A Framework for Relative Biosignature Yields from Future Direct Imaging Missions

To characterize the atmospheric compositions of Earth-like planets, the optimal detection technique would be to directly image them. However, designing an instrument capable of blocking out a star’s light to allow observations of the very faint light reflected off a planet’s surface is a major engineering challenge. The direct imaging method of exoplanet detection favors larger planets at high separations from their stars, and all planets found to date using direct imaging have been longer period, giant planets. To extend the capacities of direct imaging to be able to observe Earth sized planets in the habitable zones of their stars, one requires a large next generation telescope and a sophisticated coronagraph with a sufficiently small inner working angle.

For the 2020 Astrophysics Decadal Survey, NASA has solicited four Large Mission Concept studies, two of which aim to directly image habitable Earth-sized planets. The Large UV/Optical/Infrared Surveyor (LUVOIR) is a concept for a large telescope capable of observing wavelengths ranging from the UV to the infrared (The LUVOIR Team, 2019). LUVOIR has two potential architectures: the 15 meter diameter LUVOIR-A and 8 meter LUVOIR-B. Both of these architectures use an internal coronagraph to block out a star’s light, but they differ with regards to estimated cost and the design of the coronagraph. The other main direct imaging mission concept under consideration is the Habitable Exoplanet Observatory (HabEx) (Gaudi et al., 2020). While the HabEx telescope has a 4 meter diameter, smaller than the LUVOIR telescope, it makes up for it by having a 52m starshade external to the telescope. In addition to the starshade, HabEx has an alternative vector vortex coronagraph on the primary spacecraft and a UV spectrograph.

For future direct imaging missions, it is important to gain a sense for the estimated number of Earth-like planets for which we will be able to obtain spectra. Stark et al. (2019) have calculated the estimated yield of Earth-like exoplanets for different mission architectures. They suggest that internal coronagraph-based missions may have yields that increase steadily with telescope diameter, while starshade missions appear to have diminishing returns past a certain aperture size. Such yield estimates are dependent on a variety of factors. For instance, to calculate the number of Earth-sized planets one expects to image, one needs to have prior knowledge of $\eta_{\earth}$, the occurrence rate of Earth-like planets. Given the large uncertainties in current estimates of $\eta_{\earth}$, exoplanet yield estimates are similarly uncertain. Furthermore, yield calculations need to consider the exposure times required to obtain enough light to construct a quality spectrum (Stark et al., 2015). Exposure times will depend on the brightness of the host star and planet, the amount of exozodiacal background light in the system, and the required signal to noise ratio. One must also consider the time required to maneuver the telescope to view each target star, especially in the case of a starshade mission, and the fuel requirements of the spacecraft (Stark et al., 2016).

To maximize the yield of habitable Earth sized exoplanets or biosignatures, future missions will need to select an optimal sample of target stars. Such a list of target stars has not been finalized yet due in part to the fact that many or most of the planets that these missions will characterize have likely not been discovered. Preliminary lists are limited to bright nearby stars because of the faintness of the reflected light from exoplanets and the angular separations required to resolve a planet separately from its star. The stars which a mission chooses to observe depends on the observational strategy employed. If one utilizes a strategy of “statistical comparative planetology” as described by Bean et al. (2017), one aims to obtain observations of a few key properties of a large number of planets to test our models of planetary habitability. Alternatively, a complementary strategy is to focus on a smaller number of exoplanets most likely to host biosignatures and thoroughly characterize them. Regardless of the approach one takes, one must design missions around a coherent strategy that considers which stars are most likely to host habitable planets. In this paper, we will develop a framework to rank stars according to their potential to host planets with observable biosignatures. This will serve as a basis for future studies aiming to assess the biosignature yield of direct imaging missions.

A mission to search for biosignatures should have the ability to image planets in the habitable zones of their stars. While the habitable zone is not a specific or complete description of planetary habitability, it has proved to be an invaluable tool in the search for habitable exoplanets because it provides a fiducial target for the discovery of truly Earth-like planets. Defined as as the region around a star where a rocky planet could support liquid water on its surface, given a $\text{CO}_{2}$-$\text{H}_{2}\text{O}$-$\text{N}_{2}$ atmosphere, the conventional habitable zone provides astronomers with a general location for where to search for Earth-like planets (Maunder, 1913; Huang, 1959; Kasting et al., 1993; Lorenz, 2020). Given the flux constraints for the inner and outer edges, it is straightforward to calculate the habitable zone around any star of known luminosity and temperature.

