Radial velocity monitoring of Kepler heartbeat stars
Avi Shporer, Jim Fuller, Howard Isaacson, Kelly Hambleton, Susan E. Thompson, Andrej Prsa, Donald W. Kurtz, Andrew W. Howard, Ryan M. O'Leary
DDraft version August 12, 2016
Preprint typeset using L A TEX style emulateapj v. 04/17/13
RADIAL VELOCITY MONITORING OF KEPLER HEARTBEAT STARS † Avi Shporer
1, 2 , Jim Fuller
3, 4 , Howard Isaacson , Kelly Hambleton
6, 7 , Susan E. Thompson
8, 9 , Andrej Prˇsa ,Donald W. Kurtz , Andrew W. Howard , Ryan M. O’Leary Draft version August 12, 2016
ABSTRACTHeartbeat stars (HB stars) are a class of eccentric binary stars with close periastron passages. Thecharacteristic photometric HB signal evident in their light curves is produced by a combination oftidal distortion, heating, and Doppler boosting near orbital periastron. Many HB stars continue tooscillate after periastron and along the entire orbit, indicative of the tidal excitation of oscillationmodes within one or both stars. These systems are among the most eccentric binaries known, andthey constitute astrophysical laboratories for the study of tidal effects. We have undertaken a radialvelocity (RV) monitoring campaign of
Kepler
HB stars in order to measure their orbits. We presentour first results here, including a sample of 22
Kepler
HB systems, where for 19 of them we obtainedthe Keplerian orbit and for 3 other systems we did not detect a statistically significant RV variability.Results presented here are based on 218 spectra obtained with the Keck/HIRES spectrograph duringthe 2015
Kepler observing season, and they have allowed us to obtain the largest sample of HB starswith orbits measured using a single instrument, which roughly doubles the number of HB stars withan RV measured orbit. The 19 systems measured here have orbital periods from 7 to 90 d andeccentricities from 0.2 to 0.9. We show that HB stars draw the upper envelope of the eccentricity –period distribution. Therefore, HB stars likely represent a population of stars currently undergoinghigh eccentricity migration via tidal orbital circularization, and they will allow for new tests of higheccentricity migration theories.
Keywords: binaries: general — techniques: radial velocities INTRODUCTION † The data presented herein were obtained at the W.M. KeckObservatory, which is operated as a scientific partnership amongthe California Institute of Technology, the University of Cali-fornia and the National Aeronautics and Space Administration.The Observatory was made possible by the generous financialsupport of the W.M. Keck Foundation. Jet Propulsion Laboratory, California Institute of Technol-ogy, 4800 Oak Grove Drive, Pasadena, CA 91109, USA NASA Sagan Fellow TAPIR, Walter Burke Institute for Theoretical Physics,Mailcode 350-17, Caltech, Pasadena, CA 91125, USA Kavli Institute for Theoretical Physics, Kohn Hall, Univer-sity of California, Santa Barbara, CA 93106, USA Department of Astronomy, University of California, Berke-ley CA 94720, USA Department of Astrophysics and Planetary Science, Vil-lanova University, 800 East Lancaster Avenue, Villanova, PA19085, USA Jeremiah Horrocks Institute, University of Central Lan-cashire, Preston, PR1 2HE, UK NASA Ames Research Center, Moffett Field, CA 94035, USA SETI Institute, 189 Bernardo Avenue Suite 100, MountainView, CA 94043, USA Institute for Astronomy, University of Hawaii, 2680 Wood-lawn Drive, Honolulu, HI 96822, USA JILA, University of Colorado and NIST, 440 UCB, Boulder,80309-0440, USA
Heartbeat stars are an exciting class of stellar bina-ries which have been discovered in large numbers onlyrecently by the
Kepler photometric survey (e.g., Thomp-son et al. 2012; Beck et al. 2014). Their name originatesfrom the characteristic light curve signal seen once perorbital period, induced by the close periastron passageof a highly eccentric binary star system, a signal whoseshape resembles that of a heartbeat in an electrocardio-gram.The photometric signal seen at periastron results froma combination of several processes, including tidal distor-tion, heating, and Doppler boosting. In addition, manyof the systems exhibit tidally excited stellar pulsationsthat maintain constant amplitude throughout the orbit.They result from near-resonances between the multiplesof the orbital frequency and stellar oscillation modes(Cowling 1941; Zahn 1975; Kumar et al. 1995; Fuller &Lai 2012; Burkart et al. 2012). Another attractive qual- a r X i v : . [ a s t r o - ph . S R ] A ug Shporer et al. ity of heartbeat stars is that they show a photometricsignal whether or not the system shows eclipses. There-fore, heartbeat stars (hereafter HB stars) are astrophysi-cal laboratories for the study of tidal interactions in stel-lar binaries.Since the first discoveries of HB stars in
Kepler data(Welsh et al. 2011; Thompson et al. 2012) the number ofknown HB stars has substantially increased and is cur-rently at 173 (Kirk et al. 2016). However, the
Kepler light curves alone are not sufficient for taking advantageof the scientific opportunities HB stars offer. We haveundertaken a radial velocity (RV) monitoring campaignof
Kepler
HB stars using Keck/HIRES (Vogt et al. 1994)in order to measure their orbits. Our observations wereperformed during the 2015
Kepler observing season andwe report our first results here.We use our results to study the eccentricity – perioddiagram, which is important for testing tidal circulariza-tion theory. Such a study can only be done with a largesample of HB stars, as we have characterized here, as op-posed to a few individual systems. We show that heart-beat stars generally lie near the upper extreme of theeccentricity distribution as a function of orbital period.We infer that the lack of systems with higher eccentric-ity is a result of prior orbital circularization, and thatheartbeat stars represent systems that are likely under-going slow tidal orbital circularization. Therefore, cir-cularization timescales for heartbeat stars are likely tobe comparable to their ages, and testing this suppositionwith detailed analyses of these systems will yield valuableconstraints on tidal and orbital evolution theories.The paper is arranged as follows. We describe theKeck/HIRES observations, data analysis, and orbit fit-ting in Sec. 2, and in Sec. 3 we describe the results. InSec. 4 we discuss our results, our attempts to constrainthe companion mass, the study of the eccentricity – pe-riod relation, and some future prospects. We concludewith a summary in Sec. 5. OBSERVATIONS AND DATA ANALYSIS
Target selection
We have selected our targets from the list of 173 known
Kepler
HB stars, all flagged with the “HB” flag in the
Kepler eclipsing binary (EB) online catalog (Prˇsa et al. http://keplerEBs.villanova.edu K p ≤ K p is the Kepler magnitude) to keep the exposure timeshort. In addition, using the stellar effective temperatureand surface gravity (Huber et al. 2014) we have tried toinclude only main sequence stars and avoid giant stars,as the latter were already the focus of the work of Beck etal. (2014). Next we prioritized the systems according toa combination of several criteria, including (1) the stel-lar effective temperature, as hotter stars are more chal-lenging for RV measurements, (2) the presence of tidalpulsations or rotational modulation (due to stellar activ-ity) in the
Kepler light curve, making the system moreinteresting scientifically, and (3) target brightness. Thelist of targets we observed is given in Table 1 along withthe stellar parameters from Huber et al. (2014), includ-ing a total of 22 targets. Although we did not analyzethese light curves here we show them in Appendix B forcompleteness. As can be seen in Appendix B figures,the relative flux variation during periastron passage hasa typical full amplitude from 10 − to 10 − . Some ofthe systems show visually identifiable tidal pulsations,with an amplitude of up to several 10 − in relative flux(e.g. KID 8164262). In addition, a few of the systemsshow eclipses (e.g. KID 5790807). As described in detailbelow, for 19 targets we measured the RV orbit and for3 targets we did not detect a statistically significant RVvariability. Keck/HIRES observations and data analysis
The Keck/HIRES data analyzed and presented hereincludes 218 exposures obtained during 43 nights fromMay to October 2015. The access to a relatively largenumber of nights while using a small amount of telescopetime per night was critical for the success of this program.
