KOI-54: THE KEPLER DISCOVERY OF TIDALLY EXCITED PULSATIONS AND BRIGHTENINGS IN A HIGHLY ECCENTRIC BINARY^*

KOI-54 has been observed by Kepler almost continuously from 2009 May 2 during commissioning observations (Q0) though 2010 March 22 "Quarter 4" (Q4). The light curve exhibits two remarkable features: a periodic brightening spike of ∼0.7% occurring every 42 days and a ∼0.1% "beat pattern" of pulsations in phase with the brightening events. The beat pattern arises from the interference of two pulsations with periods near 11 hr. The pulsations continue during the brightening and repeat from event to event—they are strictly in phase with the brightening.

An initial hypothesis that the repeating brightening events might follow from near-field microlensing (Sahu & Gilliland 2003) was pursued. Such a large event would result from a black hole of several solar masses orbiting and transiting an A star. Motivated by this hypothesis, a 1958 s duration Swift X-Ray Telescope (XRT) observation was made on 2010 April 25, during one of the brightening events. The source was not detected in the 0.3–10 keV bandpass, but we estimate the 5σ upper limit on the X-ray flux to be F < 1.23 × 10⁻¹³ erg cm⁻² s⁻¹. Using the Hipparcos parallax of 3.14 mas, the upper limit to the luminosity is L < 1.5 × 10³⁰ erg s⁻¹. Given the null detection, we make no further discussion of these observations.

Radial velocity (RV) observations quickly ruled out the transiting black hole hypothesis because no large rapidly changing reflex motion of the A star was observed. Also, no ready explanation of the oscillations existed in this scenario, and when examined in detail the 42 day brightenings deviate from the expected shape of microlensing events. Instead, as we show below, KOI-54 is a highly eccentric binary star system, and the brightening events are caused by tidal distortion and irradiation of the two stars during their close periastron passage. The two dominant pulsations producing the beating seen in the light curve have frequencies that are exactly 90 and 91 times the orbital frequency, and hence result from tidally driven pulsation modes.

The Kepler CCDs are read out every 6.54 s (6.02 s live time) and co-added on board. In Short Cadence (SC) mode the signal is combined to achieve approximately 58.8 s sampling cadence, and in Long Cadence (LC) mode the data are binned to 29.424 minute cadence. KOI-54 is heavily saturated in these 6 s exposures, and blooming affects approximately 30 pixels of the 74 pixel aperture (approximately 7 pixels in diameter plus two columns of 25 pixels). However, the electrons are not lost, and because of Kepler's stability, superb relative photometry is achievable—see Gilliland et al. (2010) for a discussion. For more details on the design and performance of the Kepler photometer see Koch et al. (2010) and Jenkins et al. (2010a, 2010b).

Each Kepler pixel spans ∼4'' so the very large photometric aperture includes a few very faint (by comparison) background stars. The brightest star (KIC 8112007) has Kp = 17.6 (4850 times fainter than KOI-54), and ground-based images with 1'' seeing show no star brighter than the 18th magnitude within 5''. We conclude that background starlight contaminates the light curve by less than 250 ppm (0.025%), and thus variations in the light curve are intrinsic to KOI-54.

In addition to gaps due to the scheduled quarterly spacecraft rolls, safe-mode events and other spacecraft anomalies are present; cosmic rays and other noise sources also contaminate the signal. A simple automated sigma–threshold rejection method could not be used because of the presence of the complex oscillation signal, so the light curve was carefully examined and obvious outliers were omitted by hand: 158 points were removed, leaving a total of 14,277 observations. This corresponds to a duty cycle of an astounding 92.5% over the span of 10.6 months.

The observations span 321.7 days, nearly eight complete 41.8 day orbital cycles. For each spacecraft roll, the target is on a different CCD with pixels of different sensitivity, and thus jumps in the light curve occur from quarter to quarter. Mean fluxes over the four quarters span (1.01–1.07) × 10¹⁰e⁻/cadence. To correct for these changes, and the more troublesome safe-mode-related, pointing-adjustment, and other medium-timescale discontinuities, the "RAW" pipeline-processed light curves²¹ were detrended in the following way. First, the brightening events were masked in orbital phase from 0.9 to 1.1 (i.e., 20% of the light curve). Then using gaps in the time series to define sections, the sections were fit with a low-order polynomial (typically cubic, though linear or a constant were used if appropriate). The polynomial was then subtracted from the time series, and the remainder divided by the mean of the polynomial. A value of +1.00 was then added to this quotient giving a relative flux with a mean of unity. The pipeline uncertainty estimates were boosted by a factor of six, based on the rms scatter of the residuals of an early model fit. This scaling factor was chosen for simplicity, though each quarter should have its own scaling (e.g., values of 5.0, 5.5, and 7.3 for three different quarters were found). The scatter in the residuals was significantly in excess of what was expected just from the error bars, but in hindsight this was due to using too few sinusoidal components in our initial modeling (see below). Slightly overestimating the uncertainties has no adverse affect on the estimation of the stellar and orbital parameters, and helps account for systematic noise (e.g., photometric trends after a safe mode or other anomaly) that is not included in the statistical uncertainties.

