CALCULATING THE HABITABLE ZONE OF BINARY STAR SYSTEMS. II. P-TYPE BINARIES

As mentioned in the Introduction, to determine the boundaries of the HZ, we adopt the same model and methodology as in Paper I. We consider a circumstellar HZ to be an annulus around a star where an Earth-like planet with a CO₂/H₂O/N₂ atmosphere and sufficient amount of water can permanently maintain liquid water on its solid surface. We also assume that similar to Earth, geophysical cycles in this planet regulate the amount of its atmospheric CO₂ and H₂O, and as a result, the locations of the inner and outer boundaries of its HZ will be associated with CO₂– and H₂O–dominated atmospheres, respectively. On such a planet, the concentration of atmospheric CO₂ varies inversely with the surface temperature. In this paper, we adopt the model recently proposed by Kopparapu et al. (2013a, 2013b) as the model for the Sun's HZ without cloud feedback. In this model, a narrow HZ is defined as the region with an inner boundary at the Runaway Greenhouse limit and an outer boundary at the distance where Earth will develop its Maximum Greenhouse effect. Note that the inner and outer boundaries of a narrow HZ are also without cloud feedback. We use the distances presented by these authors as Recent Venus and Early Mars for an empirical (nominal) HZ. These boundaries have been derived from the fluxes received by Mars at 3.5 Gy and by Venus at 1.0 Gy, at the times when each planet does not show indications for liquid water on its surface (Kasting et al. 1993).

The locations of the boundaries of the HZ, and therefore the capability of a planet in maintaining conditions for habitability, depend on the total flux received at the top of the planet's atmosphere. Since the atmosphere converts stellar insolation to temperature structure and surface temperature of a planet, its interaction with the stellar radiation plays an important role in considering whether a planet can maintain liquid water on its surface and allow for detectable habitable conditions.

This interaction strongly depends on the stellar SED, implying that stars with different energy distributions will contribute differently to the absolute incident flux at the top of the planet's atmosphere. To determine the contribution of a star taking into account its SED, we define the quantity spectral weight factor W_i(f, T_i), where i = (Pr, Sec) denoting the primary and secondary stars, respectively, T_i is the star's effective temperature, and f represents the atmosphere's cloud fraction. As shown in Paper I and following Kopparapu et al. (2013a),

$$\begin{equation} {W_i}(f,T_i)=\left[1+\alpha _{\rm x}({T_i}) l_{\rm x-Sun}^2\right]^{-1}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} {\alpha _{\rm x}}({T_i})= {a_{\rm x}}{T_i}+{b_{\rm x}}{T_i^2}+ {c_{\rm x}}{T_i^3}+{d_{\rm x}}{T_i^4}. \end{equation} \tag{ 2 }$$

In these equations, x = (in,out) denotes the inner and outer boundaries of the HZ, l_x-Sun = (l_in-Sun, l_out-Sun) are in AU, and T_i(K) = T_Star(K) − 5780. The values of coefficients a_x, b_x, c_x, and d_x depend on the boundaries of the Sun's HZ and are given in Table 1 (Kopparapu et al. 2013b). The graphs of the corresponding spectral weight factors of these models are shown in Figure 2.

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Given the concept of spectral weight factor as introduced in the previous section, the contribution of each star of the binary to the total flux at the top of the planet's atmosphere can be written as W_i(f, T_i) F_i(f, T_i). Here ${F_i}={L_i}/{r_{{\rm Pl}-i}^2}$ denotes the stellar flux and L_i is the luminosity of each star of the binary. The total flux received by the planet is therefore given by

$$\begin{equation} {W_{\rm Pr}}(f,{T_{\rm Pr}})\, {{{L_{\rm Pr}}({T_{\rm Pr}})}\over {r_{\rm Pl-Pr}^2}}\,+\, {W_{\rm Sec}}(f,{T_{\rm Sec}})\, {{{L_{\rm Sec}}({T_{\rm Sec}})}\over {r_{\rm Pl-Sec}^2}}\,, \end{equation} \tag{ 3 }$$