Formulations of the habitable zone vary based on assumptions about planetary atmospheric physics and composition. Some models consider only $\text{CO}_{2}$ and $\text{H}_{2}\text{O}$ to be the prominent greenhouse gasses in planetary atmospheres, while others extend the definition of the habitable zone, including dry planets with low $\text{H}_{2}\text{O}$ concentrations, and considering alternative sources of greenhouse heating such as $\text{H}_{2}$ atmospheres (Kasting et al., 2014; Kopparapu et al., 2013; Abe et al., 2011; Pierrehumbert & Gaidos, 2011). Given the large variability in the habitable zone boundaries, future missions aim to observe the planets around the general location of the habitable zone, and determine empirically where habitable planets occur.

While the habitable zone gives a range of distances where an Earth-like planet may be habitable, it is not the only location where habitable planets may be found. For instance, in our own solar system, moons of giant planets such as Europa and Titan may host habitable environments. However, given the challenges in assessing the habitability of such locations in our own solar system, assessing the habitability of analogous environments around other stars may be insurmountably difficult, and biosignatures in these locations may not be detectable.

Knowledge that a planet exists in its star’s habitable zone does not necessarily mean that it is habitable. Rather than depending on the bulk properties of a planet, such as mass, radius, and orbital period, habitability is dependent on the climatic and geological history of a planet (Lenardic et al., 2016). Planets exhibiting the same bulk properties, may be vastly different in terms of their potential for habitability. One of the the key influences on the climatic history of a planet is that stars do not stay constant in time. Therefore, as the luminosity and effective temperature of a star evolve in time, so too does its habitable zone.

To characterize the habitability of an exoplanet we need the following properties of its host star. The star’s luminosity and effective temperature have a fairly straightforward effect on a planet’s habitability in that they define the current day instantaneous habitable zone (IHZ). The mass and age of a star are also required to assess the habitability of its planets. These have a less obvious effect on habitability, determining the evolutionary history of a star and how its luminosity and temperature, and thus its habitable zone, evolve in time (Gallet et al., 2019). Whereas the luminosity and effective temperature of a star can be determined via direct observations, the mass and age of a star require a stellar model to be fit to the observed properties of a star. Masses and ages are therefore harder to obtain and typically have much larger uncertainties. In addition to these stellar properties, other factors such as metallicity, helium abundance and rotation rates play an important role in the evolution of stars and their habitable zones (Danchi & Lopez, 2013; Valle et al., 2014). Metallicity from a scaled solar abundance pattern may not be enough to describe habitable zone evolution, as stellar composition and the abundances of specific chemical species may have a significant effect on stellar evolutionary tracks (Truitt et al., 2015; Truitt & Young, 2017). It is clear that in order to tell whether a planet is habitable and how long it has been habitable, one needs to be familiar with many of the properties of its host star.

For a planet to be a potential host for biosignatures, it must have maintained habitable conditions long enough for life to originate and develop. Planets that currently reside in the habitable zone, but haven’t been there for long are sub-optimal targets for direct imaging missions, as life wouldn’t have time to develop. In the case of the Earth, life is believed to have evolved very early in the Earth’s lifetime (Schopf, 2006). However, it wasn’t until 2.4 - 2.1 Gyr ago, during the Great Oxidation Event (GOE), that oxygen from photosynthesis appeared as a potentially detectable biosignature (Kasting, 1993; Lyons et al., 2014). Even after the GOE, oxygen levels in the Earth’s atmosphere may have been comparatively much lower than today, lasting up until the end of the Proterozoic eon, roughly 600 Myr ago (Crockford et al., 2018). It is unclear whether or not oxygen would have been observable to a hypothetical observer prior to this period. If one assumes that life on other planets may evolve on similar timescales to life on Earth, to detect observable biosignatures, one would need to look at an Earth analog that has been in the habitable zone for at least 2 Gyr (Truitt et al., 2015). Thus, the present day habitable zone alone is insufficient information to determine a planet’s likelihood to host biosignatures.

