Habitable zones with stable orbits for planets around binary systems

Abstract

1 INTRODUCTION

The discovery of hundreds of extrasolar planets has hurled the topic of planetary studies into centre stage: planet formation, planetary dynamics, planetary geology, etc.; among these studies, the question of planet habitability is of great interest, even if no hard data beyond Earth exist, yet. The quest for life on other planets started long ago when in the 1960, Frank Drake used the 85-foot radius telescope in West Virginia, hoping to detect an extraterrestrial signal (Drake 1961). However, extraterrestrial life, if it indeed exists in the neighbourhood of the Sun, is most likely of a basic type (e.g. microbial). Although a new emerging view is that planets very different to Earth may have the right conditions for life, which increases future chances of discovering an inhabited world (Seager 2013), it is a good first step along this endeavour to find out if conditions propitious to Earth-like planetary life (the only type we know for sure so far) exists.

Even in the case of Earth, life exists in many diverse environments, from scorching deserts to the eternal darkness of the ocean depths, and even deep within Earth's crust. Confronted with this bewildering diversity of environments, we must look for the most basic ingredients, like liquid water. On the other hand, our observational limitations to detect the most important habitability indicator: water vapour on terrestrial-like exoplanets, reduces our possibilities of finding habitable planets. Up to now, we have been able to characterize habitable zones mainly around single stars, consequently, the quest for habitable worlds has been mostly limited to this type of stars. But given the fact that most stars appear to be part of binary and multiple systems, we must extend the habitable-zone concept to those cases.

The condition for the existence of liquid water on the surface of a planet (Bains 2004) is the usual defining condition for what is now called the circumstellar habitable zone (CHZ). Hart (1979) studied the limits of the CHZ in late type stars and concluded that K dwarfs have a narrow CHZ and later dwarfs have none at all. However, other studies that include among other things, atmospheric radiative transfer modelling, have come with a more optimistic and, rather complicated outlook (e.g. Doyle, Billingham & Devincenzi 1998). In this case, habitability seems to be a very planet-specific matter (Seager 2013). For example, the habitable-zone calculations defined for a dry rocky planet, with a minimum inner edge of about 0.5 au, for a solar-like host star (Zsom, Seager & de Wit 2013), out to 10 au for a planet with an H2 atmosphere and no interior energy, around a solar-like host star (Pierrehumbert & Gaidos 2011), and even possibly out to free-floating habitable planets, with no host star, for planets with thick H2 atmospheres (Stevenson 1999).

For the case of planets in binary stellar systems, the usual radiative condition used to define habitable zones around single stars must be extended to the case of two sources of illumination whose relative positions change in time. Additionally, a condition of dynamical stability must be added, as the more complex and time varying potential turns unstable whole swaths of phase-space. For a planet to be a possible adobe of life, both conditions must be satisfied, and for a long time, enough for life to develop.

In this work, we will use Kopparapu et al. (2013) definition of habitable zone around single stars and extend it to the case of binary stars. As such, this is a simple definition of the habitable zone based on stellar flux limits. Some of the more complicated (and realistic) effects that pertain to particular situations can be incorporated into this simple criterion by the use of ‘corrective factors’ applied to the stellar fluxes that define the limits of the habitable zones. An example of this is provided by e.g. Kaltenegger & Haghighipour (2013), who introduce a factor called ‘spectrally weighted flux’.

Studies like the previous ones have been applied traditionally to single stars or brown dwarfs; however, most low-mass main-sequence stars are members of binary or multiple systems (Duquennoy & Mayor 1991; Fischer & Marcy 1992), and in particular in the solar neighbourhood, the fraction goes up to 50 per cent (Abt 1983; Raghavan et al. 2010). This suggests that binary formation is at least as probable as single star formation processes (Mathieu 1994). Additionally, several types of planets have been discovered in circumstellar orbits, even in close binary systems, where the effect of the stellar companion might be of great importance (Queloz et al. 2000; Hatzes et al. 2003; Zucker et al. 2004; Correia et al. 2005; Muterspaugh et al. 2010; Chauvin et al. 2011; Dumusque et al. 2012), and also in circumbinary discs (Doyle et al. 2011; Orosz et al. 2012a,b; Welsh et al. 2012; Schwamb et al. 2013). For a review on the subject see Haghighipour et al. (2010) and Kaltenegger & Haghighipour (2013).

