RADIUS CONSTRAINTS FROM HIGH-SPEED PHOTOMETRY OF 20 LOW-MASS WHITE DWARF BINARIES

All 20 of the compact binaries discussed here were discovered through the ELM Survey, whose binary parameters are outlined in Table 1 in order of increasing orbital period. The effective temperature and surface gravity determinations have been updated to reflect fits to the latest one-dimensional (1D) model atmospheres, as detailed in Gianninas et al. (2014a).

The mass estimates use the most recent evolutionary sequences of He-core WDs from Althaus et al. (2013). We update the merger times to reflect these new primary mass estimates using the formalism of Landau & Lifshitz (1958). Since these are all single-lined spectroscopic binaries, K₁ refers to the radial velocity (RV) semi-amplitude of the visible low-mass WD; we follow the convention that the primary is the star that is visible, even though the primary is usually the lowest-mass component. We adopt a systematic uncertainty of ±0.02 M_☉ for M₁ in all cases.

The targets in this sample range in SDSS-g magnitude from 15.7 to 20.2 mag. They are predicted to be strong sources of gravitational wave radiation, as all of them will merge within 5.5 Gyr; the 10 shortest-period systems have P_orb < 1.5 hr and will merge in less than 160 Myr. These binaries all reside within 0.2–3.8 kpc.

The majority of our high-speed photometric observations were obtained at the McDonald Observatory in the three yr between 2010 June and 2013 May. For these observations, we used the Argos instrument, a frame-transfer CCD mounted at the prime focus of the 2.1 m Otto Struve telescope (Nather & Mukadam 2004). Observations were obtained using a BG40 filter to reduce sky noise, which covers a wavelength range of roughly 3000–7000 Å and is centered at 4550 Å.

We performed weighted, circular aperture photometry on the calibrated frames using the external IRAF package ${ccd\_hsp}$ (Kanaan et al. 2002). We divided the sky-subtracted light curves by the brightest comparison stars in the field to correct for transparency variations, and applied a timing correction to each observation to account for the motion of the Earth around the barycenter of the solar system (Stumpff 1980; Thompson & Mullally 2009).

Additional observations of J0106−1000 were obtained using the GMOS-S instrument mounted on the 8.1 m Gemini-South telescope at Cerro Pachón over 4.3 hr in 2011 September and October under program GS-2011B-Q-52. The first 2.1 hr of observations were obtained through an SDSS-g filter, followed by 2.2 hr of observations through an SDSS-r filter. We performed aperture photometry using DAOPHOT (Stetson 1987) and used 12 photometrically constant SDSS point sources in our images for calibration. For these data, we used the IDL code of Eastman et al. (2010) to apply a barycentric correction.

Extended time-series photometry for J0751−0141 was obtained using the Puoko-nui instrument (Chote et al. 2014) mounted at the Cassegrain focus of the 1.0 m telescope at Mt. John Observatory. These observations were also obtained through a BG40 filter and were reduced using identical procedures as the Argos data from McDonald Observatory.

For two of the faintest targets in our sample, J0755+4906 and J2338−2052, we obtained data during a commissioning run using a science camera mounted at the f/5 wavefront sensor of the 6.5 m MMT telescope. This marks some of the first data published taken with this camera. J0755+4906 was also observed using the DIAFI instrument mounted on the 2.7 m Harlan J. Smith telescope at McDonald. Both the MMT and DIAFI data were obtained through an SDSS-g filter, and we performed aperture photometry with ${ccd\_hsp}$. As with J0106−1000, we applied a barycentric correction to these observations using the IDL code of Eastman et al. (2010).

For each system, the total duration of our photometric observations (T_obs) can be found in the second-to-last column of Table 1.

There are six major effects that can cause photometric variability in the primary of a binary system: eclipses, reprocessed light from the secondary (reflection), ellipsoidal variations, Doppler beaming, pulsations, and RV shifts into and out of a narrow-band filter (Robinson & Shafter 1987). We have been fortunate to see all of these effects, save for reflection, in our photometry of ELM WDs.

We proceed to constrain the variability using a harmonic analysis, outlined in Hermes et al. (2012a). In short, we phase our observations to the spectroscopic conjugation and group the data into 100 orbital phase bins. We perform a simultaneous, nonlinear least-squares fit with a five-parameter model that includes an offset and defines the amplitude of the (co) sine terms for Doppler beaming (sin ϕ), ellipsoidal variations (cos 2ϕ), reflection (cos ϕ), and the first harmonic of the orbital period (sin 2ϕ). The values are detailed in Table 2 and the stated uncertainties arise from the covariance matrix of this fit.

