# Self-discharge by streaming cosmic rays

Yutaka Ohira Department of Earth and Planetary Science, The University of
Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan<EMAIL_ADDRESS>
###### Abstract

A new nonthermal phenomenon caused by streaming cosmic rays (CRs) in the
universe is proposed. The streaming CRs drive the return current of thermal
electrons to compensate for the CR current. Then, electric fields are induced
by the resistivity of the return current. It is shown that the resistive
electric fields can accelerate secondary electrons generated by the streaming
CRs. This is the self-discharge by streaming CRs. In this work, the self-
discharge condition and the condition for runaway acceleration of secondary
electrons are presented. The self-discharge makes high-energy secondary
electrons, resulting in enhancements of ionization and nonthermal emission
including K$\alpha$ emission line of neutral irons. After the self-discharge,
the return current of thermal electrons is replaced by the electric current of
secondary electrons. Since some magnetic field generations and amplifications
are driven by the return current of thermal electrons, the self-discharge can
significantly influence them.

Cosmic rays (329), Secondary cosmic rays (1438), Plasma astrophysics (1261),
Cosmic magnetic fields theory (321), Astrophysical magnetism (102), Ionization
(2068), Molecular clouds (1072)

##  1 Introduction

Thanks to many studies about cosmic-ray (CR) interactions with neutral gases
and plasmas, it is known that CRs play various roles in different
environments. Since the energy of CRs is much higher than the energy scale of
any astrophysical objects, CRs can ionize and heat gas in the Universe
(Hayakawa et al., 1961; Fujita et al., 2007; Fujita & Ohira, 2011). In
addition, the CR interactions generate light nuclei (e.g. Li, Be,
B)(Tatischeff & Gabici, 2018), gamma rays (Hayakawa, 1952), and x rays
(Tatischeff et al., 2012). Furthermore, the CR pressure is not negligible in
some environments, so that CRs affect the dynamical evolution of astrophysical
phenomena (Drury & Völk, 1981; Jubelgas et al., 2008). Since CRs consist
mainly of protons with a positive charge, streaming CRs make an electric
current, so that a return current of thermal electrons is driven to neutralize
the CR current. The return current of thermal electrons in nonuniform systems
is expected to generate seed magnetic fields in the early universe (Miniati &
Bell, 2011; Ohira, 2020, 2021). In the current universe, it is known that the
electron return current drives the nonresonant (Bell) instability (Bell,
2004), amplifying magnetic field fluctuations, which is expected to be
important for CR acceleration (Bell, 2004) and propagation (Blasi et al.,
2015).

Interactions of CRs with Earth’s atmosphere provide high-energy particles in
the inner Van Allen belt (Li et al., 2017). In addition, CRs generate high-
energy secondary electrons in Earth’s atmosphere. There are electric fields in
thunderclouds of Earth’s atmosphere, which is generated not by CRs, but by the
charge separation due to friction (Takahashi, 1978). Low-energy electrons
cannot be accelerated by the electric fields because the friction with the
atmosphere is large. On the other hand, high-energy secondary electrons can be
accelerated in thunderclouds and generate further secondary electrons,
resulting in electron avalanche (Dwyer et al., 2012). This discharge induced
by CRs in thunderclouds is expected to be the origin of some types of gamma-
ray flares from thunderclouds (Dwyer et al., 2012).

Streaming CRs in the universe induce resistive electric fields by the Coulomb
interaction of the return current of thermal electrons (Miniati & Bell, 2011;
Silsbee & Ivlev, 2020). In this work, we first investigate a condition that
secondary electrons generated by CRs are accelerated by the resistive electric
field induced by streaming CRs themselves, that is, the self-discharge
condition. This is a new discharge mechanism and a new acceleration mechanism
for secondary electrons in the universe. Furthermore, we show that the return
current of thermal electrons is replaced by the electric current of high-
energy secondary electrons after the self-discharge. Since the resistivity of
high-energy secondary electrons is smaller than that of low-energy electrons,
the resistive electric field becomes small after the self-discharge.
Therefore, the self-discharge inhibits the magnetic field generation and
amplification by the return current of thermal electrons. Moreover, the self-
discharge enhances ionization and nonthermal emission by secondary electrons.