V monitoring of
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HB stars Table 1
Properties of the heartbeat systems and the primary stars for the 22 systems studied here.KID P a A HBb K p T effc log g c R M [d] [ppm] [mag] [K] [R (cid:12) ] [M (cid:12) ]4659476 58.99637 ± +155 − +0 . − . +0 . − . +0 . − . ± +155 − +0 . − . +0 . − . +0 . − . ± +224 − +0 . − . +0 . − . +0 . − . ± +67 − +0 . − . +0 . − . +0 . − . ± +162 − +0 . − . +0 . − . +0 . − . ± +234 − +0 . − . +0 . − . +0 . − . ± +77 − +0 . − . +0 . − . +0 . − . ± +182 − +0 . − . +0 . − . +0 . − . ± +228 − +0 . − . +0 . − . +0 . − . ± +199 − +0 . − . +0 . − . +0 . − . ± +237 − +0 . − . +0 . − . +0 . − . ± +201 − +0 . − . +0 . − . +0 . − . ± +174 − +0 . − . +0 . − . +0 . − . d ± +170 − +0 . − . +0 . − . +0 . − . ± +144 − +0 . − . +0 . − . +0 . − . ± +202 − +0 . − . +0 . − . +0 . − . d ± +172 − +0 . − . +0 . − . +0 . − . ± +149 − +0 . − . +0 . − . +0 . − . d ± +75 − +0 . − . +0 . − . +0 . − . ± +151 − +0 . − . +0 . − . +0 . − . ± +169 − +0 . − . +0 . − . +0 . − . ± +237 − +0 . − . +0 . − . +0 . − . Photometric period, taken from the
Kepler
EB catalog (Kirk et al. 2016). b Photometric amplitude of the heartbeat signal at periastron, defined as the full flux variation, in ppm, of the phase folded and binnedlight curve (see Appendix B). The typical uncertainty is a few percents. c Parameters taken from the revised KIC (Huber et al. 2014). d RV non-variable star.
It allowed us to sample the entire orbital phase of ourtargets, and sample the periastron phase more intenselysince it is that phase where most of the RV variabilitytakes place for eccentric binary systems. For each of thesystems presented here we have obtained at least 7 RVmeasurements, in order to fit a Keplerian orbital modelthat in our case includes 5 fitted parameters, since theorbital period is already precisely known from
Kepler photometry (see more details in Sec. 2.3).We used the Keck/HIRES instrumental setup of theCalifornia Planet Search as described in Howard et al.(2009). At the beginning of each observing night we useda Thorium-Argon lamp to align the spectral format towithin one-half pixel of the historical position, where onepixel represents 1.3 km s − . This careful setup is thefirst step is calculating the RVs presented here. Eachspectrum was acquired with the C2 decker (angular size of 0.87 arcsec × R ≈ , . − . −
20 per pixel.As a first step of the spectral data analysis we ob-tained the wavelength solution (assigning a wavelengthvalue for every pixel) with a precision of 0.1 pixels using aThorium-Argon calibration lamp spectrum taken at thebeginning of each night’s observing. To derive the RVmeasurements we used the method described in Chubaket al. (2012), using the telluric A and B absorption bands(7 , − ,
621 ˚A and 6 , − ,
884 ˚A respectively) dueto absorption by molecular Oxygen in the Earths atmo-sphere.We chose the reference B-type star HD 79439 to serve
Shporer et al. as the telluric lines wavelength zero-point, and measurethe position of the target stars’ telluric lines relative tothose of the reference star. This zero-point offset correctsfor drift in the CCD position throughout the night andobserving variables such as non-uniform illumination ofthe spectral slit. We subtract this offset from any mea-sured shift in the position of the target stars’ spectrallines in order to determine their true Doppler shift.We measure the position of the target stars’ spec-tral lines using four wavelength segments rich in stel-lar absorption lines. Those four segments are located at6 , − ,
867 ˚A, 7 , − ,
146 ˚A, 7 , − ,
489 ˚A, and7 , − ,
593 ˚A, which are adjacent to but not overlap-ping with the telluric A and B bands. The four wave-length segments of the target star are cross-correlatedwith a HIRES spectrum of Vesta, which serves as a solarproxy reference spectrum. The mean and RMS of thefour RV measurements serve as the RV value and uncer-tainty respectively. This raw RV measurement betweenthe target star and Vesta is then corrected for barycentricmotion of the reference star and the target, determinedby the JPL Solar System ephemeris . This way, anycontributions to the RV measurement that are not dueto the radial motion of the target star relative to the ref-erence star have been accounted for by the telluric linesand the barycentric corrections.Finally, each RV measurement is set to the RV scaleof Nidever et al. (2002) and Latham et al. (2002), byusing an offset determined by the observations of 110stars in the overlap of the samples of Chubak et al. (2012)and Nidever et al. (2002). All 218 RV measurements arelisted in Appendix A. The method we used calculatesthe RVs in an absolute scale, and the typical errors forslowly rotating stars (with rotation periods at the levelof 10 d or longer) are at the 0.1 km s − level. As themajority of the heartbeat stars observed here have anincreased rotation rate (with rotation periods at the levelof 1 d) the resulting RV errors are typically at the rangeof 0.1–1.0 km s − (for six systems the RV precision isat the level of a few km s − , see Appendix A), sufficientto measure the RV variations of the stellar components’orbital motion. http://ssd.jpl.nasa.gov Keplerian orbit fitting
We have fitted a Keplerian orbit model to the RV mea-surements using the adaptive MCMC approach describedin Shporer et al. (2009). In this approach the width of thedistribution from which the step sizes are drawn is ad-justed every 10 steps, which we refer to as a minichain,in order to keep the step’s acceptance rate at 25% (Gre-gory 2005; Holman et al. 2006). This adaptive approacheliminates a possible dependence of the fitted parameterson the width of the distribution from which the step sizesare drawn. Each chain consists of 100 minichains, or 10 steps total, and we ran 5 chains for each system. Wethen generated the posterior probability distribution ofeach parameter by combining the 5 chains while ignoringthe initial 20% steps of each chain. We took the distri-bution median to be the best-fit value and the values atthe 84.13 and 15.87 percentiles to be the +1 σ and -1 σ confidence uncertainties, respectively.The Keplerian orbital model includes six parameters,the period P , periastron time T , RV semi-amplitude K ,system’s center of mass RV γ (commonly referred to asRV zero point), orbital eccentricity e , and argument ofperiastron ω . Since the Kepler photometric data con-strain the orbital period significantly better than the RVmeasurements we adopted the photometric orbital pe-riod value, P phot , and its uncertainty from the Kepler
EB catalog, and used them as the mean and width, re-spectively, of a Gaussian prior distribution on P . Weimplemented that prior by drawing a value at randomfrom the Gaussian prior distribution in each step of theMCMC analysis. Therefore, our fitted model included 5free parameters. Those 5 parameters were fitted by 7 –12 RVs per system.When stepping through the five-dimensional parame-ter space we used the parameters √ e cos ω and √ e sin ω instead of e and ω , following, e.g., Eastman et al. (2013).We made the parameter conversion at each step andwhen the chain reached a position where e ≥ χ to infinity to make sure the step is not accepted.We set χ to infinity also when T reached a positionwhere it is more than P phot /2 away (in absolute value)from the initial T position, so T was allowed to varywithin a span of P phot , or the full orbital phase. Theinitial T position was arbitrarily set, taken to be the V monitoring of
