HABITABLE ZONES OF POST-MAIN SEQUENCE STARS

We compute the post-MS luminosities for a grid of stars from A5 to M1 (Figure 1) using the corresponding stellar masses: 1.9 M_Sun, 1.5 M_Sun, 1.3 M_Sun, 1 M_Sun, 0.75 M_Sun, and 0.5 M_Sun based on the Padova stellar evolutionary tracks for the high-mass stars (Bertelli et al. 2008, 2009) (1.9, 1.5, 1.3, and 1 M_sun) and the corresponding Darmouth models for the lower mass stars in our grid (Dotter et al. 2007, 2008) (0.75, and 0.5 M_Sun). All calculations start at the beginning of the RGB with stellar luminosities increasing until the tip of the RGB. For stars of solar mass and greater, these luminosities decrease coincident with the helium flash, before again increasing along the AGB (Figure 1). For less massive stars, stellar winds during the RGB phase reduce their masses below ∼0.5 that of the present Sun, too low to undergo the AGB phase (see, e.g., Iben 2013, p. 616).

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The maximum stellar luminosities of the grid stars are between 40,000 and 1000 times their values at the Zero Age Main Sequence (∼1000, 4500, 40,000 for the F1, Sun, and M1 star model, respectively, see Figure 1). The changing stellar effective temperature, T_eff, and SED throughout the post-MS impacts the relative contribution of absorption and scattering of incoming light in the planetary atmosphere. That, in turn, changes the effective stellar flux at the top of the atmosphere of the planet, S_eff. Therefore, both S_eff as well as the orbital distance of the post-MS HZ boundaries, change through time (see Figure 2).

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The inner edge of the empirical HZ (Kasting et al. 1993) is defined by the stellar flux received by Venus when we can exclude the possibility that it had standing water on the surface (about 1 Gyr ago), equivalent to a S_eff value of 1.77. The outer edge is defined by the stellar flux that Mars received at the time that it may have had stable water on its surface (about 3.8 Gyr ago), which corresponds to S_eff = 0.32 in our solar system. We also show (Figures 2–3) a more conservative inner edge of the HZ (dashed lines) based on 3D models that calculate when a runaway greenhouse scenario would occur on a planet based on current Earth-based atmosphere and cloud models, equivalent to S_eff ∼ 1.11 in our own solar system (Leconte et al. 2013a). Planets become too hot and get devolatilized for distances inside the inner edge of the HZ. Note that the HZ limits assume Earth-sized planets with an initial water inventory equal to that of Earth for a 1-bar nitrogen atmosphere that is water-dominated at the inner edge and CO₂-dominated on the outer edge (following Kasting et al. 1993).

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For Earth-like planets, the runaway greenhouse state (complete ocean evaporation) would be triggered when the planet's surface temperature reaches the critical temperature for water (647 K) (Kasting et al. 1993; Ramirez et al. 2014b), see Section (3.3). For planets with smaller water inventories, the runaway greenhouse will be triggered at lower temperatures (see, e.g., Kasting & Ackerman 1986). We do not consider HZ limits for extremely dry (Abe et al. 2011, Zsom et al. 2014), or hydrogen-rich atmospheres (Pierrehumbert & Gaidos 2011; see Kasting et al. 2013 for critical discussion of both limits) in this paper.

We calculate the effective stellar flux post-MS HZ limits (Figure 2; Table 1) and the corresponding post-MS HZ distances through the RGB and AGB (Figure 3; Table 2). Our 1D climate model calculates the limits for a grid of stars with effective temperatures between 2600 and 10,000 K (the model for 2600–7200 K is discussed in detail in Kasting et al. 1993; Kopparapu et al. 2013, 2014; Ramirez et al. 2014b). We use Bt-Settl modeled spectra (Allard 2003; Allard et al. 2007) to expand the temperature range of the climate model to 10,000 K for HZ calculations (following Kopparapu et al. 2013).