##  2 Self-discharge condition

We first consider thermal electrons, protons, hydrogen atoms, and streaming
CRs to derive the discharge condition. In the proton rest frame, thermal
electrons have a drift velocity, $V_{\rm e}=(n_{\rm CR}/n_{\rm e})V_{\rm CR}$,
to satisfy the current neutrality condition, where $n_{\rm CR},n_{\rm e}$, and
$V_{\rm CR}$ are the CR density, electron density, and the streaming velocity
of CRs. The CR charge is assumed to be the positive elementary charge, $e$.
The return current of thermal electrons feels the resistivity due to the
Coulomb interaction or interaction with neutral hydrogens. For gases with the
electron temperature of $T_{\rm e}=10~{}{\rm K}$, the Coulomb resistivity is
dominant as long as the electron fraction is larger than $\sim 5\times
10^{-8}$. The time scale for the Coulomb scattering is

$t_{\rm C}=1.9\times 10^{-1}~{}{\rm sec}\left(\frac{n_{\rm e}}{1~{}{\rm
cm}^{-3}}\right)^{-1}\left(\frac{T_{\rm e}}{10~{}{\rm
K}}\right)^{\frac{3}{2}}.$ (1)

Then, the resistive electric field before discharge is given by Ohm’s law,

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle-\frac{m_{\rm e}n_{\rm
CR}V_{\rm CR}}{en_{\rm e}t_{\rm C}}$ $\displaystyle=$ $\displaystyle-9.5\times
10^{-14}~{}{\rm V~{}cm^{-1}}$ $\displaystyle\times\left(\frac{V_{\rm
CR}}{c}\right)\left(\frac{n_{\rm CR}}{10^{-9}~{}{\rm
cm}^{-3}}\right)\left(\frac{T_{\rm e}}{10~{}{\rm K}}\right)^{-\frac{3}{2}},$

where $m_{\rm e}$ is the electron mass. The Coulomb resistive electric field
induced by streaming CRs does not depend on the density of thermal particles
but depends on the CR current and the electron temperature.

The typical energy of secondary electrons generated by the CR interaction with
hydrogen atoms is $\varepsilon\sim 10~{}{\rm eV}$ (Rudd et al., 1992). The
secondary electrons lose their energy due to the excitation, ionization, and
Coulomb losses. If the energy gain from the electric field, $E$, is larger
than the energy loss, the secondary electrons can be accelerated by the
electric field, resulting in the discharge. Therefore, the discharge condition
is given by

$\frac{{\rm d}\varepsilon}{{\rm d}t}=-eEv+\dot{\varepsilon}_{\rm loss}>0,$ (3)

where $v$ is the velocity of secondary electrons and $\dot{\varepsilon}_{\rm
loss}$ is the energy loss rate. If the discharge condition is satisfied at
$\varepsilon\sim 10~{}{\rm eV}$, almost all secondary electrons are
accelerated.

Figure 1: Energy gain and loss rates for electrons. The dashed and dotted
lines show the loss rates, $|\dot{\varepsilon}_{\rm loss}|$, for the
excitation and ionization processes in $n_{\rm H}=1~{}{\rm cm}^{-3}$ Dalgarno
et al. (1999), and for the Coulomb interaction in $n_{\rm e}=3\times
10^{-3}~{}{\rm cm}^{-3}$ Swartz et al. (1971), respectively. The thin and bold
solid lines show the energy gain by the resistive electric field, $-eEv$, for
cases below and above the runaway acceleration condition, Equation (6),
respectively.

Figure 1 shows energy gain and loss rates as a function of the electron energy
(Dalgarno et al., 1999). The dashed and dotted lines show loss rates for the
excitation and ionization processes in $n_{\rm H}=1~{}{\rm cm}^{-3}$, and for
the Coulomb interaction in $n_{\rm e}=3\times 10^{-3}~{}{\rm cm}^{-3}$. These
energy loss rates linearly depend on hydrogen and electron densities. For an
electron fraction of $n_{\rm e}/n_{\rm H}\leq 8.3\times 10^{-4}$, excitation
and ionization are the dominant loss process for electrons of $\varepsilon\geq
10~{}{\rm eV}$. Then, from Equation (2) and the energy loss rates at
$\varepsilon=10~{}{\rm eV}$ of $\dot{\varepsilon}_{\rm loss}=-6\times
10^{-8}~{}{\rm eV~{}s^{-1}}(n_{\rm H}/1~{}{\rm cm}^{-3})$, the self-discharge
condition (3) is reduced to the following inequality for the CR flux,

$n_{\rm CR}V_{\rm CR}>3.2\times 10^{-1}~{}{\rm
cm^{-2}~{}s^{-1}}\left(\frac{T_{\rm e}}{10~{}{\rm
K}}\right)^{\frac{3}{2}}\left(\frac{n_{\rm H}}{1~{}{\rm cm}^{-3}}\right).$ (4)