One month of SC data was obtained during Quarter 3; for the characteristics of Kepler SC data, see Gilliland et al. (2010). These observations contain 43,990 points and span 30.035 days. Examination of these SC data, and their power spectrum at higher frequencies, did not reveal any features not already well resolved in the LC data; therefore only the LC data are used for our analyses.

Finally, the relative timing precision is good to 0.5 s, but the absolute timing is uncertain by as much as 6.5 s, and this systematic uncertainty should be added to the statistical uncertainty in the epoch of periastron T_p in Table 2—see Kepler Data Release 8 Notes (Machalek & Christiansen 2010).

A variety of telescopes and instruments were used to provide moderate to high resolution spectroscopy for RV and spectral modeling. The spectra show double sets of absorption lines revealing the binary star nature of the system, and in Figure 2 we show a section of a Keck HIgh Resolution Echelle Spectrometer (HIRES) spectrum where pairs of lines are clearly exhibited. The stars are of similar spectral type and luminosity. To determine the stellar parameters we carried out an analysis of 11 relatively clean Fe i and Fe ii lines using the LTE spectral synthesis code MOOG (Sneden 1973), modified for binary star analysis. Using an input line list and two separate model atmospheres, the code computes a synthetic spectrum for each stellar component. Given a velocity separation and luminosity ratio, the code then overplots the resulting spectra onto the observed spectrum. We then varied the stellar parameters (temperature, gravity, metallicity, and microturbulent velocity) until the best overall match to our selected iron lines was found. Using a Hobby–Eberly Telescope (HET) High Resolution Spectrograph (HRS) spectrum taken very near periastron (when the stars were cleanly separated in Doppler velocity by 25.5 km s⁻¹), we obtained the following information about the stars. The stellar temperatures T₁ and T₂ are 8500 and 8800 K ± 200 K, with log g of 3.8 and 4.1 ± 0.2, and a luminosity ratio L₂/L₁ of 1.22 ± 0.04 for the wavelength range covered in the spectral analysis, ∼4500–6500 Å. Both stars are ∼2–3 times more metal-rich than solar with [Fe/H] = 0.4 ± 0.2. Thus the stars are both A-type near-main-sequence stars and potentially pulsating δ Scuti variables—see Aerts et al. (2010) for a thorough discussion of δ Scuti and other pulsating stars.

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Estimates of V_rotsin i depended on the spectra and method used, but all indicate a low projected velocity ranging from ≲5 to 10 km s⁻¹. Given the difficulty in measuring V_rotsin i we adopt a value of 7.5 ± 4.5 km s⁻¹. When corrected for the very low inclination of the system (discussed in Section 3) the true V_rot can be ≲50–100 km s⁻¹, and this is still relatively low for an A-star. This low rotation rate, combined with the metal-rich abundance, suggests that these are chemically peculiar Am or possibly Ap stars. An inspection of the Nd iii 6145 Å and Pr iii 6160 Å lines show that they are not strong, as they usually are in roAp stars.

A total of 51 pairs of radial velocities were obtained in 2010 from six different telescopes+spectrographs, and these are listed in Table 1. To minimize any potential systematic offset between the velocities acquired with different instruments, all RV observations were calibrated using the same procedure and all velocities measured using the TODCOR technique (Zucker & Mazeh 1994). Every velocity measurement is referenced to a common RV standard, HD 182488, that was observed every night, following standard practice for Kepler follow-up observations of targets of interest. Although not simultaneous with the Kepler photometry presented in this paper, these observations provided the key to understanding the nature of KOI-54: the stars are on a highly eccentric orbit, e = 0.83, with periastron passage at the times of the brightenings. Thus, the mutual interaction of the stars when closest together produces the brightening events seen in the Kepler photometry, and this is discussed in detail in Sections 3 and 5.1. Figure 3 shows the radial velocities and Keplerian fits, and Table 2 includes the orbital elements from the RV-only fit. The ratio of K-velocities gives a mass ratio of 1.034, again confirming the similarity of the two stars. Note: the TODCOR methodology we employed for measuring the radial velocities does not provide uncertainty estimates on the radial velocities, so the mean of the residuals from the RV fit to the observations was assigned as the uncertainty to all the velocities—0.31 km s⁻¹. Given the relatively good fit to the RV data over most of the orbit, this approximation is justifiable. Concerns over systematic errors, as seen in the residuals of the fit, are discussed in Section 5.1.3.