The concept of a continuously habitable zone (CHZ) has been used to define a region around a star where planets remain habitable over an extended period of time. There is no one definition for a continuously habitable zone, but definitions usually range between having the planet remain in the habitable zone for 2 Gyr, to 4 Gyr, to the star’s entire main sequence lifetime (Truitt et al., 2015; Buccino et al., 2006; Valle et al., 2014). Past studies have also considered the amount of time which a planet spends in the habitable zone over the course of its star’s main sequence evolution, which has been referred to by different names by different studies, such as the habitable zone lifetime or duration of transit of the habitable zone (Rushby et al., 2013; Danchi & Lopez, 2013; Valle et al., 2014; Waltham, 2017).

For the purpose of life detection, it bears little consequence if a planet will spend a large amount of time in the habitable zone in the future; only the character and duration of its past habitability matters. Therefore, for this study, we use a definition of the CHZ as the region around a star that has remained in the star’s habitable zone from 0.2 Gyr after formation, roughly the age for which we have the earliest evidence of Earth supporting a solid surface and liquid water (Wilde et al., 2001), up until present. Past studies, such as that of Hamano et al. (2013), suggest that analogous terrestrial planets should have surfaces that solidify after around 100 Myr at the most, so the use of the Earth-based timescale for a solid surface is a conservative upper estimate for the beginning of habitability.

Figure 1 shows how the habitable zone of a sun-like star changes as a function of time. Here we use the Kopparapu et al. (2013) formulation of the habitable zone and show the evolution of both the conservative and optimistic definitions of the inner and outer edges. The CHZ as it is defined in this paper is shown in green, using the conservative definition of the habitable zone, where the inner and outer edges are defined by the moist greenhouse limit and maximum greenhouse heating respectively. The stellar model shown here is a MESA model described in Appendix A, using approximately solar properties, with a mass of $M=1.00M_{\odot}$, initial helium and metal mass fractions of $Y_{i}=0.28$ and $Z_{i}=0.02$, and an age of $4.57$ Gyr (Bahcall et al., 1995).

As we are concerned with the emergence of biosignatures on habitable exoplanets, the amount of time that a planet has spent in the habitable zone prior to the current day is of principal interest. We next define the habitable duration, $\tau$, as the time that a hypothetical planet at a fixed orbital radius has spent in the habitable zone before present. Habitable duration is similar to the “habitable lifetime” of Waltham (2017) and the “duration of transit” of the habitable zone of Danchi & Lopez (2013); Valle et al. (2014). These other definitions are typically used in a more theoretical context, and consider the amount of time a planet would spend in the habitable zone over the entire main sequence and post-main sequence evolution of its host star, whereas our habitable duration considers only the past history of a given star.

In Figure 2, we plot the habitable duration as a function of orbital distance from a star. This plot is constructed using the same model for stellar evolution of a sun-like star seen in Figure 1, and the habitable duration can be thought of as the span on the time axis bounded by the inner and outer habitable zone boundary curves and the present age and minimum age for habitability. There are 4 dotted vertical lines in Figure 2. The first and third lines from the left represent the inner and outer boundaries of the habitable zone when the star was on the zero age main sequence, while the second and fourth line from the left give the habitable zone boundaries at the present time. The red region in this plot represents regions which are currently outside of the habitable zone, and the plateau in the center represents the CHZ. The portion of the curve in the red region between the first and second line represents planets that were formerly in the habitable zone but have since become too hot. On the right of Figure 2, the downward sloping region of the curve between the third and fourth lines are planets that were outside of the habitable zone, but have entered it as the star’s luminosity increases and the habitable zone has expanded. These are the so called Cold Start planets addressed in Kasting et al. (1993).¹¹1This should not be confused with the term “cold start” used in the context of giant planet formation, an unrelated concept.

The habitability of Cold Start planets, originating outside the habitable zone, but entering it later in their lifetime, remains dubious, or uncertain at best. Different studies disagree as to whether these planets could thaw and become habitable. The high albedo of a globally glaciated planet would require a large stellar flux to thaw, perhaps greatly exceeding that of the outer habitable zone boundary (Kasting et al., 1993). Yang et al. (2017) suggest that flux required to thaw a fully glaciated planet may result in a sudden shift to a moist greenhouse atmosphere, resulting in significant water loss and hydrogen escape. Alternatively, Wolf et al. (2017) suggest that if a Cold Start planet were to support present day Earth levels of CO₂, there would be a narrow range of stellar fluxes where it could support a habitable climate for an extended period of time. However, it remains unclear whether the these globally glaciated Cold Start planets would have sufficient outgassing to allow large enough concentrations of greenhouse gases in their atmospheres. Cold Start planets are an area which warrants future research, and yet are often naively assumed to be habitable. With only knowledge of the instantaneous habitable zone of a star, one is unable to tell whether a given planet has been continuously habitable, or if it belongs to this class of Cold Starts. Future direct-imaging missions will need to choose to target or avoid Cold Start planets, so we will perform our analysis for both cases.

