CALCULATING THE HABITABLE ZONE OF BINARY STAR SYSTEMS. I. S-TYPE BINARIES

The locations of the boundaries of the HZ depend on the flux of the star at the orbit of the planet. In a binary star system where the planet is subject to radiation from two stars, the flux of the secondary star has to be added to that of the primary (planet-hosting star) and the total flux can then be used to calculate the boundaries of the binary HZ. However, because the response of a planet's atmosphere to the radiation from a star depends strongly on the star's SED, a simple summation of fluxes is not applicable. The absorbed fraction of the absolute incident flux of each star at the top of the planet's atmosphere will differ for different SEDs. The planet's Bond albedo increases with the star's effective temperature (T_Star) because for stars with higher values of T_Star, more stellar photons are deposited at the top of planet's atmosphere in the short wavelengths where Rayleigh scattering in a planet's atmosphere is very efficient (see the definition of the spectral weight factor in Section 2.2). That increases the amount of reflected stellar light for hotter stars. As a result, the absolute incident stellar flux at the top of the planetary atmosphere that leads to a similar absorbed stellar flux and surface temperature for similar planetary atmospheres is larger for hotter stars (see Section 2.2 and Figure 2). Therefore in order to add the absorbed flux of two different stars and derive the limits of the binary HZ, one has to weight the flux of each star according to the star's SED. The relevant flux received by a planet in this case is the sum of the spectrally weighted stellar flux, separately received from each star of the binary, as given by

$$\begin{eqnarray} {F_{\rm Pl}}(f,{T_{\Pr }},{T_{\rm Sec}})&=& {W_{\rm Pr}}(f,{T_{\rm Pr}})\, {{{L_{\rm Pr}}({T_{\rm Pr}})}\over {r_{\rm Pl-Pr}^2}}\nonumber\\ &&+ {W_{\rm Sec}}(f,{T_{\rm Sec}})\, {{{L_{\rm Sec}}({T_{\rm Sec}})}\over {r_{\rm Pl-Sec}^2}}\,. \end{eqnarray} \tag{ 1 }$$

In this equation, F_Pl is the total flux received by the planet, L_i and T_i (i = Pr, Sec) represent the luminosity and effective temperature of the primary and secondary stars, f is the cloud fraction of the planet's atmosphere, and W_i(f, T_i) is the binary stars' spectral weight factor. The quantities r_Pl-Pr and r_Pl-Sec in Equation (1) represent the distances between the planet and the primary and secondary stars, respectively. In using Equation (1), we normalize the weighting factor to the flux of the Sun.

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From Equation (1), the boundaries of the HZ of the binary can be defined as distances where the total flux received by the planet is equal to the flux that Earth receives from the Sun at the inner and outer edges of its HZ. Since the planet revolves around one star of the binary in an S-type system, we determine the inner and outer edges of the HZ with respect to the planet-hosting star. As mentioned before, it is customary to consider the primary of the system to be stationary, and calculate the orbital elements with respect to the stationary primary star (see bottom panel of Figure 1). In the rest of this paper, we will follow this convention and consider the planet-hosting star to be the primary star as well. In that case, the range of the HZ of the binary can be obtained from

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

In Equation (2), the quantity l_x represents the inner and outer edges of the HZ with x = (in,out). As mentioned earlier, the values of l_in and l_out are model-dependent and change for different values of cloud fraction, f, and atmosphere composition.

