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		<p>The past decades have seen a great number of discoveries of planets around stars other than our Sun ( Schneider et al. 2011). As some of these planets are of terrestrial nature, the hope of identifying Earth analogs has lead to considerable advances toward the detection of possibly habitable worlds (Borucki 2011;Fordetal.2012). Even though quite frequent in the solar neighborhood (Kiseleva-Eggleton &amp; Eggleton 2001), not many attempts have yet been made to specifically target binary stars in this endeavor. Nonetheless, more than <measure type="value"><num>60</num></measure> planets have already been found in and around such systems (Haghighipour 2010;Doyle et al. 2011;Welsh et al. 2012;Roell et al. 2012;O r o s z et al. 2012a, 2012bDumusque et al. 2012). Although several P-type (circumbinary) planets orbiting both stars of a close binary have also been discovered ( Doyle et al. 2011;W e l s h et al. 2012;Orosz et al. 2012aOrosz et al. , 2012b), most planets are in the so-called S-type ( Rabl &amp; Dvorak 1988) configuration where the planet orbits only one of the binary's stars. A prominent example of an S-type system is α Centauri AB which hosts a terrestrial planet around the fainter binary component, α Cen B ( Dumusque et al. 2012).</p>
		<p>The reason for the general reluctance to include binary systems in the search for terrestrial, habitable planets lies in the assumption that the additional interactions with a massive companion will make planets harder to find. That is primarily because the gravitational interaction between the second star and a planet may alter the planet's orbit significantly and complicate the task of interpreting the planetary signal. One aim of this work is therefore to show that changes in the planet's orbit can actually enhance its detectability (see Section 5). Of course, the orbit of a binary as well as its stellar parameters have to be well determined in order to be able to identify signals from additional terrestrial planets. Sensing the need for a better understanding of binary star systems, efforts have been intensified to improve physical as well as orbital data for nearby binaries (e.g., Torres et al. 2010) and to evolve existing data analysis methodologies ( Chauvin et al. 2011;Haghighipour 2010;Pourbaix 2002;Pourbaix et al. 2002).</p>
		<p>Understanding the complex interactions between a stellar binary and a planet is essential if a system's potential habitability is to be evaluated. For instance, one of the main assumptions of classical habitability, as introduced by Kasting et al. (1993), is that the planet moves around its host star on a circular orbit. This may not be a valid assumption for planets in a binary star system where the gravitational perturbation of the secondary can excite the eccentricity of the planet's orbit (Marchal 1990;Georgakarakos 2002;Eggl et al. 2012). Eggl et al. (2012) found that except for S-type systems where the secondary star is much more luminous than the planet's host star, variations in planetary orbit around the planet-hosting star are the main cause for changes in insolation. Even though Eggl et al. (2012)g a v ea n analytic recipe for calculating the boundaries of the habitable zones (HZs) in S-type binaries, it remains to be seen whether an Earth-like planet in the HZ of a system with <measure type="value"><num>two</num></measure> Sun-like stars will in fact be detectable.</p>
		<p>In order to answer this question, we consider <measure type="value"><num>three</num></measure> techniques, namely, radial velocity (RV), astrometry (AM), and transit photometry (TP), and discuss whether the current observational facilities are capable of detecting habitable planets in such systems. We provide analytical formula for estimating the strength of RV and AM signals for habitable, Earth-like planets, and show that the planet-binary interaction can enhance the chances for the detection of these objects.</p>
		<p>The rest of this article will be structured as follows. In Sections 2 and 3 analytic estimates of the maximum and root mean square (rms) of the strength of an RV and an AM signal that an Earth-like planet produces in an S-type binary configuration will be derived. Section 4 will deal with the consequences of such a setup for TP. We will then briefly recall the different types of HZs for S-type binaries established in Eggl et al. (2012), and use their methodology to identify similar habitable regions in the α Centauri system (Section 5). This system has been chosen because first, it has inspired many studies on the possibility of the formation and detection of habitable planets around its stellar components (Forgan 2012;Guedes et al. 2008;Thébault et al. 2009) and second, Dumusque et al. (2012) have already discovered an Earth-sized planet in a short-period orbit around its secondary star. Therefore, we will compare our RV estimates to the actual signal of α Cen Bb, and study its influence on an additional terrestrial planet presumed in α Cen B's HZ. Finally, in Section 6, the projected RV, AM, and TP trace that terrestrial planets will leave in the HZ of the α Centauri system are analyzed, and the results are discussed within the context of the sensitivity of the current observational facilities.</p>