For a lager electron fraction ($n_{\rm e}/n_{\rm H}\geq 8.3\times 10^{-4}$),
the secondary electrons of $\varepsilon\sim 10~{}{\rm eV}$ lose their energy
by the Coulomb loss. Then, the self-discharge condition is reduced to

$n_{\rm CR}V_{\rm CR}>3.9\times 10^{2}~{}{\rm
cm^{-2}~{}s^{-1}}\left(\frac{T_{\rm e}}{10~{}{\rm
K}}\right)^{\frac{3}{2}}\left(\frac{n_{\rm e}}{1~{}{\rm cm}^{-3}}\right),$ (5)

where $\dot{\varepsilon}_{\rm loss}=-2\times 10^{-4}~{}{\rm
eV~{}s^{-1}}(n_{\rm e}/1~{}{\rm cm}^{-3})(\varepsilon/1~{}{\rm eV})^{-0.44}$
is used for the Coulomb loss (Swartz et al., 1971). As one can see, the self-
discharge condition is satisfied more easily in gases with lower temperatures
and lower densities because the lower temperature makes the resistive electric
field larger and the lower density makes the energy loss of the secondary
electrons smaller.

##  3 Acceleration of secondary electrons

Once the self-discharge occurs, secondary electrons are accelerated by the
electric field. If the electric field is not sufficiently large (see the thin
solid line in Figure 1), the acceleration of secondary electrons stops at
several $10~{}{\rm eV}$ because the energy loss due to the ionization and
excitation becomes larger than the energy gain by the electric field. However,
for larger electric fields, that is, larger CR fluxes (see the bold solid line
in Figure 1), secondary electrons continue to be accelerated until the
electric field disappears. The runaway acceleration condition is

$n_{\rm CR}V_{\rm CR}>2.4~{}{\rm cm^{-2}~{}s^{-1}}\left(\frac{T_{\rm
e}}{10~{}{\rm K}}\right)^{\frac{3}{2}}\left(\frac{n_{\rm H}}{1~{}{\rm
cm}^{-3}}\right).$ (6)

If both the discharge and runaway acceleration conditions are satisfied, the
secondary electrons could be accelerated to very high energies. However, if
the return current of the secondary electrons compensates for the CR current,
the return current of thermal electrons and the resistive electric field
become small. To estimate the maximum energy of secondary electrons, we have
to understand the evolution of the electric field. In this work, we consider a
simple one-zone system as the first step. The time evolution of the number
density of secondary electrons is given by

$\displaystyle n_{\rm 2nd}$ $\displaystyle=$ $\displaystyle n_{\rm
CR}\left(\frac{t}{t_{\rm ion}}\right)+n_{\rm 2nd,0}~{}~{},$ (7) $\displaystyle
t_{\rm ion}$ $\displaystyle=$ $\displaystyle 2\times 10^{8}~{}{\rm
sec}~{}\left(\frac{n_{\rm H}}{1~{}{\rm cm}^{-3}}\right)^{-1},$ (8)

where $n_{\rm 2nd,0}$ is the density of background secondary electrons that
are generated by isotropic background Galactic CRs and $n_{\rm 2nd,0}\sim
10^{-10}-10^{-8}\ {\rm cm}^{-3}$ tipically (Ivlev et al., 2021), and $t_{\rm
ion}$ is the ionization time scale of CRs with the energy of $1~{}{\rm GeV}$.
The equation of motion for the secondary electron fluid is

$m_{\rm e}\frac{{\rm d}}{{\rm d}t}(n_{\rm 2nd}V_{\rm 2nd})=-eEn_{\rm 2nd},$
(9)

where $V_{\rm 2nd}$ is the mean velocity of secondary electrons and the
resistive force is ignored here because the runaway acceleration condition is
now satisfied. From the current neutrality condition, $n_{\rm CR}V_{\rm
CR}=n_{\rm e}V_{\rm e}+n_{\rm 2nd}V_{\rm 2nd}$, the equation of motion for the
thermal electron fluid is given by

$m_{\rm e}\frac{{\rm d}}{{\rm d}t}(n_{\rm e}V_{\rm e})=-m_{\rm e}\frac{{\rm
d}}{{\rm d}t}(n_{\rm 2nd}V_{\rm 2nd}).$ (10)

From Equations (7), (9), (10), and Ohm’s law, $E=-m_{\rm e}V_{\rm e}/et_{\rm
C}$, we can obtain the following equation for the electric field,

$\frac{{\rm d}E}{{\rm d}t}=-\frac{n_{\rm CR}t+n_{\rm 2nd,0}t_{\rm ion}}{n_{\rm
e}t_{\rm ion}t_{\rm C}}E.$ (11)

