ON THE HARD Γ-RAY SPECTRUM OF THE POTENTIAL PEVATRON
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On the hard γ-ray spectrum of the potential PeVatron
supernova remnant G106.3+2.7
Yiwei Bao1 and Yang Chen1, 2
arXiv:2103.01814v3 [astro-ph.HE] 19 Jul 2021
1
Department of Astronomy, Nanjing University, 163 Xianlin Avenue, Nanjing 210023, China
2
Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing, China
(Received; Revised; Accepted)
Submitted to ApJ
ABSTRACT
The Tibet ASγ experiment has measured γ-ray flux of supernova remnant G106.3+2.7
up to 100 TeV, suggesting it being potentially a “PeVatron”. Challenge arises when
the hadronic scenario requires a hard proton spectrum (with spectral index ≈ 1.8),
while usual observations and numerical simulations prefer a soft proton spectrum (with
spectral index ≥ 2). In this paper, we explore an alternative scenario to explain the γ-
ray spectrum of G106.3+2.7 within the current understanding of acceleration and escape
processes. We consider that the cosmic ray particles are scattered by the turbulence
driven via Bell instability. The resulting hadronic γ-ray spectrum is novel, dominating
the contribution to the emission above 10 TeV, and can explain the bizarre broadband
spectrum of G106.3+2.7 in combination with leptonic emission from the remnant.
Keywords: ISM: supernova remnants — ISM: individual objects (G106.3+2.7) — dif-
fusion — (ISM:) cosmic rays
1. INTRODUCTION
Galactic cosmic rays (CRs) are mostly charged particles (mainly protons) with energy up to the
so called “knee” (∼ 1–10 PeV). Based on energetic arguments, supernova remnants (SNRs) are usu-
ally believed to be the accelerators of Galactic CRs. However, although a large number of SNRs
have been detected in γ-rays, none of them has been confirmed to be a PeV particle accelerator,
as called “PeVatron” (see e.g., Bell et al. 2013). Discovered in the DRAO Galactic-plane survey
(Joncas & Higgs 1990), G106.3+2.7 is a cometary SNR with a tail in the southwest and a com-
pact head (containing PSR J2229+6114) in the northeast, at a distance of 800 pc away from Earth
(Kothes et al. 2001). Recently, HAWC and the Tibet ASγ experiments have reported the γ-ray spec-
trum of SNR G106.3+2.7 above 40 TeV, arguing that the remnant is a promising PeVatron candidate
(Albert et al. 2020; Tibet ASγ Collaboration 2021). In particular, Tibet ASγ Collaboration (2021)
for the first time measured the γ-ray flux up to 100 TeV, finding that the centroid of γ-ray emissions
Corresponding author: Yang Chen
ygchen@nju.edu.cn2 Bao & Chen
deviates from the pulsar at a confidence of 3.1σ, and is well correlated with a molecular cloud (MC).
The offset of the γ-ray emission centroid from PSR J2229+6114 is measured to be 0.44◦ (∼ 6 pc at
a distance of 800 pc).
Both leptonic and hadronic models have been proposed to give a plausible explanation to the spec-
tral energy distribution (SED) of the SNR. In the leptonic scenario, the electrons are suggested to
be transported to its current position from the pulsar wind nebula (PWN, Tibet ASγ Collaboration
2021; Liu et al. 2020), or be accelerated by the blast wave directly (Tibet ASγ Collaboration
2021; Ge et al. 2021); in the hadronic scenario, the protons are suggested to be accelerated by
the blast wave in earlier ages, or re-accelerated by the PWN adiabatically (Ohira et al. 2018;
Tibet ASγ Collaboration 2021). The large offset between PSR J2229+6114 and the γ-ray emis-
sion centroid indicates that the γ-rays are more likely to be mainly contributed from the particles
accelerated by the blast wave in the southwestern “tail” region (Tibet ASγ Collaboration 2021). In
order to figure out the origin of the γ-rays, Ge et al. (2021) separate the PWN-dominated X-ray
emitting region in the northeast from the other (the “tail”) part of the SNR using XMM-Newton and
Suzaku observations. They found that a pure leptonic model can hardly fit the radio, X-ray and γ-ray
spectral data of the “tail” region simultaneously. Therefore, there must be a hadronic component
in the γ-ray spectrum, and the SED can only be explained with a hadronic (with a proton spectral
index ≈ 1.8, Tibet ASγ Collaboration 2021) or a leptonic-hadronic hybrid (with a proton index
≈ 1.5, Ge et al. 2021) model. However, challenge arises when hard proton spectrum is required in
both hadronic and leptonic-hadronic hybrid models, while numerical simulations show that diffusive
shock acceleration can only give rise to a soft proton spectrum (with spectral index ≥ 2, see e.g.,
Caprioli et al. 2020).
