Dynamical modelling of CXOGBS J175553.2−281633: a 10 h long orbital period cataclysmic variable Free

ABSTRACT

We present modelling of the long-term optical light and radial velocity curves of the binary stellar system CXOGBS J175553.2−281633, first detected in X-rays in the Chandra Galactic Bulge Survey. We analysed 7 yr of optical I-band photometry from Optical Gravitational Lensing Experiment and found long-term variations from year to year. These long-term variations can most likely be explained with by either variations in the luminosity of the accretion disc or a spotted secondary star. The phased light curve has a sinusoidal shape, which we interpret as being due to ellipsoidal modulations. We improve the orbital period to be P = 10.34488 ± 0.00006 h with a time of inferior conjunction of the secondary star T₀ = HJD 2455260.8204 ± 0.0008. Moreover, we collected 37 spectra over 6 non-consecutive nights. The spectra show evidence for an evolved K7 secondary donor star, from which we obtain a semi-amplitude for the radial velocity curve of K₂ = 161 ± 6 km s⁻¹. Using the light-curve synthesis code xrbinary, we derive the most likely orbital inclination for the binary of i = 63.0 ± 0.7 deg, a primary mass of M₁ = 0.83 ± 0.06 M_⊙, consistent with a white dwarf accretor, and a secondary donor mass of M₂ = 0.65 ± 0.07 M_⊙, consistent with the spectral classification. Therefore, we identify the source as a long orbital period cataclysmic variable star.

1 INTRODUCTION

Cataclysmic variables (CVs) are binary star systems composed of a white dwarf primary accreting matter from a main sequence or evolved secondary star (Patterson 1984; Warner 1995; Kalomeni et al. 2016). Low-mass X-ray binaries (LMXBs) are analogous systems where the primary is either a black hole or a neutron star, instead of a white dwarf (Remillard & McClintock 2006). There are only about 20 dynamically confirmed black hole X-ray binaries known in the Milky Way (e.g. Casares & Jonker 2014). Finding and modelling CVs and LMXBs allows us to better understand the formation of compact objects and test binary evolution models (Jonker & Nelemans 2004; Repetto & Nelemans 2015).

The Chandra Galactic Bulge Survey (GBS) is a survey tasked with finding more quiescent LMXBs. Towards this goal the GBS covered a total of 12 deg² near the Bulge of our galaxy and found 1640 X-ray sources (Jonker et al. 2011, 2014). Subsequent studies have identified counterparts to these sources in multiple wavelengths; from radio (Maccarone et al. 2012) and near-infrared (Greiss et al. 2014) to optical (Hynes et al. 2012; Udalski et al. 2012; Britt et al. 2014; Wevers et al. 2016b, 2017). Some of these counterparts have been deemed likely accreting binaries, motivating further photometric and spectroscopic follow up (Ratti et al. 2013; Wevers et al. 2016a; Johnson et al. 2017).

In this work, we focus on one of these objects, CXOGBS J175553.2−281633 (hereafter CX137). The optical counterpart to CX137 was first identified by Udalski et al. (2012) and classified as an eclipsing binary with a spotted donor star and an orbital period of 10.345 h. Subsequent spectroscopic and photometric follow up by Torres et al. (2014) revealed broad Hα emission and an orbital period consistent with that of Udalski et al. (2012). Based on the properties of the Hα emission line and an X-ray luminosity of L_x > 5.8 × 10³⁰ erg s⁻¹, Torres et al. (2014) classified the source as a potential low-accretion rate CV or quiescent LMXB with a G/K-type secondary, supporting the ellipsoidal light-curve interpretation.

In this work, we build on the analysis from Torres et al. (2014) by including two extra years of I-band photometry, where the sinusoidal shape of the light curve can be explained by ellipsoidal modulations. Additionally, we see long-term variations in the shape of the light curve, these are consistent with either changes in the luminosity of an accretion disc or a spotted secondary star. In this work, we aim to settle the true nature of the object by performing a dynamical study and find that CX137 is a CV with a K7 secondary star and an orbital period of P = 10.34488 ± 0.00006 h, in agreement with previous studies. The source shows no outbursts in our seven years of optical photometry.

This paper is organized as follows: in Section 2, we describe the OGLE (Optical Gravitational Lensing Experiment) photometry and the optical spectroscopy obtained for this study. In Section 3, we provide an analysis of the data; where we determine the orbital period, generate a radial velocity curve for the secondary star, and describe the spectral features. In Section 4, we present our light-curve models, fitting routines, and resulting output parameters. We finally outline our discussion in Section 5 and conclusion in Section 6. All quoted errors in this paper represent 1σ uncertainty, unless otherwise stated.