Table 2. Light Curve Analysis of 20 Merging Low-mass WD Binaries

Object	Porb, fold	DBexp	sin (ϕ)	cos (2ϕ)	cos (ϕ)	sin (2ϕ)	$\chi ^2_{\rm red}$	T2	i	M2	u1	τ1
	(minutes)	(%)	(%)	(%)	(%)	(%)		(kK)	(deg)	(M☉)
J0651+2844	12.7534424	0.47	0.56(07)	4.01(07)	0.02(07)	0.01(07)	...	8.7(0.5)	84.4(2.3)	0.50	0.39	0.566
J0106−1000	39.10406	0.29	0.23(12)	1.76(12)	0.19(12)	0.02(12)	1.00	<21	<76.6	>0.41	0.40	0.552
J1630+4233	39.830	0.22	0.32(09)	0.02(09)	0.00(09)	0.02(09)	0.91	<17	<82.8	>0.66	...	...
J1053+5200	61.286	0.20	0.22(13)	0.35(13)	0.02(13)	0.15(13)	0.92	<21	<82.0	>0.27	...	...
J0056−0611	62.466700	0.35	0.04(06)	0.53(06)	0.01(06)	0.05(06)	1.02	<13	<81.8	>0.47	0.46	0.700
J1056+6536	62.654	0.17	0.68(29)	0.34(29)	0.00(29)	0.19(28)	0.95	<36	<84.9	>0.34	...	...
J0923+3028	64.8482	0.21	0.43(03)	0.08(02)	0.07(02)	0.16(02)	0.64	<25	<84.6	>0.37	...	...
J1436+5010	65.95	0.25	0.35(05)	0.12(05)	0.01(05)	0.07(05)	0.67	<23	<84.4	>0.46	...	...
J0825+1152	83.7936	0.18	0.43(14)	0.04(14)	0.04(14)	0.19(14)	0.92	<52	<84.8	>0.50	...	...
J1741+6526	87.9984	0.54	0.50(07)	1.30(07)	0.11(07)	0.00(07)	1.01	<31	<84.4	>1.12	0.54	0.791
J0755+4906	90.749	0.38	0.36(29)	1.05(29)	0.82(30)	0.22(30)	0.84	<58	<90.0	>0.81	...	...
J2338−2052	110.07	0.10	0.00(16)	0.30(15)	0.19(16)	0.01(16)	0.98	...	<83.8	>0.15	...	...
J0849+0445	113.2013	0.40	0.78(13)	0.41(13)	0.00(13)	0.03(12)	0.66	<24	<85.7	>0.66	...	...
J0022−1014	115.04	0.10	0.05(35)	0.01(35)	0.00(37)	0.12(36)	1.03	...	<86.0	>0.21	...	...
J0751−0141	115.21814	0.33	0.25(03)	3.20(03)	0.20(03)	0.00(03)	...	<45	85.4(9.4)	0.97	0.41	0.581
J2119−0018	124.949	0.42	0.71(15)	1.44(15)	0.14(15)	0.08(15)	1.02	<27	<82.8	>0.76	0.52	0.265
J1234−0228	131.66	0.07	0.07(06)	0.13(06)	0.09(06)	0.12(06)	0.91	...	<71.5	>0.10	...	...
J0745+1949	161.9298	0.15	0.00(04)	1.49(04)	0.07(04)	0.01(04)	0.98	...	<72.5	>0.11	0.58	0.310
J0112+1835	211.55545	0.33	0.00(03)	0.32(03)	0.04(03)	0.00(03)	1.02	<24	<85.3	>0.63	0.51	0.826
J1233+1602	217.30	0.33	0.07(21)	0.61(22)	0.22(22)	0.19(22)	0.90	<57	<90.0	>0.85	...	...

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We generate point-by-point photometric uncertainties using the formalism described in Everett & Howell (2001), which we adopt in the final binned light curves in Figures 1–4. However, this generally underestimates the uncertainties by roughly 40%—there are eight systems for which we do not detect any significant variations at the orbital or half-orbital period, and we find that these eight systems have, on average, $\chi ^2_{\rm red}=\chi ^2/\rm {d.o.f.}=1.92$. We rescale the uncertainties for all systems so that $\chi ^2_{\rm red}=1.0$ with the expectation that these eight systems only have variability at the orbital period commensurate with the RV-predicted Doppler beaming signal (see Section 3.1).