Tibet ASγ Collaboration (2021) suggest that the very hard proton spectrum can be formed in very
efficient acceleration (which seems to be an extreme case) and after a very slow particle diffusion. We
here explore an alternative self-consistent scenario in which the γ-ray spectrum can be explained more
naturally. The magnetic field in the upstream of the blast wave is believed to be amplified via the
non-resonant Bell instability (Bell 2004; Amato & Blasi 2009): protons escaping from the upstream
of the shock can drive non-resonant turbulence whose scale is smaller than the Lamour radius of
the escaped particles, and the relatively-low-energy particles are thus diffused by the magnetic field
amplified by the non-resonant turbulence (for reviews, see e.g., Blasi 2013). Based on the numerical
simulations (Bell 2004; Bell et al. 2013), we calculate the escape process from the first principle
instead of a model with apriori phenomenological assumptions. The resulting proton spectrum is
novel and can explain the hard γ-ray spectrum well in combination with leptonic emission from the
SNR. The model is described in §2 and the SED of the remnant is fitted in §3, discussion is presented
in §4 and the conclusion is drawn in §5.
2. MODEL DESCRIPTION
For simplicity, we follow the approximation that the global maximum energy Emax,global is reached
at the beginning of the Sedov phase in which Rsh ∝ t2/5 (see e.g., Ohira et al. 2012). The maximum
proton energy can be given by (Bell et al. 2013)
η vsh 2 R
sh
Eesc (t) = 230 n1/2
e 4
TeV, (1)
0.03 10 kms −1 pcOn the hard γ-ray spectrum of the supernova remnant G106.3+2.7 3
where η is the acceleration efficiency, ne the electron number density of the interstellar medium (ISM,
we assume that the ISM near the SNR are fully ionized), vsh the velocity of the shock, and Rsh the
radius of the shock.
Following Cardillo et al. (2015), we calculate the spectrum of escaped CR protons in the Sedov
phase. In the upstream, the protons are scattered by the wave generated via escaping of the higher-
energy protons, and therefore there is no wave upstream to scatter the protons with the highest
energy Eesc (t) (Bell 2004; Bell et al. 2013). Hence, the latter escape the system quasi-ballistically at
a speed ∼ c, inducing a current jCR = nCR evsh at the shock (Bell 2004, where nCR represents the
number density of the CRs). The differential number of the escaped protons with energy Eesc (t) can
thus be evaluated via
dNesc (Eesc ) 2
e dEesc = 4πRsh jCR dt. (2)
dEesc
We further assume that the CR pressure at the shock is a fixed fraction ξ of the ram pressure
3
eξρvsh
jCR = nCR evsh = , (3)
E0 Ψ(Eesc )
where
(Eesc /E0 )ln(Eesc /E0 ) α=2
Ψ= α−1 (4)
α−1 Eesc α−2
[1 − (E0 /Eesc )] α > 2,
α−2 E0
α is the power-law index of the parent CR spectrum at the shock, and E0 the minimum energy of
protons.
