# Self-modulation of cosmic rays in molecular clouds: Imprints in the radio
observations

V. A. Dogiel  I. E. Tamm Theoretical Physics Division of P. N. Lebedev
Institute of Physics, 119991 Moscow, Russia  D. O. Chernyshov  I. E. Tamm
Theoretical Physics Division of P. N. Lebedev Institute of Physics, 119991
Moscow, Russia  A. V. Ivlev  Max-Planck-Institut für extraterrestrische
Physik, 85748 Garching, Germany  A. M. Kiselev  I. E. Tamm Theoretical Physics
Division of P. N. Lebedev Institute of Physics, 119991 Moscow, Russia  A. V.
Kopyev  I. E. Tamm Theoretical Physics Division of P. N. Lebedev Institute of
Physics, 119991 Moscow, Russia A. V. Ivlev<EMAIL_ADDRESS>
###### Abstract

We analyze properties of non-thermal radio emission from the Central Molecular
Zone (CMZ) and individual molecular clouds, and argue that the observed
features can be interpreted in the framework of our recent theory of self-
modulation of cosmic rays (CRs) penetrating dense molecular regions. For
clouds with gas column densities of $\sim 10^{23}$ cm-2, the theory predicts
depletion of sub-GeV CR electrons, occurring due to self-modulation of CR
protons and leading to harder synchrotron spectra in the sub-GHz range. The
predicted imprints of electron depletion in the synchrotron spectra agree well
with the spectral hardening seen in available radio observations of the CMZ. A
similar, but even stronger effect on the synchrotron emission is predicted for
individual (denser) CMZ clouds, such as the Sgr B2. However, the emission at
frequencies above $\sim$ GHz, where observational data are available, is
completely dominated by the thermal component, and therefore new observations
at lower frequencies are needed to verify the predictions.

cosmic rays – turbulence – radio continuum: ISM – radiation mechanisms: non-
thermal – Galaxy: center

##  1 Introduction

Ivlev et al. (2018) and Dogiel et al. (2018) developed a self-consistent model
of cosmic ray (CR) interactions with molecular clouds, where CR penetration
into the clouds is governed by the self-generated MHD turbulence. The
resulting modulation of CR flux occurs in the diffuse envelopes around the
clouds, before CRs penetrate into denser regions. Inside the dense regions
MHD-fluctuations are damped by strong ion-neutral friction and particle
propagate ballistically along the local magnetic field lines.

Both the CR spectrum and spectrum of the turbulence obey the excitation-
damping balance, where the growth rate of magnetic fluctuations induced by CR
flux in the envelope is compensated by damping of MHD waves due ion-neutral
collisions (Ivlev et al., 2018). As a result, the CR spectrum inside the cloud
develops a break at a certain energy $E_{\rm ex}(N_{{\rm H}_{2}})$, which is
an increasing function of the cloud column density $N_{{\rm H}_{2}}$ (and
whose value is determined by the shape of the interstellar spectrum). While
the CR spectrum above $E_{\rm ex}$ remains unchanged, at lower energies the
self-modulation leads to significant depletion of CR density, and can generate
a universal (i.e., independent of the interstellar spectrum) flux of CRs
entering the clouds.

The CR depletion is expected to affect the gamma-ray emission due to proton-
proton collisions, produced in dense molecular clouds of Central Molecular
Zone (CMZ) as well as in molecular clouds in the vicinity of the Solar System
(see de Boer et al., 2017; Dogiel et al., 2018; Tibaldo et al., 2021).
Unfortunately, the value of $E_{\rm ex}$ turns out to be relatively small for
the parameters of such clouds, typically varying between $0.3-3$ GeV (Dogiel
et al., 2018). As a result, the gamma-ray emission may only be affected at
sub-GeV energies, where the resolution of Fermi-LAT becomes poor (Dogiel et
al., 2018).

The MHD disturbances produced by dominant CR species, protons and nuclei,
naturally also affect penetrating CR electrons. The main difference between
elections and hadrons of given energy is that electrons are able to produce
gamma-ray emission at a higher energy, since bremsstrahlung photons take away
a significant fraction of electron’s energy. Therefore, if the electron
bremsstrahlung was the dominant source of gamma-ray emission from a cloud, the
effect of CR depletion would be seen at higher emission energies and, thus,
would be likely detectable. Indeed, Yusef-Zadeh et al. (2013) pointed out that
diffuse gamma-ray emission from the Galactic center at $\sim$ GeV energies may
be due to electron bremsstrahlung, as derived from a measured non-thermal
spectrum of radio emission (see Figure 12 of Yusef-Zadeh et al., 2013). On the
other hand, it is generally believed that gamma-ray emission at such energies
must be dominated by proton-proton collisions (see, e.g., Owen et al., 2021,
and references therein).

