PERIODIC VARIABILITY OF LOW-MASS STARS IN SLOAN DIGITAL SKY SURVEY STRIPE 82

We searched for periodicity in all Stripe 82 light curves having more than 10 epochs using the 1024-node "Athena" computing cluster at the University of Washington. We used the variable-span Supersmoother⁴ algorithm of Reimann (1994) for period estimation (see also Oh et al. 2004). For a given period estimate, Supersmoother implements running linear smooths of the data at multiple span lengths, using a localized cross-validation to determine the optimal span. Period estimates are then ranked by the sum of absolute residuals about each optimally smoothed model, with the "best" period yielding the smallest residuals. Supersmoother is able to uncover a variety of light curve shapes because it makes no explicit assumptions about the underlying shape of the curve, except that when folded it be smooth and continuous. Since the algorithm does not return an explicit uncertainty (or false alarm probability) on the period, we used the fact that we have multiple constraints on the true period from the g-, r-, and i-band light curves (which typically have the smallest photometric errors). We addressed each passband's data separately, applying Supersmoother to the entire calibrated light curve. The derived periods in the g, r, and i data are referred to as P_g, P_r, and P_i. We only considered the best period, as determined by the sum of absolute residuals, for each of the three passbands.

We used concordance between the periods returned in each of the three passbands to make an assessment as to the true period (if any) in the data. We required one of the following three matching criteria be met before we considered the object a periodic candidate.

• 1.  
  The standard deviation of P_g, P_r, and P_i is less than 10⁻⁵ times their average value.
• 2.  
  A similar test as the one above but applied to each period modulo the minimum of the three periods (to account for period aliasing).
• 3.  
  The standard deviation of two of the three values of P_g, P_r, and P_i is less than 10⁻⁴ times their average value. This allowed us to diagnose cases where Supersmoother returned an aberrant period for one of the light curves, or when the "correct" period was reported as the second or third best for one of the light curves.

As an additional step, we rejected those light curves where the matched period was longer than 365 days, or was within 0.01 days of the single-day sampling alias. These criteria resulted in a manageable number of periodic candidates (10³) that could be investigated through visual inspection.

We have chosen to focus here on the extraction of periodic variability for low-mass stellar systems. The photometric calibration of our database allows us to accurately select low-mass systems based upon their mean colors.⁵ We outline the two sets of color selection criteria below.

2.2.1. M Dwarfs

2.2.2. White Dwarfs

For each system that passed our period criteria, we folded the light curve at that period and visually inspected the phased data. This helped to reject spurious systems, as well as to classify the nature of the variability. We used the continuity of the folded light curves, with respect to the photometric error bars, to ascertain if the system is periodic.

During this process we noted that some returned periods were obvious aliases of the true period; for example, an eclipsing system with two different depth minima folded at half its period will have a bimodal light curve in eclipse. We therefore report here the periods that resulted in two maxima and minima in the folded light curves. We thus expect that our periods represent the maximum system period.

Recently Bhatti et al. (2010) published a catalog of the 0 hr < R.A. <4 hr portion of Stripe 82, with particular attention paid to periodic variability. They find 32 eclipsing or ellipsoidal binaries in this region of sky (Table 2; Bhatti et al. 2010), only 11 of which pass our color-selection criteria. The remaining 21 are not included in our analysis due to their average colors. However, to compare with the Bhatti et al. (2010) results, we manually performed our Supersmoother analysis on these 21 objects. Table 3 compares our period estimates for both classes of objects—those that pass our color cuts, and those that do not. We list our derived periods along with ratios of the periods found in the two studies (recall that periods here have been aliased to result in light curves with two maxima/minima and are expected to represent the maximum likely period for the system).

