Detailed Analysis of the TeV γ-Ray Sources 3HWC J1928+178, 3HWC J1930+188, and the New Source HAWC J1932+192

Figure 3(a) shows the HAWC significance map of the region for a point-like source hypothesis and assuming a power-law spectrum with an index of −2.5, which are the standard parameters used to produce HAWC maps (Albert et al. 2020). Superimposed in blue and green are the initial and fitted positions of the two components previously described in Section 4. The width of the green circle represents the 1σ uncertainty on the size of the Gaussian. The fitted model is displayed in Figure 3(b) and the residual map and its distribution in Figure 3(c). The orange line is a fit to the distribution with a Gaussian function. After this first iteration, excesses of 4σ and 6σ significance are found in the residual map near the pulsar PSR J1932+1916 and at the location of J1928, respectively. To account for this, two components are added to the model at the locations of the excesses:

• 1.  
  A point-like source is initialized at (R. A. = 29307, decl. = 1940), near PSR J1932+1916, with a simple power law as a spectral model. This component will be simply called J1932.
• 2.  
  An extended source is initialized at (R. A. = 29208, decl. = 1779), with an initial size σ = 01, with a simple power law as a spectral model. This is the new component for J1928.

The previous extended component from the initial model will now be called J1928-EXT, and it is given as the initial position and size the output from the first fit. The position, size, index, and flux normalization are again set free. A fit is performed again with the four components. The outputs of the second fit are summarized in Table 1. The corresponding maps are displayed in Figure 4, using the same color code as in Figure 3. The spectra for the four components are shown in Figure 5. The lower edges of the spectra are fixed to 1 TeV, as the median energy of analysis bin 4 minus an error of 1σ. To determine the upper edge, individual fits are performed for each of the four components, using a power law with an exponential cutoff for that component only, with the amplitude and cutoff energy as the only free parameters, and the three other components being modeled by a simple power law with all parameters fixed. Then, the cutoff energy is fixed as well, so that only the flux normalization remains as a free parameter, and the cutoff energy is set to decreasing values until ΔTS = 2. In Table 1, all of the fitted parameters are given with statistical and systematic uncertainties. To calculate the systematic uncertainties, the same fit was performed again using different instrument response files.

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Table 1. Input Values and Fitted Values (Subscripts i and f, Respectively) for Each Component of the Best Model Representing the Region of Interest

	J1930	J1932	J1928	J1928-EXT
Hypothesis	Point-like	Point-like	Extended	Extended
posi	PSR J1930+1852	(293.07, 19.40)	(292.08,17.79)	(292.20,18.18)
posf (ra °)	292.53 ± 0.05 ± 0.004	292.99 ± 0.05 ± 0.002	292.15 ± 0.04 ± 0.001	292.05 ± 0.15 ± 0.05
(dec °)	18.84 ± 0.05 ± 0.001	19.36 ± 0.04 ± 0.001	17.90 ± 0.04 ± 0.001	18.10 ± 0.17 ± 0.05
sizei (°)	⋯	⋯	0.10	0.9
sizef (°)	⋯	⋯	0.18 ± 0.04 ± 0.003	1.43 ± 0.17 ± 0.05
indexi	−2.5	−2.5	−2.5	−2.5
indexf	−2.93 ± 0.20 ± 0.01	−2.46 ± 0.24 ± 0.01	−2.09 ± 0.16 ± 0.04	−2.60 ± 0.08 ± 0.01
fluxi	10.0	10.0	10.0	10.0
fluxf	2.46 ${}_{-0.47}^{+0.58}$ ± 0.72	1.95 ${}_{-0.49}^{+0.62}$ ± 0.50	4.23 ${}_{-1.10}^{+1.49}$ ± 1.30	40.34 ${}_{-4.11}^{+4.47}$ ± 1.93
Energy range (TeV)	1–118	1–43	1–178	1–10

Note. Each value is followed by the statistical uncertainty and the systematic uncertainty. The fit is performed in two steps. The initial model has two components, representing J1928 and J1930. A point-like component and an extended Gaussian component are added at the location of significant excess in the residual map. The flux normalization is given at 10 TeV in units of 10⁻¹⁵ TeV⁻¹ cm⁻² s⁻¹. The spectral energy distributions are plotted in Figure 5. The position of the pulsar PSR J1930+1852 is (29263, 1887).