The analytical solution is given by

$E=E_{0}\exp\left\\{-\left(\frac{t}{t_{\rm d,1}}\right)-\left(\frac{t}{t_{\rm
d,2}}\right)^{2}\right\\},$ (12)

where $E_{0}$ is the initial electric field given by Equation (2), and the
characteristic decay times, $t_{\rm d,1}$ and $t_{\rm d,2}$, are defined by

$\displaystyle t_{\rm d,1}$ $\displaystyle=$ $\displaystyle\frac{n_{\rm
e}}{n_{\rm 2nd,0}}t_{\rm C}$ $\displaystyle=$ $\displaystyle 1.9\times
10^{8}~{}{\rm sec}\left(\frac{n_{\rm 2nd,0}}{10^{-9}~{}{\rm
cm}^{-3}}\right)^{-1}\left(\frac{T_{\rm e}}{10~{}{\rm
K}}\right)^{\frac{3}{2}},$ $\displaystyle t_{\rm d,2}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2n_{\rm e}t_{\rm ion}t_{\rm C}}{n_{\rm CR}}}$
$\displaystyle=$ $\displaystyle 2.8\times 10^{8}~{}{\rm sec}\left(\frac{n_{\rm
CR}}{10^{-9}~{}{\rm cm}^{-3}}\right)^{-\frac{1}{2}}\left(\frac{n_{\rm
H}}{1~{}{\rm cm}^{-3}}\right)^{-\frac{1}{2}}\left(\frac{T_{\rm e}}{10~{}{\rm
K}}\right)^{\frac{3}{4}}.$

After the decay time ($t>\min(t_{\rm d,1},t_{\rm d,2})$), the electric current
of secondary electrons becomes sufficiently large to compensate for the CR
current. Then, to satisfy the current neutrality condition, the return current
of thermal electrons becomes small, so that the resistive electric field
becomes small.

The maximum energy of accelerated secondary electrons can be estimated by
solving the equation of motion for the first secondary electron, ${\rm
d}p/{\rm d}t=-eE(t)$. From Equation (12), the maximum momentum is given by

$p_{\rm max}=p_{0}-\frac{\sqrt{\pi}}{2}eE_{0}t_{\rm d,2}{\rm erfcx}(t_{\rm
d,2}/2t_{\rm d,1}),$ (15)

where $p_{0}$ is the initial momentum of secondary electrons, and ${\rm
erfcx}(x)=(2/\sqrt{\pi})e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}dt$ is the scaled
complementary error function.

In dense molecular clouds, many secondary electrons are generated by streaming
CRs before background secondary electrons are accelerated by the resistive
electric field. Then, the effect of the background secondary electrons is
negligible, resulting in $t_{\rm d,1}>t_{\rm d,2}$. In this case, the maximum
momentum is given by

$\displaystyle p_{\rm max}$ $\displaystyle\approx$ $\displaystyle
p_{0}-\frac{\sqrt{\pi}}{2}eE_{0}t_{\rm d,2}$ (16) $\displaystyle\approx$
$\displaystyle m_{\rm e}V_{\rm CR}\sqrt{\frac{\pi n_{\rm CR}t_{\rm
ion}}{2n_{\rm e}t_{\rm C}}}$ $\displaystyle\approx$ $\displaystyle
1.3~{}m_{\rm e}c\left(\frac{V_{\rm CR}}{c}\right)\left(\frac{n_{\rm CR}/n_{\rm
H}}{10^{-9}}\right)^{\frac{1}{2}}\left(\frac{T_{\rm e}}{10~{}{\rm
K}}\right)^{-\frac{3}{4}}.$

If the background CR density is large, the number density of background
secondary electrons becomes large. Moreover, the resistive electric field is
stronger in lower temperature regions. Then, the electric current of
background secondary electrons quickly cancels the CR current before new
secondary electrons are generated by streaming CRs, resulting in $t_{\rm
d,1}<t_{\rm d,2}$. In this case, the maximum momentum is given by

$\displaystyle p_{\rm max}$ $\displaystyle\approx$ $\displaystyle
p_{0}-eE_{0}t_{\rm d,1}$ (17) $\displaystyle\approx$ $\displaystyle m_{\rm
e}c\left(\frac{V_{\rm CR}}{c}\right)\left(\frac{n_{\rm CR}}{n_{\rm
2nd,0}}\right).$

In the following, we consider two specific environments in the universe as
examples.