We provide in this work a general theoretical formulation to calculate the equivalent of the Kopparapu et al. (2013) habitable zone for single stars, but for binary systems (see Section 2). This formulation is expressed in formulae and interpolating tables that the reader can apply to any specific case. Our work also considers binary systems with no restriction on the eccentricity of the binary orbits and guarantees that orbits within the computed habitable zones remain so, at all binary orbital phases. We only consider approximately circular habitable zones. For the dynamical stability condition, we use the work of Pichardo, Sparke & Aguilar (2005) and Pichardo, Sparke & Aguilar (2009), who found the extent of stable, non-intersecting orbits around binary systems, based on the existence of sturdy structures in the extended phase-space of the system (the so-called invariant loops). We have then applied both conditions to a sample of main-sequence binary systems in the solar neighbourhood with known orbital parameters and present our results. These constitute regions where we think it is most likely to find planets suitable for life in these binary systems.

Several other interesting papers have been submitted on this subject recently, for circular binary systems (Cuntz 2014), and for the general case of elliptical binary systems (Haghighipour 2010; Eggl et al. 2012, 2013; Haghighipour & Kaltenegger 2013; Kaltenegger & Haghighipour 2013; Kane & Hinkel 2013; Müller & Haghighipour 2014). However, all these formulations take as the stability criterion that of Holman & Wiegert (1999). The empirical approximation of Holman & Wiegert to the stability problem in binaries was an excellent and fundamental first approximation to the solution. Their work was based on a detailed trial and error technique to find the most stable (long-term) orbits in discs around binaries. The authors defined as stable orbits mostly those that would keep in their orbits for about 10⁴ binary periods (in general not enough to ensure life emergency and development though). On the other hand, and as the authors point out explicitly in their work, their formulation for stability regions breaks towards higher eccentricity binaries due to the rapidly increasing difficulty to recover stable orbits from this methodology.

In this paper, we rely on the invariant loops theory, which by definition are invariant structures in the extended phase-space of the system. The existence of invariant loops guarantees the stability of orbits for all times and not just during the integration span, as long as the binary parameters (stellar masses, semimajor axis and orbital eccentricity) do not change. Their stability is given by the constants of motion that support them. The orbit integration is only used to identify these structures in extended phase-space (for details see Arnold 1984; Lichtenberg & Lieberman 1992; Maciejewski & Sparke 1997, 2000; Pichardo et al. 2005). Because of its nature, this formulation has no restrictions in eccentricity or mass ratio. With this criterion we search for the intersection between the stable regions for planets and a short, straightforward, and also general formulation for habitability. Additionally, it is relevant to mention here that the use of the invariant loops criterion shows, for example in the circumbinary discs case, important differences in the position of stable regions (Pichardo et al. 2009) with respect to previous work, that should be taken into account in habitability calculations. We explain this in detail along this paper.

In the second part of this work, we also provide a sample of binary systems of the solar neighbourhood (the whole sample of binary stars, in the main sequence, with all orbital parameters known at the present time in literature), with their habitable zones computed using the approach in this work for habitability and planetary orbital stability.

This paper is organized as follows. In Section 2, the radiative condition for habitability is extended from the usual circumstellar case, and a detailed description of the formulation used to calculate the radiative safe zones is provided. In Section 3, we present the dynamical stability condition for habitability used in this work. In Section 4, the combined conditions are used to compute habitability zones for particular cases and the construction of a table of binaries, with known orbital parameters, for stars in the main sequence. Finally, our conclusions are given in Section 5.

