Rapid Evolution of the White Dwarf Pulsar AR Scorpii

AR Scorpii (AR Sco hereafter) is one of the most intriguing interacting binary stars known. It has been called a white dwarf "pulsar" (Buckley et al. 2017) because its bright, polarized, synchrotron flashes appear to be powered by the spin-down energy of its degenerate primary stellar component (Marsh et al. 2016; Stiller et al. 2018; Gaibor et al. 2020; Pelisoli et al. 2022). The binary consists of a low-mass red dwarf star and a rapidly spinning magnetized white dwarf (WD) orbiting over a 3.56 hr period. The WD spins (ω) with a period of 1.95 min (Marsh et al. 2016) and appears to generate two pulses per rotation, likely from two magnetic poles. In the Potter & Buckley (2018) model, the synchrotron emission is modulated by the angle between the magnetic poles of the spinning WD and the secondary star. The emission is enhanced just after a magnetic pole sweeps past the red dwarf, leading to variations seen at the spin frequency plus strong pulses at the beat frequency, ω − Ω, where Ω is the orbital frequency. Because pulses are generated from both poles, power is also seen at the first harmonic of the beat frequency (aka the double beat: 2(ω − Ω)).

Light curves of AR Sco obtained near its discovery clearly showed that the beat-pulse pairs alternated in strength with the brighter pole being about twice the amplitude of the opposite pole (Marsh et al. 2016). This alternating strength was seen in the linearly polarized flux as well (Potter & Buckley 2018). However, in a Fourier component analysis, Pelisoli et al. (2022) noted that the quality of their model fit to the observed light curves was decreasing over time. They attributed this to stochastic changes in the relative strength of the pulse pairs. Takata et al. (2021) noted that the X-ray beat pulses were single peaked in 2016 but seen as double peaked in 2020. Here, we analyze 9 yr of AR Sco optical light curves to test for any systematic evolution in the beat-pulse strengths.

We analyze light curves studied by Gaibor et al. (2020). In addition, we have obtained rapid-cadence light curves of AR Sco using the Sarah L. Krizmanich Telescope (SLKT) from 2020 through 2023 and light curves in 2023 from the South African Astronomical Observatory using the high-speed photomultiplier instrument (HIPPO; Potter et al. 2008). We analyze both the total flux and polarized flux for two epochs obtained in 2016 and 2023 using the HIPPO instrument. Only light curves covering a substantial fraction of a binary orbit are included in this study. Properties of the photometric time-series data sets are listed in Table 1.

For each photometric time series, we constructed Lomb−Scargle (L-S hereafter, Lomb 1976; Scargle 1982) periodograms implemented using the LombScargle package in astropy (Astropy Collaboration et al. 2018). For example, L-S periodograms of the ULTRAcam photometry taken in 2015 and the SAAO photometry from 2023 are displayed in Figure 1. From the strongest peaks in the periodogram (ω − Ω, 2(ω − Ω), 2ω − Ω, 3(ω − Ω), 4(ω − Ω), 2ω, 2(2ω − Ω)), we reconstructed the 2015 and 2023 light curves, and these are also shown in Figure 1. The properties of the optical light curves appear to have changed significantly between 2015 and 2023. In particular, the strength of the "secondary" beat pulse has increased relative to the main pulse over this period.

To quantify the relative strengths of the beat pulses averaged over an orbit, we use the L-S periodogram peaks at the beat frequency (A_(ω−Ω)) and the double-beat frequency ($A{{\prime} }_{2(\omega -{\rm{\Omega }})}$) to define the ratio:

$$\begin{eqnarray}&&{R}_{\mathrm{beat}}=\displaystyle \frac{A{{\prime} }_{2(\omega -{\rm{\Omega }})}}{{A}_{(\omega -{\rm{\Omega }})}+A{{\prime} }_{2(\omega -{\rm{\Omega }})}}\ .\end{eqnarray} \tag{ 1 }$$

When R_beat ≈ 0, only a single beat pulse is detected over a beat period. Alternatively, when R_beat approaches unity, the two beat pulses are similar in amplitude. Here, $A{{\prime} }_{2(\omega -{\rm{\Omega }})}={A}_{2(\omega -{\rm{\Omega }}})-{H}_{2}\,{A}_{(\omega -{\rm{\Omega }})}$ is the amplitude of the double-beat peak corrected for a contribution of the second harmonic of the beat frequency. The parameter H₂ is the ratio between the amplitudes of the beat frequency and its second harmonic. Simulated light curves show that an H₂ = 0.30 accounts for this harmonic contribution ⁷ that only becomes significant when when R_beat < 0.5, that is, when the secondary pulse is weak.

The measured beat-pulse ratios, R_beat, for 14 light curves are given in Table 1. Uncertainties on the ratio measurements were estimated by taking L-S periodograms of subsets of each light curve. The variance in R_beat was then used to calculate the error bars shown. We also estimated the beat-pulse ratio in linearly polarized flux using photometric time series obtained in 2016 and 2023, and these results are shown in Table 1.