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The component representing J1928 is found to have a size of σ = 018 ± 004_stat (39% containment), while the other extended source, J1928-EXT, has a size of σ = 143 ± 017_stat. The difference in TS between this model and the initial one is 45. Given the high number of degrees of freedom between this model and the initial model, we can use the Akaike Information Criterion (AIC; Akaike 1974), given by AIC = $2k-2{\rm{ln}}{ \mathcal L }$, with k being the number of free parameters and ${ \mathcal L }$ being the maximum value of the likelihood function. This penalizes the model with the largest number of free parameters, so that the model with the fewest parameters will be favored, unless the extra parameters actually provide a substantially better fit. The best model is the one that have a lower AIC value. In this case, the four-component model is clearly preferred to the initial model, with ΔAIC = 74. An excess of ∼3σ significance remains at the top of the region of interest, visible on map (c) of Figure 4. Adding a new component at its location improves the fit only by a ΔTS of 10, which is not significant when considering the addition of another source with 4 degrees of freedom. The ΔAIC is 12. Since there are no compelling counterparts to this excess at other wavelengths, the remaining excess may be the result of additional complexities that are not contained in our model, including spatial morphology asymmetries and more complex spectral shapes, or it may simply be due to fluctuations. However, even though it gives a similar likelihood value, using an asymmetric Gaussian shows a clear 5σ signal in the residual map at the location of J1928. Moreover, neither a power law with exponential cutoff nor a log-parabola significantly improve the fit.

The spectrum of 3HWC J1930+188 resulting from the fit of the four-component model described in the previous section is shown in green in Figure 6. The spectrum is slightly softer than the one previously published by the HAWC collaboration, using the same amount of data, analysis bins 1 to 9, and a single point-like source hypothesis, shown in gray (Albert et al. 2020), while in the present analysis it is part of a more complex model. The high number of free parameters being fitted together is responsible for the larger uncertainties. At a few TeV, the spectrum from the fit presented here is in better agreement with the VERITAS spectrum (Abeysekara et al. 2018), represented by the black dots, although the error bars are wider. The H.E.S.S. spectrum (H. E. S. S. Collaboration et al. 2018) is also shown in magenta. The spectral parameters derived in the different works cited here are gathered in Table 2.

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Table 2. Spectral Parameters and Their Statistical Uncertainties for the Spectral Energy Distributions of 3HWC J1930+188 Plotted in Figure 6

Experiment (Reference)	Reference Energy	Flux at E0	Index	Integrated Flux > 1TeV
	E0 (TeV)	(10−15 TeV−1 cm−2 s−1)		(10−12 cm−2 s−1)
HAWC (this work)	10	${2.46}_{-0.47}^{+0.58}$	−2.93 ± 0.20	1.08 ± 0.46
HAWC (Albert et al. 2020)	7	10.2 ± 0.8	−2.76 ± 0.07	1.25 ± 0.16
VERITAS (Abeysekara et al. 2018)	1	660 ± 130	−2.18 ± 0.20	0.56 ± 0.14
H.E.S.S. (H. E. S. S. Collaboration et al. 2018)	1	506 ± 124	−2.59 ± 0.26	0.32 ± 0.09

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In this section, we discuss whether the γ-ray emission of the new TeV source candidate HAWC J1932+192, potentially associated with the pulsar PSR J1932+1916, could come from the acceleration of particles in its PWN. All of the characteristics of this system have previously been gathered in Section 2, as well as in the Appendix, Table 8. The spectrum derived from 3ML under the point-like hypothesis is plotted in Figure 5.

