A CLASS OF ECCENTRIC BINARIES WITH DYNAMIC TIDAL DISTORTIONS DISCOVERED WITH KEPLER

Kepler, in its search for extrasolar planets, obtains high-precision time-series photometry of 115.6 deg² of sky (Koch et al. 2010). With this new combination of photometric quality and uninterrupted coverage of more than 160,000 stars, Kepler is discovering variability never before identified in stars. Here we present light curves that demonstrate a periodic brightening or variability on timescales of days, frequently accompanied by harmonic oscillations at shorter periods.

These light curves are reminiscent of the tidally distorted light curves theorized by Kumar et al. (1995) for eccentric orbits. KOI-54 (KIC 8112039), a highly eccentric binary system composed of two A-type stars observed by Kepler (Welsh et al. 2011), was the first system discovered whose light curve is predominantly characterized by these dynamic distortions. Binary tidal forces also induce pulsations at integer harmonics of the orbital period (Kumar et al. 1995; Willems 2003), making KOI-54 the best example of pulsations driven in this way. See Handler et al. (2002), De Cat et al. (2000), and Maceroni et al. (2009) for other examples of systems with evidence of this type of driving. The discovery of KOI-54 has provided critical observational data to constrain the theories of dynamic tidal forces on stellar binaries (Fuller & Lai 2012; Burkart et al. 2012), but more examples of these stars are required to generally test these theories.

Despite having unique light curves, we demonstrate that the 17 systems described in this paper are all tidally interacting binary systems in eccentric orbits. In Section 2 we describe the Kepler photometric time series of these systems and propose a simple model to describe their dominant features in Section 3. We fit our simple model to these systems and estimate their orbital parameters in Section 3.1, demonstrating a technique proposed by Kumar et al. (1995). In Section 4 we present spectroscopy for each system including radial velocity measurements of five systems that confirm they are in eccentric binary systems. Finally we discuss the properties of this class of stars, the effects of the tidal interactions, and the remaining work to accurately model these systems in Section 5.

We have discovered 17 objects in the Kepler data archive that demonstrate unique periodic variability. We refer to these objects as "heartbeat" stars because the shape of the variability for several is reminiscent of an echocardiogram. We show example light curves in Figure 1 that exemplify the sudden and periodic variation in brightness. Note that the heartbeat profile comes in many varieties and appears to be correlated with other periodic oscillations in the light curve of the star.

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For each system we analyzed a subset of the long cadence (one exposure taken every 29.4 minutes with a total integration time of 27.09 minutes) light curves observed during Kepler quarters 0–7 (Julian Day spanning 2,454,953.53 to 2,455,552.56). Each quarter is one-fourth of the Kepler heliocentric orbital period of 372 days, except for quarters 0 and 1 (with lengths of approximately 10 days and 33 days, respectively). Each subset contains enough data to constrain the heartbeat period (i.e., the binary orbital period) to better than 2 minutes. See Table 1 for a list of long cadence quarters used to study each object.

We used the light curves created by the Kepler simple aperture photometry module (Twicken et al. 2010; Thompson & Fraquelli 2011). These light curves are the result of performing simple aperture photometry on the calibrated pixels in the optimal aperture (Bryson et al. 2010). For more details on the calibration steps performed by the Kepler pipeline see Jenkins et al. (2010), Caldwell et al. (2010), and Fanelli et al. (2011). To remove the long-period effects that result from differential velocity aberration and thermal effects from the spacecraft (Van Cleve et al. 2011), we removed a piecewise polynomial fit from each quarter using the program wqed (Thompson & Mullally 2009). We did not adjust the light curves for contamination from other stars. The contamination is small (typically <5%) for these stars, as reported by the Kepler archive (Thompson & Fraquelli 2011). The contamination will have a small effect on the amplitude of the features we see in the light curves, but will not affect the shapes.

We measured the heartbeat period by performing a nonlinear least-squares fit to the "comb" of equally spaced frequencies found in the power spectrum of each light curve. This comb results from the periodic, yet sudden, changes in the brightness and from any harmonic pulsations present on the star. We fitted a sine wave at the heartbeat frequency found in the power spectrum as well as at a minimum of six harmonics of this frequency to establish the period (using Period04; Lenz & Breger 2005; see Figure 2). In performing the fit, the amplitude and phase of each harmonic were allowed to vary, while the frequency was fixed to an integer multiple of the heartbeat frequency. The fitted orbital periods are reported in Table 1. At this stage we do not attempt to determine what fraction of the fitted amplitudes is due to the heartbeat shape and what fraction is due to stellar pulsation; it is not required to establish the period of the heartbeat.