Kepler
HB stars
Kepler
EB catalog (Kirk et al. 2016), whichwas closest to the middle of the time period covered bythe RV measurements. For non-eclipsing systems the
Kepler
EB catalog reports the time of minimum flux,although that distinction is not important since we areusing this time only as an arbitrary starting point for T .Therefore, we used a uniform prior on T and allowed itto vary throughout the entire orbital phase, and we didnot use the eclipse time (or time of minimum flux) toconstrain the periastron time.For some of the systems we have analyzed we noticedthat the best fit χ value is significantly larger than theexpectation value (or the mean) of the χ distribution forthe given number of degrees of freedom ν (which equals n −
5, for n RV measurements and 5 fitted parameters).As this could be the result of underestimated RV errorswe have added a mechanism to our analysis to correct forthat. Once the analysis was done we checked the distancebetween the best fit χ and the expectation value of the χ distribution for ν degrees of freedom. If that distancewas larger than 2 √ ν , which is twice the χ standarddeviation for ν degrees of freedom, we repeated the anal-ysis while adding in quadrature a systematic uncertaintyto the RV measurements uncertainties. We refer to thatsystematic uncertainty as jitter, and it was set to make χ equal ν . Therefore, the analysis was iterated until thebest fit χ was close enough to the expectation value.Our analysis included also a component that testswhether a target shows no RV variability. This wasdone by calculating χ , the χ value of a constant RVmodel where for each system that constant was the RVsweighted mean. We declared a system to have no stati-cally significant RV variability if χ was smaller thanthe 99.9 percentile of a χ distribution with ν equal tothe number of RVs minus one. RESULTS
We have obtained the Keplerian orbital solution for 19systems. Table 2 lists the fitted orbital parameters, in-cluding the orbital period from the
Kepler
EB catalog,taken as a prior in the MCMC analysis, and the com-panion’s mass function, f ( m ), defined as: f ( m ) ≡ P ( K √ − e ) πG = M sin i ( M + M ) , (1) where i is the orbital plane inclination angle, and M and M are the masses of the primary star and secondarystar, respectively.Table 3 lists a few statistics describing the fitted model,including the fitted model χ , number of degrees offreedom (which is simply the number of RV measure-ments minus five, for the five fitted parameters), num-ber of analysis iterations (see Sec. 2.3), the RV system-atic uncertainty by which the RV errors were increasedin quadrature (RV jitter; defined to be zero when onlya single analysis iteration was performed), and the RVresiduals scatter. Here and throughout this paper thescatter is estimated in a robust way, using the medianabsolute deviation (MAD) where the standard deviationis calculated as 1.4826 × MAD (Hoaglin et al. 1983; Beerset al. 1990; see also Shporer et al. 2014). As shown inTable 3, a second analysis iteration, where a non-zero jit-ter was introduced was done for only 9 of the 19 systems.The RV residual scatter is at the 1.0 km s − level, simi-larly to the typical jitter value (for systems where it wasintroduced). This matches well the typical RV errors andthe expected precision for RVs derived using the telluricbands method for this population of stars that tend torotate faster than Sun-like stars (see Sec. 2.2).In Figures 1 through 5 we present the RV curves forall 19 systems with a measured orbit. Each system ispresented with two panels. In the top panel we show theRVs vs. time (black) along with the best fit model (solidred line). In the bottom panel we show the phase-foldedRV curve (black) along with a continuum of orbits thatcorrespond to a 3 σ marginalization (red). For complete-ness we present in Appendix B figures of the phase-foldedRV curves overplotted by the phase-folded Kepler lightcurves for all 22 systems studied here.We have tested our results in several ways: • We changed the stopping condition of the MCMCiterations. Instead of requiring χ to be within2 √ ν from the expectation value of the χ distri-bution for ν degrees of freedom, we required it tobe within √ ν and 4 √ ν from it, in two separateapplications of our analysis. • For each of the 19 systems we used the RV timesand injected an RV orbit identical to the fitted
Shporer et al. model and applied the same analysis. • We repeated the entire analysis, to test the repeata-bility of our results given the random componentof the MCMC analysis.In all tests above the results were fully consistent withthe original results.
V monitoring of
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HB stars Table 2
Orbital properties of the 19 heartbeat systems measured here.KID
P T K γ e ω f ( m )[d] [BJD-2457000] [km s − ] [km s − ] [rad] [ M (cid:12) ]4659476 58.99637 ± +0 . − . +0 . − . +1 . − . +0 . − . -2.884 +0 . − . +0 . − . ± +0 . − . +0 . − . -10.25 +0 . − . +0 . − . -0.779 +0 . − . +0 . − . ± +0 . − . +0 . − . -20.12 +0 . − . +0 . − . -0.470 +0 . − . +0 . − . ± +0 . − . +0 . − . -27.39 +0 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +0 . − . +0 . − . +0 . − . -1.615 +0 . − . +0 . − . ± +0 . − . +2 . − . +0 . − . +0 . − . -1.452 +0 . − . +0 . − . ± +0 . − . +1 . − . -24.38 +0 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +0 . − . -31.319 +0 . − . +0 . − . -2.475 +0 . − . +0 . − . ± +0 . − . +1 . − . +1 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +9 . − . +1 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +2 . − . +1 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +0 . − . -33.53 +0 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +3 . − . -14.3 +1 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +7 . − . +0 . − . +0 . − . -2.725 +0 . − . +0 . − . ± +0 . − . +0 . − . +0 . − . +0 . − . -0.556 +0 . − . +0 . − . ± +0 . − . +0 . − . -14.02 +0 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +0 . − . -16.93 +0 . − . +0 . − . +0 . − . +0 . − . ± +0 . − . +1 . − . -7.15 +0 . − . +0 . − . +0 . − . +0 . − . Table 3
Statistical quantities describing the RV Keplerian model fits. Columns include (from left to right): KIC ID, best fit χ , number of degreesof freedom, number of fitting iterations, the additive systematic RV uncertainty, and the RV residuals scatter.KID χ ν − ] [km s − ]4659476 4.7 2 2 1.9 2.25017127 6.3 5 2 0.63 0.405090937 4.8 6 1 – 0.615790807 4.1 7 1 – 0.215818706 5.1 6 1 – 0.315877364 10.8 5 1 – 0.385960989 9.9 5 1 – 2.66370558 3.2 2 1 – 0.0426775034 3.1 4 2 0.99 0.578027591 6.5 5 2 1.0 0.718164262 7.1 5 2 2.4 1.49016693 4.8 3 2 2.3 1.79965691 3.6 3 1 – 0.2010334122 2.4 2 2 3.6 3.011071278 5.3 3 1 – 0.2311403032 7.1 7 2 0.89 0.7811649962 4.1 5 1 – 0.3511923629 4.9 3 2 0.33 0.1612255108 10.3 6 1 – 2.4 Shporer et al.
