KOI126: A TriplyEclipsing
Hierarchical Triple
with Two LowMass Stars
The Kepler spacecraft has been monitoring the light from 150,000 stars in its primary quest to detect transiting exoplanets. Here we report on the detection of an eclipsing stellar hierarchical triple, identified in the Kepler photometry. KOI126 (A, (B, C)), is composed of a lowmass binary (masses , ; radii , ; orbital period days) on an eccentric orbit about a third star (mass ; radius ; period of orbit around the lowmass binary days; eccentricity of that orbit ). The lowmass pair probe the poorly sampled fullyconvective stellar domain offering a crucial benchmark for theoretical stellar models.
The Kepler mission’s primary science goal is to determine the frequency of Earthsize planets around Sunlike stars. To accomplish this, thousands of stars are monitored to detect the characteristic dimming of star light associated with a planet passing in front of its host (1).
The Kepler observing scheme lends itself not just to planetary transits but, additionally, to the detection of the analogous eclipses in stellar multiple systems (2). We report on one such hierarchical system consisting of two closelyorbiting lowmass stars (KOI126 B, C; days) that are together orbiting with a longer period ( days) a more massive and much more luminous star (KOI126 A). The two lowmass stars double the number of fullyconvective stars with mass and radius determinations better than a few percent; the stars in the eclipsing binary CM Draconis (3–5) are the remaining entries in this inventory.
The KOI126 system is oriented such that all three components and the Kepler spacecraft lie near a common plane and, consequently, eclipses among all three stars are observed. The unique variety of these eclipses reveal complex, informationladen dynamical content. As a consequence, the absolute system parameters for all three stars may be accurately determined from the photometry alone and are immune to the biases that plague the traditional study of lowmass eclipsing systems (e.g., as a result of stellar spots on the lowmass stars, 6).
The Kepler photometry, reported here, spans 418 days of near continuous observation of KOI126 (KIC 5897826, 2MASS J19495420+4106514, ) . The first 171 days consist of successive 29.426 min exposures, whereas the remaining 247 days were observed at both the 29.426 min (or long) cadence and at the 58.85 sec (or short) cadence (Fig. S1 shows the long cadence timeseries). We used the short cadence data (7) exclusively in our determination of system parameters.
KOI126 was originally identified by the Kepler data pipeline (8, 9) as a planetary candidate. The suspected transit events (Fig. 1) were found to be unusual upon closer inspection. The 1–3% decrements in the relative flux, occurring approximately every 34 days, exhibit a variable eclipse profile. At some epochs, two resolved transits are seen; at others, the transits are nearly simultaneous. The transitlike events have durations that vary substantially across the observed epochs and show asymmetric features about mideclipse, indicative of accelerations transverse to the lineofsight.
The periodic, superposed transits in the Kepler light curve are most readily explained as the passage of a close (or inner) binary (KOI126 B, C) across the face of a mutually orbited star (KOI126 A). At the start of each passage, KOI126 B and C are at a unique phase in their binary orbit, yielding a unique transit route and light curve shape. To first order, the duration of transit is shorter or longer depending on whether the motion of a given component of the inner binary is prograde (shorter duration) or retrograde (longer duration) relative to the orbit of the inner binary centerofmass. The shorttimescale orbital motion of the inner binary accounts for the apparent accelerations.
Preliminary modeling of the Kepler light curve predicted the secondary passage of the inner binary behind KOI126 A (shown in the bottom two timeseries in Fig. 1). A BoxLeastSquare algorithm (10) search of the Kepler light curve, excluding these secondary and transit events, revealed the relatively shallow eclipses between KOI126 B and C, occurring every days (Figs. S2, S3). Based upon these detections, KOI126 B and C were inferred to be each less luminous by a factor of 3,000–5,000, as observed in the wide Kepler bandpass (1), than KOI126 A. Both the eclipses of the inner binary pair and occultations of that pair by KOI126 A were not observed to be strictly periodic as they were absent in the data for long stretches of time. The long cadence event near (BJD) features the alignment of all three objects along the lineofsight, resulting in a short brightening in the light curve (11).