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We folded each light curve at the measured period and then smoothed by performing a running average spanning 15 minutes (see Figures 3 and 4). We set the zero phase to either the maximum or the minimum of the heartbeat feature, depending on the shape of the feature. Because of the quality and length of the Kepler data, almost all the features shown in the folded light curves are real. However, because these are folded and smoothed curves, any variations that are not harmonics of the heartbeat period are lost.

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Note that each folded light curve has a unique shape, quite different from KOI-54. Some dim before they brighten, others dim after they brighten, and others show distinct "W" or "M" shapes. Besides the heartbeat, stellar oscillations at harmonics of the heartbeat period are also apparent. For example, KIC 3230227, identified by Uytterhoeven et al. (2011) as an eclipsing δ Sct star, has short-period oscillations evident in the folded light curve and thus harmonics of the orbital frequency. Similarly, Figures 3 and 4 reveal that most of the sample have pulsations at harmonics of the heartbeat frequency.

The presence of harmonic oscillations is a clue to the nature of these systems. For KOI-54, Welsh et al. (2011) show that short-period modes that are harmonics of the orbital frequency dominate the light curve once the effects of the tidal distortion and external heating have been removed. Another hint to the nature of these objects comes from KIC 3230227, KIC 4936180, and KIC 8719324. They all show evidence of a grazing eclipse. Therefore, it is likely that the heartbeat shape is also caused by a binary companion. While several of the heartbeat systems identified here are also in the Kepler eclipsing binary star catalog (Slawson et al. 2011; Prša et al. 2011), all of those in the catalog (except KIC 3230227) show no evidence of an eclipse.

In the next section we explain how this varied collection of light curve shapes can be created with tidal distortions in an eccentric binary system.

Tidal forces change the shape of a star, causing brightness variations (Brown et al. 2011; Welsh et al. 2010; Morris 1985). For circular orbits, this results in nonspherical star-producing brightness variations at twice the orbital period, known as ellipsoidal variations. For eccentric binaries, the tidal force is phase-dependent and the largest variations in shape and brightness occur at periastron. As a result, instead of seeing the double-wave in the flux variations, as in the case of the circular orbit, the flux changes drastically when the binary system nears periastron.

As a star passes through periastron, in the static tide limit, the star's shape adjusts to the instantaneous tidal force. The shape of the star's distortion can be approximated by summing over axisymmetric and sectoral spherical degree two (ℓ = 2) modes. When observed at different inclinations, the relative contributions of the ℓ = 2, m = −2, 0, 2 modes, integrated over the observable sphere, will change. Additionally, the observed phase of these modes will depend on the angle of periastron for the binary system. As a result the shape of the flux variations due to the tidal distortions will depend on inclination, angle of periastron, and eccentricity (Kumar et al. 1995). The amplitude of these variations will depend on factors that influence the strength of the tidal forces, such as the masses of the objects and the orbital separation at periastron.

The tidal brightness variations were derived by Kumar et al. (1995) while predicting the tidal oscillations induced in a B-type star orbiting a neutron star. Even though the effect was too subtle to observe at the time, they did describe an analytic model whose shape depends on the orbital inclination, eccentricity, and angle of periastron of the binary orbit. We calculate a grid of solutions to this model for different inclinations and periastron angles and plot each solution in Figure 5 for two different eccentricities.

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A side-by-side comparison of the grid in Figure 5 to the folded light curves in Figures 3 and 4 shows that this fairly simple model can reproduce many of the features observed in these systems. For instance, the shape of the KOI-54 brightenings occur for systems at low orbital inclinations while larger inclinations can cause the light curve to first increase in brightness and then decrease (KIC 3547874), or vice versa (KIC 9790355), depending on the angle of periastron. Eccentricity has the effect of changing the duration of the heartbeat event; heartbeats in more eccentric orbits span a shorter phase interval.