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The ELC code models the binary star light curve and RV, but does not model the oscillations. Attempts to isolate the binary light curve from the pulsation light curve proved inadequate for the high precision Kepler observations, so we added the following functionality into the ELC code, now dubbed "ELCsinus." First, a trial binary star light curve is subtracted from the observed light curve. The mean is then subtracted and the Fourier transform is computed. The 15 largest peaks in the power spectrum are found (omitting sidelobes due to leakage of the two dominant peaks), and a sum of sines and cosines is made using the 15 measured amplitudes and phases. This 15 sine pulsation model is then added to the binary star model to create the light curve model. In this ELCsinus model, the pulsations modulate the average flux and are not scaled by the binary star light curve; thus they are not boosted during the brightening events. The radial velocities are unchanged. The ELCsinus model thus consists of a light curve, an RV curve for each star, and several derived parameters: L2/L1, and the log g and V_rotsin i for each star. All of these are used to compute the goodness-of-fit χ² statistic and parameter uncertainty estimates.

As shown in Figure 4, we obtain an excellent fit to the light curve, matching the amplitude, shape, and phase of the brightening events. The brightening is due to a combination of mutual irradiation and tidal distortion during periastron passage of the pair of A stars on their highly eccentric, nearly face-on orbit. The system parameters of KOI-54 are reported in Table 2. The uncertainties listed are the formal uncertainties and should be treated as lower limits, as there are many potential sources of systematic errors as discussed in Section 5.5. As an example, the mass and radius of Star 1 from the ELCsinus fit yield a log g that is 1.6σ larger than the spectroscopically measured log g (even though any deviation from the observed log g incurs a χ² penalty). Although it is very difficult to assess the size of systematic uncertainties, a doubling of the formal uncertainties seems reasonable, especially for the inclination, masses, and radii of the stars.

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Note that the star designated as Star 1 is the less massive star. Also note that the pulsations continue to occur during the brightening events and exhibit a fixed pattern: there is no discernable phase drift. The pulsations are matched well with our simple modification to ELC, indicating that the pulsations are not altered by the irradiation and gravitational distortions at periastron. Indeed, as discussed in the next section, the pulsations are driven by events at periastron passage.

KOI-54 is a remarkable binary star system. In the following section, we further discuss various properties of the system, but first we mention that if the Hipparcos parallax is reliable for the binary, the distance is 318 ± 71 pc. At apastron the stars are separated by 0.79 AU, which translates to 2.5 mas—a potentially resolvable separation.

5.1.1. Photometric Determination of e and i

The ELCsinus model produces an excellent fit to the photometry and tightly constrains the system parameters. To our knowledge, this is the first determination of stellar parameters for a non-eclipsing double-lined spectroscopic binary based on the brightening during periastron passage. This is possible because of (1) the high precision of the Kepler photometry and (2) the strong sensitivity of the brightening to periastron passage, and hence to the orbital parameters. Remarkably, we have found that the photometry alone can determine the orbital eccentricity as precisely as the radial velocities. This is a consequence of the dependence of the amplitude of the brightening to changes in eccentricity as illustrated in Figure 6. In this set of simulations, closely matched to KOI-54, the only parameter that varies is the eccentricity. In particular, the lower panel shows how one can estimate the eccentricity to ∼1% just from its amplitude, if all other parameters were known. The strong sensitivity to eccentricity is simply a consequence of the brightening being due to tidal forces and irradiation that scale as the separation of the stars cubed and squared, respectively, and the separation of the stars at periastron is linear in the eccentricity. (Note: we are assuming simple tidal distortion aligned along the line joining the center of masses; in the general asymmetric case the tidal force is much more sensitive to the separation, going as F∝r⁻⁷—see Hut 1981 for a discussion.)

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In addition to the eccentricity, the Kepler photometry is also able to constrain the orbital inclination, even though there are no eclipses and no double-humped ellipsoidal variations are present. But in fact the ellipsoidal variations are present—for such a highly eccentric orbit the humps have shifted from the usual photometric quadrature phases, 0.25 and 0.75, to the phase of periastron (defined here as phase 1.0), and the humps have merged. The two humps are not equal in height because the orbital inclination is not exactly zero and the orbital ellipses are not aligned along our line of sight (the argument of periastron is not ±90°); thus the two phases of maximum visible ellipsoidal distortion are not symmetric about periastron. In addition, the inclination and argument of periastron determine the orientation of the irradiated hemispheres of the stars to our line of sight, and this creates a small asymmetry and shift in the phase of the brightening. For the geometry of KOI-54, the larger the inclination, the more the peak would shift to earlier orbital phases, and the narrower the peak the more asymmetric it would become (brighter post-peak than pre-peak). Such effects allow the inclination to be measured.