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We use the lack of eclipses or a reflection effect to place constraints on the system inclination and the maximum temperature of the secondary, respectively, and we list these additional constraints in Table 2. To constrain this inclination angle, we use the He- and CO-core WD mass–radius models of Althaus et al. (2013) and Renedo et al. (2010) to estimate the radius of the secondary given its minimum mass, which we coarsely arrive at given i < sin ⁻¹[(R₁ + R₂)/a]; the eclipse depth, which roughly scales as (R₂/R₁)², would be detectable given our observations for all but two systems (J0755+4906 and J1233+1602). We also use this estimate for the secondary radius to constrain the temperature of the secondary using the maximum value of the cos ϕ in Table 2 and the expected amplitude of a reflection effect approximated by δf = 17/16(R₁/a)²[1/3 + 1/4(R₁/a)](T₂/T₁)⁴(R₂/R₁)⁴ (Kopal 1959).

Additionally, we have computed Fourier transforms (FTs) of our time-series photometry, which allows us to search for any variability, such as pulsations, that is not at a harmonic of the orbital period. In some cases, this Fourier analysis has allowed us to refine the orbital period of the system from the less-sampled RV observations. Usually, though, the periods are so long and the data coverage so sparse that there is too much alias structure around the peaks of interest to justifiably refine the orbital period. Since we have signals that may occur at both cos ϕ and sin ϕ, our Monte Carlo analysis yields a more reliable estimate for the amplitude of reflection and Doppler beaming, respectively. Comparing these FTs to the FT of the brightest comparison star in the field allows us to check for any coincident peaks that may be the result of atmospheric variability or instrumental effects. We have also computed significance levels based on 〈A〉, the average amplitude of the FT within a 1000 μHz region centered at 2000 μHz.

Doppler beaming introduces a detectable modulation in the flux of a binary star that is approaching or receding, with an expected amplitude primarily dictated by the RV of the source (Zucker et al. 2007). The dominant component of the effect is due to aberration of the high-velocity source, although there is also a dependence on the frequency of the emitted radiation with the source velocity (Bloemen et al. 2011).

Observing Doppler beaming in binaries is still new, but by no means is it novel. The first ground-based detection involved the WD+WD binary NLTT 11748 (Shporer et al. 2010), although there was marginal evidence for the effect in the short-period sdB+WD binary KPD 1930+2752 (Maxted et al. 2000). The effect is now routinely observed using high-quality, space-based photometry (e.g., van Kerkwijk et al. 2010).

Nine of the twenty systems in our sample display a significant modulation at the orbital period commensurate with Doppler beaming. Since we already know the RV amplitude of these systems from spectroscopy, detecting this signal yields little new information about our binaries. However, it does help us to calibrate our photometric uncertainties. We can generally predict the expected Doppler beaming signal to ±0.02% relative amplitude, which we list in Table 2, and we compare this to the results of our harmonic analysis. For all but three systems, the observed sin ϕ term matches within 3σ the expected Doppler beaming amplitude, which is given in Table 2. We use these Doppler beaming signal predictions to rescale the photometric uncertainties such that $\chi ^2_{\rm red}=1.0$.

However, this is not an entirely robust approach, since there are occasionally long-timescale transparency variations or other atmospheric variability contaminating some of the expected signals, which sometimes overlap with the orbital period, and thus the Doppler beaming signal. For example, one anomalous system (J0923+3028) has an observed sin ϕ term more than twice the expected value, but the photometry may be influenced by long-period atmospheric variability; there is a comparable signal at 1.4 hr, which is longer than the 1.08 hr orbital period.

In the cases for which we see tidal distortions, the dominant modulation occurs when the larger face comes into view twice per rotation, effectively a cos 2ϕ modulation of the rotation rate of the WD tidal bulge, which would be the orbital period for a synchronized system (ϕ = 0–2π represents one full orbit). Since tidal distortions do not cause a perfectly ellipsoidal shape, we treat the ellipsoidal variations as harmonics to the first four cos ϕ terms, as derived in Morris & Naftilan (1993). These authors showed that the ellipsoidal variation amplitude is dominated by

$$\begin{equation*} L(\phi) / L_0 = \frac{-3 \, (15 + u_1) \, (1 + \tau _1) \, q \, (R_1 / a)^3 \, \sin ^2 i}{20 \, (3-u_1)} \cos (2\phi), \end{equation*}$$

where all of the terms can be written in terms of the inclination (i), given that we can determine the mass ratio (q = M₂/M₁) and semimajor axis of the system (a) from the spectroscopy and that we can assume reasonable values for the linear limb-darkening (u₁) and gravity-darkening (τ₁) coefficients for the primary. We note that this formalism is only valid in the regime when the tidally distorted star is rotating at or near the orbital period, as shown by Bloemen et al. (2012). For now, we assume that this is valid.

