††thanks: Preprint arXiv:2210.13120, published in Chinese Phys. Lett. 40
(2023) 011401, https://doi.org/10.1088/0256-307X/40/1/011401††thanks:
Corresponding author. Email<EMAIL_ADDRESS>
# Axion-Photon Conversion of LHAASO Multi-TeV and PeV Photons

Guangshuai Zhang (张光帅)1 Bo-Qiang Ma (马伯强)1,2,3 1School of Physics, Peking
University, Beijing 100871, China
2Center for High Energy Physics, Peking University, Beijing 100871, China
3Collaborative Innovation Center of Quantum Matter, Beijing, China

###### Abstract

The Large High Altitude Air Shower Observatory (LHAASO) has reported the
detection of a large number of multi-TeV-scale photon events including also
several PeV-scale gamma-ray-photon events with energy as high as 1.4 PeV. The
possibility that some of these events may have extragalactic origins is not
yet excluded. Here we propose a mechanism for the traveling of very-high-
energy (VHE) and ultra-high-energy (UHE) photons based upon the axion-photon
conversion scenario, which allows extragalactic above-threshold photons to be
detected by observers on the Earth. We show that the axion-photon conversation
can serve as an alternative mechanism, besides the threshold anomaly due to
Lorentz invariance violation, for the very-high-energy features of the newly
observed gamma ray burst GRB 221009A.

high-energy photon; axion; axion-photon conversion; gamma ray burst

###### pacs:

14.80.Va; 11.30.Cp; 12.60.-i; 98.70.Rz

It has long been known that the Universe is opaque to very-high-energy (VHE)
photons with energy above 100 GeV and ultra-high-energy (UHE) photons with
energy above 100 TeV, since these photons may collide with low-energy
background photons, such as cosmic microwave background (CMB) photons or
extragalactic background light (EBL) photons, and produce electron-positron
pairs Gould:1966pza ; Fazio:1970pr ; Protheroe:2000hp  during their
propagation in the Universe. In other words, gamma ray photons whose energy is
above certain threshold cannot travel very long in the Universe due to the
absorption of background low-energy photons.

Suppose a UHE photon and a low-energy background photon collide with each
other and produce the electron-positron pair. For the process to be
kinematically possible, it requires that

$s>4m_{e}^{2}c^{4},$ (1)

where $s=(p_{\gamma}+p_{b})^{2}=(p_{e^{+}}+p_{e^{-}})^{2}$ is the invariant
mass of the process. It follows that, for given background photon energy, the
energy of the gamma ray photon should exceed certain threshold,

$E_{\gamma}>E_{\mathrm{th}}\equiv\frac{2m_{e}^{2}c^{4}}{\epsilon(1-\cos\theta)},$
(2)

where $E_{\gamma}$ is the energy of the gamma ray photon, $\epsilon$ is the
energy of the soft photon and $\theta$ is the angle between the three-momenta
of the two photons in the observer frame. Conversely,

$\epsilon>\epsilon_{\mathrm{th}}\equiv\frac{2m_{e}^{2}c^{4}}{E(1-\cos\theta)}.$
(3)

The cross section for the process $\gamma\gamma\rightarrow e^{+}e^{-}$ in tree
level is well-known,

$\sigma(v)=\frac{\pi}{2}\frac{\alpha^{2}}{m_{e}^{2}}(1-v^{2})\Big{[}(3-v^{4})\ln\frac{1+v}{1-v}-2v(2-v^{2})\Big{]},$
(4)

which is the so-called Breit-Wheeler formula Breit:1934zz . Here $v$ is the
speed of the outgoing electron-positron in the center of mass frame of the
process

$v(E_{\gamma},\epsilon,\theta)=\sqrt{1-\frac{2m_{e}^{2}c^{4}}{E_{\gamma}\epsilon(1-\cos\theta)}}$
(5)

and $\alpha$ is the fine-structure constant. The cross section reaches its
maximum of around $1.70\times 10^{-25}~{}\mathrm{cm}^{2}$ when $v$ is around
0.7. When $v\ll 0$ as well as $v\rightarrow 1$, the cross section reaches $0$.
In other words, suppose the collision is face to face, the cross section
reaches maximum when the energy of the background photon is around
$(512~{}\mathrm{GeV}/E_{\gamma})~{}\mathrm{eV}$. Thus in case the energy of
the gamma ray photon lies in the interval between 100 GeV and 10 TeV, the
absorption is dominated by the interaction with the optical/infrared photons
of the EBL. While if the energy of the gamma ray photon is above 100 TeV, the
absorption is dominated by the interaction with CMB photons.

Given the gamma ray photon energy, the mean free path of the gamma ray photon
due to the electron-positron pair production process in the low-energy photon
background is

$\lambda^{-1}(E_{\gamma})=\int_{-1}^{1}\frac{(1-\cos\theta)}{2}\mathrm{d}\cos\theta\int_{\epsilon_{\mathrm{th}}}^{\infty}\mathrm{d}\epsilon~{}\sigma(v(E_{\gamma},\epsilon,\theta))\frac{\mathrm{d}n(\epsilon)}{\mathrm{d}\epsilon},$
(6)

where $n(\epsilon)$ is the number density distribution of the background
photons with respect to the photon energy. The CMB spectrum can be well
approximated with that of black-body radiation with $T=2.7255~{}\mathrm{K}$
ParticleDataGroup:2020ssz . According to the Planck law, the photon number
density distribution is thus

$\frac{\mathrm{d}n(\epsilon)}{\mathrm{d}\epsilon}=\frac{8\pi}{h^{3}c^{3}}\frac{\epsilon^{2}}{e^{\epsilon/(k_{B}T)}-1},$
(7)

where $h$ is the Planck constant, $c$ is the light speed and $k_{B}$ is the
Boltzmann constant. Fig. 1 shows the numerical result of the mean free path of
gamma ray photons with energy around 1 PeV, which is attenuated by CMB
photons. The mean free path is around 10 kpc for gamma ray photons with
$E_{\gamma}=1~{}\mathrm{PeV}$.