A periodogram of the light curve, after removing eclipse events and correcting for instrumental systematics, shows an 17 day modulation with a relative amplitude of 500 partspermillion (Fig. S4 plots a representative sample of this variation). It is likely associated with the rotation of KOI126 A (see Supporting Online Material, SOM).
In addition to the Kepler photometry, we collected sixteen spectra of KOI126 over 500 days (SOM). The spectra only showed features associated with a single star, KOI126 A. The primary goal of these observations was to acquire precision radial velocity (RV) measurements of KOI126 A (Fig. S5, Table S1). An ancillary goal of the spectroscopic study was to fit model spectra to the composite spectra for KOI126 A in order to determine stellar parameters (SOM, Table 1). From this analysis, we found that KOI126 A is metal rich, relative to the Sun ([Fe/H] = ), and has an effective temperature of K and a stellar surface gravity .
We used a full dynamicalphotometric model to explain the data (SOM). Newton’s equations of motion along with a general relativistic correction to the orbit of the inner binary (12–14) were integrated to determine the positions and velocities of the bodies at a selected time. Each individual star’s position was then corrected to account for the finite speed of light (11). These corrected positions, along with the absolute object radii and relative flux contributions, were used to calculate the combined flux.
The model was fitted to the short cadence Kepler data by performing a leastsquares minimization (SOM). Only the data shown in Fig. 1 were utilized in the fit. While not used in the fit, the long cadence data are nearly exactly matched when using the bestfit parameters (Fig. S6). Subject only to the short cadence photometric data, we determine the individual masses and radii with fractional uncertainties less than and , respectively.
We included the RV data for KOI126 A in a subsequent fit (Table 1). The bestfit parameters are identical to those found using photometry alone; although, masses were determined to better than and radii to better than . The inner binary orbit is nearly circular and inclined by relative to the outer binary orbit.
The photometric data could not be fit by assuming fixed Keplerian orbits for the inner and outer binaries. This is due to the relatively rapid variation of the Keplerian orbital elements (such as the orbital inclination or the eccentricity) as a result of the gravitational interaction between the three stars (Fig. S7) (12, 15). The observational evidence of this periodicity is most easily seen with the rapid circulation of the ascending node of the inclined inner binary: the inner binary orbit precesses, like a spinning top, every days in response to the gravity of KOI126 A. This precession explains the occasional absence of the eclipses between KOI126 B and C. Similarly, the oscillation of the outer binary inclination, compounded with the measured eccentricity, explains the sparsity of occultations of the inner binary by KOI126 A.
Our final dynamical model did not include the effects of stellar spin, tidal distortion, or frictional dissipation because the data did not demand them. Nevertheless, we did investigate these contributions by extending our numerical model to include parameterized forces appropriate for each effect (SOM). The only relevant force on the observed timescale, assuming plausible rotation and dissipation rates, is caused by the mutual tidal distortion of KOI126 B and C. The controlling parameter for this force is the internal structure (or the apsidal) constant, , which is intimately related to the interior stellar density profile and provides an important constraint on stellar models (16, 17). There are no reliable constraints on for stars with masses similar to KOI126 B or C. For a fullyconvective star, approximated as a polytrope with index , is estimated to be as large as (18).
Our data constrain at 95% confidence, assuming that KOI126 B and C have equivalent apsidal constants. Although the current constraints on are modest, a fit to the predicted Kepler light curve over the remainder of the nominal mission (3.5 years in total, with 25 more transitlike events) demonstrates that will be measured with a relative precision of . In addition, masses and radii will be determined to better than .
The masses of KOI126 B and C lie below the threshold for having fully convective interiors (less than ). It has been suggested (5, 19–22) that the large (10–15%) disagreements seen between model predictions (23, 24) and measured radii for lowmass stars were confined to stars in close binaries outside of this convective domain. A sample of lowmass stars with dynamicallymeasured properties validates this claim (3, 19–22, 25, 26); however, there is little reliable information available for stars with masses under (see Fig. 2).