2 RADIATIVE CONDITION FOR HABITABLE ZONES IN BINARY STARS

The CHZ thus defined is a thick spherical shell that surrounds the star, whose thickness and size depend on the star's luminosity. However, an important fraction of stars in the solar neighbourhood are part of binary systems. The fact that planets have already been discovered within binary systems makes it necessary to extend the simple definition of the CHZ to the stellar binary case.

In this section, we extend the simple CHZ condition for single stars given by the previous equation, to the case of stellar binary systems. In this case, we have two sources of luminosity at positions that change with the binary orbital phase. It may be thought that in this case it is simply a matter of applying the equation twice, once for each star. However, this naive approach is not correct, since there may be regions where, although within the individual CHZ for each star, the combined irradiance of both stars may push the region out of the combined habitable zone. Additionally, it is fundamental to add the condition of orbital stability, as both, the correct irradiance and orbital stable regions should have a non-empty intersection during the entire binary orbital phase, for planets to be able to exist within a proper binary habitable zone (BHZ). In Fig. 1, we present a schematic figure that shows the combined concept to construct habitable zones in a binary star: the radiative safe zone in grey circles (upper half of the diagram), and the stable regions for orbits to settle down (lower half of the diagram). Note how the demand that both conditions (radiative and dynamical) are met severely restricts the resulting BHZ.

Figure 1.

Illustrative diagram for the concept of BHZ. The stars are on the x-axis (black dots whose size is proportional to their luminosity – upper half, or mass – lower half). The upper part of the diagram shows the flux isophotes (equation 2) with the boundaries of the radiative safe zone shown with thick black lines and the zone itself shaded in grey. The lower part shows the equipotentials, with the circumbinary and circumstellar zones of dynamical stability sown in grey. Our definition of BHZ is shown as the annular green region, where both conditions are met. In this case, there is a circumbinary and a circumsecondary habitable zone.

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To illustrate the concept of BHZ, we show in Fig. 1 a schematic diagram that presents the radiative condition (upper half) and dynamical condition (lower half). The stars are the black dots on the x-axis, whose size is proportional to the assumed luminosity (upper half), or mass (lower half). In the upper half, isopleths of constant combined stellar flux are shown, with those corresponding to the boundaries of the raditaive safe zone shown with thick black lines. The radiative safe zone is shaded in grey. Likewise, in the lower half of the diagram, equipotentials are shown with the circumbinary and two circumstellar dynamical safe zones in grey. Note that the saddle point between the stars for isopleths and equipotentials does not coincide, as the former is set by the relative luminosities, whereas the latter by their mass ratio.

We define our BHZ as the annular regions (shown in green) where both conditions are met. For the circumbinary habitable zone, it is an annular region centred at the barycentre of the system. For the CHZ, they are annular regions centred in the corresponding star. Note that in this particular example, there is a circumbinary and a circumsecondary habitable zones, but not a circumprimary. In this section, we will tackle the radiative condition alone, leaving the condition of dynamical stability for the next section and the combined effect for Section 4.

The edges of the radiative safe zone are set by two stellar flux isopleths and the resulting shape is more complicated than that of the CHZ. Fig. 2 illustrates a particular example: the grey region is the radiative safe zone. Dashed lines indicate possible planetary orbits that are not fit for life, while solid lines indicate safe orbits (we have approximated the planetary orbits as circles centred in either star, or the barycentre). Note that for an orbit to be safe, it must remain within the radiative safe zone at all times. In this particular case, there are safe circumprimary and circumsecondary planetary orbits, but no circumbinary ones.

Figure 2.

Safe (continuous) and unsafe (dashed) planetary orbits. The radiative safe zone is the grey region. The planetary orbits are simply circles centred on either star, or the centre of mass of the system. In this case, no circumbinary safe orbits are possible.

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In addition to the above complication, the separation between the stars will vary for the general case of binaries with elliptical orbits. First, we will tackle the simpler case of circular orbit binaries.

