BEAMING BINARIES: A NEW OBSERVATIONAL CATEGORY OF PHOTOMETRIC BINARY STARS

Transcription

1 The Astrophysial Journal, 670: , 2007 Deember 1 # The Amerian Astronomial Soiety. All rights reserved. Printed in U.S.A. BEAMING BINARIES: A NEW OBSERVATIONAL CATEGORY O PHOTOMETRIC BINARY STARS Shay Zuker, 1 Tsevi Mazeh, 2 and Tal Alexander 3,4 Reeived 2007 April 18; aepted 2007 July 4 ABSTRACT The new photometri spaeborne survey missions COROT and Kepler will be able to detet minute flux variations in binary stars due to relativisti beaming aused by the line-of-sight motion of their omponents. In all but very short period binaries (P > 10 days), these variations will dominate over the ellipsoidal and refletion periodi variability. Thus, COROT and Kepler will disover a new observational lass: photometri beaming binary stars. We examine this new ategory and the information that the photometri variations an provide. The variations that result from the observatory helioentri veloity an be used to extrat some spetral information even for single stars. Subjet headinggs: binaries: lose binaries: general binaries: spetrosopi surveys tehniques: photometri tehniques: radial veloities 1. INTRODUCTION In 2003 Loeb & Gaudi suggested a new photometri method to detet extrasolar planets, based on the minute variability of the stellar flux due to relativisti beaming indued by the star s reflex radial veloity. Rybiki & Lightman (1979) show that several fators ontribute to the beaming effet. The bolometri fators are the Lorentz transformations of the radiated energy and the time intervals, and the modifiation of the angular distribution of the radiated energy (stellar aberration). Thus, in the limit where the star s radial (line-of-sight) veloity v R is muh smaller than the speed of light, the observed bolometri flux is modified relative to the emitted bolometri flux 0 as ¼ 0 1 þ 4 v R : ð1þ or bandpass photometry, the Doppler shift of the emitted frequeny and the spetral index of the soure spetrum also have to be taken into aount, whih finally yields: h ¼ 0 1 þ (3 ) v i R ; ð2þ where and 0 are the observed and emitted flux density at frequeny, respetively, and ¼ d log /d log is the average spetral index around the observed frequeny. Note that equations (1) and (2) an also be obtained in a semilassial ontext. In any ase, sine the effet depends on the first order of v R / the veloities do not need to be highly relativisti for the effet to be deteted. Deteting beaming variability related to orbital motion requires the very high preision provided by photometri satellites. Even in the extremely high orbital veloities of the stars orbiting the massive blak hole in the Galati enter, the beaming effets annot be measured with the low photometri preision available for those stars (Zuker et al. 2006). Loeb & Gaudi (2003) made the suggestion to detet stellar periodi motion indued by an unseen planet through the beaming 1 Department of Geophysis and Planetary Sienes, Raymond and Beverly Sakler aulty of Exat Sienes, Tel Aviv University, Tel Aviv 69978, Israel. 2 Shool of Physis and Astronomy, Raymond and Beverly Sakler aulty of Exat Sienes, Tel Aviv University, Tel Aviv 69978, Israel. 3 aulty of Physis, Weizmann Institute of Siene, Rehovot 76100, Israel. 4 The William Z. and Eda Bess Novik Career Development Chair effet. They showed that the amplitude of the beaming effet produed by an extrasolar planet is of the order of miromagnitudes. Suh an effet is barely detetable by the new photometri spae missions COROT (Baglin 2003) and Kepler (Basri et al. 2005), whih are aiming to find extrasolar transiting planets, with a typial variability amplitude of 100 mag. The obvious advantage of the Loeb & Gaudi (2003) approah is its appliability to planets with almost any inlination, whereas deteting transiting planets is limited only to planets with orbital inlinations lose to 90. Here we examine the beaming effet in binary stars, and suggest a new lass of binaries: beaming binaries. These are a hybrid between spetrosopi and ellipsoidal binaries, sine the beaming binaries will be deteted by periodi photometri variations due to their orbital radial veloity. We show that a binary with an orbital period of P ¼ 100 days has a beaming variability of at least Oð100 magþ, whih is easily detetable by COROT and even more so by Kepler. We further show that for binaries with periods longer than 10 days, beaming variability dominates over the ompeting effets of ellipsoidal and refletion variability. We therefore expet the new satellites to harvest hundreds of previously unknown binaries of this new lass. Beaming variability of the order of 100 mag will also be indued by the Earth s helioentri motion relative to any observed soure, single star or binary. In spetrosopi observations, the effet of the Earth motion is orreted in order to obtain helioentri radial veloities. Here we show how the dependene of this small variability on the spetral slope at the observed bandpass an be used to probe the spetral harateristis of single and binary stars. 2. THE AMPLITUDE O THE BEAMING EECT AS COMPARED WITH THE ELLIPSOIDAL AND RELECTION EECTS The three kinds of periodi flux variations we expet to detet in binary stars are those due to the ellipsoidal tidal deformations of the stars, the refletion of the light of eah star by its ompanion, and relativisti beaming, whih is the subjet of this work. Loeb & Gaudi (2003) have presented a rough omparison of those three effets in the ontext of planet-hosting stars. We now ompare the three effets for binary stars, assuming for simpliity a irular (e ¼ 0) edge-on (sin i ¼ 90 ) orbit. Note that for systems with two stellar omponents, the observed ellipsoidal effet is the weighted average of the ellipsoidal effets of the two stars, whereas the observed beaming effet is the weighted differene between the