From the best fit, the differential flux at 10 TeV was found to be ${\left({1.95}_{-0.49}^{+0.62}\right)}_{\mathrm{stat}}\times {10}^{-15}$ TeV⁻¹ cm⁻² s⁻¹. With a spectral index equal to −2.46 ± 0.24, the integrated energy flux between 1 TeV and 43 TeV is ${F}_{\gamma \gt 1\mathrm{TeV}}\,=\,{\left(1.61\pm 0.58\right)}_{\mathrm{stat}}\times {10}^{-12}$ erg cm⁻² s⁻¹. The γ-ray luminosity is given by:

$$\begin{eqnarray}&&{L}_{\gamma }=4\pi {D}^{2}{F}_{\gamma \gt 1\mathrm{TeV}}.\end{eqnarray} \tag{ 2 }$$

Under the assumption that the distance is either 3.5 kpc (Junkes et al. 1992) or 6.6 kpc (Ranasinghe & Leahy 2017), we can calculate the γ-ray luminosity L_γ , and since the pulsar's rotational energy is $\dot{E}=4\times {10}^{35}$ erg s⁻¹, we can calculate the energy that the pulsar has to spend to produce it. Using the first distance estimation, ∼0.6% of the pulsar energy is needed to accelerate the electrons and positrons that produce the γ rays via IC scattering on ambient photons. In the case of the larger distance, this percentage goes up to ∼2%. For comparison, Di Mauro et al. (2019) found that about 1% of the spin-down energy of the Geminga pulsar has to be converted into e^± to be consistent with the γ-ray data from the Fermi-LAT and HAWC. This means that the PWN could in principle produce the observed γ-ray emission. Table 3 gathers the parameters calculated above.

Table 3. Summary of the Properties of the New Source HAWC J1932+192

Morphology Hypothesis	Point-like
Integrated energy flux Fγ>1TeV (erg cm−2 s−1)	${\left(1.61\pm 0.58\right)}_{\mathrm{stat}}\times {10}^{-12}$
Distance D (kpc)	∼3.5	6.6
γ-ray luminosity Lγ (erg s−1)	∼2.5 × 1033	∼9 × 1033
Fraction of the pulsar energy needed (%)	∼0.6	∼2

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The best fit of the data gives a size of σ = 018 ± 004_stat for 3HWC J1928+178, which represents 39% containment, given our 2D Gaussian model. The corresponding 68% containment radius is 027. The flux at 10 TeV is ${\left({4.23}_{-1.10}^{+1.49}\right)}_{\mathrm{stat}}\times {10}^{-15}$ TeV⁻¹ cm⁻² s⁻¹ and the spectral index is −2.09 ± 0.16, as reported in Table 1. The spectrum is plotted in red in Figure 7, together with the one previously published by the HAWC collaboration, in gray, using the same amount of data and analysis bins 1 to 9, but for a single point-like source hypothesis (Albert et al. 2020). As previously mentioned, the high number of free parameters being fitted together is responsible for the larger uncertainties. Both HAWC spectra are compatible with the flux point from LHAASO at 100 TeV (Cao et al. 2021) within the uncertainties. The origin of the observed TeV γ-ray emission of 3HWC J1928+178 is discussed in the next section. A classical PWN scenario is considered, as well as a possible association with a molecular cloud. Note that the presence of the large extended component J1928-EXT of angular size σ = 143 ± 017_stat may account for a large-scale galactic diffuse emission component that is absent from the model or may indicate the mismodeling of 3HWC J1928+178. In particular, J1928 and J1928-EXT may be part of the same object, if we consider a Geminga-like diffusion model. This hypothesis was considered in Jardin-Blicq (2021) and will not be treated here.