2.1 Binaries in circular orbits

2.1.1 The critical flux isopleth

Figure 3.

Various flux isopleths are shown for a binary with q = 0.5, λ = 0.1. The thick line is the critical isopleth. The plot origin is at the barycentre frame and only the positive y-axis is shown.

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Fig. 4 shows the value of the critical flux isopleth as a function of the secondary luminosity fraction. Note that |${\cal I}_{\rm c}$| does not depend on the mass ratio q. This is because the latter only shifts the isopleths on the orbital plane (see equations 3). The curve is also symmetric with respect to λ_s = 1/2. This is because λ can refer to either star (the more massive star is not necessarily the most luminous one). So we drop the s subindex from now on.

Figure 4.

Critical isopleth flux value as a function of the secondary star luminosity fraction. Note that |${\cal I}_{\rm c}$| is symmetric with respect to λ_s = 0.5.

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2.1.2 The three different radiative zone configurations

Depending on the value of the dimensionless stellar fluxes that define the CHZ (equation 1) with respect to the critical isopleth flux, the radiative safe zone may have three different configurations:

• Configuration I: (⁠|${\cal I}_{\rm c}< {\cal I}_{\rm o}$|⁠). This corresponds to two separate circumstellar radiative safe zones.

• Configuration II: (⁠|${\cal I}_{\rm o}< {\cal I}_{\rm c}< {\cal I}_{\rm i}$|⁠). This corresponds to a combined case, where we find a circumbinary radiative safe zone with two inner holes around the stars.

• Configuration III: (⁠|${\cal I}_{\rm i}<{\cal I}_{\rm c}$|⁠). This corresponds to a single radiative safe zone with a single inner hole.

These configurations are illustrated in Fig. 5.

Figure 5.

The three different radiative zone configurations. The illustration is for a binary with λ = 0.4.

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In any of these configurations, safer planetary orbits are the ones better contained within the safe radiative zone. In the most strict sense, this means that we must consider their shape, which introduces an added complexity we will not get into. Since stable orbital planetary configurations in binaries have in general low eccentricities (smaller to ∼0.3; Pichardo et al. 2005), we will define a radiative binary habitable zone (rBHZ) as the largest circular annular region that can fit entirely within the radiative safe zone. We consider two cases: circumstellar and circumbinary zones. Orbits that are safe for life will lay most likely within these zones. We should remember that these zones are defined by the radiative condition only. In next section, we will consider the dynamical condition, and in Section 4, we will combined both conditions to define the combined BHZ.

Now, a problem arises because the flux isopleths are not circles (see Fig. 3), particularly close to the critical isopleth. Our definition of BHZ implies that the largest safe orbit must be entirely inscribed within the flux isopleth that defines the outer boundary of the safe zone. Similarly, the smallest safe orbit must be entirely circumscribed outside the inner boundary flux isopleth. Note that this condition is not over the planetary orbit itself but just in order to fix the boundaries for habitability, in such way that it is possible to ensure that a planet will remains all the time inside this zone.

The study can be separated depending on the configuration of the radiative safe zones, we now examine these conditions for each case.

2.1.3 Configuration I (⁠|${\cal I}_{\rm c}< {\cal I}_{\rm o}$|⁠)

In this case, we have two separate CHZ. The orbital radius of the largest safe planetary orbit is given by the smallest distance between the outer boundary isopleth and the corresponding star. This ensures that the orbit will remain within the outer boundary isopleth.

Similarly, for the radius of the smallest, circumstellar safe orbit, we must now consider the maximum distance from either star to the corresponding inner boundary flux isopleth, so the orbit remains outside it at all times.

The circumstellar isopleths are elongated along the η_x axis in the direction of the other star, while they are squashed in the opposite direction. This means that both, the outer and inner circumstellar orbital radii (r_os and r_is, respectively) correspond to the respective boundary isopleth intersections with the η_x axis (see Fig. 6).

Figure 6.