		<p>To estimate the RV signal that an Earth-like planet produces in an S-type binary system, we will build upon the formalism presented by Beaugée ta l . ( 2007). We assume that the nonplanetary contributions to the host star's RV signal (such as the RV variation caused by the motion of the binary around its center of gravity) are known and have been subtracted, leaving behind only the residual signal due to the planet. The motion of the planet around its host star then constitutes a perturbed <measure type="value"><num>two</num></measure>-body problem, where the gravitational influence of the secondary star is still playing a role and is mirrored in the forced variations of the planet's orbit.</p>
		<p>In practice, the extraction of the planetary signal is all but a trivial task. Even after subtraction of the binary's barycentric and proper motion, the residual will contain contributions from the binary's orbital uncertainties as well as from non-gravitational sources which could be orders of magnitude larger than the star's reflex signal, such as the Rossiter-McLaughlin effect in transiting systems, for example ( Ohta et al. 2005). The discovery of α Cen Bb showed, however, that a substantial reduction of non-planetary RV interference is possible if the respective binary star has been studied in sufficient detail.</p>
		<p>The amplitude of the planet induced RV signal of the host star, V r , is given by</p>
		<p>where K is equal to</p>
		<p>In Equation (2), μ = m 1 /(m 0 + m 1 ) is the planet to star mass ratio with m 1 and m 0 being the masses of the planet and host star, respectively. The planet's mean motion, n p = 2π/P p ,i s given by n p = 񮽙 κ/a 3 with κ = G(m 0 + m 1 ), and P p and G being the planet's orbital period and gravitational constant. The quantities a, e, i, f, and ω in Equations (1) and (2) denote the planet's semimajor axis, eccentricity, orbital inclination relative to the plane of the sky, true anomaly, and argument of periastron, respectively.</p>
		<p>Our goal in this section is to identify the range of the possible peak amplitudes that a terrestrial planet in an S-type binary configuration can produce. We note that the gravitational influence of the second star causes the planet's orbital elements to vary, thus inducing additional time-dependent changes in the RV signal V r (Lee &amp; Peale 2003). While we know from secular perturbation theory that a does not change significantly with Beaugé et al. 2007). <!--encoding problem with °--><!-- The planet is at <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure> with ω = <measure type="value"><num>45</num></measure> • when e 񮽙 = <measure type="value"><num>0</num></measure> -->. Bottom: the amplitude variations of the primary's radial velocity signal due to an Earth-like planet that is subject to the gravitational perturbations of a second star. Both stars are Sun-like with a separation of <measure type="value"><num>20</num> <measure type="LENGTH" unit="AU">AU</measure></measure> and an orbital eccentricity of <measure type="value"><num>0.5</num></measure>. The planet's initial orbit was circular with a semimajor axis of <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure>. Our analytically estimated maximum amplitude V max r is also shown. (A color version of this figure is available in the online journal.) time for hierarchical systems such as the one under consideration (Marchal 1990;Georgakarakos 2003), ω becomes a function of time. We assume coplanar orbits of the planet and the binary star which result in the planet's inclination to the plane of the sky (i) to remain constant. In contrast, the planetary eccentricity will vary between <measure type="value"><num>zero</num></measure> and a maximum e max , where the latter value can be expressed as a function of the system's masses and the binary's orbital parameters ( Eggl et al. 2012). This is important, because the reflex RV signal (V r ) of a star can be increased significantly by planetary orbital eccentricities (Figure 1). Using Equation (1), we identify the global maximum of V r at f = ω = 0, when e = e max. This leads to</p>
		<p>where</p>
		<p>Equation (3) presents a fully analytic estimate of the expected maximum RV signal that a terrestrial planet produces in an S-type binary configuration. <!-- 4 --> As an example for the influence of a double star on a planetary RV signal, the induced variations in the RV of the planet's host star are presented in the bottom panel of Figure 1. The host star is a constituent of a solar-type binary with a semimajor axis of <measure type="value"><num>20</num> <measure type="LENGTH" unit="AU">AU</measure></measure> and an orbital eccentricity of <measure type="value"><num>0.5</num></measure>. Changes in the amplitude of V r are due to variations in the planet's eccentricity.</p>
		<p>Since we do not know the state of the planet's orbital eccentricity at the time of observation, we consider a range for the maximum possible amplitudes of its RV</p>
		<p>Although the range of the amplitude of the host star's RV signal, as given by Equation (5), can be used to identify the "best case" detectability limits, the maximum values of the RV signal due to the planet will be "snapshots" that are reached only during brief moments. As a result, their values for assessing the precision needed to trace fingerprints of an Earth-like planet are rather limited. In such cases, expressions for the rms of the astrometric signal are preferable.</p>
