# Axion spectra and the associated x-ray spectra of low-mass stars

Ilídio Lopes ID<EMAIL_ADDRESS>Centro de Astrofísica e
Gravitação - CENTRA, Departamento de Física, Instituto Superior Técnico - IST,
Universidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

###### Abstract

Axion particles are among the best candidates to explain dark matter and
resolve the strong $CP$ problem in the Standard Model. If such a particle
exists, the core of stars will produce them in large amounts. For the first
time, we predict the axion spectra and their associated luminosities for
several low-mass stars – one and two solar masses stars in the main sequence
and post-main sequence stages of evolution. Equally, we also compute the x-ray
excess emission resulting from the conversion of axions back to photons in the
presence of a strong magnetic field in the envelope of these stars. Hence, a
given star will have a unique axion spectrum and $L_{a}$ axion luminosity. And
if such star has a strong magnetic field in its stellar envelope, it will also
show a characteristic x-ray spectrum and $L_{a\gamma}$ x-ray luminosity. Such
radiation will add up to the x-ray electromagnetic spectrum and $L_{X}$
luminosity of the star. The present study focuses on axion models with an
axion-photon coupling constant, $5\;10^{-11}{\rm GeV}^{-1}$, a value just
below the most recent upper limit of $6.6\;10^{-11}{\rm GeV}^{-1}$ found by
CAST and IAXO helioscopes. The range of axion parameters discussed here spans
many axion models’ parameter space, including the DFSZ and KSVZ models. We
found that axions with a mass in the range $10^{-7}$ to $10^{-5}\;{\rm eV}$
and an axion-photon coupling constant of $5\;10^{-11}{\rm GeV}^{-1}$ produce
an axion emission spectra with an averaged axion energy that varies from 1 to
5 KeV, and an $L_{a}$ ranging from $10^{-6}$ to $7\;10^{-3}\;L_{\odot}$. We
also predict that $L_{a\gamma}$ varies from $5\;10^{-8}$ to
$10^{-5}\;L_{\odot}$ for stars with an averaged magnetic field of
$3\;10^{4}\;{\rm G}$ in their atmospheres. Most of these $L_{a\gamma}$
predictions are larger than the $L_{X}$ observed in some stars. Therefore,
such preliminary results show the potential of the next generation of stellar
x-ray missions to constrain several classes of axion models.

Neutrinos – Sun:evolution –Sun:interior – Stars: evolution –Stars:interiors

##  I I. Introduction

The axion or their closest relatives – the axionlike particles (ALP) are among
the most popular candidates that are being proposed to explain the existence
of dark matter. An ALP is a ultralight pseudoscalar boson $a$ predicted by
several extensions of the Standard Model (SM, e.g., Di Luzio et al., 2020).
ALPs are well-motivated particles that occur in many extra-dimension theories,
like string theory and notably the M theory (e.g., Turok, 1996; Marsh, 2016).
Nowadays, all these types of particles capable of solving the strong $CP$
problem are many times referred to as QCD axion (Dafni et al., 2019). In this
article, we opt to call all these different kinds of particles, ALPs included,
simply by "axions" if not stated otherwise.

Many of these SM extensions predict the existence of axion channels’ with
photons $\gamma$ and electrons $e$ (Peccei & Quinn, 1977; Wilczek, 1978;
Weinberg, 1978); such processes are regulated by the coupling constants
$g_{a\gamma}$ or $g_{ae}$ respectively. Two large classes of axion models are
currently quite popular in the literature, the Dine-Fischler-Srednicki-
Zhitnitsky (DFSZ, e.g., Dine et al., 1981) and the
Kim–Shifman–Vainshtein–Zakharov (KSVZ, e.g., Kim, 1979; Shifman et al., 1980)
in which the anomaly is carried by SM quarks or new colored fermions,
respectively (e.g., Kim, 2010). The two axion channels mentioned above occur
in DFSZ and KSVZ models, although the axion-electron interaction is more
common in the grand unification theories or GUTs (e.g., Irastorza & Redondo,
2018). In astrophysical scenarios both processes can exist, the dominant
spectrum will depend mostly on the magnitude of the relative values of
$g_{ae}$ and $g_{a\gamma}$ (e.g., Di Luzio et al., 2020). In this work we will
focus on the latter process.

The axion-photon channel gives rise to the Primakoff production of axions
inside stars. Indeed, Primakoff (1951) discovered that in the presence of
external electric and magnetic fields photons convert into axions. Although
many works compute the axion emission spectrum for the Sun, this has not yet
been done for other low-mass stars. A detailed calculation of the solar axion
emission spectrum is found in Andriamonje et al. (2007) and references
therein. A more detailed study by Guarini et al. (2020) found that solar
large-scale magnetic fields enhance the axion emissivity by the coherent
conversion of thermal photons to axions. It was also reported that, in the
Sun, the plasmon-axion conversion could compete with the Primakoff production
of axions (O’Hare et al., 2020). Equally, Caputo et al. (2020) found that,
depending on the strength and nature of the solar magnetic field, the plasmon-
axion conversion dominates over Primakoff production for energies lower than
$200\;eV$.

Here, we are interested in studying for the first time the axion spectrum
emitted inside low-mass stars and the electromagnetic spectrum excess produced
by the axion-to-photon conversion in the atmosphere of the same stars.
Accordingly, we will compute the Primakoff production of axions inside stars
and their transformation back to x-ray radiation by inverse Primakoff
production in the stellar surface. Since we compute the spectra of axions in
the energy range from 1 to 15 KeV, the contribution of addition mechanisms
mentioned above and in the Refs. (Guarini et al., 2020; O’Hare et al., 2020;
Caputo et al., 2020) is negligible. This study is of interest for the next
generation of stellar x-ray observational missions, for which the high
precision measurement of the electromagnetic field will make it possible to
put a stringent constraint on several classes of axion models. Low-mass stars
with masses varying from 1 to 2 solar masses are good observational targets to
study the impact of axions on the stellar evolutions. Among other reasons, we
highlight the following ones: (i) these stars are very abundant in the
Universe; (ii) the internal structure of these stars, including our own Sun,
is very well known (by means of helio- and asteroseismology), much better than
for more massive stars (which are much less abundant in the Universe); (iii)
the atmosphere of these stars and their magnetic fields are well understood.

We organize the article in seven sections: In Sec. II we explain the
motivation for the axion existence in particle physics, cosmology and
astrophysics; in Sec. III, we motivated the interest of using low mass stars
to study the axion properties; in Sec. IV we explain how the axions are
produced inside low-mass stars, and compute the axion spectrum for these stars
in the main-sequence and post-main sequence phases; in Sec. V we present the
model used to calculate the x-ray spectrum of axions; in Sec. VI we analyse
and discuss the x-ray emission from axions for several stellar models. And
finally, in Sec. VII, we summarize and present our results. If not stated
otherwise, we work in natural units where the speed of light in the vacuum,
the Planck constant and the Boltzmann constant are set to unity
$\hbar=c=k_{B}=1$.

##  II II. Axion motivation in particle physics and astrophysics

The axion appeared originally on a pioneer generalization of the standard
model that was made by introducing a new global symmetry known now as the
Peccei-Quinn symmetry (Peccei & Quinn, 1977). Such transformation allows to
explain the absence of the $CP$ violation in the strong interaction (the so-
called QCD strong $CP$ problem). This first model was proposed by Peccei &
Quinn (1977), as well as by Weinberg (1978) and Wilczek (1978) and since then
is known as the Weinberg-Wilczek-Peccei-Quinn model. The axions in this model
have a symmetry-breaking scale of the order of the electroweak scale. Since
this model has been ruled out, many alternative models (usually referred to as
ALP models) are being developed where the symmetry-breaking scale becomes
arbitrary.

The success of the axion theories has motivated many experimental physicists
to propose and develop experiments to search for such particles: several
detectors around the world are actively searching for the axions using the
inverse Primakoff effect. The most well known of these detectors is the CERN
Axion Solar Telescope or CAST (i.e. Bernabei et al., 2001). The most recent
upper limit obtained CAST fixed the upper limit of the axion-photon coupling
in $6.6\;10^{-11}{\rm GeV}^{-1}$ at 95% confidence level (Anastassopoulos et
al., 2017). Moreover, a new generation of experiments is being built to
continue the CAST research, like International Axion Observatory (i.e., IAXO,
Armengaud et al., 2014) or the dielectric haloscope MADMAX (i.e., MAgnetized
Disk and Mirror Axion eXperiment , Egge et al., 2020; Knirck et al., 2018).
However many other detectors based on different experimental techniques and
strategies have been proposed during the last decade to look for axions, like
multilayer optical haloscopes (Baryakhtar et al., 2018), plasma haloscopes
(Lawson et al., 2019), dish antennae (Horns et al., 2013), and a new class of
detectors based on Antiferromagnetically doped topological insulators (TOORAD;
Marsh et al., 2019).

Axions are excellent candidates to explain many features that occur during the
formation and evolution of the Universe, including inflation, dark radiation
and dark energy (e.g., Widdicombe et al., 2018). The production and existence
of axions has been suggested in many cosmological and astrophysical contexts,
such as pure axion stars and hybrid axion stars (i.e., Chavanis, 2018;
Visinelli et al., 2018; Panotopoulos and Lopes, 2019), to mention a few. In
some scenarios, the gravitational field of a recently formed axion star can
capture neutral hydrogen leading to the appearance of a hypothetical mix of
stars (Bai, Barger, & Berger, 2016). Unlike pure axion stars, due to the
electromagnetic emission these should be easy to observe. Axions are now known
to produce many other astrophysical phenomena. For instance, they produce an
unique electromagnetic spectrum stimulated by the presence of strong magnetic
fields during the core-collapse of a supernova (Payez et al., 2015), or
trigger superradiant instability of spinning black holes (Cardoso et al.,
2018).

##  III III. Axions, stars and magnetic fields

The impact of axions on stars has already been discussed for a diverse
population of stellar objects. In general, the presence of axions inside stars
leads to the formation of efficient energy-loss mechanisms. Due to this it has
been possible to put constraints to the properties of such particles, for
instance using the Sun (van Bibber et al., 1989), red giant stars(Viaux et
al., 2013), white dwarfs (Dessert et al., 2019) and neutron stars (Perna et
al., 2012). There are already many observations of stars at different stages
of their evolution that show trends of high energy losses and unique features
in their electromagnetic spectra. In many of these cases, the standard cooling
mechanisms are unable to explain such astronomical phenomena. The only way to
describe these observations is to propose non-standard cooling channels; for
which axion emission is a very plausible solution (e.g., Isern et al., 1992;
Raffelt et al., 2011; Giannotti et al., 2016, 2017).

