RETURN OF THE KING: TIME-SERIES PHOTOMETRY OF FO AQUARII'S INITIAL RECOVERY FROM ITS UNPRECEDENTED 2016 LOW STATE

Modulations with periods between 11 and 23 minutes are present in all of our light curves, as is a longer-period variation at half of the 4.85-hour orbital period. For most of our observations, the period of the oscillations is strongly dependent on the binary orbital phase (ϕ_orb); see Figure 2. To compute ϕ_orb, we used the time of orbital minimum from Kennedy et al. (2016) and the orbital period from Marsh & Duck (1996). Between 0.9 < ϕ_orb < 1.4, the light curve shows a 22.52-minute period consisting of two humps of unequal magnitude. During the remainder of the orbital period, the two pulses become approximately equal in amplitude, so the period is halved to 11.26 minutes. This orbital-phase dependence became somewhat less obvious during a bright bump in the recovery between approximately JD 2457596 and 2457604, and the reason for this is discussed in Section 4.3.

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We computed periodograms of our dataset using both the Lomb–Scargle (Lomb 1976; Scargle 1982) and phase-dispersion-minimization (PDM; Stellingwerf 1978) techniques. To prevent high-cadence time series from dominating the power spectra, we resampled each light curve to a cadence of 75 s per image. Both Lomb–Scargle and PDM show strong signals at the spin (ω), beat (ω − Ω), and double-beat (2ω − 2Ω) frequencies, along with additional sidebands. The Lomb–Scargle method shows the most power at 2ω − 2Ω (the 11.26-minute period), followed by ω, Ω, and ω − Ω (the 22.5-minute period). The first four harmonics of Ω are all clearly detected, as are the first two harmonics of ω. The 3ω − 3Ω, 4ω − 4Ω, and ω + Ω frequencies also show some power. Although the PDM technique also favors 2ω − 2Ω as the strongest signal, it shows ω − Ω to be stronger than ω. The Lomb–Scargle periodogram of the low state is shown in the top panel of Figure 3, along with a power spectrum of high-state data for comparison.

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The power spectrum varied during the recovery, as shown in the middle two rows of Figure 3. During the bump in the long-term light curve between JD 2457596 and 2457604, the power at ω − Ω was almost negligible. In addition, after JD 2457604, the power at Ω increased significantly, as did ω in comparison to 2ω − 2Ω.

Kepler K2 observations of FO Aqr's high state (Kennedy et al. 2016) provide a useful benchmark for comparison. Since the K2 data were obtained at a one-minute cadence for 69 consecutive days, we resampled the K2 data using the following algorithm in order to approximate our own sampling rate. The K2 and low-state data were each stored in a separate array. For each observation in the K2 and low-state datasets, we calculated the time elapsed since the earliest observation in that dataset. Then, for each elapsed-time value in the low-state data, we found the closest elapsed-time value in the K2 data. The timestamp and magnitude of this K2 measurement were then included in the resampled light curve. The algorithm prevented an observation from being added to the resampled light curve more than once.

There are a number of striking differences between our power spectra and the resampled K2 power spectra, underscoring that the accretion processes have fundamentally changed in response to the diminished mass-transfer rate. In the resampled high-state data, ω was by far the strongest signal, followed by Ω. The K2 data also show ω − Ω and especially 2ω − 2Ω to be weak in comparison to ω. The signal at 2ω − 2Ω shows the most dramatic difference between the low and high states, increasing in power by well over an order of magnitude to become the dominant signal of the low-state power spectrum. In short, FO Aqr has transitioned from an ω-dominated power spectrum in its high state to one dominated by 2ω − 2Ω in the low state.

The power spectra are helpful in identifying which frequencies are present in the data, but they do not provide information on phases of the components. Thus, we model the light curves based on the periodicities observed in the power spectra.

