EXPANDING THE CATALOG: CONSIDERING THE IMPORTANCE OF CARBON, MAGNESIUM, AND NEON IN THE EVOLUTION OF STARS AND HABITABLE ZONES

Here we present an extended grid of stellar models suitable for the prediction of HZ locations. In T15, we discussed the importance of mass, metallicity, and oxygen abundance to the stellar evolution. In this paper, we focus on the variation observed in carbon and magnesium abundances, which also produce a measurable effect in the stellar evolution (albeit smaller than the effect observed for variations in oxygen) and which also exhibit substantially variable abundance ratios in neighboring stars (Takeda 2007; Mishenina et al. 2008; Neves et al. 2009; Young et al. 2014; Pagano et al. 2015). We also include a discussion on the practicality of considering neon's contribution to stellar evolution, though the range of abundance values we quote are not based directly on observational data. In this work, ratios without brackets (e.g., C/Fe) indicate the linear absolute abundance ratio in terms of mass fraction, while a bracketed ratio denotes the log of the atom number relative to the solar abundance value for that same element. The latter is the conventional [C/Fe] given by

$$\begin{eqnarray}&&[{\rm{C}}/\mathrm{Fe}]={\mathrm{log}}_{10}\displaystyle \frac{({\rm{C}}/\mathrm{Fe})}{{({\rm{C}}/\mathrm{Fe})}_{\odot }}.\end{eqnarray} \tag{ 1 }$$

We primarily quote linear ratios relative to solar (i.e., C/Fe = 1.72 C/Fe_⊙) since the range of abundance ratios is small enough to not require logarithmic notation. We use the mass fraction because this is the conventional usage for stellar evolution calculations.

We again consider the major contributors to stellar evolution: mass, metallicity (Z), and individual specific elemental abundances. Variations in Z alone are made with a fixed abundance pattern that is uniformly scaled, while the spread in carbon and magnesium values we use reflects the actual observed variations in abundance ratios in nearby stars (Bond et al. 2006, 2008; Ramírez et al. 2007; González Hernández et al. 2010; Hinkel et al. 2014; Young et al. 2014). One exception is that the range in neon values we use does not result from observed neon abundances in stars; rather, we vary neon relative to solar to create a range of values that we might reasonably expect to see in stars if neon could be measured more accurately. Changes in C/Fe_⊙, Mg/Fe_⊙, and Ne/Fe_⊙ at each metallicity are made by changing the absolute abundance of each element while holding all other metal abundances constant. The relative abundances of hydrogen and helium are adjusted in compensation to ensure the sum of mass fractions = 1.

Beyond the original grid for oxygen (discussed in T15) that comprised a total of 376 models, we now introduce an additional 240 models for both carbon and magnesium. Also, for the purposes of this work, we have produced a smaller grid of 48 models for neon that includes only end-member cases of interest, resulting in a total addition of 528 new models. The grids for C, Mg, and Ne still encompass stars of mass 0.5–1.2 M_⊙ at each 0.1 M_⊙ (which includes spectral types from approximately M0–F0 at solar metallicity), overall scaled metallicity values of 0.1–1.5 Z_⊙ at each 0.1 Z_⊙, and now abundance values of C, Mg, and Ne ranging from 0.58–1.72 C/Fe_⊙, 0.54–1.84 Mg/Fe_⊙, and 0.5–2.0 Ne/Fe_⊙.

The models included in our catalog were simulated using the stellar evolution code TYCHO (Young & Arnett 2005). As detailed in T15, TYCHO outputs information on stellar surface quantities for each time-step of a star's evolution, which we then use to calculate the inner and outer radii of the HZ as a function of the star's age. New OPAL opacity tables (Iglesias & Rogers 1996; Rogers & Nayfonov 2002) were generated at the specific abundance values needed for each enriched and depleted C/Fe, Mg/Fe, and Ne/Fe value to match the desired composition of the stellar model. The TYCHO evolutionary tracks are used as input to our HZ calculator (CHAD), which is easily upgradable to incorporate improved HZ predictions as they become available.