Cold Start planets become increasingly numerous as a star evolves. During their main sequence evolution, most stars gradually increase in luminosity, causing their habitable zones to move outwards. In Fig 3, we show how the habitable duration curve changes as a function of time for the sun-like star modelled in Figs 1 and 2. This figure shows a variation of the habitable duration curve, where regions outside of the current habitable zone are set to zero. One can observe that, as the star ages, the plateau region of the curves representing the CHZ gets progressively narrower as the inner edge recedes. On the other hand, the downward sloping region representing Cold Starts occupies an increasing portion of the habitable zone as the star ages and more planets enter the habitable zone.

To assess the potential for a star to host planets with biosignatures, we need a metric for biosignature yield that incorporates what we know about a star’s long-term habitability even in the absence of knowledge about its particular planetary system. We define the relative biosignature yield, $B$, for a given star as

where $H(a,t)$ is the probability that a planet at orbital separation $a$ from a star of age $t$ hosts detectable biosignatures, and where $\Gamma(a,R_{p})=\frac{\partial^{2}N}{\partial a\partial R_{p}}$ is the distribution of the number of planets per star in semimajor axis $a$ and planetary radius $R_{p}$. $B$ is then an expected yield for the number of biosignatures on the planets orbiting the star.

Obviously, we do not know the absolute value of $H$, but we can make reasonable inferences as to its functional form, which can guide which stars are the best targets. Past studies have made many different assumptions for $H(a,t)$ (sometimes tacitly). One may, for example, naively assume that all planets in the instantaneous habitable zone (IHZ) have an equal chance of hosting biosignatures. Studies, such as Stark et al. (2019), have calculated expected yields of Earth-like planets in the IHZ for direct imaging missions, but it should be noted that estimating the yield of Earth-like planets in the habitable zone and estimating the yield of biosignatures are not the same. If a mission to study biosignatures were to synonymize the two, then this would be equivalent to assuming:

Another choice could be to consider all planets that remain habitable beyond a given amount of time, say 2 Gyr, to have the same chance of hosting biosignatures. This would have the form of

where $\tau(a,t)$ is the habitable duration. This is the common “2 Gyr CHZ” as used by, for instance, Truitt et al. (2020).

A more general (and perhaps more realistic) case would be where there is both a constant probability of biosignature creation per unit time, $b$, and a constant probability of destruction per time, $x$. This would take the form of

In the case of small $x+b$, $H(a,t)$ can again be approximated as $H(a,t)\propto\tau(a,t)$, as in Equation 5.

For the cases where $H(a,t)$ depends on $\tau(a,t)$, there are two formulations of $\tau$ that one can use. If only planets in the CHZ are considered habitable, then $\tau$ is set to zero outside the CHZ. Alternatively, if both Cold Start planets and planets in the CHZ are assumed to be habitable, then $\tau$ is only set to zero outside the IHZ.

In this work, we will consider three of these models for $H(a,t)$: the IHZ model, and two models with constant probability of biosignature creation with time, one assuming Cold Starts are valid and one using only the CHZ. The value of $b$ in these models is, of course, unknown (just as the constant value of H(a,t) is unknown in the IHZ model), but by restricting ourselves to the relative yields expected among stars, in the small $b$ case, we need only consider the time dependence of H(a,t) and leave $b$ as a parameter to be determined from observations.

The distribution of exoplanets in semimajor axis, $\Gamma(a,R_{p})$, is also a function that is not well known for the regions of parameter space that we are interested in, but it is a quantity that future missions hope to determine.

For the purposes of this paper, our goal is not to choose the best form of $\Gamma$ to use, but to determine what effects it has on the values of the metrics. Therefore we will consider two contrasting estimates for $\Gamma(a,R_{p})$: one that is uniform in $a$ and one that is uniform in $\ln(a)$. One should note that the true form of $\Gamma$ is almost certainly dependent on more than just $a$ and $R_{p}$ of $\Gamma$ that are independent of stellar properties with the purpose of illustrating how different forms of $\Gamma$ affect expected biosignature yields.