5.1.2. Relative Contributions of Irradiation and Distortion

Both tidal distortion and the irradiation/reflection contribute to the brightening, and Figure 7 shows the separate contributions of each. For KOI-54 the non-sphericity is the larger contributor, with the reflection component roughly 75% as large. The dominance of the ellipsoidal effect is not a necessity, but rather a consequence of the specific binary system parameters, and could reverse in another system (i.e., reflection effect > ellipsoidal effect). As expected, the ellipsoidal component is more centrally peaked than the reflection component, as it is more sensitive to the separation of the stars. Note that the reflection component is slightly asymmetric, being brighter after periastron than before. This is a consequence of the inclination of the orbital plane and the orientation of the irradiated hemispheres with respect to the observer.

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5.1.3. Fit to the Radial Velocities

5.1.4. Tidal Evolution

We have begun a search of parameter space for coeval stellar evolution models that fit the stellar constraints as given in Table 2. We are using an updated version of the Iben (1963, 1965) stellar evolution code that includes the OPAL opacities (Iglesias & Rogers 1996) supplemented by the Alexander & Ferguson (1994) and D. Alexander (1995, private communication) low-temperature opacities and the Grevesse & Noels (1993) solar element abundance mixture. As a starting point, we evolved an M₁ = 2.33 M_☉ and an M₂ = 2.40 M_☉ model with solar abundances: X = 0.68 (H mass fraction), Y = 0.28 (He mass fraction), and Z = 0.04 (mass fraction of elements heavier than H and He). We find that the constraints in Table 2 are difficult to match with solar-metallicity Z = 0.02 models, especially the combination of temperature and radius: the radii are too small for the temperatures. A metal abundance higher than Z = 0.04 would allow a wider range of parameters of the evolution models to match the (M₁, R₁, T₁)–(M₂, R₂, T₂) pair, and indeed the spectroscopy does indicate enhanced metallicity. However, for this initial investigation we did not further pursue the metallicity as this would also require varying the Y abundance (beyond the scope of this investigation) and also that these stars may be metallic A stars (Am) so the surface abundances from spectroscopy may not be representative of the entire star.

In selecting models, we emphasized the constraints on the ratios of temperature, luminosity, and radius over their actual values as the ratios are better determined observationally. We discounted the spectroscopic log g measurements that suffer from large uncertainties. To be specific, we sought to match the ratios in temperature T₂/T₁ = 1.035 ± 0.034, luminosity L₂/L₁ = 1.22 ± 0.04, and in radius R₂/R₁ = 1.06 ± 0.02. As expected, a family of solutions was found; for example a pair of stars with temperatures T₁, T₂ of 8498 K, 8667 K satisfies our matching criteria, as does another pair at 8754 K, 8999 K. Looking more closely, in the cooler pair, with an age of 0.354 Gyr, the temperature, luminosity, and radius ratios (1.020, 1.26, and 1.08) are close to the measured values, but the absolute radii (R₂ = 2.52 and R₁ = 2.35 R_☉) are somewhat larger than our estimated values of 2.33 and 2.20 R_☉. In the hotter pair, with an age of 0.263 Gyr, the temperature, luminosity, and radius ratios (1.028, 1.25, and 1.06) are again close to the measured values, and the absolute radii (2.28 and 2.15 R_☉) are closer to the reported values, but the temperatures are at the high end of the uncertainty range. From this exploratory investigation, the important conclusion we can draw is that the (M₁, R₁, T₁)–(M₂, R₂, T₂) set we measure is consistent with stellar evolution models of identical age and metallicity stars.

