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 short-time scales where more environmental details are available. However, an easier method is to transform the environmental variables into habitable-space, a generalized biological "space" system (not necessarily related to actual physical space area or volume available for life).
Here we propose to use a Habitable-Space Coordinates System (HS-CS) 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 non-habitable conditions proportional to their magnitude. Habitable-space functions (HS-Function) transforms any environmental variable, for which its limits for life are known, into habitable-space units (HS-Units), a uniform analog scale proportional to habitability. The generalized HS-Function HS(x) is given by:
where x is the environmental variable of interest an xmin and xmax are its known limits for life. HS-Units are a convenient alternative to traditionally binary habitability assessments where environments are either habitable or non-habitable. The HS-Functions are just a linear transformation of environmental variables into a fuzzy logic function in metric space. The plus-minus signs are conveniently selected as negative if xmax has a more adverse effect to life than xmin, if any difference.
As a simple example, we can construct a HS-Function 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 HS-Function is taken as negative. The thermal HS-Function HS(T) for microbial life is therefore given by:
or just simply HS(T) = 4.8 - (0.015 K-1)T, by substituting the values. Note that the middle point HST = 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 HS(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 HS-Units over temperature units. The real application of HS-Functions 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 habitable-spaces in an "analog Venn diagram"). This is simply given by a habitable-distance function (HD-Function) of the two HS-Functions as:
where HD(T) is the HD-Function 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 HD(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 HD-Function combines HS-Functions 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 HS-Functions, match some of the conditions. Larger numbers represent non-habitable conditions. As an analog scale the HD-Functions can be used to easily rank habitable environments from best to worst. It is given by a generalized euclidian-distance:
where the xi are the environmental variables of interest and n is the number of HS-Functions. In many cases the HS-Functions will depend on more than one variable because the limits can depend on other factors too and not being fixed.
An HD-Function provides a simple quantitative screening tool for habitable environments from microbial to planetary scales. Many of the traditional operations in euclidean-space are also meaningful in habitable-space (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 Habitable-Space Coordinates System (HS-CS), which provides a simple and general framework for quantitative habitability assessments from microbial to planetary scales.
Quick Steps for Habitability Assessments using Habitable-Space Metrics