This suggests that it may be difficult to find the effect of CR depletion in
the gamma-ray range, and therefore it is worth shifting to lower emission
energies. Indeed, features of the synchrotron emission from CR electrons
modulated at $\sim$ GeV energies should be observable at radio frequencies in
$\sim$ GHz range. Due to a very high resolution of radio telescopes, it is
possible to observe individual clouds and, thus, to disentangle cloud emission
from the background. The aim of this paper is to describe and quantify
fingerprints of CR self-modulation that are expected in radio observations of
molecular gas.

The paper is organized as follows. In Section 2 we discuss available radio
observations from dense molecular clouds, in Section 3 we present a concise
summary of the CR self-modulation theory and derive the expression for a
modulated spectrum of CR electrons inside the clouds, in Section 4 we obtain
the expected synchrotron emission from the CMZ and compare the theoretical
results with observations, and in Section 5 we summarize our conclusions.

##  2 Observed radio emission from the CMZ region and from individual
molecular clouds

One of the most promising targets for studies of the CR modulation is the CMZ
– a region located near the Galactic center, with dimensions about 500 pc
$\times$ 200 pc in the Galactic plane and 30 pc in height (Ferrière et al.,
2007). The gas is mostly concentrated in cold molecular clouds (see, e.g.,
Launhardt et al., 2002; Ferrière et al., 2007; Ginsburg et al., 2016; Mills et
al., 2018) with the volume filling factor of $\sim 10\%$ and the average
volume density of $\sim 10^{4}$ cm-3 (Mills, 2017; Mills et al., 2018). The
diffuse medium surrounding the clouds has an average gas density of $\lesssim
50$ cm-3 (Oka et al., 2005, 2019; Riquelme et al., 2018). Thus, the average
column density of the CMZ, $N_{{\rm H}_{2}}^{\rm CMZ}$, is dominated by the
contribution of dense clouds. Its value can be estimated as a product of the
gas density averaged over the CMZ volume, $\sim 10^{3}$ cm-3, multiplied with
the CMZ height, which yields $N_{{\rm H}_{2}}^{\rm CMZ}\sim 10^{23}$ cm-2. The
mean magnetic field strength in the Galactic center is estimated as $B\sim
0.1$ mG (Crocker et al, 2010), while in the clouds it can reach $B\sim 1$ mG
(Ferrière, 2009).

Figure 1: Diffuse non-thermal radio emission measured by Yusef-Zadeh et al.
(2013) from the inner $2^{\circ}\times 1^{\circ}$ region around the Galactic
center (bullets, statistical error bars are smaller than the symbols). The
solid line shows the spectrum of synchrotron emission predicted by our model
(see Section 4).

Yusef-Zadeh et al. (2013) found a strong spatial correlation between
distributions of nonthermal radio sources and of molecular clouds in the CMZ,
and concluded that the nonthermal radio continuum is produced by relativistic
electrons. It was estimated that contribution of the thermal component into
the total flux is less than 25%. The resulting spectral index of the
integrated radio flux from the CMZ, defined as $\beta=-\Delta(\log
S_{\nu})/\Delta(\log\nu)$, is $\beta^{325~{}{\rm MHz}}_{1.4~{}{\rm
GHz}}=0.17\pm 0.01$, $\beta^{1.4~{}{\rm GHz}}_{4.5~{}{\rm GHz}}=0.58\pm 0.01$,
and $\beta^{4.5~{}{\rm GHz}}_{8.5~{}{\rm GHz}}=1.14\pm 0.01$. The measured
spectrum $S_{\nu}$ is showed in Figure 1. These measurements can be naturally
interpreted as a result of depletion of CR electrons in the CMZ, whose
spectrum below the break is much harder than, e.g., in the Galactic local
medium (see Bisschoff et al., 2019). The solid line shows the synchrotron
spectrum predicted by our theoretical model, as described below in Section 4.

Figure 2: The spectrum of radio emission reported by Protheroe et al. (2008)
from the central region of Sgr B2 (bullets). One can see a
$S_{\nu}\propto\nu^{2}$ scaling at lower frequencies, indicating thermal
nature of the emission. For comparison, we also plot the synchrotron emission
spectrum (solid line) calculated for the central region of Sgr B2 from our
model (see Section 4).