The fact that the structure of low-mass stars is much better known than other
stars also results from the high-quality data already available and possible
to be obtained by the current (Michel et al., 2008; Koch et al., 2010) and
future (Rauer et al., 2014) asteroseismology missions. Asteroseismology probes
the internal structure of these stars with unprecedented precision. These
high-quality observations allow to obtain a detailed knowledge of the
thermodynamical structure of many low-mass stars at different stages of
evolution: main-sequence, sub-giant and red-giant phases. Such a unique set of
asteroseismic data combined with high-precision spectroscopic observations
make such stars a unique precision laboratory of experimental physics study to
the properties of new particles like axions.

Particularly relevant for this study is the existence of magnetic fields in
these stars. In the last seven decades powerful techniques based on the Zeeman
effect (e.g., Babcock, 1947; Donati and Landstreet, 2009) and
spectropolarimetry (e.g., Donati et al., 1997) have made possible the
detection and measurement of magnetic activity of several star’s types across
the HR diagram. Recent measurements found that main sequence and sub-giants
stars with masses below 2.5 $M_{\odot}$ have surface magnetic fields that vary
between $0.5$ and $40\;{\rm kG}$ (e.g., Berdyugina, 2009; Kochukhov et al.,
2019; Kochukhov, 2019). Classical examples of magnetic main-sequence stars are
HD 215441 ($B\sim 3.4\times 10^{4}\;{\rm G}$), HD 154708 ($B\sim 2.45\times
10^{4}\;{\rm G}$), HD 137509 ($B\sim 2.9\times 10^{4}\;{\rm G}$), and HD 75049
($B\sim 3\times 10^{4}\;{\rm G}$) (see Ferrario et al., 2015, and references
therein). Such stellar activity usually is accompanied by a significant x-ray
emission, i.e., a large luminosity in the x-ray energy band of the
electromagnetic spectrum. See et al. (2016) found that for main-sequence stars
with masses smaller than 1.5 $M_{\odot}$, $L_{\small X}$ varies from $10^{-6}$
to $10^{-4}\;L_{\odot}$, where emissions have a sinusoidal behavior with
periods varying from a few days to a month. This intense x-ray radiation is
particularly prominent on the chemically peculiar $A_{p}$ and $B_{p}$ stars
(Robrade, 2016). Since dynamo processes are at the origin of magnetic fields
in the outer layers of all these stars (e.g., Lopes et al., 2014), for which
their intensities change from the equator to the poles of the star and over
long stellar cycles (Nagovitsyn et al., 2017; Savanov, 2019). Baliunas et al.
(1995) reported for the first time in a systematic manner the existence of
sun-like magnetic cycles on low mass stars by measuring the chromospheric
variations on these stars. Consequently, most of the x-ray emission in these
stars will be strongly time-dependent. For some of these stars it is even
possible to observe such x-ray time variability. Recently the XMM-Newton
satellite has discovered such a long-term X-ray magnetic cycle in
$\epsilon$-Eri (Coffaro et al., 2020). Such x-ray time dependence results from
the variability of the stellar magnetic field. As we will discuss later, such
magnetic field time dependence induces a variation on the x-ray spectrum
resulting from axion conversion to photons in the stellar atmosphere.

##  IV IV. Axion Primakoff emission spectrum in stars

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Figure 1: The radial distribution of the axion energy loss rate
$\mathbf{(d\Phi/dE)_{a}}$ in the core of the star for the models: top panels
\- $A_{\odot}$ (Sun) and $A_{1}$; lower panels \- $A_{2}$ and $B_{3}$. The
description of the models is on Table 1. The contours correspond to the
variation of $\mathbf{(d\Phi/dE)_{a}}$ (as a percentage) between the centre
and the surface of the star. In this representation, $\mathbf{(d\Phi/dE)_{a}}$
is independent of $g_{a\gamma}$.

The phenomenology commonly used to describe all axions (including ALPs) result
from these being very light particles with relatively low masses and very weak
interactions. These particles interact so weakly with other particles that
they have in the past been referred to as invisible (e.g. Kim, 1979; Dine et
al., 1981). Axions behave similarly to neutrinos, like them they provide an
additional cooling mechanism in stars, their impact on the evolution of a star
is well documented (e.g., Raffelt, 2008; Archidiacono et al., 2015).

The interior of stars is a relatively well understood weakly coupled plasma
which permits the precise calculations of axion production reactions (Kim,
2010). Although many different types of axion reactions occur inside stars,
the most crucial axion reaction is the axion-to-photon coupling – Primakoff
reaction, that drives the production of axions in photon collisions with
charged particles $(Z)$ in the stellar plasma: $\gamma+Z\rightarrow a+Z$
(e.g., Raffelt, 1988; Irastorza & Redondo, 2018). This process that produces
(or annihilates) axions reads

$\displaystyle{\cal
L}_{a\gamma}=-\frac{g_{a\gamma}}{4}F_{\mu\nu}F^{\mu\nu}a=g_{a\gamma}\mathbf{E}\cdot\mathbf{B}\,a,$
(1)

where $a$ is the axion field, $F$ and $\tilde{F}$ the electromagnetic field-
strength tensor and its dual, $\mathbf{E}$ and $\mathbf{B}$ are the electric
and magnetic fields, and $g_{a\gamma}$ the axion-photon coupling constant.
Such interaction implies the conversion of $\gamma$ into $a$ (and its reverse)
in the presence of external electric and magnetic fields.

The hot plasma of the interior of most stars is an excellent source for the
production of axions with an energy of several keV, by the transformation of
thermal x-ray photons into axions in the electric fields of the charged
particles (Raffelt, 1988). The transition rate for a photon of energy $E$ into
an axion of the same energy by the Primakoff effect in a stellar plasma is

$\displaystyle\Gamma_{\gamma
a}=\frac{g_{a\gamma}^{2}Tk_{s}^{2}}{32\pi}\left[\left(1+\frac{k_{s}^{2}}{4E^{2}}\right)\ln{\left(1+\frac{4E^{2}}{k_{s}^{2}}\right)}-1\right],$
(2)

where T is the temperature and $k_{s}$ the screening scale. $k_{s}$ in the
Debye-Hückel approximation is

$\displaystyle
k^{2}_{s}=\frac{4\pi\alpha_{em}}{T}\left(n_{e}+\sum_{nuclei}Z_{j}^{2}n_{j}\right),$
(3)

where $\alpha_{em}$ is the fine-structure constant, $n_{e}$ and $n_{j}$ are
the number densities of electron and of the jth ion of charge $Z_{j}$. Thus,
the axion flux spectrum produced by the star up to a certain dimensionless
radius ($0\leq r\leq 1$) is obtained from

$\displaystyle\left(\frac{d\Phi}{dE}\right)_{a}=2\pi\int_{0}^{r}\varphi_{a}(r,E)\;rdr,$
(4)

where $r=R/R_{\star}$ and $R_{\star}$ is the total stellar radius.
$\varphi_{a}(r,E)$ is the stellar axion luminosity at the dimensionless radius
(Andriamonje et al., 2007), accordingly

$\displaystyle\varphi_{a}(r,E)=\frac{R_{\star}^{3}}{2\pi^{3}D_{\rm
AU}^{2}}\int_{r}^{1}\frac{Ek\Gamma_{\gamma
a}}{e^{E/T}-1}\frac{xdx}{\sqrt{x^{2}-r^{2}}},$ (5)

where $D_{\rm AU}$ is the fiducial distance from the Earth to the star. The
wave number $k$ is

$\displaystyle k^{2}=E^{2}-\omega_{pl}^{2},$ (6)

where $\omega_{pl}$ is the electron plasma frequency. This last quantity is
given by

$\displaystyle\omega_{pl}=\sqrt{\frac{4\pi\alpha_{em}n_{e}}{m_{e}}},$ (7)

where $m_{e}$ is the electron mass. The total axion flux at the 1 A.U. is

$\displaystyle\Phi_{a}=2\pi\int_{0}^{1}\int^{\infty}_{\omega_{\rm
pl}}\varphi_{a}(r,E)\,dE\,rdr,$ (8)

the mean axion energy is

$\displaystyle\langle
E_{a}\rangle=\frac{2\pi}{\Phi_{a}}\int_{0}^{1}\int^{\infty}_{\omega_{\rm
pl}}\varphi_{a}(r,E)\,E\,dE\,rdr,$ (9)

and the total axion luminosity is $L_{a}=4\pi D_{\rm AU}^{2}\Phi_{a}$.

In Table 1 we summarized the main characteristic of the stellar models
considered in this study. Figure 1 shows the contour plot of the axion
luminosity inside the star [Eq. (4)] as a function of the axion energy E and
the dimensionless radius for some stellar models. Figure 2 shows the axion
emission spectra of the same stellar models present in Table 1. For
convenience we consider that all stars are located at a fiducial distance of 1
A.U.

The stellar models were computed with the release version 12115 of the stellar
evolution code MESA (Paxton, et al., 2011, 2013, 2015, 2018, 2019). For the
computation of these stellar models, we assume the input physics used for the
standard solar model (SSM) but with the necessary adaptation for these stars.
The SSM is a reference solar model: a one-dimensional stellar evolution of one
solar mass star allowed to evolve in time until the present-day solar age,
$4.57$ Gyrs, having been calibrated to the values of luminosity and effective
temperature of the present Sun, respectively, $3.8418\times 10^{33}$ erg s-1
and $5777$ K, as well as the observed abundance ratio at the Sun’s surface:
(Z${}_{\text{s}}$/X${}_{\text{s}}$)${}_{\odot}=0.0181$, where $Z_{s}$ and
$X_{s}$ are the metal and hydrogen abundances at the surface of the star
(Turck-Chieze and Lopes, 1993; Bahcall, et al., 1995, 2006). The details of
this SSM in which we use the AGSS09 (low-Z) solar abundances (Asplund, et al.,
2009) can be found in Lopes and Silk (2013) and Capelo and Lopes (2020). All
the generated one and two-solar mass models start in the pre-main sequence,
assuming the star was initially chemically homogeneous and fully convective.

The SSM corresponds to the model $A_{\odot}$ in Table 1. In Figs. 1 (top-left
panel) and 2 (top and lower panels – blue light curve) is shown the solar
axion luminosity contours and the axion solar spectrum, respectively. In the
first line of columns 7 and 8 of Table 1 are shown some characteristics of the
axion emissivity for the Sun. In particular, we check that for an axion model
with $g_{a\gamma}=10^{-10}\;{GeV}^{-1}$, the total solar axion flux at the
Earth is $3.75\;10^{11}\;{\rm cm^{-2}s^{-1}}$ near the value predicted
originally by van Bibber et al. (1989), which corresponds to a luminosity
$1.9\;10^{-3}L_{\odot}$. The properties of this axion model are consistent
with the axion emissivity contours and emission spectrum found in the
literature for the Sun, for instance by Andriamonje et al. (2007) and Arik et
al. (2009), respectively.