Figure 1.
RV curves of KID 4659476, KID 5017127, KID 5090937, and KID 5790807. RV measurements are shown in black, includingerror bars which are typically smaller than the marker size. Each system is shown in two panels, top panel shows the RVs as a functionof time and the bottom panel the phase-folded RV curve with periastron at phase 0.5. The fitted Keplerian model is shown as a red solidline in the top panels, and by a 3 σ contour plot in the bottom panels. The title for each plot lists (from left to right) KIC ID, T eff (K),log g , P (d), K (km s − ), and e . V monitoring of
Kepler
HB stars Figure 2.
Similar to Fig. 1 for KID 5818706, KID 5877364, KID 5960989, and KID 6370558. The title for each plot lists (from left toright) KIC ID, T eff (K), log g , P (d), K (km s − ), and e . Shporer et al.
Figure 3.
Similar to Fig. 1 for KID 6775034, KID 8027591, KID 8164262, and KID 9016693. The title for each plot lists (from left toright) KIC ID, T eff (K), log g , P (d), K (km s − ), and e . V monitoring of
Kepler
HB stars Figure 4.
Similar to Fig. 1 for KID 9965691, KID 10334122, KID 11071278, KID 11403032. The title for each plot lists (from left toright) KIC ID, T eff (K), log g , P (d), K (km s − ), and e . Shporer et al.
Figure 5.
Similar to Fig. 1 for KID 11649962, KID 11923629, and KID 12255108. The title for each plot lists (from left to right) KICID, T eff (K), log g , P (d), K (km s − ), and e . V monitoring of
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HB stars Table 4
Heartbeat systems showing no radial velocity variability.KID P − ] [km s − ]9972385 58.42 9 -0.55 0.4711122789 3.24 20 -19.73 1.4211409673 12.32 10 -5.44 0.88 For three other targets we monitored we did not detectany appreciable RV variability. All three were identifiedby measuring a χ smaller than the 99.9 percentile ofthe χ distribution (see Sec. 2.3). In fact, all three had χ smaller than the 99.0 percentile, while all other 19systems were above the 99.999 (= 100 − − ) percentile.The three non-variable systems are listed in Table 4,their RV measurements are included in Appendix A, andtheir RV curves are plotted in Appendix B. The RV scat-ter of these systems is in the range of 0.47 – 1.42 km s − ,which compares well with the residuals RV scatter of the19 systems with a fitted Keplerian orbit (see Table 3). DISCUSSION
The 19 HB stars whose RV orbits were measured hereconstitute the largest sample to date where the RVs weremeasured with a single instrument. Considering the or-bital period range covered here, P ≤
90 d, this new sam-ple roughly doubles the number of HB stars with RV-measured orbits (Maceroni et al. 2009; Welsh et al. 2011;Hambleton et al. 2013; Beck et al. 2014; Hareter et al.2014; Schmid et al. 2015; Smullen & Kobulnicky 2015;Hambleton et al. 2016) listed in Table 6.The stellar parameters listed in Table 1 show that oursample contains stars hotter than the Sun, of spectraltypes F and A, and effective temperatures in the rangeof 6,200 – 8,100 K. A minor caveat is that these stellar pa-rameters are taken from Huber et al. (2014), who revisedthe
Kepler input catalog (KIC; Brown et al. 2011) stellarproperties of stars observed by
Kepler . Therefore, theseparameters might be less precise than spectroscopically-derived parameters, and could be affected by light fromthe binary stellar companion and/or other stars on thesame line of sight (whether or not they are gravitationallybounded to the HB system). Nonetheless, this charac-teristic of our sample is not likely to significantly changewith more precise parameters.The fact that HB stars tend to have relatively hot pri- maries (hotter than the Sun, as noted above) is at leastpartially an observational bias. Since we focus here onmain sequence stars, hotter stars are larger in radius andhave lower surface gravity. Hence they have a larger tidaldistortion for the same tidal force, leading to a largerphotometric signal observed by
Kepler during periastron.This is supported by the correlation between stellar T eff and the photometric amplitude of the HB signal at peri-astron, identified by Thompson et al. (2012). It is there-fore easier to detect HB systems with hot primaries. Inaddition, hot stars typically have a lower level of stellaractivity than cool stars, making it easier to detect theHB photometric signal for hot stars. However, the sam-ple of heartbeat stars studied here has been shaped byseveral subjective selection criteria (see Sec. 2.1). Hence,this sample on its own is not appropriate for investigat-ing the temperature distribution or circularization timescales of eccentric binaries.As can be seen in Figures 1–5 and Tables 2–3, someof the orbital solutions are of better quality than oth-ers, where the quality is quantified by how well the fit-ted parameters are constrained, the residual scatter, andthe fitted model χ . In general, the quality of the or-bital solutions worsens with decreasing number of RVsper target and with increasing eccentricity. The high ec-centricity systems tend to have longer periods with only1–2 observable periastron events during the observingseason, making the observations more time critical anddifficult to schedule. Those orbits can be refined in thefuture with additional RVs, especially during periastronpassage. RV Non-variable Stars
As already noted in Sec. 3, three of the targets weobserved show no RV variability at the 1 km s − level (seeTable 4 and Appendix B). The reason for the RV non-variability is unclear. These three targets have brightnessand stellar parameters similar to the 19 systems withmeasured orbits (see Table 1). The non-variability couldbe the result of a few possible scenarios, some of themsimilar to the false positive scenarios of eclipsing andtransiting systems (Bryson et al. 2013; Coughlin et al.2014; Abdul-Masih et al. 2016).One possible scenario is a triple or higher multiplicitysystem where the spectrum is dominated by lines from a4 Shporer et al. bright star with no RV variability, which may or may notbe bound to the binary heartbeat system and is locatedon the same line of sight. A more detailed study of allspectra obtained here, including searching for additionalsets of lines, is ongoing and will be reported in a futurepublication.In a similar scenario, the
Kepler photometric heartbeatsignal does not originate from the star whose RVs weremonitored but from another nearby star that is at leastpartially blended with the target on
Kepler ’s pixels thatare 4 arcsec wide. High angular resolution imaging isneeded to further study this scenario.In a third scenario the target is a single variable starwith variability mimicking that of a HB signal. Themechanism inducing the photometric variability can befor example stellar pulsations, or rotation and stellarspots. High quality spectra combined with detailed
Ke-pler light curve analysis are required to further study thisscenario.In fact, we have identified KID 11409673 as a rapidlyoscillating peculiar A star (roAp). These strongly mag-netic stars are oblique pulsators, pulsating in high radialovertone p-modes with their pulsation axis inclined to therotation axis and closely aligned to their magnetic axis(e.g., Kurtz 1982; Holdsworth et al. 2016). We believethe 12.3 d photometric periodicity is the rotation period,and the photometric variability arises from a combina-tion of stellar rotation and persistent stellar spots. Afull study of this roAp star will be presented in a futurepublication.Finally, it is possible that our RV measurements arenot sensitive enough to detect the binary companion.This could occur if the system has a low orbital incli-nation or the companion mass is small. For a systemto have a stellar-mass companion and an RV amplitudeat or below the 1 km s − level, the orbital inclinationneeds to be exceptionally small, with i (cid:46) ∼ M J level to avoidRV detection, and at that mass level it is not expected togenerate a photometric HB signal at the observed ampli-tudes. In addition, if the binary companion induces anRV amplitude at the level of the RV scatter of the three non-variable systems then we would expect that scatterto be close to the low end of the RV amplitude distri-bution of the 19 systems with a fitted orbit. However,the latter has a range of 13 – 65 km s − (see Table 2),which seems to be distinct than the RV scatter of thenon-variable systems, of 0.5 – 1.4 km s − (see Table 4).Therefore, this scenario is considered unlikely. Companion Mass
For the 19 systems with a measured orbit, we calcu-lated the companion mass, M , using Eq. 1 that can berearranged into a cubic polynomial in M :sin iM − f ( m ) M − f ( m ) M M − f ( m ) M = 0 , (2)which has only one real root. The polynomial coefficientsin Eq. 2 are composed of f ( m ), which we measured di-rectly from the orbital solution (Table 2), M , for whichwe use the values of Huber et al. (2014, see Table 1), andsin i . To derive the companion mass values and uncer-tainties we generated an M distribution by solving forthe polynomial roots for a distribution of f ( m ) and M .To address our lack of knowledge of the orbital planeinclination angle, and in turn of the sin i coefficient inEq. 2, we have chosen three approaches.In the first approach, we assumed that sin i = 1 whichcorresponds to an edge-on system, providing the compan-ion’s minimum mass.In the second approach, we used the median value ofthe sin i distribution, which equals 0.6495 . Therefore,the M estimate derived in this approach reflects ourcurrent knowledge of f ( m ) and M and shows the likelyvalue of M , and the uncertainty we can hope to achieveonce the inclination angle is estimated in the future, forexample from modeling the Kepler light curve.In the third approach we used the entire sin i distri-bution, so the results are an accurate reflection of ourcurrent knowledge of M .In the second and third approaches above, we have as-sumed i is distributed uniformly in sin i since the orbitalangular momentum axis has no preferred direction. Onesubtlety here is that systems with high inclination an-gles have larger RV amplitudes, hence are detected more Given the highly asymmetric nature of this distribution wechose to use the median instead of the distribution expectationvalue (the mean), which equals 0.5890.
