Binary star detectability in Kepler data from phase modulation of different types of oscillations
Douglas L. Compton, Timothy R. Bedding, Simon J. Murphy, Dennis Stello
aa r X i v : . [ a s t r o - ph . S R ] M a y MNRAS , 1–7 (2015) Preprint 9 April 2018 Compiled using MNRAS L A TEX style file v3.0
Binary star detectability in
Kepler data from phasemodulation of different types of oscillations
D. L. Compton , ⋆ , T. R. Bedding , , S. J. Murphy , , D. Stello , Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
Accepted 2016 May 4. Received 2016 May 1; in original form 2016 March 22
ABSTRACT
Detecting binary stars in photometric time series is traditionally done by measuringeclipses. This requires the orbital plane to be aligned with the observer. A new methodwithout that requirement uses stellar oscillations to measure delays in the light arrivaltime and has been successfully applied to δ Scuti stars. However, application to othertypes of stars has not been explored. To investigate this we simulated light curves witha range of input parameters. We find a correlation between the signal-to-noise of thepulsation modes and the time delay required to detect binary motion. The detectabilityof the binarity in the simulations and in real
Kepler data shows strong agreement,hence, we describe the factors that have prevented this method from discovering binarycompanions to stars belonging to various classes of pulsating stars.
Key words: stars : oscillations - stars : variables - stars : binaries - techniques :radial velocities
The primary science goal of
Kepler was to find stars withEarth-like exoplanet companions by observing transits inthe photometric time series (Koch et al. 2010; Borucki et al.2010). Large numbers of eclipsing binaries were also discov-ered (Prˇsa et al. 2011; Kirk et al. 2016), some of which showstellar oscillations (e.g. Southworth 2015). These oscillationscan also be used to find binaries without transits or eclipses,provided the pulsation modes are stable enough to act likea ‘clock’. That is the subject of this paper.As a star and its companion orbit each other, the lighttravel time from the host star to the observer will vary. Thevariation is a periodic function that is related to the integralof the radial velocity variation. Telting et al. (2012) usedthe differential arrival time of pulsation modes to confirmthe presence of a companion to an sdB star. Murphy et al.(2014) developed the phase modulation (PM) method, us-ing changes in the pulsation phases of δ Sct stars to find bi-nary companions. This method complements the frequencymodulation (FM) method by Shibahashi & Kurtz (2012),wherein the binary motion modulates the oscillation fre-quencies, causing multiplets in the Fourier transform. TheFM method is suitable for data sets that are much longerthan the orbital period of the binary. Hence, for the wideorbits the frequency splitting can approach the frequencyresolution of the pulsation spectrum. In contrast, the PM ⋆ E-mail: [email protected] method favours wider orbits because the light travel timeacross the orbit is larger. The PM method has the benefitof providing a visualisation of the orbit by tracking the timedelays as a function of orbital phase. It also allows the sig-nals from different pulsation modes to be combied straight-forwardly. This method has been used on known δ Scutibinaries in the
Kepler field (e.g. Balona 2014). Additionally,many new binary systems have been discovered using thePM method (e.g. Murphy et al. 2014). Several giant plan-ets have been discovered around pulsation sdB stars usingtiming variations (e.g. Silvotti et al. 2007; Lee et al. 2009;Qian et al. 2009; Geier et al. 2009).The goal of this paper is to apply the PM method toother types of pulsating stars to detect the presence of a bi-nary companion. We start by introducing the PM methodand apply it to artificial light curves (Section 2). We adopta Monte Carlo method by mass-producing light curves andextracting relevant parameters from the time-delay spec-trum. The resulting distributions are analysed to diagnosedetectability and then compared with real
Kepler binariesin Section 3. Finally the implications of this research arediscussed in Section 4.
