γ-Ray Emission from Classical Nova V392 Per: Measurements from Fermi and HAWC

HAWC is sensitive to γ-rays with energy above 300 GeV. Based on the timing and locations of the PMTs struck by the shower, we reconstructed the location on the sky of the particle that initiated the shower. For this analysis we used the R.A. and decl. for the J2000 epoch (Albert et al. 2020b). A key parameter for this analysis is fHit, the fraction of PMTs that are struck during the shower event. This quantity can be used to parameterize the angular resolution and the γ–hadron selection criteria, and is sensitive to the energy of the initiating particle as described in Albert et al. (2020b) and Abeysekara et al. (2017a, 2017b).

In the remainder of this section, we show the statistical significance of the HAWC observations, and report the best-fit flux and CLs assuming unbroken PLs. In Section 5 we set limits on the maximum (TeV) energy to which the Fermi-LAT SED could extend and be compatible with HAWC data; this method is applied to the nova for the first time, to our knowledge. In Section 6 we provide HAWC limits in different bins of true energy, without imposing the assumption of an unbroken PL as the SED shape; this method is new, to the best of our knowledge. In Section 7 we assess systematic uncertainties on the HAWC limits.

The time frame chosen for the main HAWC investigation of V392 Per covers 40 days, beginning 7 days before the optical discovery of the nova.

For each day of the observation, we made a significance map of the region of interest for each of the nine bins of fHit as described in Albert et al. (2020b). Throughout this paper, we measure significance in units of standard deviations (σ). The same shocks that produce GeV γ-rays are also generally expected to be the source of any TeV radiation, so we analyzed HAWC data assuming the same Γ = 2.0 PL index as observed for the Fermi-LAT data.

We defined three periods within this time range. The "On" period covers 7 days starting from 2018 April 30 (MJD 58,238 to 58,245), the same as the Fermi-LAT 8 day active period excluding the last day, when we had power issues at the HAWC site. The "Before" period is 7 days starting from 2018 April 23 (MJD 58,231 to 58,238), before the "On" period. This includes 1 day of optical activity during which Fermi-LAT was not observing due to a solar panel problem. The "After" period is 7 days after the end of the "On" period, starting on 2018 May 8 (MJD 58,246 to 58,253). In addition, we defined a 7 day "On − 1 yr" period on the same days as the "On" period, but a year before V392 Per's eruption, in order to represent a period when no signal is expected.

For each period, we performed forward-folded fits of a Γ = 2.0 PL spectral model to the nine HAWC pixel-level data maps for each fHit bin, centered at the V392 Per location as in Albert et al. (2020b). Throughout the rest of the paper we report best-fit values for the SED point at E = 1 TeV ($S={E}^{2}\ {dN}/{dE}$), its uncertainty (dS), or the corresponding 95% upper CL (S₉₅); all are in units of erg s⁻¹ cm⁻². Results are for the "On" period whenever no specific period is given. We used the method described in Albert et al. (2018) for setting 95% CLs. The SED points and SED 95% CL values in this paper were calculated using the HAL ³⁸ (HAWC Accelerated Likelihood) plugin (Younk et al. 2016; Abeysekara et al. 2021) to the 3ML multimission analysis framework (Vianello et al.2016).

We also calculated the statistical significance of the normalization of the Γ = 2.0 PL SED compared to zero TeV emission. A significance map is this calculation as a function of sky position. Figure 3 shows the significance map during the "On" period contemporaneous with the Fermi-LAT GeV detection. There is a mild excess of 1.6σ significance near the nova location.

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Figure 4 shows the significance at the nova position for each day during the study period, with the "On" period indicated between the black lines. Some transits are missing when electrical storms or power outages interfered with HAWC data taking.

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Table 1 shows the limits from a HAWC SED fit and the resulting significance for a Γ = 2.0 PL spectral model for all the time periods, as well as the best Fermi-LAT SED fit. While there is a weak 1.6σ suggestion of TeV emission during the "On" period, and an even weaker hint during the "After" period, the best-fit HAWC flux and 95% upper limit on the flux are far less than would be expected for a continuation into the HAWC TeV regime of the Γ = 2.0 PL seen by Fermi-LAT in the GeV regime.

