Shocks and dust formation in Nova V809 Cep

During the second and third month of the outburst, the nova produced a flare of low-frequency radio emission that brightened more quickly than expected for freely expanding thermal ejecta and reached a brightness that exceeded expectations for thermally emitting material expanding at the speed of the principal optical absorbing system, suggesting that the flare was non-thermal in origin. The first piece of evidence in support of non-thermal emission is the rapid brightening at low frequencies. Between Epochs 2 and 3, the 4.6-GHz emission increased roughly as $t^{6}$, where $t$ here is the time since $t_{0}$. Even if the material that generated the early-time flare was not ejected from the region of the white dwarf until four days after the start of the eruption (Munari et al., 2014), the 4.6-GHz flux density increased faster than $t^{4}$. During the same interval, the 7.4-GHz brightness increased as $t^{4}$. Between Epochs 3 and 4, the 4.6-GHz emission continued to increase faster than $t^{2}$. For a freely expanding blackbody with constant temperature, the brightness of thermal radio emission can only increase as fast as $t^{2}$, proportional to the increasing area of the source on the sky. Although it would be possible for the size of a thermally emitting source to increase faster than $t^{2}$ if an ionization front was rapidly propagating through the ejecta, or if the electron temperature of the emitting material was increasing, the high radio brightness during Epochs 3 and 4 (see discussion below and Fig. 3) disfavors either of these possibilities.

The second piece of evidence in support of non-thermal emission is the strength of the low-frequency radio emission during Epochs 3 and 4. We parameterize the radio flux density in terms of a brightness temperature, $T_{b}$ – the minimum electron temperature required for the observed radio emission to arise from thermal Bremsstrahlung. Brightness temperature is a measure of surface brightness. If a uniform temperature, Bremsstrahlung-emitting medium is optically thick over its full area on the sky, the brightness temperature corresponds to its physical temperature; however, if it is even partially optically thin, or if the radio emission emanates from a region that is smaller than the size of the putative spherical ejecta, then the brightness temperature only provides a lower limit for the physical temperature. Assuming a spherically expanding shell, $T_{b}$ is given by:

$$T_{b}(\nu,t)=\frac{S_{\nu}(t)c^{2}D^{2}}{2\pi k_{b}\nu^{2}(v_{ej}t)^{2}},$$

(1)

where c is the speed of light, $k_{b}$ is the Boltzmann constant, $t$ is the time since $t_{0}$, and $v_{ej}$ is the maximum expansion velocity. Using a distance of D = 6.5 kpc and a maximum ejection velocity of 1200 km/s (corresponding roughly to the faster of the two principal absorption systems seen in optical spectroscopy; Munari et al., 2014), the brightness temperature at 4.6 GHz reached $1.3\pm 0.2\times 10^{5}$ K (Epoch 4 (day 76); see Figure 3). For the early-time radio flare to have had a thermal origin would therefore require that on day 76 (Epoch 4), the ejecta included a shell of material with an electron temperature of at least $10^{5}$ K, that was at least as extended on the sky as the 1200 km/s flow, and that was dense enough to be optically thick at 4.6 GHz. If the radio emission emanated from knots, or a portion of the ejecta that subtended a smaller solid angle on the sky than the 1200 km/s flow, the required electron temperature for the early-time flare to have been thermal is even higher. We argue below that such a large, hot, dense ejecta on day 76 (Epoch 4) seems unlikely.

Crucially, given that $T\sim 10^{4}$ K is the approximate equilibrium temperature for a plasma cooled by forbidden emission lines (Weston et al., 2016a), photoionized ejecta from novae are typically expected to have electron temperatures of around this value. In fact, Cunningham et al. (2015) found that for a wide range of white dwarf masses, ejecta masses, and ejecta speeds, the temperature of the photoionized ejecta generally does not exceed a few times $10^{4}$ K at any point. It is possible that the peak brightness of the early-time radio flare in V809 Cep could have been generated by the most spatially extended flow associated with the so-called diffuse enhanced optical spectroscopic system (which had speeds of up to 2000 km/s, Munari et al., 2014) if it had a temperature of $4\times 10^{4}$ K and was completely optically thick. However, the low-frequency spectral index on day 76 (Epoch 4) was $\alpha=0.5\pm 0.1$, much lower than the $\alpha\approx 2$ expected for optically thick Bremsstrahlung, indicating that the radio-emitting ejecta were not optically thick at that time.