Table 1 shows five solar-mass star models at different stages of evolution
from the pre-main sequence up to the red giant phase, and another four models
for two solar-mass stars at identical evolution stages. Conveniently, we
choose the fiducial value of $g_{a\gamma}=5\;10^{-11}\;{\rm GeV}^{-1}$; this
value is slightly lower than the current upper limit found by CAST helioscope
(Anastassopoulos et al., 2017). Figure 1 also shows that the axion emission
always occurs in the stellar core, within a radius smaller than
$0.5R_{\star}$. The magnitude and energy range of the axions emitted is
strongly dependent on the physics of the stellar core [cf. Eqs. (2) and (3)].
Specifiably, the core’s temperature and density for these stars varies from
$5.2\,10^{6}$ to $2.5\,10^{7}$ ${}^{o}K$ and from $3$ to $1.5\,10^{3}$
$g\;cm^{-3}$.

This dependence of the axion emissivity on the structure of the star leads to
quite distinct axion spectra shapes in comparison with the Sun (see Fig. 2).
For instance, one of the two solar-mass stars has an axion spectrum that is 15
times larger than the solar one (compare $A_{\odot}$ and $B_{1}$ models in
Fig. 2). Moreover, the average axion energy $\left\langle E_{a}\right\rangle$
and axion luminosity $L_{a}$ vary significantly between models (see Table 1).
Equally, the axion luminosity changes by several orders of magnitude from
$10^{-3}$ to $10^{-6}L_{\odot}$ (see Table 1), and the shape of the axion
spectra changes slightly with the star; the more massive stars have a more
intense axion emission spectrum as shown in Fig. 2.

Figure 2: Axion emission spectrum $(d\Phi/dE)_{a}$ due to Primakoff axion production: We compare the axion emission fluxes for stars with 1 and 2 solar masses obtained from up-to-date stellar evolution models. The different curves correspond to models shown in Table 1: top panel: $A_{1}$ ($0.01$ Gyr, red curve); $A_{2}$ ($10.61$ Gyr, green curve); $A_{3}$ ($11.37$ Gyr, blue curve); $A_{4}$ ($11.58$ Gyr, cyan curve); $A_{5}$ ($12.16$ Gyr, brown curve); lower panel: $B_{1}$ ($0.02$ Gyr, red curve);$B_{2}$ ($0.7$ Gyr, gree curve); $B_{3}$ ($0.98$ Gyr, blue curve); $B_{4}$ ($1.05$ Gyr, cyan curve); The Sun ($A_{\odot}$, $4.6$ Gyr, orange curve) is plotted in both panels. For the calculation of the axion spectrum, we use an axion–photon coupling of $g_{a\gamma}=5\times 10^{-11}\;{\rm GeV^{-1}}$. Star | Mass | Age | Radius | Lum. | Temp. | $\left\langle E_{a}\right\rangle$ | $L_{a}$ | $L_{a\gamma}$
---|---|---|---|---|---|---|---|---
| $M_{\odot}$ | Gyr | $R_{\odot}$ | $L_{\odot}$ | $10^{3}{\rm{}^{o}K}$ | ${\rm KeV}$ | $L_{\odot}$ | $L_{\odot}$
$A_{\odot}$ | $1.0$ | $4.6$ | $1.0$ | $1.0$ | $5.87$ | $4.2$ | $4.6\;10^{-4}$ | $3.7\;10^{-7}$
$A^{*}_{\odot}$ |  |  |  |  |  | $4.2$ | $1.8\;10^{-5}$ | $5.9\;10^{-10}$
$A^{**}_{\odot}$ |  |  |  |  |  |  |  | $4.9\;10^{-8}$
$A_{1}$ | $1.0$ | $0.01$ | $1.41$ | $0.59$ | $4.27$ | $1.6$ | $1.1\;10^{-5}$ | $1.6\;10^{-8}$
$A_{2}$ | $1.0$ | $10.6$ | $1.45$ | $1.92$ | $5.64$ | $4.8$ | $6.8\;10^{-4}$ | $1.1\;10^{-6}$
$A_{3}$ | $1.0$ | $11.4$ | $1.75$ | $2.10$ | $5.3$ | $4.7$ | $4.4\;10^{-4}$ | $1.1\;10^{-6}$
$A_{4}$ | $1.0$ | $11.6$ | $2.01$ | $2.25$ | $5.0$ | $4.4$ | $2.4\;10^{-6}$ | $1.1\;10^{-7}$
$A_{5}$ | $1.0$ | $12.2$ | $6.0$ | $15.3$ | $4.7$ | $1.5$ | $4.2\;10^{-6}$ | $2.1\;10^{-7}$
$B_{1}$ | $2.0$ | $0.02$ | $1.63$ | $16.1$ | $9.1$ | $5.3$ | $6.9\;10^{-3}$ | $1.4\;10^{-5}$
$B^{*}_{1}$ |  |  |  |  |  | $5.3$ | $2.8\;10^{-4}$ | $2.2\;10^{-8}$
$B^{**}_{1}$ |  |  |  |  |  |  |  | $1.0\;10^{-6}$
$B_{2}$ | $2.0$ | $0.7$ | $2.42$ | $19.9$ | $7.84$ | $5.0$ | $5.1\;10^{-3}$ | $2.3\;10^{-5}$
$B_{3}$ | $2.0$ | $1.0$ | $3.50$ | $28.8$ | $7.15$ | $5.2$ | $1.1\;10^{-6}$ | $2.1\;10^{-7}$
$B_{4}$ | $2.0$ | $1.1$ | $16.2$ | $101.9$ | $4.56$ | $0.7$ | $1.1\;10^{-6}$ | $2.0\;10^{-7}$
Table 1: Comparison of the axion properties of several stellar models,
including an up-to-date standard solar model of the Sun (Capelo and Lopes,
2020). The table shows different stellar quantities such Mass, Age, Radius,
Luminosity (Lum.), Temperature (Tem.), Axion Luminosity and x-ray axion
Luminosity (see the main text). The axion has a mass of $m_{a}=10^{-7}\;{\rm
eV}$ and axion–photon coupling $g_{a\gamma}=5\;10^{-11}\;{\rm GeV^{-1}}$. The
magnetic conversion layer in the star has a thickness of $\Delta
R_{\star}=0.15R_{\star}$ and an average magnetic field of $B=3\;10^{4}\;{G}$.
In the calculation of the model pairs: $A^{*}_{\odot}$ and $B^{*}_{1}$, and
$A^{**}_{\odot}$ and $B^{**}_{1}$, we assume $g_{a\gamma}=10^{-11}{\rm
GeV}^{-1}$, and $m_{a}=10^{-5}{\rm eV}$ respectively.

##  V V. Inverse Primakoff interaction

In some regions of the atmosphere of most stars, the stellar plasma has the
conditions to naturally stimulate the conversion of axions to photons (Zioutas
et al., 2009). The inverse Primakoff reaction occurs when the energy of the
photons is sufficiently far above all resonances [Eq. 6]. Since we are
concerned with axions with energy in the KeV range this condition
automatically satisfies (van Bibber et al., 1989). Typically, such an axion
process occurs in a region made of a plasma of low density and metallicity
immersed in a strong magnetic field. To that end, we will assume that the
inverse Primakoff reaction occurs in a magnetized region of the star with a
low density. This condition is satisfied in the outer layers of most main-
sequence and post-main sequence stars since most of the stellar mass locates
in the internal region of the star. For instance for a star like the Sun, 98%
of its mass is below $0.7$ of its radius (Capelo and Lopes, 2020). The
external layer of these stars with very low densities are known to be
responsible for the generation of a strong magnetic field by dynamo action
when in the presence of differential rotation and meridional flows (Lopes et
al., 2014). Accordingly, as predicted by many stellar dynamo models, we assume
the existence of a magnetic layer located between the upper layers of the star
and lower part of the stellar atmosphere, where the near-surface convection
influences the local magnetic fields. This most external layer is responsible
for the magnetic activity observed in many stars (e.g., Passos and Lopes,
2008; Beeck et al., 2015; Braithwaite and Spruit, 2017). Such large
concentrations of a magnetic field emerging on the stellar surface lead to the
formation of active regions encompassing magnetic features, such as dark spots
and bright faculae (Solanki et al., 2006; Shapiro et al., 2020).

Under the approximation that all axions travel along radial trajectories in
the stellar atmosphere, and the hypothesis that the stellar magnetic field
varies in a length scale much larger than the photon and axion wavelengths, we
can derive an analytic expression for the axion conversion to photons using a
WKB approximation (Raffelt and Stodolsky, 1988). Accordingly, the propagation
of the photon and axion in the radial direction with a Energy $E$ is given by

$\displaystyle\left[i\partial_{r}+E+\mathbf{\hat{A}}\right]\cdot\mathbf{\hat{V}}=\mathbf{0},$
(10)

where $\mathbf{\hat{V}}$ is a 3 component vector and $\mathbf{\hat{A}}$ is a
$3\times 3$ symmetric matrix that defines the interaction of the photon-axion
pair with the magnetic field $\mathbf{B}$. The vector $\mathbf{\hat{V}}$ reads

$\displaystyle\mathbf{\hat{V}}=\begin{pmatrix}V_{\perp}\\\ V_{\parallel}\\\
V_{a}\\\ \end{pmatrix},$ (11)

where $V_{\parallel}(r)$ and $V_{\perp}(r)$ denotes the vector potential in
the plane normal and perpendicular to the direction of the propagation and
parallel to the external magnetic field, while $V_{a}(r)$ is the axion field.
The matrix $\mathbf{A}$ reads

$\displaystyle\mathbf{\hat{A}}=\begin{pmatrix}\Delta_{\perp}&0&0\\\
0&\Delta_{\parallel}&\Delta_{B}\\\ 0&\Delta_{B}&\Delta_{a}\\\ \end{pmatrix}.$
(12)

Here, we follow the notation of Raffelt and Stodolsky (1988). The quantities
$\Delta_{\perp}(r)=4/2E\;\xi\sin^{2}{\phi}$,
$\Delta_{\parallel}(r)=7/2\;E\xi\sin^{2}{\phi}$ and
$\Delta_{B}(r)=(g_{a\gamma}/2)B(r)\sin{\phi}$ are terms that define the
interaction of the axion with the magnetic field, and
$\Delta_{a}=-m_{a}^{2}/(2E)$ is a term that incorporates the axion mass
$m_{a}$ and is responsible for the small difference momenta between axion and
photon states. $B(r)$ is the strength of the magnetic field at radius $r$ and
$\phi$ is the angle between the magnetic field and the photon momentum, and
$\xi(r)=(\alpha_{em}/45\pi)[B(r)/B_{c}]^{2}$ where $B_{c}=m_{e}^{2}/e$ is a
critical magnetic field strength.