2 OBSERVATIONS

2.1 Gaia

Gaia provides precise coordinates for the optical component of CX137 at RA = 17^h55^m53^s.26, Dec. = −28°16′33″.84 (ICRS), in addition to proper motion components of μ_RA = 1.139 ± 0.108 mas yr⁻¹, and μ_Dec. = −6.977 ± 0.087 mas yr⁻¹. The parallax of the source was measured by Gaia DR2 to be π = 1.116 ± 0.069 mas, which corresponds to a distance of |$d = 879^{+59}_{-52}$| pc (Bailer-Jones et al. 2018; Gaia Collaboration et al. 2018).

2.2 OGLE photometry

The optical counterpart of CX137 was observed during the fourth phase of the OGLE project with the 1.3-m Warsaw telescope at Las Campanas Observatory (Udalski, Szymański & Szymański 2015). OGLE provided us with 7 yr of I-band photometry, from 2010 to 2016. The typical cadence of these observations ranges from 20 min to nominally once a night with exposure times of 100 s. There is a three month period in each year when the source is not visible. The photometry was obtained using the difference image analysis method outlined in Wozniak (2000). The individual photometry has typical errors of <0.01 mag, see Table 1 for a log of observations.

2.3 Optical spectroscopy

We observed CX137 with the Intermediate dispersion Spectrograph and Imaging System (ISIS; Jorden 1990) on the 4.2-m William Herschel Telescope (WHT) at the Roque de los Muchachos Observatory on La Palma, Spain, during five different observing runs between 201 7June and August. The ISIS spectrograph has a dichroic that splits the spectra into a red and blue arm, allowing for a wide spectral range to be observed simultaneously. For the blue arm, we used gratings R158, R300, and R600; and for the red arm, we used gratings of R158 and R600. We also obtained one high-resolution spectrum with the Inamori-Magellan Areal Camera and Spectrograph (IMACS; Dressler et al. 2011) on the Magellan Baade 6.5 m Telescope at Las Campanas observatory with the R1200 grating. We provide a log of spectroscopic observations and specifications of each grating in Table 2. The spectral resolutions provided in the table were approximated by measuring the width of spectroscopic lines in an arc lamp spectrum taken with each grating.

We reduced the spectra using standard iraf¹ routines. The data were bias-subtracted and flat-fielded, sky emission subtracted, the spectra were optimally extracted and wavelength calibrated using an arc lamp taken after each spectrum. We determine the zero-point of the wavelength calibration of our spectra by measuring the positions of bright sky lines in each spectrum, and apply the corresponding shift to each individual spectrum such that the wavelength of the sky lines match between all the spectra. For the ISIS spectroscopy, we analysed the data taken in both the red and blue arms using the same procedure, but treat them as individual spectra.

2.4 Spectral templates

Throughout this work, we make use of spectral templates from the X-shooter library (Chen et al. 2011). We selected spectra from 71 M, 33 K, and 23 G stars of varying luminosity classes and evolutionary stages. All templates were taken with a 0.7 arcsec slit with the VIS arm of X-shooter and a nominal resolution of R ∼10 000, equivalent to ∼30 km s⁻¹ at a wavelength of 8600 Å.

All the spectra of CX137 and templates were subsequently processed using molly.² First, we apply a heliocentric velocity correction to all spectra using the hfix task. We then use vbin to bin all the data to a uniform velocity scale so the dispersion of the templates matches that of the CX137 spectra. We then normalize each spectrum by dividing it by a fit to the star’s continuum. To estimate the continuum we fit a second-order polynomial to each spectrum, masking out regions with strong emission lines or telluric bands.