The expected temperature for photoionized ejecta and the $\alpha=0.5$ spectral index thus both argue against a thermal origin for the early-time flare.

The high $T_{b}$ is also unlikely to be due to a shell of shock-heated, $10^{5}$ K gas between the forward and reverse shocks. Steinberg & Metzger (2018) explore the possibility of forward and reverse shocks in the ejecta producing a shell of cool ($\sim 10^{4}K$), dense gas bounded by a shell of $T\sim 10^{6}$ K gas. We define $M_{hot,needed}$ as the mass of $10^{5}$ K gas that would be needed to make a shell of width $L_{cool}$ optically thick at radio frequencies and $M_{hot,expected}$ as the mass of $10^{5}$ K gas theoretically expected to be present. Below, we compare $M_{hot,expected}$ and $M_{hot,needed}$.

The expected geometrical thickness of the shell of hot gas can be approximated as the cooling length ($L_{cool}$; Steinberg & Metzger, 2018):

$$L_{cool}=v_{shock}t_{cool}/4.$$

(2)

Because it is not clear whether the early-time radio flare was the result of the forward or reverse shock, we take $v_{shock}$ to be the velocity of the forward shock. The reverse shock was directed into the fast moving, low density ejecta and thus had a higher velocity than the forward shock  which was directed into higher density ejecta. This lower forward-shock velocity ends up being a more conservative estimate for $v_{shock}$ (see Fig. 4 for a comparison between $M_{hot,needed}$ and $M_{hot,expected}$ as a function of $v_{shock}$). The maximum velocity of the principle absorption system seen in optical spectroscopy was 1200 km/s, and the lowest expansion velocity observed was 600 km/s (Munari et al., 2014); therefore, we assume a shock velocity between 600 km/s and 1200 km/s. As gas behind the shock cools radiatively, $t_{cool}$ is given by (Steinberg & Metzger, 2018):

$$t_{cool}=\frac{\bar{\mathbb{\mu}}(3/2)k_{b}T_{shock}}{\mu_{p}\mu_{e}n_{shock}\Lambda},$$

(3)

where the mean molecular weight is $\bar{\mathbb{\mu}}=0.74$ for a typical composition of a fully ionized classical nova ejecta (Vlasov et al. 2016; Sokolovsky et al. 2020, $\mu_{p}=1.39$, and $\mu_{e}=1.16$. From Steinberg & Metzger (2018), we estimate the value of the cooling function ($\Lambda$) for $10^{5}$ K gas to be $10^{-20}$ cm³/s. Assuming $\bar{\mathbb{\mu}}=0.74$, the temperature of the shock, $T_{shock}$, is related to the shock velocity as (Steinberg & Metzger, 2018):

$$T_{shock}\approx 1.4\times 10^{7}\;{\rm K}\left(\frac{v_{shock}}{10^{3}\,{\rm km\;s}^{-1}}\right)^{2}.$$

(4)

Multiplying the mean molecular weight by the expected shell thickness and number density at the location of the shell on day 76 – for the appropriate range of shock velocities, a $1/r^{2}$ density profile (where $r$ is the distance from the white dwarf), a number density ($n_{shock}$) at $R_{shock}=v_{shock}t$ from the standard shock jump conditions (Steinberg & Metzger, 2018) $n_{shock}=4n$, and a total ejecta mass determined from the late-time radio evolution – we find that $M_{hot,expected}$ likely fell between $10^{-10}$ and $10^{-8}M_{\odot}$ (see Fig. 4).