In the following, we discuss the conversion of photon to axion mixing as a
photon propagates through a highly magnetized medium. Following Raffelt and
Stodolsky (1988), if the magnetic field is homogeneous, the subsystem of
equations $(V_{\parallel},V_{a})$ on the system of equations (10) [also the
second and third lines of (11) and (12)] can be computed separately from the
equation $V_{\perp}$ (also the first line of 11 and 12). Therefore, the
leading stationary wave Eq. (10) reduces to two decoupled systems of first-
order differential equations (e.g., Fortin and Sinha, 2019; Perna et al.,
2012). Accordingly, the subsystem of Eqs. $(V_{\parallel},V_{a})$ reduces to
the simplified stationary wave equation

$\displaystyle\left[i\partial_{r}+E+\mathbf{A}\right]\cdot\mathbf{V}=\mathbf{0}$
(13)

where $\mathbf{V}$ is a vector (similar $\mathbf{\hat{V}}$) with only two
component that defines the amplitudes of photon and axion states, $\mathbf{A}$
is a $2\times 2$ symmetric matrix (similar $\mathbf{\hat{A}}$) that defines
the interaction of the axion with the magnetic field. Accordingly,
$\mathbf{V}$ reads

$\displaystyle\mathbf{V}=\begin{pmatrix}V_{\parallel}\\\ V_{a}\\\
\end{pmatrix}$ (14)

and $\mathbf{A}$ takes the form

$\displaystyle\mathbf{A}=\begin{pmatrix}\Delta_{\parallel}&\Delta_{B}\\\
\Delta_{B}&\Delta_{a}\\\ \end{pmatrix}.$ (15)

Fortin and Sinha (2019) have made a detailed study of the conversion of
axionlike particles to photons in a electromagnetic background have shown that
Eq. (13) is valid also in the high-magnetization limit where the electron
cyclotron frequency $\omega_{c}=\sqrt{\alpha_{e}}B/(m_{e}c)$ is much larger
than electron plasma frequency $\omega_{pl}$ [see Eq. 7] and $E$
($\omega_{c}\gg E$ and $\omega_{pl}$). Equation (13) can be diagonalized by
rotating the fields, $\mathbf{V^{\prime}}=\mathbf{R(-\theta)}\cdot\mathbf{V}$,
where $\mathbf{R(\theta)}$ is the two dimensional unitary rotation matrix.
$\mathbf{R(\theta)}$ is given by

$\displaystyle\begin{pmatrix}\cos{\theta}&-\sin{\theta}\\\
\sin{\theta}&\cos{\theta}\\\ \end{pmatrix},$ (16)

where $\theta$ is a mix angle satisfying the relation

$\displaystyle\tan{(2\theta)}=\frac{2\Delta_{B}}{\Delta_{\parallel}-\Delta_{a}}.$
(17)

Accordingly, the wave Eq. (13) becomes diagonal in the $\prime$ referential of
the rotation vector $\mathbf{V^{\prime}}$, thus

$\displaystyle\left[i\partial_{r}+E+\mathbf{A^{\prime}}\right]\cdot\mathbf{V^{\prime}}=\mathbf{0}.$
(18)

Here the matrix $\mathbf{A^{\prime}}$ reads

$\displaystyle\mathbf{A^{\prime}}=\begin{pmatrix}\Delta_{\parallel}^{\prime}&0\\\
0&\Delta_{a}^{\prime}\\\ \end{pmatrix}.$ (19)

where $\Delta_{\parallel}^{\prime}=\lambda_{1}$ and
$\Delta_{a}^{\prime}=\lambda_{2}$ with
$\lambda_{j}=1/2\left[\Delta_{\parallel}+\Delta_{a}-(-1)^{j}(\Delta_{\parallel}-\Delta_{a})/\cos{(2\theta)}\right]$
($j=1,2$).

Following Raffelt and Stodolsky (1988) the previous calculation is greatly
simplified if we define phases relative to the unmixed component, and neglect
a common phase. The solution can be found by first performing a matrix
rotation to an eigenstate basis where the propagation matrix is diagonal,
propagating the two eigenstates independently, and then rotating back to the
photon-axion basis. This yields an evolution equation for the mixing
components in the original referential as

$\displaystyle\begin{pmatrix}V_{\parallel}\\\ V_{a}\\\ \end{pmatrix}(r)={\cal
M}(r)\cdot\begin{pmatrix}V_{\parallel}\\\ V_{a}\\\ \end{pmatrix}(0)$ (20)

where ${\cal M}$ is

$\displaystyle{\cal M}=\mathbf{R(\theta)}\cdot{\cal
M}_{D}\cdot\mathbf{R(-\theta)}$ (21)

and ${\cal M}_{D}$ is given by

$\displaystyle{\cal
M}_{D}=\begin{pmatrix}e^{-i(\Delta_{\parallel}^{\prime}-\Delta_{\parallel})r}&0\\\
0&e^{-i(\Delta_{a}^{\prime}-\Delta_{\parallel})r}\\\ \end{pmatrix}.$ (22)

The probability for an axion-photon transition amplitude (van Bibber et al.,
1989; Irastorza & Redondo, 2018) is computed from the off-diagonal terms in
${\cal M}_{(12)}$ of Eq. (21). Therefore, the probability of an axion being
converted to a photon is

$\displaystyle
P_{a\gamma}=\;\sin^{2}{(2\theta)}\;\sin^{2}{\left[\frac{\Delta_{B}r}{\sin{(2\theta)}}\right]}.$
(23)

The strength and geometry of the magnetic field in the atmosphere of a main
sequence and post main sequence star varies in a complicated manner, in such a
way that the extraction of the averaged poloidal and toroidal components of
$B$ is quite diverse between these stars. Moreover, it was found that the
degree of complexity of the magnetic field appears to be independent of the
stellar mass. For instance in these stars the geometry surface magnetic fields
varies from purely poloidal to non-poloidal (complicated) geometries
(Kochukhov et al., 2019). In principle, we can consider more complex magnetic
field configurations to compute $P_{a\gamma}$, but the estimation obtained is
largely identical to the one computed with Eq. (23). Moreover, to simplify the
calculation of the x-ray spectrum produced by this inverse Primakoff reaction,
we will assume the process occurs in the presence of an averaged magnetic
field $B$ located in a fiducial layer of thickness (of a fixed percentage of
the stellar radius) $\Delta R_{\star}$ in the exterior of star.

In real stars, the axion magnetic conversion layers will have a very complex
structure, with varying density, chemical composition and magnetic fields.
Nevertheless, our model allows us to obtain for the first time an estimation
of the excess of x-ray radiation due to inverse Primakoff reaction. Therefore,
we assume the magnetic layer has a thickness of $\Delta R_{\star}=0.15$ and is
crossed by constant magnetic field $B$.

As we will find out in the next section, the electromagnetic emission coming
from these axions will appear as an additional photon source in the x-ray
energy range of the electromagnetic spectrum above the stellar background.

Under the previous assumptions, and since we are interested in the weak-mixing
axion regime where $\theta\ll 1$, from Eq. (17) we compute
$\theta\approx\Delta_{B}/(\Delta_{\parallel}-\Delta_{a})$, and by using this
result on Eq. (23) we obtain

$\displaystyle P_{a\gamma}\approx
4\frac{\Delta_{B}^{2}}{\Delta^{2}_{a}}\,\sin{\left(\frac{\Delta_{a}\Delta
R_{\star}}{2}\right)}$ (24)

with
$\Delta_{B}^{2}/\Delta^{2}_{a}=g_{a\gamma}^{2}B^{2}E^{2}/m_{a}^{4}\;\sin{\phi}$.
The derivation of the previous expression is possible, since for axions with a
very low mass, we have $|\Delta_{a}|\gg\Delta_{\parallel}$ and
$\theta\approx\Delta_{B}/\Delta_{a}$.

Figure 3: The conversion probability of an axion to photon $P_{a\gamma}$ as a
function of the energy. The $P_{a\gamma}$ is obtained by varying some of the
parameters of the reference axion model (see end of this caption): (a) red and
yellow curves correspond to Eqs. (23) and (24) with $m_{a}=10^{-7}\;{\rm eV}$
(continuous curve), $10^{-6}\;{\rm eV}$ (dashed curve) and $10^{-5}\;{\rm eV}$
(dotted-dashed curve); (b) blue and cyan curves correspond to an axion model
identical to (a) with $m_{a}=10^{-4}\;{\rm eV}$ and $\Delta
R_{\star}=0.005\;R_{\odot}$; (c) green and yellow curves correspond to an
axion model identical to (a) with $m_{a}=10^{-3}\;{\rm eV}$ and $\Delta
R_{\star}=0.001\;R_{\odot}$; (d) dashed green and yellow curves correspond to
a axion model identical to (c) with $\phi=10^{-3}\pi$; Reference axion model
$g_{a\gamma}=5\times 10^{-11}\;{\rm GeV^{-1}}$, $m_{a}=10^{-7}\;{\rm GeV}$,
$B=3\times 10^{4}\;{G}$, $\phi=\pi/2$ and $\Delta R_{\star}=0.15\;R_{\star}$
(see main text). For comparison, we also show $P_{a\gamma}$ for the conversion
of axions to photons occurring inside an experimental axion detector (with a
magnetic field $B$ and dimension $D_{m}$): (e) red curve corresponds to an
axion with $m_{a}=1\times 10^{-3}\;{\rm eV}$, and a detector with $B=9\times
10^{4}\;{G}$ and $D_{\rm m}=10\;{\rm m}$ (Zioutas et al., 2009); (f) blue
curve corresponds to conversion to an axion with $m_{a}=8\times 10^{-4}\;{\rm
eV}$ and a detector with $B=3\times 10^{-1}\;{G}$ and $D_{\rm m}=6\times
10^{5}\;{\rm m}$; $D_{\rm m}$ is identical to $\Delta R_{\star}$ for a star
(see main text)(Fraser et al., 2014; Sikivie, 1983).