To calculate the spectral weight factor W(f,T) for each star of the binary depending on their SEDs, we calculate the stellar flux at the top of the atmosphere of an Earth-like planet at the limits of the HZ, in terms of the stellar effective temperature. To determine the locations of the inner and outer boundaries of the HZ of a main-sequence star with an effective temperature of 2600 K < T_Star < 7200 K, we use Equation (3) (see Kopparapu et al. 2013a)

$$\begin{equation} {l_{\rm x-Star}}\,=\,{l_{\rm x-Sun}}\, \left[{{L/{L_{\rm Sun}}}\over {1+{\alpha _{\rm x}} ({T_i})\,{l_{\rm x-Sun}^2}}} \right ]^{1/2}. \end{equation} \tag{ 3 }$$

In this equation, l_x = (l_in, l_out) is in AU, T_i(K) = T_Star(K) − 5780, and

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

where the values of coefficients a_x, b_x, c_x, d_x, and l_x-Sun are given in Table 1 (see Kopparapu et al. 2013b). From Equation (3), the flux received by the planet from its host star at the limits of the HZ can be calculated using Equation (5). The results are given in Table 1;

$$\begin{equation} {F_{\rm x-Star}}(f, {T_{\rm Star}})\,=\,{F_{\rm x-Sun}}(f,{T_{\rm Star}})\, \left [1\,+\,{\alpha _{\rm x}}({T_i})\,{l_{\rm x-Sun}^2}\right]. \end{equation} \tag{ 5 }$$

From Equation (5), the spectral weight factor W(f,T) can be written as

$$\begin{equation} {W_i}(f,{T_i})\,=\,\left [1\,+\,{\alpha _{\rm x}}({T_i})\, {l_{\rm x-Sun}^2}\,\right ]^{-1}. \end{equation} \tag{ 6 }$$

Table  2 and Figure 3 show W(f,T) as a function of the effective temperature of a main sequence planet-hosting star for the narrow (top panel) and empirical (bottom panel) boundaries of the HZ. As expected, hotter stars have weighting factors smaller than one whereas the weighting factors of cooler stars are larger than one.

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To use Equation (2) to calculate the boundaries of the HZ, we assume here that the orbit of the (fictitious) Earth-like planet around its host star is circular. In a close binary system, the gravitational effect of the secondary may deviate the motion of the planet from a circle and cause its orbit to become eccentric. In a binary with a given semimajor axis and mass-ratio, the eccentricity has to stay within a small to moderate level to avoid strong interactions between the secondary star and the planet and to allow the planet to maintain a long-term stable orbit (with a low eccentricity) in the primary's HZ. The binary eccentricity itself is constrained by the fact that in highly eccentric systems, periodic close approaches of the two stars truncate their circumstellar disks depleting them from planet-forming material (Artymowicz & Lubow 1994) and restricting the delivery of water-carrying objects to an accreting terrestrial planets in the binary HZ (Haghighipour & Raymond 2007).

This all indicates that in order for the binary to be able to form a terrestrial planet in its HZ, its eccentricity cannot have large values. In a binary with a small eccentricity, the deviation of the planet's orbit from circular is also small and appears in the form of secular changes with long periods (see, e.g., Eggl et al. 2012). Therefore, to use Equation (2), one can approximate the orbit of the planet by a circle without the loss of generality.

The habitability of a planet in a binary system also requires long-term stability in the HZ. For a given semimajor axis a_Bin, eccentricity e_Bin, and mass-ratio μ of the binary, there is an upper limit for the semimajor axis of the planet beyond which the perturbing effect of the secondary star will make the orbit of the planet unstable. This maximum or critical semimajor axis (a_Max) is given by (Rabl & Dvorak 1988; Holman & Wiegert 1999)

$$\begin{eqnarray} {a_{\rm Max}}&=&{a_{\rm Bin}}\left (0.464 - 0.38\, \mu - 0.631\, {e_{\rm Bin}} + 0.586 \, \mu \, {e_{\rm Bin}}\right. \nonumber\\ &&\left.+ 0.15 \, {e_{\rm Bin}^2} - 0.198\, \mu \,{e_{\rm Bin}^2}\right). \end{eqnarray} \tag{ 7 }$$

In Equation (7), μ = m₂/(m₁ + m₂) where m₁ and m₂ are the masses of the primary (planet-hosting) and secondary stars, respectively. One can use Equation (7) to determine the maximum binary eccentricity that would allow the planet to have a stable orbit in the HZ (l_out ⩽ a_Max). For any smaller value of the binary eccentricity, the entire HZ will be stable. We will present a detailed discussion of this topic in Sections 3.2 and 3.3, where we calculate the boundaries of the HZ of the α Centauri and HD 196885 systems, respectively.