where r_Pl-Pr and r_Pl-Sec are the distances between the planet and the primary and secondary stars, respectively. Defining HZ as a region where the total flux received by an Earth-like planet at that top of its atmosphere is equal to that of Earth received from the Sun, the boundaries of the HZ of the binary can be calculated from

$$\begin{equation} {W_{\rm Pr}}(f,{T_{\rm Pr}})\, {{{L_{\rm Pr}}({T_{\rm Pr}})}\over {l_{\rm x-Bin}^2}}\,+\, {W_{\rm Sec}}(f,{T_{\rm Sec}})\, {{{L_{\rm Sec}}({T_{\rm Sec}})}\over {r_{\rm Pl-Sec}^2}}\,=\, {{L_{\rm Sun}}\over {l_{\rm x-Sun}^2}}\,. \end{equation} \tag{ 4 }$$

In deriving this equation, we have assumed that the primary star is at the center of the coordinates system (see the lower panel of Figure 1) and the boundaries of the HZ of the binary (l_x-Bin) are measured with respect to this star.

As shown by Equation (4), calculations of the boundaries of HZ require calculating the distance between the secondary star and the Earth-like planet (r_Pl-Sec). As shown by the bottom panel of Figure 1, this quantity can be written as

$$\begin{equation} {r_{\rm Pl-Sec}^2}\,=\, {r_{\rm Pl-Pr}^2}\,+\,{r_{\rm Bin}^2}\,-\, 2\,{r_{\rm Pl-Pr}}\, {r_{\rm Bin}}\, \cos (\nu -\theta)\,, \end{equation} \tag{ 5 }$$

where ν is the true anomaly of the binary, ${r_{\rm Bin}}={a_{\rm Bin}}(1-{e_{\rm Bin}^2})/1+{e_{\rm Bin}}\cos \nu$, and r_Pl-Pr = l_x-Bin. In a circumbinary system, where the planet is subject to the gravitational forces of two stars, the angle θ can be calculated using planet's equation of motion. Recently, Leung & Lee (2013) have presented an elegant analytical treatment of the orbit of a planet in a P-type system. While the methodology and solutions presented by these authors can be used to calculate θ, we present here an approach that is more efficient when used in numerical computations.

In general, the equation of motion of a planet in a circumbinary orbit, as shown in the lower panel of Figure 1, can be written as

$$\begin{equation} {{d^2}\over {d{t^2}}}\,{\boldsymbol {r_{\rm Pl-Pr}}}\,=\, -G\,{{M_{\rm Pr}}\over {r_{\rm Pl-Pr}^3}}\,{\boldsymbol {r_{\rm Pl-Pr}}}\, -G\,{{M_{\rm Sec}}\over {r_{\rm Pl-Sec}^3}}\,{\boldsymbol {r_{\rm Pl-Sec}}}. \end{equation} \tag{ 6 }$$

In Equation (6), ${\boldsymbol {r_{{\rm Pl}-i}}}$ are the vectors connecting the planet to the binary stars, M_i represents stellar masses, and G is the gravitational constant. Considering a plane–polar coordinates system (which would be consistent with the co-planarity of Kepler circumbinary planets), Equation (6) can be written as

$$\begin{equation} {P_r}={\dot{r}}, \end{equation} \tag{ 7 }$$

$$\begin{equation} {P_\theta }={r^2}{\dot{\theta }}, \end{equation} \tag{ 8 }$$

$$\begin{equation} {{\dot{P}}_r}={{{\dot{P}}_\theta }\over {r^3}}-{1\over {r^2}}- {\varepsilon \over {|{\boldsymbol {r-{r_{\rm Bin}}}}|^3}} [r-{r_{\rm Bin}}\cos (\nu -\theta)], \end{equation} \tag{ 9 }$$

$$\begin{equation} {{\dot{P}}_\theta }=-\varepsilon {{r\,{r_{\rm Bin}}}\over {|{\boldsymbol {r-{r_{\rm Bin}}}}|^3}} \sin (\nu -\theta), \end{equation} \tag{ 10 }$$

where ${\boldsymbol {r}}={\boldsymbol {{r_{\rm Pl-Pr}}}}$, ε = M_Sec/M_Pr and we have set GM_Pr = 1. The boundaries of the HZ of the binary can be calculated by simultaneously solving Equations (4) and (7)–(10), substituting for r_pl-Sec from Equation (5).