Figure 3 shows Eqs. (23) and (24) in comparison with other expressions in the
literature. We start by noticing that for all parameters considered in this
study, the two expressions are equivalent. In this figure, we choose a set of
axion parameters that better shows the similitude of the inverse Primakoff
reaction occurring in the external layers of stars and on axion detectors. The
numerical results obtained with our analytical expressions are identical to
others found in the literature to determine the conversion of axions to
photons in experimental detectors, for instance Zioutas et al. (2009) and
Sikivie (1983). $P_{a\gamma}$ [see Eq. 24] function is strongly dependent on
the axion parameters (mass, energy and axion-photon coupling constant) and the
magnitude of the magnetic field, as shown by the approximative Eq. (24) and
confirmed in Fig. 3. Some of $P_{a\gamma}$ results show a oscillatory pattern
(see cases (a) , (b)  and (c)  in Fig. 3), as well as the $P_{a\gamma}$ of
Zioutas et al. (2009). This behavior is related to the mass of the axion
$m_{a}$ and $\Delta R_{\star}$ (see Eq. 24). In an axion detector, $D_{\rm m}$
replaces $\Delta R_{\star}$, where $D_{\rm m}$ is the dimension of the region
where $B$ is applied Zioutas et al. (2009). We also observe that $P_{a\gamma}$
depends on the phase $\phi$ as shown by the analytical approximation (24),
leading in some cases to the $P_{a\gamma}$ functions varying by a few orders
of magnitude (see cases (c) and (d) in Fig. 3). Nevertheless, without loss of
generality, we opt to choose $\phi=\pi/2$ or $\sin{\phi}=1$ if not stated
otherwise.

##  VI VI. Axion electromagnetic inverse Primakoff spectrum

Here, we calculate the electromagnetic spectrum related with the inverse
Primakoff reaction. Such spectrum is obtained as the product of the axion
emission spectrum coming from the stellar interior [Eq. 4] and the axion-to-
photon conversion probability $P_{a\gamma}$ [Eq. 23]. Accordingly, the flux of
axion-induced x-ray photons at 1 A.U. is calculated as

$\displaystyle\left(\frac{d\Phi}{dE}\right)_{a\gamma}=P_{a\gamma}\;\left(\frac{d\Phi}{dE}\right)_{a},$
(25)

where $({d\Phi}/{dE})_{a}$ is given by Eq. (4) for $r=R_{\star}$.

The total axion flux conversion to x-ray flux at an 1 A.U. distance, the
correspondent luminosity, and the mean x-ray axion energy is computed in a
similar way to Eqs. (8) and (9), but now modified by the conversion
probability $P_{a\gamma}$ [Eq. 23]. Accordingly

$\displaystyle\Phi_{a\gamma}=2\pi\int_{0}^{1}\int^{\infty}_{\omega_{\rm
pl}}\varphi_{a}(r,E)\,P_{a\gamma}\,dE\,rdr,$ (26)

the corresponding x-ray luminosity is $L_{a\gamma}=4\pi D_{\rm
AU}^{2}\Phi_{a\gamma}$. The predictions of the x-ray axion luminosities for
the stars studied in this work are shown in column 9 of Table 1.

Figure 4 show the typical x-ray spectra generated by the magnetic layer
located in the external layers of the stars (see Table 1). All these x-ray
axion spectra are a few orders of magnitude smaller that the axion spectra
coming from the core of the star, accordingly the x-ray axion luminosity
varies by the same other of magnitude (see column 9 of Table 1).

We found that some of the x-ray axion spectra have a unique sinusoidal shape
that is related to the mass of the axion, as illustrated in Fig. 4 [panels (b)
and (d)]. Such periodic behavior is more pronounced for axions with a lower
mass, the variation of the mass of the axion leads to spectra with similar
sinusoidal shapes but different periodicities.

Since such stars’ magnetic field is time-dependent like the stellar magnetic
cycle (Baliunas et al., 1995; Lopes et al., 2014), the x-ray spectrum produced
from the interaction of axions with the stellar magnetic fields (see section
III and references therein) will also vary in time.

The spectra calculated here are initial estimates of the averaged x-ray
emission rate of axion conversion in the atmosphere of a star. To make
reliable predictions of the x-ray emission spectra is necessary to include a
more detailed account of the stellar atmosphere structure. Indeed, the
atmospheres of low-mass stars, among others, present a large diversity in
terms of thermodynamics properties, chemical composition and intensity and
topology of the magnetic fields. Only by including a detailed description of
the atmosphere of such stars, it is possible to make a reliable prediction of
the x-ray emission coming from converting axions to x-rays in stellar
atmospheres.

It is worth mentioning that axion spectra and the x-ray spectra of the axion
emission are strongly dependent on the axion-photon coupling constant and the
mass of the axion. For instance, if the value of the axion-photon coupling
constant changes from $5\times 10^{-11}\,{\rm GeV}^{-1}$ to $10^{-11}\,{\rm
GeV}^{-1}$, $L_{a}$ for the Sun and a two-solar mass star decreases by an
order of magnitude, while $L_{a\gamma}$ decreases by 3 orders of magnitude
(compare model $A^{*}_{\odot}$ with $A_{\odot}$, and $B^{*}_{1}$ with $B_{1}$
in Table 1). If similarly, when we vary $m_{a}$ from $10^{-7}\;{eV}$ to
$10^{-5}\;{eV}$, then $L_{a\gamma}$ decreases by one order of magnitude
(compare model $A^{**}_{\odot}$ with $A_{\odot}$, and $B^{**}_{1}$ and $B_{1}$
in Table 1).

Nevertheless, we can find a combination of parameters, such as
$g_{a\gamma}=5\times 10^{-11}\,{\rm GeV}^{-1}$ and $m_{a}=10^{-7}\,{eV}$ (as
show in Table 1) for which the x-ray axion luminosities of these stars vary
between $10^{-5}$ and $10^{-7}\,L_{\odot}$. Such luminosities are comparable
or even larger than the x-ray luminosity currently measured for many low-mass
stars: $10^{-4}$ to $10^{-7}\;L_{\odot}$ (See et al., 2016). This result shows
the potential of x-ray stellar astronomy to put constraints in low mass axion
models.

|
---|---
|
Figure 4: x-ray emission spectrum $(d\Phi/dE)_{a\gamma}$ due to inverse
Primakoff photon production: Comparison spectra calculated for stellar models
of one (top panels: a, b) and two (lower panels: c, d) solar masses shown in
Fig. 2 and Table 1. The mass of the axion is equal to: $10^{-7}\;{\rm eV}$ (a)
and $10^{-5}\;{\rm eV}$ (b) for models $A_{\odot},A_{1},\cdots,A_{5}$; and
$10^{-7}\;{\rm eV}$ (c) and $10^{-5}\;{\rm eV}$ (d) for models
$A_{\odot},B_{1},\cdots,B_{4}$. In all these models the axion magnetic
conversion layer has a thickness of $0.15\,R_{\star}$ and an averaged magnetic
field of $3\;10^{4}\;{G}$.

##  VII VII. Conclusion

Stars are commonly used to test new particles and interaction channels, mostly
because many of such interactions occurring inside stars are strongly
dependent on temperature, density and chemical composition of the stellar
plasma. By taking advantage of similar behaviors for the axion, we study how
such particles are generated inside main-sequence and post main-sequence
stars. We have confirmed previous computations done for the present Sun and
found that for other low-mass stars the axion spectrum changes in intensity
and shape by a few orders of magnitude, depending significantly on the mass
and age of the star.

We found that stars with external magnetic fields could have an excess of
radiation in the x-ray energy range of the electromagnetic spectrum resulting
from the conversion of axions into photons by their atmospheric magnetic
field. We also establish that for stars with an averaged magnetic field above
$30$ kG for some axion models, this can lead to the production of an x-ray
excess in their electromagnetic spectra. These stars will have an excess of
x-ray electromagnetic radiation above $10^{-5}\;L_{\odot}$, if the axions have
an axion-photon coupling constant above $5\;10^{-11}{\rm GeV}^{-1}$ and an
axion mass of the order of $10^{-7}{\rm eV}$. These values correspond to a
region of the axion parameter space $m_{a}$ – $g_{a\gamma}$ that is not been
probed by current axion detectors including the CAST collaboration as reported
in Anastassopoulos et al. (2017). We predict that main sequence and post-main
sequence stars can have x-ray luminosities due to axion emission above
$10^{-5}\;L_{\odot}$ which is larger or comparable to the x-ray luminosity
measured in some low-mass stars (See et al., 2016).

Therefore, this work shows that if high precision stellar x-ray observations
are made available by the next generation of satellites, it will be possible
to constrain the $m_{a}$ – $g_{a\gamma}$ parameters using another axion
mechanism operating low-mass stars, rather than the classical axion energy-
loss that is known to affect the evolution of the Sun and horizontal branch
stars. By taking advantage of a large number of stars expected to be observed
by the next generation of x-ray missions, this novel method should allow us to
enhance the current axion constraints by using measurements of stellar x-ray
luminosities. Moreover, by using a large ensemble of stars including more
massive stars such result can possibly be extended for axion models with
masses varying from $10^{-7}$ to $10^{-5}{\rm eV}$, a region of the axion
parameter space not continuously probed by current detectors. In that way,
such study could complement the few but important measurements in that axion
mass range made by haloscopes.

Moreover, this study could also contribute and complement other methods to put
constraints using the x-ray emission resulting from the interaction of stellar
axions with the galactic magnetic field between the source of axion and the
Earth. Dessert et al. (2020) using x-ray data from the Quintuplet and
Westerlund 1 super star clusters, were able to found upper limits to the
axion-photon coupling and axion mass with a $95\%$ confidence level:
$g_{a\gamma}\lesssim 3.6\times 10^{-12}\;{\rm GeV}^{-1}$ and $m_{a}\lesssim
5\times 10^{-11}\;{\rm GeV}$. This constraint is obtained under the usual
assumption that the magnetic field is homogeneous between the location of
stellar clusters and Earth.

The existence of axions is complicated to establish mainly since the axion is
a low mass particle with a weak interaction with standard ones. For this
reason, many classic phenomena can mask the astrophysical signature of the
axion. As in many other situations in physics, a possible discovery of axions
will only be possible if we can find different astrophysical scenarios that
can be explained by the same axion particle. Therefore the study of the x-ray
spectra produced by the interaction of the axion with the magnetic fields,
either in the atmosphere of the stars or in the galaxy, can contribute
significantly to resolve this problem.