Finally, Nesc can be expressed as (see Cardillo et al. 2015, for derivation)
h i
dNesc (Eesc ) 2
4πRsh jCR dt 2
4πξρvsh 2
Rsh dR dΨ ρv 2 R3 E −2 1+ln(Eesc /E0 ) α=2
sh sh esc ln(Eesc /E0 )2
= = ∝ (5)
dEesc e dEesc E0 Ψ dΨ dEesc ρv 2 R3 (E /E )−α α > 2.
sh sh esc 0
3. APPLICATION TO SNR G106.3+2.7
3.1. γ-ray spectroscopic luminosity
Since kinematic/physical signature of direct MC-SNR contact seems inconclusive yet (Liu et al.
2021), the MC may only be illuminated by the escaped protons. For such a scenario, the MC is
sometimes approximated as a truncated cone (see e.g., Li & Chen 2012; Celli et al. 2019). Here the
MC is assumed to subtend a solid angle Ω at the SNR center with an inner radius R1 and an outer
radius R2 . The diffusion coefficient near the SNR is adopted to be D(E) = D100 TeV (E/100 TeV)δ ,
where D100 TeV is the diffusion coefficient for particles at 100 TeV, and δ the energy dependence index.
The diffusion equation for the protons with energy Eesc escaping the shock when the shock radius
was Rsh at time Tesc writes
∂ D(Eesc ) ∂ 2 ∂
f (Eesc , r, t) = r f (Eesc , r, t) + Q, (6)
∂t r 2 ∂r ∂r4 Bao & Chen
where
1 dNesc (Eesc )
Q= 2
δ(r − Rsh )δ(t − Tesc )
4πRsh dEesc
(7)
h i
2
CQ ρvsh 3
Rsh (Eesc /E0 )−2 1+ln(Eesc /E0 ) α=2
ln(Eesc /E0 )2
= 2
δ(r − Rsh )δ(t − Tesc ) ×
4πRsh (α − 2) (E /E )−α+2 α>2
esc 0
is the injection term, f the proton distribution function, and CQ a free parameter which is propor-
tional to the total proton energy.
As is shown in Appendix A, the solution of Equation 6 is (see also Atoyan et al. 1995; Celli et al.
2019)
( " 2 # " 2 #)
dNesc (Eesc )/dEesc r − Rsh r + Rsh
f (Eesc , r, t) = exp − − exp − , (8)
4rπ 3/2 Rsh Rdif Rdif Rdif
p
where Rdif = 2 D(Eesc )(t − Tesc ) is the diffusion length scale.
At time Tage , the differential number of protons with energy Eesc lying inside the conic MC shell
can thus be calculated to be Z R2
NCR,MC = f (E, r, t)Ωr 2 dr. (9)
R1
Finally, the hadronic γ-ray spectroscopic luminosity Φγ (Eγ ) is evaluated via the cross section pre-
sented in Kafexhiu et al. (2014)
dσ
Z
Φγ (Eγ ) = cnMC (E, Eγ )NCR,MC (E)dE, (10)
dEγ
where dσ/dEγ is the γ-ray differential cross section.
In addition to the hadronic component, we add a leptonic component contributed by the electrons
accelerated by the blast wave. We approximate the spectrum of electrons in the SNR to be a broken
power-law
dNe E −α1 E ≤ Eb
∝ (11)
dE E −α2 E < E < min (E
b max,global , Eloss ),
where dNe /dE the differential number of electrons, and Eloss the maximum energy determined by
energy loss; α1 and α2 are the power-law indices in the low energy and high energy band, respectively;
Eb , the break energy of the electron spectrum, can be constrained well (∼ 9 TeV, assuming a
magnetic field of 6 µG, Ge et al. 2021) by the radio (Pineault & Joncas 2000) and X-ray (Ge et al.
2021; Fujita et al. 2021) emissions, which are both dominated by the SNR. The electron spectrum
above the break is quite soft, and the leptonic γ-ray emissions are dominated over by the hadronic
emissions above 10 TeV. Hence, the total γ-ray spectrum is insensitive to the electron spectrum above
Eb .
