# Gamma rays from Nebulae around Recurrent Novae

W. Bednarek & J. Sitarek  University of Lodz, Faculty of Physics and Applied
Informatics, Department of Astrophysics, 90-236 Lodz, ul. Pomorska 149/153,
Poland<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Novae were discovered to emit transient $\gamma$ rays during the period of
several days to a few weeks after initial explosion, indicating presence of
acceleration processes of particles in their expanding shells. In the case of
recurrent novae, electrons can be in principle accelerated in the nova shells
for the whole recurrence period of nova producing delayed $\gamma$ ray
emission as considered in Bednarek (2022). Here we extend the ideas presented
in this article by considering the fate of electrons which diffuse out of the
shells of novae supplying fresh relativistic electrons to the recurrent nova
super-remnants during the whole active period of nova ($\geq 10^{4}$ yrs). We
develop a model for the acceleration of electrons and their escape from the
nova shells. The electrons within the recurrent nova super-remnants produce
$\gamma$ rays in the comptonization process of the radiation from the red
giant companion and the Cosmic Microwave Background Radiation. As an example,
the case of a symbiotic nova RS Oph (with the recurrence period estimated on
$\sim$10-50 yrs) is considered in more detail. Predicted $\gamma$-ray emission
from the nova super-remnant around RS Oph is discussed in the context of its
observability by satellite experiments (i.e. Fermi-LAT) as well as current and
future Cherenkov telescopes.

novae, cataclysmic variables — binaries: symbiotic — radiation mechanisms:
non-thermal — gamma-rays: stars

##  1 Introduction

Novae are thermonuclear explosions in a layer of matter accumulated on the
surface of a white dwarf (WD) as a result of the accretion process from a
companion star in the WD binary system. If the companion of the WD is a main
sequence star then the nova is called a classical nova. In the rare case of a
Red Giant (RG) companion, the nova is called symbiotic. It is expected that
the recurrence time scale of nova explosions depends on the mass of the WD and
the accretion rate. If the nova appears many times during a human life time in
this same binary system, then it is called a recurrent nova. In this article
we concentrate on the case of the symbiotic novae with a short recurrence
periods (such as recently observed RS Oph).

The material expelled during recurrent nova explosion is slowed down due to
the interaction with the surrounding medium forming a nebula, dubbed nova
super-remnant (NSR). The structure of such NSR has been recently investigated
by applying hydrodynamical simulations in the case of a specific recurrent
nova M31N 2008-12a (in the Andromeda Galaxy) by Healy-Kalesh et al. (2023). It
is found that in subsequent eruptions a NSR is formed with the radius of tens
of pc. In fact, such optical NSR has been recently detected with a projected
size of at least 134 by 90 parsecs in the case of the most rapidly recurring
nova M31N 2008-12a (Darnley et al. 2019).

Novae have been recently established as a new type of GeV and sub-TeV
$\gamma$-ray sources (Abdo et al., 2010; Ackermann et al., 2014; H. E. S. S.
Collaboration et al., 2022; Acciari et al., 2022), indicating that particles
(electrons, hadrons) are efficiently accelerated in their explosions. The GeV
$\gamma$-ray emission is observed for several days to a few weeks after the
nova explosion (e.g. Ackermann et al. 2014). However, for how long the process
of acceleration of particles in nova shell is active is still an open issue.
The observed transient $\gamma$-ray emission has been proposed to be likely
produced by hadrons in collisions with the matter expelled in a nova explosion
(e.g. Tatischeff & Hernanz 2007, Abdo et al. 2010, Sitarek & Bednarek 2012,
Martin & Dubus 2013, Metzger et al. 2015, Ahnen et al. 2015, Acciari et al.,
2022; H. E. S. S. Collaboration et al., 2022, Cheung et al. 2022, Zheng et al.
2022). However, relativistic electrons can also contribute (see e.g. Abdo et
al. (2010), Sitarek & Bednarek 2012, Martin & Dubus 2013, Vurm & Metzger 2018,
Martin et al. 2018, Bednarek 2022), due to efficient energy losses on the IC
process in the radiation from the nova photosphere or the companion RG star
(in the case of symbiotic novae).

Recently, Bednarek (2022) considered the model in which the electrons can be
accelerated in nova shells for a much longer period of time than indicated by
the $\gamma$-ray observations. In fact, provided that acceleration of
particles occurs within the nova shell, the acceleration process could
continue during the time scale corresponding to the recurrence period of the
recurrent novae, at least in the part of the shell which propagates in the
polar region of the nova binary system, where the shell is not significantly
decelerated. In terms of such a model, the time dependent $\gamma$-ray
emission produced in the Inverse Compton Scattering (ICS) of the radiation
from the photosphere and the RG, as observed in the case of recurrent Nova RS
Oph, has been calculated and compared with the sensitivities of the
$\gamma$-ray telescopes. In Bednarek (2022) it is concluded that with the
future CTA this transient $\gamma$-ray emission from electrons in the shells
should be detected during the period of the years.

In this follow-up work, we consider the fate of the relativistic electrons,
which were accelerated in the nova shells but after some time escaped from
them into the interstellar space. We suppose that such escaping electrons
deposit in NSR around the nova. In the case of recurrent novae such NSRs
should be able to accumulate large amount of relativistic electrons from
multiple explosions. These electrons are expected to produce $\gamma$ rays by
scattering mainly the Cosmic Microwave Background Radiation (CMBR) and thermal
radiation from RG.

As in Bednarek (2022), we consider the recurrent nova RS Oph which shows one
of the shortest average recurrence period (on average $\sim$14.7 yrs, with the
spread between 8.6 to 26.6 yrs), among the known Galactic recurrent novae see
Tab. 21 in Schaefer (2010). During its latest outburst in 2021 the source
showed $\gamma$-ray emission from GeV energies (Cheung et al. 2022, Zheng et
al. 2022) up to hundreds of GeV (H. E. S. S. Collaboration et al., 2022;
Acciari et al., 2022), becoming the first nova detected in very-high-energy
band. The RS Oph binary is composed of a White Dwarf (WD) and a RG with masses
1.2 – 1.4$M_{\odot}$ and 0.68 – 0.80 $M_{\odot}$, respectively (Brandi et al.,
2009). The same authors report an orbit with the period of 453.6 days and the
inclination of 49∘ – 59∘. The surface temperature of the RG is estimated on
3600 K and its radius on $67^{+19}_{-16}$ R⊙ (Dumm & Schild 1998), where
$R_{\odot}$ is the radius of the Sun . The mass-loss rate of the RG is
estimated on $5\times 10^{-7}M_{\odot}/\textrm{yr}$ (Booth, Mohamed, &
Podsiadlowski, 2016), however $3.7\times 10^{-8}-10^{-6}M_{\odot}/\textrm{yr}$
values have been also suggested (Iijima, 2008; Schaefer, 2009). The wind
concentrates around the equatorial plane of the binary system. The wind
velocity is estimated on 40 km s-1 (Wallerstein 1958). The upper limit on the
mass expelled during nova explosion is constrained by the above mass loss rate
of the RG and the recurrence period of the nova ($\sim 15$ yrs), on $5.6\times
10^{-7}-1.5\times 10^{-5}M_{\odot}$. This material forms very complicated
structure around the nova. The observations of the nova RS Oph a few hundred
days after explosion in 2006 with the Hubble Space Telescope shows two-
component flow, composed of a low-velocity high density equatorial region and
a high-velocity low density polar region. The polar region expended with the
velocity of $(5600\pm 1100)$ km s-1 (Bode et al. 2006). On the other hand, at
about one day after the nova explosion, the expansion velocity was
$\sim(4000-7500)$ km s-1 (Buil 2006). Therefore, the polar expansion seems not
to decelerate significantly. In the case of the outburst in 2021, the radio
structure of the nova shell is very similar to that observed in 2006 (Munari
et al. 2022). Also the velocity of the nova shell, averaged over the first 34
days, equal to $\sim$7550 km-1 s-1 is generally consistent with that observed
in 2006. The initial velocity of the shell is an important parameter in our
modelling since it determines the ballistic time scale of the shell.