Tidal distortion is not the only physical effect that can change the observed brightness of an eccentric binary system. Irradiation, tidally induced pulsations (Kumar et al. 1995; Willems & Aerts 2002), Doppler boosting (Bloemen et al. 2011; van Kerkwijk et al. 2010; Loeb & Gaudi 2003), and eclipses must also be considered in order to obtain the best fit to the light curves. Fitting an extensive set of physical effects to each system, as done for KOI-54 (Welsh et al. 2011; Burkart et al. 2012), is complex and beyond the scope of this discovery paper. We do, however, show that the tidal distortions alone can fit the dominant features of most of the light curves. By doing so we also approximate the orbital parameters of these systems.

3.1. Fitting Eccentric Tidal Distortions

We use the analytic model of Kumar et al. (1995, Equation (44)) for the tidal oscillations induced in eccentric binaries to obtain the orbital parameters for each of the heartbeat stars. Our fit to the fractional flux (δF/F) contains six parameters: eccentricity (e), orbital inclination (i), angle of periastron (ω), an amplitude scaling factor (S), a fractional flux offset (C), and time of periastron. The period and the semimajor axis (a) were fixed to one when performing the fit to the folded light curves. R(t) describes the distance between the two stars as a function of time and ϕ(t) gives the angular position of the ellipse measured from periastron (i.e., true anomaly) as a function of time.

$$\begin{equation} \frac{\delta F}{F} = S\cdot \frac{1-3\sin ^2(i) \sin ^2(\phi (t)-\omega)}{(R(t)/a)^{3}} + C. \end{equation} \tag{ 1 }$$

Inclination is defined such that the plane of the orbit is perpendicular to the line of sight at an inclination of zero. Because the stellar distortion is symmetric along the axis of the elongation, the angle of periastron is reported from −90° to +90°; ω = 0 when looking down the minor axis of the ellipse and increases in the direction of the orbiting primary. The fit was performed using a Levenberg–Marquardt chi-square minimization.

The results of our fits can be found in Figures 6 and 7 and Table 1. Since we fit folded light curves with extremely small uncertainties, and did not fit all the features present in the light curve, our fits result in large reduced chi-square values with small error bars. To compensate, the 1σ uncertainties on the fitted parameters presented in Table 1 represent the error on the fitted parameters when the uncertainties on the folded light curve are increased so that the reduced chi-square is equal to one. The error bars vary greatly depending on the chi-square of the fit.

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We have not included several physical parameters known to affect the light curves of such highly eccentric systems. These effects will have a different influence depending on the properties of the stars in the system. In many cases the harmonic pulsations are large and may significantly alter the results of the fit. As an example, KIC 4248941 shows large harmonic pulsations that could easily sway the results of our fits. Also, the exclusion of orbital effects such as irradiation (and to a lesser degree Doppler boosting) can prevent us from obtaining a good fit for the true orbital parameters. For instance, KIC 4372379, KIC 4936180, and KIC 7622059, all show a heartbeat shape but are not fit well by our simple tidal distortion model. Adding the additional physical effects may account for the discrepancy.

While incomplete, we note that the Kumar et al. (1995) model can give a good first estimate of the orbital parameters even in cases where the irradiation is significant and has the advantage of not requiring knowledge about the properties of the stars to make this estimate. For KOI-54, where the irradiation makes up almost half of the change in flux, the fit to the tidal distortion model gives a reasonable estimate of the orbital parameters. According to the detailed modeling of Welsh et al. (2011), the orbital parameters of KOI-54 are e = 0.8315(32), ω = 3946(51), and i = 552(10). Our fit is presented in the last panel of Figure 7 and Table 1. While our model does not include irradiation, the pulsations, or the information from radial velocity measurements, our eccentricity, 0.8180 (80), angle of periastron, 52° (31), and inclination, 107 (2.9), all match within the 2σ error bars.

The ability to fit KOI-54 well with only the tidal distortion model is in part because of its inclination. At low inclinations the shape of the contribution from irradiation closely resembles that of the tidal distortion. For higher inclinations this will not always be the case, and so when the irradiation is significant it can appreciably alter the shape of the heartbeat (see Burkart et al. 2012). Once knowledge of the companion can be obtained, the effects of irradiation can be included accurately in the model of the light curve.

We have three obviously eclipsing systems: KIC 3230227, KIC 4936180, and KIC 8719324. We fit these systems with inclinations of 4279, 232, and 7578, respectively. Our model for KIC 4936180 is obviously incomplete and so the fitted inclination is likely incorrect. Both KIC 8719324 and KIC 3230227 have very good fits, but rather small inclinations for eclipsing systems. It is possible to achieve an eclipse at these inclinations if the distance at periastron for these highly eccentric systems is very small (⪅3 R_☉). This is possible if the secondary has a rather low mass (⪅0.5 M_☉). A more likely possibility is that these are cases where the irradiation has a significant effect on the shape of the light curve, causing us to estimate the inclination at a lower value.

