A Detailed Observational Analysis of V1324 Sco, the Most Gamma-Ray-luminous Classical Nova to Date

Nova outbursts have been detected in the GeV gamma-ray regime with the Fermi Gamma-ray Space Telescope and its Large Area Telescope (LAT; see, e.g., Cheung et al. 2010, 2012b, 2012a, 2015; Hays et al. 2013; Ackermann et al. 2014; Cheung et al. 2016). They show gamma-ray luminosities (>100 MeV) of ${10}^{34}\mbox{--}{10}^{36}$ erg s⁻¹, lasting for 2–8 weeks around optical maximum (Ackermann et al. 2014; Cheung et al. 2016).

The presence of gamma-rays implies that there are relativistic particles being generated in the nova event. There are two potential classes of processes for producing gamma-rays from relativistic particles: leptonic and hadronic. In the leptonic class, electrons are accelerated up to relativistic speeds and produce gamma-rays via inverse Compton and/or relativistic bremmstrahlung processes (Blumenthal & Gould 1970; Vurm & Metzger 2016). In the hadronic process, it is ions that are being accelerated to relativistic speeds; these particles collide with a dense medium to produce ${\pi }^{0}$ mesons, which then decay to gamma-rays (Drury et al. 1994). The likely source of the accelerated particles is strong shocks, which can accelerate particles to relativistic speeds via the diffusive shock acceleration mechanism (Blandford & Ostriker 1978; Bell 1978; Metzger et al. 2014).

The first nova detected by Fermi/LAT was V407 Cyg, and it received considerable attention (Abdo et al. 2010; Aliu et al. 2012; Chomiuk et al. 2012a; Esipov et al. 2012; Mukai et al. 2012; Nelson et al. 2012; Orlando & Drake 2012; Shore et al. 2012; Martin & Dubus 2013). Given that V407 Cyg has a Mira giant secondary with a dense wind (a member of the symbiotic class of systems), a model to explain the gamma-rays was proposed wherein a shock was generated as the nova ejecta interacted with the dense ambient medium. A similar model was proposed to explain the inferred presence of relativistic particles in another star with a giant wind, RS Oph (Tatischeff & Hernanz 2007).

These models, however, could not explain the subsequent novae detected by Fermi/LAT: V1324 Sco, V959 Mon, V339 Del (Ackermann et al. 2014), and V1369 Cen and V5668 Sgr (Cheung et al. 2016). These systems do not have a detectable red-giant companion (see, e.g., Hornoch 2013; Munari et al. 2013; Munari & Henden 2013; Finzell et al. 2015; although note that the underlying binary of V5668 Sgr has yet to be identified). Although it is theoretically possible for these novae to have high-density circumstellar material despite not having a red-giant companion (Spruit & Taam 2001), no evidence has yet been found for substantial circumstellar material around cataclysmic variables (Harrison et al. 2013). Therefore, the non-detection of red-giant companions implies that these novae have main-sequence companions with low-density circumstellar material.

It is in fact much more likely that the shocks are being produced within the ejecta, due to different components of the ejecta colliding with one another (internal shocks). There has already been long-standing evidence for internal shocks in classical novae from X-ray observations (e.g., O'Brien et al. 1994; Mukai & Ishida 2001). One idea for generating internal shocks in novae was put forward by Chomiuk et al. (2014a), supported by radio imaging. First, the binary interacts with the puffed-up nova envelope, resulting in a relatively slow flow with density enhancement along the binary orbital plane. Later, a separate, fast wind is launched from the white dwarf. When these two outflows collide with one another, they produce shocks. Progress has been made on the theoretical front by constraining the shock conditions necessary to explain both the thermal and non-thermal emission observed in gamma-ray-detected novae and finding them consistent with the conditions expected in novae (Metzger et al. 2014, 2016; Vlasov et al. 2016; Vurm & Metzger 2016).

The goal of this paper is to use a multiwavelength analysis of the gamma-ray-detected nova V1324 Sco in order to assess the causes and energetics of the shocks that yield GeV gamma-rays. V1324 Sco was originally discovered at optical wavelengths by the Microlensing Observations in Astrophysics collaboration (MOA; Wagner et al. 2012). From their high-cadence monitoring of the Galactic bulge, MOA noticed a transient appear on 2012 May 22.8 UT at R.A. = 17^h50^m5390 and decl. =$-32^\circ 37^{\prime} 20\buildrel{\prime\prime}\over{.} 46$ (J2000). Between 2012 May 22–2012 June 1, this source brightened only gradually, and its brightness was modulated with a periodicity of ∼1.6 hr. Starting on 2012 June 1, the source brightened much more rapidly (see Section 2), and a spectrum obtained on 2012 June 4 identified the transient as a classical nova (Wagner et al. 2012). In retrospect, the gradual brightening over the week prior to June 1 is likely a nova "precursor" event; similar events have been observed in a handful of other novae but remain poorly understood (Collazzi et al. 2009). The precursor event is outside the scope of this paper and will be the subject of R. M. Wagner et al. (2017, in preparation), although it can be seen in Figure 1.

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Shortly thereafter, the transient was found to be associated with GeV gamma-rays, as detected by Fermi/LAT in its all-sky survey mode (Cheung et al. 2012a). The gamma-rays were detected during the time range 2012 June 15–July 2 with an average flux of $\sim 5\times {10}^{-7}$ photons cm⁻² s⁻¹ (>100 MeV; Ackermann et al. 2014). Ackermann et al. also point out that the spectrum of V1324 Sco may extend to higher energies than the other gamma-ray-detected novae, although the spectral analysis suffers from poor statistics above a few GeV.

Multiwavelength observations were initiated during the period of gamma-ray detection, including observations at radio (Chomiuk et al. 2012b; Section 3), infrared (Raj et al. 2012), and X-ray wavelengths (Page et al. 2012; Page & Osborne 2013; Section 5). High-resolution spectra observed during the nova outburst allowed our team to measure absorption features, enabling an estimate of reddening and placing a lower limit on the distance (Finzell et al. 2015). We estimate $E(B-V)\,=1.16\pm 0.12$. Three-dimensional reddening maps of the Galaxy (Schultheis et al. 2014) imply that V1324 Sco is >6.5 kpc away (see also Munari et al. 2015 for similar results). V1324 Sco's location near or beyond the Galactic bulge implies that it is significantly more gamma-ray luminous than other gamma-ray-detected novae (${L}_{\gamma }\gtrsim 2\times {10}^{36}$ erg s⁻¹ in the energy range 100 MeV–10 GeV, exceeding other novae by ≳order of magnitude; Ackermann et al. 2014; Finzell et al. 2015). We can also use this distance estimate to infer the nature of the nova host system; no underlying binary is detected in the VVV survey down to ${m}_{K}\lt 16.6$ mag (Minniti et al. 2010), implying that the binary contains a dwarf star and not a red giant (Finzell et al. 2015).

While previous work has established the context of V1324 Sco, our goal here is to present a detailed picture of the nova outburst itself to understand how gamma-ray-producing shocks formed. Section 2 gives an overview of the outburst, as traced by UV/optical/near-IR (UVOIR) photometry. In Section 3, we pres-ent our radio data and discuss a first peak in the radio light curve that is a strong indicator of shocks. We then use the second peak in the radio light curve to estimate the ejecta mass and kinetic energy of the outburst. In Section 4.1, we present the optical spectra and use them to constrain the kinematics and filling factor of V1324 Sco's outburst. In Section 5, we detail the X-ray limits and discuss how they are consistent with observations at other wavelengths. In Section 6, we summarize that V1324 Sco is—in all non-gamma-ray observations—a classical nova. We discuss how classical novae can produce a range of gamma-ray luminos-ities and the conditions that might lead to the highest gamma-ray luminosity observed from a nova to date. Finally, we conclude the paper in Section 7 by summarizing our findings for V1324 Sco.

V1324 Sco falls within one of the fields that the MOA Collaboration continually observes with the MOAII 1.8 m telescope at Mt. Johns Observatory in New Zealand. V1324 Sco was initially detected in 2012 April by their high-cadence I-band photometry (Wagner et al. 2012). The initial detection showed a slow monotonic rise in brightness between April 13 and May 31, followed by a very large increase in brightness starting June 1 (Figure 1; Wagner et al. 2012). For the rest of this paper, we take 2012 June 1 to be day 0, or the start of the nova outburst. We also adopt the convention throughout this paper that all dates with − or + denote days before or after 2012 June 1, respectively.

All initial high-cadence observations, taken as part of the regular MOA program, were taken in the I broad band and were reduced using standard procedures (see Bond et al. 2001 for details). The MOA survey emphasizes rapid imaging of the Galactic bulge fields; on a clear night, an individual field will be imaged every ∼40 minutes. The result of this high time cadence photometry can be seen in Figure 1. It should be noted that the primary purpose of the high-cadence observations is difference imaging; as a result, the individual values should only be used to measure changes, not as an absolute measurement (Bond et al. 2001).

After the steep optical rise, a follow-up campaign was triggered by the MicroFUN group,³² who believed that the transient was a potential microlensing event. Apart from the standard I broadband filter, the MicroFUN follow-up observations also used V and I Bessel filters. Other observations were made in B, V, and I filters using the Small & Moderate Aperture Research Telescope System (SMARTS) 1.3 m telescope and Auckland Observatories.

Along with the MOA and MicroFUN data, we also present multicolor photometry from Fred Walter's ongoing Stony Brook/SMARTS Atlas of (mostly) Southern Novae (see Walter et al. 2012 for further information on this data set), as well as data from the American Association of Variable Star Observers (AAVSO).³³ The SMARTS data use the ANDICAM instrument on the 1.3 m telescope and provide both optical (B, V, R, I) and near-IR (J, H, K) filters going from day +35 to day +124, while the AAVSO data use optical (V, B, R) filters and go from day +7 to day +445.