By way of brief introduction, linear perturbation theory can be used to determine the pulsation frequencies in stars, given their internal structure. Stellar pulsations are generally of two classes, the p-modes or g-modes depending upon whether pressure or gravity is the local restoring force. Being acoustic waves, p-mode characteristics are dependent on the local sound speed, and are important in the envelope and outer portion of the star. Their motion is primarily vertical and they have periods of the order of hours and less. The g-modes are set up by the competition between gravity and buoyancy deep within the star. Their wave motion is primarily horizontal and they have periods of the order of one day (for non-degenerate stars). The g-modes cannot extend through convective zones as the convective instability means there is no local restoring force. The modes are classified according to three spherical harmonic integers: the number of radial nodes (n), the angular degree (ℓ), and azimuthal order (m). ("Nodes" are the locations where the standing wave pattern, caused by the interference of the pulsations, has zero amplitude.) The degree ℓ gives the number of nodal circles on the sphere, and |m| is the number of nodal lines that cross the equator (or equivalently, go through the poles). Radial modes do not change the spherical symmetry of the star during the pulsation cycle and are characterized by ℓ = m = 0. The frequencies of non-radial modes of degree ℓ are split by rotation or magnetic fields. Rotation lifts the degeneracy in m in such a way that each mode of degree ℓ results in 2ℓ + 1 multiplet components in the frequency spectrum, with m = −ℓ, ..., 0, ..., +ℓ. The classical κ (opacity valving) mechanism caused by He ionization in the stellar envelopes is predicted to drive p-mode pulsations in A- to early F-type main-sequence stars, while g-mode pulsations are not expected in these types of stars. For a complete reference see Aerts et al. (2010).

The periodic tidal forces experienced by the stars at periastron passage are expected to induce stellar oscillations, including g-modes which otherwise would not be present. To explore such tidal excitation, we assume one of the stars (M₁) to rotate rigidly and the other star (M₂) to be well described as a point source. Whenever the ratio of the external tidal force to the self-gravity at the star's equator is small, i.e., the dimensionless parameter ε_T ≡ (R₁/a)³M₂/M₁ ≪ 1, the tide-generating potential in the eccentric orbit can be expanded as a series in terms of spherical harmonics and of the companion's mean motion. We follow the method outlined in detail in Willems & Aerts (2002), who have shown that the spherical harmonic with degree ℓ = 2 is by far the dominant term in the expansion of the tide-generating potential, and only components with m = −2, 0, 2 occur. The values of these components depend on the orbital inclination and eccentricity, as well as on the mean and true anomaly of the companion in its relative orbit. Whenever the forcing frequency, σ_T = kΩ_orb + mΩ_rot, matches the star's free gravity oscillation modes, the tidal action exerted by the companion gets enhanced and the particular oscillation mode gets resonantly excited with the forcing frequency of the dynamic tide (Smeyers et al. 1998). Thus, we calculated the ℓ = 2, m = 0 g-mode eigensolutions for each of these stars using the procedure and non-adiabatic non-radial pulsation code of Pesnell (1990) as reported in Guzik et al. (2000). We find that the g-mode spectrum for stars of this type is very dense, with period spacings of the order of 0.03 days. These are damped oscillations that would die out without the periodic tidal "pumping," but the damping times are very much longer than the orbital period (on the order of 1000 years), so once a mode is excited, it is certainly not expected to change amplitude during an orbital period—or the lifetime of the Kepler Mission.

While still adhering to the constraints on the stars discussed above, we tweaked the tidal model parameters to find a star whose ℓ = 2, m = 0 spectrum has one mode that coincides exactly with F1 (i.e., 90 times the orbital frequency). This model has a mass of 2.33 M_☉, T_eff = 8663 K, and R = 2.15 R_☉, and most importantly, its g₁₄ mode of ℓ = 2, m = 0 has frequency F1, proving that standard stellar models exist that can generate a resonant tidal excitation. (The g₁₄ mode has n = 14 radial nodes in the interior of the star and the ℓ = 2, m = 0 is a quadrupole that distorts the star in the correct directions in response to tidal forcing.) Even though we did not tune the predictions of the tidal excitation theory to fit for KOI-54 in this discovery paper, we expect a frequency pattern as in the example treated in Willems & Aerts (2002), i.e., a pattern of resonances at spaced values of the rotational and orbital frequency as kΩ_orb + mΩ_rot. We applied this to all the frequencies detected in KOI-54 and their splitting values, which allowed us to deduce that Ω_rot ∼ 0.133 day⁻¹ (1.54 μHz) from the occurrence of m = 1 and m = 2 tidal splittings. This translates into a mean rotation period of 7.5 days and an equatorial rotation velocity of some 15 km s⁻¹. If the orbital and rotational axes are aligned, as assumed in the tidal theory, this leads to a prediction of V_rotsin i of 1.5 km s⁻¹. This rotation velocity is certainly low compared to the observed V_rotsin i, being ∼1.7σ from our adopted value, but we remark that the V_rotsin i measurements were made in the standard way and did not take into account pulsational broadening of the lines. We therefore tentatively accept the size of the tidal splitting of the lines as compatible with the measured V_rotsin i. Furthermore, it is unlikely that the rotation inside the star will be rigid, and so this value of Ω_rot should be interpreted as an average value over the depth of the star, effectively negating concerns that this average rotation velocity is too low compared to the observed surface V_rotsin i. In summary, we have found a model consistent with the observations that can explain all but one of the 30 largest observed pulsations as either harmonics of the orbital frequency or m = 1 and m = 2 tidal splittings of those harmonics. (F25 remains unaccounted for in this model.) Despite the very attractive features of this model, it is still very exploratory in nature and we do not claim uniqueness of the solution. Future observations will lead to much better frequency resolution, allowing a much more refined stellar model. We point out that slight deviations of the observed frequencies of the splittings from their exact predicted frequencies is quite interesting, and probably related to the nonrigidity of the stellar rotation. When several years of Kepler observations are available, we have the exciting prospect of using the observed tidal splittings to allow for an asteroseismic determination of the internal rotation law of the star.