Previously, CM Draconis (3–5) provided the only precise constraints on stellar models for stars below the convective mass boundary. In its case, theoretical models seemed to underestimate the stellar radii at the 5–7% level, a disagreement less than that seen with more massive stars but still consistent with increased activity attributed to fast rotation as a result of tidal spinup (5). In comparison, the radii of KOI126 B and C are also underestimated by the models; however, this disagreement is smaller (2–5%; Fig. 2).
In addition to accurately measured masses and radii, the metallicities and ages of KOI126 B and C are approximated by the values estimated for KOI126 A, if we assume all components were coevolved and formed from the same protostellar nebula. In this case, KOI126 B and C have a supersolar metallicity ([Fe/H] ) which can be compared to the poorly determined, subsolar metallicity estimated for CM Dra (5). The enhanced metallicity of KOI126 relative to solar may partially resolve the discrepancy between the observed and predicted radii of KOI126 B and C, although we are not aware of any models that properly account for this enhancement. We compared the mass and radius of KOI126 A with stellar models in a wellcalibrated mass range (27) and estimated a system age of Gyr.
We are unable to measure the spin periods of KOI126 B and C, but, it is likely that synchronization has occurred and that the spin periods are nearly equivalent to the orbital period of the inner binary (SOM). These spin periods are slower than the expected orbitsynchronized spin periods for CM Dra A and B ( days) by nearly 0.5 days. This fact may partially account for the differences in radii between the similarmass stars CM Dra B and KOI126 C – CM Dra B may have increased magnetic activity relative to KOI126 C owing to its faster rotation (5).
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Funding for this Discovery mission is provided by NASA’s Science Mission Directorate. J. A. C. and D. C. F. acknowledge support for this work was provided by NASA through Hubble Fellowship grants #HF51267.01A and #HF51272.01A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 526555. J. A. C. is very grateful for helpful discussions with P. Podsiadlowski, S. Rappaport, G. Torres, A. Levine, J. Winn, S. Seager, and T. Dupuy.
Figure 1: Bestfit dynamicalphotometric model to the fitted data for KOI126. The top eight panels are data observed during the passage of B and C in front of A from the perspective of the Kepler spacecraft. The bottom two panels are data observed during the passage of B and C behind A (binned by a factor of 10). In all cases, the solid black line gives the bestfit model. The inlaid diagrams show, to scale, the orbits of B (green) and C (blue) relative to A (yellow star) at times corresponding to those in the associated timeseries. The dashed orbit is that of the centerofmass of B and C. The numbered circles give the locations of B and C at the times indicated by the vertical dotted lines in the associated timeseries, corresponding to the numeric index (0–2). The circles are to scale for the radii of B and C. The specific values for each respective panel, reading from left to right and top to bottom, are (in BJD) 2455102.815, 2455136.716, 2455170.465, 2455204.267, 2455238.207, 2455271.751, 2455305.713, 2455339.496, 2455259.000, 2455326.506.
Figure 2: Masses and radii of known lowmass stars with dynamicallyestimated masses and radii measured to better than 3% fractional accuracy (22, 28). The black curves correspond to the theoretical stellar isochrones by Baraffe et al. The dashed, dotted, solid and dashdotted curves correspond to 1 Gyr, 2 Gyr, 4 Gyr and 5 Gyr solar metallicity isochrones, respectively. The blue points correspond to CM Dra A, B and the red points correspond to KOI126 B, C. The inset panel corresponds to the region in the larger plot enclosed by the dashed rectangle.