Rupen, Mioduszewski, & Sokoloski (2008) estimated the distance to RS Oph on
$(2.45\pm 0.37)$ kpc (see also Bailer-Jones et al. 2021). This value is also
very similar to the distance derived with the Gaia DR3-derived parallax:
$2.68_{-0.15}^{+0.17}$ kpc (see also Schaefer 2022, Munari et al. 2022.
Multiple, partially contradicting distance measurements are available (see the
discussion in Acciari et al., 2022).

##  2 Injection of electrons from the Nova

In Bednarek (2022) we discuss the scenario in which electrons are assumed to
be continuously accelerated in the nova shell long after nova explosion.
Electrons, confined within the expanding shell, produce $\gamma$-rays by
scattering RG and nova soft radiation. here, we extend this model by assuming
that electrons are finally realized from the shell into the nova surrounding.
They diffuse in the NSR around recurrent nova producing persistent
$\gamma$-ray emission.

###  2.1 Electrons within the shell

It is assumed that electrons are accelerated within the shell region for a
certain period of time, $t_{\rm inj}$, after the nova explosion. They are
confined for some time within the expanding shell of the nova, losing energy
on different radiation processes (i.e. synchrotron, IC, Bremsstrahlung). The
shells move sub-relativisticly through the complex material surrounding the
nova. They are finally decelerated in the interaction with the interstellar
matter. Relativistic electrons in the shell can be responsible for the
extended $\gamma$-ray emission from novae on a time scale of years. The
$\gamma$-ray emission produced by electrons within the shell on a time scale
of years has been recently discussed in Bednarek (2022).

Figure 1: Schematic picture of the NSR around a recurrent nova (not to scale).
In the case of the Nova RS Oph, every recurrent explosion injects a shell in
which electrons are accelerated (2 shells are shown). The shells move with
different velocities in the equatorial and polar regions of the binary system.
A part of the sphere covered by the equatorial region is marked by the
parameter $\Omega$. In the equatorial region the shells are effectively
decelerated by entraining the matter from the RG wind. Shells in both regions
are decelerated on the sub-parsec distance scale by entraining the matter from
the surrounding interstellar space. The electrons escape from the shells to
the surrounding medium due to the (Bohm) diffusion process. These electrons
form a nebula around the recurrent binary system, subsequently producing
$\gamma$-ray by ICS of the CMBR. The emission is expected from a region with a

size of the order of a few parsec at sub-TeV $\gamma$ rays.

Up to our knowledge, there is no any specific model for the acceleration of
particles in novae on a time scale of the nova recurrence period. All the
presently considered models concentrate on the time scale after the nova
explosion corresponding to the observed time scale of the $\gamma$-ray
emission already detected by the Fermi-LAT telescope, i.e. of the order of a
month. Therefore, we introduce a general model for the acceleration of the
electrons, their energy losses, and their escape from the shell which is able
to consider these processes on the time scale of the recurrence period of the
nova. For the acceleration and propagation of electrons in the nova shell, we
exploit the time-dependent model recently considered by Bednarek (2022). Here
we remind the main aspects of model. Two regions around the nova binary system
with different proprieties are considered. Following other works (Chesneau et
al., 2007; Walder, Folini, & Shore, 2008; Montez et al., 2022), we assume that
the wind from the RG is concentrated in the equatorial region of the binary
system which extend is defined by its part of whole sphere (i.e. the parameter
$\Omega$). A part of the nova shell, propagating in the equatorial region, is
effectively decelerated due to the entrainment of the matter from the RG wind.
On the other hand, the (1 - $\Omega$) part of the nova shell, propagating in
the polar region, expends freely up to the distances at which the matter
entrained from the interstellar space decelerates the shell. In fact such two
component structure of the nova shell has been observed in the X-rays in the
case of previous explosion of the Nova RS Oph see Montez et al. (2022), in
which case the fast shell takes about $\sim 0.2$ of the whole sphere and the
slow shell $\sim 0.8$. As a result of propagation in two different regions,
their physical conditions (such as magnetic field strength, density of matter,
radiation field, etc …) differ significantly. These conditions determine the
acceleration process of the electrons, their energy losses in the shell and
also the escape conditions from the shell into the NSR.

The electrons are assumed to be injected into the nova shell with a power-law
spectrum and an exponential cut-off during initial time interval starting from
the nova explosion. We develop a simple model which allows to determine the
basic physical parameters of the expending shell in the long period after
explosion such as, its expansion velocity and density of matter, the magnetic
field strength, and different radiation fields (see for details Sect. 3 in
Bednarek 2022). The magnetic field is assumed to be at some level of
equipartition with the kinetic energy of the shell. We take into account the
radiation field from the RG and also from the nova photosphere at the early
time after explosion. Then, we define the acceleration process of electrons,
as a function of time, and their energy losses. The cut-off energies in the
electron spectrum are obtained by balancing the acceleration time scale with
their time scale for energy losses or the dynamical time scale of the shell.
The total power of the electrons is normalized to the initial kinetic energy
of the shell. Since the electrons lose energy on different radiation
processes, we numerically calculate the evolution of the equilibrium spectrum
of the electrons within the shell at an arbitrary moment after explosion (for
details see Sect. 4 in Bednarek 2022). Due to the time-dependent conditions
within the shell, we follow the fate of electrons with different energies
applying the time step method to a single shell. With $\Delta t$ being the
time step, the location of the shell at the time $t_{\rm n}$ is calculated
from, $R_{\rm n}=R_{\rm n-1}+v_{\rm sh}(t_{\rm n-1})\cdot\Delta t$. The
thickness of the shell is related to its radius according to $\Delta R=\beta
R$, with $\beta<1$. In such a model, we define the condition for the escape of
the electrons from the shell into the interstellar region. We assume that the
diffusion process of the electrons is well described by the Bohm’s
prescription. However, since the physical conditions in the propagating shell
vary in time, we calculate the characteristic diffusion distance of electrons
in the shell from

$\displaystyle R^{\rm dif}_{\rm n}=R^{\rm dif}_{\rm(n-1)}+\sqrt{D_{\rm
B}/(2t_{\rm n})}\cdot\Delta t,$ (1)

where the Bohm diffusion coefficient is $D_{\rm B}=cR_{\rm L}/3$, and $R_{\rm
L}=E/(eB)\approx 3\times 10^{16}E_{10}/B_{\mu G}$ cm is the Larmor radius,
$E_{\rm e}=10E_{10}$ TeV is the electron energy, and $B=10^{-6}B_{\rm\mu G}$ G
is the magnetic field strength at the time $t_{\rm(n-1)}$. The electrons can
escape from the shell into the interstellar medium when the diffusion distance
becomes comparable to the thickness of the shell, i.e. $R_{\rm dif}=\beta R$.

