Estudio de la habitabilidad de los exoplanetas a partir de la extensión de la función de Cobb-Douglas

Abstract

This work, part of the undergraduate program in physics, aimed to extend the habitability model based on the Cobb–Douglas function by incorporating exoplanet orbital elements. Using a mathematical, computational, and astrobiological approach, a tool was developed to more accurately assess the habitability of exoplanets. The habitability of an exoplanet depends on several physical factors simultaneously, such as radius, mass, etc. The Cobb–Douglas function allows these parameters to be combined in a multiplicative and weighted manner, reflecting their relative contribution to a single numerical index. This study proposes a quantitative model for assessing the habitability of exoplanets using a Cobb–Douglas function of multiple planetary variables, grouped into three dimensions: interior (radius and density), surface (escape velocity and temperature), and periphery (eccentricity and orbital period). Each subscript was normalized to Earth units and weighted according to its physical relevance: 0.4 for interior, 0.5 for surface, and 0.1 for periphery. The function was applied under the optimization process in terms of parameters under constraints called decreasing returns to scale (DRS) and constant returns to scale (CRS), using the fmincon algorithm, ensuring its concavity and global optimization. Twenty-nine potentially habitable exoplanets were identified under constant yields, of which 21 coincide with the Habitable Worlds Catalog, and 47 under decreasing yields, with 30 coincidences. In addition, a strong correlation (r = 0.89) was observed between mass and escape velocity, confirming their gravitational link. The model was validated using the k-NN algorithm (k = 5), showing local consistency between similar planets.