We fit the SED as is plotted in Figure 1, with the parameters listed in Table 1. We find that the
parameters are insensitive to X-ray flux, since the different sets of X-ray flux data from Ge et al.
(2021) and Fujita et al. (2021) can be fitted with very similar parameters.On the hard γ-ray spectrum of the supernova remnant G106.3+2.7 5
In Figure 2, we plot the proton spectrum insides the MC. As can been seen, the protons with energy
< 75 TeV are still trapped by the turbulence driven by the escaping protons via Bell instability, absent
in the MC. Hence, only protons with energies ranging from 75 TeV to 280 TeV can reach the MC
within the relatively short lifetime of the SNR and contribute to a novel hadronic γ-ray spectrum.
In Figure 3, we explore the dependence of hadronic γ-ray spectrum on α and δ and find that the
choice of α (in the range 2.0–2.6) or δ (in the range of 0–1) does not impact the results significantly.
The reason is that the protons which can hit the MC within the relatively short lifetime of the SNR
at a narrow energy range (75–280 TeV) is almost mono-energetic (as shown in Figure 2). Hence, the
energy dependence index δ and the power-law index α can hardly affect the proton spectrum in the
MC. Celli et al. (2019) have also proposed a similar phenomenological escaping scenario which can
lead to a hard hadronic γ-ray spectrum in middle-aged SNRs, while in our case, we model a younger
SNR from the first principle.
Since the hadronic γ-ray luminosity Φγ ∝ CQ nMC Ω (where nMC is the number density of the gas
in the MC, a free parameter), we only list CQ nMC (Ω/4π) as a single parameter in Table 1. Once
CQ nMC Ω is obtained from data fitting, the total proton energy Ep,tot (including that in the GeV
protons trapped near the shock) can be calculated via
Z Emax,global −2 h 1+ln(Eesc /E0 ) i
2 3 E ln(Eesc /E0 )2
α=2
Ep,tot ≈ E dE · CQ ρvsh (Tage )Rsh (Tage ) × (12)
E0 E0 (α − 2) (E /E )−α+2 α > 2.
esc 0
If we adopt nMC = 100 cm−3 , Ep,tot will be 1.6 (4π/Ω) × 1049 erg, which is quite reasonable.
As shown in Figure 1, the leptonic component accounts for the radio (Pineault & Joncas 2000) and
X-ray (Ge et al. 2021 for Model A; Fujita et al. 2021 for Model B) emission and dominates the γ-ray
flux below 500 GeV. Meanwhile, the hadronic γ-rays (the yellow lines) have a very hard spectrum
(dNγ /dEγ ∝ Eγ−1 ) below 500 GeV, which stems from the proton cutoff at 75 TeV and the low-energy
tail of the pp interaction cross section. The spectrum above 10 TeV is dominated by the hadronic
component, and the whole γ-ray spectrum can be explained by the hybrid model naturally.