The aforementioned metrics primarily apply to the case where a target star is not known to host planets, for instance in the early stages of mission design before planets are discovered. These metrics then serve to estimate how likely a given star is to host long-term habitability and biosignatures based on planet occurrence rates and the evolution of the star’s habitable zone. However, some future mission designs prioritize nearby stars known to host exoplanets. As such, it would be useful to have a variation of our biosignature yield metric for target selection among known planet hosts.

Rather than using the distribution $\Gamma(a,R_{p})$ for planet occurrence rates, we can instead use a function $\phi(a,R_{p})$ for the measured posterior distributions of a known planet’s semimajor axis and radius. If the detailed distribution of planetary distances and radii aren’t known, then $\phi$ could be easily be approximated as a 2D Gaussian with a diagonal covariance matrix using the squared uncertainties in $a$ and $R_{p}$. With the substitution of $\phi$ instead of $\Gamma$, the biosignature yield can be reformulated as

where here we consider the sum over $N$ planets in the system, where $\phi_{i}(a,R_{p})$ is the distribution in parameters for each planet.

Note that comparing the biosignature yields between known planet hosts and stars without known planets depends heavily on the extent to which a future mission prioritizes planet host stars. Further consideration is also needed for the case where only some of a star’s planets are known. For example systems such as 55 Cancri, which are known to host giant planets, but may contain undetected Earth-sized planets in their habitable zones, the probability of planet occurrence is presumably influenced in complex ways that lie outside the scope of this work. Because the purpose of this paper is to illustrate how to calculate biosignature yields, and because most direct-imaging target planets haven’t been discovered yet, we will restrict our analysis to the formulation of $B$ in Section 2.3.

As a first step for comparing our metrics for relative biosignature yields between stars, we would like to know how they vary as a function of time when all other stellar properties are held constant. We shall use the model of a sun-like star used in Figures 1 – 3 and calculate our metrics for it as it evolves in time. For each time step in the model evaluation, we shall calculate the different $B$ values using the evolutionary history of the star’s luminosity and effective temperature. We continue the star’s evolution beyond the subgiant phase, up until shortly before the helium flash.

In Figure 3, we illustrated how the habitable duration, $\tau$, changed as a function of time. This helps to inform how we would expect our metrics to vary for stars of different ages. If one uses a form of $\Gamma$ uniform in $a$, then the metrics that use $H(a,t)$ that is linear in $\tau$ would be directly proportional to the area under the curve for $\tau$. One can see that Figure 3 could be used to infer how the metric with $\Gamma$ uniform in $a$ and $H$ including Cold Start planets changes in time. It would appear that, in the age regime that the figure covers, the area under the curve is continuously increasing. If one only considers the plateau regions of the $\tau$ curve representing the CHZ, then it is less clear how the area, and hence the value for $B$ would change in time, as the region under the curve would get taller, but narrower.

We graph the evolution of our metrics in time in Figure 4. The left panels shows the time evolution of metrics with 3 different assumptions for $H(a,t)$, and the 2 assumptions for $\Gamma$ in the upper and lower subplots. The right panel shows the star’s evolution on the HR diagram starting on the zero age main sequence (ZAMS). The different maxima of the $B$ curves for the $\Gamma$ uniform in $a$ case are marked on the evolutionary track and the $B$ evolution curves. One can observe that the blue and orange curves, representing formulations of $H(a,t)$ that are dependent on the habitable duration, appear very similar for both assumptions of $\Gamma$, though those for the uniform in $\ln(a)$ case appear to peak a bit earlier. Looking at the blue curves, one can see the metrics which consider Cold Start planets to be habitable continue to increase until they peak for older stars near the end of the main sequence. Interestingly, they appear to have secondary maxima on the subgiant branch, which likely corresponds to the onset of degeneracy in the core. After this point, the star’s luminosity increases quickly and regions that enter the habitable zone don’t remain habitable for long, causing B to drop off sharply.

The metrics which only consider the CHZ in $H$ behave similarly to the Cold Start metrics, but peak around 2 Gyr earlier. It still appears that older stars are preferred by these metrics, but not quite as old as those of the Cold Start metrics. As the star reaches the subgiant branch, there comes a point where the inner edge of the CHZ recedes to an extent where none of the planets in the current habitable zone have been continuously habitable since their formation. The CHZ metric goes to zero here and remains there.