A stronger effect of CR depletion would be expected for very dense individual
clouds. One of the suitable candidates is the giant molecular cloud complex
Sgr B2 (e.g., Hüttemeister et al., 1995). While its average column density is
about $N_{{\rm H}_{2}}^{\rm CMZ}$, in the central region of the Sgr B2 within
$5~{}{\rm pc}\times 2.5~{}{\rm pc}$ we have $N_{{\rm H}_{2}}^{\rm
B2}=10^{24}-10^{25}$ cm-2 (Schmiedeke et al., 2016), i.e., 1–2 orders of
magnitude higher than the average CMZ value.

Protheroe et al. (2008) (see also Jones et al., 2011; Meng et al., 2019) found
that the radio emission from the central region of Sgr B2 is dominated by
thermal emission of HII gas at frequencies above $\sim 0.3$ GHz, as shown in
Figure 2. Thus, unlike the CMZ, no indications of synchrotron-like emission
from the Sgr B2 region were observed. We note that the same is true for
several other very dense molecular clouds, located outside of the Galactic
center: the complexes G333.125–0.562 and G333 (Jones et al., 2008) and
IRAS15596 (Jones, 2014). On the other hand, we point out that the radio
spectra from the Sgr B2 region reported in Yusef-Zadeh et al. (2007, 2016)
indicate a transition to non-thermal emission at the lowest observed
frequencies of $\sim 0.3$ GHz.

Below we are going to find out the values of parameters required for the radio
emission from modulated spectrum of electrons to match the observations of the
CMZ. We are also going to investigate the ratio between thermal and non-
thermal emission components for individual molecular clouds, to determine the
feasibility to detect the non-thermal component.

##  3 Spectrum of CR electrons inside dense clouds

Ivlev et al. (2018) and Dogiel et al. (2018) developed the model of CR
modulation in a diffuse envelope of a dense molecular cloud. It was assumed
that energy losses in the envelope can be neglected and the that the cloud
absorbs a flux of CR protons111For simplicity, here we only consider CR
protons; heavier nuclei can be straightforwardly added, see Dogiel et al.
(2018). with a certain velocity $u_{p}$. For the proton spectrum $f_{p}(p)$ in
the momentum space (normalized such that $\int f_{p}(p)dp$ is the total number
density of CRs), this gives the boundary condition at the cloud edge in the
following form:

$S_{p}(p)=u_{p}f_{p}^{\rm(c)}(p)\,,$ (1)

where $S_{p}(p)$ is the flux of CR protons into the cloud, $p$ is their
momentum and $f_{p}^{\rm(c)}(p)$ is their spectrum at the cloud edge. The flux
velocity $u_{p}$ exceeds the Alfvén velocity $v_{\rm A}$, which is a necessary
condition of excitation of MHD waves by CRs. In the Appendix we calculate
$u_{p}$ for different propagation regimes in the cloud.

Given Equation (1), the flux of CR protons entering the cloud was estimated as
(Dogiel et al., 2018)

$S_{p}(p)=\frac{v_{\rm A}f_{p}^{(0)}(p)}{1-(1-v_{\rm
A}/u_{p})e^{-\eta_{p}(p)}}\,,$ (2)

where $f_{p}^{(0)}(p)$ is the proton spectrum in the ISM, which is assumed to
decrease with $p$ faster than $\propto p^{-1}$. The “diffusion depth”,

$\eta_{p}(p)=\int\limits_{0}^{z_{0}}\frac{v_{\rm A}dz}{D_{p}(z,p)}\,,$ (3)

characterizes the relative importance of CR advection and diffusion in the
envelope (Ivlev et al., 2018): diffusion determines the flux as long as
$\eta_{p}\ll 1$, whereas for $\eta_{p}\gtrsim 1$ advection dominates and
$S_{p}\approx v_{\rm A}f_{p}^{(0)}$. The value of $\eta_{p}(p)$ depends on the
proton diffusion coefficient $D_{p}(z,p)$, which is determined by the spectrum
of self-generated turbulence. The integration limit $z_{0}(p)$ is the
(momentum-dependent) outer border of the diffusion zone in the envelope (see
Figure 2 of Ivlev et al., 2018).

If only a small fraction of penetrating protons is attenuated in a cloud, we
can assume that their density in the cloud (where the self-generated
turbulence is absent) remains almost constant. Therefore, Equations (1) and
(2) allow us to estimate an average spectrum of any CR species inside the
cloud, provided $D(z,p)$ is known.