2.3.1. Stellar Mass Loss Model

We calculate stellar mass loss (${{\dot{M}}}_{{\rm{star}}}$ in M_Sun yr⁻¹) using the Baud & Habing (1983) parameterization (1) for the AGB and a modified Reimers Equation (2) for the RGB (Reimers 1975; Vassiliadis & Wood 1993):

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{star}\_\mathrm{RGB}}=\displaystyle \frac{4}{3}\times {10}^{-13}\displaystyle \frac{L}{{gR}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{star}\_\mathrm{AGB}}=4\times {10}^{-13}\displaystyle \frac{{M}_{i}}{M}\displaystyle \frac{L}{{gR}}.\end{eqnarray} \tag{ 2 }$$

Where L, g, and R, M_i, and M are the stellar luminosity, gravity, radius, initial stellar mass at the start of the AGB, and stellar mass (in solar units), respectively. Resultant stellar masses, luminosities, and stellar ages are given in Table A1.Stellar mass loss during the RGB phase reduces the remaining mass of the two smallest grid stars below 0.5 solar masses, too low to undergo the AGB phase (see, e.g., Iben 2013, p. 616). Therefore we end our calculations at the end of the RGB, as predicted by the evolutionary models, or once a minimum mass of ∼0.5 solar masses is reached (e.g., Sweigart & Gross 1978; Iben 2013, p. 616), whichever comes first.

Table 3.  Time (in Gyr) a Planet Spends in the Post-MS HZ*

Star	Max time in HZ	Mars	Jupiter	Saturn	Kuiper Belt
M1	9.00	⋯	5.80	2.1	0.50
K5	2.10	⋯	1.00	0.45	0.06
Sun	0.5	0.1	0.37	0.21	∗
F5	0.39	0.35	0.23	0.03	∗
F1	0.07	0.07	0.06	0.02	∗
A5	0.21	0.20	0.035	0.01	∗

Note. Star (∗) indicate that those planets do not spend more than 1 Myr in the post-MS HZ,

Dashes (-) indicate that these planets are not located in the post-MS HZ during the RGB and AGB phase of the star.

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Studies have shown that Reimers' equation overestimates mass loss rates in the RGB, (Sweigert et al. 1990; Vassiliadis & Wood 1993) leading Vassiliadis & Wood (1993) to suggest multiplying Reimers' Law by one-third in order to obtain mass losses consistent with the HB morphology of RGB stars (Fusi Pecci & Renzini 1976, 1978; Renzini & Fusi Pecci 1988), see Equation (1). Although Reimers' equation was originally derived from a compilation of M-star data, G- and K-stars also exhibit similar mass loss rates (Reimers 1977; Sweigert et al. 1990; Vassiliadis & Wood 1993). Mass loss equations for A and F stars during the RGB are not available (e.g., Cranmer & Saar 2001), therefore we assume here that those mass loss rates also follow Reimers' equation.

The Baud & Habing (1983) parameterization, see Equation (2), includes the superwind near the end of the AGB, which predicts larger stellar mass losses than that those computed by Reimers' for this stellar phase in accordance with observations (ibid). The alternative Schröder & Cuntz (2005) parameterization gives similar answers to those given by the modified Reimers' formula but is only valid for a relatively narrow T_eff range (3000–4500 K; Schröder & Cuntz 2005). The Rosenfield et al. (2014) variant of Schröder and Cuntz's equation (2005) also yields stellar mass loss rates that are too strong for both the RGB and the early stages of the thermally pulsating AGB phase and is therefore not considered here. Recent models attempting to improve upon Equations (1) and (2) (e.g., Cranmer & Saar 2001) were not used because they require knowledge of unknown stellar parameters like surface X-ray fluxes (see discussion).

Table A1(a) shows that most of the mass loss for intermediate mass stars (∼1–2 solar masses) occurs during the AGB (see also, e.g., Villaver & Livio 2007) and stars below solar mass would have too little mass (∼0.5 Solar masses) to undergo the AGB phase (see, i.e., Sweigart & Gross 1978). Our K5 model reaches that mass at the end of the RGB whereas our added stellar mass loss causes our M1 model to achieve that mass slightly before the end of the RGB phase predicted by the Dartmouth models (Table 2). Note that most of the RGB mass loss for our M1 occurs in the last 1% of the RGB time frame (according to Equation (1)), therefore the exact time the RGB evolution of the M1 star ends does not influence the time a planet can spend within the HZ.