Finally, this study is of interest for the future search of axion signatures
in stellar astrophysics scenarios, namely when looking for signatures of the
production of axions in stars. It is also useful to take advantage of a large
amount of high-quality observational data that will be made available by some
present, and future astronomical observatories and astrophysical space
missions. As shown in this work, a prominent axion signature to look for is an
excess of radiation in the x-ray energy band of the electromagnetic spectrum
(that cannot be explained by the standard magnetism of stars) – the next
generation of x-ray satellites will be a powerful tool to test such class of
axion models. Among others, we mention the Nuclear Spectroscopic Telescope
Array mission or NuSTAR (this is a x-ray telescope launched in 2012, operating
in the band from 3 to 79 keV Harrison et al., 2013), and the Time-Resolving
Observatory for Broadband Energy x-rays (STROBE-X; Ray et al., 2019) mission.
The latter is a future x-ray satellite expected to observe in the energy
range: 2 to 60 KeV, exactly the energy range where axion models are expected
to have the maximum emission.

In this preliminary study, like in others found in literature, we do not
consider the diversity of spectral properties of the stellar atmosphere and
magnetic fields found in such low-mass stars (and many others across the HR
diagram) in our calculations. We also neglect the time variability observed in
the magnetic fields of the Sun and many other stars. However, we expect that a
detailed prediction of the x-ray radiation emitted resulting from axion
interaction with the stellar magnetic field should take such effects into
account. Nevertheless, we believe that the overall result found in this study
should remain valid.

###### Acknowledgements.

The author is grateful to the MESA team for having made their code publicly
available. The author also thanks the anonymous referee for the revision of
the manuscript. I.L. thanks the Fundação para a Ciência e Tecnologia (FCT),
Portugal, for the financial support to the Center for Astrophysics and
Gravitation (CENTRA/IST/ULisboa) through the Grant Project No. UIDB/00099/2020
and Grant No. PTDC/FIS-AST/28920/2017.

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<EMAIL_ADDRESS>
# Investigating CO2 streamer inception in repetitive pulsed discharges

S. Mirpour and S. Nijdam  Department of Applied Physics, Eindhoven University
of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands

###### Abstract

In this study, we investigate the responsible species and processes involved
in repetitive pulsed streamer inception in CO2. We applied a 10 kV high-
voltage pulse with a repetition frequency of 10 Hz and pulse width of 1 ms to
a pin electrode which is placed 160 mm apart from the grounded plane
electrode. We measured the inception times (delay between the rising edge of
the high-voltage pulse) by a photo-multiplier tube for 600 high voltage
cycles. We observed one peak in the histogram of inception times with a median
of 1.2 $\upmu$s. To identify the source of this peak, we applied a negative or
positive LV pulse before the main HV pulse to manipulate the leftover space
charges. Three different phenomena are observed: 1) drift, 2) neutralization,
and 3) ionization in the LV pulse. At low LV amplitude and pulse width, the
peak starts to drift toward the faster and slower inception times under a
positive and negative LV pulse, respectively. However, under the same LV pulse
configuration for positive and negative LV pulse, the observed shift in
inception times is not the same. We present a hypothesis to explain this
asymmetry based on the difference of the detachment processes between air and
CO2.

Keywords CO2, repetitive pulse, discharge inception, streamer, delay

submitted to Plasma Sources Science and Technology, Version

##  1 Introduction

Streamers are widely explored in different gases because of their importance
in various fields such as high-voltage engineering and atmospheric electricity
[1, 2, 3, 4, 5, 6, 7, 8]. Recently, CO2 gas has been considered as a possible
candidate for high-voltage switchgear, since the gas currently used for
insulation and switching, SF6, is an extremely strong greenhouse gas [9, 10].
Also, there is strong evidence that lightning might exist in the Martian
atmosphere which mostly consists of CO2 [11, 12]. Hence, it is important to
understand the physics behind streamer inception and development processes in
this gas. Streamer inception voltage and time have been mostly studied in air
[13, 14, 15, 16]. There are not many studies of discharge inception in CO2
(see below), and the current understanding of the streamer inception process
is very limited. This study investigates the streamer inception process in CO2
in more detail.

By applying positive high-voltage pulses to an electrode, electrons start to
drift opposite to the electric field. When such an electron reaches an area
where the ionization rate is higher than the attachment rate, it can start to
replicate, although this is of course a stochastic process. The electron
density then grows until the formed space-charge becomes large enough to
transform into a streamer. The number of electrons required for this
transition can be derived from the so-called Meek criterion [17, 18]. In our
previous work [13], we hypothesized that in synthetic air, the very first
electron to trigger the streamer inception process can come from three
different sources: free electrons, O${}_{2}^{-}$ detachment, or from reactions
of neutral species. We showed that, in repetitive discharges, the leftover
charges from the streamers likely build up an in-homogeneous distribution of
ions in front of the anode, which produces a peak in the streamer inception
distribution times histogram. By adding a low-voltage (LV) pre-pulse before
the main high-voltage (HV) pulse we could manipulate and drift this in-
homogeneous ion cloud towards or away from the anode, depending on the LV
pulse polarity. Thus, the ions will be at a new position at the beginning of
the HV pulse which results in a shift of the observed peak in the time
distribution histogram. From these shifts, we can derive the mobility of ions
and identify the source. Photoionization is another well-known source of
initial seed electrons in air. However, no efficient photoionization mechanism
is known for CO2 due to the rapid absorption of photons in this gas [19].
Therefore, in CO2, detachment of electrons from negative ions is the most
probable source of initial free electrons.

The statistical time lag for CO2 discharges has been studied by Seeger et al.
[20]. They showed that, at pressures higher than ambient pressures, by
increasing the applied voltage, the discharge inception time lag decreases and
reaches as low as 100 ns. They found a fairly good agreement between the
theoretical prediction of streamer inception fields (considering the
ionization integral as 13) and actual observations. However, that study offers
little in the way of examining which species are playing an important role in
inception.

Wang et al. [21] investigated the critical breakdown field for CO2 discharges
at elevated temperatures. They showed that at room temperature the reduced
breakdown electric field in CO2 is about 86 Td. Increasing the temperature
leads to CO2 dissociation, which leads to more collisions between CO and O-
creating electrons through associative detachment. This will influence the
effective ionization rate, contributing to a significant drop in the critical
breakdown field. At room temperature this mechanism is insignificant due to
the low dissocation degree. Furthermore, O- is turned into a relatively stable
ion, CO${}_{3}^{-}$, by the cluster stabilization process. Hence, the main
electron source and sink are electron impact ionization and dissociative
attachment to CO2 through e- \+ CO2 $\rightarrow$ CO + O-.

This study aims to clarify the role of different species in the streamer
inception process in CO2 by adding a pre-pulse to manipulate the charge
distribution before applying the main HV pulse. The effects on these on
inception time histograms can give great insights in the relevant processes.
This paper is organized as follow: In section 2 we describe the method,
section 3 discusses the results and gives an hypothesis based on the different
LV pulse configurations. Finally, in section 4 we summarize the paper and
provide the conclusion.

##  2 Set-up and Methods

###  2.1 Experimental conditions

All experiments in this study were performed in a point-to-plan geometry
(figure 1) in which the powered electrode, (anode with a tip radius of about
100 $\upmu$m), is separated 160 mm from the grounded electrode. The anode is
powered by a circuit comprising an HV solid state switch (Behlke, HTS
651-15-SiC-GSM) and a 1 nf capacitor. With this configuration, we can produce
repetitive pulsed waveforms with an amplitude of 10 kV, pulse widths of 1 ms,
rise times of 40 ns, and repetition rates of 10 Hz. The reason we chose this
repetition rate and pulse duration is to have only one discharge within the
duration of an HV pulse. We did not observe any dependency of streamer
inception on pulse duration at this frequency. The background pressure was
$10^{-6}$ mbar, and for each experiment the vessel was filled to 300 mbar with
CO2 (purity level of 99.9999 %).

Figure 1: a) Schematic view of the experimental setup with HV power supply
connected to the anode (not to scale). b) ICCD image of the CO2 discharge at
300 mbar with HV amplitude of 10 kV and repetition frequency of 10 Hz.

To investigate the influence of residual charged species on streamer
inception, an LV pre-pulse was applied before each HV pulse. To apply the LV
pre-pulse, the negative input of the HV pulser was connected to a second
pulser, comprised of another solid-state HV switch (Behlke, HTS 181-01-HB-C)
and a 1 nF capacitor. This could produce a pre-pulse which attaches to the
main HV pulse and has an amplitude $V_{\rm LV}$ between 0 and 8 kV and a pulse
width $t_{\rm LV}$ between 0 and 100 ms (c.f. figure 2).

Figure 2: a) Scheme of applied HV waveform with 1. positive and 2. negative
attached LV pulse. b) Typical applied HV and LV voltage with PMT waveform
output.

###  2.2 Measuring inception time $t_{\rm inc}$

To measure the streamer inception time ($t_{\rm inc}$), we used a photo-
multiplier tube (PMT,H10720-1 10 Hamamatsu) placed behind a window of the
vessel. The response time of the PMT is less than 2 ns. The output signal was
transferred to a 12-bit HD 6104 Teledyne Lecroy oscilloscope with a sample
rate of 1 GS/s. We repeated this measurement for 600 HV cycles per setting.
Note that we consistently observed only one discharge inception event per 1 ms
HV pulse. For each cycle, $t_{\rm inc}$ was determined as the temporal delay
between the HV pulse reaching 10 % of its maximum and the PMT waveform
reaching 10 % of its maximum (see figure 2). The total error in $t_{\rm inc}$
is estimated at less than 10 ns. Streamer inception is here defined as an HV
pulse in which the PMT signal shows a peak three times above the average
background noise.

From the measured $t_{\rm inc}$ values, a histogram is made using a
logarithmic binning function, MATLAB function:logspace(log10(0.01),
log10(max(data)), 700), which divides the data into 700 bins starting from 10
ns. The acquisition window was 1 ms for all experiments. In the histogram
figures, we indicate the inception probability ($Prob$), which is the number
of pulses for which streamer inception was detected, divided by the total
number of HV pulses (600). An intensified CCD (ICCD, Stanford Computer Optics
4QuickE) with a nanosecond time gate and a Nikkor UV 105 mm lens f/4.5 was
used to image the streamers. The streamers in figure 1.b are rendered in a
false-colour scale for clarity. We found that in each case that a peak was
detected by the PMT, the ICCD image always showed a developed streamer.