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γ -ray emission. From the fitted size of 3HWC J1928+178, σ = 018, the diameter d and volume V can be calculated assuming a spherical geometry. All the properties derived hereafter are summarized in Table 4. With the pulsar being located at a distance of D = 4.3 kpc, 39% and 68% of the emission are contained in regions of sizes d ≃ 27 pc and 41 pc, respectively. The integrated energy flux between 1 TeV and 178 TeV is ${F}_{\gamma \gt 1\mathrm{TeV}}\,=\,{\left(3.45\pm 1.22\right)}_{\mathrm{stat}}\times {10}^{-12}$ erg cm⁻² s⁻¹. The γ-ray luminosity, given by Equation (2), is L_γ = 7.7 × 10³³ erg s⁻¹.

Table 4. Summary of the Properties of the Fitted Source J1928 in the Hypothesis Where IC Scattering on CMB Photons is the Dominant Radiation Process

Angular size θ (°)	0.18 (39%)
	0.27 (68%)
Distance D (kpc)	4.3
Size (68%) d (pc)	∼41
Volume V (pc3)	∼ 3.7 × 104
Integrated energy flux Fγ>1TeV (erg cm−2 s−1)	(3.45 ± 1.22) × 10−12
γ-ray luminosity Lγ (erg s−1)	∼7.7 × 1033
Total energy WIC (erg)	∼2.9 × 1046
Energy density IC (eV cm−3)	∼0.04

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The emission observed in PWNe at TeV energies is dominated by radiation processes involving electrons scattering on ambient photons: IC scattering. In the Thomson regime, the γ-ray spectral energy distribution due to electrons with energy E_e peaks at

$$\begin{eqnarray}&&{E}_{\gamma }\simeq 33{E}_{e}^{2}{k}_{B}T\quad {\rm{TeV}},\end{eqnarray} \tag{ 3 }$$

where E_γ and E_e are in TeV, k_B is the Boltzmann constant, T is the temperature of the photon field, and k_BT is in eV (Hinton & Hofmann 2009). Hence, ${E}_{e}\simeq 11\sqrt{{E}_{\gamma }}\,\mathrm{TeV}$ and a 1 TeV γ-ray photon is produced via IC scattering of an electron of energy of ∼10 TeV on cosmic microwave background (CMB) photons. The electron cooling time for IC scattering in the Thomson regime is given by

$$\begin{eqnarray}&&{\tau }_{\mathrm{IC}}=\displaystyle \frac{{E}_{e}}{{{dE}}_{e}/{dt}}\simeq 3.1\times {10}^{5}\displaystyle \frac{1}{{U}_{\mathrm{rad}}}\displaystyle \frac{1}{{E}_{e}}\quad {\rm{yr}},\end{eqnarray} \tag{ 4 }$$

where E_e in is TeV and U_rad is the radiation energy density in eV cm⁻³ (Hinton & Hofmann 2009). For electrons of energy E_e = 10 TeV scattering on CMB photons, k_B T = 2.35 × 10⁻⁴ eV and U_rad = 0.26 eV cm⁻³, so the electron cooling time is τ_CMB ≃ 120 kyr. For far-IR (FIR) photons, k_B T = 3 × 10⁻⁴ eV and U_rad = 0.3 eV cm⁻³, so τ_FIR ≃ 100 kyr. The total energy is the product of the γ-ray luminosity and the cooling time W = τ_IC L_γ , equal to W_CMB ≃ 2.9 × 10⁴⁶ erg, using τ_IC = τ_CMB. Finally, dividing by the volume, the energy density is simply _W = W/V. Assuming a spherical geometry and a diameter of 41 pc, the energy density is _IC ≃ 0.04 eV cm⁻³. This is much smaller than the energy density of the interstellar medium (ISM) _ISM ≃ 1 eV cm⁻³. Given the age of the pulsar of 82 kyr, this is consistent with an old PWN, where the electrons have started to cool and diffuse away from their source. Note that for electrons with E_e > 300 TeV, the Klein Nishina regime starts. Adapting Equations (3) and (4) to the Klein Nishina regime for CMB photons only gives 300 TeV electrons producing 230 TeV photons, with the cooling time becoming τ_CMB ≃ 30 kyr. However, the total energy and the energy density are of the same order of magnitude as what was calculated in the Thomson regime.