Finally, we incorporate the UV data taken contemporaneously with the X-ray observations. The UV data come from the Ultraviolet/Optical Telescope (UVOT; see Roming et al. 2005 for further details) on board Swift. Each observation was taken using the UVM2 filter, which is centered on 2246 Å and has an FWHM of 498 Å (Poole et al. 2008). These observations were taken at the same time as the X-ray observations (see Section 5), stretching from day +22 to day 520; however, we only include observations where V1324 Sco was detected.

A portion of the UVOIR data set is presented in Table 1; the entire data set can be found in the machine-readable version of the table. Multiband photometry is plotted in Figure 2. Note that no attempt has been made to standardize the photometry from different observatories.

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Table 1.  Table of Photometric Data

Observation Date	JD	$t-{t}_{0}$ a	Filter	Mag	Mag Error	Observer/Group	Telescope/Specific Filterb
		(Days)
2012 Apr 13	2456030.07502	−48.92499	I	18.700	0.150	MOA	MJUO-Ibroad
2012 Apr 13	2456030.95591	−48.04410	I	18.770	0.090	MOA	MJUO-Ibroad
2012 Apr 13	2456030.95714	−48.04287	I	18.590	0.090	MOA	MJUO-Ibroad
2012 Apr 13	2456030.99663	−48.00338	I	18.800	0.110	MOA	MJUO-Ibroad
2012 Apr 14	2456031.05194	−47.94807	I	18.760	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.06304	−47.93697	I	18.900	0.110	MOA	MJUO-Ibroad
2012 Apr 14	2456031.07414	−47.92587	I	18.850	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.08652	−47.91348	I	18.860	0.110	MOA	MJUO-Ibroad
2012 Apr 14	2456031.09762	−47.90238	I	19.030	0.110	MOA	MJUO-Ibroad
2012 Apr 14	2456031.10976	−47.89024	I	18.660	0.070	MOA	MJUO-Ibroad
2012 Apr 14	2456031.12211	−47.87789	I	18.930	0.120	MOA	MJUO-Ibroad
2012 Apr 14	2456031.13321	−47.86679	I	18.890	0.100	MOA	MJUO-Ibroad
2012 Apr 14	2456031.14435	−47.85566	I	18.830	0.100	MOA	MJUO-Ibroad
2012 Apr 14	2456031.15670	−47.84331	I	18.860	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.16781	−47.83220	I	18.950	0.110	MOA	MJUO-Ibroad
2012 Apr 14	2456031.17893	−47.82108	I	18.890	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.19127	−47.80874	I	18.910	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.20237	−47.79764	I	18.880	0.120	MOA	MJUO-Ibroad
2012 Apr 14	2456031.21348	−47.78653	I	18.850	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.22585	−47.77416	I	18.880	0.100	MOA	MJUO-Ibroad
2012 Apr 14	2456031.23825	−47.76176	I	18.750	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.25182	−47.74818	I	18.870	0.090	MOA	MJUO-Ibroad
2012 Apr 14	2456031.96000	−47.04001	I	18.610	0.080	MOA	MJUO-Ibroad
2012 Apr 14	2456031.96123	−47.03877	I	18.700	0.080	MOA	MJUO-Ibroad
2012 Apr 14	2456031.99908	−47.00093	I	18.880	0.120	MOA	MJUO-Ibroad
2012 Apr 15	2456032.05184	−46.94817	I	18.740	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.07405	−46.92596	I	18.840	0.130	MOA	MJUO-Ibroad
2012 Apr 15	2456032.08742	−46.91258	I	18.920	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.09855	−46.90146	I	18.810	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.10966	−46.89035	I	18.770	0.090	MOA	MJUO-Ibroad
2012 Apr 15	2456032.12200	−46.87801	I	18.880	0.150	MOA	MJUO-Ibroad
2012 Apr 15	2456032.13311	−46.86690	I	18.740	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.14421	−46.85580	I	18.850	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.15655	−46.84346	I	18.740	0.090	MOA	MJUO-Ibroad
2012 Apr 15	2456032.16869	−46.83132	I	18.780	0.090	MOA	MJUO-Ibroad
2012 Apr 15	2456032.17989	−46.82012	I	18.910	0.090	MOA	MJUO-Ibroad
2012 Apr 15	2456032.19224	−46.80777	I	18.960	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.20335	−46.79666	I	18.890	0.080	MOA	MJUO-Ibroad
2012 Apr 15	2456032.21445	−46.78556	I	18.800	0.080	MOA	MJUO-Ibroad
2012 Apr 15	2456032.22683	−46.77317	I	18.840	0.070	MOA	MJUO-Ibroad
2012 Apr 15	2456032.23920	−46.76081	I	18.840	0.100	MOA	MJUO-Ibroad
2012 Apr 15	2456032.25030	−46.74971	I	18.820	0.080	MOA	MJUO-Ibroad
2012 Apr 15	2456032.26264	−46.73736	I	18.960	0.190	MOA	MJUO-Ibroad

Notes. All of these data, as well as data from AAVSO and Walter et al. (2012), can be found in the machine-readable table.

^aTaking t₀ to be 2012 June 1.0. ^bWe abbreviate the different facilities used by the MOA and MicroFUN groups as MJUO: Mt. John University Observatory; AUCK: Auckland Observatory; CTIO: SMARTS 1.3 m Telescope.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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We present an overview of the different phases in the evolution of the optical light curve to help orient the reader to the different qualitative variations. These different phases come from the classification scheme laid out in Strope et al. (2010)—with the exception of the early-time rise. Throughout this overview, we will reference Figures 1 and 2; note that while Figure 2 features multiple bands, it has a significantly lower time resolution than Figure 1.

2.2.1. Early-time Rise (Days −49 to 0)

2.2.2. Onset of the Steep Optical Rise (Days 0 to +10)

2.2.3. Flattening of the Optical Light Curve (Days +10 to +45)

2.2.4. Dust Event (Days +45 to +157)

2.2.5. Power-law Decline (Days +157 to End of Monitoring)

In the optical regime, V1324 Sco is photometrically a D (Dusty) class nova (Strope et al. 2010), because of the extraordinary dust event that took place between days +46 to +157. Other D-class novae include FH Ser, NQ Vul, and QV Vul (Strope et al. 2010). Among D-class novae, the speed of V1324 Sco's photometric decline is quite typical. V1324 Sco's t₂ value—that is, the time for a nova to decline by 2 mag from maximum in the V band—is ${t}_{2}\approx 24$ days. This is consistent with other D-class novae, all of which are of order tens of days (see Strope et al. 2010 and references therein).

In the case of V1324 Sco, the dust event includes a drop in flux all the way out to the near-IR. This suggests that the change in temperature from optical maximum was significant and that the dust photosphere was cold. A fit to the near-IR colors at the epoch closest to the I-band minimum suggests that the dust photosphere was $\lt 1000$ K. While dust events are quite common in novae—Strope et al. (2010) give 16 examples of other such novae—there are only a few novae with dust dips showing comparably cool photospheres (e.g., QV Vul and V1280 Sco; Gehrz et al. 1992; Sakon et al. 2015).

If the shocks in novae are dense and radiative (as predicted by Metzger et al. 2014, 2015), then they are ideal locations for dust formation (Derdzinski et al. 2017). Radiative shocks can also explain the observed gamma-ray luminosity and non-detection in X-rays (Section 5; Metzger et al. 2014). When compared to other gamma-ray-detected novae in the AAVSO database (Kafka 2016), V1324 Sco had an unusually dramatic dust event. For example, there is no sign of dust formation in V959 Mon (Munari et al. 2015), and V339 Del showed signatures of dust formation at infrared wavelengths, but the event did not have a profound effect on the optical light curve (Gehrz et al. 2015; Evans et al. 2017). Derdzinski et al. (2017) find that some of these variations could be attributable to viewing angle, if dust preferentially forms along the orbital plane, as would be expected in the geometry suggested by Chomiuk et al. (2014a). In addition, Evans et al. (2008) point out that novae on CO white dwarfs are more likely to produce dust than novae on ONe white dwarfs. Observations of dust (or lack thereof) in these three gamma-ray-detected novae can be reconciled if V1324 Sco is viewed at an edge-on inclination and hosts a CO white dwarf, while V959 Mon's binary hosts an ONe white dwarf viewed at high inclination (Page et al. 2013; Shore et al. 2013), and V339 Del hosts a CO white dwarf but is observed at low inclination (Schaefer et al. 2014; Shore et al. 2016).

The origin of "flat tops" in novae remains something of an open question. Inluminous red novae (LRNs) like V1309 Sco (which eject several orders of magnitude more mass than classical novae), Ivanova et al. (2013) proposed that a plateau around maximum is caused by a recombination front, much as in Type IIP supernovae (Chugai 1991). In this case, the light curve flattening near maximum is explained as a photosphere radius that does not change substantially in Eulerian coordinates (but shrinks in Lagrangian coordinates) and has roughly constant temperature, due to the fact that the ejecta are cooling and recombining.

Another possible explanation for flat-topped light curves was developed in the case of T Pyx and proposed by Nelson et al. (2014) and Chomiuk et al. (2014b), where there is multiwavelength evidence that the bulk of the ejecta remained in a quasi-hydrostatic configuration around the binary until the end of the "flat top" period. In this nova, it appears that 1–2 months pass before the ejecta are accelerated to their terminal velocity and are expelled from the environs of the binary, although the physical origin of the delay remains a mystery (it is, perhaps, attributable to binary interaction with the quasi-static envelope).

It should be noted that, of the gamma-ray-detected novae, at least two—V1369 Cen and V5668 Sgr—had a similar flattening of the optical light curve near maximum (Cheung et al. 2016), though both exhibited large (${\rm{\Delta }}V\gt 1$ mag) fluctuations in brightness during their period of flattening (unlike V1324 Sco; Figure 2). In both systems, the evolution of optical spectral line profiles around the maximum imply that several episodes of mass ejection transpire over the course of the variegated plateau (Walter et al. 2012). These systems support the idea that flat-topped light curves in novae may be a signature of complex, prolonged mass loss—the sort of mass loss that will produce shocks and gamma-rays.