###  2.2 Electrons released from the shell into the NSR

In the present work we consider the fate of electrons which were able to
escape from the nova shells. We argue that these relativistic electrons
accumulate in nova surrounding. The electrons are confined around the nova for
a certain time, diffusing slowly in the outward direction into the NSR. We
investigate the $\gamma$-ray emission produced by the electrons in the IC
scattering of the radiation from the RG (a companion of the WD), and the CMBR.
We include the energy losses of the electrons within the shell on the ICS of
the thermal radiation from the nova photosphere during its initial propagation
for the period of about 1 month. However, averaged over the whole recurrence
period of RS Oph, this photosphere radiation field is negligible when compared
to the RG thermal radiation, hence it is not taken into account when computing
the persistent $\gamma$-ray production in the nova super-remnant. Our interest
in the existence of relativistic electrons in the NSR we focus on the case of
the recently detected in the sub-TeV range Nova RS Oph (Acciari et al., 2022;
H. E. S. S. Collaboration et al., 2022). In the case of RS Oph six –
nine111Three out of those nine outbursts were only indirectly observed.
explosions have been documented with the recurrence period of the order of 15
years. Those observations indicate that electrons can be accelerated to TeV
energies at least in the early period after the nova explosion. The schematic
picture of the scenario considered here is shown in Fig. 1.

Following the method described in the previous subsection for the evaluation
of processes within the nova shell, we calculate the spectra of the electrons
escaping from the shell into the interstellar medium for the parameters of the
nova RS Oph and reasonable parameters of the acceleration model of the
electrons. We consider the electron injection spectra with the spectral index
-2 (Bell 1978). In Fig. 2 we show how the maximum energies of the electrons
evolve in time for a few different values of the magnetization of the shell.
The magnetization parameter of the shell, $\alpha$, which is obtained assuming
some level of the equipartition between the kinetic energy density of the
shell and energy density of the magnetic field in the shell, i.e. $\alpha
n_{\rm sh}m_{\rm p}v_{\rm sh}^{2}/2=B^{2}/(8\pi)$, where $n_{\rm sh}$ is the
density of the matter in the shell, $m_{\rm p}$ is the proton mass, $v_{\rm
sh}$ is the shell velocity, and $B$ is the magnetic field strength within the
shell (see for details Bednarek 2022). Both cases, with and without
deceleration in the RG wind, are considered. The maximum energies of the
electrons are limited either by the energy losses (at the early time) or the
dynamical time scale of moving shell (at the latter time). They become larger
at latter time due to decreasing magnetic field within the shell. These
maximum energies are also significantly lower in the case of drastically
decelerated shell since the acceleration parameter is assumed to depend on the
shell velocity (see Eq. 3 in Bednarek 2022). In this section we consider a
relatively slow acceleration process of the electrons in the shell with the
energy gain proportional to $\xi\sim(v_{\rm sh}/c)^{2}$ (where $v_{\rm sh}$
and $c$ are speed of the shell and light respectively), which is
characteristic for the second order Fermi acceleration process. When
calculating the $\gamma$-ray emission from nova super-remnants, we will
discuss also more efficient acceleration process.

Figure 2: The maximum energies of the electrons in the shell as a function of
propagation time of the shell. The electrons are accelerated within the nova
shell in the case of its propagation in the polar region of the binary system,
the so-called “no deceleration” case, (see panel on the left) and in the case
of propagation in the equatorial wind of the RG (deceleration case) which is
confined within the part of the whole sphere defined by $\Omega=0.1$ (in the
centre) and 0.5 (on the right). The magnetization of the nova shell is defined
by $\alpha=10^{-4}$ (dot-dashed curve), $10^{-3}$ (dashed), 0.01 (dotted), and
0.1 (solid). The initial mass of the nova shell is $M_{\rm
sh}^{0}=10^{-6}~{}M_{\odot}$ and its initial velocity is $v_{0}^{\rm sh}=6000$
km s-1. The mass loss rate of the RG wind is $M_{\rm RG}=10^{-7}$ M⊙ yr-1 and
its velocity $v_{\rm RG}=40$ km s-1.

We apply a simple model for the initial acceleration of the electrons since
our main goal is to consider a radiation model for the nova super-remnant
(formed by electrons escaping from the shell) in the late stages. A few more
detailed models for the high energy processes in the initial stage after a
nova explosion have been considered already. For example, Tatischeff & Hernanz
(2007) consider non-linear acceleration of particles at the initial blast wave
and show that protons can reach TeV energies already at days time scale after
explosion. Also Martin & Dubus (2013) develop a model for the shock
acceleration and radiation of electrons and protons in the shocked plasma
during initial month time scale after the nova explosion. Martin et al. (2018)
consider detailed model for the acceleration of particles in collisions of the
wind from the WD with the initial shell of the nova during first month after
explosion. They show that $\gamma$-ray emission comes mainly from hadronic
interactions with the dense matter downstream of the shock. Similar general
model, but limited to the equatorial region of the nova binary system, has
been also considered by Metzger et al. (2015). Detailed calculations of the
$\gamma$-ray emission in terms of the hadronic and leptonic scenario, during
initial stage of nova, are presented in Vurm & Metzger (2018). All these
models have been developed in order to explain the $\gamma$-ray emission from
the initial stage of the nova explosion as observed by the Fermi-LAT during
the time scale of the order of a month. Here we consider a model for possible
$\gamma$-ray emission from novae at long time after explosion during the whole
activity period of the recurrent nova. We argue that electrons, escaping from
the shell can accumulate for a long time in the vicinity of the recurrent
nova. They produce $\gamma$-ray nebula similar to those observed around
rotation powered pulsars.

We investigate the dependence of the spectrum of the escaping electrons on the
basic parameters of the model, assuming the likely parameters of the nova RS
Oph. In Fig. 3 we show the spectra of the electrons escaping from the polar
part of the shell, assuming that they are accelerated only during short time
(30 days) after the explosion of the nova or during the whole recurrence
period of RS Oph (the average period of 15 yrs is assumed). The shorter time
scale (of the order of a month) is consistent with the observed by the Fermi-
LAT telescope time scale for the initial GeV gamma-ray emission from Nova RS
Oph (Cheung et al. 2022). We suppose that the acceleration process of
electrons lasts at least for the period of the initial $\gamma$-ray production
phase as observed by the Fermi-LAT. Note also, that such order of the time
scale for acceleration of particles is expected in the model by Martin et al.
(2018) in which the acceleration process is active during the time of
collision of the fast wind from the white dwarf with a slower moving initial
shell of the nova. The longer time scale (of the order of the recurrence
period of the Nova RS Oph 15 yrs) is motivated by the fact that at this time
scale the nova shell is expected to significantly decelerate due to the
entrainment of the interstellar matter. In order to estimate the deceleration
distance of the shell we introduce a model for the velocity of the shell as a
function of distance from the nova. We assume a conservation of the momentum
of the shell interacting with the surrounding medium, $M^{\rm sh}_{\rm
0}v^{\rm sh}_{\rm 0}=v_{\rm sh}(R)[M_{\rm sh}+M_{\rm cos}(R)]$, where $M^{\rm
sh}_{\rm 0}$ and $v^{\rm sh}_{\rm 0}$ are the initial mass of the nova shell
and its initial velocity, $M_{\rm cos}=(4/3)\pi R^{3}n_{\rm cos}m_{\rm p}$ is
the mass entrained from the interstellar medium, $R$ is the radius of the
shell, $v_{\rm sh}(R)$ is the velocity of the shell at the distance $R$, and
$n_{\rm cos}$ is the density of the interstellar medium. For reasonable
densities of the interstellar medium (in the range $n_{\rm cos}\approx 0.1-10$
cm-3), the shell starts to significantly decelerate at the distance of the
order of $\sim 10^{17}$ cm (see the results of calculations presented in Fig.
6 in Bednarek 2022). This distance scale is passed by the shell during the
time similar to the recurrence period of the Nova RS Oph. Our simple model for
the velocity profile generally agrees with the results of observations for the
outburst of nova RS Oph during the early expansion phase reported in
Tatischeff & Hernanz (2007, see Fig. 1). In our model (described by the solid
curve in Fig. 6a, Bednarek 2022) the velocity of the forward shock, moving in
the polar region, drops by a factor of the order 2 during the first $\sim$30
days as also observed in the case of the outburst in RS Oph during 2006
(Tatischeff & Hernanz 2007).

