# Detection of the extended $\gamma$-ray emission from the high Galactic
latitude Calvera’s SNR candidate

Yuliang Xin Xiaolei Guo  School of Physical Science and Technology, Southwest
Jiaotong University, Chengdu 610031, China
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We report the extended GeV $\gamma$-ray emission that spatially associated
with the high Galactic latitude supernova remnant (SNR) candidate - Calvera’s
SNR with the Pass 8 data recorded by the Fermi Large Area Telescope. The
$\gamma$-ray spectrum of Calvera’s SNR between 100 MeV and 1 TeV shows an
evident ($\sim$ 3.4$\sigma$) spectral curvature at several tens of GeV. The
multi-wavelength data can be fitted with either a leptonic model or a hadronic
one. However, the leptonic model exhibits the inconsistent between the flat
radio spectrum and the hard GeV $\gamma$-ray spectrum of Calvera’s SNR. For
the hadronic model, the spectral index of protons should be harder than 1.6.
And the total energy of protons is fitted to be more than one order of
magnitude higher than the explosion energy of a typical supernova, which also
challenges the hadronic model. The evident spectral curvature and the absence
of non-thermal X-ray emission from Calvera’s SNR makes it to be an interesting
source bridging young-aged SNRs with bright non-thermal X-ray emission and
old-aged SNRs interacting with molecular clouds.

gamma rays: general - gamma rays: ISM - ISM: individual objects (Calvera) -
radiation mechanisms: non-thermal

##  1 Introduction

Supernova remnants (SNRs) are widely believed to be the dominant accelerators
of Galactic cosmic rays (CRs; Baade & Zwicky, 1934; Liu et al., 2022). Up to
now, nearly three hundred of SNRs have been clearly identified mainly through
the radio observations (Green, 2014). And among them, only ten percent is
detected with $\gamma$-ray emission (Zeng et al., 2019, 2021). Thanks to the
quick developments of the ground Cherenkov and space telescopes, especially
the Large Area Telescope (LAT) on board the Fermi satellite, more and more
SNRs are detected with good spatial and spectral measurements in the
$\gamma$-ray range during the past decade (Yuan et al., 2014; Federici et al.,
2015; Xiang & Jiang, 2021; Acero et al., 2022), revealing a variety of
$\gamma$-ray spectra (Zeng et al., 2019). Determining the nature of the
$\gamma$-ray emission plays an essential role in evaluating contributions of
SNRs to the flux of Galactic cosmic rays, which requires studying individual
remnants to determine both if and how they accelerate cosmic rays. Here we
report on the $\gamma$-ray detection of a recently discovered SNR associated
with the high Galactic latitude Calvera pulsar.

Calvera (also known as 1RXS J141256.0+792204 or PSR J1412+7922) is a high
Galactic latitude ($b$ $\sim$ 37∘) soft X-ray source, which was first
discovered in the ROSAT All-Sky Survey and then identified to be an isolated
neutron star (Haberl, 2007; Rutledge et al., 2008). Using XMM-Newton data,
Zane et al. (2011) discovered X-ray pulsations with the period of P = 59.2 ms,
which confirms the neutron star nature of Calvera. The spectral analysis shows
that the X-ray emission from Calvera is thermal, which can be well reproduced
by a two-component model composed of either two hydrogen atmosphere models or
two blackbodies in the range of 0.1 - 0.25 keV (Zane et al., 2011). The
following Chandra and NICER observations determined the period derivative of
Calvera to be $\dot{P}=(3.19\pm 0.08)\times 10^{-15}$ s s-1, corresponding to
the characteristic age of $\tau_{c}$ = $P/2\dot{P}$ = 294 kyr (Halpern et al.,
2013; Halpern & Gotthelf, 2015; Bogdanov et al., 2019). The spin-down
luminosity and the surface dipole magnetic field strength are calculated to be
$\dot{E}=6.1\times 10^{35}$ erg s-1 and $B_{s}=4.4\times 10^{11}$ G,
respectively. The deep search failed to detect the radio pulsations at the
known period for Calvera, which makes it to be a radio-quiet X-ray pulsar
(Hessels et al., 2007; Zane et al., 2011). With the Fermi-LAT data, Zane et
al. (2011) claimed to detect the $\gamma$-ray pulsations from Calvera at $>$
100 MeV. However, the further analysis of additional Fermi-LAT data did not
confirm the $\gamma$-ray detection, and an $\gamma$-ray upper limit was
derived to be at least two orders of magnitude below the typical $\gamma$-ray
luminosities of pulsars with comparable spin-down luminosity (Halpern, 2011;
Halpern et al., 2013).