Parameter  Value 

Masses  
Radii  
Average Densities (g cm)  
Surface gravities (logarithms in cgs units)  
Observed relative fluxes  
“Outer” binary [(A, (B, C))] elements on 2,455,170.5 (BJD)  
Period, (day)  
Semimajor Axis, (AU)  
Eccentricity,  
Argument of Periapse,  
Mean Anomaly,  
Skyplane Inclination,  
Longitude of Ascending Node,  
“Inner” binary [(B, C)] elements on 2,455,170.5 (BJD)  
Period, (day)  
Semimajor Axis, (AU)  
Eccentricity,  
Argument of Periapse,  
Mean Anomaly,  
Skyplane Inclination,  
Longitude of Ascending Node,  
Star A parameters from spectroscopy  
Effective Temperature, (Kelvin)  
Metallicity, [Fe/H]  
Projected Rotational Velocity, (km s) 
Supporting Online Material
Online data
All data used in this analysis have been made available at
http://archive.stsci.edu/prepds/kepler_hlsp/
Spectroscopy
Six of the sixteen spectra were collected using the Tull Coude Spectrograph on the 2.7m Harlan J. Smith telescope at the McDonald Observatory in west Texas, which has a resolving power of R=60,000 and a wavelength range of 375010000 angstroms. The remaining ten spectra were obtained using the Tillinghast Reflector Echelle Spectrograph (TRES; S1) on the 1.5m Tillinghast Reflector at the Fred L. Whipple Observatory on Mt. Hopkins, AZ. They were taken with the medium fiber, corresponding to R=44,000 and a wavelength range of 38509100 angstroms.
The spectra were extracted and analyzed according to an established procedure (S2). We used multiorder crosscorrelations to obtain precise relative velocities separately for the TRES and McDonald datasets. Spectral orders containing contaminating atmospheric lines were rejected, along with orders in the blue with low signaltonoise and orders in the red with known reduction problems. In total, we used 18 spectral orders in the McDonald crosscorrelations, covering the wavelength range of 45006680 angstroms, and 18 spectral orders in the TRES crosscorrelations, covering the wavelength range of 45806520 angstroms. The TRES velocities were shifted to an absolute scale using the weighted mean offset from the single order velocities and the known TRES zeropoint (determined from the long term monintoring of IAU RV standards). The McDonald offset was fit for in the final joint solution.
We shifted and coadded each dataset and classified the combined spectra by crosscorrelating against the grid of CfA synthetic spectra, which are based on Kurucz models calculated by John Laird and rely on a linelist compiled by Jon Morse. Interpolation between gridpoints using the correlation peak heights yields the bestfit parameters, and the RMS scatter of the results from individual spectral orders provides an internal error estimate. Because of degeneracies in the stellar spectrum between , , and [Fe/H], these parameters are highly correlated, and an error in one parameter would likely result in systematic errors beyond the internal error estimates for the other parameters. For this reason, we double the errors to account for possible systematic effects. With fixed at 3.95, we find =5875 100 K, [Fe/H]=+0.15 0.08, and =4.6 0.9 km/s.
We infer the rotational period of KOI126 A to be 226 days, which is supersynchronous. We note that the observed 17 day periodicity in the Kepler light curve is comparable to the inferred rotational period, which suggests a possible association with the rotational modulation of surface features on KOI126 A. The variation is very roughly sinusoidal and out of phase with the eclipse events. While the period of this variation is almost exactly half the period of the orbit of KOI126 (B, C) about KOI126 A, it is unlikely, given the amplitude and phase, that it is associated with the ellipsoidal distortion of A by the tidal field of (B, C) (S3).
DynamicalPhotometric Model
Positions and velocities. A hierarchical (or Jacobi) coordinate system is used when calculating the positions of the three bodies. In this system, is the position of C relative to B and is the position of A relative to the center of mass of (B,C). We may specify and in terms of osculating Keplerian orbital elements (period, eccentricity, argument of pericenter, inclination, longitude of the ascending node, and the mean anomaly: , , , , , , respectively).
Newton’s equations of motion, which depend on , and the stellar masses, may be specified for the accelerations and (S4, S5). An additional term may be added to the acceleration of due to the postNewtonian potential of the inner binary (S6). These are the only accelerations used in the fit giving the parameters listed in Table 1. We worked in units such that .
Further perturbing accelerations may be added to the acceleration of corresponding to the nondissipative equilibrium tidal potential between B and C and the potential associated with the rotationallyinduced oblate distortion of B and C (S4). In this approximation, the axial spins of B and C follow the evolving orbit, staying normal to the orbit and spinning at a rate synchronous with the orbit. Both the accelerations due to tides and rotations depend on , , , , , and . The acceleration due to rotation also depends on the angular axial spin rate of both B and C. The spin rates and apsidal constants are assumed to be the same for both B and C. We do not model the tidal or rotational distortion of A as their contributions to the total accelerations are negligible (S4).