As we have shown above, the efficiency of the acceleration process of
electrons, $\xi$, depends in our model on the velocity of the shell.
Therefore, it has to drop significantly at the time scale of the order of the
recurrence period of the Nova RS Oph. Interestingly, there are observational
evidence for shocks appearing at late stages after nova explosion. Babul et
al. (2022) reported enhancement of radio emission starting 1.5 months after
the eruption, and lasting at least for a month more. It is interpreted as
synchrotron emission due to internal shocks within the ejecta. Moreover, at
least 25% of novae observed by Chomiuk et al. (2021) show non-thermal
synchrotron emission. Two of them, V5589 Sgr and V392 Per, appear to be on the
bridge between classical and symbiotic novae. Another example of the late non-
thermal synchrotron emission (months time scale) is the symbiotic recurrent
nova V3890 Sgr (Nyamai et al. 2023).

The cases of slow and fast acceleration of the electrons within the shell are
considered. They depend on the velocity of the shell. In the case of fast
acceleration the acceleration coefficient is assumed to be $\xi\sim(v_{\rm
sh}/c)$. Specific spectra of the escaping electrons depend on the
magnetization parameter of the shell which determines the diffusion process of
the electrons in the shell and one of the main energy loss processes (i.e.
synchrotron emission). We conclude that long acceleration process of the
electrons results in their spectra extending to higher energies. In the case
of fast acceleration scenario, the electrons are able to escape into NSR with
multi-TeV energies. In the case of slow acceleration, typical energies of the
escaping electrons are one-two orders of magnitude lower. The spectra of the
electrons show interesting dependence on the magnetization parameter of the
shell. For extended acceleration (during 15 yrs), the spectra of the electrons
extend to larger energies for strong magnetization. This is due to the fact
that the maximum energies of the electrons are determined in this case by the
dynamical time scale of the shell. For acceleration limited to the early stage
of the shell propagation, the dependence of the electron spectra on the
magnetization is just the opposite. This is due to the fact that, soon after
the nova explosion, the maximum energies of the accelerated electrons are
determined by the synchrotron energy losses. This complicated behaviour is
also in line with the dependency of the maximum electron energies inside the
shell shown in Fig. 2.

Figure 3: The differential spectral energy distribution of the electrons
escaping from the nova shell into the NSR assuming that their acceleration
process operates only during the short period after the nova explosion
(assumed to be 30 days) or during the whole recurrence period of the nova
(i.e. 15 yrs). The electrons are accelerated within the shell to a power-law
spectrum with a spectral index $-2$ and the exponential cut-off at $E_{\rm
max}$. The cut-off energy of $E_{\rm max}$ is defined by two acceleration
prescriptions, slow and fast (see text for details). The magnetization of the
nova shell is defined by $\alpha=10^{-5}$ (solid curve), $10^{-4}$ (dotted),
$10^{-3}$ (dashed), and $10^{-2}$ (dot-dashed). The shell propagates in the
polar region of the nova binary system (no deceleration case). The initial
mass of the nova shell is $M_{\rm sh}^{0}=10^{-6}~{}M_{\odot}$ and its initial
velocity is $v_{0}^{\rm sh}=6000$ km s-1.

Figure 4: As in Fig. 3 but for the part of the shell propagating in the wind
of the RG expelled in the equatorial region of the nova binary system. Then,
the shell decelerates according to the prescription defined in Sect. 6 in
Bednarek (2022). A part of the sphere in which the shell propagates is fixed
on $\Omega=0.3$.

In Fig. 4, we investigate the spectra of the escaping electrons in the part of
the shell which propagates in the equatorial region of the nova binary system.
Since this part of the shell decelerates fast, due to the entrainment of the
dense RG wind material, the energies of the accelerated electrons are lower by
one-two orders of magnitudes. Moreover, the electrons in the shell have much
more time to lose energy on radiation processes already during the confinement
within the shell. Therefore, their energies are usually limited to the sub-TeV
range. The electrons escaping with such energies into the nova super-remnant
mainly contribute to the $\gamma$-ray emission in the GeV range.

Note that recurrent novae of the RS Oph type, are expected to produce many
explosions in a relatively short period. In fact, the electrons escaping from
the last shell should pass through the content of the previous shells. This
process is difficult to consider due to the unknown structure and
magnetization of the previous shells at some time after slow down. We expect
that such “old” shells affects the magnetic field strength within the inner
part of the NSR. We consider this effect by studying much stronger magnetic
field within the NSR (than expected from the values of the interstellar
magnetic field strength) in the further sub-section.

It is assumed that nova shells decelerate completely and merge with the
interstellar medium at the distance $R_{\rm inj}=2\times 10^{17}$ cm, which is
a typical distance for the deceleration of the shells in the medium with
density of the interstellar space, equal to 1 particle cm3 (see Fig. 6 in
Bednarek 2022). Thus, all the electrons that are still present in the shell at
this time feed the NSR. For simplicity we assume that the electrons that were
able to escape earlier as well diffuse in the interstellar medium to $R_{\rm
inj}=2\times 10^{17}$ cm, hence the injection of all the electrons into the
NSR occurs at the same distance. In most of the cases this is a reasonable
assumption. However, the electrons that have energies $\lesssim 10$ GeV and
escaped closer than $3\times 10^{16}$ cm would not be able to diffuse in the
magnetic field of 3 $\mu$G up to $R_{\rm inj}$ due to IC energy losses on RG
radiation field. Nevertheless such low energy electrons are also not likely to
diffuse away out of the shell having still strong magnetic field, and would
more likely by trailed by it up to dispersion of the shell at $R_{\rm inj}$.
Due to Klein-Nishina effect higher energy electrons suffer smaller energy
losses during the diffusion and e.g. 1 TeV electrons can keep most of their
energy even if they escape from the shell at the distance of $3\times 10^{15}$
cm.

##  3 Gamma rays from NSR around a recurrent nova

As we have shown above, the acceleration process of the electrons depends on
the conditions within the nova shell. In the case of recurrent novae, many
shells are expected to provide relativistic electrons to the medium
surrounding the nova super-remnant. The electrons finally escape from the nova
shell into the surrounding medium. They produce $\gamma$ rays in the IC
process by scattering the CMBR and optical radiation from the RG. We
distinguished two main regions in which a nova shell propagates, the
equatorial wind region and the polar region. The physical conditions for the
electrons in these two regions differ significantly. Electrons are injected
into the NSR from these two regions producing $\gamma$-ray emission with
features which can differ significantly. In this section we investigate the
$\gamma$-ray spectra (and their detectability) produced by the electrons from
these two regions as a function of the free parameters of the model. As an
example, we consider the physical parameters of the nova RS Oph and its binary
system (see Introduction). We assume that this recurrent nova explodes with
the average recurrence period of 15 yrs for the last $t_{\rm max}=10^{4}$ yrs
or $10^{5}$ yrs. This time scale for the activity period of the recurrent nova
determines the total energy transferred to the relativistic electrons in the
nova super-remnant.