2.3.2. Planetary Atmospheric Mass Loss Model

The mixing layer formalism of Cantó & Raga (1991) shows that a flow with sound speed, v_w, and density, p_w, and a tangential velocity of the same order has an entrained mass flow $\dot{M}\approx \alpha {p}_{w}{v}_{w}$ (given entrainment efficiency, α). For a planetary atmosphere, we multiply the entrained mass flux by the surface of the leading hemisphere to estimate the planetary atmospheric mass loss per unit time, ${{\dot{M}}}_{a}$, entrained by the stellar wind (following Zendejas et al. 2010):

$$\begin{eqnarray}{\dot{M}}_{a} & \approx & 2\pi {R}_{p}^{2}{\rho }_{w}{v}_{a};\\ {v}_{a} & = & \alpha {v}_{w}.\end{eqnarray} \tag{ 3 }$$

Here, the atmospheric outflow velocity, v_a, is a function of the atmospheric entrainment efficiency, α (Cantó & Raga 1991). The above expression can be combined with the relation ${{\dot{M}}}_{\mathrm{star}}=4\pi {D}^{2}{\rho }_{w}{v}_{w}$ (where D is the planetary orbital distance) to obtain Equation (4):

$$\begin{eqnarray}&&{{\dot{M}}}_{a}={\left(\displaystyle \frac{{R}_{p}}{D}\right)}^{2}\displaystyle \frac{{{\dot{M}}}_{\mathrm{star}}\alpha }{2}\end{eqnarray} \tag{ 4 }$$

Thus, the rate of planetary atmospheric mass loss is a function of both the orbital distance and the stellar mass loss rate. This expression can be combined with Equations (1) and (2) to yield the planetary atmospheric mass loss as a function of stellar mass loss for the RGB phase Equation (5) and AGB phase Equation (6):

$$\begin{eqnarray}&&{\dot{M}}_{a\_\mathrm{RGB}}=\displaystyle \frac{2}{3}\times {10}^{-13}{\left(\displaystyle \frac{{R}_{p}}{D}\right)}^{2}\displaystyle \frac{L}{{gR}}\alpha \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\dot{M}}_{a\_\mathrm{AGB}}=2\times {10}^{-13}\displaystyle \frac{{M}_{i}}{M}{\left(\displaystyle \frac{{R}_{p}}{D}\right)}^{2}\displaystyle \frac{L}{{gR}}\alpha .\end{eqnarray} \tag{ 6 }$$

By comparing laboratory experiments of plane, turbulent mixing layers Cantó & Raga (1991) determined that α = 0.03. Bauer & Lammer (2004, p. 151) suggested an upper value of α = 0.3 for Venus. Planets with Earth-like atmospheres should be less turbulent than Venus, thus we use an intermediate value for α of 0.2 for our calculations and explore the sensitivity of the results to this parameter (see discussion and Appendix B). We terminate our calculations either when planetary surface pressures become smaller than 0.25 bar, or at the end of the AGB, whichever comes first.

2.3.3. Planetary Orbital Radius Expansion Model

We assume that planets are far enough away from the expanding star that orbital dissipation arising from tidal forces are negligible. As long as the planet's orbital distance is larger than the stellar radius of the expanding star, and tidal disruption effects are negligible, the orbital variation can be approximated by Equation (7) (e.g., Zahn 1977, Villaver & Livio 2007):

$$\begin{eqnarray}&&\displaystyle \frac{1}{D}\displaystyle \frac{{dD}}{{dt}}=-\displaystyle \frac{1}{{M}_{*}}\displaystyle \frac{{{dM}}_{*}}{{dt}}.\end{eqnarray} \tag{ 7 }$$

Here, M_* is the mass of the star at the end of the main-sequence. This expression is integrated to give

$$\begin{eqnarray}&&D(t)={D}_{o}\displaystyle \frac{{M}_{*}}{{M}_{*}(t)},\end{eqnarray} \tag{ 8 }$$

where D_o is the initial orbital distance and M_*(t) is the time-dependent stellar mass loss during the RGB and AGB and D(t) is the orbital distance of the planet at time t.