To approximate the beat (ω − Ω) and double-beat (2ω − 2Ω) signals, we add two periodic functions, one shifted in phase by π relative to the other. Because we do not know the true shape of the pulses, we have tested several possible periodic functions including $F={(\cos (\theta )+1)}^{2}/4$ and $F=\cos {(\theta )}^{3}$, but the choice makes little difference in matching the observations. We also include a spin pulse and an orbital modulation. The full model is:

$$\begin{eqnarray*}{F}_{\mathrm{tot}} & = & {A}_{s}{F}_{s}(t,\omega ,{\phi }_{s})+{A}_{b1}{F}_{b}(t,\omega -{\rm{\Omega }},{\phi }_{b})\\ & & +{A}_{b2}{F}_{b}(t,\omega -{\rm{\Omega }},{\phi }_{b}+\pi )\\ & & +{A}_{o}({F}_{o}(t,{\rm{\Omega }},{\phi }_{o})+{A}_{o2}{F}_{o}(t,{\rm{\Omega }},{\phi }_{o}+\pi )),\end{eqnarray*}$$

where F_s, F_b, and F_o are all periodic functions describing the spin pulse, beat pulse, and orbital modulations seen in the light curve. When fitting the light curves, we fix the orbital and spin periods at 0.20205976 day and 1254.3401 s respectively. We then calculate the least-squares difference between the data and the model by varying the component amplitudes (A_s, A_b1, A_b2, A_o) and the phases (ϕ_s, ϕ_b, ϕ_o). A_o2 allows freedom in fitting the amplitudes of the 2Ω component, and appears close to unity in our light curves.

Because of their very high signal-to-noise ratio, we used the MDM data to estimate the model parameters, and the resulting fit is shown in Figure 2. The model works quite well in matching the brightness and times of the peaks. The ratio between the model amplitudes (A_s:A_b1:A_b2) is 1.0:2.4:1.5. While the components are roughly comparable in amplitude, the spin varies in phase with respect to the double-humped beat signal. In the first half of the orbit, the spin pulse is nearly in phase with the stronger beat maximum, making the ω − Ω variation stand out. In the second half of the orbit, the spin pulse occurs at a similar phase to the lower amplitude beat maximum, creating a prominent 2ω − 2Ω variation. Because of the relation between the orbital, spin, and beat periods, this pattern repeats each orbit. Our modeling shows that the spin pulse is approximately in phase with the stronger beat maximum at ϕ_orb ∼ 0.22 and with the weaker one at ϕ_orb ∼ 0.72.

The drop in brightness is likely a result of a major decrease in the mass-transfer rate in the system. This can cause a decrease in the disk luminosity as well as the brightness of the accretion curtains and the shock energy at the WD accretion poles. Livio & Pringle (1994) suggested that the cause of low states in magnetic CVs could be starspots on the secondary passing over the inner Lagrangian point. The paucity of low states in the long-term light curve of FO Aqr implies that magnetic activity on its secondary is weaker than in other magnetic CVs.

The rate at which a CV fades to a low state or brightens to a high state is often quantified by its e-folding time, which is the time required for the system's flux to change by a factor of e. Kafka & Honeycutt (2005) studied 43 low states in five magnetic CVs and found that the typical transition e-folding timescale was 20 days, or a rate of −0.05 mag day⁻¹. This is significantly faster than the measured e-folding time for FO Aqr's recovery (115 ± 7 days), which we found using an outlier-resistant linear regression of the low-state data shown in Figure 1. Indeed, the e-folding time for FO Aqr's recovery is over twice as long as the slowest single-sloped transition from Kafka & Honeycutt (2005). Although some transitions occur in two steps with different recovery rates, the slowest e-folding time reported by Kafka & Honeycutt (2005) during the faintest part of a recovery is just 66 days. We conclude, therefore, that the recovery rate from the low state of FO Aqr has been unusually slow.

The two ASAS-SN observations of FO Aqr in an intermediate state in 2016 April, in conjunction with the earliest ASAS-SN observation of the low state in 2016 May, enable us to constrain the upper limit of the decline's e-folding time. We assume that the two measurements from April captured the original decline from the high state and that the decline was smooth and single-sloped. Since these data are contaminated by the variability at ω, Ω, and the associated sidebands, we used Monte Carlo simulations to simulate the effect of these modulations on the decline's e-folding time. We find that for a single-sloped transition, the e-folding time must have been less than ${17}_{-3}^{+5}$ days. Since the rebrightening was underway just two weeks after the two April observations (Figure 1), recovery from the low state must have started almost immediately after the end of the initial fade to minimum brightness.