Relativistic electrons, which were able to escape from the nova shells,
diffuse into the surrounding medium forming NSR. It resembles well known
nebulae around rotation-powered pulsars. During the diffusion process
electrons lose energy on the synchrotron and the Inverse Compton processes.
They produce $\gamma$ rays by scattering the CMBR and RG radiation. In order
to follow the fate of the electrons in the nova super-remnant we modified the
Monte Carlo code which has been originally developed for the diffusion of
electrons within globular clusters (see Bednarek & Sitarek 2007, and Bednarek
et al. 2016). The code follows the diffusion process of electrons through the
globular cluster and their energy losses on the synchrotron process and the
inverse Compton process in the optical radiation from the stars and the CMBR.
In the code, we replaced the radiation field from the stars in the globular
cluster by the radiation field of the RG. In this way we are able to calculate
the $\gamma$ ray (and also synchrotron) spectra produced by the relativistic
electrons during the activity time of the recurrent nova.

###  3.1 Contribution from the polar parts of the shells

In Fig. 5, we show the $\gamma$-ray spectra produced in the nova super-remnant
by the electrons accelerated in terms of the two acceleration scenarios. They
are defined by different prescriptions for the acceleration coefficient which
defines the energy gain rate of the electrons, $\xi=v_{\rm sh}/c$ (fast
acceleration, see figures (a),(b), (e) and (f)) and $\xi=0.1(v_{\rm
sh}/c)^{2}$ (slow acceleration, see figures (c), (d), (g) and (h), see also
Sect. 3, Bednarek 2022). It is assumed that the electrons are injected
continuously during the recurrence period of the nova. The spectra are shown
for different magnetization parameter of the nova shells, $\alpha$. Two
activity periods of the recurrent nova are considered, $T_{\rm active}=10^{4}$
yrs and $10^{5}$ yrs (see Fig. 5). In the case of fast acceleration, the
$\gamma$-ray spectra clearly extend through the multi-TeV energy range, with
slow dependency on the value of $\alpha$. They should be easily detected by
the CTA. The extensive observations with the present Cherenkov telescopes such
as H.E.S.S., MAGIC, or VERITAS can also significantly constrain the allowed
range of considered parameter space of the model. As RS Oph has a rather low
Galactic latitude (Galactic coordinates $l=19.8^{\circ}$, $b\sim
10.37^{\circ}$), where the sensitivity of GeV instruments strongly varies, we
compare the expected emission with two bounding, publicly-
available222https://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm
sensitivities: for latitudes $b=0^{\circ}$ and $30^{\circ}$. It appears that
the sensitivity line for the nova RS Oph location should stay somewhere in the
middle between these two (marked) lines. Therefore, we conclude that in the
case of the older nova super-remnants (with the activity time equal to $T_{\rm
active}=10^{5}$ yrs, see Figs. 5b,d), the $\gamma$-ray emission is expected to
be also constrained by the Fermi-LAT telescope at GeV $\gamma$-ray energy
range. In the case of the slow acceleration model for electrons, the
$\gamma$-ray emission is limited to sub-TeV energies (see Fig. 5cd). In this
case, $\gamma$ rays can hardly be detected by the CTA. But, the allowed
parameter space of the model is expected to be likely constrained by the
observations with the Fermi-LAT.

Figure 5: The $\gamma$-ray spectral energy distribution produced in the IC
scattering of the RG and the CMBR by relativistic electrons which are
accelerated in the polar region of the RG wind. The electrons are accelerated
during the initial period after the nova explosion equal to $t_{\rm max}=15$
yrs (upper panel) and $t_{\rm max}=30$ days (bottom panel), in the sequence of
nova explosions with the recurrence time $t_{\rm rec}=15$ yrs. They lose
energy on the synchrotron process in the NSR magnetic field assumed to be
$B_{\rm NN}=3$ $\mu$G. The electrons were accelerated in the $1-\Omega=0.7$
fraction of the shells propagating in the polar region of the RG wind (without
deceleration), taking $10\%$ of the kinetic energy of the shells. The
acceleration process of the electrons is defined by the acceleration
efficiency $\xi$ and the magnetization parameter of the shell $\alpha$: (a and
e) $\gamma$-ray spectra for $\xi=v_{\rm sh}/c$, $T_{\rm active}=10^{4}$ yrs
and different magnetization $\alpha=10^{-5}$ (solid curve), $10^{-4}$
(dotted), $10^{-3}$ (dashed), and $10^{-2}$ (dot-dashed); (b and f) as in (a
and e) but for $T_{\rm active}=10^{5}$ yrs. (c, d, g and h) as in (a, b, e and
f) but for the slow acceleration model, i.e. $\xi=0.1(v_{\rm sh}/c)^{2}$. The
50 hr sensitivity of the CTA South is marked by the thin dashed curve (see
Fig. 2 in Maier et al. 2017). The dot-dot-dashed lines show the 10 yrs Fermi-
LAT sensitivity (Ajello et al., 2021) for Galactic longitude $l=0^{\circ}$ and
latitude $b=0^{\circ}$ (upper curve) or $b=30^{\circ}$ (lower curve)

In the bottom panel of Fig. 5, we show the $\gamma$-ray spectra for these same
parameter space as in the upper panel of Fig. 5 but assuming that the
electrons are accelerated in the shells only during their initial stage of
propagation after nova explosion, i.e. within the first $t_{\rm max}=30$ days.
This time scale corresponds to the typical period of the GeV $\gamma$-ray
emission detected from novae (e.g. Ackermann et al. 2014). In this case, only
the $\gamma$-ray emission from the nova super-remnant to which the electrons
are injected from the weakly magnetized shells with the fast acceleration
model are within the sensitivity limit of the CTA (see Fig. 5ef).
$\gamma$-rays produced in the slow acceleration model have the chance to be
observed only by the satellite telescopes of the Fermi-LAT type.

###  3.2 Contribution from the equatorial part of the shells

We also calculate the $\gamma$-ray spectra produced within the nova super-
remnant which are expected from the electrons escaping from the ($\Omega$)
part of the shells which propagate within the region of the RG wind (Fig. 6).
Surprisingly, the shape of these $\gamma$-ray spectra look quite similar to
those expected from the electrons escaping from the polar regions of the
shells. This is due to two counter-working effects. We argued that the shell
in the RG wind decelerates due to entrainment of the wind. As a result, the
acceleration coefficient of the electrons drops resulting in lower maximum
energies of the accelerated electrons. But, from another site, lower velocity
of the shell allows more efficient escape of electrons from the shell to the
NSR. As a result, the effect of decelerating shells becomes compensated.

The $\gamma$-ray spectra from the NSR, produced by the electrons escaping from
the decelerating parts of the shells are typically on the level of only a
factor of $\sim$2-3 lower than those expected from the polar regions of the
shell. For the extreme values of the considered model parameters predicted
$\gamma$-ray spectra can be still within the 50 hr sensitivity of the CTA.

Figure 6: As in Fig. 5 but for the part of the shell propagating within the
equatorial region of the RG wind with $\Omega=0.3$. In this case the nova
shells are significantly decelerated due to the entrainment of the RG wind
(see the main text for details).

###  3.3 Dependence of gamma-ray emission on other model parameters

We also study the effect of other model parameters on the $\gamma$-ray
emission from NSR. All the calculations of the $\gamma$-ray emission from the
NSRs have been performed assuming that the relativistic electrons escape from
the shells of novae at distances not larger than $R_{\rm inj}=2\times 10^{17}$
cm. $R_{\rm inj}$ corresponds to the characteristic distance at which nova
shells start to significantly decelerate due to interaction with the
interstellar space. Its value depends on the local density of interstellar
space. In Fig. 7b, we investigate the effect of $R_{\rm inj}$ on the
$\gamma$-ray spectrum from NSR. We show the spectra for $R_{\rm inj}=10^{16}$
cm – $10^{18}$ cm. We conclude that the $\gamma$-ray spectra do not depend
significantly on the injection distance $R_{\rm inj}$ at the TeV energies.
However, lower energy electrons, injected at large distances from the nova
binary system, are not able to cool efficiently in a already weak radiation
field of the RG companion star. Their energy losses on the CMBR are also
inefficient.