Recently, a ring of low surface brightness radio emission around the Calvera
pulsar was detected with the data from LOFAR Two-metre Sky Survey (LoTSS;
Shimwell et al., 2017; Arias et al., 2022). Together with the low-significance
emission at 325 MHz by the Westerbork Northern Sky Survey (WENSS; Rengelink et
al., 1997) and 1.4 GHz by the NRAO VLA Sky Survey (NVSS; Condon et al., 1998),
the spectral index of the ring is calculated to be 0.71 $\pm$ 0.09. Arias et
al. (2022) considered three possible interpretations for the radio ring,
including an HII region, a SNR, or an Odd Radio Circle (ORC). The flat radio
spectrum makes an HII region disfavored, and the difference in size between
the ring and the other known ORCs discards the possibility that the ring is an
ORC. Considering the positional coincidence among the ring, the Calvera pulsar
and the X-ray-emitting non-equilibrium ionisation plasma around this region,
the SNR interpretation is favored. If the ring is indeed a SNR associated with
Calvera pulsar, Calvera’s SNR (SNR G118.4+37.0) will be one of few SNRs in the
Galactic halo. And the large heights above the Galactic plane also makes it to
be an interesting source to study the SNR evolution in diffuse environments
and probe the interstellar medium (ISM) of the Milky Way halo (Arias et al.,
2022).

In the present work, we carry out a detailed analysis of the GeV $\gamma$-ray
emission around the region of Calvera with the Fermi-LAT Pass 8 data. And this
paper is organized as follows. In Section 2, we describe the data analysis
routines and results, including the spatial and spectral analysis. A
discussion about the physical origin of the $\gamma$-ray emission is presented
in Section 3 based on the multi-wavelength observations, followed by the
conclusion of this work in Section 4.

##  2 Fermi-LAT Data Analysis

###  2.1 Data Reduction

We use the latest Pass 8 version of the Fermi-LAT data recorded from 2008
August 4 (Mission Elapsed Time 239557418) to 2022 March 24 (Mission Elapsed
Time 669772805) to analysis the $\gamma$-ray emission around Calvera. This
analysis is performed within a $20^{\circ}\times 20^{\circ}$ square region
centered at the position of Calvera pulsar. For the spectral analysis, the
energy range adopted is 100 MeV - 1 TeV, while the spatial analysis is carried
out in the energy range of 1 GeV - 1 TeV in consideration of the improved LAT
resolution at higher energies. The events with zenith angles greater than
$90^{\circ}$ are excluded to reduce the contamination from Earth Limb. The
standard ScienceTools software package
111http://fermi.gsfc.nasa.gov/ssc/data/analysis/software/ with the binned
analysis method, together with the instrumental response function (IRF) of
“P8R3_SOURCE_V3”, are adopted. The Galactic and isotropic diffuse background
emissions are modeled according to gll_iem_v07.fits and
iso_P8R3_SOURCE_V3_v1.txt
222http://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html,
respectively. The sources in the incremental version of the fourth Fermi-LAT
source catalog (4FGL-DR3; Abdollahi et al., 2020, 2022), together with the two
diffuse backgrounds, are included in the model, which is generated by the
user-contributed software
make4FGLxml.py333http://fermi.gsfc.nasa.gov/ssc/data/analysis/user/. For the
fitting procedure, the spectral parameters and the normalizations of sources
within $10^{\circ}$ around Calvera are set to be free, as well as the
normalizations of the two diffuse backgrounds.