3.2. Age of SNR G106.3+2.7
There remains some uncertainty about the age of SNR G106.3+2.7: although the characteristic
age of PSR J2229+6114 (≈ 104 yr) is usually adopted to be the age of the remnant, sometimes the
remnant is suggested to be very young (∼ 1 kyr, see e.g., Albert et al. 2020). With an adiabatic
expansion (Rsh ≈ 1.2(ESN /ρISM )1/5 t2/5 , where ρISM is the density of the ISM), the estimated ESN ≈
7 × 1049 erg (Kothes et al. 2001) is unusually low if the age of remnant is ≈ 10 kyr. However, as is
estimated in Cardillo et al. (2015), Equation 1 indicates that the global maximum energy Emax,global ∝
ESN , and SNRs with a low ESN can hardly accelerate CRs to ∼ 102 TeV. In order to reconcile the
dilemmas, in this paper we consider a spherically symmetric scenario in which ESN = 1051 erg (the
canonical value) and Tage = 1000 yr. Such a remnant age is plausible for a hosted pulsar with a
characteristic age of ∼ 104 yr. Assuming a canonical braking index n:=ν ν̈/ν̇ 2 = 3 (where ν is the
spin frequency of the pulsar), the real age of the pulsar writes Tage = τc [1 − (P0 /Pnow )2 ], where τc
is the characteristic age of the pulsar, P0 the initial spin period of the pulsar, and Pnow the spin
period of the pulsar at present. For PSR J2229+6114, Pnow = 50 ms (Halpern et al. 2001), and Tage
can be as small as ∼ 1 kyr if P0 is appropriately close to Pnow . Such value of P0 is allowed, since
in pulsar evolution models P0 is usually suggested to be in a wide range from ∼ 4 ms to ∼ 400 ms6 Bao & Chen
(see e.g., Arzoumanian et al. 2002; Faucher-Giguère & Kaspi 2006; Popov et al. 2010). Instances in
which P0 ≈ Pnow and Tage ≪ τc can also be found in the catalog: (a) CCO 1E 1207.4−5209 inside
SNR G296.5+10.0 has P0 ≈ Pnow = 424 ms, leading to τc > 27 Myr, which exceeds the age of the
SNR by 3 orders of magnitude (Gotthelf & Halpern 2007), and (b) PSR J1852−0040 is suggested
to have P0 ≈ Pnow ≈ 102 ms, giving rise to τc ≈ 2 × 108 yr, 4 orders of magnitude larger than
Tage ∼ 5 × 103 yr measured from the observation of the associated SNR (Gotthelf et al. 2005). On
the other hand, as can be seen from Equation 1, Eesc is determined by the Rsh and vsh which can be
constrained by observations directly, irrespective of Tage . Ge et al. (2021) showed that vsh is at least
3000 km s−1 in the southwestern tail region based on the non-thermal X-ray spectrum up to 7 keV
without a clear spectral cutoff, while Rsh ≈ 6 pc can be estimated via radio observations (see e.g.,
Tibet ASγ Collaboration 2021). Although Tage does affect Rdif , its impacts can be compensated
by the free parameter of D100 TeV . Hence, for simplicity we only consider a spherically symmetric
remnant evolving in uniform medium following the well-known Truelove & McKee (1999) model1 .
3.3. Other issues
A sharp cutoff in the novel proton spectrum is an interesting characteristic in our particle escaping
scenario with a spherically symmetric morphology applied, which can well fit the Tibet ASγ data in
this SNR. Meanwhile, instead of a sharp cutoff, smoother break can be predicted if an asymmetric
morphology is considered. In that case, the shock radius Rsh and velocity vsh both vary in different
2
directions, and thus Eesc ∝ vsh Rsh also varies in different direction accordingly, and thus the spatial
variation of Eesc may turn the sharp cutoff into a softer break. Since the data available at present
can not distinguish the asymmetric model, here we adopt a symmetric model to explain the γ-ray
spectrum in the zeroth order. Further observation carried out by LHASSO may help to distinguish
these models. During the lifetime of SNRs, the shock may break and the CRs can thus escape
with continuous power-law spectra, which is the case as previously adopted (see e.g., Li & Chen
2010; Ohira et al. 2011). In the SNR catalog2 (Ferrand & Safi-Harb 2012), there are only a few
SNRs with γ-ray emissions in . 10 TeV which are confirmed to be of hadronic origin, and our
novel hadronic spectrum may be expected to be found in more SNRs with the help of LHASSO
(see e.g., Aharonian et al. 2021), Tibet ASγ, HAWC, CTA (see e.g., Abdalla et al. 2021) and ASTRI
Mini-Array (see e.g., Pintore et al. 2020) experiments in future.