We obtained sensitive radio observations of V1324 Sco between 2012 June 26 and 2014 December 19 with the Karl G. Jansky Very Large Array (VLA) through programs S4322, 12A-483, 12B-375, 13A-461, 13B-057, and S61420. Over the course of the nova, the VLA was operated in all configurations, and data were obtained in the C (4–8 GHz), Ku (12–18 GHz), and Ka (26.5–40 GHz) bands, resulting in coverage from 4 to 37 GHz. Observations were acquired with 2 GHz of bandwidth and 8 bit samplers, split between two independently tunable 1 GHz wide basebands. The details of our observations are given in Table 2.

Table 2.  VLA Observations of V1324 Sco

Julian	Date	$t-{t}_{0}$	Config.	4.5 GHz Fluxa,b	7.8 GHz Flux	13.3 GHz Flux	17.4 GHz Flux	27.5 GHz Flux	36.5 GHz Flux
($245000+$)	(UT)			(mJy)	(mJy)	(mJy)	(mJy)	(mJy)	(mJy)
6104.3	2012 Jun 26	25.0	B	0.165 ± 0.023	0.149 ± 0.021	–	–	–	–
6106.1	2012 Jun 28	27.0	B	–	–	–	–	–	<0.822
6144.1	2012 Aug 5	65.0	B	0.605 ± 0.115	2.074 ± 0.160	–	–	–	–
6147.1	2012 Aug 8	68.0	B	0.817 ± 0.124	2.720 ± 0.191	–	–	–	–
6151.2	2012 Aug 12	72.0	B	1.385 ± 0.100	2.803 ± 0.171	–	–	5.133 ± 0.636	5.647 ± 0.766
6159.2	2012 Aug 20	80.0	B	–	–	2.574 ± 0.307	2.690 ± 0.364	3.246 ± 0.489	2.639 ± 0.533
6167.0	2012 Aug 28	88.0	B	1.036 ± 0.092	1.128 ± 0.089	–	–	–	–
6171.1	2012 Sep 1	92.0	B	–	–	1.169 ± 0.144	1.250 ± 0.165	1.760 ± 0.261	2.370 ± 0.377
6177.9	2012 Sep 7	98.0	BnA	0.502 ± 0.055	0.431 ± 0.038	–	–	–	–
6182.1	2012 Sep 12	103.0	BnA	–	–	0.810 ± 0.128	1.080 ± 0.165	1.738 ± 0.290	2.126 ± 0.392
6215.0	2012 Oct 15	136.0	A	<0.218	0.338 ± 0.049	0.843 ± 0.121	1.201 ± 0.174	2.085 ± 0.332	2.460 ± 0.415
6243.8	2012 Nov 12	164.0	A	0.228 ± 0.055	0.484 ± 0.050	1.297 ± 0.174	1.955 ± 0.256	4.153 ± 0.582	5.667 ± 0.838
6297.6	2013 Jan 5	218.0	A	0.353 ± 0.039	0.701 ± 0.051	1.699 ± 0.198	2.539 ± 0.301	4.240 ± 0.574	6.230 ± 0.983
6357.5	2013 Mar 6	278.0	D	0.761 ± 0.075	1.300 ± 0.121	2.877 ± 0.337	4.071 ± 0.469	5.660 ± 0.725	6.821 ± 0.917
6402.3	2013 Apr 20	323.0	D	0.963 ± 0.216	1.352 ± 0.153	2.810 ± 0.313	3.637 ± 0.403	4.677 ± 0.543	5.089 ± 0.605
6445.4	2013 Jun 2	366.0	DnC	1.037 ± 0.186	1.507 ± 0.139	2.855 ± 0.322	3.283 ± 0.379	3.290 ± 0.429	3.680 ± 0.538
6501.3	2013 Jul 28	422.0	C	1.041 ± 0.105	1.446 ± 0.118	2.318 ± 0.270	2.337 ± 0.286	3.017 ± 0.659	3.160 ± 0.811
6584.0	2013 Oct 19	505.0	B	0.937 ± 0.084	1.440 ± 0.093	1.933 ± 0.218	1.773 ± 0.206	1.682 ± 0.222	1.486 ± 0.213
6649.7	2013 Dec 23	570.0	B	0.963 ± 0.106	1.120 ± 0.087	1.350 ± 0.182	1.350 ± 0.186	1.027 ± 0.205	1.098 ± 0.304
6719.5	2014 Mar 3	640.0	A	0.719 ± 0.068	0.841 ± 0.064	0.899 ± 0.114	0.843 ± 0.119	0.696 ± 0.143	0.529 ± 0.132
7009.8	2014 Dec 18	930.0	C	0.368 ± 0.058	0.293 ± 0.034	0.268 ± 0.044	0.244 ± 0.044	<0.282	<0.365

Notes. Taking t₀ to be 2012 June 1.

^aDetections are defined as flux $\gt 5\sigma$. Non-detections are given as the $5\sigma$ upper limits. ^bIf no observations were taken for a given frequency, it is denoted by –.

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At the lower frequencies (C band), the source J1751–2524 was used as the complex gain calibrator, while J1744–3116 was used for gain calibration at the higher frequencies (Ku and Ka bands). The absolute flux density scale and bandpass were calibrated during each run with either 3C 48 or 3C 286. Referenced pointing scans were used at the Ku and Ka bands to ensure accurate pointing; pointing solutions were obtained on both the flux calibrator and gain calibrator, and the pointing solution from the gain calibrator was subsequently applied to our observations of V1324 Sco. Fast switching was used for high-frequency calibration, with a cycle time of ∼2 minutes. Data reduction was carried out using standard routines in AIPS and CASA (Greisen 2003; McMullin et al. 2007). Each receiver band was edited and calibrated independently. The calibrated data were split into their two basebands and imaged, thereby providing two frequency points.

An observation in A configuration (the most extended VLA configuration) from 2012 December 16 suffered severe phase decorrelation at higher frequencies. Despite efforts to self-calibrate, we could not reliably recover the source, and we therefore do not include these measurements here.

In each image, the flux density of V1324 Sco was measured by fitting a Gaussian to the imaged source with the tasks JMFIT in AIPS and gaussfit in CASA. We record the integrated flux density of the Gaussian; in most cases, there was sufficient signal on V1324 Sco to allow the width of the Gaussian to vary slightly, but in cases of low signal-to-noise ratio, the width of the Gaussian was kept fixed at the dimensions of the synthesized beam. Errors were estimated by the Gaussian fitter and added in quadrature with the estimated calibration errors of 5% at lower frequencies (<10 GHz) and 10% at higher frequencies (>10 GHz). All resulting flux densities and uncertainties are presented in Table 2. V1324 Sco appeared as an unresolved point source in all observations.

Next, we discuss the different phases of the radio light curve evolution. The radio emission is shown in Figure 3 (radio light curve) and Figure 4 (radio spectral energy distributions).

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V1324 Sco was detected during the first radio observation (day +25), coincident with the end of the gamma-ray emission. In subsequent radio observations, the light curve rose steeply to a first maximum, peaking on day +72. This initial peak is in contrast with the second maximum, which peaked around day +300 (see Figure 3).

The radio spectrum on the rise to initial maximum started out consistent with flat (albeit with a large error bar): $\alpha \,=-0.3\pm 0.7$ on day +25 (where α is defined as ${f}_{\nu }\propto {\nu }^{\alpha }$, ${f}_{\nu }$ is the flux density, and ν is the observing frequency; see Figure 4). The radio spectrum then rapidly transitioned to $\alpha =2.0\pm 0.3$ on day +65 (the time of initial maximum), implying optically thick emission. The spectrum then flattened out again ($\alpha =0.6\pm 0.1$ on day +72).

During this initial radio peak, the light curve rises steeply, as ${f}_{\nu }\propto {t}^{2.9}$, assuming day 0 is 2012 June 1. This is steeper than expected for the expansion of an optically thick isothermal sphere (${f}_{\nu }\propto {t}^{2};$ Seaquist & Bode 2008) and could indicate that this maximum is dominated by thermal emission increasing in temperature or non-thermal emission.

To further investigate the nature of this initial radio maximum, we can use the brightness temperature, which is a proxy for surface brightness. Brightness temperature parameterizes the temperature that would be necessary if the observed flux originated from an optically thick thermal blackbody. The equation for brightness temperature is given by

$$\begin{eqnarray}&&{T}_{b}(\nu ,t)=\displaystyle \frac{{S}_{\nu }(t){c}^{2}{D}^{2}}{2\pi {k}_{b}{\nu }^{2}{({v}_{\mathrm{ej}}t)}^{2}},\end{eqnarray} \tag{ 1 }$$

where S_ν is the observed flux, D is the distance, t is the time since explosion, ${v}_{\mathrm{ej}}$ is the ejecta velocity, ν is the observing frequency, and k_b is the Boltzmann constant. Typical brightness temperatures of thermal emission from novae are $\sim {10}^{4}$ K (Cunningham et al. 2015). If the measured brightness temperature is far in excess of $\sim {10}^{4}$ K, it is a solid indication of synchrotron emission.

Figure 5 shows the estimates of the brightness temperature as a function of time, using the distance lower limit of 6.5 kpc from Finzell et al. (2015) and a velocity of 1000 km s⁻¹ (the velocity of the slow flow as estimated from optical spectroscopy; Section 4.3). Two observation epochs (days 80 and 92) were removed due to the lack of low-frequency observations, which usually set the maximum brightness temperature. Note that a larger distance would increase these values, while a larger ${v}_{\mathrm{ej}}$ would decrease them. If we instead assume a velocity of 2600 km s⁻¹ (the velocity of the fast flow; Section 4.3), the brightness temperature estimates will decrease by a factor ∼7.

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During the initial radio maximum, we see that the brightness temperature substantially surpasses 10⁴ K, registering at $5\,\times {10}^{5}$ K (assuming 1000 km s⁻¹ outflow velocity and still $\sim {10}^{5}$ K if we assume the faster outflow). Such brightness temperatures are very difficult to produce with thermal emission alone and can be most easily explained as synchrotron emission (Taylor et al. 1987; Weston et al. 2016b).