3 DATA ANALYSIS

3.1 Photometric periodicities

We use all 7 yr of OGLE I-band photometry to determine the orbital period of the binary. For this, we employ the gatspy python package (VanderPlas & Ivezić 2015), which provides an implementation of the Lomb–Scargle periodogram to find periodicities in the photometric data. The strongest peak of the periodogram is at a period of P = 5.17244 h. When the data are phase-folded at this period we see large scatter in the light curve, which is due to the fact that the maxima expected from ellipsoidal modulations at phase 0.25 and 0.75 have different strengths (see Fig. 1). Fig. 1 shows the light curve phase-folded at a period of twice that of the corresponding strongest peak, P = 10.34488 ± 0.00006 h, consistent with the period found in Udalski et al. (2012) and Torres et al. (2014). Motivated by the fact that spin periods in the range of 0.1–10 per cent of the orbital period have previously been observed in magnetic CVs (Norton, Wynn & Somerscales 2004), we search for periodicities in the 100–20 000 s range with null results. We detect no measurable change in the orbital period over our 7 yr baseline. On the other hand, we detect a possible long-term trend at a period of ∼796 d. Since the full span of the light curve is only three times this period, more data are required to confirm if this is a real periodicity or just a temporary artefact. The data phase-folded at this period is shown in the bottom panel of Fig. 1.

3.2 Spectral type of the secondary

Fig. 2 shows the blue and red normalized ISIS spectra of CX137 averaged in the rest frame of the secondary star. The spectra are mostly dominated by absorption lines from the secondary, with additional Balmer emission lines from an accretion disc. We detect the Mg triplet absorption lines from the secondary, and interstellar Na D lines. We see a weak contribution from TiO bands of the secondary in the ∼6100–6300 Å range, and no evidence for He i emission lines, which are common in CVs (e.g. Rodríguez-Gil et al. 2009; Ratti et al. 2013). This might be due to the lines being veiled by a large flux contribution from the secondary. We can set an upper limit to the absolute equivalent width of He i λ7065 to be <1.6 Å, and <1.2 Å for He ii λ4686.

To estimate the temperature of the secondary star, we first average all the CX137 ISIS data taken with the R600 grating to use as a high S/N reference. We compare this spectrum to that of the X-shooter templates described in Section 2.4. First, we corrected each template spectrum for the systemic velocity of each star and broaden it by convolving it with a Gaussian function to match the spectral resolution of the ISIS data. We subtract each template to the normalized CX137 spectrum in the 5580–6150 Å wavelength range (masking out telluric lines and emission lines not associated with the secondary), and search for the template star that produces the lowest residuals, allowing for a varying multiplicative f factor, which represents the fractional contribution of the template star from the total flux. We find that the spectrum of CX137 best matches that of HD79349, a K7IV star with a temperature of 3850 ± 30K, and a systemic velocity of 47.12 ± 0.15 km s⁻¹ (Gaia Collaboration et al. 2018; Arentsen et al. 2019). We find a best fit for the average optimum factor of f = 0.52 ± 0.06.

3.3 Radial velocities

To measure the radial velocity of the secondary in each spectrum we use the xcor task in molly to cross-correlate the CX137 spectra with the spectrum of the K7IV star HD79349, the template star that best matches the spectra of CX137 (described in Section 2.3). The actual choice of template star does not have a noticeable effect in the measured radial velocities. We correct the template star’s spectrum for its systemic velocity and broaden it by convolving it with a Gaussian function to match the spectral resolution of the CX137 spectra. We consider the wavelength range listed in Table 2 for each CX137 spectrum, masking out telluric features and emission lines not associated with the secondary before cross-correlating them. We calculate the radial velocities from both the red and blue arms data of the ISIS spectrograph independently. The resulting radial velocity curve is shown in Fig. 3, with the individual measurements provided in Table A1. We note that the radial velocities measured near phase 0.25 have a large scatter due to noisy spectra taken in poor weather conditions.

3.4 Rotational broadening of the secondary star

To estimate the rotational broadening of the secondary we compare the set of spectral templates described in Section 2.4 to the high-resolution IMACS spectrum of CX137 taken near photometric orbital phase 0. We normalize the IMACS and X-shooter spectra by dividing them by a second-degree polynomial fit to their respective continuum (masking out absorption features) in the 8500–8900 Å range. We scale down the resolution of the X-shooter templates to match that of the IMACS spectrum by convolving them with a Gaussian function. We then broaden the templates by a range of velocities from 20 to 200 km s⁻¹ in steps of 1 km s⁻¹ using the rbroad task in molly. This task takes the input spectrum and broadens it through convolution with the rotational profile of Gray (1992), where we adopt a limb-darkening coefficient of 0.75. Finally, we subtract the broadened templates from the CX137 spectrum, following the same procedure as described in Section 2.3. Through χ² minimization we find a best fit of vsin (i) = 101 ± 3 km s⁻¹ to the rotational velocity of the secondary star in CX137. We find the best match to be comparably good for a K4III, K3.5III, K2III, and K7IV template star. G and M stars produce statistically worst fits. The individual vsin (i) measurements are shown in Table 3.