##  4 Application to molecular clouds in the vicinity of supernova remnants in
the current universe

In the current universe, supernova remnants (SNRs) are believed to inject
about $U_{\rm CR}=10^{50}~{}{\rm erg}$ in CRs. Since the size of SNRs, $R_{\rm
SNR}$, is typically from a few pc to 10 pc, the number density of GeV CRs is
expected to be $n_{\rm CR}\sim 4.4\times 10^{-6}~{}{\rm cm}^{-3}(U_{\rm
CR}/10^{50}~{}{\rm erg})(R_{\rm SNR}/5~{}{\rm pc})^{-3}$ around SNRs. Gamma-
ray observations show that CRs interact with molecular clouds in the vicinity
of SNRs (Ohira et al., 2011; Ackermann et al., 2013), supporting $U_{\rm
CR}=10^{50}~{}{\rm erg}$. For a molecular cloud with $T_{\rm e}=10~{}{\rm
K},n_{\rm H}=10^{3}~{}{\rm cm}^{-3},n_{\rm e}=1~{}{\rm cm}^{-3}$, the
discharge (Equation (5)) and runaway acceleration (Equation (6)) conditions
are satisfied when the drift velocity of CRs, $V_{\rm CR}$, is larger than
$3\times 10^{-3}~{}c$ and $1.8\times 10^{-2}~{}c$, respectively. The drift
velocity of CRs is at least on the order of the shock velocity of SNRs, which
is $10^{-3}~{}c$ for middle-aged SNRs and $10^{-2}~{}c$ for young SNRs.
Therefore, both conditions are expected to be satisfied marginally. Escaping
CRs have the drift velocity faster than the shock velocity of SNRs, although
the CR density becomes smaller than the above estimate. The dilution of CR
density depends on the diffusion coefficient around SNRs and the shape of
magnetic field lines, which are open problems and actively investigated. If
any signatures of the self-discharge are observed, we can obtain information
about the CR diffusion.

For the above parameter set, $t_{\rm d,1}$ is larger than $t_{\rm d,2}$. From
Equation (16), the maximum momentum of runaway secondary electrons becomes
$p_{\rm max}=2.7~{}m_{\rm e}c(V_{\rm CR}/c)$. Therefore, the accelerated
secondary electrons due to the self-discharge could enhance MeV gamma-ray and
6.4 keV Fe line productions, and the ionization rate in molecular clouds.

##  5 Application to the first CRs in the early universe

According to our recent study, the first CRs are accelerated by SNRs of the
first stars at the redshift of $z\approx 20$ (Ohira & Murase, 2019). The mean
number density of the first CRs is estimated to be $n_{\rm CR}=3\times
10^{-14}~{}{\rm cm}^{-3}$ at $z\approx 20$. The drift velocity is almost the
speed of light because there are no or very weak magnetic fields in the early
universe, so that the mean CR flux is $10^{-3}~{}{\rm cm^{-2}~{}s^{-1}}$. On
the other hand, the hydrogen and electron number densities and temperature are
$n_{\rm H}\sim 10^{-3}~{}{\rm cm}^{-3},n_{\rm e}\sim 10^{-7}~{}{\rm cm}^{-3}$
and $T_{\rm e}\sim 1~{}{\rm K}$ at $z\approx 20$. Therefore, both discharge
(Equation (4)) and runaway acceleration (Equation (6)) conditions are
satisfied. Since there are no background isotropic CRs in the early universe,
from Equation (16) ($t_{\rm d,1}>t_{\rm d,2}$), the maximum momentum of
accelerated secondary electrons becomes $p_{\rm max}=1.7~{}m_{\rm e}c(V_{\rm
CR}/c)$. The accelerated secondary electrons could heat and ionize gases. In
addition, the return current of thermal electrons quickly decays in the time
scale of $t_{\rm d,2}\sim 6.7~{}{\rm kyr}$. Therefore, some generation
mechanisms of magnetic fields by the return current of thermal electrons
(Miniati & Bell, 2011; Ohira, 2020) can work not for the cosmological
timescale, but only for $t_{\rm d}$. However, inhomogeneity of the return
current of secondary electrons can generate magnetic fields (Ohira, 2020).
Moreover, the Biermann battery driven by the streaming CRs (Ohira, 2021) can
work for the cosmological timescale by the gradient of the electron pressure,
which is generated during the decay time, $t_{\rm d}$. Either way, the self-
discharge significantly influences the magnetic field generations by streaming
CRs.