Next we considered the effect of changing the PL index. Figure 5 shows the SEDs corresponding to the S₉₅ limit for various PL indices. Also shown is the best fit to the Fermi-LAT flux assuming an unbroken PL extending to very high energies. Softer PLs (larger indices) produce less restrictive limits at low energy. The upper envelope of the lines in Figure 5 can be thought of as an SED limit as a function of energy, independent of the actual value of the PL index—at least within the family of PL spectrum shapes (Surajbali 2021). All the limits are statistical only; we discuss systematic uncertainties in Section 7.

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We now present limits in bins of energy, assuming a Γ = 2.0 PL index within each energy bin. In Figure 6 we show the Fermi-LAT SED for V392 Per and the S₉₅ HAWC upper limits. This analysis uses maps binned in fHit, and its energy resolution effects are reasonably matched by half-decade energy bins (e.g., 1–3.16 TeV, 3.16–10 TeV, etc.). HAWC energy estimators could provide better resolution at higher energy, but the additional event selection criteria would reduce sensitivity to a transient source such as a nova.

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The method used to find limits in true energy bins, using data consisting of maps binned in fHit, is as follows. First, we perform a forward-folded fit of the energy spectrum assumed, a point-source model, and the detector response including the point-spread function to the set of data maps for only the normalization $\hat{k}$ of a PL of the form E⁻², where E is in TeV. Then for each true energy bin j we perform a second fit for the normalization k_j of the Γ = 2.0 PL, but with the contribution of energy bin j removed from the original unrestricted PL, in a way that retains the best-fit contributions of all other energy bins as determined by $\hat{k}$ from the original fit. Specifically, we fit to the data the form

$$\begin{eqnarray}&&{dN}/{dE}=\hat{k}\,[\ {E}^{-2}-\mathrm{bin}(E,j)\ {E}^{-2}\,\,]\,+{k}_{j}\ \mathrm{bin}(E,j)\ {E}^{-2}\end{eqnarray} \tag{ 3 }$$

where bin(E, j) = 1 when E falls between the lower and upper edges of energy bin j. Finally, the normalization of the limit is determined by increasing the value of k_j until the fit log-likelihood increases by an amount (2.71/2) appropriate for a 95% CL. This method allows us to report a limit separately in each individual energy bin, without assuming an overall PL SED, as the normalization of each energy bin is determined separately. Because now each energy bin contains less data than the whole of the data, these limits are less restrictive than those in Section 4, where a single unbroken PL is assumed for the underlying SED.

The limit from the lowest-energy HAWC bin is compatible with the continuation of the Fermi-LAT SED, but higher-energy bins are incompatible at the 95% CL. The limit from fitting a single Γ = 2.0 PL across the full HAWC energy range is considerably more restrictive, placing a 95% upper limit at ${E}^{2}\tfrac{{dN}}{{dE}}=4.0\times {10}^{-12}$ erg s⁻¹ cm⁻² (Figure 5).

We simulate by Monte Carlo the expectations for the energy-dependent limits for each energy bin under the hypothesis of no physical flux (only Poisson fluctuations of the background). The distribution of expected limits is shown in Figure 7. The inner (green) and outer (yellow) bands cover 68% and 95% of the simulated limits, respectively, and the central dashed (red) line shows the median of the expected limits in each energy bin. The observed limits (from Figure 6) are shown here in a black solid line to allow comparison with the expected distribution of limits assuming no flux. The observed limits for bins above 3 TeV are typically 1–2 standard deviations above expectation, consistent with either a modest statistical fluctuation or weak TeV emission.

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There have been two previous TeV observations of novae detected by Fermi-LAT. Both observations were made by imaging air Cherenkov telescopes (IACTs). IACTs have better point-source sensitivity than HAWC, but IACTs can only observe sources that fall into their limited field of view (a few degrees). This typically requires specific pointing, a source visible at night, and good weather. As a result, it is harder for IACTs to observe contemporaneously with a Fermi-LAT observation. In contrast, HAWC observes two-thirds of the sky daily.

The first search for TeV nova emission was conducted by the VERITAS collaboration on V407 Cyg (Aliu et al. 2012). VERITAS began observations 9 days after the beginning of the Fermi-LAT detection, and extended them over a week of continued Fermi-LAT detection. VERITAS was unable to detect significant flux above 0.1 TeV, and set 95% limits as shown in Figure 8. Figure 8 also shows the Fermi-LAT SED reported in Abdo et al. (2010). Because of the curvature of the Fermi-LAT SED, VERITAS analyzed their data with a Γ = 2.5 PL. The VERITAS limit is quoted at energies of 1.6–1.8 TeV, where the limit and assumed PL are least correlated (depending slightly on which of two analysis methods was used). To roughly compare this with the HAWC sensitivity, the HAWC differential limits on V392 Per are shown, but reanalyzed with the same Γ = 2.5 PL. These limits are calculated as described above, but instead of Γ = 2.0, using Γ = 2.5.