This type of initial radio bump has been seen in several other novae, including QU Vul (Taylor et al. 1987), V1723 Aql (Krauss et al. 2011; Weston et al. 2016a), and V5589 Sgr (Weston et al. 2016b), and we are beginning to develop theories to explain such behavior. These previous works, along with theoretical analysis by Metzger et al. (2014) and Vlasov et al. (2016), postulate that the initial radio maxima could be either synchrotron emission or (unusually hot) thermal free–free emission.

To explain the initial maximum as thermal emission, rather extraordinary conditions are needed. Both Taylor et al. (1987) and Metzger et al. (2014) invoke a strong shock as a means of generating hot, free–free emitting gas. Note that the gas would not only need to be hot ($\gt {10}^{5}$ K) but also dense, as it would need to be optically thick to produce the initial maximum. Such large amounts of high-temperature gas would also produce significant X-ray emission, which was not observed in V1324 Sco. This could be explained if there is a high column density of material absorbing the X-ray-emitting region (see Section 5). However, for low-velocity (${v}_{\mathrm{sh}}\lesssim 1500$ km s⁻¹) shocks expanding into dense media, like internal shocks in novae, cooling is very efficient and drives the post-shock gas temperature to $T\sim {10}^{4}$ K (Metzger et al. 2014; Derdzinski et al. 2017). This makes it difficult to achieve the $\sim {10}^{5}\mbox{--}{10}^{6}$ K gas necessary for the initial radio bump to be explained by thermal emission.

A non-thermal explanation for the initial radio maximum is preferred by Vlasov et al. (2016), as synchrotron emission is an elegant explanation for brightness temperatures substantially in excess of 10⁴ K. A peak in the radio light curve could be produced by synchrotron emission suffering free–free absorption (or perhaps via the Razin–Tsytovich effect; see also Taylor et al. 1987). In this scenario, on the rise to maximum, the spectral index is predicted to be $\alpha =2$, the light curve peaks when the optical depth is of order unity, and during the optically thin decline from maximum, the spectral index would be $\alpha =-0.5$ to −1.0 (Vlasov et al. 2016). While such evolution of the spectral index is widely seen in supernovae (e.g., Chevalier 1982; Weiler et al. 2002), the spectral index evolution during V1324 Sco's first radio maximum looks quite different. The spectral index never drops below $\alpha \approx 0.2$ (day +88; Figure 4). See the top panel of Figure 14 in Vlasov et al. (2016) for an illustration of how free–free absorbed synchrotron emission provides a problematic fit to the radio spectral energy distribution during the decline from V1324 Sco's initial radio maximum.

A similar radio spectral index evolution, combined with high brightness temperatures, has now been seen in several other novae, and a synchrotron explanation is favored over a thermal one (Taylor et al. 1987; Krauss et al. 2011; Vlasov et al. 2016; Weston et al. 2016b). However, the physical explanation for a relatively flat (non-negative) spectral index on the decline from initial maximum, when the emission is expected to be optically thin, remains a mystery. Perhaps yet-unexplored physics is affecting the energy spectrum of non-thermal leptons in novae, making the spectrum flatter than predicted by models of diffusive shock acceleration (Bell 1978; Blandford & Ostriker 1978). Regardless of a thermal or synchrotron origin, the initial radio maximum in V1324 Sco is a clear indication of shocks in the months following outburst.

After this initial radio bump, a second radio maximum occurred, starting sometime around 2012 September 15 (day +106). It first appeared at high frequencies and progressed toward lower frequencies. During this second radio maximum, V1324 Sco peaked at 6.8 mJy at high frequency (36 GHz) on day +278 and at ∼1.0 mJy for low frequency (4.5 GHz) on day +422.

The evolution of the second radio maximum is consistent with the "standard" picture of radio emission in novae—namely, thermal emission from the 10⁴ K expanding ejecta (Seaquist & Palimaka 1977; Hjellming et al. 1979). This portion of V1324 Sco's radio light curve is similar to the other novae that have been studied in the radio (e.g., Seaquist & Palimaka 1977; Hjellming et al. 1979; Chomiuk et al. 2012a; Nelson et al. 2014; Weston et al. 2016a).

The spectral index is steep on the rise to second maximum, reaching $\alpha =1.6$ on day +164 (once there has been time for the initial radio bump to fade away). By day +323, there is clear evidence that the radio spectrum is flattening at higher frequencies (Figure 4). This spectral turnover cascades to lower frequencies, until by day +640, the radio spectrum is consistent with optically thin free–free emission ($\alpha =-0.1$). This spectral index evolution is consistent with expectations for free–free emission from expanding thermal ejecta (Seaquist & Bode 2008).

The power-law rise to second maximum is also consistent with expectations for an isothermal sphere expanding at constant velocity (${f}_{\nu }\propto {t}^{2};$ Seaquist & Bode 2008). The rise to second maximum at 7.5 GHz is well approximated by a power law with index 2 (assuming that day 0 is 2012 June 1). The rise to second maximum at 17.5 GHz is a bit shallower (power-law index of 1.7), and this difference is likely attributable to the more substantial effect of the first radio maximum on the light curve between days +100–200 (Figure 3).

We therefore modeled the second radio bump as thermal emission from the expanding nova ejecta. We fit the radio data observed after day +106 using the standard model of Hjellming et al. (1979). Specifically, we utilize a homologously expanding "Hubble flow" model, where the fastest ejecta are found at largest radii and throughout the ejecta $v\propto r$. The ejecta are bounded at an inner and outer radius, and we refer to the ratio between these as ξ. In between these inner and outer radii, we estimate an ${r}^{-2}$ density profile (for more details on this model, see Seaquist & Palimaka 1977). The other physical quantities that go into the Hubble flow model are ejecta mass, maximum ejecta velocity, filling factor, temperature, and distance. More details on this radio light curve model and interplay between these variables can be found in Appendix A.

Figure 6 shows the fit to the second peak in the radio light curve using this model. The reduced chi-squared value fit for this model is ${\chi }^{2}/\nu =3.36$. The fitting scheme was error weighted, which partially explains why the highest and lowest frequencies are not fit as well. Further, by construction, the model has a spectral index of $\alpha =2.0$ during the rise, as this is the spectral index of optically thick thermal emission in the Rayleigh–Jeans tail. As can be seen in Figure 4, we never observe a spectral index this high; during the rise to second optical maximum, we observe $\alpha =1.0\mbox{--}1.6$. This discrepancy between observed and predicted spectral index is common among novae (Roy et al. 2012; Chomiuk et al. 2014a; Nelson et al. 2014; Wendeln et al. 2017), and currently lacks a suitable explanation. Clearly, V1324 Sco is another data point illustrating that this discrepancy requires more attention.

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Despite discrepancies in the spectral index on the rise, the flux density and timescale of the second peak reveal important information on the mass and energetics of the explosion. We can derive physical parameters for the ejecta—ejected mass (${M}_{\mathrm{ej}}$) and total ejecta kinetic energy (${\mathrm{KE}}_{\mathrm{ej}}$), as well as the distance—from the model fit, with some assumptions. We assume the canonical temperature of photoionized gas—10⁴ K (Osterbrock 1989; Cunningham et al. 2015). This ejecta temperature is not only theoretically predicted but has been observed in resolved radio images of novae (e.g., Taylor et al. 1988; Hjellming 1996). We also take the maximum ejecta velocity to be $2600\pm 260\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (see Section 4.2) and a volume filling factor of ${f}_{V}\,=$ $(2.1\pm 0.7)\times {10}^{-2}$ (see Appendix B).

The physical quantities derived from the light curve fit are

$$\begin{eqnarray}&&D=14.8\pm 1.6\,{\rm{k}}{\rm{p}}{\rm{c}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{M}_{{\rm{e}}{\rm{j}}}=(1.8\pm 0.6)\times {10}^{-5}\,{M}_{\odot },\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{\rm{K}}{\rm{E}}}_{{\rm{e}}{\rm{j}}}=(3.8\pm 2.0)\times {10}^{45}\,{\rm{e}}{\rm{r}}{\rm{g}},\end{eqnarray} \tag{ 4 }$$

where the uncertainties quoted are 1σ values. The uncertainties in the filling factor have been propagated through this estimate and are included in the error bars. For both the derived distance and the total kinetic energy, the dominant source of uncertainty comes from ${v}_{max};$ ${\mathrm{KE}}_{\mathrm{ej}}$ has a very strong dependence on the maximum velocity (${\mathrm{KE}}_{\mathrm{ej}}\propto {v}_{\max }^{4})$. The uncertainty in the derived mass is dominated by the uncertainty in the filling factor measurement (although the uncertainty in ${v}_{\max }$ is still non-negligible).

Let us now consider how these derived values depend on our assumptions. In Equation (17), we see that for a fixed T_e and ${v}_{\max }$, ${M}_{\mathrm{ej}}/\sqrt{{f}_{V}}$ is also fixed. The filling factor can therefore be understood as a factor that only affects the derived ejecta mass and has no other effect on the radio light curve. If the filling factor were to decrease by an order of magnitude, it would decrease the derived ejecta mass by a factor of ∼3.

We now consider how ejecta mass depends on distance. Equation (16) implies that at least one of D, ${v}_{\max }$, and T_e needs to be left free to vary in order to provide a suitable fit to a light curve. If we fix D and instead let ${v}_{\max }$ vary, we find

$$\begin{eqnarray}&&{v}_{\max }=D{{\rm{\Psi }}}^{1/2}{T}_{e}^{-1/2}.\end{eqnarray} \tag{ 5 }$$

(see Appendix A for a discussion of Ψ). We can fix D at the minimum possible distance, $D=6.5\,\mathrm{kpc}$ (Finzell et al. 2015), and maintain ${T}_{e}={10}^{4}$ K; then, the implied maximum ejecta velocity is 1150 km s⁻¹ (consistent with the velocity of the P Cygni absorption trough in early spectra; Figure 7). The lack of observed velocities >2600 km s⁻¹ implies that V1324 Sco is not located farther than 15 kpc away, unless its thermal ionized ejecta are somehow substantially cooler than 10⁴ K (which we consider very unlikely; e.g., Cunningham et al. 2015). A velocity of 1145 km s⁻¹ in turn implies an ejecta mass almost an order of magnitude lower, $2.3\times {10}^{-6}$ ${M}_{\odot }$.