Ivlev et al. (2018) obtained the expression for $\eta_{p}(p)$ assuming the
excitation-damping balance for MHD waves, which is valid as long as
$\eta_{p}\lesssim 1$. It was shown (Dogiel et al., 2018) that CR protons give
the major contribution to the self-generated turbulence, while the
contribution of other CR species can be largely ignored. The growth rate of
MHD waves is proportional to the diffusion component of the flux (Ivlev et
al., 2018), and hence the excitation-damping balance can be written as

$\frac{\pi^{2}e^{2}v_{\rm A}}{m_{p}c^{2}\Omega}\>p\left[S_{p}(p)-v_{\rm
A}f_{p}^{(0)}(p)\right]=\nu\,,$ (4)

where $\Omega=eB/(m_{p}c)$ is the proton gyrofrequency scale, $B$ is the
magnetic field strength, $e$ is the elemental charge, $m_{p}$ is the mass of
the proton, and $\nu$ is the damping rate of MHD waves due to ion-neutral
collisions. Using Equation (2), this can be solved for $\eta_{p}$, which
yields

$\eta_{p}(p)=\ln\left[\left(1+\frac{\pi^{2}e^{2}v_{\rm
A}^{2}}{m_{p}c^{2}\Omega\nu}\>pf_{p}^{(0)}(p)\right)\delta_{p}\right]\,,$ (5)

where $\delta_{p}=1-v_{\rm A}/u_{p}>0$. Since $\eta_{p}\geq 0$ by definition,
the maximum momentum of modulated protons, $p_{\rm ex}$, is determined from
condition that the argument of the logarithm is equal to unity (see Dogiel et
al., 2018). Only protons with $p<p_{\rm ex}$ are able to resonantly excite MHD
waves in the envelope (which leads to their efficient scattering), while
protons with larger momenta propagate through the envelope freely, without
scattering.

Substituting Equation (5) into Equation (2) and assuming $\delta_{p}\approx
1$, one can estimate the average proton spectrum inside a cloud,

$u_{p}f^{\rm(c)}_{p}(p)\propto\left\\{\begin{array}[]{ll}p^{-1}\,,&{\rm
if}~{}p_{\rm A}\leq p\leq p_{\rm ex}\,,\vspace{.2cm}\\\ f_{p}^{(0)}(p)\,,&{\rm
if}~{}p\geq p_{\rm ex}\,.\end{array}\right.$ (6)

In the Appendix we demonstrate that $u_{p}$ does not practically depend on
$p$. Therefore, the spectrum has a universal form $f^{\rm(c)}_{p}(p)\propto
p^{-1}$ for $p_{\rm A}\leq p\leq p_{\rm ex}$, with $p_{\rm A}$ estimated from
the condition $\eta_{p}(p_{\rm A})\sim 1$. We note that the universal
dependence $S_{p}(p)\propto p^{-1}$, occurring for $0<\eta_{p}\lesssim 1$
(where $S_{p}$ is dominated by diffusion), follows directly from the
excitation-damping balance, Equation (4). For $p\lesssim p_{\rm A}$, where
$\eta_{p}$ exceeds unity and according to Equation (2) the proton flux
converges to the advection component $v_{\rm A}f_{p}^{(0)}(p)$, we again
obtain $f^{\rm(c)}_{p}(p)\propto f^{\rm(0)}_{p}(p)$.

Now we can estimate the electron spectrum inside the cloud. Indeed, the
diffusion coefficient for both protons and electrons is calculated as

$D(p)\approx\frac{vB^{2}}{6\pi^{2}k^{2}W(k)}\,,$ (7)

where $v(p)$ is the particle velocity, $k(p)=eB/(pc)$ is the resonant
wavenumber, and $kW(k)$ is the total energy density of the proton-generated
turbulence per unit $\log k$ range (i.e., $8\pi kW(k)/B^{2}$ is the
dimensionless energy density). Therefore, the electron flux $S_{e}(p)$ is
described by Equation (2) where the proton interstellar spectrum and the flux
velocity are replaced with the respective electron values, $f_{e}^{(0)}$ and
$u_{e}$, while the electron diffusion depth is

$\eta_{e}(p)=\eta_{p}(p)\frac{v_{p}(p)}{v_{e}(p)}\,,$ (8)

as follows from Equation (3). Assuming for simplicity that both electrons and
protons are relativistic, $p\gg m_{p}c$, and setting $\delta_{e}\approx 1$, we
obtain the electron spectrum inside the cloud,