For the early-time flare to have been due to thermal emission from gas enclosed within $L_{cool}$, that gas must either have been optically thick with a temperature in excess of $T_{b}=1.3\times 10^{5}$ K or had an even higher temperature. In fact, the mass of hot gas that would have been needed to make the hot shell optically thick, $M_{hot,needed}$, is more than an order of magnitude greater than $M_{hot,expected}$. To estimate $M_{hot,needed}$, we assume the hot shell expanded with velocity $v_{shock}$. We derive the number density of electrons needed to make a $T_{e}\sim 10^{5}$ K shell optically thick at 4.55 GHz by setting $\tau_{v}=1$ in the expression for the emission measure and taking a constant density profile within the hot shell. The emission measure is $EM=\int_{r_{o}}^{r_{i}}n_{e}^{2}dl$, where $n_{e}$ is the electron density. Following Bode & Evans (2008), we calculate the emission measure needed to make the hot shell optically thick ($\tau_{\nu}=1$) from:

$$\tau_{\nu}=1=8.235\times 10^{-2}\left(\frac{T_{e}}{K}\right)^{-1.35}\left(\frac{\nu}{GHz}\right)^{-2.1}\frac{EM(\tau_{\nu})}{cm^{-6}pc}.$$

(5)

Figure. 4 shows both $M_{hot,expected}$ and $M_{hot,needed}$ enclosed in a shell of thickness $L_{cool}$ as a function of of shock velocity.

For $v_{shock}$ between 600 km/s to 1200km/s, $M_{hot,needed}$ exceeded $M_{hot,expected}$ by at least an order of magnitude. For $v_{shock}$ of 900km/s, $M_{hot,needed}=1.5x10^{-8}M_{\odot}$ and $M_{hot,expected}=8.5x10^{-10}M_{\odot}$. It is thus implausible for the high brightness temperature to have been the result of thermal emission from a shock-heated shell. This finding further supports our assertion that the high brightness temperature is most likely the result of non-thermal emission produced in a shock.

Non-thermal emission from novae is widely accepted to be the result of internal shocks in the ejecta (Chomiuk et al., 2020a; Vlasov et al., 2016; Finzell et al., 2018; Weston et al., 2016a; Franckowiak et al., 2018; O’Brien & Lloyd, 1994). The collision of the fast and slow flows results in shocks that are responsible for accelerating particles to relativistic speeds. Novae V1723 Aql (Weston et al., 2016a), V5589 Sgr (Weston et al., 2016b) and Nova 1324 Sco (Finzell et al., 2018) had brightness temperatures of greater than $10^{5}$ K, interpreted as evidence for radio synchrotron emission from shocks. We similarly propose that the bright radio emission that the VLA detected on April 19 2013 (Epoch 4) was the result of shocks in ejecta from V809 Cep. V809 Cep thus joins six other classical novae with radio evidence for shocks: V906 Car (Aydi et al., 2020a), V1324 Sco (Finzell et al., 2018), V1723 Aql (Weston et al., 2016a), 5589 Sgr (Weston et al., 2016b), V959 Mon (Chomiuk et al., 2014a) and QU Vul (Taylor et al., 1987). Non-thermal radio emission has also been found in the eruptions of closely related objects including the helium nova V445 Pup (Nyamai et al., 2021) and the symbiotic system V407 Cyg (Giroletti et al., 2020).

The radio spectral evolution during the rise to maximum of the early-time flare in V809 Cep had some resemblance to theoretical predictions for synchrotron from internal shocks in novae by Vlasov et al. (2016). Those authors suggested that photo-ionized gas ahead of the shock initially absorbs radio synchrotron emission, causing the spectral index ($\alpha$) to evolve from approximately 2.0, as expected for optically thick thermal emission, to 2.5 when the photo-ionized gas ahead of the shock becomes transparent and optically thick synchrotron emission begins to dominate the spectrum. Vlasov et al. (2016) predicted that the spectral index then gradually decreases toward the $\alpha=-0.7$ expected for optically thin synchrotron (the exact value depends on the energy spectrum of the emitting particles, e.g. Pacholczyk 1970). Later, the spectral index is expected to once again rise as the radio emission becomes dominated by optically thick thermal emission from the bulk of the expanding, photoionized ejecta. The peak of the early-time flare occurs as the synchrotron transitions from optically thick to optically thin (Vlasov et al., 2016).

At low frequencies (below 7.4 GHz), the radio emission during the early-time flare in V809 Cep was consistent with theoretical expectations – if the VLA began detecting the flare when photoionized gas ahead of the shock had already become partially transparent. During the flare, the low-frequency spectral index transitioned from a high value suggestive of optically thick synchrotron emission (2.8 $\pm$ 0.7 in Epoch 2) to the more modest value of 0.5 $\pm$ 0.1 at the peak of the flare (during Epoch 4). Though the error bars on the low frequency spectral index at Epoch 2 are large, in combination with the high brightness temperature, the high spectral index implies optically thick synchroton emission.