4 LIGHT-CURVE MODELLING

We proceed to model the optical light curve of CX137 using xrbinary,³ a light-curve synthesis code developed by Robinson. This code assumes a binary system composed of a compact primary and a co-rotating secondary star that fills its Roche Lobe and is transferring mass via an accretion disc. The code models the tidal distortion of the secondary (responsible for the ellipsoidal modulations), and accounts for irradiation of the surface of the secondary from the bright accretion disc. The accretion disc is assumed to be optically thick and to emit as a multitemperature blackbody. The disc’s temperature profile as a function of radius is given by an equation of the form T⁴∝R⁻³(1 − (R_in/R)^0.5), where R_in is the inner disc radius. In order to account for the observed symmetries in the light curve, we model the photometry of CX137 with three different models: (i) a model with a Roche Lobe filling secondary and an accretion disc that is allowed to vary in luminosity and eccentricity, (ii) a similar model, but with a circular accretion disc where the temperature of the edge of the disc can have a hot and a cool side, and (iii) a model with a circular accretion disc and an edge of uniform temperature, but with two spots on the surface of the secondary. For all models we fit the light curve using the emcee Markov Chain Monte Carlo (MCMC) sampler (Foreman-Mackey et al. 2013).

The relevant parameters of the model are: the inclination of the system i; an orbital phase shift of the photometric T₀ with respect to the spectroscopic T₀, ϕ; the temperature of the secondary star T₂; the temperature of the edge of the accretion disc T_E; the mass ratio q = M₂/M₁; the argument of periastron of the disc ω_D; the outer disc radius R_D; the disc luminosity L_D; the height of the accretion disc H_D; the eccentricity of the accretion disc e_D; the temperature ratio between the hot and cool sides of the disc edge T_h; the width of the hot side of the disc edge W_h; the location of the centre of the hot edge of the disc θ_h; the polar coordinates of the first and second spots on the surface of the secondary ϕ_S1, θ_S1, ϕ_S2, and θ_S2, respectively; the temperature ratio between the spots’ temperature and the secondary temperature T_S1, and T_S2 respectively, and the size of the spots R_S1, and R_S2. Only the relevant parameters are included in each of the three versions of the models described in the following section.

In all models we fix the semi-amplitude of the radial velocity of the secondary to K₂ = 161 km s⁻¹ (derived in Section 3.3). We use wide uniform priors for ϕ, T₂, T_E, ω_D, R_D, e_D, W_h, and all the parameters pertaining to the spots. For L_D we use a prior that is flat in log space to allow for even sampling of the parameter space across orders of magnitude. For i we use a prior that is flat in cos (i). We implement a Gaussian prior on the mass ratio centred at q = 0.79 ± 0.06 (derived in Section 3.4). We restrict the accretion disc to be larger than the circularization radius R_c = (1 + q)(0.5 − 0.227log (q))⁴ (Frank, King & Raine 2002). Finally, apply a flat prior on the temperature of the secondary of T₂ = [3500, 4100], based on the temperature of the template star that best matches the spectra of CX137 (derived in Section 2.3). The xrbinary code interpolates the temperature from a table of Kurucz models, therefore the measurement of the temperature of the secondary is not very precise (±125 K), we report only the statistical model uncertainties in Table 5.

In order to account for the year-to-year variations in the light curve we separate the photometry into eight epochs, nominally one for every year of data. Dividing the photometry into eight epochs allows us to roughly track the evolution of the system, assuming the parameters of the system are approximately constant in the ∼8 months of data each epoch spans (see Fig. 1). We see the shape of the light curve does remain fairly constant within each epoch, except for the 2016 epoch, which we therefore split into two epochs of equal time span named 2016a and 2016b, each of which do have a stable light-curve shape. Subdividing the epochs further proved to be too computationally expensive.

4.1 Model 1: variable disc

For the first model, we allow the accretion disc to vary in luminosity and eccentricity, but do not include any spots on the disc or the secondary. For all epochs, we keep i, ϕ, T₂, K₂, T_E, and q constant but allow the parameters that define the accretion disc ω_D, R_D, L_D, and e_D, H_D to vary from epoch to epoch. The temperature of the edge of the disc T_E could conceivably change from epoch to epoch, but since this parameter has little to no effect on the output light curve we constrain it to always be the same for computational purposes.