$u_{e}f^{\rm(c)}_{e}(p)\propto\left\\{\begin{array}[]{ll}{\displaystyle
p^{-1}\frac{f_{e}^{(0)}(p)}{f_{p}^{(0)}(p)}\,,}&{\rm if}~{}p_{\rm A}\leq p\leq
p_{\rm ex}\,,\vspace{.2cm}\\\ f_{e}^{(0)}(p)\,,&{\rm if}~{}p\geq p_{\rm
ex}\,,\end{array}\right.$ (9)

with the same $p_{\rm A}$ and $p_{\rm ex}$ as in Equation (6). For
relativistic CRs, their interstellar spectra are well described by power-law
dependencies, $f_{p,e}^{(0)}(p)\propto p^{-\gamma_{p,e}}$, with
$\gamma_{p}\approx 2.7$ and $\gamma_{e}\approx 3.2$ (see e.g. Bisschoff et
al., 2019). Thus, the electron spectrum inside a cloud scales as
$f_{e}^{\rm(c)}(p)\propto p^{-1+\gamma_{p}-\gamma_{e}}\approx p^{-1.5}$
(again, for $u_{e}=$ const) within the “universal” momentum range of $p_{\rm
A}\leq p\leq p_{\rm ex}$.

##  4 Expected radio spectrum from dense clouds

Using the results of previous section as well as the expressions for $u_{p}$
and $u_{e}$ presented in the Appendix, we evaluate the electron spectrum
$f^{\rm(c)}_{e}(p)$ inside the cloud. This allows us to compute the total
synchrotron emissivity $\mathcal{P}_{\nu}$ of ultra-relativistic electrons per
unit volume (Ginzburg & Syrovatskii, 1965),

$\mathcal{P}_{\nu}\approx\frac{\sqrt{3}e^{3}B_{\perp}}{m_{e}c^{2}}\int\limits
dp\>f^{\rm(c)}_{e}(p)\frac{\nu}{\nu_{*}}\int\limits_{\nu/\nu_{*}}^{\infty}dx\>K_{5/3}(x)\,,$
(10)

where $K_{\alpha}(x)$ is the McDonald function and

$\nu_{*}(p)=\frac{3eB_{\perp}}{4\pi
m_{e}c}\left(\frac{p}{m_{e}c}\right)^{2}\,,$ (11)

is determined by the plane-of-the-sky magnetic field strength $B_{\perp}$. The
emissivity scales as $\mathcal{P}_{\nu}\propto
B^{(\gamma+1)/2}\nu^{-(\gamma-1)/2}$ for a power-law electron spectrum
$f^{\rm(c)}_{e}(p)\propto p^{-\gamma}$, with the contribution of individual
electrons peaked at a frequency of $\nu(p)\approx 0.3\nu_{*}(p)$ (Ginzburg &
Syrovatskii, 1965). This indicates that (i) the synchrotron emission is
primarily generated in denser regions, where the magnetic field is stronger,
and (ii) the emission spectrum exhibits a break due to CR modulation.
According to Equation (9), at higher frequencies, corresponding to unmodulated
electrons with $p\gtrsim p_{\rm ex}$, the emission varies as
$\mathcal{P}_{\nu}\propto\nu^{-(\gamma_{e}-1)/2}\approx\nu^{-1.1}$, while at
lower frequencies the spectrum becomes harder,
$\mathcal{P}_{\nu}\propto\nu^{-(\gamma_{e}-\gamma_{p})/2}\approx\nu^{-0.25}$.

For a quantitative comparison with available observations from the CMZ (Yusef-
Zadeh et al., 2013), we assumed $n_{{\rm H}_{2}}=50$ cm-3 for the density of
molecular hydrogen in the diffuse envelope, $B=0.1$ mG for the average
magnetic field (we note that this value does not practically affect the
resulting radio spectrum), and used the solar abundance of carbon ions to
calculate the Alfvén velocity and the ion-neutral damping in the envelope. The
cloud column density, which determines the value of $p_{\rm ex}$, was set to
$N_{{\rm H}_{2}}^{\rm CMZ}=10^{23}$ cm-2. For the proton or electron spectra
outside the envelope we used the model form suggested in Ivlev et al. (2015),