Parent particle population. The parent population of the electrons responsible for the observed γ-ray emission can be obtained using the naima ³⁶ python package (Zabalza 2015). This provides models for nonthermal radiative emission from homogeneous distributions of relativistic particles. The contributions of nonthermal radiative processes, IC scattering in this case, can be computed given a shape for the particle energy distribution, and the model can be used to fit the observed nonthermal spectra through a Markov Chain Monte Carlo procedure. In the present case, the emission is assumed to be produced by electrons upscattering CMB photons, with a temperature T = 2.72 K and an energy density of 0.26 eV cm⁻³, and FIR photons, with a temperature T = 20 K and an energy density of 0.3 eV cm⁻³. Since the γ-ray spectrum of 3HWC J1928+178 has been represented by a power law, the population of the electrons is also chosen to follow a simple power law. The fit is performed using this model for the electrons and the γ-ray spectrum from the HAWC observations from the best fit with 3ML. The best fit for the energy distribution of the electrons has a differential energy at 70 TeV F_70TeV = (1.91 ± 0.2) × 10⁴¹ erg⁻¹ and an index of −2.55 ± 0.1. The total energy of the electrons above 1 TeV is ${W}_{e}\,=\,{4.6}_{-1.2}^{+2.2}\times {10}^{46}$ erg. Given that the spin-down of the pulsar is $\dot{E}\,=\,1.6\times {10}^{36}$ erg s⁻¹, assuming that it is constant over the life of the pulsar, which is 82 kyr, gives a lower limit for the total energy released by the pulsar of 4.1 × 10⁴⁸ erg. Hence, an upper limit of ∼1% can be set on the amount of energy that the pulsar could have transferred to the electrons above 1 TeV. This is again compatible with previous estimations for the Geminga pulsar (Di Mauro et al. 2019).

Hypotheses for this association. Most of the interstellar gas in our Galaxy is molecular hydrogen H₂, contained in GMCs. These massive clouds of gas and dust have a typical size that ranges from 50 to 200 pc and a mass ranging between 10⁴ and 10⁶ solar masses. They are the sites of star formation. In addition, they are the source of most of the diffuse galactic γ-ray emission (Hunter et al. 1997). The dominant processes by which cosmic rays interact with the ISM and produce γ rays, are high-energy electron bremsstrahlung, IC interactions with low-energy photons, and nucleon–nucleon interactions. For the latter, in particular, molecular clouds are favorable environments. Hence, it is interesting to compare the galactic gas distribution and the γ-ray emission detected by HAWC, in order to assess whether the components of the molecular cloud, mainly hydrogen, could be a target for relativistic protons, producing observed γ-rays via pion decay (Albert et al. 2021).

CO as a tracer for molecular clouds. H₂ is not easily observable, because this molecule has no electric dipole moment. For this reason, it does not emit radiation from neither vibrational nor rotational transitions. However, CO emits radiation through a rotational transition (J = 1 → 0) when excited by collisions with hydrogen molecules. Hence, CO emission is used to trace H₂ molecular clouds. The abundance of CO is typically about 7.2 × 10⁻⁵ for one hydrogen molecule. Two isotopes are mainly used: ¹²CO and ¹³CO. The main difference is that ¹²CO is optically thick, while ¹³CO is optically thin, the first one being on average ∼60 times more abundant than the second one (Lucas & Liszt 1998). ¹³CO is both a good quantitative and qualitative tracer of molecular gas, being related to the column density of H₂. It can probe deep in the cloud without saturating, and it provides more accurate velocity and kinematic distances because of its narrower line. Therefore, ¹³CO is more suited to deriving the column density of the cloud, under the hypothesis of local thermodynamic equilibrium. The emission line corresponding to the CO de-excitation gives the mean velocity of the CO molecules in the cloud, and the width of this line gives the velocity dispersion associated with the cloud. Under the virial equilibrium hypothesis, and assuming uniform density within the cloud, the width of the line scales linearly with the size of the cloud.