		<p>Since rms values are by convention time-averaged, we substitute f by the mean anomaly M = n p t in all corresponding functions using the equation of the center expansion up to the sixth order in planetary eccentricities (see Appendix A) and average over M and ω. The vastly different rates of change of these quantities ( ˙ M ≫˙ ω) make it possible to consider ω to remain constant during <measure type="value"><num>one</num></measure> cycle of M, so that independent averaging can be performed. In order to eliminate short-term variations in the RV signal, we first average over M. Averaging over ω as well might be desirable if for example the initial state of ω is unknown, or if observations stretch beyond secular evolution timescales of the planets argument of pericenter. We, therefore, define <measure type="value"><num>two</num></measure> different types of rms evaluations for a square-integrable function F:</p>
		<p>and</p>
		<p>Using Equations (6) and (7), the rms values of V r are then given by <!-- 4 --> Larger signals are possible, if the terrestrial planet has a considerable initial eccentricity after its formation and migration phase. Yet, due to the eccentricity dampening in protoplanetary disks, this seems unlikely (Paardekooper &amp; Leinhardt 2010).</p>
		<p>with</p>
		<p>Here we have considered 񮽙a񮽙 M = a since ˙ a ≃ 0 (Marchal 1990; Georgakarakos 2003). Also</p>
		<p>as indicated in Appendix B. It is noteworthy that the averaging over ω causes the rms value of the RV signal to become independent of e so that its difference with the peak signal in the circular case (V circ r ) becomes a mere factor of <measure type="value"><num>1/ √ 2</num></measure>. Thanks to their intricate relation to power spectra, rms values can also be valuable for orbit fitting. The choice of singly or doubly averaged rms relations for this purpose will depend on how many planetary orbital periods are available in the data set. In the case of α Cen Bb, there are order-of-magnitude differences in the rates of change of the mean anomaly ( ˙ M) and the argument of pericenter ( ˙ ω). It would therefore make more sense to assume ω to be constant and add it as a variable in the fitting process. If stronger perturbations or additional forces act on the planet, the periods can be considerably shorter, so that the fully averaged equations might come in handy.</p>
		<p>In order to derive the maximum and rms values for an astrometric signal, we will use the framework presented in Pourbaix (2002). We again assume that the non-planetary contributions have been subtracted from the combined signal of the host star and planet. The projected motion of the planet on the astrometric plane is then given by</p>
		<p>where x E and y E are the Cartesian coordinates of the projected orbit, e is the planet's orbital eccentricity, E is the eccentric anomaly, and A, B, F, and G are the modified Thiele-Innes constants given by</p>
		<p>In these equations, d is the distance between the observer and the observed system in units of the planetary semimajor axis a. We can rewrite Equations (10) in terms of the true anomaly f as</p>
		<p>In these equations, r = a(1 − e 2 )/(1 + e cos f ) represents the planet's radial distance to its host star. Because the motion of the planet itself cannot be traced, we translate these equations into the apparent motion of the host star by the application of Newton's third law. That is</p>
		<p>Here, X and Y are the projected coordinates of the center of mass of the planet-star system, and μ denotes the planet-star mass ratio as defined for Equation (2). Assuming without the loss of generality that the barycenter of the star-planet system coincides with the origin of the associated coordinate system, the distance of the projected stellar orbit to the coordinate center will be equal to</p>
		<p>The right-hand side of Equation (14) is independent of Ω and has a global maximum at f = π, ω = <measure type="value"><num>0</num></measure> when e = e max .This translates into a maximum astrometric amplitude given by</p>
		<p>where</p>
		<p>The planetary maximum AM signal will again lie between ρ circ and ρ max. A remarkable feature of ρ max and ρ circ is their independence of the system's inclination i. This is visualized in Figure 2. The same figure also shows the time evolution of the AM signal due to an Earth-like planet orbiting α Cen B at a distance of <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure>.</p>
		<p>The astrometric rms values are given by</p>
		<p>and</p>
		<p>Details regarding the derivation of Equations (17) and (18) can be found in Appendix B. In contrast to the doubly averaged equations for the rms of an RV signal, Equation (18)s h o w sa dependence on the binary's eccentricity. In cases where the planetary inclination i coincides with the inclination of the binary itself, analytic expressions for 񮽙e 2 񮽙 M,ω are available (Georgakarakos 2003(Georgakarakos , 2005). <!-- 5 5 --> The analytic expressions given in these articles are also averaged over initial phases, i.e., different relative starting positions of the planet and the binary stars.</p>
		<p>, due to an Earth-like planet orbiting its Sunlike host star. <!--encoding problem with ° at the end of this sentence--><!-- The planet's orbital elements are a = <measure type="value"><num>1</num><measure type="LENGTH" unit="AU">A U</measure></measure> ,e = e max = <measure type="value"><num>0.5</num></measure>, ω = <measure type="value"><num>0</num></measure>, Ω = 111 • -->. As shown here, the maximum distance from the origin of the coordinate system is independent of the system's inclination with respect to the plane of the sky (i). Bottom: evolution of the astrometric signal (x f ,y f ) caused by an Earth-like planet in a binary star system. The planet is orbiting α Cen B at a distance of <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure>. The evolution of the astrometric signal is shown for <measure type="value"><num>3750</num></measure> periods of α Centauri AB. Since the system is coplanar, the changes in orientation and shape of the projected ellipse are due to variations in the planet's eccentricity (e) and argument of pericenter (ω).</p>