Finally, in Fig. 7c, we investigate the dependence of the $\gamma$-ray
spectrum on the activity period of the NSR, Tactive. The $\gamma$-ray emission
is on a comparable level at multi-TeV energies, independent on the age of the
NSR for the considered range of ages, i.e. $10^{4}-10^{7}$ yrs. In fact, the
collision time of the electrons with the CMBR can be estimated on

$\displaystyle\tau_{\rm CMBR}\sim(n_{\rm CMBR}\sigma_{\rm T}c)^{-1}\approx
4.4\times 10^{3}~{}~{}~{}{\rm yrs},$ (2)

where $n_{\rm CMBR}$ is the density of the CMBR and $\sigma_{\rm T}$ is the
Thomson cross section. $\tau_{\rm CMBR}$ is clearly shorter than considered by
us activity period of the recurrent Nova. The scattering process of the CMBR
contributes mainly to the $\gamma$-ray spectrum at GeV energies. This part of
the $\gamma$-ray spectra saturate for the large activity time of the recurrent
nova (see dashed and dot-dashed curves in Fig. 7c). We conclude that the high
energy electrons in the NSR can lose efficiently energy during the time
$T_{\rm active}$.

Figure 7: The $\gamma$-ray spectral energy distribution as a function of: (a)
the strength of the magnetic field within the NSR $B_{\rm NSR}=3$ $\mu$G
(solid), 10 $\mu$G (dashed), 30 $\mu$G (dotted), 100 $\mu$G (dot-dashed), and
300 $\mu$G (dot-dot-dashed) (a). The other parameters are $\alpha=10^{-4}$,
$R_{\rm inj}=2\times 10^{17}$ cm, $T_{\rm rec}=15$ yrs, $T_{\rm
active}=10^{5}$ yrs, and $\xi=0.01$; (b) the injection distance of
relativistic electrons within the NSR $R_{\rm inj}=10^{16}$ cm (dotted curve),
$10^{17}$ cm (dashed), and $10^{18}$ cm (solid). The other parameters are as
above and $B_{\rm NSR}=3$ $\mu$G; and (c) as a function of the activity stage
of the nova $T_{\rm active}=10^{4}$ yrs (solid), $10^{5}$ yrs (dotted),
$10^{6}$ yrs (dashed), and $10^{7}$ yrs (dot-dashed). The other parameters are
the same as in (a) and (b).

###  3.4 Synchrotron emission from electrons in NSR

All the above calculations have been performed under assumption that the
magnetic field strength is of the order of that observed in the interstellar
space (i.e. $B_{\rm NSR}=3\mu$G). In principle, the magnetic field in the
surrounding of the recurrent nova might build up to larger values as observed
in the pulsar wind nebulae. For example, this stronger magnetic field can be
provided to the inner parts of the NSR with the multiple nova shells. In Fig.
7a, we show the $\gamma$-ray spectra calculated for much stronger values of
the magnetic field $B_{\rm NSR}$. As expected, the TeV part of the
$\gamma$-ray spectrum is strongly suppressed for stronger magnetic fields due
to the efficient synchrotron energy losses of the electrons. However, the GeV
$\gamma$-ray emission is still on a comparable level provided that $B_{\rm
NSR}<100\mu$G. For very strong magnetic fields, i.e. $B_{\rm NSR}>100\mu$G,
the GeV $\gamma$-ray emission becomes also strongly suppressed. Therefore, we
conclude that strongly magnetised NSRs around novae have still a chance to be
detected at GeV $\gamma$-ray energies but not at TeV energies.

The TeV $\gamma$-ray emission from NSR is expected to be very sensitive on the
strength of the magnetic field within the NSR. If $B_{\rm NN}$ is larger than
100 $\mu$G, then the TeV $\gamma$-ray emission is below sensitivity of the CTA
even for the most favourite parameters of the model (see Fig. 7a). However,
the $\gamma$-ray emission in the GeV energy range is quite stable, provided
that $B_{\rm NSR}<100$ $\mu$G. If the magnetic field within NSR is
significantly stronger than 3 $\mu$G, then the synchrotron emission from TeV
electrons is expected in the range from the infrared to the X-rays,
$\varepsilon_{\rm syn}=m_{\rm e}c^{2}(B/B_{\rm cr})\gamma_{\rm e}^{2}\sim
0.05B_{\mu G}E_{\rm TeV}^{2}$ eV. We show the expected synchrotron spectra for
the example parameters of the model (as used in Fig. 7a), as a function of the
magnetic field within the NSR in Fig. 8. As expected the synchrotron emission
starts to dominate over the $\gamma$-ray IC emission from the NSR for stronger
magnetic fields. We note that this synchrotron emission in the optical range
is a few orders of magnitudes below the optical emission from the RG. The
thermal X-ray emission, detected in quiescence from RS Oph (e.g. Nelson et al.
2011) or the extended emission observed in the polar regions of the RS Oph
during the outburst in 2006 (Montez et al. 2022) comes from a relatively very
small region (of the order of arc seconds) around the nova RS Oph. This
emission cannot constrain synchrotron X-ray emission expected in our model
which comes from much more extended $\gamma$-ray NSR (see Eq. 7).

Figure 8: The IC (thick curves) and the synchrotron (thin) spectra calculated
for different strength of the magnetic field within the NSR expected around RS
Oph: $B_{\rm NSR}=3$ $\mu$G (solid), 10 $\mu$G (dashed), 30 $\mu$G (dotted)
and 100 $\mu$G (dot-dashed). The other parameters are as in Fig. 7a.

##  4 Estimate of the nova super-remnant size

The propagation process of charged particles in the vicinity of compact
sources is not clear. Therefore, we cannot definitely conclude on the size of
the NSR. In the calculations above, we applied the Bohm diffusion
prescription. But in fact, two limiting scenarios can be considered. If the
magnetic field around the nova is ordered (e.g. magnetic field mainly
perpendicular to the radial direction from the binary system), then
$\gamma$-rays produced in the IC process can be arbitrarily confined forming a
point like source. On the contrary, the magnetic field can be random,
resulting in a rapid diffusion process, e.g. consistent with the Bohm
diffusion prescription. In this second limiting case, we can simply estimate
the maximum size of the NSR provided that injected electrons lose energy on
the IC process by scattering CMBR and RG radiation.

We estimate the size of the $\gamma$-ray region from the NSR for the electrons
with energies $E_{\gamma}=1E_{\rm TeV}$ TeV. Such electrons produce
$\gamma$-ray photons with typical energies within sensitivity of the Fermi-LAT
telescope, $E_{\gamma}\sim\varepsilon_{\rm CMBR}\gamma_{\rm e}^{2}\sim
2.8E_{\rm TeV}^{2}$ GeV as a result of comptonization of the CMBR, where
$\varepsilon_{\rm CMBR}\sim 7\times 10^{-4}$ eV. They also produce
$\gamma$-rays by scattering the RG radiation with energies comparable to the
energy of the electrons, i.e. typical for the Cherenkov telescopes. The
cooling time scale of electrons (in the Thomson regime) with energy equal to 1
TeV in the CMBR is

$\displaystyle\tau_{\rm IC}^{\rm CMBR}\approx 3.2\times 10^{13}/E_{\rm
TeV}~{}~{}~{}{\rm s}.$ (3)

and in the RG radiation

$\displaystyle\tau_{\rm IC}^{\rm RG}\approx 3\times 10^{10}R_{17}^{2}/E_{\rm
TeV}~{}~{}~{}{\rm s}.$ (4)

where $R=10^{17}R_{17}$ cm is the distance measured from the RG, that
determines the radiation field.