We adopt limb-darkening coefficients calculated by Gianninas et al. (2013), which we list in Table 2. For all WDs with T_eff >10, 000 K, we assume that the surface is purely radiative and calculate the gravity-darkening coefficients using the formalism outlined in Morris (1985), where β = 0.25. This assumption is likely valid given our theoretical (and growingly empirical) blue edge for pulsating ELM WDs found in Figure 5 of Hermes et al. (2013b); these pulsations are driven by a hydrogen partial-ionization zone, which coincides with the onset of a deepening surface convection zone. Our adopted gravity-darkening coefficients are included in Table 2.

Eight systems in our sample show a significant cos (2ϕ) variation in the light curve. For each system, we have also calculated the amplitudes of the third- and fourth-cosine harmonics, and find that they are insignificant within the uncertainties, as we would expect from the predicted amplitude ratios of Morris & Naftilan (1993). For all systems, we do not observe these higher-order harmonics to have amplitudes 2σ above the least-squares uncertainties; in fact, we do not expect these harmonics to have amplitudes above 0.13% relative amplitude for any of our systems.

We can rewrite the equation of Morris & Naftilan (1993) characterizing the amplitude of the ellipsoidal variations (A_EV) by recasting the semimajor axis using Kepler's third law:

$$\begin{equation} A_{\rm EV} = \frac{3 \pi ^2 (15 + u_1)(1 + \tau _1) M_2 R_1^3 \sin ^2{i}}{5 P_{\rm orb}^2 (3 - u_1) G M_1 (M_1+M_2)}. \end{equation} \tag{ 1 }$$

Additionally, we have placed spectroscopic constraints on the system. Dynamically, we know the mass function from previous time-series spectroscopic observations:

$$\begin{equation} f_1(M_2) = \frac{P_{\rm orb} K_1^3}{2 \pi G} = \frac{M_2^3 \sin ^3{i}}{(M_1+M_2)^2}. \end{equation} \tag{ 2 }$$

We also know the primary surface gravity (g₁) from atmosphere models fit to its summed spectrum:

$$\begin{equation} g_1 = \frac{G M_1}{R_1^2}. \end{equation} \tag{ 3 }$$

Thus, we have three equations and four unknowns (M₁, M₂, R₁, and i). To draw out the line of mass–radius constraints for each ELM WD, we perform 10, 000 Monte Carlo simulations. We draw a random inclination using the proper distribution of random orientations, as well as a random value from within the measured probability distribution for P_orb, K₁, log g₁, and A_EV, and solve the system of three equations. We reject any solutions that have M₁ > 1.4 M_☉ or M₂ > 3.0 M_☉but do not impose any inclination constraints.

Our Monte Carlo simulations draw out a series of allowed values along the M₁ − R₁ plane given the observed ellipsoidal variations. As an example, in Figure 5, we display the full output for J0056−0611 as light gray points. To further constrain the radius estimate, we highlight only those points within 0.02 M_☉ (our adopted systematic uncertainty) of the mass adopted by matching the T_eff and log g to the models of Althaus et al. (2013). The distribution of radii from our Monte Carlo simulations are symmetric for each star constrained within this mass range, so the adopted radius estimates listed in Table 3 are found from a Gaussian fit to this distribution of radii.

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Table 3. Parameters Constrained from Monte Carlo Simulations Using the Observed Ellipsoidal Variations

