Long-term photometric behavior of the eclipsing Z Cam-type dwarf nova AY Psc

In Figure 2 we show the long-term AAVSO light curve of AY Psc during 2010 August–2016 November. The data are divided into seven segments because a lot of data are missing. A detailed view of the light curves reveals three main features: normal outburst, standstill and stunted outburst. During normal outburst, the duty cycle of AY Psc approaches 100%. This behavior is thought to be due to ${\dot{M}}_{2}\approx {\dot{M}}_{\mathrm{crit}}$ (Lin et al. 1985; Warner 1995). The system rises from $V\approx 17.1$ at minimum to $V\approx 14.6$ at maximum in 6 – 8 d, exhibiting a mean outburst amplitude of $\sim 2.5(\pm 0.1)$ mag.

Figure 4 displays a best outburst data set for AY Psc covering 169 d (2016 Jun 14–2016 Nov 29). A sine curve fit was applied to the eight outburst cycles in Figure 4, giving a recurrence time of $\sim 19.03(\pm 0.04)$ d. However, the full light curve contains at least four outburst segments. To verify this result, analysis of all these outburst segments is required.

Figure 5 shows all outburst data sets and their associated power spectrums. The power spectrums correspond to the periods between 17.7 and 18.5 d, which is smaller than the period from the sine fitting ($\sim 19.03$ d). This implies that the outburst in AY Psc is not strictly periodic but rather quasi-periodic.

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Kukarkin & Parenago (1934) first noted that there was a relation between the outburst amplitude (A_n) and the outburst recurrence time (T_n) for DNe and recurrent novae. After that, this relation has been revised and improved many times to constrain its application range. Finally, a general correlation was found by analyzing normal outbursts in DNe (van Paradijs 1985). The most recent version of the Kukarkin-Parenago (K-P) relation from Warner (1995) is as follows

$$\begin{eqnarray}&&{A}_{n}=0.7(\pm 0.43)+1.9(\pm 0.22)\mathrm{log}{T}_{n}.\end{eqnarray} \tag{ 3 }$$

This is an empirical relation. Moreover, to explore if this relation is model-dependent, a theoretical K-P relation was derived by Kotko & Lasota (2012) using the disk instability model

$$\begin{eqnarray}&&{A}_{n}={C}_{1}+2.5\mathrm{log}{T}_{n},\end{eqnarray} \tag{ 4 }$$

where the constant term

$$\begin{eqnarray*}&&{C}_{1}=2.5\mathrm{log}2\tilde{g}-2.5\mathrm{log}{t}_{\mathrm{dec}}+{{BC}}_{\max }-{{BC}}_{\min },\end{eqnarray*}$$

depends on the properties of the white dwarf and the viscosity parameter α. The outburst parameters of AY Psc and the K-P relation are plotted in Figure 6. The open squares represent statistical data from the AAVSO database, the red solid line refers to the empirical K-P relation and red dashed lines denote its upper and lower uncertainties. However, the observational data were not covered in this relation. Note that this relation is only significant for systems with ${A}_{n}\gt 2.5$ mag (Warner 1995). The lower uncertainty of the empirical relation follows the overall trend of observational data reasonably well because the amplitude of AY Psc falls on the boundary of 2.5 mag. Therefore, the K-P relation for AY Psc will be roughly replaced by the lower limit of the empirical relation

$$\begin{eqnarray}&&{A}_{n}=0.27+1.68\mathrm{log}{T}_{n}.\end{eqnarray} \tag{ 5 }$$

Based on those descriptions, we suggest that the K-P relation may represent a common characteristic of DN outbursts.