From their spectroscopic temperatures (T_eff ∼8500, 8800 K) both stars are too hot to be γ Dor pulsators, and the absence of any periodicities shorter than 11 hr in the Short Cadence light curve power spectrum confirms this. Why the stars do not exhibit any δ Scuti-like pulsations is not known, though it might be that the stars are just beyond the blue edge of the classical instability strip. Presumably no unstable modes (p or g) exist for the stars, and the stars are intrinsically non-pulsators. This supposition is supported by the exploratory stellar models discussed in the previous section.

The pulsations that are observed are those modes for which their free g-mode eigenfrequency is close enough to a harmonic of the orbital period that they are resonantly driven to measurable amplitudes by the periodic tidal (and possibly thermal) forces felt at periastron. Thus while very many pulsation modes are possible, only those that are integer multiples of the orbital frequency and phase-locked with the binary orbit are present.

Some challenges posed by the pulsations in KOI-54 include: (1) Why are the 90th and 91st harmonics of the orbital frequency the strongest pulsations? (2) Why should one particular mode be strongly excited while many adjacent ones are not, i.e., why is orbital harmonic 91 very strong but 92 completely absent? (3) What limits the highest frequency excited mode and why is there no significant power present at higher harmonics? (4) Why are some modes tidally split but not others? and (5) If only one of the stars is pulsating, which one is it, and why is the other not pulsating?

Interestingly, if we accept the slow stellar rotation inferred from the tidal splitting of the pulsations, that would mean the star is rotating at only a third of the pseudosynchronous rate. This would seem unlikely in a simple tidal evolution scenario, but the very large value of orbit-to-spin angular momentum ratio means that exchanges of angular momentum are possible and episodes of spin-up and spin-down can take place. Or, as mentioned earlier, the stars are possibly Ap stars that may have been born rotating very slowly. A factor-of-two improvement in the measured V_rotsin i would be very valuable; the precision is currently available, but the accuracy is not.

Given that the stars are similar in mass, radius, and temperature, have identical ages and metallicities, and experience identical driving frequencies, a total lack of pulsations from one of the stars would be puzzling. And yet the matching of the pulsation frequencies with a single pulsating star, and the lack of any apparent double set of peaks in the power spectrum, does suggest that one star could be responsible for all the pulsations. In principle, high spectral resolution, time-resolved spectroscopy can determine which of the stars is pulsating (or if both are). If the spectra are obtained near periastron, the component lines will be well separated and pulsations in one of the sets of lines immediately tells us which star is pulsating. However, the intensity amplitude of the pulsations is very small, ∼0.1%, making this rather challenging. Velocity modulations causing line profile shape changes are much more likely to be observable, though again the expected variations will be small. Further complicating the matter is the need to cover at least one full oscillation cycle, thus requiring >12 hr, and perhaps a multi-longitude campaign. Fortunately, the star is bright (Kp = 8.380) and the integrations can be long and still temporally resolve the pulsations. The reward for such efforts is high: given that the two stars are very similar, if only one star is pulsating, it means that the excitation mechanism must be very fine-tuned to drive the oscillations in one star but not the other. Thus, there is the potential for a precision investigation of internal stellar structure, given stars of identical age and metallicity and nearly identical masses.

While we are confident that the periastron-pumped pulsating binary star model is correct, given the complexity of both the physics and the data calibration, several sources of systematic error are potentially present. In this section, we describe several of these, with an assessment of their effects when possible. These are checks of the robustness of the overall solution, not detailed investigations that could reveal small changes at the few-σ level. Toward that goal, we used a linear limb-darkened blackbody approximation instead of stellar atmospheres and locked the stellar rotation to the pseudosynchronous spin frequency. We begin with a look at the data calibration.