The PM method involves measuring the phase, Φ( t ), of apulsation mode over time, t . We generalise the light curve as c (cid:13) Compton et al. a periodic function, f ( t ). The phase of a pulsation mode withfrequency ν can be calculated using the Fourier transformover a time interval δt , which isF( t ; ν, δt ) = Z t + δt/ t − δt/ f ( t ′ ) e − πiνt ′ dt ′ . (1)The argument of the complex quantity F in Equation 1 givesthe phase:Φ( t ; ν ) = tan − (cid:18) Im(F( t ; ν, δt ))Real(F( t ; ν, δt )) (cid:19) . (2)The length of the intervals used by Murphy et al. (2014) was δt = 10 days.The phase of a pulsation signal is sensitive to changesin distance to the source. Extracting the pulsation phasefor each time interval produces a series of phases that aremodulated by orbital motion, allowing us the measure thebinary period, P orb , and projected semi-major axis a sin ( i ),where i is the inclination of the orbital plane with respectto the observer.The phase variations can be converted to time delaysby dividing by the angular frequency of the pulsation mode: τ ( t ) = ∆Φ( t )2 πν . (3)Here, ∆Φ = Φ( t ) − h Φ( t ) i , which sets the mean of the timedelays to be zero.Figure 1 shows the main steps of the PMmethod for a known Kepler binary δ Sct starKIC 11754974 (Murphy et al. 2013b): (a) the Fouriertransform of the time series around a prominent oscillationmode, (b) time-delay series, and (c) time-delay spectrum.Binary motion produces a peak in the Fourier transformof the time delays (henceforth time-delay spectrum) at theorbital frequency, f orb = 1 /P orb . The amplitude of this peakis the projected light travel time across the orbit, which is a sin ( i ) divided by the speed of light c . A more completeset of orbital parameters can be extracted from fittingdirectly to the time delays (see Murphy & Shibahashi2015). To evaluate the performance of the PM method, we gener-ated monoperiodic
Kepler time series, which simulate thebinary motion. The simulated light curves were a linearcombination of two components: a single oscillation modeand a noise term. Binarity was simulated by adding a time-dependent phase shift to the pulsation.The flux variation due to a single mode in the time seriesis given by f ( t n ; A, ν, φ ) = A cos (2 πν ( t n + τ ( t n )) + φ ) , (4)where A and ν are the amplitude and frequency of the mode,and φ is the phase relative to an arbitrary fixed startingpoint. The time stamps, t n , for the simulations were basedon the Kepler long cadence sampling (∆ t = 29 . τ ( t n ) describes the time delays induced by the binarymotion. To simplify our analysis we only considered circu-lar orbits, i.e. eccentricity e equals zero. Shibahashi & Kurtz Figure 1.
Application of the phase modulation method to the
Kepler δ Sct binary star KIC 11754974. (a) The close-up viewof a single mode at 189 . µ Hz in the Fourier transform in thephotometric time series. (b) time delays calculated from 10 d sub-series at the frequency of the chosen mode. (c) Fourier transformof the time delays (time-delay spectrum). The peak at 0 . − in the time-delay spectrum is caused by the binary motion of the δ Sct star. (2012) expressed the phase modulation function as τ ( t n ; τ max , P orb , ψ ) = τ max sin (cid:18) πt n P orb (cid:19) + ψ, (5)where τ max is the amplitude of the time-delay variations, P orb is the orbital period, and ψ describes the phase of theorbit. For a circular orbit, τ max only depends on the pro-jected semi-major axis. In the absence of noise, the maxi-mum time delay is equivalent to the projected light travel MNRAS , 1–7 (2015) inary star detectability in Kepler data Table 1.