Object	M1	R1	M2	i	dPEV/dtGR	τdetect	T0, ELV
	(M☉)	(R☉)	(M☉)	(deg)	(10−13 s s−1)	(yr)	(BJDTDB)
J0651+2844	0.252	0.040 ± 0.002	$0.50^{+0.04}_{-0.01}$	$82.7^{+7.3}_{-8.4}$	$-39.8^{+2.5}_{-0.7}$	<1	2455955.1734648(37)
J0106−1000	0.191	0.063 ± 0.008	$0.50^{+0.43}_{-0.11}$	$60.3^{+28.7}_{-19.5}$	$-9.6^{+5.6}_{-2.0}$	11	2455533.57568(11)
J0056−0611	0.174	0.056 ± 0.006	$0.80^{+0.63}_{-0.30}$	$49.7^{+22.3}_{-12.8}$	$-2.9^{+14.0}_{-0.4}$	37	2455891.62845(42)
J1741+6526	0.170	0.076 ± 0.006	$1.16^{+0.41}_{-0.05}$	$78.3^{+11.7}_{-15.8}$	$-2.1^{+0.3}_{-0.1}$	41	2455686.79210(21)
J0751−0141	0.194	$0.138^{+0.012}_{-0.007}$	$1.02^{+0.38}_{-0.05}$	$77.3^{+12.7}_{-17.2}$	$-1.3^{+0.4}_{-0.1}$	34	2455960.660518(54)
J2119−0018	0.160	0.103 ± 0.016	$0.80^{+0.44}_{-0.10}$	$75.1^{+14.9}_{-20.6}$	$-0.8^{+0.3}_{-0.1}$	150	2455769.84065(66)
J0745+1949	0.164	$0.176^{+0.090}_{-0.025}$	$0.14^{+0.13}_{-0.07}$	$63.2^{+26.8}_{-32.4}$	$-0.14^{+0.10}_{-0.06}$	180	2456245.94304(51)
J0112+1835	0.161	0.088 ± 0.009	$0.70^{+0.45}_{-0.11}$	$70.3^{+19.7}_{-19.2}$	$-0.32^{+0.14}_{-0.04}$	470	2455808.79023(89)

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The most precise constraints on the radii of low-mass WDs come from detached eclipsing systems. Fortunately, there are now six known low-mass WDs in eclipsing binaries: NLTT 11748 (Steinfadt et al. 2010; Kilic et al. 2010; Kaplan et al. 2014a), CSS 41177 A&B (Parsons et al. 2011; Bours et al. 2014), GALEX J1717+6757 (Vennes et al. 2011), J0651+2844 (Brown et al. 2011; Hermes et al. 2012b), and J0751−0141 (Kilic et al. 2014b). We include CSS 41177B (Bours et al. 2014) and the two higher-mass solutions for NLTT 11748 (Kaplan et al. 2014a) as black squares in Figure 5; the other systems either do not have stringent enough constraints on the WD radius or have a mass too high to display in this figure.

Two of the systems exhibiting ellipsoidal variations (J0651+2844 and J0751−0141) are also eclipsing, and in Figure 5 we include black squares corresponding to the radius values derived from light curve models to the eclipses. There is excellent agreement between the results of our Monte Carlo simulations and the light curve fits, which helps to validate our method.

There are four additional low-mass WDs in eclipsing binaries with A-star companions, all discovered in the Kepler field. These WDs are KOI81B (van Kerkwijk et al. 2010; Rowe et al. 2010), KOI74B (van Kerkwijk et al. 2010; Bloemen et al. 2012), KHWD3 (Carter et al. 2011), and KHWD4 (Breton et al. 2012). We mark these low-mass WDs as green squares in Figure 5 to differentiate between the other eclipsing WDs, because the A-star companions could contribute to inflating the radius of the WD (Carter et al. 2011).

We also include in Figure 5 the theoretical mass–radius relations of He-core WDs from Althaus et al. (2013). These tracks generally show that the WD radius increases with increasing T_eff and decreasing mass. Note that such low-mass WDs (<0.18 M_☉) are expected to quiescently burn hydrogen and are not theoretically predicted to undergo CNO flashes; for example, the large jump in radius for the low-mass end of the 12, 880 K isotherm in Figure 1 demonstrates the expectedly larger radius for a 0.1762 M_☉ model, which does not undergo CNO flashes and thus has a more massive residual hydrogen layer.

Our tidally distorted ELM WDs are among the lowest-mass WDs with radius constraints, so our observational results fill an important and untested region of the mass–radius relation. Six of our eight radius measurements are generally consistent with the models of Althaus et al. (2013). In some cases (notably J0106−1000 and J0651+2844), the observed radii are slightly larger than expected given their spectroscopic mass and temperature.

However, two outliers have radii significantly larger than expected from the He-core WD models: J0751−0141 and J0745+1949. It is possible that these two WDs are not on their final cooling track, but are instead in another part of their evolution, perhaps recently undergoing a CNO flash. Assuming that the surface of J0745+1949 is radiative and adopting a larger gravity-darkening coefficient for this 8380 K WD cannot explain this discrepancy; adopting τ₁ = 0.967 still yields $R_1 = 0.153^{+0.055}_{-0.020}$ R_☉.