We have recently implemented improved low temperature (∼2400 K) opacity tables in TYCHO, and we are now able to more accurately simulate evolutionary tracks, particularly for very low mass stars. The new low temperature opacities are based on Ferguson et al. (2005) and Serenelli et al. (2009) and include dust grain opacity. Ultimately, it will be extremely important to include M-stars in our catalog due to the high probability that they may host a habitable world (Borucki et al. 2010, 2011; Batalha et al. 2013). We will explore the ramifications of variable stellar composition in a grid of M-stars in a future paper. We have recalculated the original oxygen grid that was discussed in T15; we provide updated oxygen values alongside data for carbon, magnesium, and neon for certain parameters of interest.

Table 1 shows the MS lifetimes (in gigayears) for standard and end-member abundance values for all elements of interest (carbon, updated oxygen values, magnesium, and neon), as well as end-member metallicity values, for all masses in our grid. When considering how a star's specific chemical composition translates to its MS lifetime, we would expect that a star with higher metallicity (or enriched elemental abundances) would live longer than a star of the same mass with lower overall opacity. Surprisingly, this is not what we see for some of the carbon models in our grid. Upon close inspection of the listed table values (particularly for the 1.5 Z_⊙ cases), an unexpected trend emerges; specifically, it appears that some of the depleted carbon cases (0.58 C/Fe_⊙) actually have longer MS lifetimes than the associated enriched carbon cases (1.72 C/Fe_⊙). With further examination of the lifetimes given for the other elements, it is clear that the MS lifetimes do not exhibit the same inverted lifetime expectancies for these models as they do for some of the carbon cases.

In order to understand the puzzling behavior of the carbon models, we have examined two possibilities. First, since discrepancies in the expected stellar ages are sufficiently small compared to the overall calculated MS lifetimes, numerical uncertainties in the code that determine where TYCHO terminates the MS may be larger than the variability that we actually measure for the MS lifetimes. TYCHO determines the Terminal Age Main Sequence (TAMS) by stopping the code when the abundance of hydrogen in the innermost model zone drops below 1 part in 10⁶. Rezoning in TYCHO is adaptive, so minor differences in the size of the innermost zones and diffusion/convection across those zones can cause small (i.e., <1%) variation in the output value of the TAMS. Second, because of compositional normalization that is applied when creating opacity tables, the depleted carbon (and magnesium) models start out with slightly more hydrogen to ensure that the total mass fraction = 1, which may allow for "extra" MS lifetime if that hydrogen becomes available for core burning. For the highest mass stars in our sample that develop convective cores, the extent of the convective core changes slightly due to the change in electron fraction (the convective core is high enough in temperature to be dominated by electron scattering opacity) and the energy generation by the CNO cycle with a different amount of catalysts. Additional carbon also shifts the position of the second peak in the opacity versus temperature relationship in the OPAL tables, impacting the location of the convection zone base. Each of these are very small effects; it turns out that the variations in lifetime from carbon are also relatively small. Given that the effect is seen preferentially at higher metallicity and higher mass, the dominant effects are a combination of slightly increased hydrogen mass fraction and central zoning, which enhance convective transport and the CNO catalysts that play a role in the more massive stars.

To further interpret how differences in the stellar models are caused by variations in the specific elemental abundance values, notably carbon, it is also important that we distinguish between the effects from the stellar opacity and the effects from nuclear reactions. Our models (at solar metallicity) encompass the mass range wherein the transition from pp-chain to CNO-dominated hydrogen burning occurs (≤1.1 M_⊙). The energy generated by the CNO cycle depends on the mass fractions of hydrogen and total CNO catalysts X_p∗X_CNO. For models of the same mass but at much lower opacity (i.e., Z = 0.1 Z_⊙) and relatively higher temperatures, it might seem that the transition between pp-chain and CNO burning should therefore shift to a lower mass value. However, since CNO burning is extremely temperature sensitive and the transition is dependent on the amount of CNO catalysts (X_p/X_CNO)^1/12.1 (Arnett 1996), the lower metallicity stars in our sample do not actually transition to CNO burning at a significantly lower mass. The same consideration can be made for changes in the opacity due to enhanced or depleted specific elemental abundances. Changes to carbon, in particular, will alter the amount of energy generated via the CNO cycle, so the effect is more pronounced in the highest mass stars in our sample. Overall, however, the contribution from carbon is relatively insignificant.