Set of fixed parameters used in the simulations. Y isa uniform random-number-generating-function between 0 and 1.Each simulation used a different random seed.Parameter Description Simulation Value P orb Orbital period 100 d e Orbital eccentricity 0 ψ Orbital phase 2 πY i N sub Number of sub-series 150 A Pulsation amplitude 1 . ν sim Pulsation frequency 250 µ Hz φ Pulsation phase 2 πY j time across the orbit, i.e. τ max = a sin ( i ) /c . In this paper τ max will be used as the input maximum time delay, whereas a sin ( i ) /c is the empirical maximum time delay extractedfrom the simulated time series using the PM method.We simulated the white noise using the equation W ( t n ; σ ) = X t n ( σ rms ) , (6)where X t n is a random number taken from a Gaussian dis-tribution with a mean of zero and a standard deviation of σ rms .The relationship between the scatter in the time se-ries, σ rms , and the mean noise level in the amplitude spec-trum, σ amp , is σ amp = r πN σ rms . (7)Here, N is the number of data points in the time series(Kjeldsen & Bedding 1995). The relationship between A and σ rms gives the signal-to-noise ratio of the oscillation mode,S / N = Aσ amp = A p πN σ rms . (8)The signal-to-noise ratio is a convenient quantity because itcombines the oscillation amplitude and noise into one scalarquantity and it can be measured straight-forwardly from theFourier transform of the light curve.The uncertainty of the pulsation phase measurementdepends on this signal-to-noise. For a given binary orbit,Equation 3 indicates that the phase modulation of a partic-ular pulsation mode is proportional to the mode frequency.It follows that the uncertainties of the phases are lower forhigher pulsation frequencies.The randomness of our simulations gives a distributionof measured maximum time delays, a i sin ( i ) /c , and orbitalperiods, P obs . Non-varying asteroseismic and binary param-eters were marginalised by setting them as constant acrossall simulations, as shown in Table 1.The amplitude, A , was kept constant and the noiselevel, σ rms , was adjusted to control the signal-to-noise of themode. We split each time series into N sub = 150 sub-series,which corresponds to approximately ten days for most Ke-pler light curves. If the effective length of a sub-series wasless than eight days, it was discarded. Ten days is shortenough to sample the orbit for any orbital period above20 d and long enough to resolve individual modes with aminimum frequency separation of approximately 3 µ Hz (seeChristensen-Dalsgaard 2008).In general, high-frequency pulsations and longer orbital periods are advantageous. However, upper limits exist forboth these quantities. A full sample of the orbit through-out the time series (i.e. P orb . . µ Hz. Therefore, care must be taken to avoid aliases(e.g. Murphy et al. 2013a).
To determine the limits of binary detectability we generatedsimulated
Kepler time series with a grid of input parame-ters. We initially simulated time series without binarity tounderstand the influence of the pulsation mode signal-to-noise on the time-delay spectrum. We simulated 1000 lightcurves for each value of signal-to-noise in a grid spanning5 < S / N < τ max · ν or a sin ( i ) /c · ν .An example of the distribution of maximum time de-lay from 1000 simulations of purely white noise time seriesis shown in Fig. 2a. A Gaussian fit to the histogram givesthe typical maximum time delay caused by the white noisescatter. The maximum time delay of a binary must exceedthis value for a given signal-to-noise ratio to be considereddetectable, i.e. above the points in Fig. 2b. At low S / N weobserved excessive phase-wrapping in the time-delay sub-series, that is the point-to-point scatter in Fig. 1b exceededthe co-domain of Equation 2 ( | Φ( t ) | > π ). We fitted a power-law to the points above a signal-to-noise ratio of 50, whichwe considered to be a soft lower bound on the binary de-tection threshold for a monoperiodic star due to the phasewrapping. We extended our analysis by adding phase modulation( τ max > τ max and pulsation mode S / N. Arrays of S / N and τ max were loga-rithmically spaced to make a 40x40 grid of input parameters.For each grid point another 1000 monoperiodic time serieswere constructed with different random seeds. Using the PMmethod, the maximum time delay, a sin ( i ) /c , was extractedfrom the time-delay spectra. Note that we scaled the timedelays to a phase unit, a sin ( i ) /c · ν , to allow easy com-parison between stars in different binary systems or havingdifferent oscillation frequencies.We then looked at the distribution of maximum timedelays of each of the sets of 1000 simulated time series in MNRAS , 1–7 (2015)
Compton et al.