Finally, for gamma ray from a source at distance $L$ and with luminosity
$L_{\mathrm{source}}$, the luminosity observed on Earth is

$L_{\mathrm{obs}}=\exp(-L/\lambda)L_{\mathrm{source}}.$ (8)

Figure 1: Mean free path of the gamma-ray photons with energy around 1 PeV due
to CMB photon absorption.

Recently, the Large High Altitude Air Shower Observatory (LHAASO) reported the
detection of 12 UHE $\gamma$-ray sources with statistical significance $\geq
7\sigma$ Aharonian:2021pre ; Cao:2021pre1 . Among the gamma-ray-photon events
detected, the event with the highest energy has an energy as high as
$1.4~{}\mathrm{PeV}$, which is by far the most energetic gamma-ray-photon
event ever been observed. Except Crab Nebulae, the origins of these PeV scale
gamma-ray photon events have not yet been fully identified. Thus there still
exists the possibility that some of these UHE photons come from origins that
are much further than 10 kpc or even outside the Milky Way. The threshold
anomaly of electron-positron production Li:2021tcv ; LI2021a ; Li2021b due to
Lorentz invariance violation HeMa has been proposed to permit the propagation
of these multi-TeV-scale and PeV-scale photons from extragalactic origins to
reach observers on the Earth.

The existence of axion-like particles (ALP) is suggested by several extensions
of the standard model, including four-dimensional ordinary and supersymmetric
models Coriano:2006xh , Kaluza-Klein theories Chang:1999si as well as
superstring theories Svrcek:2006yi ; Arvanitaki:2009fg . ALPs are very light
pseudo-scalar bosons, which are usually represented with $a$. ALP is a sort of
generalization of the axion, the pseudo-Goldstone boson associated with
Peccei-Quinn symmetry, which is proposed as a natural solution to the strong-
CP problem Peccei:1977hh . The key feature of ALPs is the coupling between one
ALP and two photons, $a\gamma\gamma$, of which the Lagrangian is

$\mathcal{L}=\frac{1}{4}g_{a\gamma}\tilde{F}_{\mu\nu}F^{\mu\nu}~{}a=g_{a\gamma}\mathbf{E}\cdot\mathbf{B}~{}a.$
(9)

The coupling constant $g_{a\gamma}$ and the mass of the ALP $m_{a}$ are
usually considered as independent parameters. One of the remarkable
consequences of the $a\gamma\gamma$ coupling is the photon-ALP oscillation,
which states that it is possible that a photon is transformed into an ALP when
the former travels through transverse magnetic field and vice versa
Sikivie:1983ip ; Masso:1995tw .

The ALP-photon oscillation scenario could help the UHE photons to travel
through distance much longer than 10 kpc avoiding absorption of CMB photons
through the following mechanism. Suppose the environment the UHE photons are
produced is surrounded by magnetic field that is strong enough, which is the
common case among typical high-energy cosmic photon factories such as pulsar,
supernova remnant and active galactic nucleus (AGN), then these UHE photons
may be transformed into ALPs. The resulting ALPs could travel through
cosmological distance without interacting with background low-energy photons.
After reaching the Milky Way the ALPs may be transformed into UHE photons
again due to interaction with the Milky Way magnetic field and then be
detected by observers on Earth. The scenario described here was firstly
proposed to explain the excess of very-high-energy gamma ray which should
undergo attenuation of the EBL photons by Ref. DeAngelis:2007dqd and later by
Refs. DeAngelis:2008sk ; Sanchez-Conde:2009exi ; Mirizzi:2009aj ;
Dominguez:2011xy with source B-field or the intergalactic magnetic fields
(IGMF), and then was developed in Ref. referee-ref2 to include the Milky Way
magnetic field. The are also papers involving extragalactic UHE photon-ALP
conversion referee2ref2 ; referee2ref3 ; referee2ref4 , which is feasible with
the LHAASO detection ability of UHE photon events.