The solutions of Equations (4) and (7)–(10) present the actual motion of a planet in a circumbinary orbit. In the context of calculating the binary's HZ, the latter means that unlike the usual practice of constraining the orbit of the fictitious Earth-like planet to a circle, the orbit of this planet will be affected by the dynamics of the binary and may not stay circular. As a result, as shown in the next section, depending on the contribution of each star and the binary eccentricity, the shape of the (instantaneous) HZ may also deviate from a circle and may vary during the motion of the binary.

The binary eccentricity will also play an important role in the formation and stability of the planet. As shown by Artymowicz & Lubow (1994), the interaction between the binary and a disk of circumbinary objects causes the disk to be truncated and lose some of its planet-forming material. As a result of the disk truncation, the inner edge of the disk will move out to the location of the boundary of the stability given by (Dvorak 1986; Dvorak et al. 1989; Holman & Wiegert 1999)

$$\begin{eqnarray} {a_{\rm Min}}&=& {a_{\rm Bin}} \left(1.60 + 5.10 {e_{\rm Bin}}+ 4.12 \mu - 2.22 {e_{\rm Bin}^2}\right.\nonumber\\ &&\left.-4.27 \mu {e_{\rm Bin}}-5.09 {\mu ^2}+ 4.61 {\mu ^2}{e_{\rm Bin}^2}\right). \end{eqnarray} \tag{ 11 }$$

In this equation, μ is the binary mass-ratio and is equal to M_Sec/(M_Pr + M_Sec). As shown by Equation (11), the location of the stability region is strongly affected by the eccentricity of the binary. In binaries with large eccentricities, this region moves to large distances implying that (depending on the spectral types of the stars of the binary) the system may not be able to form potentially habitable planets. One can use Equation (11) to determine the range of the binary eccentricity for which a planet in the HZ of the binary will be stable. For instance, in the case of the narrow HZ for a binary with a given mass-ratio μ, the lower (upper) limit of the eccentricity corresponds to a value for which the boundary of stability coincides with the inner (outer) edge of the HZ.

The Kepler 16 binary system (Doyle et al. 2011) consists of a 0.69 solar-mass (M_☉) primary with a radius of 0.65 solar radii (R_☉) and an effective temperature of 4450 K. The secondary of this system is an M dwarf with a mass of 0.20 M_☉ and radius of 0.23 R_☉. The binary semimajor axis in this system is 0.22 AU and its eccentricity is 0.16. We calculated the luminosity of the primary star using

$$\begin{equation} {{L_\ast }\over {L_\odot }}\,=\,\left ({{R_\ast }\over {R_\odot }}\right)^2\, \left ({{T_\ast } \over {T_\odot }}\right)^4, \end{equation} \tag{ 12 }$$

where L_*, R_*, and T_* represent the stellar luminosity, radius, and effective temperature, respectively. Using T_☉ = 5780 K as the Sun's effective temperature, the luminosity of Kepler 16A is equal to 0.148 L_☉. To calculate the luminosity of the secondary star, we used the mass-luminosity relation $L\, {\sim }\, 0.23\, {M_\ast ^{2.3}}$ where M_* is the mass of the star (Duric 2004). From this relationship, the secondary has a luminosity of 0.0057 L_☉, which, combined with Equation (12), results in an effective temperature of ∼3311 K for this star.

Table 2 shows the values of the spectral weight factors of the stars of Kepler 16 binary system and boundaries of its HZ. As shown here, the outer boundaries of the narrow and empirical HZs are larger than 0.7 AU. As a point of comparison, the semimajor axis of the currently known circumbinary planet of this system is ∼0.705 AU (Doyle et al. 2011) and the limit of its planetary orbit stability (i.e., the distance beyond which the orbit of a circumbinary planet will be stable) obtained from Equation (11) is at 0.63 AU. Given the almost circular orbit of this planet (e ∼ 0.007), this implies that the giant planet of this system has a long-term stable orbit close to the outer edge of its HZ.

