Hypothesis Paper

Hypothesis Paper

Habitability of Super-Earth Planets around Other Suns: Models including Red Giant Branch Evolution

W. von Bloh, M. Cuntz, K.-P. Schröder, C. Bounama, S. Franck

Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany

email: bloh@pik-potsdam.de, bounama@pik-potsdam.de, franck@pik-potsdam.de

Department of Physics, University of Texas at Arlington, Arlington, TX 76019, USA

email: cuntz@uta.edu

Department of Astronomy, University of Guanajuato, 36000 Guanajuato, GTO, Mexico

email: kps@astro.ugto.mx

Submitted to  Astrobiology

Submitted  August 19, 2019

Running Title:  Habitability of Super-Earths including RGB evolution

Key words: Extrasolar terrestrial planets — Habitable zone — Planetary atmospheres — Modelling studies.



Corresponding Author:

Dr. Werner von Bloh

Potsdam Institute for Climate Impact Research

Box 60 12 03

14412 Potsdam

Germany

Phone: +49-331-288-2603

Fax: +49-331-288-2600

E-mail: bloh@pik-potsdam.de

ABSTRACT


The unexpected diversity of exoplanets includes a growing number of super-Earth planets, i.e., exoplanets with masses of up to several Earth masses and a similar chemical and mineralogical composition as Earth. We present a thermal evolution model for a 10 Earth mass planet orbiting a star like the Sun. Our model is based on the integrated system approach, which describes the photosynthetic biomass production taking into account a variety of climatological, biogeochemical, and geodynamical processes. This allows us to identify a so-called photosynthesis-sustaining habitable zone (pHZ) determined by the limits of biological productivity on the planetary surface. Our model considers the solar evolution during the main-sequence stage and along the Red Giant Branch as described by the most recent solar model. We obtain a large set of solutions consistent with the principal possibility of life. The highest likelihood of habitability is found for “water worlds”. Only mass-rich water worlds are able to realize pHZ-type habitability beyond the stellar main-sequence on the Red Giant Branch.

Introduction


A pivotal part in the ongoing search for extra-solar planets is the quest to identify planetary habitability, i.e., the principal possibility of life. In a previous paper, Kasting et al. (1993) presented a one-dimensional climate model to define a zone of habitability (HZ) around the Sun and other main-sequence stars that assumed as basic premise an Earth-like model planet with a CO/HO/N atmosphere and that habitability requires the presence of liquid water on the planetary surface.

In the meantime, other definitions of habitable zones have been proposed such as the Galactic HZ, the UV-HZ, and the photosynthesis-sustaining HZ (pHZ). The Galactic HZ (Lineweaver et al., 2004) caters to the requirement that a sufficient amount of heavy elements (notably those contained in carbon and silicate compounds) must be present for the build-up of planets and life, a condition easily met in the solar neighborhood. The UV-HZ (Buccino et al., 2006; Cuntz et al., 2008) is based on the premise that no lethal amounts of stellar UV flux is produced (regarding life forms assuming carbon-based biochemistry), a condition that tends to favor the environment of old main-sequence stars and giants (Guinan and Ribas, 2002) as well as planets with appreciable atmospheres, notable significant ozone layers (Segura et al., 2003).

Another definition of habitability first introduced by Franck et al. (2000a, b) is associated with the photosynthetic activity of the planet, which critically depends on the planetary atmospheric CO concentration. This type of habitability is thus strongly influenced by the planetary geodynamics, encompassing climatological, biogeochemical, and geodynamical processes (“Integrated System Approach”). This concept has previously been used in studies of fictitious planets around 47 UMa (Cuntz et al., 2003; Franck et al., 2003) and 55 Cnc (von Bloh et al., 2003), as well as detailed studies of observed super-Earth planets in the Gliese 581 system (von Bloh et al., 2007b). The latter investigation showed that Gliese 581c is clearly outside the habitable zone, since it is too close to the star, whereas Gliese 581d located near the outer edge of the habitable zone is probably habitable, at least for certain types of primitive life forms (see also Selsis et al., 2007). Moreover, von Bloh et al. (2007a) have used this type of model to compile a detailed ranking of known star-planet systems regarding the principal possibility of life, which by the way led to the conclusion that the Solar System is not the top-tier system (“Principle of Mediocrity”).