4. SUMMARY
Although the standard theory of non-linear diffusive shock acceleration can predict hard proton
spectrum with α < 2 (Caprioli et al. 2020), it is noted that observations and recent numerical simu-
lations seem to prefer a soft hard proton spectrum (Caprioli et al. 2020). In this paper, we explore
an alternative plausible scenario to explain the bizarrely hard γ-ray spectrum of SNR G106.3+2.7
within the current acceleration theory which predicts soft (α ≥ 2) proton spectra. Apart from the
common leptonic component which can be constrained by radio and X-ray observations, we invoke a
novel hadronic component. This component arises from the escape scenario proposed by Bell (2004)
and Cardillo et al. (2015), in which the CRs at the shock are diffused by the turbulence that is gen-
erated by the escaping CR particles with higher energies. Consequently, at a given time, only CRs
with the highest energy can escape upstream, while CRs with lower energies are confined. Therefore,
1
See Leahy & Williams (2017) for a fast python calculator on SNR evolution.
2
http://snrcat.physics.umanitoba.ca/SNRtable.phpOn the hard γ-ray spectrum of the supernova remnant G106.3+2.7 7
only protons with energies between 75–280 TeV can reach the MCs at an age 1 kyr. The cutoff of
the proton spectrum at 75 TeV then gives rise to a very hard hadronic γ-ray spectrum below 10 TeV
because of the low-energy tail of the pp cross section. The hadronic γ-ray spectrum, which dominates
above 10 TeV, together with the leptonic component, can explain the bizarrely hard γ-ray spectrum
of SNR G106.3+2.7 well.
In our model treatment, the γ-ray spectrum is calculated in zeroth order, assuming that a spherically
symmetric SNR expands in a uniform medium. There are seven free parameters in the hadronic
component, namely, nISM , D100TeV , R1 , R2 , Mej , η, and nMC CQ (Ω/4π). Neither of parameters α
and δ has strong correlation to the hadronic γ-ray spectrum, because the protons which can hit the
MC within the relatively short lifetime of the SNR in the narrow energy range (75–280 TeV) are
approximately mono-energetic.
APPENDIX
A. ANALYTICAL SOLUTION OF DIFFUSION EQUATION
In order to solve Equation 6 analytically, we define a new function F := rf (Eesc , r, t), and then
Equation 6 reads
∂ ∂2F
F = D(Eesc ) 2 + Q(Eesc , r, t)r, (A1)
∂t ∂r
and the boundary condition is ∂f /∂r = 0 and F |r=0 = 0. Defining the differential operator
∂ ∂2
L := − D 2,
∂t ∂r
Equation A1 can be further written as LF = Qr. We firstly discuss a simpler equation LG(r, ζ) =
δ(r − ζ) with the boundary condition G|r=0 = 0. The solution is apparently
( " 2 # " 2 #)
1 r−ζ r+ζ
G(r, ζ) = √ exp − − exp − . (A2)
πRdif Rdif Rdif
Then
R we multiply Equation A2 by h(ζ) = Q(ζ)ζ on both sides, integrate over ζ, and exchange L and
, i.e.,
Z ∞ Z ∞ Z ∞
L G(r, ζ)Q(ζ)ζ dζ = LG(r, ζ)Q(ζ)ζ dζ = δ(r − ζ)Q(ζ)ζ dζ = Qr = LF. (A3)
0 0 0
R∞
Hence, 0
G(r, ζ)Q(ζ)ζ dζ = F , and we thus have
1 ∞
Z
f= Q(ζ)ζG(Eesc, t, Tesc ; r, ζ)dζ
r 0
( " 2 # " 2 #) (A4)
dNesc (Eesc )/dEesc r − Rsh r + Rsh
= exp − − exp −
4rπ 3/2 Rsh Rdif Rdif Rdif8 Bao & Chen
10210
pp (Model A)
E2dN/dE [erg cm22−21]
pp (Model B)
10211 sync (Model A)
sync (Model B)
IC (Model A)
10212 IC (Model B)
total (Model A)
total (Model B)
10213 HAWC
XMM
Suzaku
10214 CGPS
MILAGRO
Fermi
10215 −6 VERITAS
10 10−2 102 106 1010 1014 ASg
Energ0 [eV]
Figure 1. SED of the emission from SNR G106.3+2.7. The CGPS data are taken from Pineault & Joncas
(2000), XMM-Newton data from Ge et al. (2021, for Model A), Suzaku data from Fujita et al. (2021, for
Model B; see also Table 1), Fermi-LAT data from Xin et al. (2019), VERITAS data from Albert et al.