2 PHOTOMETRIC BEAMING BINARIES 1327 TABLE 1 Stellar Parameters for Simulation Parameter 0 G0 K0 Referene Mass (M ) Radius (R ) Effetive temperature (K) Gravity darkening oeffiient in V Referenes. (1) Cox 2000 (2) Morris beaming variabilities of the two stars, as their effets are exatly in opposite phase. In this respet, beaming is similar to the refletion effet. To a good approximation (see below), the magnitude of the beaming effet an be alulated under the assumption that the two stars radiate as blakbodies. or a blakbody soure of temperature T ea the spetral index is () ¼ 3 e x e x 1 x; ð3þ where x ¼ h/kt ea. The binary s orbital separation is given by Kepler s third law, a ¼ M 1 þ M 1=3 2 P 2=3 AU; ð4þ M 1yr where M i are the two masses, with the subsript 1 referring to the primary and 2 to the seondary. The amplitude of the primary s radial veloity variation is then M 2 a 1=229:8 K 1 ¼ kms 1 : ð5þ M 1 þ M 2 1AU A orresponding expression is obtained for the seondary by interhanging the subsripts 1 and 2. The peak-to-peak amplitude of the total expeted relative flux variation from the binary due to beaming is then beaming¼ 1 K 1 ½3 1 () Š ;1 K 2 ½3 2 () Š ;2 : ;1 þ ;2 ð6þ Note that beause the observed effet is the differene between the effets of the two stars, the beaming effet vanishes for an equalmass binary, in whih the spetral harateristis are also idential for the two omponents. In order to estimate the ellipsoidal variability, we use the expression presented by Morris & Naftilan (1993) for the peak-topeak ellipsoidal variability of the primary: ;1 0:3 (15 þ u 1)(1 þ 1 ) M 2 R 3 1 : ð7þ ;1 ellips 3 u 1 M 1 a Here 1 is the gravity-darkening oeffiient of the primary and u 1 is its limb-darkening oeffiient. We alulate a similar expression for the seondary and then weight them by the expeted blakbody fluxes in order to obtain the total relative variation of the binary. Morris & Naftilan (1993) also provide a presription for alulating the amplitude of the refletion effet. They assume that eah star absorbs some of the bolometri flux of its ompanion, whih heats the stellar hemisphere faing the ompanion, induing an asymmetri inreased emission. Assuming a blakbody radiation law, Morris & Naftilan define a luminous-effiieny fator by f k ¼ T 4 2 e x 2 1 T 1 e x ; ð8þ 1 1 where T 1 and T 2 are the temperatures of the two omponents. Like the beaming effet, the ontributions of the refletion effets of the two stars are in opposite phase, and the total magnitude of the effet is the weighted differene of the two. Keeping only the leading order terms in the radii (expressed in terms of the orbital separation), we obtain recet¼ 2 3 (R 2 =a) 2 f 1 k ;1 (R 1 =a) 2 f k ;2 ;1 þ ;2 : ð9þ Note that the ellipsoidal and refletion variabilities were alulated assuming tidal loking and a irular orbit. urthermore, we use only the leading order terms in the frational radii. Thus, we might be overestimating the amplitude of those effets. However, these expressions suffie as onservative estimates for omparing with the beaming effets. We use equations (6), (7), and (9) to ompare the three effets for three typial binaries and for a range of periods. Table 1 presents the parameters assumed for the stellar omponents. or the purpose of this simple omparison, we assume u 1 ; u 2 ¼ 0:6 for our hypothetial stars. We alulate the gravity-darkening oeffiients using the presription in Morris (1985). Table 2 ompares the three effets for the three binaries, observed in the V band, for periods of 10 and 100 days. In all ases the beaming variability dominates over the other two effets. igure 1 shows the three effets for an 0 K0 binary for a range of periods. The orresponding plots for the other ases were very similar. The dependene of the effets on the orbital separation is expliit in the expressions above, and we an use it to understand the dependene on the orbital period. While ellipsoidal variability dereases with period as P 2, and the refletion variability as P 4/3,thebeaming variability only dereases as P 1/3, and we expet it to beome dominant for long enough periods. In the three ases we examined the ellipsoidal variability dominates for periods shorter than 8 days, while for periods longer than 10 days the beaming variability beomes dominant. Remarkably, the three lines interset at about the same period, and the refletion effet is almost never dominant. urthermore, the amplitude TABLE 2 The Three Periodi Photometri Effets for Sample Binary Configurations P = 10 days P = 100 days Primary Seondary Ellipsoidal Refletion Beaming Ellipsoidal Refletion Beaming 0 G0 3.9 ; ; ; ; ; ; K0 3.4 ; ; ; ; ; ; 10 4 G0 K0 1.9 ; ; ; ; ; ; 10 4