It is notable that J0745+1949 is one of the most metal-rich WDs known (Gianninas et al. 2014b), which could possibly be the result of mixing induced by a recent CNO flash. If so, then the mass determined from the T_eff and log g may not accurately represent the WD mass, which was adopted assuming that the WD was on its terminal cooling track. Higher-mass models indeed cross the same position in T_eff and log g space while undergoing CNO flashes before their terminal cooling track. There is mounting evidence that many low-mass WDs do not appear to be on a terminal cooling track, especially those that are companions to millisecond pulsars (e.g., Kaplan et al. 2013, 2014b). There are likely still unexplained complexities to the evolution of low-mass WDs.

If we do not restrict our Monte Carlo simulation analysis of J0745+1949 by the primary mass of 0.164 M_☉, we instead find $M_1 = 0.38^{+0.28}_{-0.22}$ M_☉, $R_1 = 0.25^{+0.10}_{-0.07}$ R_☉, and $M_2 = 0.19^{+0.16}_{-0.09}$ M_☉. However, this would suggest our log g estimate from spectroscopy is off by more than 0.8 dex, which is highly unlikely. The radius of a 0.363 M_☉ model WD in the throws of a CNO flash can change by more than 0.15 R_☉ in less than a year (Althaus et al. 2013), which would cause a clear change in the amplitude of the ellipsoidal variations, so follow-up photometry of J0745+1949 could constrain this scenario.

The ellipsoidal variations in the shortest-period systems in our sample provide a unique opportunity to act as a stable clock that can be used to monitor any changes to the system as a result of orbital decay from the emission of gravitational wave radiation. In each case, the ellipsoidal variations show that the tidal bulge of the primary is synchronized with the orbital period, to the limit of our uncertainties. As that orbital period shrinks with the emission of gravitational waves, this tidal bulge will spin up, and the period of the ellipsoidal variations will decrease, which we can detect by monitoring the arrival times of the ellipsoidal variations.

Some of these systems are so compact that it is possible to detect the influence of gravitational waves within a decade or less. We have already established that such a monitoring campaign is possible: we have used the times of minimum ellipsoidal variations in the 12.75 minute binary J0651+2844 as an independent clock to detect the rapid orbital decay due to gravitational wave radiation (Hermes et al. 2012b).

Our Monte Carlo simulations provide additional constraints on the most likely distribution of system inclinations and companion masses, which we include in Table 3. These parameters are found by fitting a lognormal probability density function, arising from the geometric mean and the 2σ (95.5%) inner and outer bounds.

The second most compact binary in our sample that displays ellipsoidal variations is the 39.1 minute J0106−1000. This system is a strong source of gravitational wave radiation; at i = 603, we expect the emission of gravitational waves to cause the orbit to decay at roughly dP/dt = −1.9 × 10⁻¹² s s⁻¹ (−0.06 ms yr⁻¹), which will produce a change in the half-orbital period of dP/dt = −9.6 × 10⁻¹³ s s⁻¹.

We have constructed an (O − C) diagram of the times of minimum ellipsoidal variations, guided by the period of the highest peak in the FT of our Argos data set, 39.104063 minutes. We find dP/dt = (0.3 ± 6.4) × 10⁻¹⁰ s s⁻¹, consistent with no change in period, as expected with less than a single year of coverage. Significantly, this effect accumulates with time squared, so these times of minima will change by more than 10 s within 7 yr of our initial observations in 2010 December. Roughly 30 hr of 2 m class telescope photometry in an observing season yield a phase uncertainty of roughly 8 s, so it is possible to obtain a 3σ detection of the spin-up of the tidal bulge due to the emission of gravitational waves within barely a decade of monitoring J0106−1000, since the arrival times of the ellipsoidal variations will deviate by >25 s after the first 11 yr.

We have made a similar set of calculations for the seven other systems with ellipsoidal variations, and include the results in Table 3. We include the calculated time it would take to make a 3σ detection of the period change given the phase uncertainty of 30 hr of 2 m class photometry, listed as τ_detect. It is possible to decrease this detection timescale, since more observations can increase the accuracy with which we can measure the phase of the minima of the ellipsoidal variations. We also include the T₀ from the first epoch of observations, which can be used in the future to construct an updated (O − C) diagram with more coverage.