The flattening of the low-frequency spectral index at Epoch 4 suggests that the spectrum was indeed transitioning from one indicative of optically thick to optically thin synchrotron emission at that time, as predicted by Vlasov et al. (2016). At Epoch 5, the low-frequency spectral index increased again, to 1.58 $\pm$ 0.05, as expected for optically thick thermal emission from the main, expanding ejecta beginning to dominate over the early-time synchrotron emission.

At high frequencies, the radio emission during the early-time flare can also be understood within the framework of Vlasov et al. (2016). Synchrotron emission from an expanding medium is expected to become optically thin, and the spectral index to decrease, at the highest frequencies first (29 GHz and above, in our case). The high-frequency spectral indices at Epochs 2 and 3 of 0.1 $\pm$ 0.4 and -0.3 $\pm$ 0.2, respectively, thus follow the spectral evolution modeled in Vlasov et al. (2016) if the synchrotron emission had already become fairly optically thin at those frequencies by Epochs 2 and 3. Moreover, because the high-frequency radio spectrum was already quite flat when the VLA first detected the source in Epoch 2, with the high-frequency flux densities well below an extrapolation of the low-frequency spectrum at that time, synchrotron emission at frequencies higher than 7.4 GHz likely peaked before it peaked at low frequencies, and even prior to the first radio detection. Further supporting the possibility that any synchrotron peaked at high frequencies before low frequencies, at 7.4 GHz, the brightness temperature peaked during Epoch 3, an epoch before the peak at 4.6 GHz. Brightness temperature represents a lower limit on the physical temperature of an optically thin gas; therefore, we expect the brightness temperature to peak at high frequencies prior to peaking at low frequencies as the high frequencies become optically thin first. Figure 3 displays a peak first a Epoch 3 (Day 61) at 7.4 GHz and later at Epoch 4 (Day 76) at 4.6 GHz. We thus posit that the early-time flare peaked at high frequencies prior to the first radio detection.

Absorption by photo-ionized gas ahead of the shock would initially obscure the non-thermal emission and produce a delay between the early-time radio maxima at progressively lower frequencies (Vlasov et al., 2016). As the photo-ionized gas ahead of the shock begins to diffuse and become optically thin at lower frequencies, the non-thermal emission from the shock becomes increasingly dominant first at higher frequencies and finally at lower frequencies. So, hints that the synchrotron flare peaked at higher frequencies first (rather than at all frequencies simultaneously) suggest that non-thermal emission from internal shocks in the V809 Cep ejecta was initially absorbed by photo-ionized gas ahead of the shock. However, even if any synchrotron emission was initially somewhat absorbed by photoionized gas, the high spectral index of $2.8\pm 0.7$ during Epoch 2 (Day 48) would indicate that we caught the flare when the putative synchrotron emission was already peeking through at low frequencies.

The radio spectral evolution during Stage 2 (Epochs 5-9) further supports our assertion that the spectrum during stage 1 was attributable to an emission component that was physically distinct from the main photoionized ejecta. As the early emission component thinned, the radio spectrum became increasingly dominated by optically thick thermal emission. Indeed, during Epochs 4 (Day 76), the spectral index at high frequencies rose to 1.5 $\pm$ 0.1, indicative of optically thick thermal emission. During Epoch 5 through 9 (Day 110 - 206) , the spectral index for all frequencies increased to 1.58 $\pm$ 0.05 and remained around 1.5 as the ejecta expanded and the radio emission brightened for the remainder of stage 2. Although the stage 2 spectral indices were lower than the expected 2.0 for optically thick thermal emission from a medium with an infinitely sharp outer edge, values of 1.5 are typical for actual nova remnants. Therefore, the radio spectral evolution was consistent with emission from an increasingly optically thin synchrotron source and a gradually brightening optically thick thermal component. During stage 3 and 4, the radio spectrum was typical of a thermal source becoming increasingly optically thin, with the spectrum flattening at progressively lower frequencies as gas began to become optically thin first at high frequencies ($>$ 29 GHz) and by Epoch 20 (Day 726) at all frequencies.