		<p>In TP, signal strength is equivalent to the relative depth of the dint the planet produces in the stellar light curve during its transit. Assuming that the star-planet configuration allows for occultations, and excluding grazing transits, the fractional depth of the photometric transit (TD) produced by an Earth-like planet is simply given by the proportion of the luminous area of the disk of the star that is covered by the planet as the planet moves between the observer and the star. Ignoring limb darkening, that means, TD ≃ R 2 p /R 2 ⋆ where R p is the radius of the planet and R ⋆ is the stellar radius. The overall probability to observe a transit Table 1 Physical and Orbital Parameters of the α Centauri ABb System ( Kervella et al. 2003;Guedes et al. 2008;Pourbaix et al. 2002;Dumusque et al. 2012)</p>
		<p><!-- 1.519 0.500 --></p>
		<p><measure type="interval"><num type="base">231.65</num> ± <num type="range">0.076</num></measure></p>
		<p><measure type="interval"><num type="base">204.85</num> ± <num type="range">0.084</num></measure></p>
		<p>is given by (Borucki &amp; Summers 1984)</p>
		<p>In Equation (19), r T is the radial distance of the planet to the star during the transit. For an eccentric planetary motion, the planet-star distance during transit can be expressed as et al. 2008), where ¯ ω denotes the argument of pericenter measured from the line of sight. <!-- 6 --> In analogy to Sections 2 and 3, the maximum and averaged transit probability for a planet perturbed by the secondary star in a planar configuration can be calculated by substituting for r T in Equation (19) and averaging over ¯ ω. This will result in</p>
		<p>and</p>
		<p>Equations (20) and (21) indicate that the increase in the eccentricity of the planet due to the perturbation of the secondary increases the probability of transit. In deriving these equations, we have ignored the occultation of the planet by the second star. However, depending on the period ratio between the secondary and the planet, such conjunctions are either scarce or short-lived. Consequently, their contribution to the probability of witnessing a planetary transit is negligibly small.</p>
		<p>In this section, we will show that the previously derived analytic expressions produce results that are in good agreement with the current observations of α Cen Bb. We will also present numerical evidence that the presence-or the absence-of an additional terrestrial planet in the HZ of α Cen B cannot be derived easily from the orbit evolution of α Cen Bb. Consequently, we argue that an independent detection of additional Notes. Coplanarity of the system was assumed. The formal uncertainties have been derived assuming Gaussian error propagation of the uncertainties given in Table 1. The maximum predicted planetary eccentricity for α Cen Bb is e max = <measure type="value"><num>0.003</num></measure>. When taking general relativity into account, however, the orbit of α Cen Bb will remain practically circular (see Figure 6). terrestrial companions might be difficult, but more promising. For this purpose, we will determine the HZ of α Cen B, as well as the RV, AM, and TP signatures of an Earth-like planet orbiting in the HZ of α Cen B. Since there is no a priori reason why the brighter component of α Centauri could not be hosting a terrestrial planet as well, we perform a similar analysis for α Cen A. We will also study the behavior of Equations (1) (21) for a broad range of binary eccentricities.</p>
		<p>The planet discovered around α Cen B offers a perfect opportunity to compare the RV amplitude predictions derived in Section 2 with actual measurements. The planet's known orbital parameters are given in Table 1. In Table 2, we present the analytic estimates of Section 2 applied to the α Centauri ABb system. <!--encoding problem + unit inside <num>--><!-- Assuming the system to be coplanar (i ≃ 79. • 2), the predicted RV amplitude for circular planetary motion (V circ r )is very close to the observed RV amplitude. --> This is not surprising, since the planetary parameters were derived from an RV signal using the same methodology in reverse. While still well within measurement uncertainties, the deviation of the maximum RV amplitude (V max r ) from the observed value is larger than that of V circ r. On the one hand, this might indicate that the planet is currently in an orbital evolution phase where its eccentricity is almost <measure type="value"><num>zero</num></measure>. On the other hand, the planet may be too close to its host star for our model to predict V max r correctly. In fact, we show in Section 5.3 that the latter explanation is more likely, since the influence of general relativity (GR) cannot be neglected in this case. Estimates based on Newtonian physics exaggerate the actual eccentricity of α Cen Bb. Its orbit remains practically circular despite the interaction with the binary star (see Section 5.3 for a detailed discussion). This justifies the assumption of a circular planetary orbit made by Dumusque et al. (2012).</p><commit_msg>correction file hal-01078051<commit_after><tei xmlns="http://www.tei-c.org/ns/1.0">
