# EARTH SIMILARTIY INDEX (ESI)

## Introduction

The Earth Similarity Index (ESI), or the "easy scale," is an open multiparameter measure of Earth-likeness for solar or extrasolar planets as a number between zero (no similarity) and one (identical to Earth) (Schulze-Makuch et al., 2011). Similarity measures are used in many fields to solve many pattern recognition problems, such as classification, clustering, and retrieval problems (Cha, 2007). The ESI can be used to simplify planet comparisons, extract planets of interest from large databases, select objects of interest for follow-up observations, or perform statistical analyses on the occurrence of Earth-like planets.

The ESI is not a direct measure of habitability but a fuzzy comparison, using a distance metric, between a selected set of planetary properties of a planet and Earth. The more properties that are used in the ESI calculation, the better the comparison with Earth (e.g., identify the 'Earth-like' cluster in a large database). The search for Earth-like planets is synonymous with the search for planets with ESI values closer to one. There are many ways that an ESI can be mathematically constructed depending on the needs and available data. Once constructed, the next important thing is how to interpret its values.

Figure 1. Artistic representation of potentially habitable worlds sorted by the Earth Similarity Index (ESI), a measure of their similarity to Earth's size and insolation.

## ESI for Exoplanets

(from stellar flux, radius, or mass)

The Habitable World Catalog (Figure 1) uses an ESI expression for exoplanets as a function of the planet's stellar flux, radius, or mass, since there is no information about their surface temperature. The ESI is given by a distance metric:

where S is stellar flux, R is radius, S⊕ is Earth's solar flux, and R⊕ is Earth's radius. This ESI expression uses the quadratic mean as the distance metric, which is very convenient to interpret statistically (e.g., as a chi-squared distribution) (Figure 2). This expression can be used for those planets detected by the transit method (e.g., Kepler, K2, TESS, Plato, etc.) where only the planet's radius is known. It is also extendable to planets detected by the radial velocity method, where only the mass is known by assuming that R = M1/3, where M is the planet's mass (or minimum mass). A mass-radius relationship can also be used for this conversion, but it is not usually necessary since it gives similar results within the interest values (i.e., close to ESI = 1.0).

Figure 1. Phase space of an Earth Similarity Index (ESI) as a function of the planet's stellar flux and radius. Black dots represent known planets. Vertical dashed lines enclose the habitable zone (HZ) for Sun-like stars. The horizontal dashed lines enclose planets with 0.5 to 2.5 Earth radii.

## ESI for Solar System Objects

(from radius, density, escape velocity, and surface temperature).

An ESI formulation given a planet's radius, density, escape velocity, and surface temperature provides the simplest and best combination of parameters to compare planets with Earth (Schulze-Makuch et al., 2011). Using this ESI formulation, any planetary body with an ESI value over 0.8 can be considered an Earth-like planet. This only means that the planet is likely rocky and supports a temperate atmosphere, not necessarily habitable. Planets with ESI values in the 0.6 to 0.8 range (e.g., Mars) might still be habitable since habitability depends on many other factors. The ESI is given then by:

where xi is a planetary property (e.g. surface temperature), xio is the corresponding terrestrial reference value (e.g., 288 K), wi is a weight exponent, n is the number of planetary properties, and ESI is the similarity measure. The weighting exponents are used to adjust the sensitivity of the scale and equalize its meaning between different properties. In practice, a simpler and more limited form of the ESI formulation is used for exoplanets (using only stellar flux, and mass or radius) since this is the only data available of them.

The parameters for the ESI equation for mean radius, bulk density, escape velocity, and surface temperature are shown in Table 1. The ESI can be divided for convenience into an Interior ESI, based on the mean radius and bulk density, and a Surface ESI, based on the escape velocity and surface temperature. Both the interior and surface ESI are combined into a global ESI. The ESI is rather a surface than a subsurface indirect indicator of habitability due to its Earth-centric definition (Table 1).

Table 1. Reference values and weight exponents for the four planetary properties used to define the ESI. The scale is much more sensitive to surface temperature than to the other planetary properties.

Note: Eu = Earth’s units

## References

Schulze-Makuch, D., Méndez, A., Fairén, A. G., von Paris, P., Turse, C., Boyer, G., Davila, A. F., Resendes de Sousa António, M., Irwin, L. N., and Catling, D. (2011) A Two-Tiered Approach to Assess the Habitability of Exoplanets. Astrobiology 11(10): 1041-1052. [paper link]

Cha, S.-H. (2007). Comprehensive Survey on Distance/Similarity Measures between Probability Density Functions. International Journal of Mathematical Models and Methods in Applied Sciences. 1(4). [Paper Link]