$f_{e,p}^{(0)}(p)={\rm const}\frac{E^{\alpha}}{(E_{0}+E)^{\beta}}\,,$ (12)

where $E(p)$ is the proton or electron kinetic energy, $E_{0}=500$ MeV (same
for both species), $\alpha=-0.8~{}(-1.5)$ and $\beta=1.9~{}(1.7)$ for protons
(electrons). These values of $E_{0}$, $\alpha$, and $\beta$ were selected in
Ivlev et al. (2015) to provide the upper bound for the available observational
data on the CR ionization rate in diffuse gas (Indriolo & McCall, 2012).
Constant factors were chosen to ensure that the spectrum of relativistic
protons is the same as in the local interstellar medium (Acero et al., 2016),
while the electron interstellar spectrum was enhanced by a factor of 10. The
synchrotron emission flux was calculated by assuming that the radiation mainly
comes from dense regions occupying $\sim 10$% of the total CMZ volume, $V_{\rm
rad}\sim 0.1V_{\rm CMZ}\sim 10^{61}$ cm3. The magnetic field in the dense
regions was set 5 times stronger than the average field, $B_{\rm dense}=0.5$
mG. The total emission flux at a distance $R_{\rm CMZ}\approx 8$ kpc is then
obtained as $S_{\nu}=\mathcal{P}_{\nu}V_{\rm rad}/(4\pi R_{\rm CMZ}^{2})$.

Furthermore, to explore the role of CR absorption in dense clouds, we
considered a model case of the central region of Sgr B2, corresponding to the
observations reported by Protheroe et al. (2008). In this case, we set the
cloud column density to $N_{{\rm H}_{2}}^{\rm B2}=10^{24}$ cm-2, keeping the
other parameters unchanged.

Figure 3: Modulated spectra of CR electrons and protons inside CMZ clouds. The
solid lines represents the spectra inside an average CMZ cloud with the column
density of $N_{{\rm H}_{2}}^{\rm CMZ}=10^{23}$ cm-2, the dashed lines
correspond to the central region of Sgr B2 with $N_{{\rm H}_{2}}^{\rm
B2}=10^{24}$ cm-2. For comparison, the dotted lines show the interstellar
(unmodulated) spectra. For the shown energy range, the densities of secondary
electrons (not displayed) are negligible compared to the respective modulated
electron densities.

Figure 3 shows the results of our calculations, displaying self-modulated
spectra of CR protons (grey lines) and the resulting modulated electron
spectra (black lines). We see that the modulation for an average CMZ cloud
(solid lines) leads to a depletion of sub-GeV electrons, but the depletion
magnitude is less than a factor of $\sim 3$ for $pc\gtrsim 0.2$ GeV. For the
above parameters, this modulated energy range approximately corresponds to the
frequency range between $\sim 0.3$ GHz and $\sim 3$ GHz, i.e., up to the
spectral break seen in the radio data depicted by the bullets in Figure 1.
Therefore, the solid line in Figure 1, representing the calculated synchrotron
flux, provides a good agreement with the data, both in terms of the position
of the spectral break and the slopes above and below the break.

The depletion of CR electrons in denser clouds, such as the central region of
Sgr B2, starts at higher energies, as shown in Figure 3 by the dashed line.
The depletion magnitude in this case is up to a factor of $\sim 20$ (for the
same energy range). The corresponding synchrotron flux, plotted by the solid
line in Figure 2, is too low to compete with the thermal emission. Note that
the secondary electrons were estimated to have a negligible ($\lesssim 10\%$)
contribution to the above results.

We point out that the predicted modulation of CR electrons can also be
reflected in the continuum gamma-ray emission. A complete sky survey of the
gamma-ray emission below 100 MeV was performed by COMPTEL imaging telescope
(Schönfelder et al., 1993). The spectrum was analyzed by Strong et al. (1996),
who concluded that the bremsstrahlung radiation in this range must be
primarily produced by Galactic CR electrons. Therefore, the emission may
indeed be affected by the electron depletion, but the spatial and energy
resolution of the COMPTEL data is too poor to observe the effect. The spectral
break is expected at around $0.1-0.3$ GeV, both for proton- and electron-
generated emission, and thus next-generation gamma-ray telescopes operating in
this energy range, such as AMEGO (McEnery et al., 2019) and e-ASTROGAM (De
Angelis et al., 2021), may be able to observe the depletion effect. The
necessary condition for that would be their ability to resolve individual
molecular clouds. In case of the CMZ, the required spatial resolution is about
$1^{\circ}$.