Bohm diffusion distance of these electrons during the cooling time scale on
the CMBR, $t=\tau_{\rm IC}^{\rm CMBR}$, is

$\displaystyle x_{\rm dif,CMBR}=\sqrt{2D_{\rm B}t}\approx 29/B_{\rm NSR,\mu
G}^{1/2}~{}~{}~{}{\rm pc},$ (5)

where $B_{\rm NSR,\mu G}$ is $B_{\rm NSR}$ in units of $\mu G$. During the
cooling time scale on the RG radiation the nominal diffusion distance is

$\displaystyle x_{\rm dif,RG}\approx 0.3R_{17}/B_{\rm NSR,\mu
G}^{1/2}~{}~{}~{}{\rm pc},$ (6)

Note that while $x_{\rm dif,RG}<x_{\rm dif,CMBR}$ would suggest the dominant
role of the RG radiation field in the cooling and limiting the extend of the
diffusion this is not the case. For a broad range of parameters ($B_{\rm
NSR,\mu G}<100\mu G$), Eq. 6 states that $x_{\rm dif,RG}>R=10^{17}R_{17}$ cm.
Therefore as the electrons diffuse they experience steeply falling radiation
field from the RG (as $R^{-2}$), that is not able to cool them down. We
conclude that the IC cooling of electrons on the RG radiation do not constrain
the size of the NSR.

In Fig. 9 we present numerical calculations of the energy loss of the
electrons of different energies during their diffusion through the NSR. While
in the early stages after the escape from the shock into the NSR the energy
loss on RG dominates, it is not fast enough to cool down the electrons
completely. As the electrons diffuse farther on, the CMBR losses start to
dominate. This effect is particularly important at higher energies, where the
Klein-Nishina effect further suppresses the energy losses on the RG radiation
field.

Figure 9:  Energy loss of the electrons on the radiation field of RG (dashed
lines) and on CMBR (solid lines, merging together) in their diffusion through
the magnetic field of 3$\mu$G. Electrons are injected at the distance of
$2\times 10^{17}$ cm with energies 0.1 TeV (red), 1 TeV (green) and 10 TeV
(blue). Assumed RG with the luminosity of 100$L_{\odot}$ and temperature of
3600 K.

The size of the electron NSR around Nova RS Oph in case of the Bohm diffusion
and the IC energy losses on the CMBR is then estimated on,

$\displaystyle x_{\rm dif}/D=0.67/B_{\rm\mu G}^{1/2}~{}~{}~{}{\rm degree}.$
(7)

For the magnetic field of $3\mu G$, the size of the NSR around RS Oph is of
the order of $\sim$0.4 degree, comparable to Fermi-LAT angular resolution at a
few GeV. Note that this predicted size is of the order of the size of super-
remnant optically detected in the case of M31N recurrent nova in M31 (see
Darnley et al. 2019). The inner radius of the super-remnant shell in M31N is
52 pc which is not surprising since these recurrent novae clearly differ in
basic parameters (e.g. the recurrence period of M31N is about ten times
shorter which results in larger amount of energy provided by the nova into its
super-remnant.

To validate those order of magnitude calculations we performed numerical
tracking of electrons taking into account energy losses via IC on CMBR and RG
radiation field (including also Klein-Nishina effect) as well as modification
of the diffusion coefficient as the particle is losing energy. Those
calculations confirm that the diffusion process is limited by the CMBR rather
than RG radiation field (however 5 GeV electrons still lose about 25% of their
energy in the latter process). The obtained observed size of the $\gamma$-ray
emission from those calculations is only slowly depending on the energy: $\sim
0.24^{\circ}$ for $E_{\gamma}\lesssim 0.1$ GeV and $\sim 0.28^{\circ}$ for
$E_{\gamma}\gtrsim 1$ TeV. Thus we conclude that the NSR around RS Oph is
predicted to be a point source for the Fermi-LAT and at most only moderately
extended source for the Cherenkov telescopes.

At 200 GeV the angular resolution of CTA is $\sim 0.08^{\circ}$ Maier et al.
(2017). Hence in this Bohm diffusion limiting case, the drop of the
sensitivity of CTA for observations of such slightly extended emission will be
by a factor of $\lesssim 3$. Therefore the extension of the emission might
prevent the detection in the cases when the predicted emission is close to the
point-like sensitivity limit of CTA. However in most of the investigated cases
with long activity time of the source and fast acceleration, the detection is
still within the reach of CTA, even despite the possible extension.

##  5 Conclusion

We investigated the consequences of the hypothesis that electrons are
accelerated in the shells ejected in novae for a long period after their
explosions (see Bednarek 2022). We consider the models which cover a broad
range of possible parameters for the acceleration process of the electrons
within the shell, i.e. slow and fast acceleration and also short (or long)
period injection of electrons after the nova explosion. As an example of the
short period injection we consider the period of the order of a month after
nova explosion (corresponding to the typical time scale of the GeV
$\gamma$-ray emission observed by Fermi-LAT, and typically obtained in a
number of nova acceleration models). The long period injection model assumes
acceleration of electrons in the shell during the recurrence period of the
nova RS Oph. Gamma-ray observations so far were unable to probe the
acceleration over such long time scales. Two regions in the expending nova
ejecta are also distinguished: (1) the equatorial region of the nova binary
system in which the shell is effectively decelerated due to the entrainment of
matter from the RG wind, and (2) the polar region in which the shell expands
freely up to the moment of entrainment of the matter from the interstellar
medium. As an example, electrons are assumed to be injected within the shell
with the power law spectrum (spectral index -2, Bell 1978) and normalization
to the $10\%$ of the kinetic energy of the nova shell. Such an order of the
energy conversion efficiency from the kinetic energy of the shell to particles
is expected in the case of supernova remnants responsible for the cosmic ray
content in the Galaxy (e.g. Schlickeiser 2002). With such a general
acceleration model we likely cover a broad range of possible specific
acceleration scenarios which will in the future consider in more detail the
evolution of the nova shell in long time after explosion. The $\gamma$-ray
emission predicted above, for the range of models with considered by us
boundary parameters, constrains possible emission expected from such future
specific models.

Electrons lose energy on radiation processes in evolving environment of the
shell. Those, accelerated within the shell, can finally escape from the shells
into the nova environment forming so called nova super-remnant. We concentrate
on the case of the recurrent novae which, in such scenario, can periodically
supply fresh relativistic electrons into the NSR. These electrons produce
$\gamma$ rays via ICS of mainly optical radiation from the RG and the CMBR.
The level of this $\gamma$-ray emission depends naturally on the activity
period of the recurrent nova.

We show that in the case of fast and efficient acceleration of the electrons
and weak magnetization of the shell, the $\gamma$-ray emission from the NSR
around the Nova RS Oph is expected to extend to TeV energy range. The
predicted emission is within the sensitivity of the CTA. In more optimistic
cases (acceleration for the whole recurrence period and total activity period
of at least $10^{5}$ yrs), there is a chance to detect the GeV $\gamma$-ray
emission from NSR with Fermi-LAT. We note however, that the NSR is predicted
to have the size of even $\sim 0.3$ degree at GeV and TeV energies (if the
electrons diffuse in the nova surrounding according to the Bohm diffusion
prescription and the average magnetic field strength is $3\mu$G, i.e.
comparable to the interstellar space). So, NSR is expected to be moderately
extended for $\gamma$-ray instruments. However, the magnetic field in the NSR
might be stronger, due to its supply with the incoming shells of nova. This
should essentially limit the size of the electron NSR.