V monitoring of
Kepler
HB stars ∼ i = 5 . ± .
10 deg (Welsh et al.2011).The results of the three approaches are listed inTable 5, where for completeness we list also f ( m ) and M . Examining the values of M and M shows thatfor the majority of systems it is likely that M < M .However, for a few systems the companion mass may becomparable to or larger than the primary mass, althoughthe current uncertainties are large (e.g., KID 4659476,KID 9016693). This raises the possibility that in thosesystems the secondary is not a main sequence star be-cause in that case we would expect it to dominate thespectrum. Therefore, those systems are candidates for acompact object companion and are interesting targets forfurther study. Although we should note that those sys-tems could have a large orbital inclination angle, wherethe companion’s mass is close to the minimum mass (firstapproach above), making the companion a main sequencestar with mass close but smaller than the primary mass.This is supported by the detection of HB systems withbinary mass ratios close to one (Smullen & Kobulnicky2015). In such systems it might be possible to identifyin the spectrum the spectral lines of the secondary. Adetailed study of the spectra collected here, including asystematic search for spectral lines of the secondary, isbeyond the scope of this work (and will be a subject ofa future publication) as here we focus on RV measure-ments of the primary and measurement of the orbit fora large sample of HB stars. The Eccentricity-Period Relation
Fig. 6 shows the eccentricity – period ( e − P ) diagram.In the top panel we show the 19 systems whose orbitwas measured here marked in red, and for comparisonwe show in gray Kepler
EBs where the eccentricity wasderived through analysis of their eclipse light curve (Prˇsaet al., in prep.). A visual examination of the figure showsthat the eccentricity of most HB systems analyzed here is close to the high end of eccentricity range of simi-lar orbital period systems. In other words,
HB systemsdraw the envelope of the e − P distribution . The figurealso shows that our sample of 19 HB systems encompassmost of the period range across which tidal orbital circu-larization takes place. This is reflected by the wide rangeof eccentricity of the systems observed here (0.2 – 0.9),spanning almost the entire eccentricity range.In the bottom panel of Fig. 6 we added all other HBstars with orbits measured using RVs and with orbitalperiods within 200 d (Maceroni et al. 2009; Welsh et al.2011; Hambleton et al. 2013; Beck et al. 2014; Hareteret al. 2014; Schmid et al. 2015; Smullen & Kobulnicky2015; Hambleton et al. 2016). We list those systemsin Table 6. The dashed gray curves show lines of con-stant orbital angular momentum with an e − P relationof e = (cid:112) − ( P /P ) (2 / . We plotted lines with circu-larization period P (e.g., Mazeh 2008) of 4, 7, and 11d. This was in an attempt to match the envelope of the e − P distribution, although it can be seen that no sin-gle curve can match the envelope throughout the entireperiod range. This suggests that HB systems are bornwith a range of angular momenta and eventually tidallycircularize to a range of periods P , with P typicallybelow 10 d for main sequence binaries. Another possibleexplanation is that the population of HB systems studiedhere has a wide age range, since as shown by Meibom &Mathieu (2005) the circularization period and the shapeof the e − P distribution depends on the population age.Longer period systems, beyond P = 90 d, were notincluded in our targets since we wanted to monitor theentire orbit in one observing season. At short periods,the orbital eccentricity grows smaller and the heartbeatsignal becomes less concentrated near periastron and in-stead appears as typical ellipsoidal modulations. There-fore, the occurrence of systems classified as HB systemsmay decrease at short orbital periods, although the clas-sification becomes somewhat arbitrary.Measuring the orbits of additional HB stars, and otherhigh-eccentricity systems, will better shape the e − P up-per envelope. Still, a close visual examination of Fig. 6bottom panel shows that there are several systems witheccentricity well beyond that of similar period systems,raising the possibility they do not belong to the same dis-tribution, which in turn suggests they could be impacted6 Shporer et al.
Table 5
Masses of the two stars of the heartbeat systems measured here. The primary mass is taken from the KIC and the secondary is estimatedfrom the orbital properties.KID M f ( m ) M M M [ M (cid:12) ] [ M (cid:12) ] [ M (cid:12) ] [ M (cid:12) ] [ M (cid:12) ]4659476 1.31 +0 . − . +0 . − . +0 . − . +0 . − . +1 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +1 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +1 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +1 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +1 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +1 . − . Minimum mass estimate, assuming sin ( i ) = 1 meaning a completely edge on configuration with i = 90 deg. b Assuming the median value of sin ( i ), of 0.6495, where i is distributed as sin( i ). c Using the distribution of sin ( i ), where i is distributed as sin( i ). Table 6
Known heartbeat systems with RV measured orbit with
P <
200 d.Reference Name
P e a [d]Beck et al. 2014 KID 8912308 20.17 0 . ± . − KID 2697935 b . ± . − KID 8095275 23.00 0 . ± . − KID 2720096 26.70 0 . ± . − KID 9408183 49.70 0 . ± . − KID 5006817 94.81 0 . ± . − KID 2444348 103.50 0 . ± . − KID 9163796 121.30 0 . ± . − KID 10614012 132.13 0 . ± . − KID 8210370 153.50 0 . ± . − KID 9540226 175.43 0 . ± . . ± . . ± . . ± . . ± . . ± . c KID 3230227 7.05 0 . ± . − KID 4248941 8.65 0 . ± . − KID 8719324 10.24 0 . ± . − KID 11494130 18.97 0 . ± . . ± . a Values and errors are as given by the relevant paper. b See also Lillo-Box et al. (2015). c Two other systems included in that work are not listed in the table: For KID 9899216 the orbital parameters are not well constrained,and for KID 3749404 we adopt the parameters given by Hambleton et al. 2016.