For treatments of $H$ that do not depend on the habitable duration, but only depend on the IHZ, the time evolution is much different. Between the upper and lower left subplots there is a substantial difference between the IHZ metrics depending on assumptions for $\Gamma$. The metric with the instantaneous $H$ and $\Gamma$ uniform in $a$ depends primarily on the width of the habitable zone, so it roughly traces the luminosity evolution of the star. As the star gets gradually brighter on the main sequence the metric value also increases gradually. When it reaches the subgiant branch and the luminosity of the star increases dramatically, so too does the IHZ metric. Given that this metric is primarily related to the luminosity of the host star, this would imply that giant stars are the best candidates to detect biosignatures around. Since we know that giant evolved stars pose major obstacles to habitability, this suggests that this metric is a poor proxy for habitability. On the other hand, the metric for instantaneous habitability with $\Gamma$ uniform in $\ln(a)$ is almost constant in time. This tells us that this metric is insensitive to the age of the target stars, which may also limit its usefulness in a search for biosignatures. In the following section, we shall see that not only is this metric insensitive to stellar ages, it is also barely dependent on stellar masses. This severely limits the utility of such a metric, as it gives similar values for very different stars.

We will next observe how our biosignature yield metrics vary for a wide variety of stars. The key variables that affect the values of our metrics are the stellar mass and age, so we modelled stars with a range of masses and ages to see how their relative biosignature yields, $B$, varied. We used a simple stellar model in MESA (Paxton et al., 2011, 2013, 2015, 2018, 2019), described in Appendix A, and varied the stellar masses and ages, holding the metallicity and helium mass fractions constant at roughly solar values. Using the outputs from these MESA models, we calculated values for the expected biosignature yield for different assumptions about $\Gamma$ and $H(a,t)$.

Figure 5 demonstrates how our values for the metrics vary over a grid of masses and ages. For these subplots, the columns represent assumptions about the form of $\Gamma$, the distribution of exoplanets. The rows indicate assumptions about the functional form of $H(a,t)$ the probability of a planet hosting biosignatures at a given semimajor axis. Given the differences in colormaps between the different subplots, it is clear that different assumptions can greatly affect whether a star of given mass and age is a good candidate to host biosignatures. Recall that because we are studying relative biosignature yields, we can only compare between stars that share a set of assumptions about $H$ and $\Gamma$, and that we have assumed that $\Gamma$ is independent of stellar mass and age. The color scales in Figure 5 should not be compared between the different subplots. The color scale on each of the subplots goes from zero in blue to the maximum value of metric in red.

Various assumptions about $\Gamma$ and $H$ result in different locations for the maxima of the metrics, $B$. If one considers Cold Start planets to be habitable and assumes that $\Gamma$ is uniform in $a$, then the maximum values of the metric hug the terminal age main sequence (dashed black line in Fig 5), with a maximum corresponding to older G stars. If we keep the assumption that Cold Starts can be habitable and change $\Gamma$ to be uniform in $\ln(a)$, then the maximum values of the metric are for low mass stars with large ages. Choosing $H$ to only consider planets in the CHZ slightly modifies the shape of the colormaps for both assumptions about $\Gamma$. In the CHZ only, $\Gamma$ uniform in $a$ case, the highest $B$ values still seem to be close to the terminal age main sequence, but the maximum is towards lower stellar masses than the Cold Starts case, leaning towards older late-G or early-K stars. For the CHZ only, $\Gamma$ uniform in $\ln(a)$ case, the highest metric values are a bit more focused towards lower mass older age values than in the Cold Starts, $\Gamma$ uniform in $\ln(a)$ case. This preference towards lower stellar masses for the Only CHZ $H$ formulation is expected, as lower mass stars evolve less quickly on the main sequence, resulting in a lower proportion of Cold Start planets.

The metrics that consider only the IHZ of a star at a given age vary dramatically from the other assumptions about $H$, as they don’t give any consideration to the evolutionary history of the stars, and only consider the current location of their habitable zones. Using the IHZ $H$ and $\Gamma$ uniform in $a$, the maximum value of the metric is towards high mass stars near the end of the main sequence. This makes sense as this metric favors habitable zone width and more luminous stars would have wide habitable zones. The IHZ, $\Gamma$ constant in $\ln(a)$ metric is fascinating in that it does not vary as much as the other metrics. As such, one can see that the colormap seems much redder than the other plots, with a maximum towards lower stellar masses, and not much age dependence.