For four of the objects listed in Table 3, our analysis found a period that was not an exact match to, or an alias of, the Bhatti et al. (2010) period. Figure 1 shows the folded r-band light curves of these four objects: SDSS J003042.11+003420.2, SDSS J035300.50+004836.0, SDSS J000845.39+002744.2, and SDSS J023621.96+011359.1. For each panel, the light curve resulting from the periods derived in this analysis is shown on the top, and the light curve folded at the Bhatti et al. (2010) period is shown below with a +0.5 mag offset added for clarity. Both data sets appear to show the overall coherence expected of a periodic light curve folded at its correct period. However, the top set of light curves (resulting from our periods) show fewer outliers. We quantify this by undertaking a Fourier decomposition of the light curves using Equation (1) (defined in Section 4.2) and comparing the χ² between the model fits to the data sets folded at the two periods. This comparison yields χ²_Bhatti − χ²_Becker = [7; 119; 355; 217] for r-band Fourier decompositions of SDSS [J003042.11+003420.2; J035300.50+004836.0; J000845.39+002744.2; J023621.96+ 011359.1], respectively. This verifies that our periods are ones best supported by our data.

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For five of the objects, our analysis failed to find consistent periods in the g-, r-, and i-band data. For three of these, the Bhatti et al. (2010) period is found as the second-best period. For the other two, we do not find the Bhatti et al. (2010) period in the top three periods, for any passband. This includes the eclipsing binary from Becker et al. (2008), which was originally discovered in data from the Two Micron All Sky Survey (Skrutskie et al. 2006) and also found in the Stripe 82 catalog of Bhatti et al. (2010). Our corresponding light curve only shows one data point per eclipse minimum, and thus we were unable to derive an orbital period. This is evidence that the Bhatti et al. (2010) data and ours differ both in calibration and in content.

Finally, we examine those objects not contained in the Bhatti et al. (2010) sample. With our color cuts, we find 60 periodically variable objects between 0 hr < R.A. < 4 hr. Of these, only eight are found by Bhatti et al. (2010). We have checked that the 52 missed candidates do not correspond to stars identified by Bhatti et al. (2010) as RR Lyrae or δ Scuti, and have verified from our light curves that the objects appear truly periodic. We display the folded g-, r-, and i-band data for four of these objects in Figure 2. Bhatti et al. (2010) estimate ∼55% efficiency at recovering known input periods. Since we find a factor of 7–8 times more periodic variables in the same sample that they analyzed, its likely that the discrepancy arises due to differences in internal photometric calibration procedures, along with differences in our respective period-finding algorithms. We estimate our own period recovery efficiencies below.

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To assess the uncertainties on Supersmoother's best-fit periods given the particular properties of Stripe 82 data, including sampling rate and photometric error bars, we undertook a Monte Carlo simulation to determine our period recovery efficiency. We first defined brightness bins between r = 18 and r = 22 in steps of 0.5 mag to assess the impact of source brightness and photometric error on period recovery. Within each magnitude bin, we randomly selected 100 stars from the database whose intrinsic light curves have a reduced χ² < 1, where this random selection allows us to define an average light-curve sampling across the Stripe 82 region.

We next defined a range of shapes, periods, and amplitudes to study. Given the breadth of light-curve shapes allowed in eclipsing systems, we decided to limit our analysis to sinusoidal light curves for its computational simplicity, meaning our analysis is likely optimistic for the detached sample, especially those light curves spending small fractions of their duty cycle in eclipse. We investigated sinusoidal light curves having a range of periods P between −2.15 < log(P) < 0.445 (0.007 to 2.786) days in steps of Δlog(P) = 0.1. We also investigated a range of amplitudes A for the light curves with −2 < log(A) < 0.2 (0.01–0.63) mag, in steps of Δlog(A) = 0.2. For each object, we modified the r-band light curve with the variability imprint defined by each combination of period and amplitude, defined as Δr = Asin(2πϕ + φ). Here ϕ is derived from the epoch of observation T as ϕ = T/P − T//P, where the // operation generates the integer portion of T/P. The variable φ was generated randomly and represents a shifting of the zero point of the light curve.