(2020), HAWC data from Albert et al. (2020), and Tibet ASγ data from Tibet ASγ Collaboration (2021).
The hadronic γ-ray spectrum below 500 GeV is contributed by 75–280 TeV protons via a small cross section.
Proton S ectrum in the MC
10−9
E2dN/dE [Arbitrary Unit]
10−10
10−11 4
10 105 106
Energy [GeV]
Figure 2. Proton spectrum in MC.On the hard γ-ray spectrum of the supernova remnant G106.3+2.7 9
10−10 10−10
pp (α = 2.0) pp (δ = 0.0)
E2 N/ E [erg2cm−2s−1]
E2dN/dE [erg m−2s−1]
pp (α = 2.1) pp (δ = 0.2)
10−11 pp (α = 2.2) 10−11 pp (δ = 0.3)
pp (α = 2.3) pp (δ = 0.5)
pp (α = 2.4) pp (δ = 0.7)
10−12 pp (α = 2.5)
10−12 pp (δ = 0.8)
pp (α = 2.6) pp (δ = 1.0)
10−13 HAWC 10−13 HAWC
MILAGRO MILAGRO
Fermi Fermi
10−14 VERITAS 10−14 VERITAS
ASg ASg
10−15 10−15
1010 1012 1014 1010 1012 1014
Energy [eV] Energy [eV]
Figure 3. Dependence of the hadronic γ-ray spectrum on the α (left panel, with δ = 1/3) and δ (right
panel, with α = 2.3).
We are in debt to Pasquale Blasi for fruitful discussion, and to an anonymous referee, Yutaka Fujita,
Ruo-Yu Liu, Qiang Yuan and Xiangdong Li for helpful comments. This work is supported by the
National Key R&D Program of China under grants 2017YFA0402600, NSFC under grants 11773014,
11633007, 11851305 and U1931204.
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Table 1. Fitting Parameters
Parameter Model A Model B
Tage (yr) 1000 ...
Rnow (pc) 5.5 ...
Emax,global (TeV) 2.8 × 102 ...
E0 (GeV) 1.0 ...
α 2.3 ...
D100 TeV (cm2 s−1 ) 6.6 × 1027
δ 1/3 ...
R1 (pc) 7 ...
R2 (pc) 10 ...
ESN (erg) 1.0 × 1051 ...
d (pc) 800 ...
Mej (M⊙ ) 1.0 ...
ne (cm−3 ) 0.6 ...
η 0.04 ...
nMC CQ (Ω/4π) (erg−2 cm−3 ) 320 ...
Eb (TeV) 12 10
α1 2.3 ...
α2 3.8 3.45
B (µG) 4.6 ...
We (erg) 6.8 × 1047 ...
TCMB (K) 2.73 ...
uCMB (eV cm−3 ) 0.25 ...
TFIR (K) 25 ...
uFIR (eV cm−3 ) 0.2 ...
TOPT (K) 3000 ...
uOPT (eV cm−3 ) 0.3 ...
∗ d is the distance to the SNR, Rnow the radius of the
SNR at present, ESN the explosion energy of the SNR,
Mej the ejecta mass, B the magnetic field strength,
and We the total electron energy. TCMB , TFIR , and
TNIR are the temperatures of the CMB, far infrared,
and near infrared photons, respectively; uCMB , uFIR ,
and uNIR are energy densities of the CMB, far infrared,
and near infrared photons, respectively.On the hard γ-ray spectrum of the supernova remnant G106.3+2.7 11
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