The most significant of the systematic errors in the data arise from the difficulty in removing small and abrupt changes in the flux level due to changes in pointing, cosmic ray events, safe modes, etc. While such breaks are usually visible in Kepler light curves, the pulsations in KOI-54 hide these. The residuals of the ELCsinus fit exhibit small but significant meanderings and tilts on longish timescales (days to weeks) which are almost certainly due to detrending problems. Figure 4 shows the residuals of the data minus the ELCsinus fit scaled by a factor of five to be more visible. The residuals are not white noise, indicating correlations due to the detrending, but also due to the fact that the model includes only 15 sinusoids while at least 30 are present. The largest set of correlated residuals occurs at the brightening event near day BJD 2,455,105 (see the middle left panel in Figure 4). Because the determination of the eccentricity and inclination is so sensitive to the amplitude and shape of the brightenings, even small tilts or jumps could potentially affect these estimates. We were concerned about the inclination in particular. Normally (e.g., for eclipsing binaries) a small change in inclination does not significantly affect the mass estimates for the stars, as the slope of the sine function near 90° is flat. But for KOI-54 the inclination is small so changes in sin i go linearly with i. Because the masses depend on sin ⁻³i a change from 5° to 6° results in a decrease of 58% in mass. To see if detrending imperfections could affect the accuracy of our model parameters (as opposed to precision), we carried out two investigations. First, we masked out the region with the largest residuals and re-fitted. Second, we used the power spectrum feature in ELCsinus to measure, then remove, low-frequency power in the residuals, on timescales of 10 days and longer. While the χ² of the models was significantly better (by construction), there was no significant change in the system parameters.

The determination of the eccentricity depends on the relative flux increase of the brightening event. Thus any systematic error in normalization will affect the eccentricity. The light curve is in fact contaminated by background starlight contained in the same aperture used for KOI-54, and the relative amplitude of the brightening is thus biased low; hence the derived eccentricity is potentially biased low. To check the effect this may have, we subtracted 250 ppm from the light curves and re-fitted. In addition, we included a small "third light" dilution factor into the model and fitted for this parameter. In both cases, there was no significant change in the model parameters. Furthermore, such a bias would not effect the RV-only eccentricity estimate, which agrees remarkably well with the photometric estimate.

The "wings" of the brightening event (i.e., the upward curvature of the light curve) extend all the way to apastron at phase 0.5; there is no phase where the light curve is truly flat. This is a potential problem in the calibration: we detrended the light curve assuming it was flat between phases 0.1 and 0.9. From our binary-star only light curve, the maximum difference in relative flux between phases 0.1 through 0.9 is 99 ppm, roughly 1.7 times the adopted uncertainty per point. However the light curve is flat to within 25 ppm from phases 0.2–0.8, so the neglect of curvature in the calibration is benign.

In our model, we have neglected several physical effects which we address below. Systematic errors in the model are potentially more serious than those of data calibration, as we have eight brightening events that can help average over any calibration errors. For example, as is standard practice, we have ignored any consideration of magnetic fields. However, for these Am or Ap stars, such neglect at periastron passage may not be entirely justified and warrants further investigation.

We have not included relativistic Dopper boosting of the light curves. For this nearly face-on binary this boosting effect will be very small, and a quick test showed that including the boosting produced a change in total (not reduced) χ² of ∼0.1, a completely negligible amount. Nevertheless, in general, future models should include this effect as a modestly higher orbital inclination would result in a measurable difference in the light curve, given Kepler's remarkable precision.

In general, an irradiated star is hotter than an un-irradiated star, and so the light that impinges back onto the irradiating star is greater than if the irradiated star were isolated. Thus the reflection effect should be iteratively computed, and this is the standard method championed by Wilson (1990). However, for KOI-54 this is a very small effect as the total brightening is less than 1%, and tests with iteration showed almost no difference compared to a single calculation assuming a point-source irradiator (which is exact for perfectly spherical stars). Given the large computational cost of full geometry tile-by-tile iteration and the negligible benefit, we used the point-source irradiation approximation. Our models have a maximum non-sphericity of 0.7% at periastron (ratio of minimum-to-maximum radii: pole radius to L1-direction radius), justifying the spherical approximation for irradiation. And despite the relatively close periastron passage of 0.065 AU (∼6.4 stellar radii), the stars only fill ≲ 30% of their Roche lobe radii.