V monitoring of
Kepler
HB stars Period [d]2 5 10 20 50 100 200 E cc en t r i c i t y This workPrsa et al. 2016 (in prep.; Kepler EBs)
Period [d]2 5 10 20 50 100 200 E cc en t r i c i t y This workPrsa et al. 2016 (in prep.; Kepler EBs)Beck et al. 2014 (red giants)Hambleton et al. 2013 (KIC 4544587)Hambleton et al. 2016 (KIC 3749404)Hareter et al. 2014 (HD 51844)Maceroni et al. 2009 (HD 174884)Schmid et al. 2015 (KIC 10080943)Smullen & Kobulnicky 2015Welsh et al. 2011 (KOI-54)
Figure 6.
Orbital eccentricity vs. orbital period. In both panels the 19 HB systems with orbits measured here are shown in red, andin gray we mark
Kepler
EBs where the eccentricity was derived through analysis of the eclipse light curves (from Prˇsa et al. in prep.).The top panel shows how the HB stars are typically positioned at the top envelope of the eccentricity-period distribution. In the bottompanel we add all known HB stars with orbits measured using RVs and
P <
200 d (see legend and Table 6). The dashed gray lines mark aneccentricity-period relation of e = (cid:113) − ( P /P ) (2 / , which is the expected functional form assuming conservation of angular momentum.The three curves use P of 4, 7, and 11 d, showing that it is difficult to use a single curve to match the upper envelope of the distributionthroughout the entire period range. See Sec. 4.3 for further discussion. Shporer et al. by other physical processes in addition to tidal circular-ization. Those systems include for example KID 4544587(Hambleton et al. 2013), some of the systems studied bySmullen & Kobulnicky (2015), and also two of the sys-tems studied here at P ∼
10 d and e ∼ . Kepler field stars). Therefore,systems with increased eccentricity compared to similarperiod systems provide an opportunity to study theseprocesses.It is also worth noting that the red giant HB systemsstudied by Beck et al. (2014, marked in black in Fig. 6bottom panel) show smaller eccentricities than other HBsystems at the same period. This is likely the result ofthe larger radii R of red giant stars, as the circulariza-tion time scale is proportional to R − in tidal theorieswith a constant lag angle or lag time. The red giantsystems also have systematically longer periods than theheartbeat stars we monitored with RVs, which is likelyrequired for these systems to have retained significant ec-centricity. However, the substantial eccentricity of thesered giant HB systems suggests that their tidal circular-ization timescales are not extremely short, and we spec-ulate that the occurrence of these eccentric red giant sys-tems is evidence against highly efficient tidal dissipationin sub-giants as has been suggested by Schlaufman &Winn (2013). A more detailed study examining the stel-lar radius and semi-major axis distribution of red giantsystems is required for a firm conclusion. Tidal Circularization
Tidal friction will act to circularize the orbits of HBstars, although the mechanisms and time scales of tidalcircularization remain poorly understood. Pioneeringworks such as Zahn (1975, 1977) have suggested thattidal friction is more efficient in stars with convectiveenvelopes where an effective turbulent friction can dampthe equilibrium tidal distortion. In stars without convec-tive envelopes, tidal dissipation can still occur via dis-sipation of dynamical tides (e.g., gravity waves) in theradiative envelope, but may be less efficient. Our samplecontains stars on both sides of the convective/radiative transition, and could be used to constrain or revise exist-ing tidal theories. In many HB stars (e.g., KID 9016693in Fig. 10), tidally excited oscillations are present in thelight curve, and can be used to study tidal dissipationvia dynamical tides. Although a detailed investigationis beyond the scope of this work, we examine here somebasic tidal parameters for our HB systems.Since the HB stars’ orbital eccentricities show a strongcorrelation with orbital period, we investigate the perioddependence of two other parameters that are closely re-lated to the tidal force acting on the primary star. Thefirst is the periastron distance: a peri = a (1 − e ) , (3)where a is the orbital semi-major axis. As shown in thetop panel of Fig. 7, a peri does not show a correlation withperiod, and has a roughly constant value of 0.080 au witha scatter of 0.021 au. For comparison, the mean andscatter of the semi-major axis (not shown) are 0.26 auand 0.14 au, respectively.The second parameter we investigate is the tidal forceacting on the primary star at periastron divided by thestar’s surface gravity: F tide F gravity = (cid:32) GR M a (cid:33) (cid:18) GM R (cid:19) − = (cid:18) R a peri (cid:19) M M , (4)where R is the primary star’s radius (see Table 1) and G the gravitational constant. Here we used the M valuesderived from the second approach described in Sec. 4.2,where in Eq. 2 sin ( i ) is replaced by the distributionmedian of 0.6495 (see Table 5 second column from theright). As can be seen in Fig. 7 bottom panel, this pa-rameter also does not correlate with period, and the typ-ical error bar is comparable to the scatter. The reasonfor the latter is that this ratio depends strongly on stel-lar parameters ( M , M , R ) which are usually less con-strained than orbital parameters. The errors on a peri are in comparison much smaller since that parameterdepends weakly on the masses of both stars and morestrongly on the period and eccentricity.The relatively small range of a peri and tidal forcingamplitudes for HB stars in Fig. 7 likely reflects the sen-sitivity of tidal circularization time scales to the am-plitude of tidal forcing (see e.g., Zahn 1975, 1977; Hut V monitoring of
Kepler
HB stars a peri are very rare dueto short circularization time scales. Systems with larger a peri are abundant, but exhibit smaller photometric vari-ations and have not been flagged as HB systems. Thisis also likely to be the cause of the somewhat narrowrange in angular momentum per unit mass (which scalesas P / ) of HB systems in Fig. 6.The system with the largest a peri (KID 10334122, P =37.95 d, a peri = 0.131 ± F tide /F gravity (see Fig. 7),since its lower eccentricity (compared to systems withsimilar period) results in a larger a peri and a weakertidal force. It is also not surprising to find this sys-tem positioned below the envelope in the e − P diagram(see Fig. 6). That system has a large uncertainty on a peri because of the large uncertainty on its eccentric-ity ( e = 0 . +0 . − . ), and it has a small uncertainty on F tide /F gravity resulting from relatively low uncertaintiesof M and R (see Table 1).In Fig. 8 we investigate how the two parameters men-tioned above relate to the photometric amplitude of theHB signal during periastron ( A HB ; see Table 1). The Y-axes of the two panels in Fig. 8 are the same as in Fig. 7,while the X-axis is the photometric amplitude, and themarkers’ radii are linear in the primary star T eff (SeeTable 1). The data in both panels do not show a clearcorrelation, although A HB is expected to increase withdecreasing periastron distance and increasing tidal force(Kumar et al. 1995; Thompson et al. 2012). This sug-gests that our sample is incomplete and suffers from ob-servational bias, and/or, that our understanding of A HB is incomplete. The data do show, however, that hotterstars have larger A HB , perhaps indicating that tidal cir-cularization is less efficient in hotter stars in our sample,although the cause of this correlation is presently unclear. Higher Multiplicity Systems
Many of the HB stars examined here may be membersof triple or higher multiplicity systems. Several works(e.g., Duquennoy & Mayor 1991; Meibom & Mathieu2005; Tokovinin et al. 2006; Raghavan et al. 2010) havefound that the fraction of higher multiplicity systemsamongst short period and highly eccentric binaries is veryhigh, exceeding 90% for binaries with
P < P , more than several times longer than the HBperiod (depending on its mass, inclination, and eccen-tricity) in order to allow long term dynamical stability(Kiseleva et al. 1994). If a typical HB progenitor systemwas born with an orbital period P (cid:38)
30 d before havingits orbital eccentricity excited, we expect P (cid:38) P in mostcases. Although the orbital RV amplitude induced bysuch a third body can be several km s − , we expect onlya small RV change over the ∼
100 d baseline of our obser-vations, and therefore this motion can be safely includedas a constant RV offset. We are currently pursuing afollow-up survey on many of these HB systems that willreveal whether any of them have tertiary companionswith orbital periods P (cid:46) Future Prospects
The orbits measured here comprise a relatively largesample obtained with a single instrument, and they canfacilitate several follow-up scientific studies in additionto that of tidal circularization and the shape of the e − P distribution (Sec. 4.3). Other examples include: • Stars in binary systems with highly eccentricorbits are expected to rotate near a “pseudo-synchronous” rotation period, where the stellar ro-tation is synchronized with the orbital motion closeto periastron which is faster than the mean or-bital motion along the entire orbit. The pseudo-synchronous rotation rate depends on eccentric-ity and orbital period (Hut 1981). However, thepseudo-synchronous rotation period also depends0
Shporer et al.