In case of Earth-mass planets (1 ), a detailed investigation of geodynamic habitability was presented by Franck et al. (2000b) with respect to the Sun as well as stars of somewhat lower and higher mass as central stars. Franck et al. found that Earth is rendered uninhabitable after 6.5 Gyr as a result of plate tectonics, notably the growth of the continental area (enhanced loss of atmospheric CO by the increased weathering surface) and the dwindling spreading rate (diminishing CO output from the solid Earth).

This implies that there is no merit in investigating the future habitability of Earth during the post–main-sequence evolution of the Sun, as in the framework of pHZ models, the lifetime of habitability is limited by terrestrial geodynamic processes. However, this situation is expected to be significantly different for super-Earth planets due to inherent differences compared to Earth-mass planets (e.g., Valencia et al., 2007a). A further motivation for this type of work stems from the ongoing discovery of super-Earths in the solar neighborhood with the Gliese 876 (Rivera et al., 2005) and Gliese 581 (Udry et al., 2007) systems as prime examples.

In the following, we discuss the definition of the photosynthesis-sustaining habitable zone, including the relevant geodynamic assumptions. Next, we describe the most recent model of solar evolution that is used as basis for our study. Thereafter, we present our results including comparisons to previous work. Finally, we present our summary and conclusions.

A New Model of Solar Evolution


A key element of the present study is to consider a star akin to the Sun as the central object of the star-planet system. Schröder and Smith (2008) recently obtained a new model of solar evolution that will be adopted in the following. This model is based on a well-tested stellar evolution code that allows us to follow the change of solar properties at the main-sequence (MS), along the Red Giant Branch (RGB) and beyond. It is the Eggleton evolution code in the version described by Pols et al. (1995, 1998), which has updated opacities and an improved equation of state.

Among other desirable characteristics, the code uses a self-adapting mesh and a proper treatment of “overshooting” that has been tested and calibrated with giant and supergiant stars in eclipsing binary systems. The code also considers a detailed description of the mass loss following Schröder and Cuntz (2005) that has been tested based on a set of well-observed stars (Schröder and Cuntz, 2007). Thus it permits an accurate description of the time-dependent solar luminosity along the RGB (see Fig. 1). A further consequence of the steadily increasing mass loss is the increase of the orbital distances of any putative planets, given as with as solar mass owing to the conservation of the orbital angular momentum of the planet.

The solar evolution model by Schröder and Smith (2008) suggests an age of the Sun as 4.58 () Gyr, and the RGB-tip is reached after 12.167 Gyr, which is also the point in time, where our computations are suspended. This model also confirms some well-established facts: (1) The MS-Sun has already undergone significant changes, i.e., the present solar luminosity exceeds the zero-age value by 0.30 , and the zero-age solar radius has been 11% smaller than the present value. (2) There was an increase of effective temperature from 5596 K to 5774 () K. (3) The present Sun is increasing its average luminosity at a rate of 1% in every 110 million years, or 10% over the next billion years. All these findings are consistent with established solar models like the one by Gough (1981) and subsequent work.

During the solar MS, the consequences of evolution for Earth-type planets (as well as other types of planets) are extremely slow, compared to natural or human-driven climate changes on Earth. Nonetheless, solar-type evolution will force global warming upon any planet, which has been the subject of detailed previous investigations both concerning the climatic HZ (e.g., Underwood et al., 2003; Jones et al., 2005) and the photosynthesis-sustaining habitable zone of the Sun (e.g., Franck et al., 2000b). According to the evolution model by Schröder and Smith (2008), the tip-RGB evolution will be reached with a luminosity of 2730 , an effective temperature of 2602 K, and a radius of 256 . At that time, the Sun will have lost 0.332  of its initial mass. There is an ongoing debate at what point in time a planet originally located at 1 AU, equating 215 , will be engulfed as a consequence. Contrary to the previous model by Sackmann et al. (1993) that is based on a less accurate description of the solar mass loss, Schröder and Smith (2008) concluded that such an engulfment will happen during the late phase of the solar RGB evolution. In fact, the minimal orbital radius for a planet able to survive is found to be about 1.15 AU.