##  3 Results and discussion

###  3.1 Baseline experiment and general phenomenology

To gain an overview of phenomenology, we applied an HV pulse (10 kV amplitude
with a duration of 1 ms) and an LV pre-pulse (with a duration of 25 $\upmu$s
and varying $V_{\rm LV}$) to the electrode. We repeated this 600 times at a
repetition frequency of 10 Hz. Figure 3 shows the histogram of $t_{\rm inc}$
on a logarithmic scale where the size of the time bins is scaled as log $t$.
Figure 3.a shows the baseline experiment with no LV pulse where $t_{\rm inc}$
forms a single peak around 1-2 $\upmu$s. Application of a positive LV pulse
before the main HV pulse led to three observations. For $V_{\rm LV}$ up to 500
V (Drift, figure 3.a-c) the $t_{\rm inc}$ peak started to shift to lower
values, below 70 ns. Increasing $V_{\rm LV}$ more (Neutralization, figure
3.d-g) caused the peak to shift to higher values around 150 ns. Finally, an LV
pulse of 8 kV (Ionization during LV pulse, figure 3.h) again shifted the peak
to 50 ns. Note that this phenomenon depends on an interplay between $V_{\rm
LV}$ and $t_{\rm LV}$ and therefore we introduce $S$= $t_{\rm LV}{}\cdot
V_{\rm LV}{}$ as the main parameter in the phenomenology. Below, each of the
observed phenomena is elaborated in a separate section.

Figure 3: Histograms of discharge inception time $t_{\rm inc}$ for 600
discharges produced a) for no LV pulse and b-h) by applying a 25 $\upmu$s
duration LV pulse with the indicated voltage before a 10 kV HV pulse of 1 ms
with a repetition frequency of 10 Hz. Inception probabilities are 100 % for
each case, as indicated.

###  3.2 Drift due to applying an LV pulse

Since we mostly observe one peak in the $t_{\rm inc}$ histogram, the median
values ($\tilde{t}_{\rm inc}$) were calculated and are plotted against $V_{\rm
LV}$ in figure 4). To test the role of leftover charges on the inception, we
applied both positive and negative LV pulses immediately before the main HV
pulse. Figure 4 shows the effects of these LV pulses on $\tilde{t}_{\rm inc}$,
where for negative the LV pulses, a pulse width $t_{\rm LV}$ of 10 ms was
used, while for the positive LV pulses this was 10 $\upmu$s, a factor 1000
lower. In the electric field imposed by a positive LV pulse, the negatively
charged species will drift towards the electrode and thereby initiate
discharges quicker after the HV pulse is applied. The results show that for
every 100 V of applied positive LV pulse, we observed a 94 ns shift in
$\tilde{t}_{\rm inc}$.

We would expect to have the same drift magnitude in opposite direction for a
negative LV pulse. Interestingly however, a negative LV pulse with a 10
$\upmu$s pulse width leads to a negligible shift in $\tilde{t}_{\rm inc}$. As
can be seen in Figure 4, to have a 90 ns shift in $\tilde{t}_{\rm inc}$ we
need to apply a 10 ms, 100 V negative LV pulse instead of a 10 $\upmu$s, 100 V
pulse. Note that $\tilde{t}_{\rm inc}$ is not fully linear with $V_{\rm LV}$
so the mentioned numbers are averages.

Figure 4: Median of the delay histograms, $\tilde{t}_{\rm inc}$, produced by
applying left) negative and right) positive LV pulses with different $V_{\rm
LV}$. For the negative and positive LV pulse, $t_{\rm LV}$ is fixed and set to
be 10 ms and 10 $\upmu$s, respectively.

To investigate the effect of the duration of the LV pulse, we fixed $V_{\rm
LV}$ to 1 kV and -1 kV and varied $t_{\rm LV}$. Figure 5 shows the
$\tilde{t}_{\rm inc}$ shift when applying positive and negative LV pre-pulses.
For negative LV pre-pulses, a 1 ms increase in $t_{\rm LV}$ leads to a 86 ns
decrease in $\tilde{t}_{\rm inc}$. Similar to what we observed in the previous
experiment $\tilde{t}_{\rm inc}$ the reverse effect for positive pre-pulses is
also present but again roughly a thousand times stronger; to have a 92 ns
shift in $\tilde{t}_{\rm inc}$ we need to apply a 1 kV positive pre-pulse with
$t_{\rm LV}{}=1$ $\upmu$s instead of 1 ms.

Figure 5: Median of the delay histograms, $\tilde{t}_{\rm inc}$, produced by
applying left) negative and right) positive LV pulses with different $t_{\rm
LV}$. For the negative and positive LV pulse, $V_{\rm LV}$ is fixed and set to
be -1 kV and 1 kV, respectively. Note the difference in time units between
left-hand and right-hand side on the horizontal axis.

Note that in our previous experiments in air [13] the effects of positive and
negative pre-pulses were nearly equal, but of course opposite in direction.
Below, we give a hypothesis to explain this large asymmetry of three orders of
magnitude between the effects of positive and negative pre-pulses in CO2.

####  3.2.1 Hypothesis

In order to explain the surprising asymmetry between the effects of positive
and negative pre-pulses in CO2 (and not in air), we hypothesize that this is
caused by the detachment process in CO2.

During an HV pulse, energetic electrons near the tip cause CO2 dissociation by
direct electron impact. The dissociative attachment reaction rate increases in
higher electric fields and causes the production of CO and O- near the
electrode [22]. Thus, in this specific region, both these species are
available and recombination of O- and CO can liberate an electron through
associative detachment at any time during and between voltage pulses. During
an HV pulse, such an electron can enter the ionization region and initiate a
discharge. If there is a ”charged species patch”: a volume containing positive
ions, negative ions and free electrons, located some distance from the tip,
like we also concluded in [13], then the electrons from this region can again
explain the peaked distribution of the baseline experiment. Note that from our
results we cannot determine the shape of the charged species patch. It could
be a thin layer, a localized patch, or a large region extending away from the
electrode tip. We only know that it has a quite sharp boundary towards the
electrode.

When applying a positive LV pulse, all negative species in the charged species
patch will drift toward the tip. Since ions are much heavier than electrons,
electron drift is more prominent and therefore the electron mobility will
determine the effect of the LV pulse on inception behaviour. A negative LV
pulse with the same $S$ value will also push away the electrons from their
initial position for roughly the same distance as the positive LV pulse.
However, the much less mobile ions can still produce electrons by associative
detachment and remain much closer to the tip than the drifted electrons due to
their lower mobility. During the main HV pulse inception times are determined
by the closest electrons and not the far-away ones. Therefore, for positive LV
pulses, electron drift dominates the effect, while for negative LV pulses,
negative ion drift does so.

To elaborate on this hypothesis, we have calculated the drift time of
electrons and O- ions in CO2 from a distance $z$ to the pointed electrode (at
$z$=0) on the axis of symmetry. The drift time was calculated by

$t_{\rm
drift}(z)=\int_{z}^{0}\frac{1}{\mu(E(z^{\prime}))E(z^{\prime})}dz^{\prime},$
(1)

where $\mu$, the mobility of the respective species, is derived from [23] for
electrons and [24] for ions, and $E(z)$, the electric field on the axis (see
figure 6), is calculated by a COMSOL Multiphysics simulation [25]. The
corresponding drift times for electrons and ions are shown in figure 7. It
should be noted that since mobility data for electrons and ions in CO2 is not
available for low electric fields, the given estimations are based on constant
mobility in these fields (dashed lines in figure 7).

Figure 6: a) The computational domain for the electric field computation with
COMSOL (included are electric field contour lines). The grounded electrode is
placed at $x=0$ and the electrode tip at $x=160\,$mm. b) The electric field on
the symmetry axis as a function of distance for the first 20 mm from the tip
for $V_{\rm HV}{}=10\,$kV and $V_{\rm LV}{}=1\,$kV. The breakdown electric
field (dashed line) is 0.66 kV/mm at 300 mbar and is taken from Bagheri et al.
[19].

Let us now assume that we indeed have a localized charged species patch
containing, amongst other species, negative ions and free electrons. Figure 7
enables us to investigate the initial and post-drift positions of the ion and
electron distributions, $\tilde{t}_{\rm inc}$ in the baseline experiment
($V_{\rm LV}$ = 0) is 1.2 $\upmu$s. Figure 7 shows that at $V_{\rm
HV}{}=10\,$kV, electrons should be generated at 25 mm from the tip to have
enough time to drift and reach the tip. Hence, we can estimate the initial
position of the boundary of the charged species patch to be about 25 mm away
from the tip. Note that a possible detachment time is not included in this
calculation. Moreover, it is assumed that an electron should reach the tip of
the electrode to initiate a discharge.

Figure 7: Drift time calculated using equation (1) on the axis of symmetry of
electrons and O- ions from a distance $x$ to the electrode at $x=0$ when a
voltage of $V_{\rm LV}{}=10\,$kV or $V_{\rm LV}{}=1\,$kV is applied across the
gap. The mobilities in the unavailable electric field ranges are assumed
constant, and the calculated drift times in those ranges are plotted as dashed
lines.

Under a positive LV pulse (figure 8.a) with $S=1\,$mVs, the corresponding
electron drift distance is around 2 mm toward the electrode tip. During the
main HV pulse, electrons with this new position require 1.01 $\upmu$s to reach
the ionization zone. Therefore, we expect to observe around 182 ns shift in
the $\tilde{t}_{\rm inc}$ peak. This is relatively close to our observation
from the right-hand side of figure 5, where the drift for every $S$ = 1 mVs
positive LV pulse was 92 ns. According to equation 1, the ion drift during
application of 1 kV and 1 $\upmu$s is less than 1 $\upmu$m.

Under a negative LV pulse with $S$=1 mVs (figure 8.b), electrons drift
outwards roughly the same distance as they drift inwards when a positive LV is
applied. Ions which are nearly immobile during $S$=1 mVs, are still re-
generating new electrons in the absence of the drifted electrons. This
procedure also continues after the LV pulse is applied and at the beginning of
the main HV pulse. The newly generated electrons are closer to the tip than
their drifted cousins. Therefore, they have more chance to initiate a
discharge. This supports our observation that no shift was measured in
$\tilde{t}_{\rm inc}$ after an LV pulse was applied with $S$ = 1 mVs. O- ions
only start to drift noticeably under a stronger electric field. Ions under a
negative LV pulse of $S$=1 Vs drift out about 0.45 mm from their initial
position (figure 8.c). At the beginning of the main HV pulse, the electrons
newly generated from the ions that are located in a new position start to
drift toward the tip and imitate the discharge. According to equation 1, the
time that electrons need to drift back 0.45 mm is 95 ns. This is very close to
our observation in figure 5 where the the $\tilde{t}_{\rm inc}$ shift under an
LV pulse with a $S$=-1 Vs is 92 ns.

