One of the main problems of measuring habitability, the suitability of an environment to support life, is how to connect the dependency of many environmental variables with life. The best method to do this is by relating the environment state to a biological proxy for habitability such as population size/mass, growth rate, carrying capacity, diversity, or productivity (check here and example with temperature and relative humidity). This will always depends of the species or community, and the spatial and temporal scale under consideration. Such formulations are usually very complex and more appropriate to local and shorttime scales where more environmental details are available. However, an easier method is to transform the environmental variables into habitablespace, a generalized biological "space" system (not necessarily related to actual physical space area or volume available for life).
Here we propose to use a HabitableSpace Coordinates System (HSCS) as a simple and consistent reference frame to measure the habitability of a complex system as a function of many environmental variables (Figure 1). In this coordinates system values from 1 to +1 correspond to habitable conditions and zero, the exact middle point, to generally the most habitable conditions. Values below 1 or above +1 describe nonhabitable conditions proportional to their magnitude. Habitablespace functions (HSFunction) transforms any environmental variable, for which its limits for life are known, into habitablespace units (HSUnits), a uniform analog scale proportional to habitability. The generalized HSFunction H_{S}(x) is given by:
where x is the environmental variable of interest an x_{min} and x_{max} are its known limits for life. HSUnits are a convenient alternative to traditionally binary habitability assessments where environments are either habitable or nonhabitable. The HSFunctions are just a linear transformation of environmental variables into a fuzzy logic function in metric space. The plusminus signs are conveniently selected as negative if x_{max} has a more adverse effect to life than x_{min}, if any difference.
As a simple example, we can construct a HSFunction for temperature. We know that microbial growth is possible between 15°C (258 K) and 120°C (393 K). Here the maximum value, higher temperatures, are assumed to be more adverse to life than lower temperatures, so the sign in the HSFunction is taken as negative. The thermal HSFunction H_{S}(T) for microbial life is therefore given by:
or just simply H_{S}(T) = 4.8  (0.015 K^{1})T, by substituting the values. Note that the middle point HS_{T} = 0 corresponds to 53°C, and not necessarily the optimum conditions for microbial life (actually near 40° for all cultured microbial life). If we just want to focus on complex life now the limits are about 0°C (273 K) to 50°C (323 K) and the function becomes H_{S}(T) = 11.9  (0.04 K^{1})T. Now the middle point is 25°C, which nicely corresponds to the optimum growth temperature for most primary producers (i.e. vegetation and phytoplankton). The previous example does not show any notable advantage of using HSUnits over temperature units. The real application of HSFunctions are their ability to easily combine the effect of different environmental variables in habitability assessments, even the same variables. Using the previous example, lets say we want to calculate the combined thermal habitability for both microbial life and complex life (the union of the two habitablespaces in an "analog Venn diagram"). This is simply given by a habitabledistance function (HDFunction) of the two HSFunctions as:
where H_{D}(T) is the HDFunction with values below one corresponding to temperatures tolerable by both microbial and complex life. We can also find the best temperature for both groups by minimizing the function, resulting in 28°C (301 K). Solving for H_{D}(T) = 1 gives the range of habitable temperatures for both groups, between 7°C (280 K) and 50°C (323 K) (note that the lower value is not 0°C as expected because the distance functions compounds the fact that this value is close to two limits, 15°C for microbial life and 0°C for complex life). The HDFunction combines HSFunctions of many environmental parameters into a single number (now a positive value) describing how far from habitable conditions is the selected environment. Values between zero and one match all the habitable criteria with zero being generally more habitable. Values between one and √n, where n is the number of HSFunctions, match some of the conditions. Larger numbers represent nonhabitable conditions. As an analog scale the HDFunctions can be used to easily rank habitable environments from best to worst. It is given by a generalized euclidiandistance:
where the x_{i} are the environmental variables of interest and n is the number of HSFunctions. In many cases the HSFunctions will depend on more than one variable because the limits can depend on other factors too and not being fixed.
An HDFunction provides a simple quantitative screening tool for habitable environments from microbial to planetary scales. Many of the traditional operations in euclideanspace are also meaningful in habitablespace (i.e. vector and tensor operations). Our first real application of this metric was to identify and rank habitable exoplanets based on basic planetary and stellar properties for our Habitable Exoplanets Catalog (HEC).
Figure 1. Summary of the HabitableSpace Coordinates System (HSCS), which provides a simple and general framework for quantitative habitability assessments from microbial to planetary scales.
Quick Steps for Habitability Assessments using HabitableSpace Metrics