The evolution of the central star, as well as its effects on planetary orbits, has significant consequences for planetary habitability. This property has previously been investigated for different types of climatic habitable zones by Kasting et al. (1993), Underwood et al. (2003), Jones et al. (2005), and others. Furthermore, a previous assessment of the spatial and temporal evolution of climatic HZs for different types of stars beyond the main-sequence has been given by Lopez et al. (2005). They showed that for a 1  star at the first stages of its post–main-sequence evolution, the temporal transit of the HZ is estimated to be several times  yr at 2 AU and about  yr at 9 AU. Lopez et al. (2005) concluded that under these circumstances life could develop at distances in the range of 2 to 9 AU in the environment of subgiant or giant stars. This view is consistent with our current understanding that terrestrial life existed at least as early as  yr after the Earth formed, which tends to imply that life may be able to form over time intervals from to  yr. The short-time window ( yr) for the origin of life is bounded by the last ocean-vaporizing impact and the earliest evidence for life on Earth ( yr ago). This window might be extended if the origin of life occurred close to  yr ago (Chyba and Hand, 2005).

The main goal of this study is to investigate habitability in the framework of the photosynthesis-sustaining HZ for stars like the Sun with special consideration of the post–main-sequence evolution. Our study will be focused on super-Earth planets, and we will consider a significant set of geodynamic processes. Our findings will also be compared with the previous work by Lopez et al. (2005).

Habitability of Super-Earths

Definition of the photosynthesis-sustaining habitable zone

To assess the habitability of terrestrial planets, including super-Earth planets, an Earth-system model is applied to calculate the evolution of the temperature and atmospheric CO concentration. On Earth, the carbonate-silicate cycle is the crucial element for a long-term homeostasis under increasing solar luminosity. On geological time-scales, the deeper parts of the Earth are considerable sinks and sources of carbon. The role of weathering for the Earth’s climate was first described by Walker et al. (1981). They found that an increase in luminosity leads to a higher mean global temperature causing an increase in weathering. As a consequence, more CO is extracted from the atmosphere, thus weakening the greenhouse effect. Overall the temperature is lowered and homeostasis is achieved.

On geological time scales, however, the deeper parts of the Earth are considerable sinks and sources for carbon. As a result, the tectonic activity and the continental area change considerably. Therefore, Tajika and Matsui (1992) have favored the so-called “global carbon cycle”. In addition to the usual carbonate-silicate geochemical cycle, it also contains the subduction of large amounts of carbon into the mantle with descending slabs and the degassing of carbon from the mantle at mid-ocean ridges. In particular, the potential of weathering to stabilize the surface temperature of a terrestrial planet by a negative feedback mechanism is also strongly modulated by the biosphere.

Our numerical model couples the solar luminosity , the silicate-rock weathering rate , and the global energy balance to obtain estimates of the partial pressure of atmospheric carbon dioxide , the mean global surface temperature , and the biological productivity as a function of time (Fig. 2). The main point is the persistent balance between the CO (weathering) sink in the atmosphere-ocean system and the metamorphic (plate-tectonic) sources. This is expressed through the dimensionless quantities

(1)

where is the weathering rate, is the continental area, and is the areal spreading rate, which are all normalized by their present values of Earth. Eq. (1) can be rearranged by introducing the geophysical forcing ratio GFR (Volk, 1987) as

(2)

Here we assume that the weathering rate depends only on the global surface temperature and the atmospheric CO concentration. For the investigation of a super-Earth under external forcing, we adopt a model planet with a prescribed continental area. The fraction of continental area relative to the total planetary surface is varied between and .

The connection between the stellar parameters and the planetary climate can be formulated by using a radiation balance equation

(3)

where denotes the stellar luminosity, the planetary distance, the planetary albedo, and the outgoing infrared flux of the planet. Following Williams (1998) has been approximated by a third order polynomial and by a second order polynomial. These approximations have been derived from runs of a radiation-convection model by Kasting and Ackerman (1986) and Kasting (1988). They are valid in a range of . The Eqs. (2) and (3) constitute a set of two coupled equations with two unknowns, and , if the parameterization of the weathering rate, the luminosity, the distance to the central star and the geophysical forcing ratio are specified. Therefore, a numerical solution can be attained in a straightforward manner.