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		<p>The past decades have seen a great number of discoveries of planets around stars other than our Sun ( Schneider et al. 2011). As some of these planets are of terrestrial nature, the hope of identifying Earth analogs has lead to considerable advances toward the detection of possibly habitable worlds (Borucki 2011;Fordetal.2012). Even though quite frequent in the solar neighborhood (Kiseleva-Eggleton &amp; Eggleton 2001), not many attempts have yet been made to specifically target binary stars in this endeavor. Nonetheless, more than <measure type="value"><num>60</num></measure> planets have already been found in and around such systems (Haghighipour 2010;Doyle et al. 2011;Welsh et al. 2012;Roell et al. 2012;O r o s z et al. 2012a, 2012bDumusque et al. 2012). Although several P-type (circumbinary) planets orbiting both stars of a close binary have also been discovered ( Doyle et al. 2011;W e l s h et al. 2012;Orosz et al. 2012aOrosz et al. , 2012b), most planets are in the so-called S-type ( Rabl &amp; Dvorak 1988) configuration where the planet orbits only one of the binary's stars. A prominent example of an S-type system is α Centauri AB which hosts a terrestrial planet around the fainter binary component, α Cen B ( Dumusque et al. 2012).</p>
		<p>The reason for the general reluctance to include binary systems in the search for terrestrial, habitable planets lies in the assumption that the additional interactions with a massive companion will make planets harder to find. That is primarily because the gravitational interaction between the second star and a planet may alter the planet's orbit significantly and complicate the task of interpreting the planetary signal. One aim of this work is therefore to show that changes in the planet's orbit can actually enhance its detectability (see Section 5). Of course, the orbit of a binary as well as its stellar parameters have to be well determined in order to be able to identify signals from additional terrestrial planets. Sensing the need for a better understanding of binary star systems, efforts have been intensified to improve physical as well as orbital data for nearby binaries (e.g., Torres et al. 2010) and to evolve existing data analysis methodologies ( Chauvin et al. 2011;Haghighipour 2010;Pourbaix 2002;Pourbaix et al. 2002).</p>
		<p>Understanding the complex interactions between a stellar binary and a planet is essential if a system's potential habitability is to be evaluated. For instance, one of the main assumptions of classical habitability, as introduced by Kasting et al. (1993), is that the planet moves around its host star on a circular orbit. This may not be a valid assumption for planets in a binary star system where the gravitational perturbation of the secondary can excite the eccentricity of the planet's orbit (Marchal 1990;Georgakarakos 2002;Eggl et al. 2012). Eggl et al. (2012) found that except for S-type systems where the secondary star is much more luminous than the planet's host star, variations in planetary orbit around the planet-hosting star are the main cause for changes in insolation. Even though Eggl et al. (2012)g a v ea n analytic recipe for calculating the boundaries of the habitable zones (HZs) in S-type binaries, it remains to be seen whether an Earth-like planet in the HZ of a system with <measure type="value"><num>two</num></measure> Sun-like stars will in fact be detectable.</p>
		<p>In order to answer this question, we consider <measure type="value"><num>three</num></measure> techniques, namely, radial velocity (RV), astrometry (AM), and transit photometry (TP), and discuss whether the current observational facilities are capable of detecting habitable planets in such systems. We provide analytical formula for estimating the strength of RV and AM signals for habitable, Earth-like planets, and show that the planet-binary interaction can enhance the chances for the detection of these objects.</p>
		<p>The rest of this article will be structured as follows. In Sections 2 and 3 analytic estimates of the maximum and root mean square (rms) of the strength of an RV and an AM signal that an Earth-like planet produces in an S-type binary configuration will be derived. Section 4 will deal with the consequences of such a setup for TP. We will then briefly recall the different types of HZs for S-type binaries established in Eggl et al. (2012), and use their methodology to identify similar habitable regions in the α Centauri system (Section 5). This system has been chosen because first, it has inspired many studies on the possibility of the formation and detection of habitable planets around its stellar components (Forgan 2012;Guedes et al. 2008;Thébault et al. 2009) and second, Dumusque et al. (2012) have already discovered an Earth-sized planet in a short-period orbit around its secondary star. Therefore, we will compare our RV estimates to the actual signal of α Cen Bb, and study its influence on an additional terrestrial planet presumed in α Cen B's HZ. Finally, in Section 6, the projected RV, AM, and TP trace that terrestrial planets will leave in the HZ of the α Centauri system are analyzed, and the results are discussed within the context of the sensitivity of the current observational facilities.</p>