First, we fit each epoch of photometry independently, we then use the posterior distribution of those MCMC chains as starting positions when fitting all eight epochs simultaneously. We run the MCMC sampler for 1600 steps with 400 walkers and discard the first 50 per cent as burn-in. We test for convergence by using the Gelman–Rubin statistic and see that the potential scale reduction factor is |$\hat{R} \lt 1.3$| (Gelman & Rubin 1992). The most likely values are shown in Table 4. We see that the posterior distribution of all the relevant parameters is mostly Gaussian.

For this model, we interpret the changes in the light curve as being due to an accretion disc of varying shape and luminosity. We see the light curves are well modelled by a disc that gets smaller and more eccentric from 2010 to 2013, and then recedes back to its original luminosity 3 yr later and circularizes into a less eccentric disc. The best-fitting parameters for each epoch are shown in Table 4. We find a best fit for the secondary temperature of 4055 ± 25 K and a secondary mass of M₂ = 0.62 ± 0.04 M_⊙, both consistent with a K7 star (Cox 2000) and in agreement with the spectral classification performed in Section 2.3. The large Roche Lobe radius of the secondary R₂ = 0.92 ± 0.09 R_⊙ implies it must be evolved in order to fill its Roche Lobe (discussed further in Section 5).

We note that some of the best-fitting eccentricity measurements are as high as e = 0.58, which is not expected for a low accretion-rate CV with a accretion disc of radius ∼R_c, and for the high mass ratio found for CX137 of q > 0.7 (e.g, Warner 1995). For this reason, we proceed to model the light curve with a disc that is forced to be circular, but with an edge that has two zones of independent temperature.

4.2 Model 2: asymmetrical disc edge brightness

In this model, we fix the eccentricity of the disc e and argument of periastron ω_D to be 0. In the previous variable disc model, we found the phase shift ϕ to be consistent with 0 with an uncertainty in phase shift of just 0.002, we therefore also fix this parameter to 0 for computational purposes. In this model, we allow the disc edge to have two different temperatures. We model this in xrbinary by using a ‘spot’ that is allowed to cover an arbitrary width of the edge of the disc, effectively creating a hot and a cool zone on the outer edge of the disc. Physically, this could be produced by the impact of the gas stream on the disc, which causes the region near the impact hotspot to be hotter than the region on the opposite side of the disc. Changes in the mass transfer rate from the secondary can affect the temperature of this ‘spot’ (e.g. SDSS J123813.73−033933.0, Zharikov et al. 2006), which could be responsible for the observed year to year variations and the ∼796 d periodicity derived in Section 3.1.

In this model, we fit for the temperature ratio between the hot and cool sides of the disc edge T_h, the width of the hot side of the disc edge W_h, and the location of the centre of the hot edge of the disc θ_h; these last two measured in degrees. θ_h is defined such that θ_h = 0 deg is the direction pointing from the primary straight away from the secondary, and θ_h = 90 deg points towards the observer at phase 0.75, when the observer sees the side of the disc where we would expect an accretion hotspot to be.

We fit the model in the same way as described in Section 4.1, in this case running the MCMC with 2000 steps and 400 walkers, and also discarding the first 50 per cent as burn-in. The resulting model has an |$\hat{R} \lt 1.4$|⁠. The most likely values are shown in Table 4. We see a correlation between W_h and T_h, since a large hot zone can produce a similar light curve to a smaller zone with a higher temperature. These parameters are also correlated with the disc height H_D, which together with W_h define the effective area of the hot zone. The best-fitting disc radius is ∼1.5R_c throughout all epochs, and a hot region that covers ≳ 100 deg of the edge of the disc. Models predict that for the best-fitting parameters of CX137, a typical hotspot would cover ≲ 5 deg of the edge of the disc (Livio 1993). From observations, Warner (1995) find spots that cover the range of 14–40 deg, much smaller than what we measure for CX137.

We find a best fit for the secondary temperature of 3814 ± 20 K and a secondary mass of M₂ = 0.68 ± 0.03 M_⊙. A cooler but more massive star is not necessarily consistent with the K7 secondary we expect from our spectral analysis in Section 2.3. Allowing the disc to be hotter effectively lowered the temperature of the secondary to the point where this model is not entirely self-consistent, and therefore disfavored. This model can help towards a better understanding of the systematic uncertainties in measuring M₁, M₂, and i. Finally, we explore a third model in which the accretion disc is circular and the disc edge has one uniform temperature, but we include two spots on the surface of the secondary.