The photosynthesis-sustaining HZ is defined as the spatial domain of all distances from the central star, e.g., the Sun, where the biological productivity is greater than zero, i.e.,

(4)

In our model, biological productivity is considered to be solely a function of the surface temperature and the CO partial pressure in the atmosphere. Our parameterization yields maximum productivity at C and zero productivity for C or C or bar (Franck et al., 2000a). A photosynthesis-based biosphere of a super-Earth may, however, use methane to produce CO, because hydrogen is less likely to escape to space. The inner and outer boundaries of the pHZ do not depend on the detailed parameterization of the biological productivity within the temperature and pressure tolerance window. Hyperthermophilic life forms can tolerate temperatures somewhat above C. However, these chemoautotrophic organisms are outside the scope of this study.

Silicate Rock Weathering

Weathering plays an important role in Earth’s climate because it provides the main sink for atmospheric carbon dioxide. The overall chemical reactions for the weathering process are

The total process of weathering embraces (1) the reaction of silicate minerals with carbon dioxide, (2) the transport of weathering products, and (3) the deposition of carbonate minerals in the oceanic crust. The available thickness of crust where CaCO is stable in the presence of silicate scales inversely with the thermal gradient and hence inversely with surface gravity. Therefore, there may be a problem for storing carbonates in the crust of super-Earth planets. Additionally, there is an exchange with the mantle via alteration of the oceanic crust.

When combining all these effects, the normalized global mean weathering rate can be calculated as

(5)

following Walker et al. (1981). Here the first factor reflects the role of the CO concentration in the soil, , with as the activity of in fresh soil-water that depends on and the global mean surface temperature . The quantities and are the present-day values for the activity and the surface temperature, respectively. The activity is itself a function of the temperature and the CO concentration of the soil. The concentration of CO in the soil water [CO(aq)] can be obtained from the partial pressure of CO in the soil according to

(6)

where is Henry’s law constant. We assume that [CO(aq)] = [HCO]. HCO dissociates in two steps which are

The corresponding concentrations can be calculated from the law of masses as

(7)
(8)

where and are (temperature dependent) equilibrium constants. An additional constraint for the concentrations is given by the charge balance

(9)

Here [H] and therefore can be derived from a simultaneous solution of Eqs. (7) to (9) as a function of . The sulfur content of the soil can be taken into account analogously. The equilibrium constants for the chemical activities of the carbon and sulfur systems involved are taken from Stumm and Morgan (1981). Note that the sulfur content of the soil also contributes to the global weathering rate, but its influence does not depend on the temperature. It can be regarded as an overall weathering attribute that has to be taken into account for the estimation of the present-day value.

For any given weathering rate, the surface temperature and the CO concentration of the soil can be calculated in a self-consistent manner. is assumed to be linearly related to the terrestrial biological productivity (see Volk, 1987) and the atmospheric CO concentration . Thus we find

(10)

where , and are again present-day values. Note that the present-day concentration of CO in the soil is ten times the present-day concentration of CO in the atmosphere, i.e., .

Thermal Evolution Model

Parameterized convection models are the simplest models for investigating the thermal evolution of terrestrial planets and satellites. They have successfully been applied to the evolution of Mercury, Venus, Earth, Mars, and the Moon (Stevenson et al., 1983; Sleep, 2000). Franck and Bounama (1995) have investigated the thermal and volatile history of Earth and Venus in the framework of comparative planetology. The internal structure of massive terrestrial planets with one to ten Earth masses has been investigated by Valencia et al. (2006) to obtain scaling laws for the total radius, mantle thickness, core size, and average density as a function of mass. Further scaling laws were found for different compositions. We will use such scaling laws for mass-dependent properties of our 10  super-Earth model as well as for mass-independent material properties given by Franck and Bounama (1995) (see Tab. 1).