		<p>To estimate the RV signal that an Earth-like planet produces in an S-type binary system, we will build upon the formalism presented by Beaugée ta l . ( 2007). We assume that the nonplanetary contributions to the host star's RV signal (such as the RV variation caused by the motion of the binary around its center of gravity) are known and have been subtracted, leaving behind only the residual signal due to the planet. The motion of the planet around its host star then constitutes a perturbed <measure type="value"><num>two</num></measure>-body problem, where the gravitational influence of the secondary star is still playing a role and is mirrored in the forced variations of the planet's orbit.</p>
		<p>In practice, the extraction of the planetary signal is all but a trivial task. Even after subtraction of the binary's barycentric and proper motion, the residual will contain contributions from the binary's orbital uncertainties as well as from non-gravitational sources which could be orders of magnitude larger than the star's reflex signal, such as the Rossiter-McLaughlin effect in transiting systems, for example ( Ohta et al. 2005). The discovery of α Cen Bb showed, however, that a substantial reduction of non-planetary RV interference is possible if the respective binary star has been studied in sufficient detail.</p>
		<p>The amplitude of the planet induced RV signal of the host star, V r , is given by</p>
		<p>where K is equal to</p>
		<p>In Equation (2), μ = m 1 /(m 0 + m 1 ) is the planet to star mass ratio with m 1 and m 0 being the masses of the planet and host star, respectively. The planet's mean motion, n p = 2π/P p ,i s given by n p = 񮽙 κ/a 3 with κ = G(m 0 + m 1 ), and P p and G being the planet's orbital period and gravitational constant. The quantities a, e, i, f, and ω in Equations (1) and (2) denote the planet's semimajor axis, eccentricity, orbital inclination relative to the plane of the sky, true anomaly, and argument of periastron, respectively.</p>
		<p>Our goal in this section is to identify the range of the possible peak amplitudes that a terrestrial planet in an S-type binary configuration can produce. We note that the gravitational influence of the second star causes the planet's orbital elements to vary, thus inducing additional time-dependent changes in the RV signal V r (Lee &amp; Peale 2003). While we know from secular perturbation theory that a does not change significantly with Beaugé et al. 2007). The planet is at <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure> with ω = <measure type="value"><num>45</num> <measure type="TEMPERATURE" unit="°">•</measure></measure> when e 񮽙 = <measure type="value"><num>0</num></measure>. Bottom: the amplitude variations of the primary's radial velocity signal due to an Earth-like planet that is subject to the gravitational perturbations of a second star. Both stars are Sun-like with a separation of <measure type="value"><num>20</num> <measure type="LENGTH" unit="AU">AU</measure></measure> and an orbital eccentricity of <measure type="value"><num>0.5</num></measure>. The planet's initial orbit was circular with a semimajor axis of <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure>. Our analytically estimated maximum amplitude V max r is also shown. (A color version of this figure is available in the online journal.) time for hierarchical systems such as the one under consideration (Marchal 1990;Georgakarakos 2003), ω becomes a function of time. We assume coplanar orbits of the planet and the binary star which result in the planet's inclination to the plane of the sky (i) to remain constant. In contrast, the planetary eccentricity will vary between <measure type="value"><num>zero</num></measure> and a maximum e max , where the latter value can be expressed as a function of the system's masses and the binary's orbital parameters ( Eggl et al. 2012). This is important, because the reflex RV signal (V r ) of a star can be increased significantly by planetary orbital eccentricities (Figure 1). Using Equation (1), we identify the global maximum of V r at f = ω = 0, when e = e max. This leads to</p>
		<p>where</p>
		<p>Equation (3) presents a fully analytic estimate of the expected maximum RV signal that a terrestrial planet produces in an S-type binary configuration. <!-- 4 --> As an example for the influence of a double star on a planetary RV signal, the induced variations in the RV of the planet's host star are presented in the bottom panel of Figure 1. The host star is a constituent of a solar-type binary with a semimajor axis of <measure type="value"><num>20</num> <measure type="LENGTH" unit="AU">AU</measure></measure> and an orbital eccentricity of <measure type="value"><num>0.5</num></measure>. Changes in the amplitude of V r are due to variations in the planet's eccentricity.</p>
		<p>Since we do not know the state of the planet's orbital eccentricity at the time of observation, we consider a range for the maximum possible amplitudes of its RV</p>
		<p>Although the range of the amplitude of the host star's RV signal, as given by Equation (5), can be used to identify the "best case" detectability limits, the maximum values of the RV signal due to the planet will be "snapshots" that are reached only during brief moments. As a result, their values for assessing the precision needed to trace fingerprints of an Earth-like planet are rather limited. In such cases, expressions for the rms of the astrometric signal are preferable.</p>