The probability that a photon is transformed into an ALP after travelling for
distance $l$ through environment with transverse magnetic field reads
Sikivie:1983ip

$P_{a\gamma}=\frac{1}{1+(E_{\mathrm{c}}/E_{\gamma})^{2}}\sin^{2}\Big{(}\frac{g_{a\gamma}B_{T}l}{2}\sqrt{1+(E_{\mathrm{c}}/E_{\gamma})^{2}}\Big{)},$
(10)

where $B_{T}$ is the transverse magnetic field and the critical energy
$E_{\mathrm{c}}$ is defined as Mirizzi:2009aj

$\displaystyle E_{\mathrm{c}}$
$\displaystyle=\frac{m_{a}^{2}-\omega_{\mathrm{pl}}^{2}}{2g_{a\gamma}B_{T}}$
(11) $\displaystyle\simeq
25\frac{|m_{a}^{2}-\omega_{\mathrm{pl}}|}{(10^{-10}~{}\mathrm{eV})^{2}}\Big{(}\frac{10^{-9}~{}\mathrm{Gauss}}{B_{T}}\Big{)}\Big{(}\frac{10^{-11}~{}\mathrm{GeV}^{-1}}{g_{a\gamma}}\Big{)}~{}\mathrm{GeV},$

with $\omega_{\mathrm{pl}}$ being the plasma frequency of the medium

$\omega_{\mathrm{pl}}=\sqrt{\frac{4\pi\alpha n_{e}}{m_{e}}},$ (12)

here $n_{e}$ is the electron number density of the medium. Eq. (10) shows that
the maximum value of the transformation probability increases with the
increasing of the energy of the photon $E_{\gamma}$. In the limit that $E\gg
E_{\mathrm{c}}$, the transformation probability $P_{a\gamma}$ oscillates
periodically in space with the maximum amplitude reaching one. Thus UHE
photons that travel through cosmological distance provide an efficient
approach for testing the existence of such scenario as well as searching for
ALPs.

The extragalactic background light (EBL) is the second, after the CMB, most
abundant part of the photon medium in the Universe. During the propagation of
VHE and UHE photons in the Universe, their interaction with low-energy EBL
photons should be also considered. The situation of the EBL background photon
distribution could be more complicated, and as a rough estimate, we can take
$\epsilon\simeq 10^{-3}~{}\text{eV}$ to $1~{}\text{eV}$ according to the
distribution of EBL, leading to the corresponding thresholds
$E_{\text{th}}\simeq 261~{}\text{GeV}$ to $261~{}\text{TeV}$ lihao ;
Li:2022wxc . A recent analysis on data suggests a photon number density
$n_{\mathrm{EBL}}=10^{4}~{}m^{-3}$ with a typical EBL photon energy of 1 eV
Brevik:2020cky . One can take this simple EBL model to estimate the EBL
influence on the VHE and UHE photon propagation in the Universe. More explicit
calculations using a realistic EBL model lihao suggest that cosmic photons
above 261 GeV are also difficulty to reach detectors on the Earth due to their
absorption by the interaction with EBL background photons if their sources are
far away.

More recently, the newly observed gamma ray burst GRB 221009A on 9 October
2022 has received extensive attention due the brightest and near-distance
features of this special GRB Zhu2022 . More strikingly, LHAASO reported
gcn32677 the observation of a large number of very-high energy photon events
with energies larger than 500 GeV including also VHE photons with energy up to
18 TeV to be associated with this GRB 22100A. Immediately afterward it is
first reported in Ref. Li:2022wxc that the observation of such high energy
photons by LHAASO is an extraordinary result since extragalactic background
light could absorb these photons severely and the flux is too weak to be
observed. Then it is proposed in Ref. Li:2022wxc that Lorentz invariance
violation induced threshold anomaly of electron-positron production Li:2021tcv
; LI2021a ; Li2021b , i.e., new physics beyond special relativity, can serve
as an explanation of this observation. There have been also a number of
proposals to explain the high energy features of GRB 221009A by axion
interpretations Galanti:2022pbg ; Baktash:2022gnf ; space ; Nakagawa:2022wwm .
The suggestion of adopting the axion-photon conversion as a mechanism to
explain the high energy features of GRB 221009A relies on the astrophysical
environments with pronounced magnetic fields in the line of sight of the
following characteristics: (i) photon-to-axion conversion upon leaving the
host galaxy or cluster; (ii) non-significant re-conversion along the
intergalactic medium (IG) in the path to observer; (iii) axion-to-photon
conversion upon entry into the Milky Way. For (ii) to be feasible, we need to
assume that the intergalactic magnetic field is weak enough. More
specifically, this can be realized for space

$\frac{g_{a\gamma}}{10^{-11}~{}\mathrm{GeV}^{-1}}\lesssim\left(\frac{E}{255~{}\mathrm{TeV}}\right)^{-1}\left(\frac{m_{a}}{10^{-8}~{}\mathrm{eV}}\right)^{2}\left(\frac{B_{\mathrm{IG}}}{\mathrm{nGauss}}\right)^{-1}.$
(13)

In fact, the explicit parameter space for the coupling constant $g_{a\gamma}$
and the ALP mass $m_{a}$ has been constrained from various investigations
referee-ref1 ; referee-ref2 ; referee-ref3 ; referee-ref4 ; referee-ref5 ;
ALPwebsite , and the allowed region of the parameter space to explain the high
energy photons associated with the newly observed GRB 221009A is also
discussed in Ref. space , suggesting that the axion-photon conversion can
serve as a viable mechanism to explain the high energy features of GRB
221009A. Here we indicate that the axion-photon conversion may also explain
some of the multi-TeV and PeV photons already observed by LHAASO once these
photons can be associated with extragalactic origins. We notice that Carpet-2
reported the coincidence of a 251 TeV photon event with GRB 221009A Carpet-2 .
This can be considered as a supplementary support to our proposal of adopting
the axion-photon conversion as an alternative mechanism for the detection of
extragalactic multi-TeV and PeV photons.