###  2.2 Spatial Analysis

In the 4FGL-DR3 catalog, there is a $\gamma$-ray point source (4FGL
J1409.8+7921) in the region of Calvera, which has no identified counterpart
(Abdollahi et al., 2020, 2022). And the photon spectral index and integral
photon flux in the energy of 1 - 100 GeV with a power law model are given to
be 1.82 $\pm$ 0.22 and (4.64 $\pm$ 1.69)$\times 10^{-11}$ photon cm-2 s-1,
respectively. With the energy range of 1 GeV - 1 TeV, we first created a
$2^{\circ}\\!.0$ $\times$ $2^{\circ}\\!.0$ Test Statistic (TS) map with the
command gttsmap by subtracting the emission from the diffuse backgrounds and
all 4FGL-DR3 sources (except 4FGL J1409.8+7921) in the best-fit model, which
is shown in the left panel of Figure 1. The TS map shows significant
$\gamma$-ray emission around Calvera, which is marked as SrcX afterwards. To
test the spatial extension of the $\gamma$-ray emission from SrcX, we first
adopted Fermipy, a PYTHON package that automates analyses with the Fermi
Science Tools (Wood et al., 2017), to get the best-fit position of SrcX as a
point source. And the corrected coordinate of SrcX is fitted to be R.A. =
$212^{\circ}\\!.454\pm 0^{\circ}\\!.021$, Dec. = $79^{\circ}\\!.344\pm
0^{\circ}\\!.028$. With the new position, the TS value of SrcX as a point
source is 23.8. Then we adopted the extended templates to describe the
$\gamma$-ray emission from SrcX, including an uniform disk, a two-dimensional
(2D) Gaussian template, and an uniform ring. The best-fit central positions
and extensions (the radius containing 68% of the intensity; R68) of the
uniform disk and 2D-Gaussian template are given by Fermipy, which are shown in
Table 1. For the template of the uniform ring, the central position, as well
as the inner and outer radii, were adopted from the radio observation by LOFAR
at 144 MHz (Arias et al., 2022). The TS values for the different extended
templates are listed in Table 1.

We compared the overall maximum likelihood of the extended template
($\mathcal{L}_{\rm ext}$; alternative hypothesis) with that of the point
source model ($\mathcal{L}_{\rm Pt}$; null hypothesis), and defined the
significance of the extension as to be TSext = -2(log$\mathcal{L}_{\rm Pt}$ \-
log$\mathcal{L}_{\rm ext}$) (Lande et al., 2012). The alternative hypothesis
is significantly preferred to the null hypothesis only if TSext $>$ 16\. We
found that the uniform disk centered at (R.A. = $212^{\circ}\\!.649$, Dec. =
$79^{\circ}\\!.389$) with a radius of $0^{\circ}\\!.478$ can best fit the
$\gamma$-ray emission from SrcX. And the improved significance is TSext =
30.0, which corresponds to $\sim$5.5 $\sigma$ extension with one additional
degree of freedom relative to the point source model. With the extended
template of uniform disk, the TS value of SrcX is fitted to be 54.1 in the
energy range of 1 GeV - 1 TeV, corresponding to a significance level of
$\sim$6.4 $\sigma$ with five degrees of freedom.

Table 1: Spatial Analysis for SrcX between 1 GeV and 1 TeV Spatial Template | R.A., Dec. | TS Value | Degrees of Freedom | -log(Likelihood)
---|---|---|---|---
Point Source |  $212^{\circ}\\!.454\pm 0^{\circ}\\!.021$, $79^{\circ}\\!.344\pm 0^{\circ}\\!.028$ | 23.8 | 4 | 122016.00
Uniform disk |  $212^{\circ}\\!.649\pm 0^{\circ}\\!.038$, $79^{\circ}\\!.389\pm 0^{\circ}\\!.038$, R68 = $0^{\circ}\\!.392^{+0^{\circ}\\!.025}_{-0^{\circ}\\!.025}$ | 54.1 | 5 | 122001.07
2D-Gaussian |  $212^{\circ}\\!.361\pm 0^{\circ}\\!.071$, $79^{\circ}\\!.358\pm 0^{\circ}\\!.068$, R68 = $0^{\circ}\\!.394^{+0^{\circ}\\!.068}_{-0^{\circ}\\!.050}$ | 49.2 | 5 | 122003.51
Uniform Ring |  $212^{\circ}\\!.803$, $79^{\circ}\\!.388$, Rinner = $14^{\prime}\\!.2$, Router = $28^{\prime}\\!.4$ | 46.1 | 6 | 122005.03