We ran each modified light curve through Supersmoother and compared the recovered period with the known input period. We show the results for the bright end of this sample (19 < r < 19.5) in Figure 3. For each combination of P and A, this figure shows the fraction of the 100 input stars whose period was recovered to within 10⁻⁵ days (left panel) and 10⁻⁶ days (right panel). The solid contours at 5%, 50%, and 95% recovery are for comparing the best-fit period to the true period, while the dashed contours include the additional fraction of fitted periods that were two times aliases of the true period. In the latter case, an observer is likely to recognize upon period folding that the fitted period is an alias, and classify the light curve as periodic. The surface shows nearly 100% period recovery at high amplitude (A>0.2 mag) below 0.5 days. At periods longer than 0.2 days this efficiency falls rapidly when the matching criteria are tightened from 10⁻⁵ days to 10⁻⁶ days. This is understandable as at longer periods the objects go through fewer oscillations during a given temporal window. By including aliased-by-two periods, we can extend this recovery down to A>0.09 mag. We are not likely to recover any periodic light curves with amplitudes smaller than 0.02 mag at all periods.

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We examine how this matching degrades as the source brightness is decreased in Figure 4. In the left panel we plot the 5% recovery contours for five of the magnitude bins, and in the right panel the 95% recovery contours, for exact period matches. On average, if we also consider aliased-by-two periods we can extend the 5% recovery level 0.02 mag smaller in light curve amplitude, and the 95% recovery level 0.05–0.1 mag smaller in amplitude, at a given source brightness. The overall efficiency profiles suggest that the dearth of objects in our catalog with periods longer than ∼1 day may plausibly be explained by our low period recovery efficiency at these long periods, at all source brightnesses. Similarly the lack of systems fainter than r = 20.5 may be explained by a precipitous drop in efficiency at all but the largest amplitudes. Our high efficiency at bright magnitudes cannot alone explain our larger yield from the Stripe 82 data compared to the Bhatti et al. (2010) analysis.

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We found SDSS DR7 (Abazajian et al. 2009) spectra for 29 of our objects, including 27 whose spectra indicate they contain at least one M dwarf member. All but one of the M dwarf systems (26 of the 27; 96%) show signs of chromospheric magnetic activity (as traced by Hα emission). This is considerably higher than the magnetic activity rate seen in the field (24%, West et al. 2004) for a similar distribution of spectral types. This high rate of magnetic activity confirms previous results from close binaries, as compared to their field counterparts (Silvestri et al. 2006).

We next used the mean system colors, as well as the spectra when available to estimate spectral types. The results are given in Table 2. The Color Class is an estimate of the spectral type of the system using broadband colors and the color–spectral-type relations of Kowalski et al. (2009). Systems with u − g < 1.8 are labeled with a "+WD," indicating a white dwarf companion (West et al. 2004). The objects with spectra were typed using the Hammer software described in Covey et al. (2007), resulting in our Spectral Class estimation. The Color and Spectral classes typically agree to within 1 subtype.

We evaluate the light curves under the assumption that each is an eclipsing binary system with two minima in its light curve. The standard classification scheme for eclipsing binary systems uses the following categories.

• 1.  
  EW: W UMa-type (classical contact) binaries where both stars are surrounded by a common convective envelope (Lucy 1968). Primary and secondary eclipse depths are typically similar in depth due to the constant effective temperature of the envelope.
• 2.  
  EB: EW-type binaries with different eclipse depths. The archetype is β Lyrae. However this particular light-curve shape may result from a variety of physical configurations, including semi-detached binaries or thermal relaxation oscillations in EW systems (Flannery 1976).
• 3.  
  EA: detached binary stars, where both objects are contained completely within their respective Roche lobes.