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It should be noted that, during the dust event, we expect some fraction of the nova ejecta to cool, recombine, and become neutral. Since neutral particles will not emit free–free emission—or, at least for atoms with significant dipole moments, they will emit significantly less free–free emission than ionized particles—we do not expect this mass to show up in the radio emission. However, the bulk of the second radio maximum occurs after the dust event, when the ionization of the gas should be increasing from a minimum around day +70 and approaching a photoionized equilibrium with temperature, 10⁴ K (Cunningham et al. 2015).

Despite uncertainties, radio light curves remain one of the most robust ways to estimate the ejecta masses of classical novae (Seaquist & Bode 2008). We conclude that, given measured ejecta velocities in excess of 2000 km s⁻¹ and the lower limit on the distance, our measurements imply an ejecta mass for V1324 Sco of a few $\times {10}^{-5}$ ${M}_{\odot }$.

All spectroscopic observations—including date, telescope, and observer—are listed in Table 3. Spectra are shown in Figures 8–10. Note that all plots have been corrected to put spectra into the heliocentric frame.

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Table 3.  Optical Spectroscopic Observations

UT Date	$t-{t}_{0}$	Observer	Telescope	Instrument	Dispersion	Wavelength Range
	(Days)				(Å)	(Å)
2012 Jun 04.0	$+3.0$	Bensby	VLT	UVES	0.02	3700–9500
2012 Jun 08.5	$+7.5$	Bohlsen	Vixen VC200L	LISA	0.5	3800–8000
2012 Jun 14.5	$+13.5$	Bohlsen	Vixen VC200L	LISA	0.5	3800–8000
2012 Jun 18.5	$+17.5$	Bohlsen	Vixen VC200L	LISA	0.5	3800–8000
2012 Jun 20.9	$+19.9$	Buil	0.28 m Celestron	LISA	∼0.6	3700–7250
2012 Jun 21.2	$+20.2$	Walter	SMARTS 1.5 m	RC	∼5.5	3240–9500
2012 Jun 23.1	$+22.1$	Walter	SMARTS 1.5 m	RC	∼1.0	5620–6940
2012 Jun 24.9	$+23.9$	Buil	0.28 m Celestron	LISA	6.2	3700–7250
2012 Jun 25.1	$+24.1$	Walter	SMARTS 1.5 m	RC	∼1.5	3650–5420
2012 Jul 03.0	$+32.0$	Walter	SMARTS 1.5 m	RC	∼1.5	3650–5420
2012 Jul 07.1	$+36.1$	Walter	SMARTS 1.5 m	RC	∼1.0	5620–6940
2012 Jul 11.1	$+40.1$	Walter	SMARTS 1.5 m	RC	∼5.5	3240–9500
2012 Jul 15.0	$+44.0$	Walter	SMARTS 1.5 m	RC	∼1.0	5620–6940
2012 Jul 16.1	$+45.1$	Chomiuk	Clay Magellan	MIKE	0.035	3700–9200
2012 Jul 19.0	$+48.0$	Walter	SMARTS 1.5 m	RC	∼5.5	3240–9500
2013 May 20.0	$+353.0$	Wagner	LBT	MODS1	∼3.5	3420–10000
2013 Aug 04.0	$+450.0$	Chomiuk	SOAR	Goodman	∼1.0	3000–7000

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The details of the data reduction for the UVES and MIKE data can be found in Finzell et al. (2015) and Walter et al. (2012) for the RC Spectrograph data. The SOAR Goodman data were taken using a 400 lines/mm grating centered at 5000 Å and were reduced using the standard procedure in IRAF³⁴ with optimal extraction and wavelength calibration using FeAr arcs.

An optical spectrum was obtained on 2013 May 20.4 UT (day 353) using the 8.4 m Large Binocular Telescope (LBT) and Multi-Object Double Spectrograph (MODS1). Observing conditions were photometric, but the seeing as measured from two independent sources ranged from 18 to 19 at the start of the observation. MODS1 utilized a 08 entrance slit (so there was some loss of light at the entrance slit) and G400L (blue channel; 3200–5800 Å) and G670L (red channel; 5800–10,000 Å) gratings, giving a final dispersion of 05 per pixel. The com-bined spectrum covers the range 3420–10000 Å at a spectral resolution of 3.5 Å. The spectra of quartz–halogen and HgNeArXe lamps enabled the removal of pixel-to-pixel and other flat-field variations in response and provided wavelength calibration, respectively. Spectra of the spectrophotometric standard star BD+33 2642 were obtained to measure the instrumental response function and provide flux calibration of the V1324 Sco spectra. The spectra were reduced using a set of custom routines to remove the bias from the detectors and provide flat-field correction and then using IRAF for spectral extraction and wavelength and flux calibration.

In the case of the spectra taken by C. Buil and T. Bohlsen, both observers used a LISA spectrograph attached to commercially available telescopes of different sizes (0.28 m Celestron for Buil; 0.22 m Vixen VC200L for Bohlsen). More information about their observations can be found on their Web sites.^{35 ,36}

As seen in Figure 7, there were strong P Cygni absorption profiles starting at least as early as day +3. The Hα emission component peaked at $\sim -180\,\mathrm{km}\,{{\rm{s}}}^{-1}$ on day +3 and had an FWHM of $\sim 800\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The entirety of Hα, including both the emission feature and the P Cygni absorption, extended out to $-1100\,\mathrm{km}\,{{\rm{s}}}^{-1}$ in the blue, or $900\,\mathrm{km}\,{{\rm{s}}}^{-1}$ from the line center. We take the P Cygni absorption profile to be coming from the fastest material, meaning that—at this early time—the maximum expansion velocity was $\sim 900\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The second most prominent features in the early spectra—aside from the hydrogen lines—are the Fe ii lines, all of which showed P Cygni profiles. This is evident in Figures 8 and 9, which show the time evolution of the blue (3850−4950 Å) and red (5700−6400 Å) spectral regions, respectively.

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The spectrum obtained on day +13 shows the Hα line profile clearly broadening (Figure 7). Note that this is also the time when the light curve flattens out and stays at roughly constant brightness for the next month (Section 2.2). Between days +13–35, we see the emission wings of the Hα line expand to ±2600 km s⁻¹ from the line center. We also see the P Cygni absorption trough move blueward during this time. We discuss the physical implications of this line broadening further in Section 4.3.

Just a few days after the Magellan MIKE spectrum (day +45), V1324 Sco underwent a massive dust dip lasting for ∼50 days. Although the light curve did eventually rebound out of the dust dip, there was only a brief window of $\lesssim 25$ days before V1324 Sco went into solar conjunction. As a result, our spectroscopic coverage did not pick back up until 2013 May 20—355 days after outburst—well into the nebular phase. As seen in Figure 12, the strongest lines in the nebular phase are the [O iii] lines at 5007 and 4959 Å, followed by Hα and [Fe vii] at 6084 Å.

V1324 Sco is an Fe ii-type nova (Williams et al. 1991), due to the prominence of the Fe ii spectral features—second only to the Balmer features—during optical maximum. The Fe ii-type classification is common among D-type novae, including FH Ser, NQ Vul, and QV Vul (see Strope et al. 2010 and references within). It is also notable that all Fermi-detected novae to date have been of the Fe ii type—see V1369 Cen (Izzo et al. 2013), V5668 Sgr (Williams et al. 2015), V339 Del (Tajitsu et al. 2015), and V5856 Sgr (Luckas 2016; Rudy et al. 2016).

Looking at Figure 7, it is clear that the Balmer line profile changes as a function of time. This type of line profile evolution is common among novae (Payne-Gaposchkin 1957; McLaughlin 1960; for some recent examples; see, e.g., Skopal et al. 2014; Surina et al. 2014; Shore et al. 2016). The spectroscopic velocities for Hα and Hβ are plotted in Figure 11, along with the photometric light curve for comparison purposes. Velocities quoted are half-width at half-maximum (HWHM) measured for Hα and Hβ. Because a number of the spectra taken by Walter et al. were either blue (3650–5420 Å) or red (5620–6940 Å), we chose to use both of these features to maximize the number of velocity measurements. The HWHM was measured by fitting a Gaussian profile to the emission lines using the IRAF routine splot. Uncertainties in HWHM were found by adding in quadrature both the uncertainty in the line measurement—found by measuring the line multiple times in splot—and the (average) dispersion of the spectrum.

In V1324 Sco, the width of the Balmer lines increases around the time that gamma-rays are first detected (day +14). The HWHM velocity then varies, but stays at a large value ($\lesssim 1500$ km s⁻¹) over the time period when gamma-rays are observed (until day +31, shown in the top panel of Figure 11). Another spike is seen in the velocity evolution around day +40, and then the velocity appears to decline as the nova transitions to its nebular phase.

The profile evolution of the Balmer lines implies that there is relatively slow-moving material in the outer parts of the ejecta surrounding faster internal material. This conclusion is common in studies of classical novae and is by no means peculiar to V1324 Sco (e.g., McLaughlin 1960; Friedjung 1966; O'Brien et al. 1994). In V1324 Sco, P Cygni profiles apparent in early spectra imply that the outer, slow component is expelled at ∼1000 km s⁻¹ (day +3 in Figure 11). Over the next couple of weeks, a faster component of ejecta becomes visible, reaching velocities of ∼2600 km s⁻¹. The delayed appearance of this fast component implies that it must be internal to the slow component (possibly because it is launched later, or over a longer period of time). Inevitably, the internal fast component will catch up with the external slow component, producing shocks (and gamma-rays; see Section 6.2). Therefore, from the line profile evolution of V1324 Sco, we estimate that the differential velocity in the shock is ∼1600 km s⁻¹.