The thermal history and future of a super-Earth has to be determined to calculate the spreading rate for solving Eq. (1). A parameterized model of whole mantle convection including the volatile exchange between the mantle and surface reservoirs (Franck and Bounama, 1995; Franck, 1998) is applied. Assuming conservation of energy, the average mantle temperature can be obtained by solving

(11)

where is the density, is the specific heat at constant pressure, is the heat flow from the mantle, is the energy production rate by decay of radiogenic heat sources in the mantle per unit volume, and and are the outer and inner radii of the mantle, respectively. The radiogenic heat source per unit volume is parameterized as

(12)

where is the decay constant and the constant is obtained from the present heat flux of Wm for an Earth-size planet at 4.6 Gyr.

The mantle heat flow is parameterized in terms of the Rayleigh number as

(13)

with

(14)

where is the thermal conductivity, is the critical value of for the onset of convection, is an empirical constant, is the gravitational acceleration, is the coefficient of thermal expansion, is the thermal diffusivity, and is the water-dependent kinematic viscosity. The viscosity can be calculated with the help of a water fugacity-dependent mantle creep rate. It strongly depends on the evolution of the mass of mantle water , and the mantle temperature , i.e., and is parameterized according to Franck and Bounama (1995).

The evolution of the mantle water can be described by a balance equation between the regassing flux and outgassing flux as

(15)

where is the water content in the basalt layer, is the average density, is the average thickness of the basalt layer before subduction, is the areal spreading rate, is the melt generation depth and is the outgassing fraction of water. is the regassing ratio of water, i.e., the fraction of subducting water that actually enters the deep mantle. The average thickness of the basalt layer as well as the melt generation depth scale inversely with surface gravity , i.e., and . The pressure closing of cracks in the deeper parts of the basalt layer scales also inversely with and thus reduces the storage capacity of volatiles for a super-Earth planet. Therefore, the ratio is independent of . According to Eq. (15) gravity influences only the time scale of mantle water evolution. Therefore, as a first approximation the melt generation depth does not depend on mantle temperature. However, there is a temperature dependence of (McKenzie and Bickle, 1988; Langmuir et al., 1992). The regassing ratio depends linearly on the mean mantle temperature that is derived from the thermal evolution model via

(16)

The factor is adjusted to obtain the correct modern amount of surface water (one ocean mass) for an Earth-size planet and is fixed at . This value is obviously very low at the beginning of the planetary evolution because of the enhanced loss of volatiles resulting from back-arc volcanism at higher temperatures.

The areal spreading rate is a function of the average mantle temperature , the surface temperature , the heat flow from the mantle , and the area of ocean basins (Turcotte and Schubert, 1982), given as

(17)

In order to calculate the spreading rates for a planet with several Earth masses, the planetary parameters have to be adjusted accordingly. We assume

(18)

and with as planetary radius, see (Valencia et al., 2006). The total radius, mantle thickness, core size and average density are all functions of mass, with subscript denoting Earth values. The exponent of has been obtained for super-Earths (), and has already been used by von Bloh et al. (2007b) in their models of Gliese 581c and 581d. The values of , , , the density of the planet, and the other planetary properties are also scaled accordingly.

The source of CO to the atmosphere is expressed in mass of carbon outgassed at the spreading zones, . It has to be converted to an equivalent concentration of CO in the atmosphere. This can be done by the following equation

(19)

where and are the molar weights of CO and C, respectively. The mass dependent pre-factor scales as and has therefore been neglected in our study. Therefore the conversion does not depend on the planetary mass and the spreading rates can be directly used to calculate in Eq. (1).

In Tab. 1 we give a summary of the selected values for the parameters used in the thermal evolution model of the 10  super-Earth planet, while also depicting an Earth-size planet for comparison. According to Valencia et al. (2007b), we assume that a more massive planet is likely to convect in a plate tectonic regime similar to Earth. Thus, the more massive the planet is, the higher the Rayleigh number that controls convection, the thinner the top boundary layer (lithosphere), and the faster the convective velocities. This is the so-called boundary-layer limit of convection. From this limit it follows that the interior of a super-Earth is always hotter and less viscous than that of a Earth-mass planet. Nevertheless, friction is the rate-limiting process for subduction. Increasing the planetary radius acts to decrease the ratio between driving forces and resistive strength (O’Neill and Lenardic, 2007). Thus a super-sized Earth might be in an episodic or stagnant lid regime.

