A Comparative Analysis of the Cobb-Douglas Habitability Score (CDHS) with the Earth Similarity Index (ESI)

Surbhi Agrawal1, Suryoday Basak1, Snehanshu Saha1, Kakoli Bora2 and Jayant Murthy3

I. Introduction and Methods is 1.27 EU, the radius is R = M 0.5 ≡ 1.12 EU1. Accordingly, the escape velocity was calculated by V = p2GM/R ≡ 1.065 (EU), and the density by A sequence of recent explorations by Saha et al. [4] e the usual D = 3M/4πR3 ≡ 0.904 (EU) formula. expanding previous work by Bora et al. [1] on using The manuscript, [4] consists of three related Machine Learning algorithm to construct and test analyses: (i) computation and comparison of ESI planetary habitability functions with exoplanet data and CDHS habitability scores for Proxima-b and raises important questions. The 2018 paper analyzed the Trappist-1 system, (ii) some considerations the elasticity of their Cobb-Douglas Habitability on the computational methods for computing the Score (CDHS) and compared its performance with CDHS score, and (iii) a machine learning exercise other machine learning algorithms. They demon- to estimate temperature-based habitability classes. strated the robustness of their methods to identify The analysis is carefully conducted in each case, potentially habitable planets from exoplanet dataset. and the depth of the contribution to the literature Given our little knowledge on exoplanets and habit- helped unfold the differences and approaches to ability, these results and methods provide one impor- CDHS and ESI. tant step toward automatically identifying objects of interest from large datasets by future ground and Several important characteristics were introduced space observatories. The paper provides a logical to address the habitability question. [2] first ad- evolution from their previous work. Additionally, dressed this issue through two indices, the Planetary model and the methods yielding a new metric for Habitability Index (PHI) and the Earth Similarity ”Earth-similarity” paves the way for comparison Index (ESI), where maximum, by definition, is set with its predecessor, ESI [2]. Even though, CDHS as 1 for the Earth, PHI=ESI=1. proposed in [1] and extended in [4] consider identical ESI represents a quantitative measure with which planetary parameters such as Mass, Radius, Escape to assess the similarity of a planet with the Earth Velocity and Surface Temperature; the contrasts are on the basis of mass, size and temperature. But overwhelming compared to the similarities. The note ESI alone is insufficient to conclude about the hab- aims to bring out the differences and investigates itability, as planets like Mars have ESI close to the contrasts between the two metrics, both from 0.8 but we cannot still categorize it as habitable. machine learning and modeling perspectives. There is also a possibility that a planet with ESI It is worth mentioning that once we know one value slightly less than 1 may harbor life in some observable – the mass – other planetary parameters form which is not there on Earth, i.e. unknown to used in the ESI computation (radius, density and us. PHI was quantitatively defined as a measure of escape velocity) can be calculated based on certain the ability of a planet to develop and sustain life. assumptions. For example, the small mass of Prox-