Figure 1 shows the Hertzsprung–Russell Diagrams (HRDs) for evolutionary tracks from ZAMS (Zero Age Main Sequence) to TAMS for all masses in our grid. The top row is for carbon, where C/Fe = 0.58 C/Fe_⊙ (dashed), 1.0 C/Fe_⊙ (solid), and 1.72 C/Fe_⊙ (dotted), all at Z = Z_⊙. The middle and bottom rows (respectively) show the similar HRDs for magnesium and neon, where Mg/Fe = 0.54 Mg/Fe_⊙ and Ne/Fe = 0.5 Ne/Fe_⊙ (dashed), 1.0 Mg/Fe_⊙ and 1.0 Ne/Fe_⊙ (solid), and 1.84 Mg/Fe_⊙ and 2.0 Ne/Fe_⊙ (dotted), again at Z = Z_⊙. The rightward-most dotted lines are for the lowest mass star with enriched elemental values, while the leftward-most dashed line is for the highest mass star with depleted values.

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For the higher mass models (the left-hand column of Figure 1), we see evidence of the Kelvin–Helmholtz mechanism (KH "jag"), wherein a star nearing the end of its MS lifetime begins to cool and compress due to decreased internal pressure from the end of core hydrogen burning. This compression reheats the core, causing the observed fluctuations in L and T_eff. A detailed scrutiny of the figures reveals a slight crossover that occurs in the late MS for both carbon and magnesium, for the depleted (dashed line) cases relative to standard (solid line) cases. The crossover occurs due to the slightly larger core in the high C models; thus, the shift in the KH-jags for these models on the HR diagram is physical, from variability that exists in the interior structures of the stars. The total abundance of carbon in stars is, generally, a factor of several higher than for that of magnesium (e.g., Lodders 2010, p. 379); however, the abundance range of carbon (from 0.58 to 1.72 C/Fe_⊙) is smaller than that of magnesium (from 0.54 to 1.84 Mg/Fe_⊙), and magnesium contributes more opacity per gram in the stellar interior than carbon does (e.g., Morse 1940). Thus, magnesium actually makes a bigger difference to the evolution relative to its abundance in stars. Oxygen is not only much more abundant than carbon, but also has a high contribution to the opacity.

Table 2 shows Δ(L/L_ZAMS) at each mass and end-member composition for all elements. As expected, the change in luminosity over the MS is largest for less enriched compositions except in the case of the higher metallicity, higher mass stars, where the shape of the K–H jag obscures the trend. Table 3 similarly shows ΔT_eff at each mass and end-member composition for all elements. The lowest mass, lowest opacity models all exhibit the largest change in temperature over the course of their MS lifetimes, even though they do not live quite as long as higher opacity stars at the same mass. Interestingly, even though we see the highest ΔT values for depleted magnesium (0.54 Mg/Fe_⊙ at 0.1 Z_⊙), the largest change in L actually occurs for the depleted oxygen model (0.44 O/Fe_⊙, though also at 0.1 Z_⊙).

This work constitutes a sound argument for considering the contributions of neon (and, to some extent, nitrogen) to the stellar evolution. Neon would definitely be an important player in the evolution based on its opacity contributions per unit mass (similar to that of magnesium). It is difficult to assign the appropriate abundance ratio ranges for modeling, as it is challenging to measure neon in stars with much certainty, though work has been done to measure neon abundances from the X-ray spectra of cool stars (Drake & Testa 2005). For the purposes of this work, we have assigned an artificial range of neon abundances (enriched and depleted by a factor of two from the solar neon abundance, similar to the range of other low Z elements), which we can use to estimate contributions to the stellar evolution. Nitrogen is also not easily measurable in stars, but can probably be safely neglected; it is similar in opacity per gram to carbon, but relatively less abundant in stars, by a factor of about four in the Sun (Hansen et al. 2004). Thus, its contribution to the stellar evolution is likely negligible even though it is more abundant than either magnesium or neon. One exception to this would be if nitrogen is actually observed to be widely variable in stars with future measurements; if the abundance values vary a great deal more between individual stars than other elements, it could be an important consideration.

