CD-HPF:New Habitability Score Via Data Analytic Modeling Snehanshu Saha Professor., Computer Science Dept.,PESIT-BSC, Bangalore.

INTRODUCTION At present, only known habitable planet is Earth. 2 important questions to be answered – Can we have life’s existence exactly similar to Earth , somewhere else? Or Do we have life in unknown form existing somewhere else?

Introduction.. Astronomers uses mainly 2 important parameters to answer these 2 questions- Earth’s Similarity Index (ESI)- based on 4 parameters radius, surface temperature, density and escape velocity and is given as-

Introduction.. Planetary Habitability Index(PHI) which is basically to estimate 2nd question and is given as- Where S is Substrate, E is Energy, C is Chemistry of compounds and L is liquid solvent.

Introduction.. We are proposing an approach that makes use of Cobb Douglas Production Function to obtain the new habitability score for exoplanets.

Cobb Douglas Production Function (CDPF) In general, CDPF is given as- Where Y is the production function and α and β are elasticity constants which are some positive fractions , k is a positive constant and x1 & x2 are input parameters. NOTE: Function can be extended for any number of inputs.

Elasticity Constants in CDPF Sum of α & β helps us in deducing the important results about the function , If their sum is 1, function is homogeneous of degree 1, and is called as Constant returns to Scale(CRS), increase in 1 input gives increase in output in same proportion. If their sum is less than 1, it is decreasing returns to scale (DRS), where diminishing returns will set in.

Elasticity Constants in CDPF.. If the sum is more than 1, it is increasing returns to scale(IRS), here output increases with variable factors.

Use of CDPF to estimate CDHS We have formulated CDPF to estimate Cobb Douglas Habitability Score(CDHS) for exoplanets. First, we have calculated interior and surface CDHS for each exoplanet and then used a convex combination of the two to compute final CDHS. While doing this we found and proved that function is maximized for CRS & DRS case.

Use of CDPF to estimate CDHS We specify CDPF as- where above parameters are radius, density , surface temperature & escape velocity with their elasticity constraints.

Use of CDPF to estimate CDHS  

Use of CDPF to estimate CDHS Set criteria is to choose α,β,γ,δ so as to maximize Y. For CRS, where all the elasticities of different cost components are equal,Y can be specified as- and In such cases, Y is geometric mean of all inputs.

Data set For this work , we used HABCAT database, available at http://phl.upr.edu/projects/habitable-exoplanets-catalog From this database we selected 664 confirmed exoplanets for which surface temperature was known.

Our Results CDPF was applied to get CDHS based on radius & density , we called it as CDHSi and another CDHS based on escape velocity and surface temperature and we called it as CDHSs. We got data in the range from 0.8607 to 168.35 for CDHSi and from 1.01521 to 19.9395 for CDHSs.

Graphs for CDHSi & CDHSs CDHSi Here α & β values are 0.8 & 0.1 respectively.

Graphs for CDHSi & CDHSs CDHSs Here α & β values are 0.8 & 0.1 respectively.

CDHS Calculation Next we calculated final CDHS as- where w’ was considered as 0.99 & w” as 0.01, because w’+w” must be equal to 1. Values obtained for CDHS were in the range from 0.87225 to 166.87.

Classification based on CDHS After calculating CDHS , we applied KNN classification algorithm to check the number of planets belonging to Earth’s Class. According to CDHS result , 5 classes are considered with k =7, where Earth’s class was class 4th. These classes have following ranges according to classification done-

KNN Algorithm Consider k as the desired number of nearest neighbors and S:=p1,...,pn be the set of training samples in the form pi=(xi,ci), where xi is the d-dimensional feature vector (1 in our case) of the point pi and ci is the class that pi belongs to. Let T:=p1’,p2’,…..pr’ be the testing samples. for each p’=(x’,c’) in T { compute the distance d(x’,xi) between p’ & all pi belonging to S; Sort pi according to d(x’,xi); Select k closest training samples to p’ from the sorted list; Assign a class to p’ based on majority vote ; }

Class Ranges According to CDHS Class1 - From 1.98 to 166.87 Class2 - Less than 1.98 to greater than 1.680 Class3 - From 1.68 to greater than 1.443 Class4 - From 1.443 to greater than 1.23 Class5 - From 1.23 to greater than equal to 0.87225.

Result of Classification 80% of data is used as training and 20% data for testing. In Class5, we obtained 13 exoplanets. Result plot of KNN Accuracy obtained is 92.5%.

High Level Dataflow Planets highly likely to be habitable are- GJ163c Employ KNN (preliminary classification) No. of classes Load the dataset Cobb Douglas Engine Attribute Filtering Computed CDHPF Reorganized Classes Final 5 classes Class 5 contains Earth & other habitable planets Planets highly likely to be habitable are- GJ163c GJ667C c GJ832c HD40307g Kepler-186f Kepler-62e Kepler-62f etc.

Acknowledgement I sincerely thank Ms. Kakoli Bora and Ms. Surbhi Agrawal, my Ph.D. students for sincere efforts. I acknowledge Dr. Margarita Safanova , Indian Institute of Astrophysics , who has been an excellent collaborator in this project.

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