In fact, direct correlated emission in the optical and $\gamma$-ray light
curves of Nova V906 Car at early stage of the nova suggests that emission in
this different energy ranges is produced by this same population of particles,
likely relativistic electrons producing broad low energy emission in the
synchrotron process and the $\gamma$-ray emission in the IC process (Aydi et
al. 2020).

## Acknowledgments

We thank D. Green and R. Lopez-Coto for useful discussions and the Referee for
many useful comments. This work is supported by the grant through the Polish
National Research Centre No. 2019/33/B/ST9/01904.

## References

  * Abdo et al. (2010) Abdo A. A., Ackermann M., Ajello M., Atwood W. B., Baldini L., Ballet J., Barbiellini G., et al., 2010, Sci, 329, 817. doi:10.1126/science.1192537
  * Acciari et al. (2022) Acciari V. A., Ansoldi S., Antonelli L. A., Arbet Engels A., Artero M., Asano K., Baack D., et al., 2022, NatAs, 6, 689. doi:10.1038/s41550-022-01640-z
  * Ackermann et al. (2014) Ackermann, M. et al. 2014, Science 345, 554
  * Ahnen et al. (2015) Ahnen, M.L. et al. 2015 A& A 582, A67
  * Ajello et al. (2021) Ajello, M., Atwood, W. B., Axelsson, M., et al. 2021, ApJS, 256, 12. doi:10.3847/1538-4365/ac0ceb
  * Aydi et al. (2020) Aydi, E., Sokolosky, K.V., Chomiuk, L. et al. Nature Astronomy, 4, 776. DOI:10.1038/s41550-020-1070-y
  * Babul et al. (2022) Babul A.-N., Sokoloski J. L., Chomiuk L., Linford J. D., Weston J. H. S., Aydi E., Sokolovsky K. V., et al., 2022, MNRAS, 515, 3028. doi:10.1093/mnras/stac1366
  * Bailer-Jones et al. (2021) Bailer-Jones, C.A.L., Rybizki, J., Fouesneau, M., Demleitner, M., & Andrae, R. 2021, AJ, 161, 147, doi: 10.3847/1538-3881/abd806
  * Bednarek&Sitarek (2007) Bednarek, W., Sitarek, J. 2007 MNRAS 377, 920
  * Bednarek et al. (2016) Bednarek, W., Sitarek, J., Sobczak, T. 2016 MNRAS 458, 1083
  * Bednarek (2022) Bednarek, W. 2022 MNRAS 515, 1644
  * Bell (1978) Bell, A.R. 1978 MNRAS 182, 147
  * Bode et al. (2006) Bode, M.F. et al. 2006 ApJ 652, 629
  * Booth, Mohamed, & Podsiadlowski (2016) Booth R. A., Mohamed S., Podsiadlowski P., 2016, MNRAS, 457, 822. doi:10.1093/mnras/stw001
  * Brandi et al. (2009) Brandi E., Quiroga C., Mikołajewska J., Ferrer O. E., García L. G., 2009, A&A, 497, 815. doi:10.1051/0004-6361/200811417
  * Buil (2006) Buil, C. 2006, Central Bureau Electronic Telegrams, No. 403, 1
  * Chesneau et al. (2007) Chesneau O., Nardetto N., Millour F., Hummel C., Domiciano de Souza A., Bonneau D., Vannier M., et al., 2007, A&A, 464, 119. doi:10.1051/0004-6361:20066609
  * Cheung et al. (2022) Cheung, C.C., Johnson, T.J., Jean, P. et al. 2022 ApJ, 935, 44
  * Chomiuk et al. (2021) Chomiuk, L., Linford, J.D., Aydi, E. et al., 2021, ApJ, 257, 49C. doi:10.3847/1538-4365/ac24ab
  * Darnley et al. (2023) Darnley, M.J. et al., 2023, MNRAS (in press) doi:10.1038/s41586-018-0825-4
  * H. E. S. S. Collaboration et al. (2022) H. E. S. S. Collaboration, Aharonian F., Ait Benkhali F., Angüner E. O., Ashkar H., Backes M., Baghmanyan V., et al., 2022, Sci, 376, 77. doi:10.1126/science.abn0567
  * Dumm & Schild (1998) Dumm, T., Schild, H. 1998 New Astronomy 3, 137
  * Healy-Kalesh et al. (2023) Healy-Kalesh, M.W., Darnley, M.J., Harvey, E.J. et al., 2023, MNRAS (in press) doi:10.1093/mnras/stad617
  * Iijima (2008) Iijima T., 2008, ASPC, 401, 115
  * Maier et al. (2017) Maier, G. et al. (CTA Consortium) 2017, in Proc. 35th ICRC (Busan, Korea), PoS(ICRC2017), 846
  * Martin & Dubus (2013) Martin, P., Dubus, G. 2013 A& A 551,A37
  * Martin et al. (2018) Martin, P. et al. 2018 A& A 612, A38
  * Metzger et al. (2015) Metzger, B.D. et al. 2015 MNRAS 450, 2739
  * Montez et al. (2022) Montez R., Luna G. J. M., Mukai K., Sokoloski J. L., Kastner J. H., 2022, ApJ, 926, 100. doi:10.3847/1538-4357/ac4583
  * Munari et al. (2022) Munari, U., Giroletti, M., Marcote, B. et al., 2022, A&A 666, L6. doi:10.1051/0004-6361/202244821
  * Nelson et al. (2011) Nelson, T., Mukai, K., Orio, M., Luna, G.J.M., Sokoloski, J.L. 2011 ApJ 737, 7
  * Nyamai et al. (2023) Nyamai, M.M., Linford, J.D., Allison, J.R. et al. 2023, ApJ (in press) doi:10.48550/arXiv.2301.09116
  * Rupen, Mioduszewski, & Sokoloski (2008) Rupen M. P., Mioduszewski A. J., Sokoloski J. L., 2008, ApJ, 688, 559. doi:10.1086/525555
  * Schaefer (2009) Schaefer B. E., 2009, ApJ, 697, 721. doi:10.1088/0004-637X/697/1/721
  * Schaefer (2010) Schaefer B. E., 2010, ApJS, 187, 275. doi:10.1088/0067-0049/187/2/275 2010, ApJS, 187, 275. doi:10.1088/0067-0049/187/2/275
  * Schaefer (2022) Schaefer B. E., 2022 MNRAS, 517, 6150. doi:10.1093/mnras/stac2900
  * Schlickeiser (2002) Schlickeiser, R. 2002 Cosmic Ray Astrophysics, A&A Library, Springer-Verlag Berlin Heidelberg
  * Sitarek & Bednarek (2012) Sitarek, J., Bednarek, W. 2012 PRD 86, 063011
  * Tatischeff & Hernanz (2077) Tatischeff, V., Hernanz, M. 2007 ApJL, 663, L101
  * Vurm & Metzger (20180) Vurm, I., Metzger B.D. 2018 ApJ, 852, 62
  * Walder, Folini, & Shore (2008) Walder R., Folini D., Shore S. N., 2008, A&A, 484, L9. doi:10.1051/0004-6361:200809703
  * Wallerstein (1958) Wallerstein G. 1958 PASP 70, 537
  * Zheng et al. (2022) Zheng, J-H., Huang, Y-H., Zhang, H-M., Liu, R-Y., Wang, X-Y. 2022 PRD 106, 103011. 10.1103/PhysRevD.106.103011