The ¹³CO (rotation emission line J = 1 → 0 at 110 GHz) data from the Galactic Ring Survey (GRS; Jackson et al. 2006) ³⁷ were obtained using the SEQUOIA multipixel array on the Five College Radio Astronomy Observatory (FCRAO, Arny & Valeriani 1977) 14 m telescope located in New Salem, Massachusetts, between 1998 December and 2005 March. Three molecular clouds can be found at the location of the HAWC TeV emission. Figure 8(a) shows the HAWC significance map, where a region corresponding to the emission with significance >5σ is defined. From this region, the velocity distribution is extracted as a function of the brightness temperature averaged over this region, visible in Figure 8(b). Three maxima can be highlighted at ∼4.5 km s⁻¹, ∼22 km s⁻¹, and ∼46 km s⁻¹. The ¹³CO maps corresponding to each velocity are also displayed in Figure 8(c). The most intense one, at ∼22 km s⁻¹, is further studied in the next paragraph.

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Detailed study of the brightest cloud. The cloud at ∼22 km s⁻¹ has a very complicated and elongated shape. The study is restricted to the portion of the cloud within the >5σ γ-ray emission of 3HWC J1928+178, represented by the white box in Figures 8 and 9. In this region, the cloud is decomposed into two parts, which could be interpreted as two clumps of the cloud. They are represented by the two smaller magenta and cyan boxes that are labeled "1" and "2" in Figure 9. The ¹³CO maps for the peak velocity and the velocity distribution are also shown on the right-hand side of the same figure. Some basic properties can now be derived, such as the column density, the mass, and the volume of these clumps, to estimate the total cosmic-ray energy and the energy density that would be needed to explain the observed γ-ray emission.

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To do so, the clumps are assumed to have a spherical shape. The most probable distance ³⁸ for this cloud is D = 4 kpc (Reid et al. 2016), which would be compatible with the distance of the pulsar. Their diameter d and volume V can be calculated from their angular size θ. Clumps 1 and 2 have diameters of ∼12 and ∼18 pc, respectively. They are smaller than the source representing J1928, which was found to contain 68% of the emission within ∼41 pc. For each clump, the ¹³CO column density N(¹³CO) is determined using the brightness temperature T_mb, in K, and the FWHM of the velocity distribution peak Δv, in km s⁻¹, as explained in Simon et al. (2001):

$$\begin{eqnarray}&&N{(}^{13}{\rm{CO}})=8.75\times {10}^{14}{T}_{\mathrm{mb}}{\rm{\Delta }}v.\end{eqnarray} \tag{ 5 }$$

The clump mass M, in the unit of solar masses, is given by

$$\begin{eqnarray}&&M=3.05\times {10}^{-25}N{(}^{13}{\rm{CO}})\,{\theta }_{x}{\theta }_{y}{D}^{2}\quad {{\rm{M}}}_{\odot },\end{eqnarray} \tag{ 6 }$$

where θ_x and θ_y are the half-axes of the clump in arcseconds and D is the distance to the cloud in pc, which is here assumed to be 4 kpc. With the mass and the volume, the particle density in the cloud, which is a potential target for cosmic rays, can be calculated using

$$\begin{eqnarray}&&n=\displaystyle \frac{M}{\mu {{\rm{m}}}_{{\rm{H}}}V},\end{eqnarray} \tag{ 7 }$$