##  5 Conclusions

We analyzed non-thermal radio emission from dense CMZ regions, and
demonstrated that the observed spectral features can well be interpreted in
the framework of our recent theory of CR self-modulation in molecular clouds
(Ivlev et al., 2018; Dogiel et al., 2018). Our conclusions are summarized as
following:

  * •

Self-modulation of CR protons with energies below a certain threshold $E_{\rm
ex}$, determined by the column density of a cloud, generates a universal
($\propto E^{-1}$) flux of protons entering the cloud. This leads to a
significant depletion of both the proton and electron densities below $E_{\rm
ex}$. Proton-proton collisions are generally believed to be the main source of
gamma-ray emission from clouds. However, due to low spatial and energy
resolutions of Fermi-LAT at the relevant sub-GeV energies, the effect of
proton depletion cannot be observed (although next-generation gamma-ray
telescopes may potentially be able to resolve the emission). On the other
hand, thanks to a high resolution of radio telescopes in the $\sim$ GHz range,
it is possible to observe individual clouds and, thus, to detect the
synchrotron emission of modulated electrons;

  * •

Assuming power-law spectra of relativistic interstellar protons and electrons,
with the respective spectral indices $\gamma_{p}$ and $\gamma_{e}$, the
electron (momentum) spectrum inside a cloud scales as $f_{e}(p)\propto
p^{-1+\gamma_{p}-\gamma_{e}}$ for $p<p_{\rm ex}$ and $f_{e}(p)\propto
p^{-\gamma_{e}}$ for $p>p_{\rm ex}$. This break naturally leads to a spectral
break in the synchrotron emission;

  * •

The radio continuum from the CMZ is evidently non-thermal (see Yusef-Zadeh et
al., 2013) and, due to its large average column density of $N_{{\rm
H}_{2}}^{\rm CMZ}\sim 10^{23}$ cm-2, the CMZ is one of the most promising
targets for studying the CR self-modulation. The value of $E_{\rm ex}(N_{{\rm
H}_{2}}^{\rm CMZ})\sim 1$ GeV ensures the synchrotron spectral break at $\sim
1$ GHz;

  * •

The observed radio spectrum from the CMZ can be approximated by a double
power-law dependence with $S_{\nu}(\nu)\propto\nu^{-0.17}$ below $\sim 1$ GHz
and $S_{\nu}(\nu)\propto\nu^{-1.14}$ above $\sim 1$ GHz (see Figure 1). We
demonstrate that this spectrum is well described by our theory of CR self-
modulation, assuming the interstellar proton and electron spectra in the CMZ
have the shape of the local CR spectra, with the electron density enhanced by
a factor of 10;

  * •

A stronger depletion effect is expected for individual CMZ clouds, whose
column densities are substantially larger than $N_{{\rm H}_{2}}^{\rm CMZ}$.
For example, the expected depletion magnitude is a factor of $\sim 5$ higher
for the central region of Sgr B2 (see Figure 3), where the column can exceed
$N_{{\rm H}_{2}}^{\rm CMZ}$ by 1–2 orders of magnitude (see Protheroe et al.,
2008). However, the radio emission from the Sgr B2 region is evidently
dominated by the thermal component at all observed frequencies (see Figure 2),
and therefore new observations at lower frequencies are needed to verify the
depletion effect.

We would like to thank an anonymous referee for constructive and stimulating
suggestions. The authors are grateful to Farhad Yusef-Zadeh for critical
reading of the manuscript and useful comments, and to Roland Crocker for
discussions. The work is supported by Russian Science Foundation via the
Project 20-12-00047.

## CR flux velocity at the edge in dense cloud

To solve the problem of CR depletion self-consistently, one needs to evaluate
the flux velocities $u_{p,e}$ of CR protons and electrons entering dense
clouds with energies in a range from several hundreds of MeV to dozens of GeV.
For these energies, the main mechanism of energy loss is bremsstrahlung for
electrons, and pion production in nuclear collisions for protons. Both process
can well be treated as catastrophic, since CR particles lose a significant
fraction of their energy in each collision.

In the absence of scattering, CRs propagate along the magnetic field lines.
Their spectrum is then described by the following equation:

$v\mu\frac{\partial f}{\partial z}+\frac{f}{\tau}=0\,,$ (1)

with the boundary condition $f(0,p,\mu)=f_{0}(p,\mu)$. Here, $\mu$ is the
pitch angle and $\tau^{-1}=v\sigma_{\rm loss}n_{{\rm H}_{2}}$ is the collision
rate with H2 molecules, determined by the cross section $\sigma_{\rm loss}$
for the relevant catastrophic loss mechanism (bremsstrahlung or pion
production). If the CR spectrum at the cloud boundaries is isotropic, we have
$f_{0}(p)=f^{\rm(c)}(p)/2$. The resulting solution is

$f(p,z,\mu)=f_{0}(p)\exp(-\sigma_{\rm loss}n_{{\rm H}_{2}}z/\mu)\,,$ (2)

and the flux velocity at the boundaries can be estimated as

$u=\frac{1}{2}\int\limits_{0}^{1}v\mu\>d\mu-\frac{1}{2}\int\limits_{0}^{1}v\mu\exp(-\sigma_{\rm
loss}N_{{\rm H}_{2}}/\mu)\>d\mu\,,$ (3)

where $N_{{\rm H}_{2}}=n_{{\rm H}_{2}}L$ is the column density of a cloud of
size $L$. For a typical case $\sigma_{\rm loss}N_{{\rm H}_{2}}\ll 1$, after a
linear expansion of the second term we obtain