		<p>Since rms values are by convention time-averaged, we substitute f by the mean anomaly M = n p t in all corresponding functions using the equation of the center expansion up to the sixth order in planetary eccentricities (see Appendix A) and average over M and ω. The vastly different rates of change of these quantities ( ˙ M ≫˙ ω) make it possible to consider ω to remain constant during <measure type="value"><num>one</num></measure> cycle of M, so that independent averaging can be performed. In order to eliminate short-term variations in the RV signal, we first average over M. Averaging over ω as well might be desirable if for example the initial state of ω is unknown, or if observations stretch beyond secular evolution timescales of the planets argument of pericenter. We, therefore, define <measure type="value"><num>two</num></measure> different types of rms evaluations for a square-integrable function F:</p>
		<p>and</p>
		<p>Using Equations (6) and (7), the rms values of V r are then given by <!-- 4 --> Larger signals are possible, if the terrestrial planet has a considerable initial eccentricity after its formation and migration phase. Yet, due to the eccentricity dampening in protoplanetary disks, this seems unlikely (Paardekooper &amp; Leinhardt 2010).</p>
		<p>with</p>
		<p>Here we have considered 񮽙a񮽙 M = a since ˙ a ≃ 0 (Marchal 1990; Georgakarakos 2003). Also</p>
		<p>as indicated in Appendix B. It is noteworthy that the averaging over ω causes the rms value of the RV signal to become independent of e so that its difference with the peak signal in the circular case (V circ r ) becomes a mere factor of <measure type="value"><num>1/ √ 2</num></measure>. Thanks to their intricate relation to power spectra, rms values can also be valuable for orbit fitting. The choice of singly or doubly averaged rms relations for this purpose will depend on how many planetary orbital periods are available in the data set. In the case of α Cen Bb, there are order-of-magnitude differences in the rates of change of the mean anomaly ( ˙ M) and the argument of pericenter ( ˙ ω). It would therefore make more sense to assume ω to be constant and add it as a variable in the fitting process. If stronger perturbations or additional forces act on the planet, the periods can be considerably shorter, so that the fully averaged equations might come in handy.</p>
		<p>In order to derive the maximum and rms values for an astrometric signal, we will use the framework presented in Pourbaix (2002). We again assume that the non-planetary contributions have been subtracted from the combined signal of the host star and planet. The projected motion of the planet on the astrometric plane is then given by</p>
		<p>where x E and y E are the Cartesian coordinates of the projected orbit, e is the planet's orbital eccentricity, E is the eccentric anomaly, and A, B, F, and G are the modified Thiele-Innes constants given by</p>
		<p>In these equations, d is the distance between the observer and the observed system in units of the planetary semimajor axis a. We can rewrite Equations (10) in terms of the true anomaly f as</p>
		<p>In these equations, r = a(1 − e 2 )/(1 + e cos f ) represents the planet's radial distance to its host star. Because the motion of the planet itself cannot be traced, we translate these equations into the apparent motion of the host star by the application of Newton's third law. That is</p>
		<p>Here, X and Y are the projected coordinates of the center of mass of the planet-star system, and μ denotes the planet-star mass ratio as defined for Equation (2). Assuming without the loss of generality that the barycenter of the star-planet system coincides with the origin of the associated coordinate system, the distance of the projected stellar orbit to the coordinate center will be equal to</p>
		<p>The right-hand side of Equation (14) is independent of Ω and has a global maximum at f = π, ω = <measure type="value"><num>0</num></measure> when e = e max .This translates into a maximum astrometric amplitude given by</p>
		<p>where</p>
		<p>The planetary maximum AM signal will again lie between ρ circ and ρ max. A remarkable feature of ρ max and ρ circ is their independence of the system's inclination i. This is visualized in Figure 2. The same figure also shows the time evolution of the AM signal due to an Earth-like planet orbiting α Cen B at a distance of <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure>.</p>
		<p>The astrometric rms values are given by</p>
		<p>and</p>
		<p>Details regarding the derivation of Equations (17) and (18) can be found in Appendix B. In contrast to the doubly averaged equations for the rms of an RV signal, Equation (18)s h o w sa dependence on the binary's eccentricity. In cases where the planetary inclination i coincides with the inclination of the binary itself, analytic expressions for 񮽙e 2 񮽙 M,ω are available (Georgakarakos 2003(Georgakarakos , 2005). <!-- 5 5 --> The analytic expressions given in these articles are also averaged over initial phases, i.e., different relative starting positions of the planet and the binary stars.</p>
		<p>, due to an Earth-like planet orbiting its Sunlike host star. The planet's orbital elements are a = <measure type="value"><num>1</num><measure type="LENGTH" unit="AU">A U</measure></measure> ,e = e max = <measure type="value"><num>0.5</num></measure>, ω = <measure type="value"><num>0</num></measure>, Ω = <measure type="value"><num>111</num> <measure type="TEMPERATURE" unit="°">•</measure></measure>. As shown here, the maximum distance from the origin of the coordinate system is independent of the system's inclination with respect to the plane of the sky (i). Bottom: evolution of the astrometric signal (x f ,y f ) caused by an Earth-like planet in a binary star system. The planet is orbiting α Cen B at a distance of <measure type="value"><num>1</num> <measure type="LENGTH" unit="AU">AU</measure></measure>. The evolution of the astrometric signal is shown for <measure type="value"><num>3750</num></measure> periods of α Centauri AB. Since the system is coplanar, the changes in orientation and shape of the projected ellipse are due to variations in the planet's eccentricity (e) and argument of pericenter (ω).</p>