2.3.4. Super-Earths and Super-moon Models

The HZ limits evolve through the stellar post-MS both in effective stellar flux on top of the planet's atmosphere, S_eff, (Figure 2) as well as orbital distance from the star (Figure 3) due to the star's changing SED and luminosity, respectively. We derive a parameterization (Equation (9)) to compute post-main-sequence HZ distances for grid stars old enough to be currently on the post-main-sequence (Sun—A5) using the constants given in Table A1(d). Our parameterization only includes the slowly changing portion of the RGB, during which most planets retain their atmospheres.

$$\begin{eqnarray}&&D={{at}}^{4}+{{bt}}^{3}+{{ct}}^{2}+{dt}+e,\end{eqnarray} \tag{ 9 }$$

where D is the distance (in au) and t is stellar age for solar metallicity stars (in Gyr).

The effective stellar flux at the boundaries of the post-MS-HZ as shown in Figure 2 decreases during the RGB by a maximum of ∼22% for the F1 and 5% for the M1 stellar type. These two stellar types bracket the maximum and minimum luminosity change for our grid of stars, respectively. The corresponding increases of S_eff during the Helium-flash and decreases during the following AGB, respectively, are 42% and 69% for the M1 and 8% and 8% for the F1 stellar type. Note that post-MS stellar evolution timescales differ for different stars (Figures 2–3).

The orbital distance of the post-MS-HZ limits change through time (Figure 3; Table 2). The inner and outer edges of the post-MS HZ for an A5 star is initially located at 3 and 7 au respectively, at the beginning of the RGB before moving to 28 and 72 au respectively by the end of the RGB (total duration: ∼50 Myr). After the helium flash, the inner and outer edges are located at 6 and 14 au respectively at the start of the AGB, extending outward to 72 and 192 au by the end of the AGB (total duration: ∼183 Myr).

Our Sun's inner and outer edges of the post-MS HZ are initially located at 1.3 and 3.3 au before expanding outward to 46 and 123 au by the end of the RGB (total duration: ∼850 Myr). During the AGB the post-MS HZ edges move outward from 5 and 13 au to 39 and 110 au respectively (total duration: ∼160 Myr).

For the coolest grid star (M1), the post-MS HZ edges are initially located at 0.3 and 0.9 au and increase out to 13 and 34 au by the end of the RGB (total duration: ∼9 Gyr).

The maximum amount of time that a planet can remain in the post-MS HZ varies with stellar spectral class (Table 3; as well as metallicity, see Section 3.2). A planet orbiting the coolest grid star (M1) can remain in the post-MS HZ for about 9 Gyr, assuming solar metallicity (Table 3). In contrast, a planet orbiting the hottest grid star (A5) can only remain in the HZ for up to 40 million years. A planet orbiting a post-MS Sun can reside in the HZ for ∼700 million years, which is comparable to the amount of time for life to evolve on the Earth (Mojzsis & Arrhenius 1997). Note that the above quoted numbers are conservative estimates because they assume that the planet's orbital radius stays constant with time. However, these times increase as the star loses mass and the planets' orbits therefore move outwards (see Section 3.3 and Tables A1–A2. Also stellar metallicity can extend that time significantly.

Stars with higher metallicity evolve more slowly and therefore the time a planet at a certain distance remains in the HZ is longer than for a star with lower metallicity (see Danchi & Lopez 2013). Higher metallicity stars have proportionately less H to fuse, which slows down nuclear burning.