Figure 2. (a) A single distribution (S / N ≈ N = 1000 simulated timeseries of the same star with no binary component. The distribution was fitted with a Gaussian (green line). The mean and standarddeviation of the Gaussian is represented by the dotted red line and the blue bar, respectively. (b) The diamonds and attached errorbars represent the mean and standard deviation, respectively, of each distribution as a function of signal-to-noise. The blue diamondcorresponds to the distribution in (a). The solid red line is a power-law fit to the points with S / N >
50 (dashed black line), and thedotted red line is its extrapolation. the grid. These distributions were found to be Gaussian-like, as in Fig. 3, as long as the time delay due to binary wasconsistently above the maximum noise-peak. In contrast, themaximum time delay distribution of a set of noisy time serieswas found to be spread out and non-Gaussian because morefalse peaks in the time-delay spectra were extracted. Themean of the maximum time delay distribution in Fig. 3 isless than the input maximum time delay. This is causedby an undersampling of the orbit. Murphy et al. (in prep)will give a full characterisation of the effect, but with oursampling of 10 sub-series per orbit there is little impact onour results.The grid of distributions are compiled into Fig. 4 us-ing contours of constant relative uncertainty. The relativeuncertainty was calculated by dividing the standard devi-ation of each distribution by the injected maximum timedelay, τ max . The contour lines follow a power-law for largeS / N and time-delay amplitude, where binarity is most eas-ily detectable. Conversely, the contours become irregular atlarger uncertainties and the binary is harder to detect. Thelarge uncertainties correspond to a parameter space wherethe noise in the time-delay spectrum consistently exceeds τ max (red line in Fig. 4 and Fig. 2b). We concluded that thebinary peak in the time-delay spectrum could not be reliablydetected if the relative uncertainty was much greater than30%. Therefore, contours greater than this are not shown inFig. 4. We should keep in mind that our simulation only in-cluded a single pulsation mode in each time series, and thatwe should expect to do somewhat better in multiperiodicstars. We looked for phase modulation
Kepler data for in a va-riety of pulsating stars with known binaries. Examples areshown in Fig. 5, and all have coherent modes with lifetimes
Figure 3.
The distribution of maximum time delays from1000 simulations with input parameters of τ max ≈
120 s andS / N ≈ a sin ( i ) /c , and the maxi-mum time delay injected into the simulated, τ max . longer than the four years of the observations. We consid-ered two types of main-sequence pulsating stars near the in-stability strip. δ Scuti stars have high-amplitude and high-frequency p modes, which is why they were initially cho-sen when the PM method was developed. γ Doradus starshave g mode pulsations at lower frequencies. In addition,we looked at δ Sct/ γ Dor, hybrids which have both δ Sctand γ Dor pulsations. We also investigated red giant branch(RGB) and clump stars, where the coupling between pres-sure and gravity-dominated modes generates mixed dipolemodes with long lifetimes (e.g. Dupret et al. 2009). Finally,
MNRAS , 1–7 (2015) inary star detectability in Kepler data Figure 4.
The contours define lines of constant relative uncertainty in the time delays calculated from a sample of 1000 simulated lightcurves as a function of signal-to-noise ratio, S / N of the pulsation mode, and injected maximum time delay, τ max . The red line is thepower-law fit to the average maximum noise peak, shown in Fig. 2b. Each symbol represents the strongest pulsation mode of a known Kepler binary star. The red squares and purple triangles are δ Sct p modes (by Murphy et al. (2014)). Orange diamonds are sdB starsmeasured by Telting et al. (2012, 2014) (KIC 7668647 and KIC 11558725). The dotted-lined box represents the area where the most idealred giants lie (Beck et al. 2014; Gaulme et al. 2014). The green crosses and blue plusses are δ Sct/ γ Dor hybrids, p mode and g mode,respectively (see Van Reeth et al. (2015) for KIC 3952623 and Keen et al. (2015) for KIC 10080943). we also considered two classes of compact evolved stars, thesubdwarf B (sdB) stars and white dwarfs, which both havecoherent and modest-amplitude g modes (e.g. Reed et al.2011; Greiss et al. 2014) that could be suitable for detectingbinary phase modulations.We used
Kepler light curves that had been reduced us-ing the multiscale Maximum A Posteriori (msMAP) algo-rithm developed by Stumpe et al. (2012). This removes sys-tematic trends, discontinuities, outliers, and artifacts. Thelight curves were then analysed using the PM method, asoutlined in Section 2.1.Relating our simulations to the observed data requiresknowledge of the projected light travel time across the or-bit and the signal-to-noise ratios of the observed pulsations.For each star the strongest mode in the amplitude spectrumwas selected for analysis and its frequency ν i and ampli-tude A i were noted. The light travel time was calculatedas described in section 2.1. The signal-to-noise was calcu-lated using Equation 8 from the pulsation amplitude andthe mean noise level. The results are plotted as symbols in Fig. 4. The location of a star in Fig. 4 gives an estimate ofthe relative uncertainty of its maximum time delay for anindividual mode. If the relative uncertainty is greater than30% it is unlikely to be detectable. Analysing multiple modesreduces the total relative uncertainty by approximately thesquare root of the number of modes.Care must be taken when calculating the mean noiselevel, σ amp . For the highest amplitude δ Sct stars, the spec-tral window will dominate the amplitude spectrum (e.g.Murphy et al. 2013b), even when the window function isideal. To estimate the noise, we first pre-whitened the peakwith the highest amplitude in the pulsation spectrum. Themean residual amplitude within ±
10% of the mode fre-quency was taken to be the mean noise level and used tocalculate the signal-to-noise ratio. For most δ Sct stars, wenote that this overestimates the noise because variance fromother oscillation modes remains.