Figure 1: Left: $2^{\circ}\\!.0$ $\times$ $2^{\circ}\\!.0$ TSmap for photons
above 1 GeV. The TSmap is smoothed with a Gaussian kernel of $\sigma$ =
$0^{\circ}\\!.02$. The white sold circle shows the best-fit radius of the
uniform disk for the spatial template of SrcX. The black diamond reveals the
Calvera pulsar PSR J1412+7922. And the cyan cross marks the fitting position
of SrcX as a point source. Two green dashed circles describe the ring-
morphology of the diffuse radio emission (Calvera’s SNR) by LOFAR at 144 MHz
(Arias et al., 2022). Right: SED of SrcX. The black arrows indicate the 95%
upper limits and the gray histogram shows the TS value for each energy bin.
The green solid and dashed lines show the best-fit power-law spectrum and its
1$\sigma$ statistic error in the energy range of 100 MeV - 1 TeV. The red sold
line is the best-fit log-parabola spectrum.

###  2.3 Spectral Analysis

We performed a spectral analysis of SrcX in the energy range from 100 MeV to 1
TeV with the spatial template of the uniform disk. With a single power-law
(PL; $dN/dE\propto E^{-\alpha}$) spectrum, the TS value of SrcX is calculated
to be $\rm TS_{PL}=$ 44.4. And the spectral index and integrated photon flux
of SrcX are fitted to be 1.67 $\pm$ 0.10 and $(6.16\pm 2.10)\times 10^{-10}$
photon cm-2 s-1, respectively. We then test for the spectral curvature in the
spectrum of SrcX using a log-parabola (LogPb; $dN/dE\propto
E^{-(\alpha+\beta{\rm log}(E/E_{b}))}$) spectrum. And the TS value of SrcX is
fitted to be $\rm TS_{LogPb}=$ 55.8. The variation of TS values with the LogPb
model is $\rm\Delta TS$ = $\rm TS_{LogPb}$ \- $\rm TS_{PL}$ = 11.4, which
suggests an evidence of spectral curvature for the $\gamma$-ray emission of
SrcX ($\sim$3.4$\sigma$ improvement with respect to a PL model). The fitting
of LogPb model presents the spectral parameters of $\alpha=2.05\pm 0.27$ and
$\beta=0.31\pm 0.14$. And the integrated photon flux is calculated to be
$(1.45\pm 0.52)\times 10^{-10}$ photon cm-2 s-1, respectively.

Furthermore, to obtain the spectral energy distribution (SED) of SrcX, the
data from 100 MeV to 1 TeV are divided into 6 logarithmic equal energy bins,
and the same likelihood fitting analysis was performed for each energy bin.
For the energy bin with TS value of SrcX smaller than 5.0, an upper limit with
95% confidence level is calculated. The results of the SED are shown in the
right panel of Figure 1, and the global fitting with the PL and LogPb models
are also overplotted.