In a first order approximation, we assume a fixed thickness of the basalt layer and melting depth corresponding to relatively low values. Furthermore, the initial amount of water scales linearly with the planetary mass. However, his might be an underestimate because more massive planets tend to accrete more volatiles.

Results

Habitability based on the integrated system approach


In the following, we study the habitability of super-Earth planets based on the integrated system approach that has previously been used in various other planetary studies (e.g., Franck et al., 2000b; Cuntz et al., 2003; von Bloh et al., 2003; Franck et al., 2003; von Bloh et al., 2007a, b). The simulations have been carried out for a 10  mass super-Earth with a fixed relative continental area varied from to . Fig. 3 shows the behavior of the photosynthesis-sustaining habitable zone (pHZ) of the Sun for a 10  super-Earth planet. The age domain beyond 11 Gyr that also includes the post–main-sequence evolution is depicted in Fig. 4. The width of the pHZ during the main-sequence evolution is found to be approximately constant, but for higher ages, it increases over time and moves outward, a phenomenon most noticeable beyond 11.5 Gyr. For example, for ages of 11.0, 11.5, 12.0, and 12.1 Gyr, the pHZ is found to extend from 1.41 to 2.60, 1.58 to 2.60, 4.03 to 6.03, and 6.35 to 9.35 AU, respectively.

At relatively high ages, habitable solutions are identified as water worlds, if the Sun as central star has reached the RGB. The reason is that planets with a considerable continental area have higher weathering rates that provide the main sink of atmospheric CO. Therefore, such planets are unable to build up CO-rich atmospheres which prevent the planet from freezing or allowing photosynthesis-based life. This result is consistent with previous findings for Earth-mass planets around the Sun or stars of similar masses (Franck et al., 2000b; Cuntz et al., 2003).

Note that the partial pressure of carbon dioxide in the planetary atmosphere is determined by the equilibrium of sources and sinks. The sources are given by volcanic outgassing, while the sinks are given by the weathering of silicates on the continents. As previously found in studies of 1  planets (e.g., Franck et al., 2000b), the rate of outgassing is monotonously decreasing with age because of the decay of long-lived radionuclides and the loss of the initially available accretion energy at the planetary surface. This process starts just after completion of the planetary accretion both for an initially habitable and uninhabitable planet. A planet beyond the outer edge of the pHZ is completely frozen and thus no weathering will occur on the continents. Therefore, all CO is accumulated in the atmosphere. If the planet becomes habitable due to the increase of the luminosity of the central star, weathering starts and a new equilibrium of atmospheric CO is established as a consequence.

Furthermore, the interior of a planet with a relatively low mass is known to cool down more rapidly. Therefore, such a planet initially beyond the outer edge of the habitable zone will not become habitable at a later stage because of the failure to provide a sufficiently dense atmosphere. In contrast, a super-Earth planet might become habitable, depending on the relative size of the continental area. In a recent study, the importance of snowball planets as a possible source of water-rich terrestrial planets was elucidated by Tajika (2008), although the main focus of this paper was the assessment of internal oceans.

Super-Earth-type water worlds are even able to realize pHZ-type habitability beyond solar-type main-sequence evolution. Any model where mantle vents its water will end up as a water world super-Earth. The height of ridges, volcanos, and mountains scale with lithosphere thickness and hence with . As the central star evolves, its pHZ expands outward, and moves further away from the star, particularly for stellar ages beyond 11.8 Gyr (see Fig. 4). Similar to the climatic HZ (see Lopez et al. for details), the pHZ acts like a shell that sweeps progressively outward over a wide range of distances from the star. This results in a significant decrease of the duration of the transit of the habitable zone for any planet located beyond 1.5 AU (see Fig. 5). We find that for water worlds with , the duration of the transit of the pHZ at 2, 3, and 5 AU is given as 3.7, 0.25, and 0.10 Gyr, respectively, whereas for planets at 10 and 20 AU, much smaller durations of the transit are identified.