Circumstellar boundaries of the rBHZ (configuration I). The boundary isopleth is the dashed line. If this isopleth corresponds to the outer boundary, the largest safe orbit touches the isopleth on its η_x intersection opposite the other star (left solid circle). If the isopleth is the inner boundary, the smallest safe orbit touches the isopleth at the opposite η_x intersection (right solid circle). The case shown is q = 0.3, λ = 0.2 and the isopleth depicted is |$1.0045{\cal I}_{\rm c}$|⁠. The primary star, which in this example is also the most luminous, is the one on the left.

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Figure 7.

Fractional error made using the CHZ criterion for a single star, instead of r_is, as a function of the inner boundary flux value (in units of the critical flux value). The solid line is for the most luminous star (λ = 0.9), while the dashed line is for the less luminous star (λ = 0.1).

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2.1.4 Configuration III (⁠|${\cal I}_{\rm i}<{\cal I}_{\rm c}$|⁠)

The considerations that define the largest and smallest boundaries are the same as for the previous case, except that this time we are dealing with boundary isopleths that surround both stars and the relevant extremal distances are with respect to the binary barycentre.

In this case, there are two additional difficulties: the position of the isopleths with respect to the barycentre depends on the mass ratio too and their form, close to the critical one, is nowhere near circular.

For the outer boundary, the relevant distance is that from the barycentre to the closest point along the boundary isopleth. In general, this point is not along either axis (see Fig. 8).

Figure 8.

Circumbinary boundaries of the rBHZ (configuration III). The boundary isopleth is the dashed line and the barycentre is at the coordinate origin. If this isopleth corresponds to the outer boundary, that largest safe orbit touches the isopleth at the point where the barycentre to isopleth distance is minimum (inner circle). If the isopleth is the inner boundary, the smallest safe orbit touches the isopleth at the η_x intersection farthest away from the barycentre (outer circle). The binary parameters are the same as those in Fig. 5, except that the isopleth depicted is |$0.779{\cal I}_{\rm c}$|

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Equation (7) is symmetric around λ = 1/2. This is expected, since a reflexion around the η_y axis of Fig. 8 leaves the shortest barycentre to isopleth distance unchanged.

Note that the radius of the largest safe circumstellar orbit does not depend on q, whereas that of the circumbinary orbit does. As we mentioned, this is because the position of the isopleths is fixed with respect to the stars, but not with respect to the barycentre (equation 3).

For the inner boundary, the proper orbital radius is the distance from the barycentre to the farthest away η_x axis intersection of the boundary isopleth (Fig. 8). The easiest way to find this distance is to solve equation (4) for η_y = 0, which leads to the same polynomial as in case I (equation 6). But now (⁠|${\cal I}_{\rm b}<{\cal I}_{\rm c}$|⁠), the polynomial has only two real roots, which are the distances from either star to the boundary isopleth (r_⋆). To find the final inner circumbinary orbital radius, we now add the respective star to barycentre distance (equation 3) to each root and compare the resulting barycentre to isopleth distances. The largest one is r_ib.

As before, the solution is rather complicated and some values of r_⋆ are shown in Table 2. For distant boundaries (small values of |${\cal I}_{\rm i}/ {\cal I}_{\rm c}$|⁠), the isopleths become circular and converge to the CHZ criterion applied using the total binary luminosity. This can be seen in Fig. 9, where the fractional error made using the CHZ instead of the rBHZ criterion is plotted for the λ = 0.9 case. Again, for values beyond those listed in Table 2 the error is very small.

Figure 9.

Fractional error made using the CHZ criterion for a single star at the barycentre with the total binary luminosity, instead of r_ib, as a function of the inner boundary flux value (in units of the critical flux value). The λ = 0.9 case is shown.

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In this case, it may be that r_ib > r_ob, which means that no safe circular orbits centred in the barycentre exist. However, a non-circular and sufficiently elongated orbit may remain within the rBHZ, but this is beyond this study.