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Table 1. Time-series Photometry

Total Flux
Year	Start	Cadence	Length	Telescope	${R}_{{\rm{beat}}}$ a
	(MJD)	(s)	(hr)
2015	57197.9	1.3	2.8	WHT	0.42±0.03
2016	57463.0	1.0	2.3	SAAO	0.53±0.03
2016	57536.4	1.0	6.8	SAAO	0.49±0.03
2017	57891.3	2.0	4.0	SAAO	0.37±0.05
2018	58200.5	1.0	4.0	SAAO	0.58±0.03
2019	58613.4	1.0	5.0	SAAO	0.66±0.03
2020	59016.1	3.3	4.6	SLKT	0.69±0.04
2020	59037.1	2.5	4.5	SLKT	0.62±0.03
2021	59322.3	3.0	4.6	SLKT	0.65±0.03
2021	59342.2	2.6	4.6	SLKT	0.77±0.04
2022	59751.1	3.6	4.1	SLKT	0.79±0.03
2023	60090.2	3.9	4.2	SLKT	0.79±0.04
2023	60091.3	2.0	4.5	SAAO	0.85±0.03
2023	60092.4	2.4	4.9	SAAO	0.82±0.03
Linearly Polarized Flux
2016	57463.0	1.0	2.3	SAAO	0.52±0.03
2023	60091.3	2.0	4.5	SAAO	0.81±0.03

Note.

^a Beat/double-beat amplitude ratio calculated from Equation (1).

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Figure 2 displays the AR Sco beat-pulse ratio over time. While there is scatter in the ratio from night to night, the trend is for an increasing ratio consistent with the pulse pairs evolving to nearly equal strength. The R_beat parameter for the linearly polarized flux was also seen to increase significantly between 2016 and 2023.

Interpreting the changes in pulse amplitude is difficult given that the source of the relativistic electrons has not been established. Electrons may be accelerated by direct interaction between the WD magnetic field and the field of the secondary star (e.g., Takata et al. 2017; Garnavich et al. 2019). Slow changes in the secondary star's magnetic field could influence the WD field lines involved in trapping the emitting electrons.

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Katz (2017) proposed that the spin axis of the WD could be tilted relative to the orbital angular momentum vector, leading to a precession of the WD and its magnetic field configuration. Katz predicted that WD precession might impact the phase of the brightest point in the orbital modulation seen in AR Sco. Peterson et al. (2019) did not detect a shift in the orbital modulation curve using archival photometry. ⁸ However, the narrowness of the synchroton beams could provide a sensitive test of precessional motion if it results in relative changes in the viewing angles of the two poles. Katz (2017) predicted precession periods between 20 and 200 yr primarily depending on the WD mass.

3.1. Precession Model

To test if WD precession could generate the observed variation in pulse amplitudes, we constructed a simple model illustrated in Figure 3. The model parameters include the binary inclination, i, the obliquity of the spin axis, , and the angle between the magnetic dipole and the WD spin axis, χ. From this geometric model, we calculate the minimum angle, θ, between the center of the synchrotron beams and the direction to the Earth and follow its variation over a precession cycle.

Based on the estimated binary mass ratio (Marsh et al. 2016) and lack of detected eclipses, the orbital inclination is i ≈ 75° (Garnavich et al. 2019). That there are two pulses per WD spin suggests that the magnetic field axis is highly inclined to the spin axis (Geng et al. 2016); however, the symmetry of a truly perpendicular rotator (χ > 80°) would severely limit the range of pulse ratio variations over a precession cycle.

Following the beam model described in Potter & Buckley (2018), we approximate the synchrotron pulse profile by a Gaussian function with σ = 40°. Thus, the intensity of the synchrotron beam will have fallen to half its peak value when viewed at an angle of θ ≈ 47°. For simplicity, we assume that the peak intensity and profile distributions are identical for the two emitting poles.

We find that the observed change in R_beat from 0.5 to 0.8 can be achieved by the precession model using reasonable values of the parameters (Figure 4). A WD obliquity near ≈ 10° is sufficient to reproduce the observations. Interestingly, when the obliquity exceeds the co-inclination (90 − i), the secondary pulse strength can exceed the amplitude of the primary pulse, as shown in the lower panel of Figure 4.

Besides the R_beat ratio parameter, an additional observational constraint on precession models is a direct measurement of the amplitude changes of the pulses from each pole. As displayed in Figure 4, the amplitude of the pulse from the dominant pole varies by only 10% over a quarter of the precession cycle for the first set of model parameters. In general, this small amplitude change for one pole results when the obliquity is low and χ ≈ i. For this geometry, changes in θ for one of the emitting poles remains small over a precession cycle, while the second pole creates most of the variation in the amplitude ratio.

The lower panels of Figure 1 show that the beat-pulse amplitude of the dominant pole remained nearly the same between 2015 and 2023 and that most of the R_beat evolution came from strengthening in the secondary pulse. We estimate that the amplitude of the primary pulse varied by no more than 0.1 mag over the 9 yr of observation. We therefore infer that χ ≈ i for AR Sco.

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3.2. Precession Period

The AR Sco beat-pulse pairs have evolved from a strong asymmetry to become nearly equal in amplitude over a decade of observations. This evolution is supported by the changes noted in the X-ray beat pulse noted by Takata et al. (2021). We also find that the beat pulses have evolved in linearly polarized flux and that their amplitudes are now nearly equal. This suggests that the evolution results from physical changes or viewing angle variations in the highly polarized synchrotron beams.

The precession model suggests that in 2023 we were viewing the WD spin axis oriented so that the synchrotron beams are seen symmetrically, while in 2015, precession of the spin axis resulted in a tilt that favored our view of one of the magnetic poles. If precession is the origin of this evolution, then over the next 10 yr the ratio of the beat-pulse amplitudes, R_beat, will reach a maximum followed by a return to asymmetric beat pulses.

We dedicate this study to Tom Marsh. P.M.G. thanks the Krizmanich family for their generous donations for the construction and support of the Sarah L. Krizmanich Telescope.