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MAGIC searched for TeV emission from V339 Del (Ahnen et al. 2015), which is slightly fainter than V392 Per in the GeV band (Ackermann et al. 2014). They found no TeV-detected flux and produced the limits shown in Figure 9. The MAGIC analysis used a Γ = 2.3 PL index, motivated by a fit to the observed Fermi-LAT SED. Again for rough comparison, we show HAWC's V392 Per limits analyzed with this Γ = 2.3 PL. At the overlapping energies, the MAGIC results are about 30 times more constraining than our HAWC limits. It is also worth mentioning that MAGIC was able to observe one night at the beginning of the nova's GeV γ-ray detection, albeit under poor conditions; that observation produced a flux limit about a factor of 10 worse than those on their best nights of observation 9–12 days later, by which time the GeV γ-ray signal had faded, though not as much as V392 Per had faded by its second week.

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Thus, previous IACT nova observations have produced stronger constraints on TeV emission than HAWC, and started from lower energy than HAWC limits. However, they only apply to the period 9 days after the beginning of the optical nova; HAWC's observations began 2 days after the optical nova, and temporally overlap with the entire period of GeV detection with Fermi-LAT. The HAWC "After" period (days 9–15 of the optical nova) matches the time delay of the VERITAS and MAGIC observations. Table 1 suggests the "After" period places slightly more restrictive limits than the "On" period.

The modeling of V392 Per's γ-ray emission requires input parameters derived from optical data. Here we show how we derive these values.

8.1.1. Extinction from Interstellar Dust

To estimate the extinction due to interstellar dust along the line of sight to V392 Per, we rely on several interstellar absorption lines: the Na I D doublet and some diffuse interstellar bands (Figure 10). In this section and in Section 8.1.3, we make use of publicly available spectra from the Astronomical Ring for Access to Spectroscopy (ARAS ³⁹ ; Teyssier 2019). The low- and medium-resolution spectra cover the first month of the optical outburst, starting from the time of optical maximum (day 0). To measure the interstellar lines, we use a high-resolution spectrum from day 6.

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Based on the equivalent widths of the Na I D lines and the empirical relations of Poznanski et al. (2012), we derive a reddening value, E(B − V) = 0.78 ± 0.04 mag, for V392 Per. Based on the equivalent width of the two absorption lines at 5780.5 and 5797.1 Å and the empirical relations from Friedman et al. (2011), we derive E(B − V) = 1.04 ± 0.05 mag. This leads to an average reddening value of E(B − V) = 0.90 ± 0.18 mag for V392 Per. For a standard interstellar extinction law (A_V = 3.1 E(B − V); e.g., Mathis 1990), we find a V-band extinction value A_V = 2.8 ± 0.5 mag. This value is consistent with the one derived by Chochol et al. (2021), and we use it in the remainder of the paper.

8.1.2. Bolometric Luminosity

Multiband optical photometry was performed by several observers from the AAVSO (Kafka 2020) from day 0 (2018 April 29; the time of discovery of eruption and also the time of optical maximum) and throughout the optical outburst of V392 Per (see Chochol et al. 2021 for a more detailed description of the light curve). We make use of photometry in the BVRI bands to estimate the nova's total (bolometric) luminosity in the few weeks following the nova eruption. Near the optical peak, the optical pseudophotosphere of the nova reaches its maximum radius and the SED is characterized by an effective temperature of 6000–10,000 K, peaking in the BVRI bands (e.g., Gallagher & Starrfield 1976; Hachisu & Kato 2004; Bode & Evans 2008).

In order to estimate the bolometric luminosity of the nova as a function of time, we use the bolometric task that is part of the SNooPy Python package (Burns et al. 2011). This task directly integrates the flux measured by the BVRI photometry (we use method = 'direct'), which adds a Rayleigh–Jeans extrapolation in the red (extrap_red = 'RJ'), and corrects this SED for extinction from intervening dust (we use A_V = 2.8 mag; see Section 8.1.1). We plot the BVRI photometry, along with the derived bolometric luminosity, in Figure 11.

