10 TeV - 1 PeV may be
considered as an indication on the validity of the hypothesis that the knee in
the $1-4$ PeV region is most likely due to transport.

In the framework of nonclassical CRs diffusion the knee is due to the presence
of a break in the stable distribution $g_{3}^{(\alpha)}(r)$ at the value of
the argument $r\approx 2.2$. On this basis the expressions (5) and (4) for the
spectra obtained in the nonclassical diffusion model make it possible to
establish a relationship between the characteristics $(\textbf{r},t,E)$ of the
knee points of the nuclear ($n$) and leptonic ($e$) components.

For $n$ component

$N(\textbf{r},t,E)\sim
g_{3}^{(\alpha)}\left(|\textbf{r}|(D(E,\alpha)t)^{-1/\alpha}\right)\Rightarrow
r_{n}(D_{0}(\alpha)E_{n}^{\delta}t_{n})^{-1/\alpha}=2.2.$

For $e$ component

$N(\textbf{r},t,E)\sim
g_{3}^{(\alpha)}\left(|\textbf{r}|\lambda(t,E)^{-1/\alpha}\right)\Rightarrow
r_{e}(D_{0}(\alpha)E_{e}^{\delta}\hat{\lambda}(t_{e},E_{e}))^{-1/\alpha}=2.2.$

We assume that CR nuclei and high-energy electrons are accelerated by the same
types of sources (pevatrons) and their propagation in an inhomogeneous
turbulent galactic medium is characterized by the same diffusion coefficient
$D_{0}(\alpha)$. Due to this assumption

$r_{n}(D_{0}(\alpha)E_{n}^{\delta}t_{n})^{-1/\alpha}=r_{e}(D_{0}(\alpha)E_{e}^{\delta}\hat{\lambda}(t_{e},E_{e}))^{-1/\alpha}.$
(9)

It follows from the Eq. (9) that

$r_{n}=r_{e}\left[\left(\dfrac{E_{n}}{E_{e}}\right)^{\delta}\dfrac{t_{n}}{\hat{\lambda}(t_{e},E_{e})}\right]^{1/\alpha}\equiv
r_{e}\xi.$ (10)

It should be noted that the estimates obtained within the framework of the
proposed approach are almost diffusion model independent.

##  6 Results

Parameters of the nonclassical diffusion model and the technology for their
self-consistent determination using the available experimental data on
galactic CRs were discussed in our previous papers Lagutin:2021b ;
Lagutin:2023 . The main parameters of the model are given in the Table 1.
Spatiotemporal characteristics nearest galactic CR sources, which include the
Geminga, Monogem, and Vela pulsars, are given in Lagutin:2009 .

Table 1: The nonclassical diffusion model parameters Parameter | Value
---|---
$p$ | $2.85$
$\delta$ | $0.27$
$D_{0}(\alpha)$ |  $10^{-3}$ pc1.7yr-1
$\alpha$ | $1.7$
$t_{e}$, $t_{n}$ |  $10^{5}$ yr
$E_{e}$ |  $0.9$ TeV
$E_{n}$ |  $0.65$ PeV
Table 2: Space-time parameters of the most likely candidates for pevatrons according to Kobayashi:2004 ; Lozinskaya:1992  Source |  $r$, pc |  $t,10^{5}$ yr
---|---|---
Monoceros | 600 | 0.46
Cyg. Loop | 770 | 0.20
CTB 13 | 600 | 0.32
S 149 | 700 | 0.43
STB 72 | 700 | 0.32
CTB 1 | 900 | 0.47
HB 21 | 800 | 0.23
HB 9 | 800 | 0.27

The ‘‘lifetime’’ of the CR electrons is described by the function
$\tau(E,E_{0})$ from Eq. (3). It follows from this that the TeV-energy
electrons observed on Earth were produced by sources $\sim 10^{5}$ years ago.
During this time, in the superdiffusion mode $\overline{r^{2}}\sim
2D(E,\alpha)t^{3-\alpha}$ Perri:2015 , diffusion radius of electrons
$r_{e}\sim 200$ pc.

To obtain estimates of the distance to the nearest pevatrons, we use the value
of the knee energy of the proton spectrum $E_{n}=650$ TeV that has been
obtained in Bartoli:2015 ; Lagutin:2017 . From Eq. (10) we obtain that
$r_{n}=3.75r_{e}$. Thus pevatrons, which form a knee in the spectrum of the
nuclear component of CRs of about $4$ PeV, are located at distances of the
order of $0.75$ kpc from the Earth.

The most likely candidates for pevatrons are shown in the Table 2.

##  7 Conclusions

An approach that makes it possible to estimate the distance to the nearest
pevatrons, which form a knee in the spectrum of the CRs nuclear component of
about $4$ PeV, has been formulated.

It is based on the CRs spectra of nuclei and leptons obtained by the authors
in the framework of the superdiffusion model of nonclassical diffusion, which
have a knee, on the assumption that nuclei and electrons are accelerated by
the same types of sources and their propagation in an inhomogeneous turbulent
galactic medium is characterized by the same diffusion coefficient and also on
the knee in the spectrum of the leptonic component in the region of $0.9$ TeV,
established in the DAMPE and CALET experiments.

It has been established that pevatrons, which form a knee in the spectrum of
the CRs nucleon component of about $4$ PeV, are located at distances of the
order of $0.75$ kpc from the Earth.

###### Acknowledgements.

The work is supported by the the Russian Science Foundation (grant no.
23-72-00057).

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