It is unclear if the temporal correspondence between the broadening of the optical emission lines and the appearance of gamma-rays is meaningful or coincidental. The optical emission lines of novae typically broaden in the weeks following optical maximum, as they transition from the principal line profile to show diffuse-enhanced line systems (McLaughlin 1960). It is possible that the fast, internal component is present in the nova practically since the start of the outburst, but only becomes visible as the outer parts of the ejecta expand and drop in optical depth. However, the temporal coincidence between optical line broadening and gamma-ray turn-on is striking and could hint that the fast component is not launched until ∼13 days into the outburst. A similar evolution can be seen in the Hα profile of another gamma-ray nova, V339 Del. Figure 4 of Skopal et al. (2014) shows that the wings of the Hα profile began to increase on 2013 August 18 (date of the first gamma-ray detection).

We can also use the spectroscopic observations to determine the properties of the ejecta density in V1324 Sco. Specifically, we use the late-time (nebular) spectroscopy to measure density inhomogeneities (i.e., clumpiness) in the ejecta, which we parameterize in terms of the volume filling factor (f_V). Such inhomogeneities must be taken into account in order to get a proper mass estimate, and we incorporate the filling factor in our radio ejecta mass derivation in Section 3.3. For detailed calculations of V1324 Sco's filling factor, see Appendix B. We use measurements of the [O iii] lines to find a filling factor of ${f}_{V}=(2.1\pm 0.7)\times {10}^{-2}\,$. This is similar to the filling factor measured in the gamma-ray-detected nova V339 Del (${f}_{V}=0.07\mbox{--}0.2;$ Shore et al. 2016).

We also use the O i lines measured on day +45—permitted transitions at 7774 and 8446 Å, and the forbidden transition at 6300 Å—to constrain the column density (for at least some portions) of the ejecta (Kastner & Bhatia 1995; Williams 2012; see Appendix C for detailed calculations). If we assume a temperature of T_e = 10,000 K, the O i ratios are consistent with density $\mathrm{log}({N}_{e}/[{\mathrm{cm}}^{-3}])\gt 10$. Assuming that the density scales like ${t}^{-3}$, we would expect the density to be a factor of ∼10 times greater during the first X-ray observation (day +21) than it was on day +45. Combined with the fact that we expect the ejecta to have expanded to ∼a few $\times {10}^{14}$ cm, we derive a column density $\gtrsim {10}^{24}$ cm⁻³. As discussed in Section 5, such a high column density can explain the lack of hard X-ray emission.

In this section, we argue that all of the non-gamma-ray observational signatures of V1324 Sco are typical of classical novae, in the sense that all observational features have been seen in previous novae.

This point is relevant as there has been some discussion in the community that gamma-ray-luminous systems like V1324 Sco may not be classical novae at all, but may instead belong to the class of intermediate-luminosity transients often called LRNs (e.g., Blagorodnova et al. 2017). An LRN, observationally, appears with persistently redder colors than a classical nova and luminosities that range from slightly fainter than classical novae to several magnitudes more luminous (e.g., Kimeswenger et al. 2002; Smith et al. 2016). The physical interpretation of the LRN optical/IR outburst is the merger of a close binary (Tylenda et al. 2011; Ivanova et al. 2013; MacLeod et al. 2017). V1309 Sco is one of the canonical and best studied LRNs, and its optical light curve is similar to that of V1324 Sco. Both have an initial, slow, monotonic rise; both have a flattening of the optical light curve near peak; and both have a significant dust event (Tylenda et al. 2011).

We find, however, that an LRN does not fit with the other observational characteristics of V1324 Sco. Unlike in LRNs, the spectra of V1324 Sco evolve to show higher ionization species in the months following outburst (Figure 12). Additionally, V1324 Sco's ejecta velocities of a few thousand km s⁻¹ would be unusually high for an LRN, which typically shows velocities of a few hundred km s⁻¹ (Munari et al. 2002; Mason et al. 2010).

In addition, from the radio light curve of V1324 Sco, we estimate an ejecta mass of a few $\times {10}^{-5}\,{M}_{\odot }$ (see Section 3.3). This is at least three orders of magnitude smaller than what is expected for LRN events, which are thought to expel a significant fraction of a solar mass (Ivanova et al. 2013; MacLeod et al. 2017). For the radio light curve to be consistent with $\gt {10}^{-1}$ ${M}_{\odot }$, V1324 Sco would need to be much farther away than 15 kpc and expanding at a velocity $\ll 1000$ km s⁻¹. Such low velocities are implausible given the observed optical line profiles of V1324 Sco (Figure 7). In addition, when the ejecta mass is combined with the ejecta velocity, we find that the kinetic energy of the outburst of V1324 Sco was ${10}^{44}\mbox{--}{10}^{45}$ erg, typical for a classical nova. LRNs, on the other hand, have kinetic energies $\sim {10}^{47}\mbox{--}{10}^{48}$ erg (e.g., Nandez et al. 2014; MacLeod et al. 2017; Metzger & Pejcha 2017).

In addition, we detect a photometric modulation in the optical light curve during the power-law decline phase, which probably reflects the orbital period. We used the MOA data set, which had the best sampling as well as the highest cadence; we limited the data set to $\gt 5\sigma$ detections. The periodic modulation was measured using the Lomb-Scargle algorithm in the Python scientific library SciPy. We found it to be 3.8 hr, consistent with that observed in the precursor rise (Figure 13). Assuming this periodicity reflects the underlying host binary, the detection of such a modulation both before and after outburst implies that the binary was not destroyed in the nova event. Contrast this observation with measurements made for V1309 Sco, where Tylenda et al. (2011) watched the period dramatically decrease in the lead up to outburst, and then all periodic modulation disappeared. We therefore conclude that V1324 Sco is a classical nova, with host properties, ejecta mass, and kinetic energy consistent with other novae.

Having established in the previous section that V1324 Sco is a classical nova, we now discuss why this nova had such a high gamma-ray luminosity. As can be see in Cheung et al. (2016), V1324 Sco is the most gamma-ray-luminous classical nova discovered to date. Note, however, that Cheung et al. assume a distance of 4.5 kpc to V1324 Sco, while Finzell et al. (2015) showed that its distance is substantially farther (>6.5 kpc). Therefore, V1324 Sco is even more gamma-ray luminous than presented in Cheung et al. (2016), registering at ${L}_{\gamma }\gtrsim 2\times {10}^{36}$ erg s⁻¹.

In V1324 Sco, there is strong observational evidence for shock interaction, both from the gamma-rays and from the radio (in particular, the initial maximum; Section 3.2). We present here a simplified model of the ejecta that can explain the gamma-ray emission.

We can imagine the ejecta as being composed of two parts: a slow initial component and a fast secondary component. When these two components meet, there will be both a forward and reverse shock, and it is these shocks that will power the gamma-ray emission. We further assume that these shocks are radiative, so we expect there to be a layer of cold material between the forward and reverse shocks. This analysis is based on the models of Metzger et al. (2014) and Vlasov et al. (2016).

Initially in the outburst, a slow component is expelled. We model it to be an impulsive ejection—defined by Metzger et al. (2014) to have the density profile

$$\begin{eqnarray}&&{n}_{{\rm{s}}}(r)=\left(\displaystyle \frac{{\dot{M}}_{{\rm{s}}}}{4\pi {f}_{{\rm{\Omega }}}{v}_{{\rm{s}}}{r}^{2}\mu {m}_{p}}\right)\exp \left[-\displaystyle \frac{r}{{R}_{{\rm{s}}}}\right],\end{eqnarray} \tag{ 6 }$$

where ${\dot{M}}_{{\rm{s}}}$ is the slow component mass-loss rate, μ is the mean molecular weight, ${f}_{{\rm{\Omega }}}$ is the solid angle fraction that is subtended by the slow component, ${v}_{{\rm{s}}}$ is the velocity of the slow component, and ${R}_{{\rm{s}}}$ is the radius of the slow component (${R}_{{\rm{s}}}={v}_{{\rm{s}}}{t}_{{\rm{s}}}$, where ${t}_{{\rm{s}}}$ is the time since the slow component was ejected).

Thereafter, a faster component is expelled in a wind-like process. The fast component's density profile is given by

$$\begin{eqnarray}&&{n}_{{\rm{f}}}(r)\approx \displaystyle \frac{{\dot{M}}_{{\rm{f}}}}{4\pi {v}_{{\rm{f}}}{r}^{2}\mu {m}_{p}},\end{eqnarray} \tag{ 7 }$$

where ${\dot{M}}_{{\rm{f}}}$ is the fast component mass-loss rate and ${v}_{{\rm{f}}}$ is the velocity of the fast component. For V1324 Sco, we take ${v}_{{\rm{s}}}=1000$ km s⁻¹ and ${v}_{{\rm{f}}}=2600$ km s⁻¹ (Section 4.3).