4.3 Model 3: spotted secondary

Finally, we fit the light curves with a model in which the accretion disc is forced to be circular, and have an edge with a single temperature, fixing W_h = 0, θ_h = 0, and T_h = 1. We place two spots on the surface of the secondary with polar coordinates ϕ_S1, θ_S1, ϕ_S2, and θ_S2, respectively; and fix −110 deg < ϕ_S1 < 110 deg, and 70 deg < ϕ_S2 < 290 deg. This prior effectively constrains spot 1 to be on the side of the secondary facing the observer during orbital phase 0.75, and spot 2 on the opposite side of the secondary, allowing for a small overlap region of 20 deg. The spots have respective angular sizes R_S1, and R_S2, and a temperature ratio with respect to the secondary T_S1, and T_S2, which are constrained to be <1. We fit for the size and height of the disc as in the previous models, but for computational purposes we constrain them to be the same throughout all epochs. We find that the spotted secondary model requires two spots to be able to explain the fact that the brighter peak at phase 0.75 exhibits larger brightness variations than the dimmer peak at phase 0.25 (see the top panel of Fig. 1).

We fit the model in the same way as the one described in Section 4.1, running the MCMC with 2500 steps and 400 walkers, discarding the first 50 per cent as burn-in. The resulting model has an |$\hat{R} \lt 1.5$|⁠. The most likely values are shown in Table 4. We caution that the parameters of the spots are very highly correlated, a small cold spot can produce the same light curve as a large but hotter spot. Nevertheless, the relevant physical parameters such as the mass ratio and inclination appear Gaussian and mostly unaffected by the spot parameters.

We find that ∼3 per cent of the surface of the secondary is covered by the two modelled spots. For reference, Watson, Dhillon & Shahbaz (2006) find through Roche Lobe tomography that for the 9.9 h orbital period CV AE Aqr ∼18 per cent of one hemisphere of the secondary is spotted. Similarly, the 15 h orbital period CV BV Cen has ∼25 per cent of a hemisphere covered by spots (Watson et al. 2007).

We determine a best-fitting secondary temperature of 4050 ± 30 K and a secondary mass of M₂ = 0.65 ± 0.05 M_⊙; very similar to the parameters obtained from Model 1 described in Section 4.1. We show the light curve of each epoch, the corresponding most likely model, and the residuals in Fig. 4. We only include a plot of the spotted secondary model, since all three models presented here are able to reproduce the light-curve shape, and visually speaking are effectively indistinguishable. The data are shown phase-folded at the photometric ephemeris with T₀ = 2455260.8204 and orbital period P = 10.34488 h (derived in Section 3.1).

5 DISCUSSION

5.1 Stellar parameters

We calculate f for each model by measuring the relative flux fraction that the secondary contributes to the total flux of the system in the V band, the closest band to the 5580–6150 Å wavelength range used in Section 2.3 to derive f = 0.52 ± 0.06 from the spectroscopy. From the light-curve modelling, we find f-factors averaged over a full orbit for all epochs of photometry of: f = 0.50 ± 0.03 for the variable disc model, f = 0.54 ± 0.04 for the asymmetrical edge brightness model and f = 0.51 ± 0.09 for the spotted star model. Most of these are in perfect agreement with the value measured from the spectra. The f as a function of epoch is shown in Table 4.

We find best-fitting values for the primary mass of M₁ = 0.81, 0.86, and 0.83 M_⊙ for models 1, 2, and 3, respectively. The statistical uncertainties reported in Table 5 are in the order of the systematic uncertainties from assuming different models. Accounting for these, we adopt a primary mass estimate of M₁ = 0.83 ± 0.06 M_⊙, typical for white dwarfs in CVs (e.g. M_WD = 0.83 ± 0.23 M_⊙, Zorotovic, Schreiber & Gänsicke 2011), and too small for a neutron star (Özel & Freire 2016). The best estimate for the mass for the secondary is M₂ = 0.65 ± 0.07 M_⊙, consistent with the mass of a main-sequence K7 star (Cox 2000) and in agreement with the best-fitting template match found in Section 2.3. We find a best-fitting radius for the Roche Lobe of the secondary of R₂ = 0.97 ± 0.15 R_⊙. This radius is larger than expected for a main-sequence K7 star (which have typical values of R ∼ 0.65 R_⊙, Pecaut & Mamajek 2013), supporting an evolved secondary in CX137.