The most significant of the limitations of our model is the use of the Roche approximation. While quite successful at matching the observations, there is a systematic bias inherent in the use of Roche potentials in that the tidal elongations are always pointing along the axis connecting the centers of mass. In other words, the stars are treated as perfect, viscousless fluids that can instantaneously adjust their shapes to the gravitational and spin potentials.²² But real stars will have tidal bulges that do not instantly readjust to the changing external potential and are also "dragged" by the rotation of the star. Thus, the elongation will not be aligned with the center of masses except when the stars are in synchronous rotation in a circular orbit. The irradiation/reflection would also be affected, simply because of the different geometry. Furthermore, the Roche potential treats the mass distribution as a point-mass at the center of the star; this is equivalent to an n = 5 polytrope or a tidal Love number k₂ = 0. Thus for a given external force, the Roche potential yields the least possible tidal deformation. It then follows that the amplitude of the ellipsoidal variation is minimal in the Roche approximation. While it is true that the mass inside stars is highly centrally concentrated, a more realistic mass distribution would produce a larger tide for a given external force. So to match a given observed brightening amplitude, a weaker tidal force would be required. This could be achieved by either lower masses, or much more likely, a very small increase in separation at periastron due to a very small decrease in eccentricity. Hence the brightening in the Roche potential model is expected to be very slightly different than a case where a more realistic internal mass distribution is used.

Although we have measured very precise system parameters with our ELCsinus model, for the reasons discussed above the accuracy is worse than the precision. This is especially true for the eccentricity and inclination, and therefore the stellar masses which are proportional to the cube of the inclination. On the other hand, the photometric-only eccentricity agrees well with the RV-only eccentricity, indicating that the Roche approximation is valid at the level of analysis presented in this discovery investigation of KOI-54. A better treatment of the system, well beyond the scope of this paper, would involve dynamical tide theory—particularly relevant given the observed tidally driven pulsations.

This deficiency of the Roche model leads to an interesting supposition. The angle between the instantaneous centers of the stars and the axis of the tidal bulge depends on the internal structure of the star and is related to the efficiency of tidal dissipation. Measurement of this lag angle can then be used to constrain the tidal quality factor Q. If one could determine the instantaneous geometry of the system without the photometry, then the lag between the brightening and the true time of periastron passage could be measured. The geometry could in principle be determined though a great host of radial velocities, corrected for the distortions caused by the pulsations. Another possible way is related to the pulsations themselves; they provide a clock that could be used to set a fiducial time against which to measure the time of brightening. Interestingly, the time when the two largest modes (F1 and F2) are in phase is not the time of periastron as defined by the photometry, but 0.39 ± 0.07 days (=9.4 hr) earlier. This can actually be seen in the light curves themselves, as the asymmetric shoulders of the brightening and dip at the very pinnacle—see Figures 1 and 4.

While tidally excited oscillations have been studied in quite some detail from a theoretical point of view (e.g., see Aerts et al. 2010 for a discussion and references), observational evidence of their existence is scarce. Prior to KOI-54, only three such systems were known to exist. The rarity of these stars is understandable if one realizes that the forcing frequencies from the dynamical tides must come very close to the free eigenmode frequencies of one of the stars in order to resonantly excite the oscillation (e.g., Willems 2003). This naturally leads to the excitation of particular gravity modes whose frequencies are exact multiples of the orbital frequency.

Two of the previously known cases, HD 177863 (De Cat et al. 2000) and HD 209295 (Handler et al. 2002), were discovered from ground-based photometry and spectroscopy and are single-line spectroscopic binaries. HD 177863 is a slowly pulsating B star with e = 0.60, P_orb = 11.9 days with one detected gravity mode whose frequency is 10 times the orbital frequency. HD 209295 is a hybrid γ Dor/δ Sct star in a binary with e = 0.35, P_orb = 3.1 days with five g-modes having frequencies which are exact multiples of the orbital frequency, besides a free δ Sct mode. A much more interesting case is the eclipsing double-lined spectroscopic B-type binary HD 174884 discovered with the CoRoT mission (Maceroni et al. 2009). This system has e = 0.29, P_orb = 3.66 days and has pulsations with exact multiples of 2, 3, 4, 8, and 13 times the orbital frequency and very tightly constrained physical parameters. While it is unclear if the lowest-order multiples are due to imperfect prewhitening of the orbital curve (i.e., removal of the binary star contribution to the light curve) one does not expect this for harmonics 8 and 13 as all lower-order harmonics should have been found as well. Moreover, these two higher frequencies correspond exactly to those of free gravity modes of radial order ∼10, which points to tidal excitation.

With its periodic tidal brightening and rich set of pulsations (over 30 pulsations at either integer multiples of the orbital frequency or at tidally split multiples of the orbital frequency are present), KOI-54 now joins this elite set of eccentric binaries that exhibit tidally driven pulsations.