$u\approx\frac{1}{2}v\sigma_{\rm loss}N_{{\rm H}_{2}}\,.$ (4)

While CRs traveling inside dense clouds cannot normally trigger MHD waves
because of a strong ion-neutral damping (see Kulsrud & Pearce, 1969), the gas
itself may be turbulent and thus the magnetic field lines can be strongly
tangled. Propagation of CRs is then described as an “effective” diffusion (see
e.g. the review of Hennebelle & Falgarone, 2012). The energy of magnetic field
fluctuations with $\delta B\gg B_{0}$ is concentrated near the correlation
length, $l_{\rm corr}$, and the effective diffusion coefficient of
relativistic CRs is estimated as $D\sim cl_{\rm corr}/3\sim 10^{28}$ cm2 s-1,
assuming $l_{\rm corr}\approx 0.5$ pc for molecular clouds with the size of
several pc (Dogiel et al., 2015). We stress that this consideration does not
require magnetic field to be perfectly frozen into the turbulent gas. In dense
clouds, non-ideal MHD effects are dominated by a combination of magnetic
diffusion due to finite conductivity and ion-neutral friction (Dogiel et al.,
1987; Istomin & Kiselev, 2013), but turbulent gas motions generate
significantly tangled magnetic field lines also under such conditions.

In a turbulent cloud with the effective diffusion coefficient $D$, the CR
propagation is described by a simplified diffusion equation,

$D\frac{\partial^{2}f}{\partial z^{2}}-\frac{f}{\tau}=0\,.$ (5)

The boundary conditions are $\left.f\right|_{z=0}=\left.f\right|_{z=L}=f_{0}$
and the solution is

$f(p,z)=f_{0}(p)\frac{\exp\left(\frac{L-z}{\sqrt{D\tau}}\right)+\exp\left(\frac{z}{\sqrt{D\tau}}\right)}{1+\exp\left(\frac{L}{\sqrt{D\tau}}\right)}\,.$
(6)

The flux velocity is

$u=\left.\frac{D}{f_{0}}\frac{\partial f}{\partial
z}\right|_{z=L}=\sqrt{\frac{D}{\tau}}\tanh\left(\frac{L}{2\sqrt{D\tau}}\right)\approx\frac{1}{2}v\sigma_{\rm
loss}N_{{\rm H}_{2}}\,,$ (7)

where the last (approximate) equality requires the condition $\sigma_{\rm
loss}N_{{\rm H}_{2}}\ll l_{\rm corr}/L$ to be satisfied. In the opposite
limit, the velocity approaches the value of

$u\approx v\sqrt{\frac{1}{3}\frac{l_{\rm corr}}{L}\sigma_{\rm loss}N_{{\rm
H}_{2}}}\,,$ (8)

where we substituted $D=vl_{\rm corr}/3$. We conclude that $u$ is not affected
by the presence of turbulence in “thin” clouds (where the flux of CRs entering
a cloud from one side is practically compensated by the opposite flux,
arriving from the other side). On the other hand, for sufficiently large
$N_{{\rm H}_{2}}$ or/and small $l_{\rm corr}/L$, the flux velocity is smaller
if a cloud is turbulent (since the diffusive propagation naturally suppresses
the entering flux, while the compensation due to the opposite side is strongly
attenuated). It is noteworthy that Equation (7) would also describe an
(unlikely) situation if MHD fluctuations, leading to CR scattering, were
present in a cloud: in this case the CR propagation is characterized by a
certain (momentum-dependant) diffusion coefficient, and hence the above
analysis remains applicable.

We use the bremsstrahlung cross section from Blumenthal & Gould (1970) and the
pion production cross section from Aharonian & Atoyan (1996) to estimate the
flux velocity of electrons and protons, respectively. In both cases, the
condition $\sigma_{\rm loss}N_{{\rm H}_{2}}\ll 1$ is well satisfied for column
densities below $\sim 10^{25}$ cm-2. Therefore, unless $l_{\rm corr}/L$ is too
small, Equations (4) and (7) yield the same expression for $u$, irrespective
of whether the cloud is quiescent or turbulent. Furthermore, since the cross
sections only weakly depend on the energy of relativistic CRs, we conclude
that the flux velocities can be considered as constant.

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