Period [d]
10 20 50 100 t i da l f o r c e / s u r f a c e g r a v i t y R a p e r i M M P e r i a s t r on d i s t. [ au ] a p e r i [ a u ] Figure 7.
Periastron distance (top panel; a peri = a (1 − e )) and the tidal force acting on the primary star at periastron divided by thestar’s surface gravity (bottom panel; see Eq. 4), both as a function of orbital period (x-axis in log scale). A HB [ppm]
500 1000 1500 2000 2500 3000 t i da l f o r c e / s u r f a c e g r a v i t y R a p e r i M M P e r i a s t r on d i s t. [ au ] a p e r i [ a u ] Figure 8.
Periastron distance (top panel; a peri = a (1 − e )) and the tidal force acting on the primary star at periastron divided by thestar’s surface gravity (bottom panel; see Eq. 4), both as a function of the full amplitude of the photometric HB signal at periastron. Inboth panels the markers’ radius is linear in the primary star’s T eff (See Table 1). V monitoring of
Kepler
HB stars Table 7
Predicted pseudo-synchronous stellar rotation period ( P ps ;rightmost column). The table also lists the orbital period ( P ) andeccentricity ( e ).KID P e P ps [d] [d]4659476 58.83045 ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . ± +0 . − . . ± . on the tidal prescription adopted (e.g., constanttidal Q and constant time lag yield different pre-dictions). The orbital eccentricities measured here,combined with the orbital periods, allow for a cal-culation of the expected pseudo-synchronous rota-tion periods for these stars. If the rotation periodscan be directly measured, for example in HB sys-tems also showing stellar activity, this will allow fordirect tests of tidal theories. We list in Table 7 thetheoretically predicted pseudo-synchronous rota-tion period P ps derived using Eq. 42 of Hut (1981).We also list the orbital period and eccentricity inthe same table. • Combining the orbital RV model parameters con-strained here (especially e and K ) with modelsof the Kepler
HB light curve, which constrain i ,results in an improved measurement of M (seeSec. 4.2). That will allow to measure the tidal forceacting on the primary star, and in turn the relationbetween that force and the HB photometric ampli-tude. • Measurement of the orbital inclination angle (seeprevious paragraph) and the stellar spin inclina-tion angle (when possible) gives the sky-projection of the stellar obliquity, which is a key parameterin understanding binary stars formation and or-bital evolution (e.g., Naoz & Fabrycky 2014). Thestellar spin inclination angle can be constrained bymeasuring the stellar rotation period (for exam-ple from stellar activity), the sky-projected rota-tion rate (from rotational broadening of spectrallines), and stellar radius (using spectroscopy andlight curve modeling). • The measurement of the RV orbit paves the wayfor a detailed analysis of individual systems show-ing tidal pulsations (e.g., Welsh et al. 2011; Fuller& Lai 2012; Burkart et al. 2012; Hambleton et al.2013; O’Leary & Burkart 2014; Hambleton et al.2016). • Continued RV monitoring can unveil extraneousbodies in the system by looking for a long termRV trend. This can be complemented by otherdata sets, such as spectroscopy, high angular reso-lution imaging, astrometry, and searching for pul-sation phase/frequency modulation in the
Kepler data (Shibahashi & Kurtz 2012; Murphy et al.2014). Measuring the occurrence rate of a thirdobject will allow for testing formation and orbitalevolution theory (e.g. Naoz 2016). Such a com-bined approach of using several different data setsto measure the occurrence rate of a third objecthas already proven successful in the study of shortperiod gas giant planets (Knutson et al. 2014; Ngoet al. 2015; Piskorz et al. 2015). SUMMARY
We have presented here the first results from our RVmonitoring campaign of
Kepler
HB stars. Our resultsinclude a sample of 19
Kepler
HB stars in the orbital pe-riod range of 7 −
90 d for which we derived a Keplerianorbital solution from RV monitoring using Keck/HIRES.This is currently the largest sample of HB stars for whichan RV orbit was obtained using a single instrument, andit roughly doubles the number of such systems with pe-riods up to 90 d.We have shown that HB stars populate the upper en-velope of the e − P diagram, which is a distinguishing fea-ture for testing tidal circularization theories. This sam-2 Shporer et al. ple will support additional studies that require a sampleof HB systems with well measured orbits, for exampletesting pseudo-synchronization theory and examining thephysics of tidally excited stellar pulsations.We also presented three objects for which we did notdetect RV variability, and list a few possible scenariosthat can explain that observation. Those objects shouldbe studied in more detail in order to explain the RV non-variability.We plan to continue our RV monitoring in order toincrease the sample size by measuring the orbits of addi-tional HB systems, and to look for long term RV trendsindicative of a third object in the system. We willpursue the latter by complementing the RV monitoringwith spectroscopy, imaging, astrometry, and examiningthe
Kepler light curve for modulations in the pulsationphases.Finally, we note that the ongoing K2 mission (Howell etal. 2014) and the future TESS mission (Ricker et al. 2014)and PLATO mission (Rauer et al. 2014) are expected todetect many more HB systems. For K2 and TESS, thestars are expected to be typically brighter than
Kepler
HB stars and therefore more accessible to RV monitoring.They are also expected to have shorter orbital periodsdue to the shorter temporal coverage, which is usefulfor studying the transition period below which binarysystems are fully circularized.We are grateful to the referee, Maxwell Moe, for histhorough reading of the manuscript and his meticulouscomments that have helped improve this paper. Wewarmly thank Ben Fulton, Evan Sinukoff, Lauren Weiss,Lea Hirsch, Erik Petigura, and Geoff Marcy for contribu-tions to the Keck/HIRES observations. This work wasperformed in part at the Jet Propulsion Laboratory, un-der contract with the California Institute of Technology(Caltech) funded by NASA through the Sagan Fellow-ship Program executed by the NASA Exoplanet ScienceInstitute. JF acknowledges partial support from NSF un-der grant no. AST-1205732 and through a Lee DuBridgeFellowship at Caltech. The authors wish to recognizeand acknowledge the very significant cultural role andreverence that the summit of Mauna Kea has always hadwithin the indigenous Hawaiian community. We are mostfortunate to have the opportunity to conduct observa- tions from this mountain. Some of the data presentedin this paper were obtained from the Mikulski Archivefor Space Telescopes (MAST). STScI is operated by theAssociation of Universities for Research in Astronomy,Inc., under NASA contract NAS5-26555. Support forMAST for non-HST data is provided by the NASA Of-fice of Space Science via grant NNX09AF08G and byother grants and contracts. This research has made use ofNASA’s Astrophysics Data System Service. This paperincludes data collected by the
Kepler mission. Fundingfor the
Kepler mission is provided by the NASA ScienceMission directorate. We acknowledge the support of the
Kepler
Guest Observer Program.