MNRAS000
MNRAS000 , 1–7 (2015)
Compton et al.
Figure 5.
Examples of amplitude spectrum for each type of star analysed in this paper ordered from highest to lowest S / N. The leftcolumn is the full spectrum calculated from the observed
Kepler time series and the right column is a 20 µ Hz-wide close-up of the regionbetween the red dashed lines. The close-up plots show the spectra of the full time series (grey line) and a typical ten day sub-series(black line). The types of stars are as follows: (a) a δ Sct star with p-mode pulsations. (b) a γ Dor/ δ Sct hybrid star with low frequencyg mode and high frequency p mode pulsations. (c) an sdBs with g mode pulsations. (d) a red giant branch star with solar-like mixeddipole modes oscillations. The amplitude of each plot has been normalised to the height of the strongest pulsating peak.
Fig. 4 shows good agreement between the predicted de-tectability of binary stars and those with observed time-delay variations. This validates the use of the simulations asa way to determine a lower-bound of observable projectedlight travel time across the orbit for a given signal-to-noiseand pulsation frequency. For example, a pulsation mode with
S/N = 100 at ν = 210 µ Hz should typically show detectablephase modulation if the maximum time delay is greater than30 seconds ( a sin ( i ) /c · ν = 30 · · − = 0 . i = 90 ◦ ). However,the maximum time-delay for eccentric orbits depends on theargument of periapsis, ̟ . For the known binaries we exam-ined, we assumed the eccentricity of the orbit has a negligi-ble effect on the time-delays, i.e. the maximum time-delay isequivalent to a sin ( i ) /c and does not affect the detectabil-ity. The δ Sct binaries in our sample, denoted by thered squares and purple triangles in Fig. 4, were all detectedusing the PM method. Except for two, all lie above the max-imum time-delay of noise (solid red line). These two weredetected by analysing multiple modes (up to nine modesin total), which reduces the uncertainty of the time-delaysby approximately the square root of the number of modes,whereas the simulations were calculated for single modes,only.We found that γ Dor pulsations are not suitable for the PM method because the frequencies of the g modes aretoo low. The examples shown in Fig. 4 are δ Sct/ γ Dorhybrid stars that only have detectable time-delays for thep-mode pulsations. The PM analysis of the g modes did notyield a detectable signal of binary motion. In general, theperiods of the g modes are at least ten times greater thanthe p modes. This reduces the detectability of time-delayvariations by an the same factor. Additionally, the densityof modes in the Fourier transform of γ Dor stars can betoo high for them to be resolved with ten-day time intervals(see Keen et al. (2015) for an example); the time-delays areobscured by beating between other pulsation modes in thestar. Binary δ Sct stars analysed by Murphy et al. (2014)also show the effect of closely spaced beating modes on thetime-delay spectrum.From the analysis of RGB and clump stars we concludedthat
Kepler red giants will not have detectable time-delayvariations. We inferred the maximum delays using 16 redgiant binaries that have known orbital parameters reportedby Beck et al. (2014) and Gaulme et al. (2014). The idealred giants are high-frequency, lower red-giant-branch (RGB)stars like the one shown in Fig. 5d. We combined typicalmass ratios of red giant binaries with a range of possiblebinary parameters to create a best-case scenario for the lowluminosity RGB stars. The best red giants would exist inthe dotted box in Fig. 4. Therefore, we conclude that thesignal-to-noise ratios of the coherent dipole modes, even inthe best cases, are insufficient to detect time-delays causedby binarity.The companions of pulsating sdB stars are on thethreshold of being detectable, as shown by the orange di-