		<p>In TP, signal strength is equivalent to the relative depth of the dint the planet produces in the stellar light curve during its transit. Assuming that the star-planet configuration allows for occultations, and excluding grazing transits, the fractional depth of the photometric transit (TD) produced by an Earth-like planet is simply given by the proportion of the luminous area of the disk of the star that is covered by the planet as the planet moves between the observer and the star. Ignoring limb darkening, that means, TD ≃ R 2 p /R 2 ⋆ where R p is the radius of the planet and R ⋆ is the stellar radius. The overall probability to observe a transit Table 1 Physical and Orbital Parameters of the α Centauri ABb System ( Kervella et al. 2003;Guedes et al. 2008;Pourbaix et al. 2002;Dumusque et al. 2012)</p>
		<p><!-- 1.519 0.500 --></p>
		<p><measure type="interval"><num type="base">231.65</num> ± <num type="range">0.076</num></measure></p>
		<p><measure type="interval"><num type="base">204.85</num> ± <num type="range">0.084</num></measure></p>
		<p>is given by (Borucki &amp; Summers 1984)</p>
		<p>In Equation (19), r T is the radial distance of the planet to the star during the transit. For an eccentric planetary motion, the planet-star distance during transit can be expressed as et al. 2008), where ¯ ω denotes the argument of pericenter measured from the line of sight. <!-- 6 --> In analogy to Sections 2 and 3, the maximum and averaged transit probability for a planet perturbed by the secondary star in a planar configuration can be calculated by substituting for r T in Equation (19) and averaging over ¯ ω. This will result in</p>
		<p>and</p>
		<p>Equations (20) and (21) indicate that the increase in the eccentricity of the planet due to the perturbation of the secondary increases the probability of transit. In deriving these equations, we have ignored the occultation of the planet by the second star. However, depending on the period ratio between the secondary and the planet, such conjunctions are either scarce or short-lived. Consequently, their contribution to the probability of witnessing a planetary transit is negligibly small.</p>
		<p>In this section, we will show that the previously derived analytic expressions produce results that are in good agreement with the current observations of α Cen Bb. We will also present numerical evidence that the presence-or the absence-of an additional terrestrial planet in the HZ of α Cen B cannot be derived easily from the orbit evolution of α Cen Bb. Consequently, we argue that an independent detection of additional Notes. Coplanarity of the system was assumed. The formal uncertainties have been derived assuming Gaussian error propagation of the uncertainties given in Table 1. The maximum predicted planetary eccentricity for α Cen Bb is e max = <measure type="value"><num>0.003</num></measure>. When taking general relativity into account, however, the orbit of α Cen Bb will remain practically circular (see Figure 6). terrestrial companions might be difficult, but more promising. For this purpose, we will determine the HZ of α Cen B, as well as the RV, AM, and TP signatures of an Earth-like planet orbiting in the HZ of α Cen B. Since there is no a priori reason why the brighter component of α Centauri could not be hosting a terrestrial planet as well, we perform a similar analysis for α Cen A. We will also study the behavior of Equations (1) (21) for a broad range of binary eccentricities.</p>
		<p>The planet discovered around α Cen B offers a perfect opportunity to compare the RV amplitude predictions derived in Section 2 with actual measurements. The planet's known orbital parameters are given in Table 1. In Table 2, we present the analytic estimates of Section 2 applied to the α Centauri ABb system. <!--unit (•) inside <num>--><!-- Assuming the system to be coplanar (i ≃ 79. • 2), the predicted RV amplitude for circular planetary motion (V circ r )is very close to the observed RV amplitude. --> This is not surprising, since the planetary parameters were derived from an RV signal using the same methodology in reverse. While still well within measurement uncertainties, the deviation of the maximum RV amplitude (V max r ) from the observed value is larger than that of V circ r. On the one hand, this might indicate that the planet is currently in an orbital evolution phase where its eccentricity is almost <measure type="value"><num>zero</num></measure>. On the other hand, the planet may be too close to its host star for our model to predict V max r correctly. In fact, we show in Section 5.3 that the latter explanation is more likely, since the influence of general relativity (GR) cannot be neglected in this case. Estimates based on Newtonian physics exaggerate the actual eccentricity of α Cen Bb. Its orbit remains practically circular despite the interaction with the binary star (see Section 5.3 for a detailed discussion). This justifies the assumption of a circular planetary orbit made by Dumusque et al. (2012).</p>
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