2.1.5 Configuration II (⁠|${\cal I}_{\rm o}< {\cal I}_{\rm c}< {\cal I}_{\rm i}$|⁠)

This is the most complicated configuration to compute, since we now have the possibility of both, circumstellar and circumbinary safe zones. This is a mixture of the previous two cases and they have to be dealt separately.

For the circumstellar zone, we use the procedure used for case I, with |${\cal I}_{\rm b}={\cal I}_{\rm i}$|⁠, to interpolate values of r_is for each star from Table 1. For the outer edge, we use the procedure of case III with |${\cal I}_{\rm b}={\cal I}_{\rm o}$|⁠, to obtain from equation (7) a value that we will interpret as r_os. Note that for the outer circumstellar edge we are using an isopleth that is not circumstellar. This is because a safe circumstellar orbit may reach beyond the critical isopleth¹ (see Fig. 12). It is obvious that if the resulting r_os reaches all the way to the inner edge of the circumstellar safe zone of the other star, then the smaller of the two distances should be taken, since a safe circumstellar orbit should not get into the unsafe region of the other star.

For the circumbinary region, we apply the procedure of case III with |${\cal I}_{\rm b}={\cal I}_{\rm o}$| for the outer edge. For the inner edge, we have to deal again with the possibility of safe orbits that cross the critical isopleth (see Fig. 10). The inner circumbinary orbital radius is given by the largest of the two barycentre to inner boundary isopleth η_x intersections farther away in the direction to each star. These distances are given by the individual r_is for each star (which we computed already), plus their respective star distance to the barycentre.

Figure 10.

Circumbinary safe orbits in configuration II. The dashed line is the critical isopleth and the shaded area is the radiative safe zone. The extremal safe circumbinary orbits, which define the boundaries of the circumbinary rBHZ are shown (continuous line circles). The example shown is for q = 0.44, λ = 0.375 and |${\cal I}_{\rm o}/ {\cal I}_{\rm c}= 0.179$|⁠, |${\cal I}_{\rm i}/ {\cal I}_{\rm c}=1.592$|⁠.

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2.1.6 Procedure to obtain the habitability region

Tables 1 and 2, together with equation (7) and auxiliary relations, define the orbital radii of the largest and smallest circumstellar or circumbinary safe edges.

The detailed procedure to find the limits of the rBHZ is then as follows:

• From the binary components individual masses and luminosities, obtain the dimensionless mass fraction q and luminosity fraction λ, the latter for both stars.

• Get specific boundary radiative stellar flux values I_o, I_i (e.g. Kopparapu et al. 2013) and convert them to the dimensionless system we use here: |${\cal I}_{\rm i}$|⁠, |${\cal I}_{\rm o}$| (normalized to total binary luminosity per square semimajor axis).

• Given λ (either star), use equation (5) to find the value of the critical flux for this case: |${\cal I}_{\rm c}$|⁠.

• If |${\cal I}_{\rm o}>{\cal I}_{\rm c}$|⁠, proceed to configuration I; if |${\cal I}_{\rm i}<{\cal I}_{\rm c}$|⁠, proceed to configuration III; else, go to configuration II.

• Configuration I: two separate radiative CHZ. Use Table 1 to interpolate the r_os and r_is edges radii for each star using the appropriate value of |${\cal I}_{\rm b}$|⁠.

• Configuration II: One radiative circumbinary habitable zone with two circumstellar inner edges. We have to solve the circumstellar and circumbinary zones separately.

  • Circumstellar zones: use Table 1 to interpolate the r_is radii for each star using |${\cal I}_{\rm b}= {\cal I}_{\rm i}$|⁠.

  • Circumbinary zone: use equation (7) to get the outermost radius r_ob. For the smallest radii, add to the r_is previously computed for each star their respective star to barycentre distances, the largest of the two is the radius of the innermost safe circumbinary edge.