Now consider the rate of change of the luminosity (Table 4) for all masses in our grid at end-member compositions. It is especially useful to look at the change of luminosity per gigayear because some of the models undergo a larger change in L than do the higher opacity models at the same mass, but potentially over longer or shorter MS lifetimes. This could have different implications for whether the change in luminosity with time is greater or smaller for low opacity models (or if it varies), and whether that occurs during the second half of the star's MS lifetime. With few exceptions, the low opacity models at each mass and elemental composition change more in L per gigayear than their counterparts at higher opacities. Additionally, and as expected, it is clear that the higher mass models experience a significantly larger change in L over the course of their MS lifetimes.

Since we understand how the luminosity changes over time (the rate of change, as well as the total change), we see that the range of orbits in the HZ at different points in the MS evolution can vary substantially. Table 5 shows the fraction (listed as percentages) of orbital radii that only enter the HZ after the midpoint of the MS for each star. The results indicate that up to half of all orbits that are in the HZ only become habitable in the second half of the host star's MS lifetime. The effect is more pronounced at higher mass and enriched composition, at each element of interest. When considering the potential for detectability, it is wise to avoid planets that have only recently entered the HZ of the host star; not only would we potentially circumvent the issue of cold starts (discussed in Section 3.3), but we also assume that life requires enough time spent in "habitable" conditions before it would yield detectable biosignatures. This is a somewhat narrow assumption that depends on specific habitability considerations; indeed, Silva et al. (2016) introduces an alternative "atmospheric mass HZ for complex life" with an inner edge that is not affected by the uncertainties inherent to the calculation of the runaway greenhouse limit.

We produce complete evolutionary tracks for the position of the HZ as a function of time for all stellar models. With an independent age estimate for the star, as well as measurements for its mass and specific elemental composition, we can fairly accurately predict the future and past location of a given exoplanet, and whether that planet ever inhabited the parent star's HZ; furthermore, we can assess the timeline for when a planet will enter the star's HZ if it has not yet, as well as estimate how long the planet has been outside of the HZ if it has already departed. Assuming the aforementioned stellar properties are well measured, the time that an observed exoplanet may exist in the HZ can be estimated to the level of accuracy of the planetary atmosphere models that predict the HZ boundaries.

Generally, we find that a higher abundance of carbon, magnesium, or neon in the host star correlates with a closer-in HZ , because the star is less luminous, at a lower T_eff, and the MS lifetime is longer. Likewise, a lower elemental abundance value will typically produce shorter overall MS lifetimes with HZ distances that are farther away from the host star, which is the same trend that we observed for oxygen abundance ratios in T15.

Figures 2–4 show polar plots for carbon, magnesium, and neon, respectively. These figures are meant to demonstrate how the HZ varies between different kinds of stars, and what the differences would look like from a perspective perpendicular to that of a hypothetical planet's orbital plane. These figures each include stars of end-member masses 0.5 M_⊙ star (top) and a 1.2 M_⊙ star (bottom), at the lowest and highest composition cases for each element of interest. HZ boundaries are solid for the ZAMS and dashed for the TAMS. The inner and outer HZ boundaries should be clear based on their positions relative to each other. Only the high mass stars exhibit a small degree of overlap between the outer edge at the ZAMS and the inner edge at the TAMS, which might correspond to a "Continuously Habitable Zone" (see Section 3.3). Given this HZ prescription, it is clear that there are no orbits around the low mass stars that remain within the HZ for the entire MS lifetime. Ultimately, this does not matter much in the sense that the low mass stars are sufficiently long-lived that they would still provide a significantly long CHZ. However, we do eventually need to assess the variation of M-star activity with age, since an extremely long continuously HZ lifetime would not necessarily be enough to overcome a harsh radiation environment. We will explore these ideas further in a future paper. From the polar figures, we also see that the HZ can change substantially over the MS depending on the host star's specific chemical composition. Tables 6 and 7 show changes in the location of the HZ radius in au from ZAMS to TAMS for both the Runaway Greenhouse inner boundary (RGH), and the Maximum Greenhouse outer boundary (MaxGH), respectively, which are the conservative HZ limit cases discussed in Kopparapu et al. (2013, 2014).