Figs. 3 and 4 also depict various orbital distances of planets originally located between 1 AU and 5 AU. Note that these orbital distances do not change during the stellar main-sequence stage, i.e., below 10 Gyr (Schröder and Smith, 2008), owing to the lack of significant mass loss and in the absence of significant planet-planet interaction as typically encountered in multiple planetary systems. Thereafter, the orbital distances of any planet increases following with as mass owing to the conservation of planetary orbital angular momentum.

Comparison with previous results

The existence of habitability around stars that have evolved away from the main-sequence has already been the topic of previous investigations. Lopez et al. (2005) studied the behavior of the climatic HZ based on the concept of Kasting et al. (1993) for stars of different masses, including the Sun. Lopez et al. (2005) assume a HZ based on conservative limits of habitability. The inner limit of their conservative estimate is set by the lowest temperature at which the liquid-solid phase change of water occurs. The estimate of the outer limit assumes the existence of a greenhouse effect involving CO and HO gas (Kasting et al., 1993). The less conservative definition extends the outer edge of the limit of habitability to as large as 2.4 AU, largely depending on the radiative properties of the CO ice clouds; see Forget and Pierrehumbert (1997) and Mischna et al. (2000) for detailed studies.

Akin to the pHZ previously discussed, Lopez et al. (2005) found that for the Sun during its evolution the climatic HZ acts like a shell that sweeps progressively outward over a wide range of distances from the star. The duration of the transit during which the HZ passes over a planet located at 1 AU from the star was found to be on the order of 10 yr. After the star leaves the main-sequence, the climatic HZ progressively moves to 2 AU. The duration of the transit at this location is approximately 10 yr. A plateau is observed in the curve up to 9 AU (for the conservative limits) and up to 13 AU (for the less conservative limits), where the durations of habitable conditions lasts from a few to several times 10 yr. At 10 AU, the duration is smaller, about 10 yr. At 15 AU from the star the duration of habitable conditions lasts more than 10 yr, and at the largest distances considered in the study by Lopez et al. (2005) the duration gradually decreases.

Note that the model of solar evolution considered in the Lopez et al. study is that by Maeder and Meynet (1988). Nonetheless, their results would be quite similar if they had used the subsequent model by Sackmann et al. (1993) or the very recent model by Schröder and Smith (2008). The reason is that the outcome of the Lopez et al. study is much more dependent on the choices made concerning the upper and lower limits of the climatic HZ, mostly connected to the treatment of the CO atmospheres (i.e., radiative properties, cloud coverage, etc.) than to the adopted model of solar evolution.

Fig. 6 shows the comparison between the work by Lopez et al. (2005) and our current results. We find that for water worlds (), the transit times for photosynthesis-sustaining habitability (pHZ) for planets at a given reference distance from the star is relatively similar to the results obtained for the conservative climatic HZ (Kasting et al., 1993) adopted by Lopez et al., albeit the transit times in our study are typically lower by a factor of up to 1.5. For example, the durations of the transit concerning pHZ-type habitability for water worlds at 2, 3, and 5 AU are identified as 3.7, 0.25, and 0.10 Gyr, respectively, whereas for planets at 10 and 20 AU, the durations of the transit found are as low as 27 and 9 Myr, respectively.

However, especially at distances beyond 2 AU, significantly smaller transit times are encountered for planets with larger continental areas in terms of all stellar distances, a result consistent with previous findings. For , the transit time of the pHZ drops beneath 1 Gyr for planets located at 1.8 AU. For planets with a relative continental area of , also referred to as “land worlds”, no significant photosynthesis-sustaining habitability is found for planets beyond 1.5 AU.

Summary and Conclusions

We studied the habitability of super-Earth planets based on the integrated system approach that has previously been used in various theoretical planetary studies (e.g., Franck et al., 2000b; Cuntz et al., 2003; von Bloh et al., 2003; Franck et al., 2003; von Bloh et al., 2007a, b). This work is motivated by the quest to identify habitability outside the Solar System as well as the ongoing discovery of super-Earths in the solar neighborhood with the Gliese 876 (Rivera et al., 2005) and Gliese 581 (Udry et al., 2007) systems as prime examples.