Pojmanski (2002) has defined an empirical means of separating these systems into analogous classes describing the geometry of the system, through the use of a Fourier decomposition of the folded light curve. His classes are EC (contact, analogous to EW systems), ESD (semi-detached, analogous to EB), and ED (detached, analogous to EA). We performed a decomposition of the g-, r-, and i-band light curves of the form

$$\begin{eqnarray} m(\phi) & = & A_0 - \sum _{i=1}^4 A_i {\rm cos}(2 \pi i \phi + \varphi) + B_i {\rm sin}(2 \pi i \phi + \varphi).\quad \end{eqnarray} \tag{ 1 }$$

Here, A₀ represents the mean brightness of the system, ϕ corresponds to the phase, and φ is a nuisance parameter defined such that the model has minimum brightness at φ = 0. We use the Pojmanski (2002) polygons defining the EC, ESD, and ED regions in the two-dimensional space of Fourier coefficients A₂ and A₄, as displayed in Figure 5. Additionally, Rucinski (1997) provides a discriminant between contact and detached systems through the relationship A₄ = A₂ × (0.125 − A₂), shown as the dashed line in Figure 5. Objects lying above this line are more likely to be in contact, while objects lying below this are more likely to be detached. We use this boundary to provide asterisked classifications for contact (EC*) and detached (ED*) systems lying outside the polygon boundaries of Pojmanski (2002). We note that the Pojmanski (2002) regions overlap, and thus our objects may receive more than one classification per passband. Classifications per passband for all objects are given in Table 2.

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Table 2 contains 59 systems whose colors are consistent with being M dwarfs, inconsistent with having a white dwarf companion, and whose light-curves shapes in g, r, and i are consistent with the detached classification (ED or ED*) of Pojmanski (2002) and Rucinski (1997). This increases by a factor of several the number of published detached candidate systems (however see also Shaw & López-Morales 2007). Such systems are needed to examine the mass–radius relationships of low-mass stars (Section 1).

Only 11 stars in this subsample have orbital periods greater than 1 day, with the maximum period being 2.76 days for J003841.29+010756.0. There are only two data points in eclipse for this system, and we have classified this as having a C-quality light curve. There is, however, coherent out-of-eclipse variability that leads us to include this system in our sample. Since all our M dwarf ED systems have short orbital periods, they are likely being spun-up through orbital coupling, and their radii influenced by the enhanced magnetic field. This sample also contains seven systems classified as M4 and M5, which are expected to be fully convective (Burrows et al. 1993; Mullan & MacDonald 2001). Their mass–radius relationship is expected to be less affected by convection-related orbital coupling effects, but may still be affected by enhanced spot coverage (e.g., Morales et al. 2010).

Twenty-eight of the M dwarf light curves have shapes that suggest they are in contact configurations (EC or EC*, as defined in Section 4.2). This is a surprising outcome given the lack of such systems in the literature. The shortest period eclipsing M dwarf system verified so far, BW3 V38 (Maceroni & Rucinski 1997) with a period of 0.1984 days, is shown to be highly distorted but not in contact. Thus we caution that while our light-curve shapes may yield a EC/EC* classification, our systems may not be physically in contact, and will require further study for any definitive statements. However, the short periods of these systems are consistent with their sinusoidal light curves, both of which indicate a compact binary configuration.

Our period distribution for EC/EC* systems demonstrates a steep cutoff near 0.22 days (log P = −0.65), a feature seen in period distributions of earlier type contact systems (e.g., Paczyński et al. 2006). There is however one notable difference: the vast majority of our binaries (24/28) are accumulated near this period cutoff, with an rms of 0.02 days. These 24 candidates all have A- or B-graded light curves, as described in Section 3. In addition, 22 of them are brighter than r = 19, where our efficiency analysis in Section 3.2 shows that the median mis-fit in period at amplitudes larger than A = 0.016 mag (a condition matched by all objects) is less than 10⁻⁶ days, meaning the presence of this peak is established at high significance.

Thus, our sample of periods has a far smaller variance than is seen in the AFGK spectral-type contact binary population (Rucinski 2006), and densely populates the short period end of the distribution. We display in Figure 6 the distribution of periods for our contact (red, solid line) and detached systems (blue, dashed line). It is clear that the majority of contact systems are found at shorter periods than the detached sample. We also show the contact binary period distribution derived by Rucinski (2007) using all-sky bright-star data (ASAS contact binary sample; black, dashed line), which spans the approximate spectral types from A to K. Compared to the earlier type, bright-star periods, our systems are found at much shorter periods on average.