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8.1.3. Expansion Velocities from Spectral Line Profiles

Figure 12 shows the spectral evolution of Hα during the first few days of the eruption of V392 Per. As noted by Wagner et al. (2018) and Chochol et al. (2021), the spectral lines initially show a P Cygni profile with an absorption trough at a blueshifted velocity of around −2700 km s⁻¹ (blue line in Figure 12). On day +1, a broader emission component emerges, extending to blueshifted velocities of around −5500 km s⁻¹ (green line in Figure 12; see also a zoomed-in view of this profile in the bottom panel of Figure 12). This indicates the presence of two physically distinct outflows: a slow one and a fast one, as described in Aydi et al. (2020b). At this time, there is another absorption component, superimposed on the broad emission, with a velocity of around 3800 km s⁻¹ (black line in Figure 12). This component, which appears around the optical peak and has an intermediate velocity between those of the slow and fast components, is the so-called "principal component" as historically classified by McLaughlin (1943) and Mclaughlin (1947). Friedjung (1987) and Aydi et al. (2020b) suggest that this intermediate-velocity component is the outcome of the collision between the initial slow flow and the following faster flow, and therefore the velocity of the intermediate component depicts the speed v_cs of the cold central shell sandwiched between the forward and reverse shocks (Metzger et al. 2014).

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Fermi-LAT detections of V392 Per were only made for 8 days following optical maximum, in one of the shortest-duration and most sharply peaked γ-ray light curve yet observed from a nova (see Figure 8 of Chomiuk et al. 2021a). We note that the turn-on of the γ-rays was not fully captured in V392 Per, as Fermi-LAT was suffering technical problems during its rise to optical maximum, so this duration is a lower limit. However, the true duration is unlikely to be substantially longer than observed, given that Fermi-LAT signals tend to first become detectable around optical maximum (e.g., Ackermann et al. 2014), V392 Per's observed optical maximum was on 2018 April 29.8 (Chochol et al. 2021), and Fermi-LAT observations resumed on April 30.

The short duration of the Fermi signal in V392 Per is perhaps not surprising, as γ-ray light curves have been observed to correlate and covary with optical light curves in novae (Li et al.2017; Aydi et al. 2020a), and V392 Per's optical light curve evolves very quickly (Figure 2). In the top panel of Figure 13, we compare the duration of Fermi-LAT γ-rays against the time for the optical light curve to decline by 2 mag from maximum (t₂) for the 15 γ-ray detected novae tabulated in Table S1 of Chomiuk et al. (2021a) (see Gordon et al. 2021 for associated t₂ values). We see that novae that are slower to decline from optical maximum generally remain γ-ray bright for longer. A Spearman rank correlation test gives p = 0.0002 (for a one-tailed test), indicating a significant correlation between the γ-ray duration and the optical decline time. With t₂ = 5.9 days, V392 Per has one of the fastest-evolving optical light curves and a similarly rapid γ-ray light curve to match.

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During its Fermi-LAT detection, the GeV γ-ray luminosity of V392 Per was on average 5 × 10³⁵ erg s⁻¹. Such a luminosity is typical among γ-ray detected novae, which show variations in Fermi-LAT luminosity of >2 orders of magnitude (see Figure S1 of Chomiuk et al. 2021a along with Franckowiak et al. 2018). The average γ-ray luminosity but short duration of V392 Per motivated us to plot γ-ray duration against total energy emitted in the Fermi-LAT band in the bottom panel of Figure 13, comparing V392 Per (in red) with data on 14 other Fermi-detected novae (Chomiuk et al. 2021a). Based on Fermi-LAT light curves of five novae, Cheung et al. (2016) found a tentative anticorrelation between these properties, with the counterintuitive implication that novae that remain γ-ray bright for longer emit less total energy in the Fermi-LAT band. Figure 13 revisits this claimed anticorrelation with three times the number of Fermi-detected novae, and we find that it no longer holds; there are many novae with relatively short γ-ray duration and relatively low total γ-ray energy, V392 Per among them.