To determine to what extent the motion of the secondary star affects the location of the HZ, we calculated the narrow and empirical HZs of the system for one complete orbital motion of the secondary. Movies of the time-dependent locations of these HZs can be found at http://astro.twam.info/hz-ptype/. As an example, we show four snapshots of the evolution of the HZ for one orbit of the binary in Figure 3. The left column in this figure shows a top view of the HZ and the right column shows the changes in the locations of the boundaries of the HZ. From top to bottom, the panels correspond to the true anomaly of the secondary equal to 0, 60°, 120°, and 180°, respectively. As shown here, both the inner and outer boundaries of the HZ extend slightly outward at closest distances to the secondary star. Figure 3 also shows (in red) the boundary of the stability of planetary orbits around the binary and (in blue) the orbit of the currently known planet of this system. As shown here, a large portion of the HZ of the Kepler 16 binary is interior to its critical distance, indicating that the most of the HZ of this system is unstable.

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The eclipsing binary Kepler 34 consists of two Sun-like stars. The primary of this system has a mass of 1.048 M_☉ with a bolometric luminosity of 1.49 L_☉ and an effective temperature of 5913 K. The mass of the secondary star is 1.021 M_☉, its bolometric luminosity is 1.28 solar, and it has an effective temperature of 5867 K (Welsh et al. 2012). The semimajor axis of this binary is ∼0.23 AU and it has an eccentricity of ∼0.521, making Kepler 34 the most eccentric planet-hosting binary star system known to date (Welsh et al. 2012). Table 3 shows the spectral weight factors for the primary and secondary stars and the locations of the inner and outer boundaries of the binary's HZ. As expected, because the stars of this binary are Sun-like, the distance of the HZ to the primary star is much larger than the stellar separation. However, due to the large binary eccentricity in this system, the effect of the orbital motion of the secondary on the locations of the boundaries of the HZ, although smaller than that in Kepler 16, is still noticeable. Figure 4 shows this for the narrow and empirical HZs and for the same values of the true anomaly as in Figure 3. The changes in the locations of the boundaries of the binary HZ over the binary's full orbit can be seen more clearly in the right panels. Figure 4 also shows the boundary of the orbital stability around Kepler 34 (at ∼0.84 AU) and the orbit of its currently known planet. As shown here, the combination of the small semimajor of the binary and that its stars are of solar type has placed the entire HZ of this system in the stable region. As shown in Table 3, only when the empirical HZ is considered, the circumbinary planet of Kepler 34, with a semimajor axis of 1.089 AU and an eccentricity of 0.18 (Welsh et al. 2012) spends a small portion of its orbit in the binary's empirical HZ.

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The Kepler 35 system is an (almost) equal-mass binary with a 0.89 M_☉ primary and a secondary star with a mass of 0.81 M_☉. The bolometric luminosity of the primary of this system is 0.94 L_☉ and it has an effective temperature of 5606 K. The secondary star in this system has a bolometric luminosity of 0.41 L_☉ and its effective temperature is 5202 (Welsh et al. 2012). The semimajor axis of the Kepler 35 binary is ∼0.18 AU and its eccentricity is 0.14. Table 4 shows the spectral weight factors and the distances of the inner and outer boundaries of the HZ of this binary system. Similar to the case of Kepler 34, the HZ of Kepler 35 is at distances much larger than the separation of its two stars. This, combined with the small eccentricity of this binary, has caused the effect of the orbital motion of its secondary on the displacement of the inner and outer boundaries of its HZ to be small. Figure 5 shows this in more detail. In this figure, the narrow and empirical HZs of the system are shown when the secondary star is at its periastron and apastron distances. Figure 5 also shows that, as in the system of Kepler 34, the stability limit for circumbinary planetary orbits is at a much closer distance (∼0.51 AU), indicating that planets in the HZ of Kepler 35 will have stable orbits. The circumbinary planet of this system, with its semimajor axis equal to 0.6 AU and its eccentricity equal to 0.04 (Welsh et al. 2012), although stable, is not in the binary HZ.