• Configuration III: one radiative circumbinary habitable zone whose inner edge surrounds both stars. For the outer boundary, use equation (7) to compute r_ob. For the inner boundary, use Table 2 to compute the distances of each star to its closest η_x intersection of the inner boundary isopleth r_⋆. To each add the barycentre to respective star distance using equation (3). The largest of the two is r_ib.

Fig. 11 shows the result for a specific example. We have a primary G2V star with 1 M_⊙ and 1L_⊙, while the secondary is a G8V star with 0.8 M_⊙ and 0.6 L_⊙. The stars are assumed to be 4 au apart. We have used the stellar flux limits of Kopparapu et al. (2013) for a G2 star: I_o = 0.53 and I_i = 1.10 L_⊙ au^{− 2}, which correspond to the CO₂ condensation and water loss limits, respectively. In this case, it is clear that safe orbits can exist within two circumstellar zones.

Figure 11.

The critical flux isopleth (thick, dashed line) and radiative binary habitability zones (shaded areas) for the first worked example. The limiting safe orbits are shown (thick, continuous lines). The frame is centred in the barycentre and the unit of length is the binary separation, which in this case is 4 au.

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Fig. 12 shows the result for the same binary, but now with a separation of 3 au. By putting the stars closer together, the circumstellar safe zones have merged into a single, case II habitable zone. In this case, the only safe orbits lie between the r_is and the outer boundary isopleth. No circumbinary safe orbits are possible.

Figure 12.

Same as Fig. 11, but in this case the binary separation has shrunk to 3 au (taken as unit length in the plot). Note that the largest circumstellar orbits reach outside the critical isopleth.

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2.2 Binaries in eccentric orbits

Once the case of the binary in a circular orbit has been solved, the eccentric orbit case is rather simple. We use the circular orbit procedure twice: at periastron and apoastron, and compare the resulting limits. For the smallest safe orbit, we take the largest of the computed values for both orbital extrema. For the largest safe orbit, we take the smallest of the respective values. This guarantees that a fixed, circular, planetary orbit will remain within the safe zone for all orbital phases. Again, if the resulting outer limit is smaller than the inner limit, no safe orbit is possible.

We must stress again that, although this procedure does not take into account explicitly non-circular orbits for planets, nor does it take into account the possible variation in their shape as a function of binary orbital phase (if any), we claim that planetary orbits in binary eccentric systems will not develop readily, highly eccentric stable orbits (i.e. e ≲ 0.3). Furthermore, these deforming effects on orbits in binary systems, may arise close to the critical isopleth; however, it does not necessarily coincides with the loop gap between circumstellar and circumbinary stable orbits.

3 STABILITY CONDITION FOR HABITABLE ZONES IN BINARY STARS

In this paper, we have employed the criteria of Pichardo et al. (2005) and Pichardo et al. (2009) for circumestellar and circumbinary stable orbits, respectively. The same approach was used in Jaime, Pichardo & Aguilar (2009), where stability zones were studied for binaries of the solar neighbourhood. We show in this paper why this stability method results better in searching habitable regions for planets.

Unlike the fundamental work of Holman & Wiegert (1999), which represents an excellent empirical approximation to the stability problem, invariant loops provide an exact solution for stable orbits in ideal (isolated) binary systems at any binary eccentricity. In this manner, rather than orbits that keep stable for some small fraction of the binary star life, to ensure life development, much more time than a small fraction of the binary life is needed. With invariant loops, numerical integrations are used to identify them in phase-space, their existence and moreover, their stability, are supported by integrals of motion in the extended phase-space. As long as the orbital parameters of the binary are not changed, the stability of the loops is guaranteed without having further numerical integrations (see Pichardo et al. 2005, section 2). Consequently, due to the methodology, Holman & Wiegert's approximation overestimates the available stable regions since their stability criteria depends on a given time of integration that the particles keep in orbit (about 10⁴ periods), which results in a very relaxed criteria when looking for strict stability, needed for life emergency and its development. Furthermore, invariant loops show in the circumbinary discs cases, that there is a shift of the stable region that can be considerable for high eccentricities, compromising the intersection between stable regions for planets and habitable regions (as is the case, for example of Kepler 16). The shift cannot be detected with the Holman & Wiegert method, this was detected theoretically and presented in Pichardo et al. 2008 paper. We consider and computed this shift in this work.