In summary, we discuss the detection of multi-TeV-scale and PeV-scale photons
by LHAASO recently, and such multi-TeV and PeV photons may meet serious
challenge due to background low-energy photon absorption if these gamma
photons come from sources farther than 10 kpc or even of extragalactic
origins. We also illustrate a mechanism that relies on the scenario of the
oscillation between photons and the hypothetical axion-like particles (ALPs).
Due to interaction with the magnetic field surrounding the source, a
considerable proportion of the VHE and UHE photons may be converted into ALPs
and then travel through cosmological distance unaffected by EBL and CMB
photons. Before reaching the Earth, a substantial fraction of these ALPs may
be converted into VHE and UHE photons by the Galaxy magnetic field and then be
detected by observers on the Earth. Therefore it is urgent to check whether
the VHE and UHE photons detected by LHAASO have an extragalactic origin or
not.

Acknowledgements.  This work is supported by National Natural Science
Foundation of China (Grant No. 12075003). We acknowledge the help discussions
with Hao Li and Luohan Wang.

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# Simultaneous VLBI astrometry of H2O and SiO Masers toward the Semiregular
Variable R Crateris

Dong-Jin Kim11affiliation: Department of Astronomy, Yonsei University, 50
Yonsei-ro Seodaemun-gu, Seoul, 03722, Republic of Korea 22affiliation: Korea
Astronomy and Space Science Institute, 776 Daedeok–daero, Yuseong–gu, Daejeon
34055, Korea<EMAIL_ADDRESS>33affiliation: Max-Planck-Institut
f$\ddot{\textrm{u}}$r Radioastronomie, Auf dem H$\ddot{\textrm{u}}$gel 69,
53121 Bonn, Germany Se–Hyung Cho22affiliation: Korea Astronomy and Space
Science Institute, 776 Daedeok–daero, Yuseong–gu, Daejeon 34055, Korea;
<EMAIL_ADDRESS>11affiliation: Department of Astronomy, Yonsei University, 50
Yonsei-ro Seodaemun-gu, Seoul, 03722, Republic of Korea Youngjoo
Yun22affiliation: Korea Astronomy and Space Science Institute, 776
Daedeok–daero, Yuseong–gu, Daejeon 34055, Korea<EMAIL_ADDRESS>Yoon Kyung
Choi33affiliation: Max-Planck-Institut f$\ddot{\textrm{u}}$r Radioastronomie,
Auf dem H$\ddot{\textrm{u}}$gel 69, 53121 Bonn, Germany Dong–Hwan
Yoon22affiliation: Korea Astronomy and Space Science Institute, 776
Daedeok–daero, Yuseong–gu, Daejeon 34055, Korea<EMAIL_ADDRESS>55affiliation:
Astronomy program, Department of Physics and Astronomy, Seoul National
University, 1 Gwanakro, Gwanakgu, Seoul 08826, Korea Jaeheon Kim44affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai
200030, China Richard Dodson66affiliation: International Center for Radio
Astronomy Research, M468, The University of Western Australia, 35 Stirling
Hwy, Crawley, Western Australia, 6009, Australia María J. Rioja66affiliation:
International Center for Radio Astronomy Research, M468, The University of
Western Australia, 35 Stirling Hwy, Crawley, Western Australia, 6009,
Australia 77affiliation: Observatorio Astronómico Nacional (IGN), Alfonso XII,
3 y 5, 28014 Madrid, Spain 88affiliation: CSIRO Astronomy and Space Science,
26 Dick Perry Avenue, Kensington WA 6151, Australia Haneul Yang22affiliation:
Korea Astronomy and Space Science Institute, 776 Daedeok–daero, Yuseong–gu,
Daejeon 34055, Korea<EMAIL_ADDRESS>55affiliation: Astronomy program,
Department of Physics and Astronomy, Seoul National University, 1 Gwanakro,
Gwanakgu, Seoul 08826, Korea Suk–Jin Yoon11affiliation: Department of
Astronomy, Yonsei University, 50 Yonsei-ro Seodaemun-gu, Seoul, 03722,
Republic of Korea

###### Abstract

We obtained, for the first time, astrometrically registered maps of the 22.2
GHz H2O and 42.8, 43.1 and 86.2 GHz SiO maser emission toward the semiregular
b-type variable (SRb) R Crateris, at three epochs (May 21st, 2015, Jan 7th and
Jan 26th, 2016) using the Korean VLBI Network. The SiO masers show a ring-like
spatial structure, while the H2O maser shows a very asymmetric one-side
outflow structure, which is located at the southern part of the ring-like SiO
maser feature. We also found that the 86.2 GHz SiO maser spots are distributed
in an inner region, compared to those of the 43.1 GHz SiO maser, which is
different from all previously known distributions of the 86.2 GHz SiO masers
in variable stars. The different distribution of the 86.2 GHz SiO maser seems
to be related to the complex dynamics caused by the overtone pulsation mode of
the SRb R Crateris. Furthermore, we estimated the position of the central star
based on the ring fitting of the SiO masers, which is essential for
interpreting the morphology and kinematics of a circumstellar envelope. The
estimated stellar coordinate corresponds well to the position measured by
gaia.

instrumentation: interferometers—masers — stars: individual (R Crateris) —
stars: late-type