To explore the maximum effect of the binary semimajor axis and eccentricity on the contribution of one star to the extent of the HZ around the other component, we consider three extreme cases: an M2-M2 (Figure 4, top), an F0-F0 (Figure 4, bottom), and an F8-M1 (Figure 5) binary. We consider the M2 and F0 stars to have effective temperatures of 3520 (K) and 7300 (K), respectively, and their luminosities to be 0.035 and 6.56 of that of the Sun for our general examples here.

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We first study the effect of a secondary on the single-star HZ around the primary star. Figures 4 and 5 show the maximum contribution of the secondary to the stellar flux at the limits of the single star's HZ of the primary, when the secondary star and a fictitious rocky planet are at their closest separation, and for different values of the binary eccentricity and semimajor axis. The color coding in these figures corresponds to the contribution of the flux received from the secondary relative to that of the primary star (note the multiplication factor on the lower right corner of each panel). Figure 4 shows secondary's flux at the inner (left) and outer (right) edge of the primary's single-star empirical HZ, for a M2-M2 binary (top) and for a F0-F0 binary (bottom). The contribution of the secondary can be several times larger than the primary's at closest approach. The contribution of the secondary is larger at the outer edge of the primary's HZ than at the inner edge and increases with decreasing periastron distance of the binary. Note that the actual flux contribution of the secondary that determines the binary HZ limits is the secondary's flux averaged over the planet's full orbit and as such it will be smaller than the secondary's flux at closest approach.

To explore the effect of the secondary for a binary with a hot and a cool star, we use an F-M system (see Figure 5). The top panels of Figure 5 show the maximum flux of the M1V secondary star at the outer edges of the F8V primary's single-star narrow (left) and empirical HZs (right) when the secondary is at its closest distance (i.e., during its periastron passage). The bottom panels show the maximum flux of the F8V at the outer edges of the single-star narrow and empirical HZ of the M1V star. As shown by these panels, the flux of the brighter star has a stronger contribution to the total flux at the outer edge of the primary's HZ (up to several times the primary's flux). Figures 4 and 5 do not consider the orbital (in)stability of the fictitious Earth-like planet. When in an individual case, the stability criterion as shown in Equation (7) is imposed, the closest distance between the secondary and planet, which ensures the orbital stability of the planet as well, will be larger than the closest distance shown in these figures, and as a result, the contribution of the secondary star to the total flux received by the planet is smaller.

As a second step, to demonstrate the effect of the secondary on the boundaries of the HZ, we calculate the binary HZ of the systems mentioned above, considering the minimum value of the binary semimajor axis for which the outer edge of the primary's empirical HZ will be on the stability limit. Figures 6 shows the results for the case of the three binaries described above, assuming circular orbits, (top) an M2-M2 (left), an F0-F0 (right) and (bottom) an F8-M1 S-type binary, with (left) the F8 star and (right) the M1 star being the planet-hosting star. The top panel of Figure 6 shows that the secondary does not have a noticeable effect on the extent of the HZ. The binary HZ around each star is equivalent to its single-star HZ for the M2-M2 and F0-F0 S-type systems when stability up to the outer region of the empirical HZ is maintained. Also as expected, the effect of the M1 star on the extension of the single-star HZ around the F8 star is negligible. However, at its closest distances, the F8 star can extent the limit of the single-star HZ around the M1 star so far out that at the binary periastron, the two HZs merge.