In our approach however, we do not consider multiple stars, nor stars that have evolved away from the main sequence, or planets in highly eccentric orbits. In the case of multiple stars, this is first because it is not straightforward to extend the loop formalism to that case, and furthermore, multiple systems result generally in unstable systems, unless they are hierarchical (e.g. triple systems where a very close binary star with a far companion as the Alpha Centauri system and Proxima Centauri seem to be, or a system with a double binary where both binary systems act like a whole binary system since they tend to be extremely separated), and in that case, these systems can be reduced to regular separated binary systems at a good approximation. On the same direction, in the general case of multiple stars, stable regions for planetary orbits would be severely compromised, and even if stable regions would exist, finding them would be a challenging task. Thus, if we are not able to calculate stable regions, it is not of importance to find the correct irradiance zones because we still would not be able to establish habitable zones (i.e. the combination of stability and stellar irradiance).

Regarding the type of star, although our method allows us to calculate radiative safe zones for any type of star, in our specific study for solar neighbourhood binaries, we only calculated habitable zones for main-sequence stars since those out of this ‘life stage’, would likely be inhospitable for life.

Finally, we do not include studies of planets on very eccentric orbits (i.e. e_p ≳ 0.3), in our experience and work with stability zones in binaries constructed out of invariant loops, we have found that their probability of survival is highest, for example, either on planets in a circumbinary disc for the case of very close binary systems (where the circumbinary material feels the system almost as a single star), or in circumstellar discs, for very open binaries (where both stars act almost as single stars). However, this is not the general case, in any other binary system in between, stable regions for planetary orbits with high eccentricities (e_p ≳ 0.3), the reduction of the available phase-space for planets to settle down, and consequently, of their possibilities to be stable for long time-scales can be considerable (this is an ongoing study that will be presented in a future paper).

It is important to stress that the orbits defined by equations (11) and (12), represent stable orbits formed by non-self-intersecting loops, where gas and planets could settle down in long-term basis.

4 THE COMBINED HABITABILITY CONDITION: SOME PARTICULAR CASES

As we have mentioned, the combination of the radiative and stability conditions is one of the goals of this paper. In order to show how this approach allows us to find candidates in the search for habitable planets, in this section we apply our method to 64 binary systems of the solar neighbourhood, with known orbital parameters, and main-sequence stars. Most of the cases are eccentric binaries, thus we apply our procedure (both, radiative and stability conditions) at periastron and apoastron. Once we have these two calculations, we compare both and take, conservatively, the most restricted circular edge. It is worth to mention that it is important to compute it in both locations because we do not know a priori in which case the binary will be located (i.e. if the habitable zone will be circumstellar or circumbinary). Given this, the restriction at periastron or apoastron acts in different ways for configurations I, II or III, and the case can even change from one to another, and this should be taken into account.

Most of binaries considered in this paper are taken from Jaime et al. (2009), which correspond to binaries with stars in the main sequence. We have applied the classical stellar luminosity–mass function Cox (2000) in order to obtain the luminosity for each star. Orbital parameters are the same as the ones considered in Jaime et al. (2009).

Another important issue that must be considered, is the circumbinary disc centre shift, produced by eccentric binary systems (equation 14). In high eccentric cases, this shift might become relevant when calculating the circumbinary orbits inside the habitable zone.

Table 3 shows the results for the 64 objects at periastron, column 1 is the Hipparcos name, column 2 is an alternative name, column 3 shows the particular habitable case resulting for the binary, columns 4 and 5 contain the inner and outer habitable radii for the primary star and columns 6 and 7 are the same but around the secondary star. Finally, columns 8 and 9 provide the inner and outer habitable radii in the circumbinary case. Table 4 shows the same as Table 3 but at apoastron.