Around the start of the early-time flare, dust production also commenced, providing evidence for a connection between shocks and dust formation. In the shock-dust model (Derdzinski et al., 2017), internal shocks in the ejecta lead to dust condensation. We would naively expect, therefore, to see evidence for the shocks prior to the onset of dust formation, which began around 2013 March 11 (day 37; Munari et al. 2014). The VLA did not observe V809 Cep between day 12 (Epoch 1) and day 48 (Epoch 2) of the eruption, so we cannot pinpoint the exact time at which synchrotron emission appeared (or peaked at high frequencies). But on day 48 (Epoch 2), 11 days past the onset of dust formation, we detected radio emission almost certainly associated with shocks within the ejecta. Therefore, both signatures of shocks and dust appeared during the one month gap between Epoch 1 (Day 12) and Epoch 2 (Day 48). Given the evidence that the early-time flare did not peak at the same time at all frequencies and therefore that any synchrotron was initially obscured, the radio observations are consistent with the shocks arising either slightly before or around the same time as the onset of dust production.

Starting about nine months into the eruption, the radio emission was reasonably consistent with Bremsstrahlung (free-free emission) from freely expanding, photoionized ejecta. We modeled the main ejecta as a Hubble-flow (Gehrz, 2008; Finzell et al., 2018; Weston et al., 2016a; Seaquist & Palimaka, 1977; Hjellming et al., 1979), in which the ejecta consists of a homologously expanding isothermal spherical shell of gas bound between inner and outer radii $r_{inner}$ and $r_{outer}$, with a $r^{-2}$ density profile (where $r$ is the distance from the white dwarf), and a volume filling factor, $f$. The filling factor is a measure of inhomogeneities in the ejecta, such as clumps (e.g., Weston et al., 2016a; Hjellming, 1996). We assume no inhomogeneities ($f=1$). If the ejecta were actually clumpy, that would reduce the ejecta mass needed to generate the observed radio emission (Weston et al., 2016a; Hjellming, 1996; Abbott et al., 1981; Leitherer & Robert, 1991; Chomiuk et al., 2014b).

At low frequencies ($<$ 7.4 GHz) during the first six or so months of the outburst, the model fit the observed flux densities poorly, and as expected, was unable to account for the early-time flare. At 13.5 and 17.0 GHz, the discrepancy between the model and the observed flux densities during the first few months of the eruption supports our contention (in Sec. 4.2) that the flare at these frequencies could have peaked before Epoch 4 (Day 76). At these frequencies, the Hubble-Flow model matched the observed flux densities well after Epoch 4 (Day 76). At Epoch 4 (Day 76), however, the observed flux density at 17.4 GHz was $0.94\pm 0.02$ mJy, 0.61 mJy in excess of the the model flux density of 0.33 mJy. At 13.5 GHz, the observed flux density was $0.789\pm 0.002$, 0.58 mJy in excess of the model flux density. That the observed 13.5- and 17.4-GHz flux densities were far in excess of the model during Epoch 4 (Day 76) indicates that any synchrotron emission was still dominant at those frequencies at that time and could therefore have peaked even earlier (the VLA did not collect data at these frequencies during Epochs 2 and 3 (Day 48 and Day 61)).

At 28.8 and 36.5 GHz, and after approximately days 200-300 (¿Epoch 9) at all frequencies, however, the model reproduces the observed flux densities moderately well. Using $D=6.5$ kpc, $v_{ej}=1200$ km s^-1, and $v_{min}=600$ km s^-1, we determined the best-fit ejecta mass and temperature by employing the Markov Chain Monte Carlo program pymc. We found $M_{ejected}=(1.1\pm 0.1)\times 10^{-4}M_{\odot}$ and $T=(0.8\pm 0.2)\times 10^{4}K$ when the hubble flow model was fit to the reduced flux densities starting from Epoch 2 (Day 48) (first detection). Figure 5 displays the observed flux densities as well as the modeled free-free emission for $D=6.5$ kpc, $M_{ejected}=(1.1\pm 0.1)\times 10^{-4}M_{\odot}$ and $T=(0.8\pm 0.2)\times 10^{4}$ K.