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Figure 7 shows the standstill signal at 15.3(±0.2) mag extracted from Figure 2 during the period from 2012 June 29 to 2015 January 1. The upper panel of Figure 8 displays a well-observed transition from standstill to outburst, where we see an unusual characteristic behavior that differs from what is exhibited by most Z Cam stars. This standstill ends with an outburst (see Fig. 8). However, the currently used model of Z Cam stars predicts that a standstill can only be terminated by a decline to quiescence rather than an outburst (Buat-Ménard et al. 2001). Before several of the the unusual Z Cam systems were discovered, only one recorded standstill in the prototype star Z Cam was observed to terminate in a rise to maximum (Warner 1995; Oppenheimer et al. 1998). Recently, this rare behavior was observed in several anomalous Z Cam-type stars such as AH Her, IW And, V513 Cas and ST Cha (Simonsen 2011; Simonsen et al. 2014). A plausible explanation for this behavior was proposed by Hameury & Lasota (2014), who indicated that mass-transfer outbursts can reproduce the observed properties of these unusual systems. This implies that the changes in mass transfer from the secondary star are responsible for changes in brightness on longer timescales. For Z Cam-type systems, a direct test is to compare the standstill and outburst brightness. In effect, there have been several studies that investigated this aspect (Lortet 1968; Oppenheimer et al. 1998; Honeycutt et al. 1998). To clearly reveal the mass-transfer variations, the mean brightness during standstills and the outbursts were calculated by averaging long-term AAVSO data. In computing the mean magnitude values, only full cycles were averaged. For outbursts, two mean values were measured per outburst cycle, overlapping by one-half cycle. For standstills, means were computed for successive intervals of several tens of days. The mean brightness values of all data are plotted in Figure 9. The red dots in Figure 9 represent the means during standstills and the black dots denote the means during outbursts. Clearly, the average magnitudes during standstills are brighter than during outburst intervals. This result is consistent with previous authors (e.g. Oppenheimer et al. 1998; Honeycutt et al. 1998). Moreover, the trend in Figure 9 also implies there are changes in ${\dot{M}}_{2}$, in agreement with the general picture of standstills. The green solid line marks the transition between outburst cycles and standstills, corresponding to ${\dot{M}}_{\mathrm{crit}}$. Frank et al. (2002) give an expression for ${\dot{M}}_{\mathrm{crit}}$

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{crit}}\simeq 3\times {10}^{-9}{({P}_{\mathrm{orb}}/3)}^{2}{M}_{\odot }{\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 6 }$$

where ${P}_{\mathrm{orb}}$ is in hours. For AY Psc, with ${P}_{\mathrm{orb}}=5.22$ h, we find ${\dot{M}}_{\mathrm{crit}}\simeq 9.07\times {10}^{-9}{M}_{\odot }{\mathrm{yr}}^{-1}$. Buat-Ménard et al. (2001) pointed out that ${\dot{M}}_{2}$ will vary by about 30% around ${\dot{M}}_{\mathrm{crit}}$. More specifically, ${\dot{M}}_{2}$ in AY Psc should be restricted to the range from $6.35\times {10}^{-9}$ to $1.18\times {10}^{-8}{M}_{\odot }{\mathrm{yr}}^{-1}$.

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More detailed inspection of the AAVSO data reveals that there are diverse behaviors during standstill. Note that the brightness of the standstill is not constant, but rather keeps fluctuating. Oscillations with an amplitude of $\sim 0.2$ mag can be seen by eye in Figures 7 and 8. Moreover, we find occasional outburst-like events during standstill, as shown in Figure 10. These events are best characterized as small amplitude outbursts, which have properties very similar to the stunted outbursts seen in some NLs (e.g. Honeycutt et al. 1995, 1998; Honeycutt 2001; Honeycutt et al. 2014; Warner 1995; Hoard et al. 2000; Ramsay et al. 2016).

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Table 2 summarizes the properties of stunted outbursts in AY Psc. The stunted outbursts in Figure 10 are visible with amplitudes of $\sim 0.5-0.9$ mag and full width at half maximums (FWHMs) of $\sim 2-14$ d. The rises are slower than the declines, implying that the outbursts are Type B (inside-out). However, to date, the origin of stunted outbursts is still uncertain. Some possible mechanisms are proposed to account for the stunted outbursts, including a disk truncated by the magnetic field of the white dwarf or a very hot white dwarf, DN outbursts being related to the standard disk instability and mass transfer modulations (Honeycutt et al. 1998; Honeycutt 2001; Honeycutt et al. 2014; Ramsay et al. 2016). For AY Psc, a truncated disk mechanism can be ruled out because the inside-out outbursts (Type B) in a truncated disk are infrequent (Honeycutt et al. 1998). During standstill, ${\dot{M}}_{2}$ from the secondary star exceeds ${\dot{M}}_{\mathrm{crit}}$, and the disk stays in steady state, without DN outbursts. In addition, Warner (1995) indicated that DN outbursts have faster rise times than decline times by a factor of $\sim 2$ (i.e. Type A outbursts), which is not compatible with the stunted behaviors of AY Psc. Therefore, the disk instability may not be the cause of stunted outbursts. It seems that a changing mass transfer is a reasonable candidate mechanism for these stunted behaviors. First, a steady-state disk allows the brightness to follow changes in mass transfer. Second, Hameury & Lasota (2014) studied the disk's response to a mass transfer outburst during standstills and reproduced an outburst with an amplitude of $\sim 0.8$ mag starting from a steady state disk. Meanwhile, the possible origins of the mass transfer outbursts were also discussed by these authors. Nevertheless, these stunted outbursts are not well understood so far.