We show the influence of metallicity on the location as well as evolution of the post-MS HZ boundaries by comparing the same star type with low (Fe/H = −0.5) and high (Fe/H = 0.5) metallicity for all grid stars (Figure 4). These metallicity values are those assumed at a stellar age of zero, which subsequently evolves as the star ages. We find that the time in the post-MS HZ is about half as long for a planet orbiting the low metallicity star as it is for one the high metallicity star (see Figure 4). Although stellar metallicity changes the parameters of the evolving star, both S_eff as well as the orbital distance at the limits of the post-MS HZ boundaries change. For stars with high metallicity, the post-MS orbital distance limits are smaller compared to host stars with lower metallicity (see Figure 4). Thus, for stars with non-solar metallicities, the parameterization for the distance of the post-MS HZ, Equation (9), as a function of the star's age has to be adapted according to their slower evolution.

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Previous studies have argued that life may also initiate during the relatively short period a planet spends in the post-MS HZ (e.g., Lopez et al. 2005; Danchi & Lopez 2013). However, the HZ is not the distance where life can evolve, but is used to remotely detect signs of life if it exists. Therefore life may have also initiated before the stellar post-MS on planets outside the MS HZ, thriving subsurface e.g., underneath an ice layer. Once the post-MS HZ moves outward, such worlds would get heated, potentially uncovering hidden ecosystems, which could result in remotely detectable atmospheric biosignatures.

Our Sun's post-MS HZ moves outward (except between the RGB and AGB) beyond the Kuiper Belt (see also Stern 1990) to the outer regions of the solar system. Melting would occur on icy outer bodies (i.e., Europa, Enceladus, Ganymede, Pluto). That could possibly trigger a water-based hydrologic cycle on these potentially habitable worlds.

The icy moons in our own solar system are not massive enough to maintain a dense atmosphere if heated. Therefore they should also not support an effective temperature transport between the planetary day and night side (e.g., Joshi et al. 1997; Joshi 2003; Leconte et al. 2013b).

But in other planetary systems, more massive moons or even Earth-mass planets could be located at equivalent distances. Therefore we use the location of Mars, Jupiter, Saturn, and the Kuiper Belt in our own solar system to calculate how long Earth-mass bodies could retain their atmospheres for all grid stars (Figures 5–6). We scale the initial atmospheric surface pressure of the planet with gravity (following Kaltenegger et al. 2013). We include the effect that planetary orbital distances increase as post-MS stars lose their stellar mass, including the erosion of the planet's atmosphere due to these resulting stellar winds (Figures 5–6; Tables A3–A4; e.g., Lorenz et al. 1997).

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We terminate our calculations (indicated by a filled blue sphere in Figures 5 and 6) when planetary atmospheres losses lead to atmospheric surface pressure of 0.25 bar or less. The orbital expansion for Earth-mass planets orbiting the grid host stars during the RGB and AGB stages are also shown in Figures 5–6, respectively. Our calculations initially put the planet at Mars-, Jupiter-, Saturn-, and Kuiper Belt-equivalent orbital distances (scaled to the stellar fluxes at 1.52, 5.2, 9.5, and 30 au in our solar system).

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More planetary atmospheres within the post-MS HZ of high mass stars survive through the stellar RGB and AGB phase than those within the post-MS HZ of lower mass stars because of the higher mass loss during the post-MS phase for the latter. Except for the stellar spectral type A5, all planets at the Mars-equivalent distance lose their atmospheres during the RGB phase. None of the planetary atmospheres at the Mars-equivalent distance are able to survive through the end of the AGB. However, planetary atmospheres located at least as far as Jupiter's equivalent distance form orbiting stars as least as massive as an F5 stellar type are able to retain their atmospheres through the end of the AGB.

Cool stars have longer post-MS phases and therefore planets around our coolest grid star, M1, spend the longest time in the post-MS HZ, 9 Gyr for solar metallicity. Planets at Jupiter- and Saturn-equivalent orbital distances remaining in the post-MS HZ for up to 5.8 Gyr and 2.1 Gyr, respectively (Table 3). For our Sun, a planet at Jupiter's distance could remain in the post-MS HZ for up to 370 Myr during the stellar post-MS, whereas equivalent planets at Mars' and Saturn's distances stay in the post-MS HZ for ∼200 Myr. Planets orbiting an A1 star only stay in the post-MS HZ for tens of millions of years.