The fast wind then impacts the slow component and shock interaction ensues. Assuming that the shock is radiative (Metzger et al. 2014), there will be a cold layer of material between the forward and reverse shocks (note that this is also the region where dust will form; Derdzinski et al. 2017). We denote the mass of this cold shell as ${M}_{\mathrm{shell}}$, which grows in time as

$$\begin{eqnarray}&&\displaystyle \frac{{{dM}}_{\mathrm{shell}}}{{dt}}={f}_{{\rm{\Omega }}}{\dot{M}}_{{\rm{f}}}\left(\displaystyle \frac{{v}_{{\rm{f}}}-{v}_{\mathrm{shell}}}{{v}_{{\rm{f}}}}\right)+{\dot{M}}_{{\rm{s}}}\left(\displaystyle \frac{{v}_{\mathrm{shell}}-{v}_{{\rm{s}}}}{{v}_{{\rm{s}}}}\right),\end{eqnarray} \tag{ 8 }$$

while the momentum grows as

$$\begin{eqnarray}&&\displaystyle \frac{d}{{dt}}({M}_{\mathrm{shell}}{v}_{\mathrm{shell}})={f}_{{\rm{\Omega }}}\dot{{M}_{{\rm{f}}}}({v}_{{\rm{f}}}-{v}_{\mathrm{shell}})+{\dot{M}}_{{\rm{s}}}({v}_{\mathrm{shell}}-{v}_{{\rm{s}}}),\end{eqnarray} \tag{ 9 }$$

where ${\dot{M}}_{{\rm{s}}}$ is evaluated at the radius r according to its value a time $t\approx r/{v}_{{\rm{s}}}$ before the onset of the fast wind. A steady state solution (i.e., ${{dv}}_{\mathrm{shell}}/{dt}=0$) is soon reached, wherein the shell gains most of its momentum from the fast wind and most of its mass by sweeping up the slow shell. In this limit ${v}_{{\rm{f}}}\gg {v}_{{\rm{s}}}$ and ${\dot{M}}_{{\rm{f}}}\lesssim {\dot{M}}_{{\rm{s}}}{f}_{{\rm{\Omega }}}^{-1}$ we find that

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{\mathrm{shell}}}{{v}_{{\rm{s}}}}\approx {\left(\displaystyle \frac{{\dot{M}}_{{\rm{f}}}{v}_{{\rm{f}}}{f}_{{\rm{\Omega }}}}{{\dot{M}}_{{\rm{s}}}{v}_{{\rm{s}}}}\right)}^{1/2}.\end{eqnarray} \tag{ 10 }$$

The radius of the shell and the accompanying shock increases with time approximately as ${R}_{\mathrm{shell}}\approx {v}_{\mathrm{shell}}t$ as it passes through the slow ejecta.

The power dissipated by the shocks is determined by the number of thermal particles swept up by the shock, which can be expressed as

$$\begin{eqnarray}&&{\dot{E}}_{{\rm{r}}}=\displaystyle \frac{9}{32}{f}_{{\rm{\Omega }}}\displaystyle \frac{{\dot{M}}_{{\rm{f}}}}{{v}_{{\rm{f}}}}{({v}_{{\rm{f}}}-{v}_{\mathrm{shell}})}^{3},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\dot{E}}_{{\rm{f}}}=\displaystyle \frac{9}{32}\displaystyle \frac{{\dot{M}}_{{\rm{s}}}}{{v}_{{\rm{s}}}}{({v}_{\mathrm{shell}}-{v}_{{\rm{s}}})}^{3},\end{eqnarray} \tag{ 12 }$$

where E_r and E_f are the power dissipated at the reverse and forward shocks, respectively. Since usually ${v}_{\mathrm{shell}}\ll {v}_{{\rm{f}}}$, the shock power (and gamma-ray luminosity) will be dominated by the reverse shock,

$$\begin{eqnarray}&&{L}_{\gamma }\propto {\dot{E}}_{{\rm{r}}}=\displaystyle \frac{9}{32}{f}_{{\rm{\Omega }}}{\dot{M}}_{{\rm{f}}}{({v}_{{\rm{f}}}-{v}_{\mathrm{shell}})}^{2}\approx \displaystyle \frac{9}{32}{f}_{{\rm{\Omega }}}{\dot{M}}_{{\rm{f}}}{v}_{{\rm{f}}}^{2}.\end{eqnarray} \tag{ 13 }$$

To determine the amount of time that the gamma-ray emission will persist—hereafter referred to as ${t}_{\gamma }$—we need to find the amount of time it will take for the shock to cross the initial slow component, i.e., ${R}_{\mathrm{shell}}\approx {v}_{\mathrm{shell}}{t}_{\gamma }={R}_{{\rm{s}}}={v}_{{\rm{s}}}{t}_{{\rm{s}}}$. Rewriting this using our expression for the shell/shock velocity (Equation (10)), we find

$$\begin{eqnarray}&&{t}_{\gamma }=\displaystyle \frac{{R}_{{\rm{s}}}}{{v}_{\mathrm{shell}}}={\left(\displaystyle \frac{{\dot{M}}_{{\rm{s}}}{v}_{{\rm{s}}}}{{\dot{M}}_{{\rm{f}}}{v}_{{\rm{f}}}{f}_{{\rm{\Omega }}}}\right)}^{1/2}{t}_{{\rm{s}}}.\end{eqnarray} \tag{ 14 }$$

Using Equation (13), we find an approximate proportionality,

$$\begin{eqnarray}&&{t}_{\gamma }\propto {\left(\displaystyle \frac{{\dot{M}}_{{\rm{s}}}{v}_{{\rm{s}}}{v}_{{\rm{f}}}}{{L}_{\gamma }}\right)}^{1/2}{t}_{{\rm{s}}}.\end{eqnarray} \tag{ 15 }$$

From the above, we see that increasing either ${\dot{M}}_{{\rm{f}}}$ or ${v}_{{\rm{f}}}$ will increase ${L}_{\gamma }$ while generally decreasing ${t}_{\gamma }$ (for fixed values of ${\dot{M}}_{{\rm{s}}}$ and ${v}_{{\rm{s}}}$). Such an inverse relationship between the gamma-ray luminosity and the duration of the gamma-ray emission has been observed by Cheung et al. (2016).

The gamma-ray luminosity is proportional to the square of the relative velocity between the fast component and the central shell (Equation (13)), while ${v}_{\mathrm{shell}}$ is itself proportional to the velocity of the slow component. Therefore, one possibility is that the differential velocity between the fast and slow com-ponents is unusually large in V1324 Sco. We crudely estimate the differential velocity as ${v}_{{\rm{f}}}-{v}_{{\rm{s}}}=(2600\mbox{--}1000)$ km s⁻¹ =1600 km s⁻¹. It is still early to compare this quantity with many of the other Fermi-detected novae, but sufficient studies have been published to consider V959 Mon and V339 Del.

The optical spectroscopy of V339 Del has been studied by Shore et al. (2016); from their Figure 2, we estimate that a slow component is visible around day +5 with a velocity ${v}_{{\rm{s}}}\,\approx$ 1400 km s⁻¹. Later on, around day +22, a fast wing appears on the Hγ profile extending out to ${v}_{{\rm{f}}}\,\approx$ 1900 km s⁻¹. In this case, both the fast component and the differential velocity are slower than in V1324 Sco, which might explain why V339 Del is a factor of $\gtrsim 4$ less luminous in gamma-rays (Cheung et al. 2016; after taking into account the distance limit on V1324 Sco from Finzell et al. 2015).

A direct comparison between V1324 Sco and V959 Mon, using the evolution of optical spectra, is impossible due to the fact that V959 Mon was in solar conjunction for the first months of its outburst. We can, however, use a combination of nebular spectroscopy and imaging of the nova ejecta to infer the velocities of the slow and fast components. Ribeiro et al. (2013) model the nebular spectrum as expanding bipolar ejecta and find a maximum velocity of 2400 km s⁻¹ (rather similar to V1324 Sco). Linford et al. (2015) imaged the expanding ejecta at multiple times and frequencies using the VLA, and combining it with Ribeiro et al.'s model, found that the slow component expands at 480 km s⁻¹. According to this analysis, the differential velocity between the fast and slow components in V959 Mon should be even larger than in V1324 Sco, but V959 Mon is a factor of $\gtrsim 10$ less gamma-ray luminous than V1324 Sco (Cheung et al. 2016; taking into account the distance estimate for V959 Mon from Linford et al. 2015).

Another independent avenue to constrain the shock velocity of novae is their radio light curves. V1324 Sco showed an unusually luminous early radio peak, ${L}_{\nu ,\mathrm{pk}}\approx 3\times {10}^{30}$ erg s⁻¹ at $\nu \,\approx$ 10 GHz (Section 3.2), compared to $6\times {10}^{27}$ erg s⁻¹ for V959 Mon and an upper limit of ${L}_{\nu ,\mathrm{pk}}\lesssim 2\times {10}^{28}$ erg s⁻¹ for V339 Del (Chomiuk et al. 2013; Chomiuk et al. 2014a; Linford et al. 2015). According to the model presented by Vlasov et al. (2016), the peak synchrotron luminosity is extremely sensitive to the shock velocity, scaling approximately as ${L}_{\nu ,\mathrm{pk}}\propto {({v}_{{\rm{f}}}-{v}_{\mathrm{shell}})}^{8}$ or steeper (their Equation (48)). Explaining the ≳1–2 order of magnitude greater synchrotron luminosity of V1324 Sco (compared to V339 Del or V959 Mon) therefore requires a shock velocity that is higher by a factor of $\gtrsim 1.3\mbox{--}1.8$. All else being equal, this velocity difference would alone result in a gamma-ray luminosity ${L}_{\gamma }\propto {({v}_{{\rm{f}}}-{v}_{\mathrm{shell}})}^{2}\gtrsim 2\mbox{--}3$ times higher for V1324 Sco than the other two events. On the other hand, V1723 Aql and V5589 Sgr showed early radio peaks similar in luminosity to V1324 Sco but were not detected at all in gamma-ray emission (Krauss et al. 2011; Weston et al. 2016a, 2016b).

Clearly, results are mixed as to how the ejecta velocities of V1324 Sco compare to other gamma-ray-detected novae. It is unclear how valid a comparison V959 Mon provides, given that ejecta velocities are derived using a wholly different method than in V1324 Sco. We require additional spectroscopic studies of gamma-ray-detected novae to provide an appropriate comparison sample with V1324 Sco.

Another possibility for the high gamma-ray luminosity of V1324 Sco is a particularly high mass-loss rate and/or total mass of the fast component, particularly in comparison to the mass in the slow component. From our radio light curve fitting, we do not see an indication that the total mass in V1324 Sco is unusually large. However, from our simple Hubble flow fits, we do not account for multiple components of ejecta and cannot make any claims about the relative mass in the fast and slow components. Because ${L}_{\gamma }\propto {\dot{M}}_{{\rm{f}}}$, the mass-loss rate of the fast component would need only be a factor of ∼5 times higher than in other Fermi-detected novae (for otherwise equal shock velocities) to explain the gamma-ray luminosity of V1324 Sco.

