The search for life on the planets outside the Solar System can be broadly classified into the following: looking for Earth-like conditions or the planets similar to the Earth (Earth similarity), and looking for the possibility of life in a form known or unknown to us (habitability). The two frequently used indices, Earth Similarity Index (ESI) and Planetary Habitability Index (PHI), describe heuristic methods to score similarity/habitability in the efforts to categorize different exoplanets or exomoons. ESI, in particular, considers Earth as the reference frame for habitability and is a quick screening tool to categorize and measure physical similarity of any planetary body with the Earth. The PHI assesses the probability that life in some form may exist on any given world, and is based on the essential requirements of known life: a stable and protected substrate, energy, appropriate chemistry and a liquid medium. We propose here a different metric, a Cobb-Douglas Habitability Score (CDHS), based on Cobb-Douglas habitability production function (CD-HPF), which computes the habitability score by using measured and calculated planetary input parameters. As an initial set, we used radius, density, escape velocity and surface temperature of a planet. The values of the input parameters are normalized to the Earth Units (EU). The proposed metric, with exponents accounting for metric elasticity, is endowed with verifiable analytical properties that ensure global optima, and is scalable to accommodate finitely many input parameters. The model is elastic, does not suffer from curvature violations and, as we discovered, the standard PHI is a special case of CDHS. Computed CDHS scores are fed to K-NN (K-Nearest Neighbour) classification algorithm with probabilistic herding that facilitates the assignment of exoplanets to appropriate classes via supervised feature learning methods, producing granular clusters of habitability. The proposed work describes a decision-theoretical model using the power of convex optimization and algorithmic machine learning.

Propagation of torsional Alfvén waves along magnetic flux tubes has been extensively studied for many years but no conclusive results regarding the existence of a cutoff frequency for these waves have been obtained. The main purpose of this paper is to derive new wave equations that describe the propagation of linear torsional Alfvén waves along thin and isothermal magnetic flux tubes, and use these wave equations to demonstrate that the torsional wave propagation is not affected by any cutoff frequency. It is also shown that this cutoff-free propagation is independent of different choices of the coordinate systems and wave variables adopted in the previous studies. A brief discussion of implications of this cutofffree propagation of torsional tube waves on theories of wave heating of the solar and stellar atmospheres is also given.

Fundamental modes supported by a thin magnetic flux tube embedded in the solar atmosphere are typically classified as longitudinal, transverse, and torsional waves. If the tube is isothermal, then the propagation of longitudinal and transverse tube waves is restricted to frequencies that are higher than the corresponding global cutoff frequency for each wave. However, no such global cutoff frequency exists for torsional tube waves, which means that a thin and isothermal flux tube supports torsional tube waves of any frequency. In this paper, we consider a thin and non-isothermal magnetic flux tube and demonstrate that temperature gradients inside this tube are responsible for the origin of a cutoff frequency for torsional tube waves. The cutoff frequency is used to determine conditions for the wave propagation in the solar atmosphere, and the obtained results are compared to the recent observational data that support the existence of torsional tube waves in the Sun.

Propagation of torsional waves along isothermal and initially-untwisted magneticflux tubes embedded in the solar atmosphere is studied analytically. Conditions for wave propagation along thin and wide magnetic-flux tubes are determined, and it is shown that the propagation along thin tubes is cutoff free; however, for wide tubes the propagation is affected by a cutoff frequency. A method to determine the cutoff frequency is presented and applied to a specific model of solar magnetic flux tubes. An interesting result is that the cutoff frequency is a local quantity in the model and that its value at a given height determines the frequency that torsional tube waves must have to propagate at this height.