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The left-hand column of Figure 5 shows the inner and outer edges of the HZ for each stellar mass for all compositions at the ZAMS, while the right-hand column shows the same information for the TAMS. The top row is for carbon, the middle row is magnesium, and the bottom row is neon. There are a smaller number of neon lines included since we only modeled the end-member scenarios for the neon cases. We find that for all elements, a higher overall opacity results in the associated HZ boundaries at radii much closer to the host star. As observed with our original grid of oxygen models, we see that with increasing stellar mass, there seems to be a widening of the overall HZ range, as well as a larger spread in the HZ distances due to compositional variation. The spread in specific abundance ratios for each element of interest (at solar metallicity value) are indicated by the elongated solid lines.

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The range in distance of the HZ edges for the each element at Z_⊙ is clearly smaller than the range that exists for the variations in overall scaled metallicity. This is expected, since the total change in opacity of the stellar material is much larger for a factor of fifteen change in total Z than a factor of about two change in each elemental abundance. This figure also similarly includes the spread in C, Mg, or Ne, calculated at each scaled metallicity value. The elongated dotted lines represent the end-member values for the spread in C, Mg, and Ne abundance (0.58 C/Fe_⊙, 1.72 C/Fe_⊙, 0.54 Mg/Fe_⊙, 1.84 Mg/Fe_⊙, 0.5 Ne/Fe_⊙ and 2.0 Ne/Fe_⊙, respectively) calculated at end-member metallicity values (0.1 and 1.5 Z_⊙). These models extend the range of HZ distance even further than do the models for elemental abundances at Z_⊙ alone. The observed difference in HZ location as a function of composition is larger for higher mass stars because the absolute change in L is larger. The outer HZ edge changes more than that of the inner edge because calculation for the Maximum Greenhouse limit is more sensitive to the spectrum of the incoming radiation and T_eff.

A higher specific elemental abundance ratio present in the host star will generally result in a closer HZ, because the star would be less luminous and at a lower effective temperature. Additionally, a star of higher opacity will live significantly longer on the MS than a star of equal mass at lower opacity, due to the higher efficiency of radiation transport in the star. Figure 6 shows the HZ distance as it changes with stellar age, for three stellar mass values (top is 0.5 M_⊙, middle is 1.0 M_⊙, bottom is 1.2 M_⊙), for five different compositions at each element of interest (carbon in left column, magnesium in middle column, neon in right column). Each color represents a different abundance value: black is solar, orange is for depleted elemental values (0.58 C/Fe, 0.54 Mg/Fe, 0.5 Ne/Fe), and green is for enriched elemental values (1.72 C/Fe, 1.84 Mg/Fe, 2.0 Ne/Fe). For comparison, the red lines represent 0.1 Z_⊙ and the blue lines represent 1.5 Z_⊙. A 1 au orbit is also indicated by the dotted line in each frame, for reference. It is clear that abundance variations within a star significantly affect MS lifetime and HZ distance. As expected, the shortest lifetime corresponds to a star with a total metallicity of Z = 0.1 Z_⊙ and the longest lifetime corresponds to Z = 1.5 Z_⊙. However, when considering only the variations in the specific elemental abundances, we see that changes in magnesium and neon make the largest difference to the evolution, followed by carbon. The total MS lifetime for a 0.5 M_⊙ star at end-member neon abundances (at Z_⊙) varies by about 7 Gyr, which can also be determined by examining Table 1. Likewise, the MS varies by about 4 Gyr for magnesium, and only about 1 Gyr for carbon.

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It should now be abundantly clear that it is an extremely useful pursuit to quantify how long any given planet would remain in a star's HZ as a function of its orbital distance; however, the instantaneous habitability of a planet alone is insufficient to determine the likelihood that it actually hosts extant life, or whether any life present would even be detectable. Groups at the Virtual Planet Laboratory at the University of Washington have worked on this problem from the perspective of viewing the Earth as an exoplanet, in order to determine the current technological limits of what "biosignatures" might be measurable in the atmospheres of real exoplanets (e.g., Harman et al. 2015; Krissansen-Totton et al. 2016). This kind of information plays an integral role in determining how we think about habitability; indeed, with more sophisticated planetary atmosphere models and a broader understanding of what might be directly observable about them (Kasting et al. 2014), we will have a better idea of how to apply the data from our stellar evolution tracks to paint a more complete picture of HZ evolution around different types of stars.