Before addressing the implications of the TeV non-detections by HAWC, we must ask whether such emission could even in principle be detected, due to absorption processes that occur close to the emission site at the shock. Of particular importance at TeV energies is attenuation due to pair creation, γ − γ → e⁻ + e⁺, on the background radiation provided by the optical light of the nova. In contrast, at the GeV energies that Fermi-LAT is sensitive to, pair creation would require X-ray target photons. Attenuation is therefore less important in the GeV range than in the TeV range, because the X-ray luminosity (and photon number density) of novae is low compared to the optical/UV luminosity during the early phases of nova eruptions when γ-ray emission is observed.

Other forms of γ-ray opacity, such as photonuclear absorption (the Bethe–Heitler process), are comparatively less important than the γ γ opacity. In particular, the Bethe–Heitler opacity increases slowly with photon energy, being only ∼3 times larger at 100 TeV than at 1 GeV (Chodorowski et al. 1992); hence, if the Bethe–Heitler optical depth through the nova ejecta is low enough to permit the escape of γ-rays detectable by Fermi-LAT, then it is unlikely to impede the escape of photons in the HAWC energy range across the same epoch, particularly considering that the optical depth of the expanding ejecta is expected to decrease rapidly with time.

Figure 14 shows the optical depth, τ_{γ γ} , as a function of time since the nova eruption, for a photon leaving the vicinity of the shock at a radius R_cs = v_cs t, where v_cs ≈ 3800 km s⁻¹ is the intermediate-component velocity estimated from the optical spectrum, thought to trace the shock's cold central shell (see Section 8.1) and hence the location of forward and reverse shocks. In calculating the value of τ_{γ γ} , we have made use of the energy density of the optical/near-infrared radiation field,

$$\begin{eqnarray}&&{u}_{\gamma }=\displaystyle \frac{{L}_{\mathrm{bol}}}{4\pi {R}_{\mathrm{cs}}^{2}c},\end{eqnarray} \tag{ 4 }$$

estimated from V392 Per's bolometric light curve L_bol(t) (Figure 11). We separately consider the cases of the optical/infrared (IR) SED having the form of a blackbody at temperature T_eff ≈ 8000 K (top panel of Figure 14) and that of free–free (bremsstrahlung) emission also at temperature T_eff ≈ 8000 K (bottom panel of Figure 14). For the effective temperature we avoid using the optical colors to derive a blackbody temperature, given that these colors are heavily affected by the emission line's evolution, and would give an overestimate of the relevant temperature. These two cases (blackbody and free–free) roughly bracket the physically expected range of optical spectral shapes, given the lack of available near-IR observations of V392 Per to provide additional guidance. For example, Kato & Hachisu (2005) argue that the nova emission can be dominated by blackbody emission at early times (during the so-called fireball phase) and later transition to being dominated by free–free emission from a wind or expanding ejecta shell.

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The optical depth for TeV photons is computed as

$$\begin{eqnarray}&&{\tau }_{\gamma \gamma }(x)={R}_{\mathrm{cs}}{\int }_{1/x}{\sigma }_{\gamma \gamma }(x,y)\,\displaystyle \frac{{{dN}}_{\mathrm{ph}}}{{dydV}}\,{dy},\end{eqnarray} \tag{ 5 }$$

where x = h ν/m _e c² and y = h ν_opt/m _e c² are the dimensionless energies of the high-energy and optical (target) photons, respectively, and σ_{γ γ} is the angle-averaged pair production cross section (e.g., Zdziarski 1988). The target photon spectrum is normalized to the total radiation energy density given by Equation (4),

$$\begin{eqnarray}&&{u}_{\gamma }=\int \frac{{{dN}}_{\mathrm{ph}}}{{dydV}}\,{m}_{e}{c}^{2}y\,{dy}.\end{eqnarray} \tag{ 6 }$$

The shape of the target spectrum follows ${{dN}}_{\mathrm{ph}}/({dydV})\,\propto {y}^{2}/[\exp ({m}_{e}{c}^{2}y/{{kT}}_{\mathrm{eff}})-1]$ for a blackbody (Figure 14, upper panel) or ${{dN}}_{\mathrm{ph}}/({dydV})\propto {y}^{-1}\exp (-{m}_{e}{c}^{2}y/{{kT}}_{\mathrm{eff}})$ for an optically thin bremsstrahlung spectrum (Figure 14, lower panel). It is worth noting that at identical energy densities, the blackbody spectrum places a smaller fraction of target photons at energies h ν_opt < kT_eff compared with other plausible physically motivated spectra. As a result, the opacity for photons at $h\nu \gt {({m}_{{e}}{c}^{2})}^{2}/{{kT}}_{\mathrm{eff}}\sim 1$ TeV is comparatively lower in the blackbody case, as those photons preferentially pair-produce on the low-energy tail of the target spectrum. Note also that at $h\nu \gg {({m}_{{e}}{c}^{2})}^{2}/{{kT}}_{\mathrm{eff}}$, the γγ opacity behaves approximately as ${\tau }_{\gamma \gamma }\propto {T}_{\mathrm{eff}}^{-2}$ and $\propto {T}_{\mathrm{eff}}^{-1}$ in the blackbody and free–free cases, respectively.