##  6 Discussion

We have compared the energy loss and acceleration by the resistive electric
field at the energy of $10~{}{\rm eV}$, which is the typical energy of
secondary electrons, to understand whether the self-discharge occurs or not.
However, secondary electrons with higher energies are sometimes generated by
CRs, where the production rate is proportional to $\varepsilon^{-1}$ (Rudd et
al., 1992). The energy loss rate becomes small for higher energy secondary
electrons. Hence, the self-discharge condition can be satisfied even though
the CR flux is below the critical values given by Equations (4) and (5), but
the event rate of the discharge becomes smaller. It should be noted that
secondary electrons can be generated in fully ionized plasmas by the Coulomb
scattering and nuclear interactions. In particular, the production rate of the
secondary electrons by the Coulomb scattering is almost the same as that of
ionization (Abraham et al., 1966). Therefore, the self-discharge by streaming
CRs can occur in various regions.

In Section 3, we have investigated the evolution of the resistive electric
field and the return current of thermal electrons when the runaway
acceleration condition is satisfied. Even though it is not satisfied, as long
as the discharge condition is satisfied, secondary electrons with energies of
several 10 eV continue to be generated until the electric current of secondary
electrons compensates for the CR current. In such a case, the number of the
secondary electrons would be much larger than that of CRs, so that a large
ionization rate would be expected. In the energy scale of several 10 eV, we
need Monte Carlo simulations because the continuous slowing-down approximation
is not applicable for electrons (Diniz et al., 2018). Furthermore, we have
applied the one-zone system as the first step. In order to accurately evaluate
the ionization rate and high energy nonthermal emissions including 6.4 keV Fe
line from accelerated secondary electrons, we have to perform more rigorous
calculations that consider the spectrum of secondary electrons, stochastic
behaviors, and spatial structures, which will be addressed in future work.

Furthermore, in Section 3, we implicitly assumed that the CR current suddenly
increases, but in reality it takes a finite injection time, $t_{\rm inj}$. If
the injection time is shorter than the decay time of the electric field, our
implicit assumption is valid. However, if the injection time becomes longer
than the decay time ($t_{\rm inj}>\min(t_{\rm d,1},t_{\rm d,2})$), before the
CR flux reaches the critical value for the runaway acceleration, the resistive
electric field is reduced by acceleration of secondary electrons. Then, the
resistive electric field cannot reach the critical value for the runaway
acceleration even though the CR flux reaches the critical flux given by
Equation (6). In this case, the runaway acceleration does not occur but many
secondary electrons with energies of several 10 eV are generated. For
molecular clouds in the vicinity of SNRs, the decay time of electric field is
$t_{\rm d,2}\sim 10^{5}~{}{\rm sec}$ for the parameter set in Section 4. The
minimum timescale of the CR injection is the acceleration time of GeV CRs,
which is $t_{\rm acc}\sim 10^{4}~{}{\rm sec}(V_{\rm sh}/3000~{}{\rm
km~{}s^{-1}})^{-2}(B/100~{}{\rm\mu G})^{-1}$ for acceleration at parallel
shocks (Drury, 1983), and $t_{\rm acc}\sim 10^{4}~{}{\rm sec}(V_{\rm
sh}/3000~{}{\rm km~{}s^{-1}})^{-1}(B/1~{}{\rm\mu G})^{-1}$ for acceleration at
perpendicular shocks (Jokipii, 1987; Kamijima et al., 2020). $V_{\rm sh}$ and
$B$ are the shock velocity and magnetic field strength, respectively. On the
other hand, the longest timescale of the CR injection is the diffusion time,
$t_{\rm diff}=L^{2}/D=10^{10}~{}{\rm sec}(L/5~{}{\rm pc})^{2}(D/10^{28}~{}{\rm
cm^{2}~{}s^{-1}})^{-1}$, where $L$ and $D$ are the distance from the CR source
and the diffusion coefficient of CRs. If the CR injection time is comparable
to the acceleration time, $t_{\rm acc}$, the runaway acceleration can occur in
the molecular clouds, but cannot occur for $t_{\rm inj}\sim t_{\rm diff}$. In
the early universe, the decay time of the resistive electric field is $t_{\rm
d,2}\sim 10^{11}~{}{\rm sec}$ (see Section 5), which is larger than the
acceleration and propagation times of CRs. Therefore, the runaway acceleration
can occur in the early universe. To understand more accurately whether the
runway acceleration occurs or not, we need to study the self-discharge process
more specifically by providing specific density structures. It should be noted
that the above argument is based on the simple one-zone system that we
considered in Section 3. In reality, the gas density is not uniform,
especially in the edge of molecular clouds. We should investigate the self-
discharge process in the nonuniform system in the future.