From Berkhuijsen (1998), the filling factor is given by Equation 1(a),

$$\begin{eqnarray}&&{f}_{V}=\displaystyle \frac{\langle {n}_{e}^{2}\rangle }{{n}_{e}^{2}},\end{eqnarray} \tag{ 20 }$$

where $\langle {n}_{e}^{2}\rangle$ is the average of the density squared. This can be rewritten using Equation (4) of the same paper,

$$\begin{eqnarray}&&\langle {n}_{e}^{2}\rangle =\displaystyle \frac{{EM}}{L},\end{eqnarray} \tag{ 21 }$$

where EM is the emission measure and L is the characteristic length of the emitting material. For our purposes, we will assume that the ejecta is spherically symmetric and say that the characteristic length is $2{v}_{\mathrm{ej}}t$.

We can determine the emission measure from hydrogen recombination lines using Equation (36) in Chapter 3 of Spitzer (1978),

$$\begin{eqnarray}&&\int {I}_{\nu }d\nu =h\nu {\alpha }_{mn}\left(\displaystyle \frac{{n}_{p}}{{n}_{e}}\right)\times 2.46\times {10}^{17}{E}_{m},\end{eqnarray} \tag{ 22 }$$

where ${\alpha }_{{mn}}$ is the effective recombination coefficient for transitions from state m to n. We modify this to be

$$\begin{eqnarray}&&\int {I}_{\nu }d\nu =\displaystyle \frac{\int {F}_{\nu }d\nu }{{\rm{\Omega }}}\approx \displaystyle \frac{{F}_{\nu }{\rm{\Delta }}\nu }{{\rm{\Omega }}}=\displaystyle \frac{{F}_{\lambda }{\rm{\Delta }}\lambda }{{\rm{\Omega }}},\end{eqnarray} \tag{ 23 }$$

where Ω is the solid angle of the source, which we approximate as ${(A/D)}^{2}=\pi {(r/D)}^{2}$. Here, r is the ejecta radius, which is just ${v}_{\mathrm{ej}}t$, and D is the distance to the source.

This leads us to the following expression for emission measure, as determined by measuring the flux in Hβ:

$$\begin{eqnarray}&&\mathrm{EM}=\displaystyle \frac{{F}_{\lambda }{\rm{\Delta }}\lambda }{h{\nu }_{{\rm{H}}\beta }{\alpha }_{42}(2.46\times {10}^{17}){\rm{\Omega }}}\,\mathrm{pc}\,{\mathrm{cm}}^{-7}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{{F}_{\lambda }{\rm{\Delta }}\lambda \pi {D}^{2}}{h{\nu }_{{\rm{H}}\beta }{\alpha }_{42}(2.46\times {10}^{17}){({v}_{\mathrm{ej}}t)}^{2}}\,\mathrm{pc}\,{\mathrm{cm}}^{-7}.\end{eqnarray} \tag{ 25 }$$

Using this expression for EM, we can rewrite Equation (21) as

$$\begin{eqnarray}&&\langle {n}_{e}^{2}\rangle =\displaystyle \frac{{F}_{\lambda }{\rm{\Delta }}\lambda \pi {D}^{2}}{2h{\nu }_{{\rm{H}}\beta }{\alpha }_{42}(2.46\times {10}^{17}){({v}_{\mathrm{ej}}t)}^{3}}\,\mathrm{pc}\,{\mathrm{cm}}^{-7}.\end{eqnarray} \tag{ 26 }$$

This expression is in terms of pc ${\mathrm{cm}}^{-7}$, so we must convert it to ${\mathrm{cm}}^{-6}$. To do this, we multiply by $\left(\tfrac{3.086\times {10}^{18}\,\mathrm{cm}}{1\,\mathrm{pc}}\right)$, which gives

$$\begin{eqnarray}&&\langle {n}_{e}^{2}\rangle =\displaystyle \frac{4{\pi }^{2}{F}_{\lambda }{\rm{\Delta }}\lambda {D}^{2}}{2h{\nu }_{{\rm{H}}\beta }{\alpha }_{42}{({v}_{\mathrm{ej}}t)}^{3}}\,{\mathrm{cm}}^{-6}.\end{eqnarray} \tag{ 27 }$$

Finally, we can determine the density by using spectroscopic line ratios. We will use the [O iii] line ratio to determine the density by using Equation (5.4) in Osterbrock (1989):

$$\begin{eqnarray}&&{R}_{[{\rm{O}}{\rm{III}}]}=\displaystyle \frac{{j}_{\lambda 4959}+{j}_{\lambda 5007}}{{j}_{\lambda 4363}}=\displaystyle \frac{7.90\exp (3.29\times {10}^{4}/{T}_{e})}{1+4.5\times {10}^{-4}{n}_{e}/{T}_{e}^{1/2}}.\end{eqnarray} \tag{ 28 }$$

This leads to our expression for n_e,

$$\begin{eqnarray}&&{n}_{e}=\displaystyle \frac{{T}_{e}^{1/2}}{4.5\times {10}^{-4}}\left(\displaystyle \frac{7.90\exp (3.29\times {10}^{4}/{T}_{e})}{{R}_{[{\rm{O}}{\rm{iii}}]}}-1\right)\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 29 }$$

Squaring the above expression and combining it with Equations (20) and (27), we can now write out our expression for the filling factor.

$$\begin{eqnarray}&&{f}_{V}=\left(\displaystyle \frac{2{\pi }^{2}{F}_{\lambda }{\rm{\Delta }}\lambda {D}^{2}}{h{\nu }_{{\rm{H}}\beta }{\alpha }_{42}{({v}_{\mathrm{ej}}t)}^{3}}\right)\\ &&\quad \times \,{\left[\displaystyle \frac{{T}_{e}^{1/2}}{4.5\times {10}^{-4}}\left(\displaystyle \frac{7.90\exp (3.29\times {10}^{4}/{T}_{e})}{{R}_{[{\rm{O}}{\rm{iii}}]}}-1\right)\right]}^{-2}.\end{eqnarray} \tag{ 30 }$$

The unknown values that we need to solve Equation (30) are electron temperature (T_e), distance (D), ejecta velocity (${v}_{\max }$), the oxygen line ratio (${R}_{[{\rm{O}}{\rm{iii}}]}$), and the ${{\rm{H}}}_{\beta }$ flux (${F}_{\lambda }{\rm{\Delta }}\lambda$). We use the LBT spectrum taken on day +353, as it is taken well into the nebular phase and has a better spectral response calibration than the SOAR spectrum. Note that the MODS1 instrument was not designed to be a spectrophotometer, and the seeing was twice the width of the slit, so we believe that $\sim 50 \%$ of the flux fell outside of the slit. This issue is negated for line ratios (discussed below), but it does affect absolute line fluxes. Therefore, we will use a fiducial value of 5% for the uncertainty of the line ratios—to account for general calibration uncertainties—and $50 \%$ uncertainty for the absolute line flux.

With this value for the uncertainty, we use the IRAF tool splot to measure a ${{\rm{H}}}_{\beta }$ flux—corrected for the throughput issue mentioned above—of $8.38\pm 4.19\times {10}^{-15}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. The line ratio ${R}_{[{\rm{O}}{\rm{iii}}]}$ is determined by ${j}_{\lambda 4959}$, ${j}_{\lambda 5007}$, and ${j}_{\lambda 4363}$. We find for these quantities:

• 1.  
  ${j}_{\lambda 4959}=66.2\pm 3.3\times {10}^{-15}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1};$
• 2.  
  ${j}_{\lambda 5007}=218.0\pm 10.9\times {10}^{-15}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1};$
• 3.  
  ${j}_{\lambda 4363}=4.8\pm 0.2\times {10}^{-15}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}.$

As these lines are meant to be a measure of the flux emitted from the source—not the flux measured—we need to make further corrections for interstellar reddening. From Finzell et al. (2015), we know that the reddening is $E(B-V)=1.16\pm 0.12$ for V1324 Sco. We use the wavelength-specific reddening extinction law of Cardelli et al. (1989; Equations (1) and (3)), with R_V = 3.1, to determine the level of extinction. Doing this, we find reddening-corrected fluxes of

• 1.  
  ${j}_{\lambda 4959}=30.5\pm 12.4\times {10}^{-13}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1};$
• 2.  
  ${j}_{\lambda 5007}=95.9\pm 38.2\times {10}^{-13}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1};$
• 3.  
  ${j}_{\lambda 4363}=4.5\pm 2.2\times {10}^{-13}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1},$

and the reddening-corrected Hβ line flux is $4.35\pm 2.84\,\times {10}^{-13}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. Note that the uncertainty on the flux values has increased, due to the inclusion of the reddening uncertainty. Using these reddening-corrected flux values, we find an ${R}_{[{\rm{O}}{\rm{iii}}]}$ value of $29.3\pm 4.4$.

For two of the remaining unknown values—T_e and D—we use the same values throughout the paper (${T}_{e}={10}^{4}$ K, $D\,=$ $6.5\,\mathrm{kpc}$). The remaining value, ${v}_{\mathrm{ej}}$, is derived using the best-fit values to the radio data.

To determine the final value for f_v, we generate distributions of the input variables and plug them into Equation (30), which gives us a distribution of values for f_v. The final value that we quote for f_v is the average of this distribution, and the uncertainty in f_v is the standard deviation of f_v.

We can utilize our distribution of velocities derived in Section 3.3 to help alleviate some of the uncertainty associated with our measured quantities. From this and using our canonical nova temperature of 10⁴ K and distance of $6.5\,\mathrm{kpc}$, we get a filling factor of

$$\begin{eqnarray}&&{f}_{V}=(2.1\pm 0.7)\times {10}^{-2}\,.\end{eqnarray} \tag{ 31 }$$

The uncertainty is dominated by both the reddening value uncertainty and the fiducial flux calibration uncertainty. Note that the filling factor depends on distance as ${f}_{V}\propto {D}^{2}$, so our lower limit on distance implies that Equation (31) is also a lower limit on the filling factor.