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Stepien (2006) has proposed a lower limit on the total mass of current-day contact systems of ∼1 M_☉, based upon the 0.22-day cutoff seen in previous survey data. Contact systems with less total mass than this are not expected to be seen, since their evolutionary timescales to Roche lobe overflow are longer than the age of the universe. Our observed accumulation of M dwarf periods near this cutoff supports evolutionary processes where angular momentum loss becomes less efficient at shorter orbital periods, which then suppresses the evolution to even shorter period systems. Interestingly, two EC/EC*-classified systems have periods less than 0.2 days, SDSS J200011.19+003806.5 (0.1455201 days) and SDSS J001641.03−000925.2 (0.1985615 days). Of these two, SDSS J001641.03−000925.2 has the higher quality light curve and a detailed study of this system is forthcoming (J. R. A. Davenport et al. 2011, in preparation).

We have matched each of these 24 objects with the nearest simulated light curve from our efficiency analysis in magnitude, period, and amplitude and find that most of the objects were expected to be detected at high (⩾90%) efficiency, with the exceptions of SDSS J035856.16+004010.2 (66%), and SDSS J202617.44−003738.6, J202247.66−002902.4, and J231823.93−004415.2 (77%). Using the efficiency per object to weight the sample, we find 0.087 EC-type M dwarf binaries per square degree over the nominal magnitude range 14 < r < 21.5.

Forty-two systems have blue colors (in u − g) consistent with having a white dwarf component, and red colors (in r − i and i − z) consistent with having an M dwarf component. Eleven of these are classified as being detached type ED/ED*. Most of these ED-classified systems have longer periods (0.4–0.9 days) than expected for current cataclysmic variable (CV) systems (e.g., Figure 9 of Howell et al. 2001). The consistency between the light-curve shapes (suggesting detached systems) and the periods (suggesting pre-CV and thus pre-contact configurations) lends support to our interpretation of the systems.

Wide (detached) CV-progenitor systems should currently have low-mass accretion rates, with heating (if any) of the accretor occurring through light particle winds as opposed to streaming accretion (e.g., Szkody et al. 2003). Multi-wavelength followup of these systems should be able to resolve the nature of any active accretion (Szkody et al. 2008).

Ten systems have colors and shapes consistent with being a contact (type EC/EC*) M dwarf/white dwarf binary. These appear to be eclipsing CV systems (e.g., Howell et al. 1997, and references therein). All but two of the 10 EC systems (J233538.33−002927.3 and J234309.23−005717.1) have periods less than 0.26 days, well within the period range expected for early CV systems (Kolb 1993). The shortest of these, J013851.54−001621.6 (A. C. Becker et al. 2011, in preparation), has a period of 0.072765 days, below the "period gap" expected from CV evolution models (Howell et al. 2001) and where the majority population of the current CVs are thought to reside (Howell et al. 1997). The EC light curves are too sparsely sampled to allow us to constrain the properties of any accretion-related effects such as hot spots. However, the average light curve color-variations in u−g for these EC-classified systems (〈Δ(u − g)〉 = 0.09 mag) is larger than in the EC systems that have no indications of white dwarf companions (〈Δ(u − g)〉 = 0.03 mag).

Four of the systems pass our white dwarf selection criteria, but do not pass our M dwarf selection criteria. Of the four, SDSS J212531.92−010745.8 has previously been studied by Nagel et al. (2006) who concluded that it has a pre-degenerate hot (PG 1159-type GW Vir) primary and faint M dwarf secondary being irradiated by the primary. Their derived orbital period is exactly 0.5 times ours, indicating that our requirement of folded light curve shapes with two minima has led to an alias in our period estimate. We note that the light curve of SDSS J025403.75+005854.2 is extremely similar, and is a promising candidate for another PG 1159-type system.