We have shown that the return current of thermal electrons is replaced by the
current of secondary electrons once the self-discharge occurs. The return
current of thermal electrons drives the nonresonant (Bell) instability in weak
magnetic field (Bell, 2004), which amplifies magnetic field fluctuations.
There are many studies about the Bell instability (Reville et al., 2008;
Niemiec et al., 2008; Riquelme & Spitkovsky, 2009; Ohira et al., 2009; Marret
et al., 2015) because the Bell instability is expected to have an important
role on CR acceleration and propagation. In principle, the current of
secondary electrons can drive the nonresonant instability, but the growth rate
and the nonlinear evolution would be modified. Furthermore, secondary
electrons would be scattered by the magnetic field fluctuations amplified by
the Bell instability, which would enhance the anomalous resistivity of the
secondary electrons and the resistive electric field. Therefore, the self-
discharge would also be influenced by the Bell instability.

##  7 Summary

We have shown that secondary electrons generated by streaming CRs can be
accelerated by the resistive electric field that is induced by the streaming
CRs themselves, that is, the self-discharge is driven by the streaming CRs.
The self-discharge and runaway acceleration conditions have been derived.
After the self-discharge, the return current of thermal electrons is replaced
by the electric current of the secondary electrons. Then, the resistive
electric field decays because the resistivity of the secondary electrons is
much smaller than that of thermal electrons. Since some magnetic field
generations and amplifications are driven by the return current of thermal
electrons, the self-discharge significantly influences them. If the runaway
acceleration condition is satisfied, the secondary electrons can be
accelerated to mildly relativistic energies. As a result of the self-
discharge, the ionization, nonthermal emissions, and 6.4 keV line of neutral
Fe are enhanced by the accelerated secondary electrons.

We are grateful to the referee for valuable comments that improved the paper.
We also thank Diniz Gabriel, Teruaki Enoto, Kazuhiro Nakazawa, and Yuuki Wada
for informative discussions on discharge in thunderclouds. This work is
supported by JSPS KAKENHI Grant Number JP19H01893, and by Leading Initiative
for Excellent Young Researchers, MEXT, Japan.

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#  A Secure and Smart Framework for Preventing Ransomware Attack

Jaspreet Kaur  PhD Scholar at CSE Department
Indian Institute of Technology Jodhpur
Jodhpur, India
<EMAIL_ADDRESS>
###### Abstract

Nowadays security is major concern for any user connected to the internet.
Various types of attacks are to be performed by intruders to obtaining user
information as man-in-middle attack, denial of service, malware attacks etc.
Malware attacks specifically ransomware attack become very famous recently.
Ransomware attack threaten the users by encrypting their most valuable data,
lock the user screen, play some random videos and by various more means.
Finally attacker take benefits by users through paid ransom. In this paper, we
propose a framework which prevent the ransomware attack more appropriately
using various techniques as blockchain, honeypot, cloud & edge computing. This
framework is analysed mainly through the IoT devices and generalized to the
any malware attack.

###### Index Terms:

Ransomware attack, Malware attack, Blockchain, IoT Device, Cloud & Edge
Computing, Honeypot.

##  I Introduction

There are lots of malware attack happen today but ransomware is one of the
most dangerous one that threaten its victim. There are various types of
ransomware as Lock-type ransomware, Crypto-ransomware etc. It usually install
by as some another malware through malicious e-mail attachments or links,
infected software applications or websites and via some connected external
storage systems. Then check the vulnerabilities of the system by which make a
connection to the C & C server and encrypt users important data or lock the
screen of the users etc. Finally attackers demand the payment as
cryptocurrency or threaten the users till they are not satisfied and delete or
hide their traces so that no one can easily track them.

There are various approaches mentioned in literature for detecting and
preventing this attack as signature based and anomaly based techniques which
takes various parameters as input- open ports, various system calls, registry
editing, accessing of files and folders, file entropy changes etc. and make
prediction of attack. Some of them use sandboxing methods and use cloud
storage for backup purpose. But all of them have some weaknesses as need large
storage for signatures, not predicting zero day attacks, high false alarm
rate, new advanced ransomware evading from sandbox by sleep mode at earlier
time, only for the user mode but what if- when root is compromised, training
database or machine learning or deep learning algorithms are compromised.
These above limitations motivate us to develop a new framework which can
efficiently detect and mitigate this attack.