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The binary star Kepler 38 consists of a moderately evolved 0.95 M_☉ star as its primary and a low-mass (0.25 M_☉) stellar companion in an approximately 18.8 day orbit as its secondary. The semimajor axis of the binary is ∼0.15 AU and its eccentricity is 0.10 (Orosz et al. 2012a). Using Equation (12) and the values of the radius (∼ 1.76 R_☉) and effective temperature (5623 K) of the primary as reported by Orosz et al. (2012a), the luminosity of this star is approximately equal to 2.77 L_☉. We calculated the luminosity of the secondary star in a similar way, using the value of its radius (0.27 R_☉) and the ratio of its effective temperature to that of the primary (0.59) as given by Orosz et al. (2012a). As expected, this star has a much smaller luminosity, equal to 0.008 L_☉. Table 5 shows the values of the spectral weight factors and the boundaries of the HZ of this system. Given the significantly smaller luminosity of the secondary star, the location and width of the HZ around this binary are mainly due to the radiation from the primary star. As shown in Figure 6, the orbital motion of the secondary does not cause noticeable changes in the inner and outer boundaries of its narrow and empirical HZs. The stability limit of the Kepler 38 binary is at 0.40 AU, implying that the HZ of this system is in the stable region. However, the orbit of the currently known circumbinary planet of this system is entirely interior to the inner boundary of its HZ.

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The binary system of Kepler 47 is a unique case study in the sense that it presents the first binary system with multiple planetary companions (Orosz et al. 2012b). The primary of this system is a solar-mass star (1.043 M_☉) with a radius of 0.964 R_☉, effective temperature of 5636 K, and luminosity of 0.84 L_☉. The secondary of this system is a 0.362 solar-mass M star with a radius, effective temperature, and luminosity of 0.3506 R_☉, 3357 K, and 0.014 L_☉, respectively (Orosz et al. 2012b). The orbit of the binary is almost circular (e_Bin = 0.02) and its semimajor axis is approximately 0.08 AU. Kepler 47 is host to two Neptune-sized planets b and c with semimajor axes of 0.29 AU and 0.99 AU, respectively (Orosz et al. 2012b).

Table 6 shows the values of the spectral weight factors and boundaries of the HZ of the system. Our calculations indicate that, similar to the case of Kepler 38, the HZ of this system is primarily due to the insolation received from the primary star and the effect of the secondary, although larger than that of Kepler 38, is still negligible. Figure 7 shows the narrow and empirical HZs of this system and the boundary of planetary stability at 0.19 AU. As shown here, the HZ of the system is dynamically stable. Figure 7 and Table 6 also show that the outer boundaries of the narrow and empirical HZs of the system are slightly larger than 1.5 AU, indicating that the majority of the orbit of planet c in this system is in the HZ. We would like to note that being a Neptune-sized planet, Kepler 47c cannot be habitable. However, it may harbor terrestrial-class habitable moons and Trojan planets that may be able to develop and sustain conditions for habitability (see, e.g., Kaltenegger 2010; Haghighipour et al. 2013 and references therein).

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Kepler 64 is an F-M binary with a semimajor axis of ∼0.17 AU and eccentricity of ∼0.21. The primary of this system (F star) has a mass of 1.528 M_☉, radius of 1.734 R_☉, and an effective temperature of 6407 K (Schwamb et al. 2013). The mass of the secondary star is 0.408 M_☉, its radius is 0.378 R_☉, and its has an effective temperature of 3561 K (Schwamb et al. 2013). From these quantities and using Equation (12), the luminosities of the primary and secondary stars of this system are 4.54 L_☉ and 0.02 L_☉, respectively. These luminosities indicate that the primary of Kepler 64 will the dominating star in establishing the location of the binary's HZ. Table 7 and Figure 8 show the values of the spectral weight factors and the boundaries of the narrow as well as empirical HZs for this system. As shown here, the orbital motion of the secondary does not have noticeable effects on the location of the HZ. Figure 8 also shows the critical distance for the orbital stability (at 0.52 AU), which indicates that the HZ of Kepler 64 system is entirely stable. A comparison between the locations of the boundaries of the HZ in this system and the orbit of its circumbinary planet (semimajor axis of 0.63 AU and eccentricity of 0.05) shows that the HZ of Kepler 64 is much farther out than the orbit of its currently known planet.

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