While starspots at the L1 point may explain the decrease in the mass-transfer rate and the concomitant fading of the disk and accretion luminosity, the reason for the slow recovery is not clear. To explain the VY Scl class of CVs, Wu et al. (1995) theorized that loss of shielding by an accretion disk can heat the secondary, thus raising the mass-transfer rate. So for FO Aqr, we expect that the partial depletion of the disk would result in extra heating of the secondary, leading to an acceleration of the recovery. Wu et al. (1995) expect mass transfer and disk building should occur on a timescale of 10⁶ s. If the mass transfer through the L1 point had recovered to its pre-2016 rate, we would have expected the luminosity rebound to be on the viscous timescale of the disk. The very slow recovery rate we observe suggests that the secondary star has yet to return to its previous mass-loss rate.

As is evident from the bottom panel of Figure 1, the rate of brightening during the recovery was not perfectly uniform. For example, between JD 2457596 and JD 2457604, FO Aqr deviated above the long-term trend by about 0.2 mag. During this "bump," flares reaching magnitude 14.3 were observed in the time-resolved photometry, suggesting irregular accretion events.

In the bright state, FO Aqr shows a dip in brightness every 4.85 hr, which is thought to be a grazing eclipse of a disk (Hellier et al. 1989). We constructed a phased light curve of our entire dataset to determine if this feature is also present in the low state. To prevent the high-cadence data from dominating this plot, we resampled each light curve into 75-s bins. We then split the resampled data into 100 non-overlapping orbital-phase bins and calculated the median magnitude in each of those bins. Although the signals at ω, ω − Ω, and 2ω − 2Ω usually overwhelm the orbital modulation in individual light curves, they are all out of phase at the orbital period. Thus, the inclusion of data from many different orbital cycles smooths out these variations, isolating the comparatively weak orbital signal.

The resulting phase plot (Figure 4) shows an eclipse near the orbital phase expected from the high-state light curve (Kennedy et al. 2016), strongly suggesting that there was still an accretion disk during our observations. The low-state eclipse depth decreased by ∼0.15 mag compared to the K2 data, as would be expected if the disk were depleted during the low state. The eclipse width is also significantly less in the low state (FWHM = 0.104 ± 0.005ϕ_orb) than in the high state (FWHM = 0.18 ± 0.007ϕ_orb), implying that the outer radius of the disk has shrunk (K. Mukai 2016, private communication).

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Power spectral analysis of an IP can serve as an important method for determining whether the system is undergoing disk-fed accretion, stream-fed accretion, or some combination of the two (Ferrario & Wickramasinghe 1999). Figure 3 clearly shows that the double-beat (2ω − 2Ω) frequency dominates the optical light curve during the low state of FO Aqr, with ω − Ω also being significantly stronger than in the high state. Moreover, the model fit to the MDM light curve shows that the beat amplitude is more than twice the strength of the spin component, while in the high state the beat amplitude is only a fifth of the spin amplitude. In accordance with the modeling in Ferrario & Wickramasinghe (1999), the increased strength of the signals at the beat and double beat suggests that a substantial fraction of the accretion during our observations was stream-fed.

Although ω is quite strong in our power spectra, Ferrario & Wickramasinghe (1999) predict that ω is not uniquely indicative of disk-fed accretion. They caution that stream-fed accretion can induce a signal at ω depending on several effects, including the optical depth of the curtains, the extent of the accretion stream, and the strength of cyclotron radiation from the accretion region. Thus, the presence of ω in the low-state power spectrum does not conclusively establish that there was simultaneous disk-fed accretion. Low-state X-ray and spectral observations should reveal more definitively whether the WD was accreting significantly from the disk (M. Kennedy et al. 2016, in preparation).

An alternate explanation for the signals at ω − Ω and 2ω − 2Ω is that they result from the irradiation of stationary structures within the binary rest frame (e.g., a disk bulge or the donor star). High-energy photons from one of the WD's accretion regions will sweep over these structures at a frequency of ω − Ω. In FO Aqr's high state, this mechanism causes a modulation at that frequency (Osborne & Mukai 1989). A difficulty with this scenario is that during the low state, the disk would have been significantly depleted by the low mass-transfer rate, as we infer from the decreased eclipse depth and width during the low state (Section 4.2). While we favor stream-fed accretion as the cause of the ω − Ω and 2ω − 2Ω variations, their origin can likely be established with greater confidence from time-resolved spectroscopy of the beat modulation (M. Kennedy et al. 2016, in preparation).