where μm_H is the mean mass of an atom in the ISM, with μ ≃ 1.4 and m_H being the mass of a hydrogen atom. Moreover, using the best-fit value for the flux found with 3ML, the luminosity L_γ above 1 TeV was calculated in the previous section using Equation (2) as L_γ ≃ 7.7 × 10³³ erg s⁻¹. Considering that, at TeV energies, the spectral energy distribution of the secondary γ rays peaks at about one-tenth of the energy of the primary proton and does not vary significantly with energy (Hinton & Hofmann 2009), a 1 TeV photon can be produced by a 10 TeV proton. The total energy of the cosmic rays above 10 TeV in the cloud is W_p = τ_p L_γ , where τ_p is now the characteristic cooling time for relativistic protons. It is derived using the proton–proton interaction cross section σ_pp, the speed of light c, and the density n:

$$\begin{eqnarray}&&{\tau }_{{\rm{p}}}=\displaystyle \frac{1}{f{\sigma }_{\mathrm{pp}}{cn}}.\end{eqnarray} \tag{ 8 }$$

In this relation, f stands for the fact that a proton loses about half of its energy per interaction, with only a third of them producing π⁰. Hence, using typical values for the inelastic cross section σ_pp ≃ 35 mb for VHE protons (Hinton & Hofmann 2009) and f = 1/6, it results in a lifetime τ_p ≃ 1.8 × 10⁸ n⁻¹ yr. Finally, the energy density is simply the ratio of the total energy and the volume: _p = W_p/V. For the cloud considered here, the total energy of the cosmic rays above 10 TeV is W_p = 7.9 × 10⁴⁷ erg and the energy density is _p ≃ 4.4 eV cm⁻³. The parameters calculated for each clump and for the total cloud are gathered in Table 5.

Table 5. Summary of the Properties of the CO Cloud

	Clump 1	Clump 2
Angular size θ (°)	0.172	0.252
Distance D (pc)	4000	4000
Size d (pc)	12.0	17.6
Volume V (pc3)	906	2851
Average brightness temperature Tmb (K)	1.875	2.44
FWHM of the velocity distribution peak Δv (km s−1)	1.5	2.56
Column density N(13 CO) (cm−3)	2.46 × 1015	5.47 × 1015
Mass M (M⊙)	1151	5482
	Total cloud
Mass M (M⊙)	6633
Volume V (pc3)	3757
Density n (particles cm−3)	50
Total energy Wp (erg)	7.9 × 1047
Energy density p (eV cm−3)	4.4

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The farthest edge of clump 2 is 22 pc away from the pulsar. Considering a sphere of radius 22 pc centered on the pulsar, containing both clumps, its volume is 15 times the sum of the volumes of both clumps together. Since the total energy in the cloud is W_p = 7.9 × 10⁴⁷ erg, the energy in the sphere centered on the pulsar should be W_R = 1.2 × 10⁴⁹erg. The pulsar releases most of its energy at the beginning of its lifetime and steadily decreases its spin-down power afterward, as described by the following equation:

$$\begin{eqnarray}&&\dot{E}=\dot{{E}_{0}}{\left(1+\displaystyle \frac{t}{{\tau }_{0}}\right)}^{-\tfrac{n+1}{n-1}},\end{eqnarray} \tag{ 9 }$$

where ${\dot{E}}_{0}$ is the initial spin-down luminosity and n is the braking index. It has been argued that up to 20% of a pulsar's energy could accelerate ions (Bucciantini et al. 2011). Assuming as a reasonable value that 10% of the pulsar's energy has been used to accelerate protons, this means that the pulsar must have released 10 × W_R = 1.2 × 10⁵⁰ erg. Considering that this is equal to the difference in rotational energy between now and when the pulsar was born gives ${\rm{\Delta }}{\rm{E}}=1.2\times {10}^{50}=I\times ({{\rm{\Omega }}}_{0}^{2}-{{\rm{\Omega }}}^{2})/2$, with I ≃ 1 × 10⁴⁵ g cm² for a typical pulsar. The pulsar considered here has a period of P ≃ 70 ms. This gives a birth period of P₀ = 2π/Ω₀ ≃ 10 ms. The maximum total energy that a pulsar with a birth period of 1 ms can release during its life is E_ROT = 1 × 10⁵³ erg, for a pulsar with a typical mass of 1.4 M_⊙ and a typical radius of 10 km (Khangulyan et al. 2018). Our result is consistent with this upper limit. Moreover, integrating Equation (9) from birth (t = 0) until now (t = T), with the braking index n = 3, and using the relation between the characteristic age of the pulsar τ_c and the age at birth τ₀ = τ_c − τ, we can derive