A final acceleration due to tidal damping may be added to the acceleration of (S5). The scale of this acceleration is set by the tidal dissipation efficiency, . This acceleration is negligible for reasonable values of ().
We used an implementation of the BulirschStoer algorithm (S7) to numerically integrate the coupled firstorder differential equations for and in order to determine and their temporal derivatives at any given time. The maximum step size in the integrator was min.
The Jacobi coordinates ( and and their derivatives) may be transformed into spatial coordinates of the three bodies, relative to barycenter.
Radial velocity of KOI126 A. The RV data for KOI126 A were compared directly to the results of the numerical algorithm after applying a systematic offset associated with peculiar and bulk Galactic motion and an additional offset between the McDonald and TRES spectra to account for calibration error. The systematic offset was measured to be km s and the additional offset was measured to be km s.
Correcting for the finite speed of light. The positions of the three stars are projected to the location of the barycentric plane (i.e., the plane parallel to the skyplane that includes the barycenter of KOI126 and that is normal to the lineofsight) at a time in order to correct for the delay resulting from the finite speed of light and the motion of the barycenter of KOI126 along the lineofsight. In detail, for each star and for a given observation (or “clock”) time , a secant line root finding algorithm is used to solve for the retarded time, , where is the star’s distance at a time from the barycentric plane, is the radial velocity of the barycenter and is the speed of light. The observed coordinates of each star at the clock time are those found at each individual body’s retarded time. For , the variation of the difference between the clock time and the retarded time is as much as a few minutes, in amplitude. We note that the lighttime effects set the physical scale of the system and allow for the determination of absolute masses and radii (S8). Our code permits the use of a nonzero barycentric radial velocity which may be measured from the radial velocity data. The effect of this parameter is to shift the model parameters with a fractional difference similar to . The only parameters for which this effect is relevant are the orbital periods of the inner and outer binary which are adjusted by and sigma, respectively, relative to the periods measured with . All remaining parameters are adjusted by less than one tenth of their respective one sigma uncertainty. Given this fact and the possibility of a systematic bias in the velocity zeropoint, we opted to report (in Table 1) the parameters found assuming .
Photometric model. The skyplane projected 2D positions of all three objects were used as inputs to a light curve generating algorithm.
All stars were assumed to be spherical. Additionally, the radial brightness profile of KOI126 A was modeled as where is the projected distance from the center of A, normalized to the radius of A, and and are the two quadratic limbdarkening parameters (S9). The fluxes of B and C were specified relative to the flux of A. The sum of the fluxes was normalized to unity. The limbdarkening coefficients for A were found to be and . These values agree with theoretical expectations for stars with temperatures, gravities and metallicities similar to those found for KOI126 A ( and for , and [M/H], S10). We investigated various limbdarkened profiles for B and C according to the same model, however, any physicallyvalid choice of parameters gave similar fit values within the quoted uncertainties.
For eclipses in which there are no threeway alignments, the loss of light is computed using an analytic prescription that depends on object separations, radii and relative fluxes (S11). The total loss of light at a given time is the sum of the losses associated with the eclipse of A by B or C, the eclipse of B or C by A, or the eclipses between B and C. When a threeway alignment occurs, the total loss of light is not given as a superposition of analyticallydefined overlaps. In this case, the loss of light is computed numerically by integrating over the skyplane.
Determining bestfit parameters, covariances and uncertainties
Fitting parameters. There were 23 free parameters in the final fit to all available data: the three masses (, , ), five orbital elements of the outer binary at (BJD) (, , , , ), six orbital elements of the inner binary at (, , , ,,), the radius of A (), the relative radii of B and C (, ), the relative fluxes of B and C (, ), the two limbdarkening parameters of A (, ), the systematic offset to the radial velocity of A () and a systematic offset between TRES and McDonald RVs (). The longitude of the ascending node of the outer binary is unconstrained and, for simplicity, has been fixed to . The longitude of the inner binary is measured relative to this orientation and does not reflect the true value.