3 1328 ZUCKER, MAZEH, & ALEXANDER Vol. 670 ig. 1. The three periodi photometri effets for an 0 K0 binary star in a range of periods. of the beaming variability stays at the detetable levels for COROT and Kepler, of mmag for periods of 100 days and more. In igure 2 we ompare the three effets for a range of mass ratios. We assumed a 10 day binary with a G0 primary and used power laws for the dependene of the seondary radius and temperature on its mass: R / M 0:8 and T ea / M 0:55. While the ellipsoidal variability is mostly sensitive to the highest mass ratios, the beaming and refletion effets are more sensitive to intermediary mass ratios. This is mainly beause at the highest mass ratios the effets from both binary omponents are aneled out. 3. DISCUSSION The radial-veloity beaming light urve an yield diretly most of the spetrosopi orbital elements, inluding the period, eentriity, and time of periastron passage. Sine these values will be obtained as the result of a well-defined, magnitude-limited photometri survey, they will provide large amounts of new data for statistial studies of spetrosopi binaries, inluding, e.g., the distribution of orbital period (Duquennoy & Mayor 1991; Mazeh et al. 2006) and the relation between orbital period and eentriity (Halbwahs et al. 2003). The only spetrosopi element that annot be obtained diretly from the light urve is the radial-veloity amplitude K 1. However, asoutlinedinequation(6),thek 1 value an be derived from the amplitude of the beaming effet through the spetral index of the primary and the relative amplitudes of the beaming effet of the two omponents of the binary. In most binaries, the seondary is faint enough that we will be able to asribe the observed beaming variability solely to the primary omponent. If the primary spetral type is known, we an derive its spetral index () and obtain K 1, thus deriving the full set of orbital elements of a single-lined spetrosopi binary. In order to estimate and alibrate the relation between the beaming amplitude and K 1, some spetrosopi follow-up observations of the deteted binaries should be performed. Sine most of the radial-veloity elements will already be known from photometry, only a small number of observations is needed per star. Multiobjet spetrographs, suh as LAMES on the Very Large Telesope (Pasquini et al. 2002) or Hydra on the WIYN telesope (Barden & Armandroff 1995), seem to offer an effiient means to obtain these observations for the deteted beaming binaries in the field. ig. 2. Same as ig. 1, but for a G0 primary in a range of mass ratios. In the few ases in whih the two omponents might have very similar magnitudes and masses, the two ontributions to the beaming variability may anel out beause of their opposite phases. In ases in whih the seondary light will be signifiant but will not anel the primary light ompletely, we will need a photometri analog of spetrosopi disentangling proedures suh as TODCOR (Zuker & Mazeh 1994). Measurements in more than one photometri band may add the onstraints needed to solve for K 1 and K 2. We note in passing that the derivation of K 1, and when possible also K 2, is sensitive to any blending of the binary image with other stars, as they depend on the relative amplitude of the beaming effet. Therefore, it would be neessary to obtain a high-resolution image of the observed field in order to spot any other possible ontributions to the binary light that might dilute the beaming effet. In fat, measurements in different bands may also serve the same purpose. In addition, we propose a simple way to alibrate the relationship between the amplitudes of the radial veloity and the beaming flux variation. Sine the satellite motion is known and is linked with the motion of the Earth, we already have a well-known radialveloity signal in the data for all stars. COROT, for example, will observe dense fields around the elipti ontinuously for almost half a year. Thus, the amplitude of this helioentri veloity signal will be lose to 60 km s 1. The beaming photometri signal assoiated with the motion of the telesope will affet all stars, binary and single alike. Measuring this signal an enable us to alibrate with the radialveloity amplitude, whih in turn an be used to interpret the beaming signal of the stellar orbital veloity, if it is a binary. or single stars is atually a piee of spetral information that reveals the loation of the passband along the blakbody radiation urve and thus provides an estimate of the stellar effetive temperature. igure 3 shows the expeted photometri variability amplitude in V for different temperatures due to helioentri motion alone, assuming a blakbody radiation law and a helioentri radial veloity amplitude of 60 km s 1. igure 4 shows a simulated light urve that demonstrates the type of signal we expet to detet for a binary. The light urve inludes the beaming variability of a 10 day period G0 K0 binary star, together with the beaming variability related to the helioentri motion. The assoiated radial veloity amplitudes are K 1 ¼ 52 and K 2 ¼ 69 km s 1. The noise inluded is only a 10 4 white noise.