##  1 Introduction

It is well known that Asymptotic Giant Branch (AGB) stars rapidly lose their
mass due to the strong stellar wind. The dense and strong stellar wind forms a
circumstellar envelope (CSE), which is the main site of SiO, H2O, and OH
masers around oxygen-rich AGB stars. Various SiO maser lines arise from a
distance of 2–4 stellar radii and show a ring-like structure with inflow or
outflow motions below a dust formation layer (Diamond et al., 1994; Diamond &
Kemball, 2003; Perrin et al., 2015). The 22.2 GHz H2O maser arises partially
in the dust formation layer and partially at greater radii above the dust
layer and represents acceleration motions of stellar winds in the CSE (Rosen
et al., 1978; Reid & Moran, 1981; Imai et al., 2003). Maps of the H2O and SiO
masers, registered to the stellar continuum, indicate that the star is at the
center of shell-like maser emission in the case of W Hya (Reid & Menten, 1990,
2007). Therefore, a combined study of the H2O and SiO masers enables us to
investigate the formation and development of stellar winds.

However, previous VLBI observations of the H2O and SiO masers have been
performed separately, due to the lack of a simultaneous observation system for
the H2O and SiO masers. The Korean VLBI Network (KVN) is equipped with a
quasi-optics system for simultaneous observations at the K (21–23 GHz), Q
(42–44 GHz), W (85–95 GHz), and D (125–142 GHz) bands (Han et al., 2008).
Therefore, the KVN Key Science Project (KSP) on evolved stars was started for
the combined study of H2O and SiO masers in the CSE
(https://radio.kasi.re.kr/kvn/ksp.php, Cho et al. (2018)). The first stage of
the KSP on evolved stars focused on nine objects for which astrometrically
registered maps for both the H2O and SiO masers have been successfully
obtained.

Here we present the astrometrically registered maps for both the H2O and SiO
maser lines at three epochs for the semiregular variable star R Crateris (R
Crt). R Crt is classified as a SRb star with the spectral type of M7 (Samus’
et al., 2017). SRc type indicates super-giant stars, and SRa type variables
show a persistent periodicity with smaller light-amplitudes ($<$2.5 mag in V)
than Mira variables. In contrast, SRb type variables show an uncertain or
superimposed periodicities, such as one or more overtones. The role of the
overtone pulsation mode on the maser properties and the mass loss process has
rarely been investigated. In this sense, R Crt is a high priority target for
the study of overtone-pulsators emitting SiO, H2O and OH masers (Etoka et al.,
2001; Kim et al., 2010). The mass loss rate of R Crt was estimated to be
8.0$\times$10-7M⊙yr-1 (Olofsson et al., 2002). It has a relatively higher mass
loss rate than that of the usual semiregular variables (Knapp et al., 1998).
The estimated distance to R Crt is some what ambiguous, with estimates ranging
from 170 to 300 pc (Knapp et al., 1998; Szymczak & Le Squeren, 1999; Ishitsuka
et al., 2001).

##  2 Observations and Data Reduction

We performed simultaneous VLBI monitoring observations of H2O 616–523
(22.23508 GHz) and SiO v=1, 2, J=1$\rightarrow$0, SiO v=1, J=2$\rightarrow$1,
3$\rightarrow$2 (43.12208, 42.82058, 86.24344, 129.36335 GHz) masers toward R
Crt with the KVN, which consists of three 21 m radio telescopes (Lee et al.,
2011). The monitoring was carried out at 11 epochs from Oct 2014 to Feb 2016.
In this paper, we present three epochs of observations, which show the
astrometrically registered maps of the H2O and SiO masers. The remaining data
will be presented in a forthcoming paper (D. J. Kim et al. in prep.). The
correlator coordinates used for R Crt were R.A.=11:00:33.850,
Dec.=–18:19:29.60. The size of the synthesized beams are typically 6/3/1.5 mas
at the K/Q/W bands, and the system temperatures were up to 220/210/450/800 K
(epoch 1), 200/300/800/1200 K (epoch 2), and 140/200/400/300 K (epoch 3) at
K/Q/W/D bands respectively. In total 16 Intermediate Frequencies (IFs) were
used (6/6/2/2 for K/Q/W/D bands), and each IF has a 16 MHz bandwidth.

The schedule consisted of alternating $\sim$2 min scans between the target
source R Crt and a continuum calibrator source J1048-1909 using simultaneous
4-band observations. The angular separation between R Crt and J1048-1909 is
3.06 degrees. J1048-1909 has a positional accuracy of 0.06 mas in R.A. and
0.09 mas in Dec. (Ma et al., 2009). A fringe finder, 4C39.25, was also
observed for 5 min every hour. Total observation time was about 7 hr for each
epoch. We used the Mark5B system for data recording and playback, which has a
maximum recording rate of 1 Gbps. The correlation was performed using the DiFX
software correlator with a spectral resolution of 512 channels per IF,
providing velocity resolutions of 0.42, 0.22, 0.11, and 0.07 km s-1 for line
observations at K, Q, W, and D bands respectively. We used the Astronomical
Image Processing System (AIPS) package for the data reductions.