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To further explore the effect of binary eccentricity in a system with a hot and cool star, we carried out similar calculations as those in Figure 6 for the F8-M1 binary, assuming the binary eccentricity to be 0.3. Figures 7 and 8 show the results for four different relative positions of the two stars to show the changing influence of the secondary over the binary's orbit. The primary and planet-hosting star is chosen to be the M star (Figure 7) and the F star (Figure 8). Figure 7 shows that a luminous secondary can have considerable effects on the shape and location of the HZ. However, a cool and less luminous secondary will not change the limits of the HZ (Figure 8). The figures also demonstrate why the exact boundaries of the binary HZ can only be calculated when the relative position of the planet to the stars is known—and the flux received by the planet over one full planetary orbit can be calculated.

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The α Centauri system consists of the close binary α Cen AB and a farther M dwarf companion known as Proxima Centauri at approximately 15,000 AU. The semimajor axis of the binary is ∼23.5 AU and its eccentricity is ∼0.518. The component A of this system is a G2V star with a mass of 1.1 M_Sun, luminosity of 1.519 L_Sun, and an effective temperature of 5790 K. Its component B has a spectral type of K1V and its mass, luminosity and effective temperature are equal to 0.934 M_Sun, 0.5 L_Sun, and 5214 K, respectively.

The recent announcement of a probable super-Earth planet with a mass larger than 1.13 Earth-masses around α Cen B indicates that unlike the region around α Cen A where terrestrial planet formation encounters complications, planet formation is efficient around this star (Guedes et al. 2008; Thébault et al. 2009) and it could also host a terrestrial planet in its HZ. Here we assume that planet formation around both stars of this binary can proceed successfully and they both can host Earth-like planets. We calculate the spectral weight factors of both α Cen A and B (Table 2) and, using the minimum and maximum added flux of the secondary star, estimate the limits of their binary HZs using Equation (2).

Table 3 shows the estimates of the boundaries of the binary HZ around each star. The terms Max and Min in this table correspond to the planet–binary configurations of (θ, ν) = (0, 0) and (0, 180°) (see Figure 1 for the definition of θ and ν) where the planet receives the maximum and minimum total flux from the secondary star, respectively. As shown here, each star of the α Centauri system contributes to increasing the limits of the binary HZ around the other star. Although these contributions are small, they extend the estimated limits of the binary HZ by a noticeable amount. This is primarily due to the high luminosity of α Cen A and the relatively large eccentricity of the binary which brings the two stars as close as ∼11.3 AU from one another (and as such making planet formation very difficult around α Cen A).

Generalizing our study, we examined the effect of increasing eccentricity in an α Cen-like system (with similar G2V and K1V stars) by calculating the critical distance around both stars of this binary for which the entire HZ will be dynamically stable (i.e., l_out ⩽ a_Max). For the G2V star, the stability limit is given by

$$\begin{equation} {a_{\rm Max}}\,=\,23.5 \left (0.2892 \,-\, 0.361446\, {e_{\rm Bin}}\,+\, 0.05892\, {e_{\rm Bin}^2}\right), \end{equation} \tag{ 8 }$$

and for the K1V star, it is equal to

$$\begin{equation} {a_{\rm Max}}\,=\,23.5 \left (0.2584 \,-\, 0.313974\, {e_{\rm Bin}}\,+\, 0.042882\, {e_{\rm Bin}^2}\right). \end{equation} \tag{ 9 }$$

The maximum value of the binary eccentricity for which an Earth-like planet around the G2V star can have a stable orbit at the outer boundary of the narrow HZ is 0.62. For the empirical HZ, this maximum eccentricity reduces to 0.59. For all values of the binary eccentricity smaller than 0.62 (0.59), the entire narrow (empirical) HZ around the G2V star will be stable. For the K1V star, the maximum binary eccentricity that allows the entire HZ to be stable is 0.73 for the narrow and 0.71 for the empirical binary HZs.

Given the eccentricity of the α Cen binary (e_Bin = 0.518), both narrow and nominal HZs for α Cen A and B are stable. Figure 9 shows the maximum effective flux of a G2V star received by an Earth-like planet in the HZ of the K1V star during the secondary's (i.e., the G2V star) periastron passage. Note that this is a short-lived flux that can be buffered by the planet's atmosphere and reduces the secondary's heating effect in its closest approach. The maximum flux of the secondary is only used here to estimate the maximum shift from the single-star HZ to the binary HZ.