In the most conservative case of the free–free target spectrum, we see that τ_{γ γ} remains larger than unity for a few days after eruption at energies ≳1 TeV. Meanwhile, τ_{γ γ} ≲ 1 at all times in the more optimistic case of a blackbody spectrum. Furthermore, insofar that near the peak of the nova optical light curve the observed emission tends to be dominated by the optically thick emission from the photosphere instead of optically thin free–free emission (e.g., from a wind above the photosphere; Kato & Hachisu 2005), we favor the interpretation that over most, if not all, of the time of Fermi-LAT detection, V392 Per is transparent to TeV photons. Still, the Fermi-LAT light curve of V392 Per is unusual among γ-ray detected novae for being sharply peaked at early times (Figure 2), and its brightest GeV flux occurs within the first ∼2 days of eruption; it is possible that TeV photons were attenuated from V392 Per at these earliest times, when the nova was brightest at GeV energies. In the next section, we consider the shock conditions that would produce very high energy photons in V392 Per.

In this section we use V392 Per bolometric luminosity and ejecta expansion velocity derived from optical data to estimate the maximum energy to which particles could be accelerated and hence the maximum γ-ray energy.

The γ-ray emission from novae is understood as nonthermal emission from relativistic particles accelerated at shocks (e.g., Martin & Dubus 2013; Ackermann et al. 2014), through the process of diffusive shock acceleration (e.g., Blandford & Ostriker 1978). A variety of evidence, from across the electromagnetic spectrum, suggests that the shocks in classical novae are internal to the nova ejecta (e.g., Chomiuk et al. 2014; Aydi et al. 2020a; Chomiuk et al. 2021a), as a fast outflow impacts a slower outflow released earlier in the nova. On the other hand, in symbiotic novae where the companion is a giant star with dense wind, the shocks may occur as the nova ejecta collides with the external wind (e.g., Abdo et al. 2010). V392 Per has an orbital period intermediate between those of cataclysmic variables and symbiotic novae with an atypical radio light curve (Munari et al. 2020; Chomiuk et al. 2021b) and hence the nature of the shock interaction—internal or external—is ambiguous. However, our discussion to follow regarding the particle acceleration properties is relatively unaffected by this distinction.

Physical models for the γ-ray emission divide into "hadronic" and "leptonic" scenarios depending on whether the emitting particles are primarily relativistic ions or electrons. Several independent lines of evidence support the hadronic scenario (Chomiuk et al. 2021a), including (a) the presence of a feature in the γ-ray spectrum near the pion rest mass at 135 MeV (e.g., Li et al. 2017), (b) the nondetection of nonthermal hard X-ray emission by NuSTAR (which should be more prominent in leptonic scenarios; Vurm & Metzger 2018; Nelson et al. 2019; Aydi et al. 2020b), and (c) efficiency limitations on leptonic scenarios due to synchrotron cooling of electrons behind the shock (Li et al. 2017). Motivated thusly, we focus on hadronic scenarios for the γ-rays.

In the hadronic scenario, relativistic ions collide with ambient ions such as protons, producing pions that decay into γ-rays:

$$\begin{eqnarray}\begin{array}{rcl}{pp} & \to & {\pi }^{0}\to \gamma \gamma \\ & \to & {\pi }^{\pm }\to {\mu }^{\pm }{\nu }_{\mu }({\bar{\nu }}_{\mu })\to {e}^{\pm }+{\nu }_{e}({\bar{\nu }}_{e})+{\nu }_{\mu }({\bar{\nu }}_{\mu }).\end{array}\end{eqnarray} \tag{ 7 }$$

Here, ≈1/3 and ≈2/3 of the inelastic p–p collisions go through the π⁰ and π^± channels, respectively (Kelner & Aharonian 2008). The π⁰ channel is expected to dominate the γ-ray luminosity (e.g., Li et al. 2017), but secondary leptons produced via π^± decay can also produce γ-rays through bremsstrahlung and inverse Compton processes. A useful rule of thumb is that it requires a proton of energy 10E to generate a γ-ray of energy E. Therefore, to produce emission up to the HAWC sensitivity range (∼1–100 TeV) requires proton acceleration up to ${E}_{\max }\gtrsim 10\mbox{--}1000$ TeV. Can nova shocks accelerate particles up to such high energies?