We applied conventional phase referencing (PR) techniques for the astrometric
measurement of the position of the 22.2 GHz H2O maser with respect to the
external reference source, J1048-1909. Next, we used the Source Frequency
Phase Referencing (SFPR) technique to attain a bonafide astrometric
registration of the multiple SiO maser lines (42.8, 43.1, and 86.2 GHz). The
combination of the simultaneous multi-frequency capability of the KVN and the
SFPR analysis results in precise absolute positions of the maser lines, when
the external reference source has precise absolute coordinates. The basis of
the SFPR calibration strategy is presented in Rioja & Dodson (2011), and the
first application to a spectral line is presented in Dodson et al. (2014). The
positions of the maser spots in the PRed and SFPRed maps were measured using
the two-dimensional Gaussian fitting task in AIPS, and artificial components
were filtered out by the requirement that they must appear in more than three
successive velocity channels in the same maser feature.

##  3 Results

As shown in Figure 1, the H2O masers toward R Crt are only distributed in the
southern part of the SiO masers and spread out over 100$\times$80 mas with
several distinct maser features. The spatial distribution of the H2O maser is
asymmetric, but SiO masers show a ring-like structure of 30 mas in size. The
129.3 GHz SiO maser map was not obtained. Figure 2 displays the total power
(solid line) and correlated flux (dotted line) spectra of each maser lines at
three epochs. The fraction of missing flux were obtained from single-dish
observations using the KVN Yonsei telescope, and they range from 20 to 80%. As
a general trend, the missing flux increases with resolution. The 86.2 GHz SiO
maser shows a higher missing flux rate (up to 80 %) and a wider velocity width
than the other maser lines. The full velocity widths of the H2O and SiO maser
spectra are comparable (from VLSR=3 to 16 km s-1) except for the 86.2 GHz SiO
maser (from VLSR=-2 to 18 km s-1). The peak flux of the H2O maser
($\sim$hundreds of Jy) is much higher than those of the SiO masers
($\sim$dozens of Jy).

Figure 1: Astrometrically registered integrated intensity-velocity maps of the
H2O and SiO masers. The peak fluxes of the 22.2/42.8/43.1/86.2 GHz masers are
684.7/7.9/19.5/13.5, 395.5/28.8/2.8/2.8 and 453.8/15.0/1.8/16.0 Jy beam-1 km
s-1 at epoch 1, 2, and 3 respectively. The contour levels are plotted with log
scale based on the peak fluxes.

Figure 2: Total power (solid) and correlated flux (dashed) spectra of the H2O
and SiO masers at the 3 epochs of observations. The total power spectra were
obtained from the KVN Yonsei telescope. The fraction of missing flux is marked
at the top-left corner. Vertical dotted lines indicate the stellar velocity of
R Crt (VLSR=10.8 km s-1)

.

Figure 3 shows the position-velocity spot maps of the masers. The majority of
maser spots are blue-shifted in both the H2O and SiO masers. The high velocity
spots of the 86.2 GHz maser appear at epoch 2 (Jan 7, 2016), and their
locations are marked with a blue dashed square in Figure 3. The high velocity
spots are blue shifted up to 13 km s-1 with respect to the stellar velocity of
R Crt (VLSR=10.8 km s-1), which was measured from the CO J=1$\rightarrow$0 and
J=2$\rightarrow$1 lines (Kahane & Jura, 1994), and they exceed the terminal
velocity of R Crt, 10.3 km $s^{-1}$ (Nyman et al., 1992). We estimated the
central position and size of the ring-like structure of the SiO masers with a
least-square minimization fitting method. There was no fit in the case of the
42.8 GHz SiO maser spots due to their insufficient number. Both 43.1 and 86.2
GHz SiO maser spots were used for determining the central star position.
Figure 4 and Table 1 show the fitting results. The deduced absolute coordinate
of the central star is RA=11:00:33.8201, Dec.=–18:19:29.618, which is the mean
value of epochs 2 and epoch 3. Epoch 1 was excluded in the averaging to
minimize uncertain factors such as proper motion and annual parallax. The
fitted radius of the ring-like structures is in the range of 13.35 to 13.84
mas for the 43.1 GHz SiO maser and 11.76 to 12.72 mas for the 86.2 GHz SiO
maser.

Figure 3: The position-velocity spot maps of the H2O and SiO masers. The size
of a spot is proportional to its flux density. Upper: Maps of the H2O maser
spots. The red crosses indicate the central positions of the SiO ring-like
structures. The black cross represents the position of R Crt measured by gaia
satellite. Lower: Maps of the 42.8 (triangle), 43.1 (square) and 86.2 (circle)
GHz SiO maser spots. The spots inside the dotted-blue squares present the
highly blue-shifted components of the 86.2 GHz SiO masers (–2 to 0 km s-1).

The H2O maser features show coincident spatial distributions over three epochs
although they show remarkable changes in their intensities. In the total power
spectra, the peak intensity of the blue-shifted components diminish, whilst
the red-shifted components increase (Figure 2). The SiO masers present rapid
changes in the number of spots, intensities, and positions during a short time
interval between epochs 2 and 3 (19 days).

Figure 4: Upper: Spatial distributions of the 42.8 (red), 43.1 (green) and
86.2 GHz SiO (grey) maser spots along with superimposed fitted rings for the
43.1 and 86.3 GHz SiO maser spots, shown by the large green continuous and
grey-dashed circles, respectively (Cho et al., 2018). The size of an
individual spot is proportional to the flux density. The red crosses mark the
central position of the rings. Lower: The number of maser spots according to
the distance from the center. Each color indicates the 42.8 (red), 43.1
(green) and 86.2 GHz SiO (grey) masers.