##  II Proposed Framework

In our proposed method (figure 1), we take a example of a smart home
environment in which various smart heterogeneous (both resource constrained
and high resource power) IoT devices are to be vulnerable from this attack. We
create a database using blockchain smart contract in which authentication,
access control of IoT devices are to be stored for removing the risk of
external device attack. This smart contract also contains the valuable system
call functions as registry editing functions, root access functions,
cryptographic encryption functions, port assessing functions, desktop
controlling functions, file read or write functions, folder deleting functions
etc. These calls are included in the basic steps of ransomware attack, when
intruder performs sequence of operations as fingerprinting, port scanning,
connection to the C & C server, accessing of important data, encrypting the
important documents, lock the user screen, demand for the payment and delete
the backup data along with traces of intruders. Some of the calls same as the
normal operations and others are not but it makes difference when executed in
sequence of ransomware attack.

But due to the complex nature of blockchain operations (mining and storage),
it cannot be stored at every IoT devices specifically resource constrained
devices. That’s why we use edge computing here, means take high power IoT
devices at edges which compute all blockchain operations and data analysis.
For security reason key management and generation are to be done at particular
device itself.

Figure 1: Proposed Approach- A Secure and Smart Framework for Preventing
Ransomware Attack

We also take some honeypot devices (very high power devices) for proper
evaluation of this attack before that attack is performed on the actual
device. For backup data of these IoT devices, we use cloud architecture which
is also decentralized, distributed in nature mean managed by blockchain. Both
of the blockchain managed by the load balancer devices or software for
handling incoming request messages. For additional security, the smart home
gateway includes the firewall for filtering malicious ports, IP addresses,
file extensions etc. to preventing ransomware attack.

##  III Working Process

Any ransomware or software come to the smart home infrastructure either via
network or through physical devices. For network attack, it first go through
the gateway that include first line of defence as firewall which filter
malicious ports, IP addresses, file extensions etc. But advanced malware can
easily by pass it. Following steps are same for both (device and network
attack). This software or malware run at 2 environment simultaneously-first
one at that device which they want to access and second one on the honeypot
devices. Blockchain smart contract check the various sequences of system call
run by that software and also check the authenticity and access control of
that external device. If that software check for searching of a file,
read/write of files, need admin privilege, edit log and registry events etc.
Then this software behaviour is same as normal user behaviour. But when it
wants to access network information as open ports, open sessions, read the
user’s browsing history and bookmarks and read kernel level information then
that smart contract alert the user because it may be the malware and require
the user permission for execute it. After some time, if that software call any
encryption function, call to make a new network connection using tor browser,
call to disable key guard and need the control over the desktop, then our
blockchain stop this software processing at all devices except the honeypot
nodes before that functions are executed and wait till software execution end
up at that honeypot devices, if that software demand the ransom then it is a
ransomware attack. Uninstall that software at all the devices and check
honeypot devices for some new rules given as feedback to the addition of
blockchain rules for more efficient detection of attacks. For secure backup
data of these IoT devices, we use blockchain cloud infrastructure with load
balancer (schedulers) and various workstations. All blockchain operations are
to be maintained at cloud but key management are done at devices themselves
for providing end point security.

##  IV Theoretical Analysis

We perform the theoretical analysis for validation of our work. Some of the
observations are as:

  1. 1.

By the use of ransomware blockchain, attacker cannot compromised our data
analysis algorithms or methods.

  2. 2.

Zero day attacks are prevented by the honeypot devices and new attacks rules
are to be added at blockchain for more efficient detection of attacks.

  3. 3.

All IoT devices authenticates by the blockchain so that no external device
attack in the network.

  4. 4.

Basic firewall rules are applied at gateway for novice intruder.

  5. 5.

For backup data securely, cloud blockchain is used.

  6. 6.

For end point security, key management are to be done at devices themselves
for both of the blockchain.

  7. 7.

For complex operations of blockchain, edge and cloud infrastructure are to be
used along with load balancer (hardware or software) for handling incoming
requests.

##  V CONCLUSION AND FUTURE WORK

Nowadays ransomware attack is one of the most dangerous attack. It can very
easily spread to any vulnerable system or device and demand the ransom from
users. Various techniques are available in literature for mitigating this
attack but every solution has problem as handling zero day attack, high false
alarm rates, security of machine learning data and methods etc. So, for
overcoming to these limitations we propose a new smart and secure method which
includes blockchain, honeypot, edge or cloud computing techniques to IoT
devices for preventing this attack. Our method is generalized means can be
extended to any malware attack detection. In future work, firstly we implement
our methodology on real world scenario and do changes in it as needed.

## References

  * [1] Al-rimy, Bander Ali Saleh, Mohd Aizaini Maarof, and Syed Zainudeen Mohd Shaid. ”Ransomware threat success factors, taxonomy, and countermeasures: A survey and research directions.” Computers & Security 74 (2018): 144-166.