Figure 8: Visual explanation of electrons and ions drift under positive and
negative LV pulses with $S=$ a) 1 mVs, b) -1 mVs, and c) -1 Vs. In each
figure, 1 shows the drift during an LV pulse and 2 shows the new position of
electrons and ions at the beginning of the HV pulse.

In our previous study [13], we showed that O${}_{2}^{-}$ in synthetic air was
responsible for one of the peaks of the $t_{\rm inc}$ histogram. This peak
showed a symmetric shift for positive and negative LV pulses. A possible
explanation for this difference between air and CO2 can be the different
electron detachment mechanisms. The detachment rate of O${}_{2}^{-}$ in air
depends strongly on the electric field (see figure 9). Therefore O${}_{2}^{-}$
must be in a high field region to detach an electron [26].

Figure 9: Rate constant of the dominant detachment processes in air and CO2.
For air data was taken from [26] and for CO2 from [27].

However, for CO2 the detachment rate decreases with electric field [27] (see
figure 9). The reason is that associative detachment in CO2 proceeds through
autodetachment of the CO2- complex in a short period. In higher fields and
energy, formation of this complex is less probable because of the shorter
collision time. Thus, the associative-detachment reaction in CO2 is more
likely to release electrons in the low fields regions far away from the
electrode tip. Therefore, during a low $S$ value LV pulse, detached electrons
can move and show up their drift in $\tilde{t}_{\rm inc}$ histogram.

One question that can be raised in this section concerns the attachment of
electrons during the drift. According to our hypothesis, electrons should
drift about 25 mm to reach the ionization zone. To calculate the attachment
time, we exert a simple analysis using the BOLSIG+ solver [28, 29] with the
reactions shown in table 1 as input. Results show that at the electric field
around 1 Td at 2.5 mm, where the initial electrons release from the charged
species patch, the attachment time is 1.5 $\upmu$s which is slightly higher
than the inception time. Therefore, this gives enough time for most of the
electrons to approach the ionization zone without being attached.

From the width of the inception time distribution in the baseline experiments
and equation (1) we can estimate that the thickness of the charged species
patch is about 0.2 cm. When we consider the charged species patch as disk of
this thickness and with a diameter of 2.5 cm, its volume is approximately 1
cm3. With the following equation

$k_{detachment}\cdot[\textrm{CO}]\cdot[\textrm{O${}^{-}${}}]=1/(N\cdot t),$
(2)

where $k_{detachment}=5.5\times 10^{-10}$ [21] , we can estimate that in
$t=1.5\,$$\upmu$s attachment time, in order to have 1 electron available in
this volume of ($N=1/$cm3), the required concentration product of
[CO]$\cdot$[O-] should be of the order of magnitude 1015 cm-6. When we assume
that the concentrations of [CO] and [O-] are roughly equal, the density of
each species should be 10${}^{7}-10^{8}$ cm-3 in order to supply the required
electrons for inception. This is a feasible number for these densities in the
charged species patch.

Table 1: List of plasma-chemical reactions used for calculation the ionization and attachment coefficients. Cross-sections are taken from Itikawa database [30]. Elastic collision | e- \+ CO2 $\rightarrow$ e- \+ CO2
---|---
Ionization | e- +CO2 $\rightarrow$ 2e- \+ CO${}_{2}^{+}$
| e- \+ CO $\rightarrow$ 2e- \+ CO+
Attachment | e- \+ CO2 $\rightarrow$ CO + O-

###  3.3 Neutralization and ionization during longer LV pulse durations

In this section we investigate higher values of $S$.

####  3.3.1 Positive LV pulse

Above a certain value of $S$, $t_{\rm inc}$ reaches its minimum value and it
is not possible to have faster inception because electrons and negative ions
will have drifted all the way to the the electrode. Figure 10 shows that
applying a 250 V and 50 ms LV pulse was enough to initiate a discharge during
the rise-time of the HV pulse. Increasing $V_{\rm LV}$ to 1.5 kV was enough to
drift all negative ions close to the tip. This can be verified by the mobility
data in figure 7. Above this $V_{\rm LV}$, ions reside around the electrode
for a long time, where they will be slowly neutralized. Moreover, the
inception probability, $Prob$, decreases with increasing $V_{\rm LV}$ which
shows the lack of initial electrons available to initiate a streamer. We also
observed this effect in our previous experiments in synthetic air [13] where
applying an LV pulse for a long time increased $t_{\rm inc}$.

Figure 10: Histograms of discharge inception time $t_{\rm inc}$ for 600
discharges produced a) for no LV pulse and by applying a 50 ms LV pulse for
$V_{\rm LV}$ = b) 0.25, c) 0.5, d) 1, e) 1.5, f) 1.75, g) 2, h) 4, i) 6, j) 8
kV before a 10 kV pulse of 1 ms with a repetition frequency of 10 Hz. $Prob$
shows the probability of inception in each configuration.

For even higher $V_{\rm LV}$ values, starting at 8 kV, we observe fast
inception again with a probability of 100 %. However, at this voltage level,
the inception peak already occurs during the LV pulse, because $V_{\rm LV}$
itself is already high enough to lead to inception. We observed the same
effect for other values of $t_{\rm LV}$. For $V_{\rm LV}$ below 8 kV, the key
parameter is $S$ such that for a shorter LV pulse duration, a higher LV
amplitude is required to achieve the same effect.

####  3.3.2 Negative LV pulse

Similar to a long positive LV pulse, we applied a negative LV pulse with a
varying $V_{\rm LV}$ and a fixed $t_{\rm LV}$ of 10 ms (figure 11). As we
already discussed (left-hand side of figure 4), for a lower $V_{\rm LV}$ we
observe outward drift in which for $S=1\,$Vs, a 90 ns shift in $\tilde{t}_{\rm
inc}$ was observed (figure 11.a1-a5). This continues until $V_{\rm LV}$ = -2
kV where a further increase of $V_{\rm LV}$ leads to a peak at low $t_{\rm
inc}$ and thereby the disappearance of the original peak in the histogram.
This transition from slow $t_{\rm inc}$ to fast $t_{\rm inc}$ occurs in a
small range of $V_{\rm LV}$, such that at $V_{\rm LV}$= -3.75 kV
$\tilde{t}_{\rm inc}$ is 30 ns (figure 11.b1-b6). Further increasing $V_{\rm
LV}$ causes a negative $t_{\rm inc}$ (figure 11.b7-b9). Such a negative
$t_{\rm inc}$ indicates that the discharge initiates during the rise time of
the LV pulse immediately before the main HV pulse no discharge was observed
during the on-time of the LV pulse.

Figure 11: Histograms of discharge inception time $t_{\rm inc}$ for 600
discharges produced a1) for no LV pulse and by applying a negative 10 ms pulse
for $V_{\rm LV}$ = a2) -0.25, a3) -0.5, a4) -1, a5) -2, (drift) b1) -2, b2)
-2.5, b3) -3, b4) -3.25, b5) -3.5, b6) -3.75, b7) -4 , b8) -6, b9) -8 kV
(neutralization and ionization in LV pulse) before a 10 kV pulse of 1 ms with
a repetition frequency of 10 Hz.

##  4 Summary and conclusion

We measured inception delays $t_{\rm inc}$ of streamers in CO2 gas to study
the inception process in repetitive discharges in this gas. We applied 10 kV
high voltage pulses with a fixed pulse duration of 1 ms and repetition
frequency of 10 Hz. This can produce some residual charges that influence
$t_{\rm inc}$ of the next pulse. To manipulate the residual charges we applied
an LV pulse with different duration $t_{\rm LV}$ and amplitude $V_{\rm LV}$
before the main HV pulses. Application of the LV pulse leads to a shift in the
$t_{\rm inc}$ histogram peak. Three main phenomena were observed, based on the
different applied $S$ ($V_{\rm LV}$$\cdot$$t_{\rm LV}$) values: drift,
neutralization, and ionization during the LV pulse.

At low $S$ values, discharges incept faster or slower for positive and
negative LV pulses respectively. However, we observed an asymmetric shift
differing three order of magnitude between positive and negative LV pulses.
For a positive LV pulse with $S=1\,$mVs, we observed a 92 ns shift in
$\tilde{t}_{\rm inc}$, the median value of $t_{\rm inc}$. To have a similar
magnitude $\tilde{t}_{\rm inc}$ shift with a negative LV pulse, we must apply
a pulse with $S=1\,$Vs. This is not consistent with our earlier observations
in synthetic air. We hypothesize that this is caused by the fundamental
differences between the detachment mechanism in CO2 and air. O${}_{2}^{-}$
ions need to drift to a high electric field region to detach an electron.
However, in the case of CO2 due to the high yield of associative-detachment in
low electric fields, O- ions can release electrons through reaction with CO.
In this case, an LV pulse with a low $S$ value drifts electrons significantly
and ions hardly while an LV pulse high $S$ value can also drift ions on top of
the electrons.

We observed that the corresponding shift in $\tilde{t}_{\rm inc}$ due to
application of LV pulse follows the mobility data and scales consistently with
$S$ value. This indicates that we can neglect the effect of space charge and
its effect on electric fields for low $S$ values. However we observed
different phenomena for higher values of $S$.

Applying a positive LV pulse for a longer time increases $t_{\rm inc}$ but
decreases the probability of inception. This is likely due to neutralization
of electrons and ions at the electrode. At $V_{\rm LV}{}=8\,$kV, the LV pulse
is high enough to ionize the gas and we observed streamers already during the
LV pulse. This is different for a negative LV pulse where ions drift away from
their initial position. For a negative LV pulse with $V_{\rm LV}{}\geq 2\,$kV,
we observe formation of a fast peak ($\tilde{t}_{\rm inc}$ = 120 ns) in the
$t_{\rm inc}$ histogram replacing the original ion-associated peak. Increasing
the voltage above 6 kV caused discharge inception during the LV pulse rising
flank.

In our calculations, electrons should drift about 25 mm to reach the
ionization zone. This seems to be a long distance since electrons can attach
through several pathways such as dissociative attachment to CO2 and O2, and
three body attachment to O2 producing O- and O${}_{2}^{-}$ [31]. Our simple
analysis on four reactions showed that attachment time is slightly longer than
inception times.

## Acknowledgement

We would like to thank Andy Martinez, Prof.Dr. Ute Ebert, and Dr. Jannis
Teunissen from Centrum Wiskunde & Informatica (CWI), who provided insight and
expertise that greatly assisted the research. This project has received
funding from the European Union’s Horizon 2020 research and innovation program
under the Marie Sklodowska-Curie grant agreement 722337.

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