The clear next steps going forward would be to apply the methods introduced in this paper to an entire stellar population, such as the list of target stars for future direct imaging missions. Such target lists have not been finalized, but preliminary lists of potential nearby direct imaging target stars are available. In calculating the total biosignature yield for a direct imaging mission, one could observe which samples of target stars are most likely to host biosignatures. One of the main difficulties for calculating the biosignatures yields for a sample of target stars would be that many stars lack sufficiently precise uncertainties in their observed properties as well as derived properties such as their masses and ages. To calculate biosignature yields for such a sample of stars would require first obtaining accurate masses and ages for each of the target stars. For the two example target stars we modelled in this paper, we obtained masses, ages, stellar evolutionary histories, and biosignature yields by fitting stellar models to observed properties using an optimizer. Despite the fact that these example stars were specifically chosen because they had well characterized stellar properties, it took hundreds of iterations for an optimizer to find a suitable model fit for each of the stars. Given that each model evaluation took around 5 minutes on the 16 core CPU used in this study, it becomes clear that applying the same methods to hundreds of target stars in a catalog would be very computationally intensive. To calculate the biosignature yields for a sample of target stars, we would either need to use a supercomputer or find a more computationally efficient means to obtain model fits.

Given the challenges involved in fitting models for an actual stellar population, it would be significantly easier, though potentially less informative, to instead calculate biosignature yields for a synthetic stellar population. In that case, the masses and ages of the stars would be known a priori, and one would only need to run one stellar model per star.

While for the purposes of this paper we only used a rough estimate for the uncertainties in $B$, in future studies we will take a more statistically robust approach. To deal with the computationally intensive nature of model fitting, we will explore solutions such as Gaussian process emulation of MESA model outputs. Alternatively, methods such as isochrone fitting would also greatly reduce the computation time, but would reduce the precision of the model fits and wouldn’t take into account all of the free model parameters and outputs. With a less computationally intensive function constructed to simulate the model outputs, it becomes feasible to use Markov Chain Monte Carlo (MCMC) methods to obtain the best model fits and uncertainties in the fit parameters.

Future work is also needed to address several additional important points. Potential direct imaging target stars have a heterogeneous and heteroskedastic suite of measured fundamental properties, including abundance patterns, asteroseismic measurements, interferometrically measured angular radii, spectral energy distributions, and parallaxes. Assessing their habitable zone histories requires incorporating this information into stellar evolution models in a consistent way. Stellar evolution models, such as MESA, are themselves subject to uncertainty, such as from incomplete treatments of rotation, convection and diffusion, and uncertainties in opacities and equations of state. These uncertainties should be explored and incorporated into uncertainties in biosignature yield. In future studies, we plan to conduct a sensitivity analysis on biosignature yields with respect to both uncertainties in observed stellar properties as well as inherent stellar modelling uncertainties.

The habitability of Cold Start planets has received scant attention, but since these planets likely compose a large fraction of all discovered exoplanets in the IHZ, they warrant further study. Such planets not only add to the total potential biosignature yield, but they are found in the outer parts of habitable zones, so excluding them has important implications for the necessary inner working angle of direct imaging missions.

The relative biosignature yield of stars also depends sensitively on the dependence of $\Gamma$ on $a$, which is poorly constrained at the radii and orbital separations where most direct imaging targets will be found. Its dependence on stellar properties such as mass, metallicity and effective temperature is also poorly known, but could have major implications for the relative merits of K versus G stars. Future studies should investigate the extent to which different functional forms of the dependence of $\Gamma$ on these properties affects the relative biosignature yields.

In this paper, we have introduced a metric for the relative yield of biosignatures for target stars of future direct imaging missions. We have seen how these metrics vary for different types of stars and outlined our plans for future studies using the biosignature yield formalism defined here. This work is focused on paving the way for future research and, as such, is not overly focused on results. That being said, through modelling the biosignature yields of stars, we have gained insight into the necessities for direct imaging mission design. We find that robust planning for future missions such as HabEx and LUVOIR, including discussions of their relative merits, will require:

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  An explicit model of biosignature appearance that incorporates the history of habitability of planets

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  An understanding of the habitability of Cold Start planets

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  Knowledge of the terrestrial planet occurrence rate with orbital separation and stellar mass

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  Precisely known stellar properties of all potential target stars, especially mass and age

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  Robust and consistent reconstructions of the evolutionary histories of all potential target stars