In agreement with previous studies, it is found that photosynthesis-sustaining habitability strongly depends on the planetary characteristics. For planets of a distinct size, the most important factor is the relative continental area. Habitability was found most likely for water worlds, i.e., planets with a relatively small continental area. For planets at a distinct distance from the central star, we identified maximum durations of the transit of the pHZ. A comparison of planets with different masses revealed that the maximum duration of the transit increases with planetary mass. Therefore, the upper limit for the duration of the transit for any kind of Earth-type planet is found for most massive super-Earth planets, i.e., 10 , rather than 1  planets, which are rendered uninhabitable after 6.5 Gyr, as previously pointed out by Franck et al. (2000b).

Our study forwards a thermal evolution model for a 10  super-Earth orbiting a star akin to the Sun. The calculations consider updated models of solar evolution obtained by Schröder and Smith (2008) with a detailed mass loss description provided by Schröder and Cuntz (2005). The latter is relevant for the change of luminosity along the Red Giant Branch as well as the increase of the orbital distances of any putative planets during that phase. By employing the integrated system approach, we were able to identify the sources and sinks of atmospheric carbon dioxide on the planet, allowing us to describe the photosynthesis-sustaining habitable zone (pHZ) determined by the limits of biological productivity on the planetary surface.

Concerning the pHZ, we identified the following properties:

(1) Geodynamic solutions are identified for different solar ages, including the RGB phase. The pHZ increases in width over time and moves outward. For example, for ages of 11.0, 11.5, 12.0, and 12.1 Gyr, the pHZ is found to extend from 1.41 to 2.60, 1.58 to 2.60, 4.03 to 6.03, and 6.35 to 9.35 AU, respectively.

(2) Habitable solutions at large ages, especially for the subgiant and giant phase, are water worlds. This also means that the possibility of water worlds in principle results in an extension of the outer edge of habitability. The reason is that planets with a considerable continental area have higher weathering rates that provide the main sink of atmospheric CO. Therefore, such planets, contrary to water worlds, are unable to build up CO-rich atmospheres that prevent the planet from freezing or allowing photosynthesis-based life.

(3) The total duration of the transit of the habitable zone is similar to the predictions by Lopez et al. (2005) based on the conservative limits of the climatic HZ obtained by Kasting et al. (1993). For water worlds with , the transit times of the pHZ at 2, 3, and 5 AU are obtained as 3.7, 0.25, and 0.10 Gyr, respectively, whereas for planets at 10 and 20 AU, much smaller transit times are found.

Our results are a further motivation to consider super-Earth planets in upcoming or proposed planet search missions such as Kepler, TPF or Darwin. Moreover, our results can also be viewed as a reminder to seriously contemplate the possibility of habitable planets around red giants, as previously pointed out by Lopez et al. (2005) and others. For central stars with a higher mass than the Sun, a more rapid evolution will occur that will also affect the temporal and spatial constraints on planetary habitability when the central stars have reached the RGB.

Acknowledgments

We would like to thank Norman Sleep and an anonymous referee for their helpful comments which allowed us improving the paper.

Parameter Value Unit Description
m Average thickness of the basalt layer
Mass fraction of water in the basalt layer
4,569 kg m Density of the basalt
Degassing fraction of water
m melting depth
J s m K Thermal conductivity
m Inner radius of the mantle
m Outer radius of the mantle
kg Initial amount of mantle water
K Initial mantle temperature
mol J Henry’s law constant at 25C
-6.3 -6.3 Equilibrium constant in Eq. (7) at 25C
-10.3 -10.3 Equilibrium constant in Eq. (8) at 25C
m s Thermal diffusivity
J m K Density specific heat
K Temperature dependence of regassing ratio
K Coefficient of thermal expansion
Empirical constant in Eq. (13)
Critical Rayleigh number
Gyr Decay constant
J s m Initial heat generation per time and volume
9.81 28.26 m s Gravitational acceleration
Table 1: Parameter values for the evolution model of the mantle temperature and water

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