All short cadence data were initially fit with a multiplicative correction that was quadratic in time, to account for outofeclipse longwavelength variability. The parameters describing these corrections correlated very weakly with the remaining 23 parameters and were therefore fixed to their bestfit values to reduce computation time.
Bestfit parameters, errors and covariances. The data were fitted using an implementation of the LevenbergMarquardt (L–M) algorithm (S12, S13). The L–M algorithm minimizes the sum of the deviates squared () by adjusting the model parameters. In addition to determining the bestfit parameters, the L–M algorithm also returns the covariance matrix of the fitted parameters, approximating the –surface as quadratic in those parameters. For properly estimated measurement uncertainty, the square root of the diagonal elements gives the formal 1–sigma statistical errors in the parameters. The statistical error in any derived parameter may be estimated using the full covariance matrix.
The measurement uncertainties for the photometric data were set equal to the rootmeansquare deviation of the bestfit residuals for each short cadence event as plotted in Fig. 1. The bestfit residuals were observed to show very little temporal correlation. The bestfit solution had a reduced of for degrees of freedom. The contribution from the RV data alone was .
We note that while the masses and are known to no better than 1.5%, their ratio is known much more precisely: . The ratio of radii are also welldetermined: and . Also, the sum of the radii of B and C are known relatively better than their difference – and . The sum and differences of the masses of B and C have an analogous structure – and .
We executed the L–M algorithm, starting from random positions in parameter space, many hundreds of times before finding the final solution. From this experience we note that a large number of local maxima ( minima) populate the likelihood landscape. In particular, solutions with inner binary periods belonging to a discrete set of aliases of the reported inner binary period satisfying , for integers , show qualitatively similar transit events. These solutions yield residual correlated structure, significantly higher and physically implausible stellar parameters for KOI126 A. Moreover, these solutions do not predict the observed occultations or inner binary eclipses. A truncated Markov chain Monte Carlo search of the likelihood surface surrounding the adopted solution uncovered no larger extrema nor any local extrema with interestinglylow .
Tidal Dissipation and Timescales
Tidal dissipation is expected to modify the orbital elements of the close binary and the spins of its components (S14). In isolation from KOI126 A, the orbital eccentricity of KOI126 (B,C) should decrease with an exponential timescale Gyr if for each of B and C, where is the standard tidal quality factor (S15). In the presence of KOI126 A, the eccentricity would damp not to zero, but to a small fixed value, with the eccentricities of the close binary and third star aligned and precessing at the same rate (S16). However, the observed eccentricity () has not yet damped to that state (Figure S5). Moreover, if the system endured more than exponential damping timescales, then in the past, in violation of dynamical stability (S17). Together with estimates of , these arguments bound .
The nearly coplanar configuration of the triple system (9.2, oscillating by 0.4 on each orbit of the third body) rules out significant eccentricity limit cycles (S18) or Kozai oscillations (S19) in the close binary. Aside from eccentricity damping, in isolation from KOI126 A, the spins of KOI126 B and C would damp quickly (S14) ( years) to an orbitaligned and orbitsynchronized state. The presence of KOI126 A causes the KOI126 (B,C) orbit to precess faster than the natural spin precession of the component stars, so the spin vectors would damp to nearly the precession axis instead (S20). Although our observations are not currently sensitive to such spinorbit misalignment, its longterm effect is to damp the mutual inclination (S19) on a timescale several to ten times longer than the eccentricitydamping timescale.
Time (BJD)  Radial velocity of A (km s)  RV Error (km s) 

2455339.898438  26.651  0.101 
2455349.863281  42.856  0.093 
2455366.925781  4.787  0.087 
2455367.765625  3.123  0.097 
2455456.621094  34.408  0.061 
2455463.695312  17.632  0.062 
2455466.714844  9.171  0.063 
2455483.679688  44.097  0.064 
2455485.667969  41.976  0.073 
2455488.613281  37.594  0.063 
2455051.753906  34.411  0.400 
2455076.828125  44.535  0.401 
2455311.859375  44.997  0.401 
2455341.812500  37.164  0.401 
2455394.703125  21.844  0.401 
2455396.714844  16.319  0.401 
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