From the spectra, we determine the ratio of the double-peak separation (DP) to the full width at half-maximum (FWHM) of the Hα emission line following the method of Casares (2016). We fit Hα with a double Gaussian function to measure the DP between the two line peaks DP = 484 ± 12 km s⁻¹ and then fit a single Gaussian to determine the FWHM = 901 ± 19 km s⁻¹. We find the average ratio to be DP/FWHM = 0.55 ± 0.02, the result of these fits are shown in Fig. 5. In Fig. 6, we show the q, and DP/FWHM of Hα plotted alongside the values for other known CVs. We confirm that our parameter estimates agree well with the q–DP/FWHM relation for CVs determined by Casares (2016). Torres et al. (2014) suggested the double-peaked structure of Hα might be due to contamination from photospheric absorption lines from the secondary (e.g. Torres et al. 2019). Nevertheless, the values we derived for CX137 agree with this trend, and strengthens the case that CX137 is a CV.

Similarly, we calculate the expected FWHM of Hα using the FWHM–K₂ relation for CVs from Casares (2015). A mass ratio q = 0.78 and an FWHM =936 ± 35 km s⁻¹, corresponds to an expected value of K₂ = 145 ± 22 km s⁻¹, where the uncertainty is dominated by the scatter from the Casares relation. Consistent within the measured value of K₂ = 161 ± 6 km s⁻¹.

We measure the systemic velocity of CX137 from the optical spectra to be γ = 54 ± 4 km s⁻¹ (Fig. 3). Given the proper motion and distance to CX137 obtained by Gaia (see Section 2.1), we can determine the space velocity of CX137 with respect to the Sun to be v = 62 ± 4 km s⁻¹, statistically consistent with other CVs (v = 51 ± 7 km s⁻¹; Ak et al. 2010).

In addition to the orbital period of the binary, we detect a tentative periodicity of ∼796 d. Stellar spots are known to live well over this amount of time and up to ∼10 yr (e.g, Hall & Henry 1994; Savanov 2014). As we saw in Model 3, it is possible that these long-term periodicity is produced by the evolution of spots on the surface of the secondary. Nevertheless, our photometry only covers a baseline three times that of this period, making its interpretation or physical origin hard to establish.

5.2 X-ray luminosity

Torres et al. (2014) provide a lower limit to the X-ray luminosity of CX137 of L_x > 5.8 × 10³⁰ erg s⁻¹ for a distance of 0.7 kpc and assuming a hydrogen column density N_H = 10²¹ cm⁻². Here, we improve this measurement by using the distance to CX137 from Gaia of |$d = 879^{+59}_{-52}$| pc (Bailer-Jones et al. 2018). In addition, we calculate the extinction in the line of sight to CX137 from the Bayestar19 3D dust maps (Green et al. 2019) to be A_V ≈ 0.58. We obtain an N_H = 1.7 × 10²¹ cm⁻² from the A_V–N_H relation from Güver & Özel (2009). We calculate a counts to unabsorbed flux conversion ratio of 5.6 × 10⁻¹⁵ erg cm⁻² s⁻¹ count⁻¹ for a 2.16 ks exposure taken with the Advanced CCD Imaging Spectrometer (ACIS-I) during Chandra Cycle 9, using a power-law spectrum with Γ = 2. This corresponds to a 0.5–10 keV unabsorbed flux of (8.4 ± 2.1) × 10⁻¹⁴ erg cm⁻² s⁻¹, or L_x = (7.8 ± 2.2) × 10³⁰ erg s⁻¹ at the distance from Gaia.

We can estimate an accretion rate following the method of Beuermann et al. (2004) using |$L_{\rm acc} = \dot{M} G M_1 (1/R_1 - 1/R_D)$|⁠, R₁ = (1.463 − 0.885(M₁/M_⊙)) × 10⁹ cm, and L_acc = (1 + α)L_x; where α is typically 0.1. We adopt our best estimate for the primary mass of M₁ = 0.83 M_⊙, and a typical disc radius of R_D = 10¹⁰ cm, as determined by our models presented in Section 4. We obtain an accretion rate estimate of |$\dot{M} \sim 10^{15}$| g s⁻¹ (10^−10.8 M_⊙ yr⁻¹).