4 No. 2, 2007 PHOTOMETRIC BEAMING BINARIES 1329 ig. 3. Amplitude of the expeted beaming variability of a blakbody radiation indued by the helioentri motion of COROT, in the V band, assuming that the star is loated lose to the elipti and the radial veloity amplitude is 60 km s 1. ig. 4. Simulated COROT light urve of a 10 day period G0 K0 binary star. The beaming effets related to the binary orbit and the helioentri motion are easily notied. The dashed line represents the pure helioentri motion beaming signal. Note that in the ase of the helioentri motion the ontributions of the two omponents are in phase and therefore do not anel out. Sine the signal related to the observatory motion is ommon to all the observed objets, it will appear as a systemati effet and may be mistakenly removed as suh. Algorithms suh as SysRem (Tamuz et al. 2005) should identify suh effets, and also the individual response of eah objet to the same effet, and are thus ideally suited to provide the required information for its orret analysis. In the light urve we show in igure 4 we have negleted stellar mirovariability. When atual light urves are analyzed, this variability should be properly aounted for. or older solar-type stars of low hromospheri ativity, we may use the solar mirovariability as the only available example. Although the solar rotational period is about 26 days, Aigrain et al. (2004) show that due to the short lifetimes of the spots and faulae, there is no lear periodi signal in this period. Instead, most of the signal is at higher frequenies, orresponding to periods shorter than 10 days. or nonsolar-type stars, variability due to hromospheri ativity may be larger than solar, and are should be taken in separating the beaming effets from the variability effets. Efforts are urrently underway to haraterize the mirovariability that we expet to observe with the high-preision photometri satellites (e.g., Aigrain et al. 2004; Lanza et al. 2006; Ludwig 2006) in order to failitate the detetion of planetary transits. However, the stellar mirovariability is not expeted to be stritly periodi, and therefore it should be possible to single out the beaming effet, speially beause the shape of the beaming modulation is known and depends only on a few parameters. Spetrosopi information obtained as part of the follow-up observations should also help to further study the hromospheri ativity of the binary andidates. In order to give an order-of-magnitude estimate of the expeted number of beaming binaries that will be deteted by, e.g., COROT, we estimate that 10% of the observed late-type stars have been disovered to be spetrosopi binaries with a threshold of K 1 > 3kms 1 and periods of less than a year or so. This estimate is based on results of the seminal works of Duquennoy & Mayor (1991) and Latham et al. (2002). During the lifetime of the mission, COROT is expeted to monitor about 60,000 stars. Thus, we roughly expet 6000 of them to be detetable as beaming binaries with 10 mag variability or higher. Sine this estimate applies to binaries with late-type primaries, and aounting for the fat that beaming is biased against equal-mass binaries, we an somewhat sale this number downward, and estimate COROT to yield at least 1000 beaming binary stars. Sine the COROT sample of beaming binaries will be disovered by a systemati, magnitudelimited survey, it will augment signifiantly our statistial knowledge of binaries. 4. CONCLUSION As we have shown in x 2, we expet the multitude of very preise light urves of COROT and Kepler to yield hundreds of new binaries through their periodi beaming variability. Thus, a new observational ategory will emerge: beaming binaries. In all types of binaries the disovery and the analysis of the binary motion strongly depend on the timing and the number of the measurements. or the satellite photometri data we expet ontinuous radial veloity data that will yield the spetrosopi orbital elements, inluding period and eentriity in partiular. One the spetral index is known, the radial veloity amplitude an be derived as well. Without the beaming binaries, COROT and Kepler are supposed to find most of the elipsing binaries. However, binaries with periods longer than a few days need very fortuitous geometrial situations to present elipses and are therefore rare in the data of photometri surveys (e.g., Mazeh et al. 2006). The beaming binaries an be deteted up to periods of a hundred days and more. Therefore, the new lass of binaries will extend our detailed knowledge of the statistial harateristis of binaries by an order of magnitude, speially beause these binaries will emerge in the ontext of a well-defined, omplete, magnitude-limited photometri searh. This approah an shed light on the distribution of orbital period (e.g., Duquennoy & Mayor 1991; Mazeh et al. 2006) and the relation between orbital period and eentriity (e.g., Halbwahs et al. 2003). This researh was supported by a grant from the German-Israeli oundation for Sientifi Researh and Development (GI) to T. M.; T. A. is supported by Minerva grant 8563 and a New aulty grant by Sir H. Djangoly, CBE, of London, UK.

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