We consider a shock generated as a fast wind of velocity v _f ≈ v₃ ≈ 5500 km s⁻¹ (see Figure 12) collides with a slower outflow of velocity v _s ≈ v₁ ≈ 2700 km s⁻¹, generating an internal-shocked shell of velocity v_cs ≃ ξ v₁ (where the dimensionless parameter ξ ≲ 2, typically; if v_cs = v₂ = 3800 km s⁻¹, then ξ = 1.4). Recent studies have shown that the values of v _f , v _s , and even v_cs may be observed directly in the optical spectra of novae (Aydi et al. 2020b), and we have taken our fiducial values here to match those inferred from V392 Per's optical spectra (Section 8.1.3).

Insofar as an order-unity fraction of the optical nova light is reprocessed thermal emission by the (radiative) reverse shock (e.g., Li et al. 2017; Aydi et al. 2020a), the nova luminosity is related to the mass-loss rate according to

$$\begin{eqnarray}&&\displaystyle \begin{array}{l}{L}_{\mathrm{bol}}(t)\sim {L}_{\mathrm{sh}}(t)\simeq \frac{1}{2}{\dot{M}}_{f}{v}_{f}^{2}\\ \approx \,8\times {10}^{36}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\frac{{\dot{M}}_{f}}{{10}^{-6}{M}_{\odot }\,{\mathrm{yr}}^{-1}}{\left(\frac{{v}_{{f}}}{5500\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{2},\end{array}\end{eqnarray} \tag{ 8 }$$

where we assume v _f ≫ v _s and treat the fast outflow as a wind of mass-loss rate ${\dot{M}}_{f}$.

In diffusive shock acceleration, as cosmic rays gain greater and greater energy E, they can diffuse back to the shock from a great upstream distance, z, because of their larger gyroradii r _g = E/eB_sh, where

$$\begin{eqnarray}&&\displaystyle \begin{array}{l}{B}_{\mathrm{sh}}\approx {\left(6\pi {\epsilon }_{B}{m}_{p}{n}_{f}{v}_{\mathrm{cs}}^{2}\right)}^{1/2}\simeq {\left(\frac{3}{2}\frac{{\epsilon }_{B}{\dot{M}}_{f}}{{v}_{f}{t}^{2}}\right)}^{1/2}\\ \,\approx \,0.07\,{\rm{G}}\ {\epsilon }_{B,-2}^{1/2}{\left(\frac{{\dot{M}}_{f}}{{10}^{-6}{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)}^{1/2}{\left(\frac{{v}_{f}}{5500\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{-1/2}\\ \,\times \,{\left(\frac{t}{1\,\mathrm{wk}}\right)}^{-1}\end{array}\end{eqnarray} \tag{ 9 }$$

is the magnetic field behind the reverse shock, for an assumed efficiency of magnetic field amplification _B = _B,−2 × 10⁻². This is commensurate with the required field amplification to accelerate ions with an efficiency ∼1% (Caprioli & Spitkovsky 2014), as inferred through application of the calorimetric technique (Metzger et al. 2015) to correlated γ-ray/optical emission in novae (Li et al. 2017; Aydi et al. 2020a). In the above, we have taken ${n}_{f}={\dot{M}}_{f}/(4\pi {m}_{p}{R}_{\mathrm{cs}}^{2}{v}_{f})$ for the density of the fast outflow at radius R_cs = v_cs t.