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In the α Cen system, where the binary eccentricity is 0.518, the stability limit around the primary G2V star is at ∼2.768 AU. This limit is slightly exterior to the outer boundary of the star's narrow and empirical HZs. Although the latter suggests that the HZ of α Cen A is dynamically stable, the close proximity of this region to the stability limit may have strong consequences on the actual formation of an Earth-like planet in this region (see, e.g., Thébault et al. 2008; Eggl et al. 2012).

HD 196885 is a close main sequence S-type binary system with a semimajor axis of 21 AU and eccentricity of 0.42 (Chauvin et al. 2011). The primary of this system (HD 169885 A) is an F8V star with a T_Star of 6340 K, mass of 1.33 M_Sun, and luminosity of 2.4 L_Sun. The secondary star (HD 196885 B) is an M1V dwarf with a mass of 0.45 M_Sun. Using the mass–luminosity relation L ∼ M^3.5, where L and M are in solar units, the luminosity of this star is approximately 0.06 L_Sun and we consider its effective temperature to be T_Star = 3700 K. The primary of HD 196885 hosts a Jovian-type planet suggesting that the mass-ratio and orbital elements of this binary allow planet formation to proceed successfully around its primary star. We assume that terrestrial planet formation can also successfully proceed around both stars of this binary and can result in the formation of Earth-sized objects.

To estimate the boundaries of the binary HZ of this eccentric system, we ignore its known giant planet and use Equation (2) considering a fictitious Earth-like planet in the HZ. We calculate the spectral weight factor W(f,T) for both stars of this system (Table 2) and estimate the locations of the inner and outer boundaries of the binary's HZ (Table 3).

As expected (because of the large periastron distance of the binary and the secondary star being a cool M dwarf), even the maximum flux from the secondary star does not have a noticeable contribution to the location of the HZ around the F8V primary of HD 196886. However, being a luminous F star, the primary shows a small effect on the location of the HZ around the M1V secondary star (Table 3).

Generalizing our study, we examined the effect of increasing eccentricity in an HD 196885-like system (with similar F8V and M1V stars) by calculating the critical distance around both stars of this binary for which the entire HZ will be dynamically stable (i.e., l_out ⩽ a_Max). Around the F8V star, the stability limit is given by

$$\begin{equation} {a_{\rm Max}}\,=\,21\left(0.36786\,-\, 0.482742\,{e_{\rm Bin}}\,+\, 0.1\, {e_{\rm Bin}^2}\right). \end{equation} \tag{ 10 }$$

Around the M1V secondary, the critical distance is at

$$\begin{equation} {a_{\rm Max}}\,=\,21 \left(0.18014 \,-\, 0.193258\, {e_{\rm Bin}}\,+\, 0.002\, {e_{\rm Bin}^2}\right). \end{equation} \tag{ 11 }$$

Using Equation (7) for the case when the planet orbits the F8V star, the maximum value of e_Bin for which the binary HZ will be stable is 0.59 for the narrow and 0.57 for the empirical HZs. This suggests that for all values of e_Bin ⩽ 0.57, the HZ of the F8V star will be stable. Similar calculations for the HZ around the M1V star indicate that the upper value of the binary eccentricity for which the narrow and empirical HZs of the M1V star will be stable are e_Bin ⩽ 0.82 and e_Bin ⩽ 0.81, respectively. Figure 10 shows the maximum effective flux of each star of the binary received by an Earth-like planet in the HZ of the other star during their periastron passages. Similar to the case of α Cen binary, this is a short-lived maximum flux that can be buffered by the planet's atmosphere and would reduce the secondary's heating effect at its closest distance. The maximum flux of the secondary is only used here to estimate the maximum shift of the HZ. In practice, the influence of a secondary in an S-type binary on the extent of the binary HZ around the other component has to be determined in a case by case basis for the specific geometry of the system.

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