##  3 Discussion

The above data analysis shows that the extension of the $\gamma$-ray emission
from SrcX is well consistent with the radio size of the Calvera’s SNR, as
shown in the left panel of Figure 1, which supports SrcX as the GeV
counterpart of Calvera’s SNR. Figure 2 shows the $\gamma$-ray spectra of
several Fermi-LAT observed SNRs, together with Calvera’s SNR. The old-aged
SNRs interacting with molecular clouds, like IC 443 and W44 (Ackermann et al.,
2013), show the spectral break at $\sim$ GeV, whose $\gamma$-ray emission are
suggested to be produced by the decay of neutral pions due to the inelastic pp
collisions (hadronic model). Another class of SNRs with hard GeV $\gamma$-ray
spectra, including RX J1713.7-3946 (H. E. S. S. Collaboration et al., 2018a)
and RX J0852.0-4622 (H. E. S. S. Collaboration et al., 2018b), shows the
spectral curvature at $\sim$ TeV. These SNRs are typically young-aged systems
with strong non-thermal X-ray emission, and the $\gamma$-ray emission from
them are suggested to be from inverse Compton scattering (ICS) of accelerated
electrons (leptonic model). The GeV $\gamma$-ray spectrum of Calvera’s SNR
shown in Figure 2 is similar to that of the young-aged SNRs. However, the
absence of the non-thermal X-ray emission makes it different from them.
Moreover, the spectral analysis in Section 2.3 shows an evident spectral
curvature at tens of GeV for the $\gamma$-ray spectrum of Calvera’s SNR, which
is much lower than that of these young-aged SNRs with the spectral curvature
at $\sim$ TeV. Therefore, Calvera’s SNR may be one class of SNRs bridging
young-aged SNRs with bright non-thermal X-ray emission and old-aged SNRs
interacting with molecular clouds.

Figure 2: Typical $\gamma$-ray SEDs of several prominent SNRs. The energy
fluxes of Calvera’s SNR are scaled upward by 40 times for comparison.

Figure 3: Modeling of the multi-wavelength SED of Calvera’s SNR. The left
panel is for the leptonic model, where the power law spectra with indices of
2.0 and 2.4 for electrons are presented by the black solid and gray dashed
lines, respectively. The radio emission is dominated by the synchrotron
component, shown as the green dashed curve. For $\gamma$-ray emission, the
contribution of the ICS emission includes three components from CMB (blue
dashed), infrared (magenta dashed), and optical (green dashed) radiation
fields. The right panel is for the hadronic model, where the PL and BPL models
are presented by the black solid and gray dashed lines, respectively. The
contribution from pion-decay process with PL model is shown as the red dashed
line. The flux of bremsstrahlung component is too low to show, and the black
solid line represents the sum of the different radiation components. The gray
dot-dashed line shows the differential sensitivity of CTA-North (50 hr;
Cherenkov Telescope Array Consortium et al., 2019).

To explore the origin of the $\gamma$-ray emission from Calvera’s SNR, both
hadronic and leptonic models are considered here. And for the leptonic model,
the $\gamma$-ray emission from the bremsstrahlung process of relativistic
electrons is also calculated, together with the ICS component. The radio and
non-thermal X-ray emission are produced by high-energy electrons via the
synchrotron process. And for ICS process, three components of the interstellar
radiation field are taken into account including the cosmic microwave
background (CMB), the infrared and optical blackbody components. The
temperature and energy density of the infrared and optical blackbody
components are (T1 = 30 K, u1 = 0.1 eV cm-3) and (T2 = 5000 K, u2 = 0.1 eV
cm-3), respectively, considering the large heights above the Galactic plane
for SrcX (Porter et al., 2006, 2008). The components of the different
radiation mechanisms are computed using the naima package (Zabalza, 2015).

The distance of Calvera’s SNR is adopted to be 3.3 kpc, which is estimated by
assuming a neutron star radius of 13 km, and fitting the thermal emission from
Calvera pulsar (Mereghetti et al., 2021). With this distance, the physical
radius of SrcX is estimated to be about 27.5 pc. For the SNRs in the Galactic
halo, the ambient densities are typical very low. Using the model of Miller &
Bregman (2013) with the distance between Calvera and the center of the Galaxy
of r = 9.5 kpc, Arias et al. (2022) estimated the ambient density in the halo
around Calvera to be ngas = $4\times 10^{-4}$ cm-3. Here, we adopt this value
to calculate the hadronic and bremsstrahlung components.