Bahramian et al. (2020) detected CX137 at a higher L_x in the Swift Bulge Survey (Shaw et al. 2020) during repeated biweekly scans of the Galactic Bulge with the Neil Gehrels Swift Observatory. They measured an average L_x = 5 × 10³¹ erg s⁻¹ over many epochs in 2017, with a peak of L_x = 3 ± 2 × 10³² erg s⁻¹, indicating a flux increase of |$38^{+33}_{-26}$| compared to the Chandra GBS measurement, which would consequently bring up the accretion rate to |$\dot{M} \sim 4\times 10^{16}$| g s⁻¹ (10^−9.2 M_⊙ yr⁻¹) during this period. van Teeseling, Beuermann & Verbunt (1996) found that the accretion rate in non-magnetic CVs is likely underestimated by a factor of ∼2 for systems with inclinations of ≳ 60 deg. This would bring the accretion rate to |$\dot{M} \sim 10^{17}$| g s⁻¹ (10^−8.8 M_⊙ yr⁻¹), more similar to the |$\dot{M}$| expected for a Roche Lobe filling subgiant with an orbital period of 10 h (King, Kolb & Burderi 1996). An accretion rate of |$\dot{M} \sim 10^{17}$| g s⁻¹ is expected for CVs with long periods ≳ 4 h, yet it is still low for a CV with a 10 h period like CX137 (Wynn, King & Horne 1997).

Using the L_x versus duty cycle correlation for dwarf novae from Britt et al. (2015), we can estimate the duty cycle for CX137 to be 0.063 ± 0.022. Accounting for observational cadence, the source should have been in outburst during 94 ± 34 d out of the 1504 days CX137 was observed by OGLE. One explanation for the lack of outbursts might be that CVs with long orbital periods tend to have shorter outbursts (Hameury & Lasota 2017). Given that the secondary star in CX137 contributes a large fraction of the total flux, an outburst would be of low amplitude, and we could have missed it if it happened when the source was not being observed. KIC 5608384 is another example of a CV with a long period (8.7 h) and a low accretion rate (⁠|$\dot{M} = 0.3 - 6.5 \times 10^{-9}$| M_⊙ yr⁻¹) that showed only one 4 d outburst in four years of Kepler photometry (Yu et al. 2019).

6 CONCLUSION

We obtained multiple spectra of the binary star CX137 and constructed a radial velocity curve from which we measure a K₂ = 161.1 ± 0.7 km s⁻¹ and a systemic velocity γ = 54 ± 4 km s⁻¹. Additionally, we modelled 7 yr of optical photometry. The optical light curve has an asymmetrical sine curve shape, which we interpret as being due to ellipsoidal modulations of a tidally distorted secondary star. We see long-term variations in the shape of the light curve, which are well fitted by a spotted secondary star (Model 3; Section 4.3). From the light-curve modelling, we derive a best-fitting inclination of i = 63.0 ± 0.7 deg, a primary mass of M₁ = 0.83 ± 0.06 M_⊙, consistent with a white dwarf accretor, and a secondary mass of M₂ = 0.65 ± 0.07 M_⊙, consistent with an evolved K7 secondary.

SUPPORTING INFORMATION

CX137_Photometry.txt

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ACKNOWLEDGEMENTS

This project was supported in part by an NSF GROW fellowship. PGJ and ZKR acknowledge funding from the European Research Council under ERC Consolidator grant agreement no. 647208. JS was supported by a Packard Fellowship. MAPT acknowledge support by the Spanish MINECO under grant AYA2017-83216-P and support via Ramón y Cajal Fellowship RYC-2015-17854. We thank Tom Marsh for the use of molly. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by National Institutions, in particular the institutions participating in the Gaia Multilateral Agreement. The ISIS spectroscopy was obtained with the WHT, operated on the island of La Palma by the Isaac Newton Group of Telescopes in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile. This research has made use of NASA’s Astrophysics Data System. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU.

Software:astropy (Astropy Collaboration et al. 2018), pyraf (Science Software Branch at STScI 2012), SAOImage DS9 (Smithsonian Astrophysical Observatory 2000), emcee (Foreman-Mackey et al. 2013), corner (Foreman-Mackey 2016), matplotlib (Hunter 2007), scipy (van der Walt, Colbert & Varoquaux 2011), numpy (Oliphant 2007), extinction (Barbary 2016), pyphot (https://github.com/mfouesneau/pyphot), molly (http://deneb.astro.warwick.ac.uk/phsaap/software/molly/html/INDEX.html), and xrbinary(http://www.as.utexas.edu/∼elr/Robinson/XRbinary.pdf).

DATA AVAILABILITY

All the optical photometry used for this work are available on the online Supporting Information version of this article. And the radial velocity data is shown in Table A1.

Footnotes

REFERENCES

APPENDIX A: RADIAL VELOCITY TABLE

This section contains a data table with all the relevant radial velocity measurements.