Facilities: Kepler, Keck:I (HIRES)
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V monitoring of
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A. RADIAL VELOCITY TABLE
Table 8 lists all 218 Keck/HIRES RV measurements obtained here. The table columns include KIC ID, mid exposuretime (BJD), RV (km s − ), and RV error (km s − ). Table 8
KID Time RV RV Error[BJD-2457000] [km s − ] [km s − ]4659476 211.105843 24 .
36 0 . − .
05 0 . − .
52 0 . − .
64 0 . .
80 0 . .
74 0 . − . . − .
38 0 . − .
197 0 . .
56 0 . .
32 0 . .
26 0 . .
14 0 . .
18 0 . − .
66 0 . − .
37 0 . − .
84 0 . − . . Shporer et al.
TABLE 8 – continuedKID Time RV RV Error[BJD-2,457,000] [km s − ] [km s − ]5090937 201.101216 − .
68 0 . .
21 0 . . . − .
90 0 . − .
28 0 . − . . . . . . .
03 0 . − . . − .
85 0 . − . . − .
26 0 . − .
78 0 . − .
73 0 . − .
65 0 . − .
22 0 . − .
00 0 . − .
08 0 . − . . − .
07 0 . − . . − .
01 0 . .
87 0 . .
71 0 . .
34 0 . − .
60 0 . .
05 0 . − .
60 0 . .
01 0 . .
40 0 . .
07 0 . .
51 0 . .
50 0 . .
58 0 . − .
29 0 . − .
11 0 . − .
33 0 . − .
46 0 . .
87 0 . .
06 0 . .
74 0 . .
17 0 . − . . . . .
58 0 . . . . . − . . − . . − . . − . . − . . − .
20 0 . − .
63 0 . − .
95 0 . − .
94 0 . − .
76 0 . − .
285 0 . − .
856 0 . .
48 0 . . . . . .
33 0 . . . .
40 0 . .
08 0 . .
80 0 . − . . V monitoring of
Kepler
HB stars TABLE 8 – continuedKID Time RV RV Error[BJD-2,457,000] [km s − ] [km s − ]8027591 179.976502 53 .
72 0 . .
865 0 . .
29 0 . − .
81 0 . − .
78 0 . − .
55 0 . .
41 0 . .
19 0 . .
21 0 . .
40 0 . . . . . . . . . .
46 0 . .
22 0 . . . − .
82 0 . − . . .
04 0 . . . . . − .
44 0 . − . . − .
37 0 . − .
90 0 . .
98 0 . − .
94 0 . .
30 0 . .
49 0 . − .
99 0 . − .
37 0 . − .
05 0 . − .
29 0 . − .
62 0 . − .
27 0 . − .
85 0 . − .
14 0 . − .
04 0 . − .
44 0 . − .
76 0 . .
06 0 . − .
23 0 . .
39 0 . − .
97 0 . .
02 0 . .
40 0 . − . . − . . − .
75 0 . . . − .
42 0 . .
26 0 . .
27 0 . .
75 0 . − .
98 0 . − .
50 0 . .
78 0 . .
14 0 . .
19 0 . − . . − . . − . . − . . − . . − . . − . . − . . − . . Shporer et al.
TABLE 8 – continuedKID Time RV RV Error[BJD-2,457,000] [km s − ] [km s − ]11122789 210.099344 − . . − . . − . . − . . − .
47 0 . − . . − .
91 0 . − . . − .
19 0 . − .
63 0 . − . . .
54 0 . .
25 0 . .
46 0 . .
60 0 . . . .
17 0 . .
56 0 . .
54 0 . .
79 0 . .
85 0 . .
94 0 . − .
24 0 . − .
48 0 . − .
27 0 . − .
14 0 . − .
79 0 . − .
43 0 . − .
62 0 . − .
42 0 . − .
44 0 . − .
09 0 . − .
73 0 . − .
32 0 . − .
80 0 . − .
86 0 . − .
12 0 . − .
80 0 . − .
90 0 . − .
44 0 . − .
98 0 . − .
86 0 . . . − .
94 0 . − .
45 0 . − .
05 0 . − .
61 0 . − .
35 0 . − .
86 0 . − .
41 0 . .
92 0 . − . . − . . − . . − . . − . . − . . − . . − . . − .
85 0 . − . . . . B. SIMULTANEOUS LIGHT CURVE AND RADIAL VELOCITY CURVE PLOTS
Figures 9 through 12 show the phase-folded RV curve (measurements in black, model in red) along with the phase-folded and binned
Kepler light curve (blue). Figures 9 through 11 show the 19 systems for which we fitted a Keplerian
V monitoring of
Kepler
HB stars
Kepler datahere and the light curves are shown for completeness.8
Shporer et al.
Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -1000100 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -40-200204060 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -50050 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -80-70-60-50-40-30-20 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -1000100 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -50050 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -50050 Phase R e l a t i v e F l u x R V [ k m / s ] -55-50-45-40-35-30-25 Figure 9: The panels show the
Kepler phase-folded relative flux light curve (blue; left Y-axis) and the phase-folded RV curve model (red;right Y-axis) with periastron at phase 0.5, for KID 4659476, KID 5017127, KID 5090937, KID 5790807, KID 58181706, KID 5877364,KID 5960989, and KID 6370558. In each panel the title lists, from left to right, the KIC ID, T eff (K), log g , P (d), K (km s − ), and e .The RV measurements are overplotted in black, including error bars although in some panels the markers are larger than the error bars.The Kepler light curves shown here were derived by applying a running mean to the
Kepler data, followed by binning with a bin size of0.0002 in phase.
V monitoring of
Kepler
HB stars Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -1000100 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -20020406080 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -50050 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -100-50050100 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -100-50050 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -80-60-40-20020 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -80-60-40-200 Phase R e l a t i v e F l u x R V [ k m / s ] -2002040 Figure 10: Same as Fig. 9 for KID 6775034, KID 8027591, KID 8164262, KID 9016693, KID 9965691, KID 10334122, KID 11071278, andKID 11403032. In each panel the title lists, from left to right, the KIC ID, T eff (K), log g , P (d), K (km s − ), and e . Shporer et al.
Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -120-100-80-60-40-20020 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -80-60-40-20020 Phase0 0.2 0.4 0.6 0.8 1 R e l a t i v e F l u x R V [ k m / s ] -80-60-40-2002040 Figure 11: Same as Fig. 9 for KID 11649962, KID 11923629, and KID 12255108. In each panel the title lists, from left to right, the KICID, T eff (K), log g , P (d), K (km s − ), and e . V monitoring of
Kepler