During the "bump" in the long-term recovery of FO Aqr, the power of the ω − Ω signal weakened, as shown in Figure 3. Outside of the bump, the two maxima of the beat modulation were asymmetric, leading to a significant signal at the beat frequency. The strength of the ω − Ω signal declined during the bump simply because the amplitudes of the beat pulses became comparable to each other, causing the power to move to the 2ω − 2Ω frequency. We expect that the beat pulses result from accretion onto both magnetic poles, with the relative brightness of each beat maximum determined by the accretion rate through the accretion curtains and by the amount of obscuration by the disk. For example, at FO Aqr's normal brightness, the 2ω − 2Ω signal is greatly suppressed, possibly because the fully engorged disk blocks our view of one pole.

Kennedy et al. (2016) noted that the phase of the spin pulse in the K2 data was mildly dependent on the brightness of FO Aqr. If we assume that ω is the pulse from a disk-fed accretion curtain,²⁷ then our analysis shows that the spin-pulse arrival times are indeed dependent on the system's overall brightness. We isolated the spin pulse in our photometry by calculating the spin phase for each observation using the ephemeris

$$\begin{eqnarray*}&&{T}_{\mathrm{spin}}[\mathrm{BJD}]=2457032.54909(1)+0.014517825(5)\times E,\end{eqnarray*}$$

based on the Kennedy et al. (2016) spin period. We sorted the photometry by spin phase before smoothing the phased light curve with a median filter. The resulting spin light curves are shown in Figure 5.

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By using this technique to measure the phase of the spin peak in the K2 data, we find that the spin pulse arrived −0.024 ± 0.007 earlier in phase during the faintest third of the K2 data than it did during the brightest third (a difference of 0.11 mag). In the low-state photometry, our light curves taken before JD 2457596 (when FO Aqr had an average unfiltered magnitude of 15.0) show that the spin pulse arrived 4.3 minutes early (−0.20 ± 0.01 in phase). This is significantly larger than the expected K2 ephemeris uncertainty of 17 s. Between JD 2457596 and JD 2457604 (when FO Aqr had brightened to an average of 14.7 mag), the spin peak arrived 3.1 minutes (−0.15 ± 0.01 in phase) earlier than the K2 ephemeris. To test whether a change in the spin period can account for these shifts, we constructed a cubic ephemeris for the spin maxima using recent data in Table 1 in Bonnardeau (2016), and we compute a total phase shift of −0.024 phase units between the K2 observations and the low state. This accounts for only a small fraction of the observed phase shift of −0.20 ± 0.01 during that span, so it is unlikely that the phase shift is the result of a change in the spin period of the WD. Indeed, we can connect our data and the K2 with an approximately linear relation between the average magnitude of the system and the phase of the spin pulse (Figure 5, bottom panel).

Hellier et al. (1997) provide a possible explanation for this dependence. They found that during an outburst of the IP XY Ari, the inner radius of the disk decreased by over ∼50%, because of the increased rate of mass transfer ($\dot{M}$) through the inner disk. Thus, they concluded that the gas in the disk was captured by different field lines than during quiescence, resulting in a pronounced shift in both the location and size of the accretion region. This scenario is directly applicable to FO Aqr. In response to the decreased $\dot{M}$ during FO Aqr's low state, there might have been an increase in the radius at which the disk is magnetically truncated, thereby altering the shape and size of the disk-fed accretion curtains. This effect could also potentially explain why the spin pulse in FO Aqr appears to become wider and more sinusoidal as the system brightens. X-ray light curves of the low state provide one possible mechanism for testing whether the accretion region has changed in comparison to the high state.

The phase shift of the spin pulses has significant implications for previous studies of FO Aqr's spin period. For decades, the spin timing of FO Aqr has been used to identify changes in the WD spin period (e.g., Patterson 1994). However, the spin pulse's dependence on system brightness might require a correction to be made in order to prevent the phase shift from masquerading as a change in the spin period. As Figure 5 shows, this effect is apparent even in the relatively minor brightness variations in the high-state K2 data. Consequently, it will be important to monitor the phase of the spin pulse as the recovery continues so that this dependence may be tested more rigorously.