In T15, our initial goal was to estimate a CHZ for each star in our catalog, which would simply include a range of orbital radii that remain in the HZ for the entire MS. The CHZ is rather straightforwardly defined by considering the boundary overlap between ZAMS and TAMS. The left-hand column of Figure 7 shows the CHZ for stars of all masses in our grid, at a composition of solar metallicity and enriched abundance values for each element of interest. The top row is for carbon, the middle is magnesium, and the bottom is neon. It is clear that the low mass stars have no CHZ for the conservative HZ limits (at all elements), which would seem to indicate that low mass stars would have a low statistical likelihood to host a long-term habitable planet; however, that is somewhat misleading due to the extremely long MS lifetimes of low mass stars. Thus, we have defined a much more useful 2 Gyr CHZ (the CHZ₂), which is the range of orbital radii that would be continuously habitable for at least 2 billion years. We use this time because it is estimated that life on Earth took approximately 2 Gyr to produce a measurable chemical change in the atmosphere (Summons et al. 1999; Kasting & Catling 2003; Holland 2006). Of course, the CHZ₂ assumes that Earth's timescale for the evolution of life with the capability to modify the entire planetary atmosphere is representative of the norm. This is not meant to imply that other suggested timescales are at all unreasonable (e.g., Rushby et al. 2013), but in order to narrow down the large pool of potentially habitable exoplanets to the ones that would have the highest potential for both long-term habitability and detectability, there is an advantage in using Earth's history as a starting point. In addition, we provide a robust consideration of the HZ evolution because we incorporate detailed stellar properties. It is fairly clear that for this particular consideration, the variations in the elemental abundances of interest do not show a significantly large difference between them, though the difference is more pronounced when compared to the associated figures from T15 that were created for the oxygen cases.

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The right-hand column of Figure 7 shows the orbits that remain habitable for at least 2 Gyr (CHZ₂). This is determined from the inner edge of the HZ at 2 Gyr after the beginning of the MS and the outer edge 2 Gyr before the TAMS. Now we see that the significantly longer MS lifetimes of the low mass stars create a higher proportion of the HZ that is included in the CHZ₂ than for the basic CHZ. Based on these results, we would be less likely to find a planet that has been in the CHZ for at least 2 Gyr orbiting a more massive star; at the very least, we would be less confident that a planet located outside of the CHZ₂ would produce detectable biosignatures than one within the CHZ₂. Table 8 shows the fraction of time a planet would spend in the CHZ₂ versus time it would spend in the HZ over its entire MS lifetime. Comparing stars of interest with chemical compositions will inform which ones we should focus on in the continued search for detectable inhabited exoplanets.

As in T15, our consideration of HZ evolution and the CHZ must also address the issue of cold starts. Our discussion until now has assumed that any planets initially beyond the boundaries of the HZ could easily become habitable as soon as the host star's HZ expanded outward to engulf them; indeed, the albedos used in the planetary atmosphere models of Kopparapu et al. (2014) are relatively low, which assumes a planet could fairly easily become habitable upon entering the HZ. However, it may be unlikely that a completely frozen planet (a "hard snowball") entering the HZ late in the host star's MS lifetime would receive enough energy in the form of stellar radiation to reverse a global glaciation, especially if the planet harbors reflective CO₂ clouds (Caldeira & Kasting 1992; Kasting et al. 1993).

Figure 8 offers an alternative scenario to that of Figure 6, wherein we presumed a cold start would be possible and thus allowed the outer HZ limit to expand with time. Instead, we now treat the outer boundary of the HZ at ZAMS as a hard limit that does not co-evolve with the star, so a particular planet would be required to exist in the HZ from the beginning of the host star's MS in order to be considered habitable over the long-term. Obviously, a planet that is in a star's HZ from very early times would not face the problem of a cold start. We defer discussion of the cold start problem as applied to the CHZ to our previous paper.

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