MNRAS , 1–7 (2015) inary star detectability in Kepler data amonds in Fig. 4. Two Kepler sdB stars in known bina-ries were analysed. The orbital periods are about 10 days,each with a white dwarf companion. Therefore, in an at-tempt to detect the phase modulation due to binarity, wetook N sub = 500 sub-series, which corresponded to a sub-series length of 2 days, which was required to sample theshort orbit adequately. Naturally, decreasing the sub-serieslength increases the uncertainty of the time-delays, furtherreducing the detectability of the binarity of the sdBs. ThePM method would succeed for sdB stars with longer orbitalperiod companions, which would give a higher maximumtime-delay. We were unable to detect binarity from the 19pulsating sdB stars that have Kepler long cadence data, andwhich are not known to be binaries (Silvotti et al. 2014).Previous work by Telting et al. (2012, 2014) analysed thesetwo sdB stars in a similar way by fitting the time delays tothe pulsation modes. They were successful in extracting themaximum time delays by using
Kepler short cadence dataand using many tens of modes. Our analysis considers onlya single mode, and is based on long-cadence data only.We also considered white dwarfs, although we note thatnone of the pulsating white dwarfs in the
Kepler field arein known binary systems. The pulsation frequency of whitedwarfs is an order of magnitude higher than the other starsconsidered in this analysis. Therefore,
Kepler short-cadencedata (one minute sampling interval) were used in the anal-ysis, which is not entirely comparable to our simulations inFig. 4. We infer from the work of Hermes et al. (2011) andGreiss et al. (2014) that the signal-to-noise ratios of whitedwarf oscillations are similar to those of red giants. Notethat the short oscillation periods of white dwarfs can causethe maximum time delay to exceed the pulsation period,in which case one must also account for phase wrappingof the binary-induced phase shifts. We concluded that thehigh frequency pulsations suit the PM method but, for thesame reasons as for the sdBs, the smaller number of pul-sating white dwarfs with
Kepler data limits the chances ofdetecting binary systems.
We attempted to extend range of stars for which the phasemodulation method can be applied, to include red giants, γ Dor stars, white dwarfs, and subdwarf B stars. We ex-plored the asteroseismic and orbital parameter space to findthe detection limits. The results from our simulations showa relationship between the signal-to-noise ratio of the pul-sation mode and the ability to detect binarity. To confirmthis, we compared the results of the simulations with ob-served
Kepler light curves. We saw a strong agreement in bi-nary detectability between the observed and simulated data.Moreover, for δ Scts star with the highest signal-to-noise ra-tios, time-delay variations as low as a few seconds shouldbe detectable. For a monoperiodic oscillator, this maximumtime-delay corresponds to a companion mass on the order of M sin ( i ) = 10 Jupiter masses, given the limits of the Kepler time series. This limit can be decreased for stars with pul-stions above the Nyquist frequency or when analysing mul-tiple modes, which can reduce the time-delay uncertainty by √ N , where N is the number of modes analysed. The limitfor γ Dor and sdBs is approximately 10 times more massive, due to the differences in mode frequency and signal-to-noiseratio. These companions would be very-low-mass stars thathave periods of over one year. The limit for red giants is 100times greater relative to the δ Sct stars, which is on the or-der of a solar mass companion. This optimistic case does nottake into account the density of g modes, which cause heavyinterference in the time-delay spectra, ultimately causing thered giants to be unrealistic candidates for the PM method.We conclude that the PM method is best suited tosearching for companions around δ Sct stars, where it shouldbe possible to reach down to planetary masses.
ACKNOWLEDGEMENTS
This research was supported by the Australian ResearchCouncil. Funding for the Stellar Astrophysics Centre is pro-vided by the Danish National Research Foundation (grantagreement no.: DNRF106). The research is supported by theASTERISK project (ASTERoseismic Investigations withSONG and Kepler) funded by the European Research Coun-cil (grant agreement no.: 267864).
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