Table 1: The radius and central position of the ring-like structure of the SiO
maser lines.

Epoch | Rest frequency | Ring radius* | Fitting error | Converted coordinate (J2000) | $\sqrt{\Delta\alpha^{2}+\Delta\delta^{2}}$
---|---|---|---|---|---
| (GHz) | (mas) | (AU) | (mas) | R.A. | Dec. | (mas)
1 | 43.122 | 13.35 | 2.27 | 0.34 |  |  |
(May 21, 2015) | 86.243 | 12.72 | 2.16 | 1.08 | 11:00:33.8204 | –18:19:29.619 | 4.1
2 | 43.122 | 13.84 | 2.35 | 0.20 |  |  |
(Jan 7, 2016) | 86.243 | 11.76 | 2.00 | 0.73 | 11:00:33.8202 | –18:19:29.621 | 3.7
3 | 43.122 | 13.45 | 2.29 | 0.52 |  |  |
(Jan 26, 2016) | 86.243 | 11.96 | 2.03 | 0.87 | 11:00:33.8201 | –18:19:29.616 | 3.9

**footnotetext: The reference coordinate of R Crt used in the observations is
R.A.=11:00:33.850 Dec.=–18:19:29.60 (J2000). A distance of 170 pc was assumed
to calculate the ring diameter by AU unit (Ishitsuka et al., 2001). Fitting
error column indicates RMS values of the fitting.

##  4 Discussion

###  4.1 Comparison among the SiO v=1, 2 J=1$\rightarrow$0 (43.1, 42.8 GHz)
and v=1, J=2$\rightarrow$1 (86.2 GHz) masers

The most striking results from our observations is the discovery that the 86.2
GHz SiO maser spots are located closer to the central star, with a wider
velocity range than those of the 43.1 and 42.8 GHz SiO masers, as shown in
Figure 4 and Table 1. This is the opposite of what hitherto has been found. In
the case of fundamental-pulsators such as WX Psc, R Leo, and $\chi$ Cyg
(Soria-Ruiz et al., 2004, 2007) the 86.2 GHz maser was distributed in the
comparable or more distant regions (up to $\sim$30%) from the central star
than the 43.1 and 42.8 GHz masers. Line overlap effects between the H2O and
SiO emission were proposed for the interpretation of 86.2 GHz maser location
(Desmurs et al., 2014). Radiative Transfer (RT) models based on the large
velocity gradient (LVG) assumption, which suppose a localized maser
amplification, predict that 86.2 GHz maser would be stronger and located at
the distant region from the central star compared to the 43.1 GHz maser with
comparable velocity ranges (Humphreys et al., 2002; Gray et al., 2009). Such
results are consistent with the previous VLBI observations for the
fundamental-pulsators (Soria-Ruiz et al., 2004, 2007), but not for R Crt.

R Crt is a SRb type variable showing superimposed periodicities in the optical
light curve and OH maser variability (Etoka et al., 2001; Pojmanski, 2003).
Our single-dish monitoring of H2O and SiO masers also presents a signature of
secondary variability (D. J. Kim et al. in prep.). SRb type variables are
mainly overtone-pulsators, which have a short optical period with superposed
periodicities, whereas other variables (SRa, SRc, and LPV) are characterised
by a fundamental pulsation mode. The overtone-pulsator would result in more
turbulent environment across the CSE, and it may produce different physical
conditions for the masers compared to the fundamental-pulsators. In addition,
Herpin et al. (1998) detected high velocity components in the 86.2 GHz maser,
which exceed the terminal velocity of the host AGB star. A statistical study
points out that the high velocity wings of the 86.2 GHz maser lines dominantly
appear in the SRb type variables (McIntosh & Indermuehle, 2015). This tendency
is not seen in SRa, SRc, and long period variables.

On the other hand, the non-local radiative transfer (RT) model occasionally
shows a slightly wider velocity range of the 86.2 GHz maser than the 43.1 GHz
maser, supporting our observational results (Yun & Park, 2012; McIntosh &
Indermuehle, 2015). The non-local RT model considers all the velocity coherent
regions along the line of sight to reflect the influence of distant maser
clumps, which can contribute to the local maser amplification. This process is
more likely to happen in overtone-pulsators rather than fundamental-pulsators,
due to their complex dynamical environment induced by short and overlapping
shock waves. Thus, the non-local RT model would be better at reproducing the
86.2 GHz SiO maser lines with a wider velocity range and maser distributions
for SRb type variables than the RT model in fundamental-pulsators based on the
LVG assumption (Humphreys et al., 2002; Gray et al., 2009).

However, the non-local RT model has never predicted that the 86.2 GHz maser
features could appear at a smaller radius than those of the 43.1 GHz maser.
The non-local RT model (Yun & Park, 2012) used the hydrodynamic solutions for
a fundamental-pulsator. Therefore, further studies of the non-local RT model
based on the hydrodynamic CSE model for overtone-pulsators are required for
interpreting our results and for evaluating the effect of different pulsation
modes on the various SiO maser features. In addition, we need to consider the
high fractional missing flux of the 86.2 GHz maser (up to 80 %) and relatively
poor spatial sensitivity of the KVN, because is is possible that the 86.2 GHz
maser features in the outer region of the 43.1 GHz maser of R Crt could have
been resolved out or undetected due to their weak intensity. Therefore,
following-up VLBI observations for R Crt and other SRb type variables are also
required to confirm the inner distribution of the 86.2 GHz SiO masers in
overtone-pulsators. Additionally adding more antennas and shorter baselines to
the KVN would clarify these questions; such an enhancement is being planned.