$$\begin{eqnarray}&&{\tau }_{0}=\displaystyle \frac{\dot{E}\,{\tau }_{c}^{2}}{{\rm{\Delta }}E+\dot{E}{\tau }_{c}}.\end{eqnarray} \tag{ 10 }$$

With ΔE = 1.2 × 10⁵⁰ erg, $\dot{E}\,=\,1.6\times {10}^{36}$ erg s⁻¹, and τ_c = 82600 yr, the age at birth is τ₀ ≃ 2700 yr. Hence, the pulsar's true age would be 79,800 yr. Finally, its spin-down power at birth would be ${\dot{E}}_{0}\,=\,1.4\times {10}^{39}$ erg s⁻¹. As a comparison, this is the same order of magnitude as the Crab, which makes this value plausible.

Conclusions for the molecular cloud association. From the study performed in this section, we can make conclusions regarding the different hypotheses:

• 1.  
  The components of the molecular cloud, mainly hydrogen, could be a target for relativistic protons from the pulsar PSR J1928+1746 and its PWN, producing neutral pions during the interaction, which emit the observed γ rays. The energy radiated by the pulsar was found to be compatible with the energy needed to produce the observed γ-ray luminosity via proton–proton interaction.
• 2.  
  Adding up the two clumps gives a mass for the cloud of ∼6600 M_⊙ and a density of ∼50 particles per cm³. However, the γ-ray emission around 1 TeV is dominated by IC scattering of electrons on the CMB for medium densities lower than ∼240 particles per cm³ (Hinton & Hofmann 2009). Hence, bremsstrahlung does not play a significant role here: the observed emission cannot be explained by the electrons and positrons from the PWN interacting with the atoms of the cloud via bremsstrahlung.
• 3.  
  The total energy in cosmic rays >10 TeV in the cloud derived from the observed γ-ray luminosity is calculated as 7.9 × 10⁴⁷ erg, leading to an energy density of ∼4.4 eV cm⁻³. This is three orders of magnitude higher than the energy density of the sea of galactic cosmic rays above 10 TeV, which is ∼1 × 10⁻³ eV cm⁻³ (Gabici et al. 2009). Hence, these cosmic rays cannot explain the TeV emission observed by HAWC via interaction with the cloud.
• 4.  
  The remaining hypothesis is that a local accelerator, for example an SNR, as yet undetected, is producing the detected VHE γ-ray emission. It is commonly assumed that an SNR releases ∼10⁵¹ erg of kinetic energy in the ISM, and that 10% of it—that is ∼10⁵⁰ erg—is used for cosmic-ray acceleration. Assuming a cosmic-ray spectrum between 1 GeV and 1 PeV, with an energy dependence E⁻², 33% of the energy flux is found above 10 TeV, which makes ∼3.3 × 10⁴⁹ erg. The ratio of the volume around the SNR, which is uniformly filled with cosmic rays, and the volume of the cloud scales like the ratio of the energy contained in each volume:

  $$\begin{eqnarray}&&\displaystyle \frac{{D}^{3}}{{r}^{3}}=\displaystyle \frac{3.3\times {10}^{49}}{7.7\times {10}^{47}}\simeq 40\quad \Rightarrow \quad D={\left(40{r}^{3}\right)}^{1/3},\end{eqnarray} \tag{ 11 }$$

  where r = 10 pc is approximately the radius of the cloud and D is the distance from the SNR to the farther edge of the clump. Thus, an SNR located within a distance of ∼40 pc from the cloud would be able to account for the cosmic rays producing the observed TeV emission via interaction with the molecules of the cloud.