The radio data of Calvera’s SNR are from Arias et al. (2022), including the
observations by LoTSS at 144 MHz, WENSS at 325 MHz and NVSS at 1.4 GHz.
Considering the absence of the non-thermal X-ray emission from Calvera’s SNR,
we adopt the thermal X-ray emission from Calvera pulsar with $\rm 1.20\times
10^{-12}$ erg cm-2 s-1 in the energy range of 0.2 - 10 keV as an upper limit
to constrain the model parameters (Mereghetti et al., 2021).

For both leptonic and hadronic models, the spectra of electrons and protons
are first assumed to be a single power-law with an exponential cutoff (PL) in
the form of

$\frac{dN_{\rm
i}}{dE}\propto\left(\frac{E}{E_{0}}\right)^{-\alpha_{i}}exp\left(-\frac{E}{E_{\rm
i,cut}}\right)$ (1)

where $\alpha_{\rm i}$ and $\rm E_{i,cut}$ are the spectral index and the
cutoff energy of particles, respectively, for i = e or p. The radio spectral
index of 0.71 $\pm$ 0.09 would suggest the spectral index of electrons to be
$\sim$ 2.4 for the synchrotron process. And it also corresponds to a
$\gamma$-ray spectral index of 1.7 for ICS component, and 2.4 for hadronic
component assuming the same index for electrons and protons. For the leptonic
model, the spectral index of electrons with 2.4 is first adopted to fit the
multi-wavelength data with other parameters listed in Table 2. And the fitting
SED is shown as the gray dashed line in the left panel of Figure 3. The
fitting result suggests a harder distribution for electrons considering the
$\gamma$-ray spectrum, especially the upper limit in the low energy range.
Then we changed the spectral index of electrons to be 2.0 considering the
statistic uncertainties of the radio data. And the magnetic field strength of
6 $\mu$G is needed to explain the radio flux of Calvera’s SNR. Considering the
upper limits in the X-ray and $\gamma$-ray bands, the cutoff energy of
electrons needs to be less than $\sim$ 10 TeV. And the value should be much
lower if the spectral curvature at tens of GeV of the $\gamma$-ray spectrum is
taken into account. With the value of B = 6 $\mu$G and Ee,cut = 10 TeV, the
synchrotron cooling timescale of electrons is estimated to be about 3.4
$\times$ 104 yrs, which is much larger than the age of $\sim$ 1400 yr for
Calvera’s SNR assuming the Sedov-Taylor phase evolution and the ambient
density of $4\times 10^{-4}$ cm-3 (Arias et al., 2022). Here the age is
recalculated with the same evolution model and a radius of 27 pc instead of 54
pc used by Arias et al. (2022). Moreover, it should be noted that the age
estimation depends on the ambient density around SNR and higher density value
would result in an older age for Calvera’s SNR. The total energy of electrons
above 1 GeV is calculated to be We = 1.8 $\times$ 1047 erg. And such value is
similar to that of other young-aged SNRs, which are of the order of 1047 \-
1048 erg (Yang et al., 2014).

Considering the evident curvature of the $\gamma$-ray spectrum likely produced
by the synchrotron cooling of electrons, the model with electron distribution
to be a broken power law (BPL) spectrum is also taken into account, with the
form of

$\frac{dN_{\rm i}}{dE}\propto exp\left(-\frac{E}{E_{\rm
i,cut}}\right)\begin{cases}\left(\frac{E}{E_{0}}\right)^{-\alpha_{i1}}\qquad\qquad\qquad\qquad;E<E_{\rm
i,break}\\\ \left(\frac{E_{\rm
i,break}}{E_{0}}\right)^{\alpha_{i2}-\alpha_{i1}}\left(\frac{E}{E_{0}}\right)^{-\alpha_{i2}}\,\,\,\,;E\geq
E_{\rm i,break}\end{cases}$ (2)

However, the break energy, Ee,break, is calculated to be about 248 TeV with a
magnetic field strength of 6 $\mu$G, by assuming the synchrotron cooling
timescale equal to the age of Calvera’s SNR of 1400 yr. And this is conflict
with the upper limits in the X-ray and $\gamma$-ray bands.