This simple model, despite being incomplete, allows us to illustrate that tidal distortions in an eccentric binary system can explain many of the features in the light curves. Fitting these features will allow us to estimate the orbital parameters of these systems. One trend noticed from fitting our simple model is that the eccentricity is smaller for shorter period systems; see Figure 8. While we are less likely to find long-period, low-eccentricity systems with observable tidally induced variations, we would expect to detect highly eccentric, short-period systems if they existed. Since shorter period orbits undergo stronger tidal forces, they are likely to circularize more quickly, leaving a dearth of highly eccentric short-period binaries. Similar trends have been seen by Prša et al. (2008) and Mathieu & Mazeh (1988) for eclipsing binaries.

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In order to measure the stellar parameters and to confirm the prediction that these are eccentric binary systems, we obtained optical spectra of the 17 different heartbeat stars at the KPNO Mayall 4 m telescope between 2011 May 14 and September 7 (UTC) using the long-slit spectrograph RCSpec and T2KA Camera. The KPB-22b grating was used in second order with a slit width of 10 to produce a dispersion of 0.72 Å pixel⁻¹ and a resolution of R ∼ 4000 (velocity resolution ∼85 km s⁻¹ at 4400 Å). Each spectrum was observed with an exposure time between 180 and 900 s, depending on brightness and conditions and each object exposure was preceded by a 30 s exposure of an FeAr arc lamp spectrum for wavelength calibration. A standard star (one of BD+40 4032, Feige 34, or Wolf 1346) was observed each night using the same method (Stone 1977; Barnes & Hayes 1982). Bias exposures and quartz lamp flat fields were taken the day before each night of observations for calibration.

We reduced the spectra using standard IRAF (Image Reduction and Analysis Facility) tools. First, the overscan bias level was subtracted from each frame. Next, a master residual bias and normalized flat-field frame were constructed from sets of bias and quartz lamp exposures and then applied to each spectrum frame to correct for any remaining two-dimensional bias patterns and pixel-to-pixel variations in detector response. The IRAF onedspec package was used to define apertures and to extract sky-subtracted spectra. The same apertures are also employed to extract an arc lamp spectrum for each object. The spectra are then resampled on a linear wavelength scale using fits to the arc lamp spectra. The reduced standards were then fit to the flux standards provided with the IRAF onedspec package to produce a sensitivity curve in order to correct each science spectrum.

These spectra suffer from degraded focus outside of the range 3800–4700 Å where the fluxes are difficult to properly calibrate. Consequently, we have restricted our spectral fits to this limited wavelength range.

We have modeled each spectrum using a library of synthetic model atmosphere spectra from Munari et al. (2005), which are based on the model atmospheres of Castelli & Kurucz (2003). The predicted spectra are resampled to a resolution to match the observed spectra and the best-fitting model, scaled in flux to best match the spectrum, is found by minimizing the rms residuals in the fit over the wavelength interval 3800–4700 Å. The grid of atmosphere models is spaced at discrete intervals on T_eff (250 K) and log g (0.5 cgs), and so the best fits have values at these grid points with uncertainties at least as large as half of the grid point spacing. See Figure 9 for individual spectra of each system overplotted with the best-fit model. The results are listed in Table 1.

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Our spectral fits assume that each target is dominated by the primary in the system. In all but one case, we only see spectral lines for one star, even when taking observations near periastron. The exception is KIC 3547874 which shows evidence of being a double-lined binary like KOI-54 (see Figure 9). The spectrum that reveals the double lines was taken at a phase concurrent with the heartbeat. From the separation in the spectral lines, the spectrum indicates that the primary has a radial velocity of 100 ± 11 km s⁻¹ relative to a spectrum taken just before the heartbeat.

Since our spectra are of moderate resolution, several systems may show evidence of double lines if spectra were taken at the correct phase with a higher resolution spectrograph. However, in order for the secondary to contribute a significant flux, it would have to have a similar temperature as the primary (or be a giant star). A blend of two stars with similar temperatures will not cause a large error in our fitted temperatures and gravities. Note that both KIC 8719324 and KIC 4248941 had spectra taken when the radial velocity shift was much larger than the spectral resolution and therefore cannot be double-line binaries. Both would have shown evidence of double lines if the companion was of a similar temperature.