The maximum energy to which particles are accelerated before escaping from the vicinity of the shock, ${E}_{\max }$, is found by equating the upstream diffusion time ${t}_{\mathrm{diff}}\sim D/{v}_{\mathrm{cs}}^{2}$ to the minimum of various particle loss timescales. These include the downstream advection time t_adv ∼ z_acc/v_cs, where z_acc is the width of the acceleration zone, and (in hadronic scenarios) the pion creation timescale ${t}_{\pi }={({n}_{f}{\sigma }_{\pi }c)}^{-1}$, where σ_π ∼ 2 × 10⁻²⁶ cm² is the inelastic cross section for p–p interactions (Kamae et al. 2006). We consider these limiting processes in turn.

Equating t_diff = t_adv, and taking D ≈ r _g c/3 as the diffusion coefficient (Caprioli & Spitkovsky 2014), one obtains (e.g., Metzger et al. 2016; Fang et al. 2020)

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{E}_{\max } & \sim & \frac{3{{eB}}_{\mathrm{sh}}{v}_{\mathrm{cs}}{z}_{\mathrm{acc}}}{c}\\ {E}_{\max } & \approx & 340\,\mathrm{TeV}\left(\frac{{z}_{\mathrm{acc}}}{{R}_{\mathrm{cs}}}\right){\left(\frac{\xi }{2}\right)}^{2}{\epsilon }_{B,-2}^{1/2}{\left(\frac{{\dot{M}}_{f}}{{10}^{-6}{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)}^{1/2}\\ & & \times {\left(\frac{{v}_{f}}{5500\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{-1/2}{\left(\frac{{v}_{s}}{2700\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{2}\end{array}\end{eqnarray} \tag{ 10 }$$

where R_cs = v_cs t is the radius of the shock.

On the other hand, equating t_diff = t_π , we obtain

$$\begin{eqnarray}\displaystyle \begin{array}{rcl}{E}_{\max } & \sim & \frac{3e}{{c}^{2}}\frac{{v}_{\mathrm{cs}}^{2}{B}_{\mathrm{sh}}}{{n}_{f}{\sigma }_{\pi }}\approx 12\pi \frac{{m}_{p}}{{\sigma }_{\pi }}\frac{e}{{c}^{2}}\frac{{B}_{\mathrm{sh}}{v}_{f}{v}_{\mathrm{cs}}^{4}{t}^{2}}{{\dot{M}}_{f}}\\ {E}_{\max } & \approx & 2\times {10}^{7}\,\mathrm{TeV}{\left(\frac{\xi }{2}\right)}^{4}{\epsilon }_{B,-2}^{1/2}{\left(\frac{{\dot{M}}_{f}}{{10}^{-6}{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right)}^{-1/2}\\ & & \times {\left(\frac{{v}_{f}}{5500\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\frac{{v}_{{s}}}{2700\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{4}{\left(\frac{t}{1\mathrm{wk}}\right)}^{2}.\end{array}\end{eqnarray} \tag{ 11 }$$

The maximum energy is given by the minimum of Equations (10) and (11), which for the system parameters of interest works out to be the former. In particular, taking our fiducial velocity values and L_opt ∼ 10³⁷–10³⁸ erg s⁻¹ on a timescale of days to weeks (Section 8.1), we see that ${\dot{M}}_{f}\,\lesssim {10}^{-5}\mbox{--}{10}^{-6}{M}_{\odot }$ yr⁻¹ (Equation (8)). Thus, from Equation (10) we infer ${E}_{\max }\lesssim 200\mbox{--}600$ TeV (z_acc/R_cs), in which case we could expect γ-ray energies up to ${E}_{\gamma ,\max }\sim 0.1{E}_{\max }\sim 20\mbox{--}60$ TeV (z_acc/R_cs).

If acceleration occurs across a radial scale of the order of the shock radius (i.e., z_acc ∼ R_cs), our estimated ${E}_{\gamma ,\max }\sim 20\,-60$ TeV would appear inconsistent with our constraints on an extension of the measured Fermi-LAT spectrum to energies ≳10 TeV (Section 5). However, various physical effects may reduce the effective extent of the accelerating layer to a width z_acc ≪ R_cs (and hence reduce ${E}_{\gamma ,\max }$), such as ion–neutral damping of the Bell (2004) instability (Reville et al. 2007; Metzger et al. 2016) or hydrodynamical thin-shell instabilities of radiative shocks (which corrugate the shock front and alter the effective portion of its surface with the correct orientation relative to the upstream magnetic field to accelerate ions; Steinberg & Metzger 2018). The maximum γ-ray energy generated by the shock could also be lower if the magnetic field amplification factor is less than the fiducial value _B = 0.01.