The fitting multi-wavelength SED with hadronic model is compiled in the right
panel of Figure 3. In this model, the spectral index of electrons is set to be
2.4, which is consistent with the flat radio spectrum of Calvera’s SNR. The
cutoff energy of electrons Ee,cut is calculated by equalling the synchrotron
cooling timescale and the age of Calvera’s SNR to reduce the number of free
parameters, which is given by Ee,cut = 1.25 $\times$ 107 t${}_{\rm
age;yr}^{-1}$ B${}_{\rm\mu G}^{\rm-2}$ TeV. The radio flux and the upper limit
in the X-ray band constrain the total energy of electrons above 1 GeV and the
magnetic field strength to be We = 1.6 $\times$ 1046 erg and B $\simeq$ 15
$\mu$G, respectively. And the corresponding cutoff energy of electrons is
calculated to be 39.7 TeV. To explain the hard GeV $\gamma$-ray spectrum of
Calvera’s SNR, the spectral index of protons is fitted to be about 1.6. The
upper limits in hundreds of GeV constrains the cutoff energy of protons to be
less than $\sim$ 20 TeV. Due to the very low ambient density in the
$\gamma$-ray region, the total energy of protons above 1 GeV, Wp, is estimated
to be about 4.6 $\times$ 1052 (ngas/4 $\times$ 10-4 cm-3)-1 erg, which is more
than one order of magnitude higher than the explosion energy of a typical
supernova ($\sim$ $10^{51}$ erg). It should be noted that the uncertainty of
the distance measured with the thermal emission comes from the whole neutron
star surface would result in the variation of the Galactic height for
Calvera’s SNR (Mereghetti et al., 2021). A lower Galactic height with higher
ambient density could be expected. And the much higher total energy of protons
in the hadronic model may attribute to the underestimate of the ambient
density around Calvera’s SNR.

Similar to the leptonic model, a broken power law spectrum for protons is also
tested considering the expectation of a break may produced by the escape of
protons at an advanced age of SNR (Ohira et al., 2010). The fitting SED is
shown as the gray dashed line in the right panel of Figure 3, with the
parameters listed in Table 2. The spectral indices below and above the break
are adopted to be $\alpha_{\rm p1}$ = 1.5 and $\alpha_{\rm p2}$ = 2.0, with
the break energy is fitted to be about 0.5 TeV. The cutoff energy of protons
with 1 PeV can not be well constrained, and the future possible detection in
the TeV band by Cherenkov Telescope Array in the northern hemisphere (CTA-
North; Cherenkov Telescope Array Consortium et al., 2019) would be helpful.

Table 2: Parameters for the models Model | $\alpha_{e}$ | $\alpha_{p}$ | Ep,break | Ee,cut | Ep,cut | We | Wp | $B$
---|---|---|---|---|---|---|---|---
|  |  | (TeV) | (TeV) | (TeV) | ($10^{47}$ erg) | ($10^{52}$ erg) | ($\mu$G)
Leptonic | $2.4$ | $-$ | $-$ | $100$ | $-$ | $6.3$ | $-$ | $1.8$
(PL) | $2.0$ | $-$ | $-$ | $10$ | $-$ | $1.8$ | $-$ | $6.0$
Hadronic(PL) | $2.4$ | $1.6$ | $-$ | $39.7$ | $20$ | $0.16$ | $4.6$ | $15.0$
Hadronic(BPL) | $2.4$ | $1.5/2.0$ | $0.5$ | $39.7$ | $1000$ | $0.16$ | $6.4$ | $15.0$