4.1. Radial Velocity Measurements

For five systems, we obtained more than one spectrum and were able to measure the relative velocities at different orbital phases. We measured the radial velocity between the spectral lines by performing a cross-correlation against one spectrum. We show relative radial velocity measurements for these systems phased with the folded light curve in Figure 10. All radial velocities have been corrected for the Earth's motion around the Sun. The error bars are estimated from the FWHM of a Gaussian fit to the peak of the cross-correlation function. Note that KIC 3547874 is a double-lined binary and so our cross-correlation technique is too simple to accurately measure the radial velocities, especially when the lines are not well resolved.

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4.2. Fitting Radial Velocity Measurements

Our collection of heartbeat stars was discovered serendipitously while searching for planets in the Kepler archive. Kepler targets were intentionally selected to look for "Sunlike" stars, and so primarily G- and K-type stars were selected. For stars brighter than 15.5 mag, 97% of the selected stars are cooler than 6500 K, with the distribution peaking near 5500 K (Batalha et al. 2010). The temperatures of the observed stars in our group of binaries range from 6000 to 9250 K. While we certainly have not found all eccentric systems with dynamic tidal distortions in the exoplanet data, we might expect this collection to be dominated by G-type stars if heartbeat stars were common across all main-sequence stars.

The dominance of hot stars in our sample is likely the result of the size of the tides on the primary star. The fractional amplitude of the tidal distortions depends on the star's surface gravity as well as the size of the tidal force from the secondary (Pfahl et al. 2008). An A5V star has a lower log g and will have approximately twice the amplitude of a G5V star undergoing the same tidal force, making it easier to detect. We may be seeing this trend in Figure 11 where we plot amplitude against effective temperature. Since all of these stars lie along the main sequence and log g is not well constrained by our spectra, temperature can act as a proxy for log g. We measured the amplitude as the absolute value of the largest deviation from zero level in the binned, folded light curves (without including the amplitude of the obvious eclipsing events). We have not adjusted this plot for effects that can affect the strength of the external tidal forces (such as eccentricity and the mass of the companion relative to the primary). We do note that no obvious trend between temperature and eccentricity exists in this collection of heartbeat stars (see Table 1). As we continue to search the Kepler data set and can fold longer data sets, we will likely find a larger population of cooler main-sequence heartbeat stars with low amplitudes.

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5.1. Circularization

5.2. Induced Pulsations

With the new level of precision and coverage afforded by the Kepler time-series data, we have discovered 17 eccentric, tidally distorted, binary stars we have called heartbeat stars. KOI-54 is a long-period, highly eccentric, example of this class. The variety of light curve shapes arises primarily from variable tidal distortions being observed from different inclinations and periastron angles. Radial velocity measurements, as well as fits to the light curves, support this model. As Kepler obtains more data, we will likely find more examples of stars in this class with lower amplitudes.

Detailed modeling of these systems will be challenging. The tidal forces are changing throughout the orbit, causing large tidal distortions and induced pulsations on possibly both stars. These effects can cause nonsinusoidal variations in both the light curve and the radial velocity measurements. Plus, large orbital velocities can cause Doppler boosting to add to the flux variations while the secondary can cause significant irradiation variations. When more complete radial velocity measurements are obtained, the parameters of the companion can be constrained and a better model can be developed for each star. With detailed models, these stars offer unique laboratories in which higher order effects can be studied, effects often easily ignored in stellar structure calculations.

This is the first opportunity to observationally study dynamic tidal interactions on a class of stars. Our first analysis of each system has already shown some of the potential of these systems to constrain theories regarding energy transfer in tidally dynamic systems. From our fits to these heartbeat systems, we confirm that highly eccentric binaries with very short periods must be rare, indicating that tidal friction may have circularized these systems. Further investigation may confirm the theories of Zahn (1975) and Khaliullin & Khaliullina (2011) regarding the timescales of circularization. Additionally, the strength and period of the tidally induced pulsations vary across these systems. With a class of such systems, we can begin to decipher how this driving mechanism excites certain oscillation modes in a diverse sample of stars.

Funding for the Kepler mission is provided by the NASA Science Mission directorate. We thank the larger Kepler team for their support and hard work. Kitt Peak National Observatory is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescope (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.