###  4.2 The stellar position and development of asymmetric structures in
maser features

The stellar position is a crucial parameter for analyzing the morphology and
dynamics of the SiO and H2O masers. Our astrometric observation scheme, using
PR and SFPR, has provided the accurate positions of the SiO and H2O maser
spots. The position of the central star is estimated by a ring-fitting method
to the SiO maser features. The fitting results and errors are listed in Table
1. The SFPR technique typically results in a relative positional error of less
than 1 mas between the H2O and SiO masers (Yoon et al., 2018). This is
dominated by the ring fitting error, which is about 1.87 mas corresponding to
three times of the mean RMS value of the fittings. The total relative
positional errors between the central star and the H2O maser spots will be
less than 3 mas, which is 8 times more accurate than the positional error of
the central star determined by the three-dimensional velocity field of the
22.2 GHz H2O maser (Ishitsuka et al., 2001).

The astrometric performance of the PR observation using the KVN has not yet
been explicitly demonstrated, as this aspect is still undergoing
commissioning. However if we extrapolate from the expected behavior as found
in Pradel et al. (2006), we obtain an astrometrical positional error, for a
3.06o calibrator-source separation, of about $\sim$2 mas. Combined with the
ring fitting errors mentioned above we expect positional accuracies of $\sim$4
mas. In Table 1, we find positional differences of 2 mas (R.A.) and 5 mas
(Dec.) between epochs 2 and 3, which must be due to measurement errors as
there is only a short time separation of 19 days. This is consistant with the
expected astrometric error. Therefore, we estimated the absolute stellar
position (R.A.=11:00:33.8201, Dec.=–18:19:29.618) as the mean position
obtained from epochs 2 and 3. The observed gaia position for their reference
epoch (Jun 2015), marked with a black cross in Figure 3, can be compared to
our epoch 1 (May 2015) data directly as the proper motions and parallax
contributions will be negligible. In this case the gaia coordinates are
11:00:33.820623, –18:19:29.6219571, which are offset from our epoch 1 position
by 3, 3 mas on the sky.

In Figure 1, the asymmetric spatial distribution of the H2O maser with respect
to the ring-like structure of the SiO maser cannot be interpreted with a
spherical expanding shell structure. The observational results of the Japanese
VLBI Network showed a possible bipolar outflow based on the three-dimensional
velocity field of the H2O maser (Ishitsuka et al., 2001). However, the H2O
maser in Figure 1 shows a possible one-side outflow toward the southern part
of the estimated position of the central star in Figure 3. Also, the H2O maser
in Figure 2 shows a significant intensity variation between May 2015 and Jan
2016. In the case of the red hypergiant star NML Cyg which shows a bipolar
outflow in the H2O maser feature, the central star is located close to the
prominent blue-shifted H2O maser features and not close to the center between
the red-shifted and blue-shifted outflow features (Zhang et al., 2012). It is
still uncertain what kind of physical process causes the development of the
asymmetric structure of the H2O maser. To investigate the development process
of the highly asymmetric one-side outflow features of the H2O maser from the
ring-like SiO maser features in R Crt, we may need to measure the proper
motion of both the H2O and SiO masers through intensive VLBI monitoring
observations.

##  5 Summary

Simultaneous VLBI monitoring observations of the H2O 616–523 and SiO v=1, 2,
J=1$\rightarrow$0 and v=1, J=2$\rightarrow$1, J=3$\rightarrow$2 masers toward
the semiregular variable R Crt were performed with the KVN from Oct 2014 to
Feb 2016. We obtained high precision “bonafide” astrometrically registered
multi-frequency maps of the 22.2 GHz H2O and 43.1/42.8/86.2 GHz SiO masers at
three epochs with the SFPR method. The SiO masers show a ring-like feature,
while the H2O masers show a very asymmetric one-sided outflow, which is
located only at the southern part of the central star. Based on these
astrometrically registered maps, we determined the position of the central
star with an accuracy of 3 mas to the H2O masers. The estimated stellar
position is consistent with gaia DR2 data. Furthermore, the SiO v=1,
J=2$\rightarrow$1 maser spots are distributed in the inner-most region, with a
15 % smaller radius, compared to those of the SiO v=1, J=1$\rightarrow$0
maser. Some maser spots of the SiO v=1, J=2$\rightarrow$1 maser also show
highly blue-shifted components which exceed the terminal velocity of R Crt. We
suggest that these features may be related with the characteristics of the
overtone pulsation mode of the SRb-type R Crt associated with complex dynamics
in its CSE. However, we need to investigate other overtone-pulsators to
confirm whether those properties are common.

This work was supported by the Basic and Fusion Research Programs (2014-2017).
We are grateful to all of the staff members at KVN who helped to operate the
array and the single dish telescope and to correlate the data. The KVN is a
facility operated by KASI (Korea Astronomy and Space Science Institute), which
is under the protection of the National Research Council of Science and
Technology (NST). The KVN operations are supported by KREONET (Korea Research
Environment Open NETwork) which is managed and operated by KISTI (Korea
Institute of Science and Technology Information).

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