Except the SNR scenario for the $\gamma$-ray emission from SrcX, we also
consider the possibility that the emission comes from a pulsar wind nebula
(PWN) powered by Calvera pulsar, although there is no diffuse radio or X-ray
emission detected around Calvera pulsar. The $\gamma$-ray PWNe detected by
Fermi-LAT are typically driven by the energetic pulsars with spin-down power
between 1036 and 1039 erg s-1 (Acero et al., 2013). However, the spin-down
luminosity of Calvera pulsar is $\dot{E}=6.1\times 10^{35}$ erg s-1, which is
slightly lower than that. In addition, the $\gamma$-ray luminosity of SrcX in
the energy range of 10 - 316 GeV is calculated to be 1.75 $\times$ 1033 erg
s-1, which is much lower than that of the typical $\gamma$-ray PWNe with above
1034 erg s-1 (Acero et al., 2013). Nonetheless, the hypothetical PWN may also
have partial contribution to the $\gamma$-ray emission from SrcX, considering
the $\gamma$-ray morphology with spot emission around Calvera pulsar, as shown
in the left panel of Figure 1. And the future observations searching for the
possible PWN in the radio/X-ray band may help to clear the $\gamma$-ray origin
from SrcX.

##  4 Conclusions

In this work, we report the detection of the $\gamma$-ray emission in the
direction of Calvera’s SNR with more than 13 yr of Pass 8 data recorded by the
Fermi-LAT. The $\gamma$-ray spatial morphology of Calvera’s SNR is significant
extended ($\sim$6.4 $\sigma$). And it can be well described by an uniform disk
with a radius of $0^{\circ}\\!.478$, which is well consistent with the radio
size of the remnant. The $\gamma$-ray spectrum of Calvera’s SNR follows a hard
power-law form with an index of 1.67 $\pm$ 0.10 in the energy range of 100 MeV
- 1 TeV. The higher TS value with a log-parabola form suggests an evidence
spectral curvature for the $\gamma$-ray emission from Calvera’s SNR. The hard
$\gamma$-ray spectrum of Calvera’s SNR makes it similar to the typical young-
aged SNRs, whose $\gamma$-ray emission comes from the ICS process. However,
there is no non-thermal X-ray emission detected around Calvera’s SNR, which is
different from these SNRs. Considering the spectral curvature at tens of GeV,
Calvera’s SNR may be one class of SNRs bridging young-aged SNRs with bright
non-thermal X-ray emission and old-aged SNRs interacting with molecular
clouds. Both leptonic and hadronic models could marginally explain the multi-
wavelength data of Calvera’s SNR. However, the flat radio spectrum seems to be
not well consistent with the hard GeV $\gamma$-ray spectrum in the leptonic
model. And for the hadronic model, the spectral index of protons should be
harder than 1.6, which is difficult to produce in the conventional diffusive
shock acceleration model of strong shocks. Moreover, Calvera’s SNR is located
in the high Galactic latitude, which makes the ambient density around it very
low. And the total energy of protons above 1 GeV is calculated to be about Wp
= 4.6 $\times$ 1052 (ngas/4 $\times$ 10-4 cm-3)-1 erg, which is more than one
order of magnitude higher than the explosion energy of a typical supernova
($\sim$ $10^{51}$ erg). The ratio of the normalization of electron
distribution to that of protons, $K_{\rm ep}$, is much lower than the locally
measured cosmic ray electron to proton flux ratio ($\sim$ 0.01).

Calvera’s SNR provides a good target to study the SNR evolution and particle
acceleration in the high Galactic latitude, and probe the interstellar medium
of the Milky Way halo. More multi-wavelength observations in future,
especially the possible non-thermal X-ray emission and very-high-energy
$\gamma$-ray emission by CTA-North, are crucial to ultimately understanding
its nature.

This work is supported by the Natural Science Foundation for Young Scholars of
Sichuan Province, China (No. 2022NSFSC1808), the Science and Technology
Department of Sichuan Province (No. 2021YFSY0031, No.2020YFSY0016), the
Fundamental Research Funds for the Central Universities (No. 2682021CX073, No.
2682021CX074, No. 2682022ZTPY013), and the National Natural Science Foundation
of China under the grants 12103040 and 12147208.

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