In all cases, objects at the Kuiper Belt-location are also eventually heated but this heating is rather short-lived (200 and 100 million years for planets around an M1 and an A1 spectral type respectively) as it coincides with the end of the RGB.

Higher stellar mass loss rates for low mass stars increase atmospheric mass loss for their planets. Although the increase in stellar mass loss would also result in more pronounced orbital radius expansion for these planets (Table A2), the maximum times a planet can spend in the post-MS HZ given in Table 3 are negligibly affected because 1) the bulk of the radial expansion occurs near the tail end of the AGB, and 2) this AGB expansion occurs once the planet is outside the HZ in most cases (Figure 6). The orbital distances to receive the same stellar flux as planets in our own solar system are correspondingly farther away for more massive stars, reducing atmospheric mass loss for their planets.

Tables A3 and A4 in the appendix show the atmospheric mass loss for planets with 0.5, 1, 5, and 10 Earth masses for the RGB and the AGB phase of their host grid star, respectively.

Super-moons are only slightly more susceptible to atmospheric loss than Earth-mass planets and super-Earths. All modeled planets at the Mars-equivalent distance lose their atmospheres at the end of the RGB for all grid stars of solar mass or smaller (Table A3). For the smallest grid star (0.5 initial solar mass), planets out to Saturn's orbit lose their atmosphere during the RGB. During the stellar AGB phase, all planets at the Mars-equivalent distance lose their atmospheres for grid stars with 1–1.5 initial solar masses (Table A4).

More massive planets also retain their atmospheres for longer periods of time than do less massive ones (see Figures 5–6). Super-Earths are resistant to atmospheric loss compared to Earth-mass planets. At larger distances, even super-moon atmospheres located at the Kuiper-belt equivalent distance survive through the entire AGB phase for all grid stars.

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We use known exoplanets to compare their orbital distance to the post-MS HZ to estimate detection capabilities for such planets. The planets currently detected at large distances are young, massive worlds that have not yet entered the post-MS HZ stage. But they show that the post-MS HZ is populated by known young planets. They also showcase the capabilities of current detection methods. We use three known exoplanet systems with detected planets located at orbital distances between 9 and 110 au and plotted their orbits on top of the post-MS HZ (see Figure 7). The planets are shown as pointed lines at their current orbital distance in the plot. Those orbital distance lines intersect with the post-MS HZ distance for all three systems. Note, that because the exact characteristics of these planets are unknown, we could not estimate the evolution of their mass and orbital distance through the stars evolution into the post-MS.

Beta pictoris (spectral class: A6) is ∼10–20 million years old and has a mean surface temperature of ∼8050 K and luminosity 8.7 times that of the Sun (Crifo et al. 1997; Zuckerman et al. 2001; Gray et al. 2006). Beta pictoris b has a mass between 4 and 11 Jupiter masses and a radius 65% larger than Jupiter's (Currie et al. 2013), orbiting at a distance of ∼9 au from Beta pictoris. Figure 7 (top) shows the post-MS HZ orbital distance for the host star.

Fomalhaut (spectral class: A3) is ∼440 million years old and has a mean surface temperature of ∼8590 K and luminosity 16.63 times that of the Sun (Mamajek 2012). Fomalhaut b orbits at a distance ∼110 au from its parent star and has a mass estimated between 0.075 and 3 Jupiter masses (Kalas et al. 2013). Figure 7 (middle) shows the post-MS HZ for the host star.

HR 8799 (spectral class: A5 ${\rm{V}};\lambda$ Boo} is ∼30 million years old and has a mean surface temperature of 7430 K and luminosity 4.92 times that of the Sun (Gray & Kaye 1999). The planets e, d, c, and b orbit ∼14.5, 24, 38, and 68 au from their parent star, with d, c, and b measured to have a planetary radius of 1.2 times that of Jupiter (Marois et al. 2010). The planets e, d, and b each have masses equal to 7 times that of Jupiter, with b having a mass 5 times that of Jupiter (Schneider 2008; Marois et al. 2010). Figure 7 (bottom) shows the post-MS HZ for the host star.