arXiv:1804.11176v1 [astro-ph.IM] 30 Apr 2018 However, evaluating PHI values for large number of ima b suggests a rocky composition. However, since planets is not an easy task. Irwin in 2014, introduced 1.27 EU is only a low limit on mass, it is still possible another parameter was to account for the chemical that its radius exceeds 1.5 – 1.6 EU, which would composition of exoplanets and some biology-related make Proxima b not rocky . Since Proxima b mass features such as substrate, energy, geophysics, tem- perature and age of the planet — the Biological Complexity Index (BCI). Here, we briefly describe 1 The Department of Computer Science, PES Univer- sity,Bangalore [email protected] the mathematical form of ESI. This will help us 2The Department of Information Science, PESIT- understand the difference in formulation of ESI and BSC,Bangalore 2The Department of Information Science, PESIT- 1http://phl.upr.edu/library/notes/standardmass- BSC,Bangalore radiusrelationforexoplanets, Standard Mass-Radius Relation 2Indian Institute for Astrophysics,Bangalore for Exoplanets, Abel Mendez, June 30, 2012. CDHS subsequently. parameters (finitely mnay). However this throws a a) Earth Similarity Index (ESI): ESI was computational challenge of local oscillations which designed to indicate how Earth-like an exoplanet has been tackled by Stochastic Gradient Ascent in might be [2] and is an important factor to initially the same paper. assess the habitability measure. Its value lies between However, we do not intend to argue that the 0 (no similarity) and 1, where 1 is the reference value, CD-HPF is a superior metric. However, we do i.e. the ESI value of the Earth, and a general rule is argue that the foundations of the CD-HPF, in both that any planetary body with an ESI over 0.8 can mathematical and philosophical sense, are solidly be considered as Earth-like. It was proposed in the grounded, and substantiated by analytical proofs. form  w x − x0 ESIx = 1 − , (1) a. The CDHS is not derived from ESI. CDHS is not x + x0 a classifier and neither is ESI – this is because we where ESIx is the ESI value of a planet for x prop- do not (yet) categorize planets based on the values erty, and x0 is the Earth’s value for that property. of either ESI or CDHS. Even though we were The final ESI value of the planet is obtained by categorical in emphasizing the contrast and recon- combining the geometric means of individual values, ciliation of two approaches (one being CDHS), we where w is the weighting component through which reiterate our baseline argument for the benefit of the sensitivity of scale is adjusted. Four parameters: the readership. CDHS contributes to the Earth- surface temperature Ts, density D, escape velocity Ve Similarity concepts where the scores have been and radius R, are used in ESI calculation. This index used to classify exoplanets based on their degree is split into interior ESIi (calculated from radius and of similarity to Earth. However Earth-Similarity density), and surface ESIs (calculated from escape is not equivalent to exoplanetary habitability. velocity and surface temperature). Their geometric Therefore, we adopted another approach where means are taken to represent the final ESI of a machine classification algorithms have been ex- planet. However, ESI in the form (1) was not in- ploited to classify exoplanets in to three classes, troduced to define habitability, it only describes the non-habitable, mesoplanets and psychroplanets. similarity to the Earth in regard to some planetary While this classification was performed, CDHS parameters. For example, it is relatively high for the was not used at all, rather discriminating features Moon. from the PHL-EC were used. This is fundamen- What’s the paradox? tally different from CDHS based Earth-Similarity The ESI and CDHS scores should be similar in the approach where explicit scores were computed. sense of being computed from essentially the same Therefore, it was pertinent and remarkable that ingredients. However, similarity in numerical values the outcome of these two fundamentally distinct may be accidental also unless more informative pic- exercises reconcile. We achieved that. This rec- ture could be provided with the additional explo- onciliation approach is the first of its kind and ration of the general relationship between ESI and fortifies CDHS, more than anything else. The CDHS scores. We explore relationships and establish manuscript, [4] captures this spirit. We maintain the contrasts in the following section. It will also be that this convergence between the two approaches established that ESI and CDHS are not related and is not accidental. We have been constantly watch- there exists no causal relationship between the two. ing the catalog, PHL-EC and scientific investiga- CDHS is NOT derived from ESI. CDHS is NOT tions in habitability of exoplanets. Please refer to a classifier, neither is ESI !!!!!!!!!! Moreover, as new the discussion section of the revised manuscript features and/or observables are added, there is no mentioned above for further details. We urge the known estimate of the behavior of ESI, numerically readership to revisit the schematic flow elucidated speaking. As ESI is not based on maximizing a in Fig. 1 of the manuscript. score, when we go on increasing the number of b. As an exercise, we tried to find the optimal point constituent parameters, the numerical value will of the ESI in a same fashion as we found it for go on diminishing. The reason for this is that CD-HPF – the finding was that a minima or a each constituent term in the ESI is a number maxima cannot be guaranteed by the functional between 0 and 1 – now as we add more terms to form of the ESI. this, the value of the score tends to zero. This c. As ESI is not based on maximizing a score, when is not the case with our model, CDHS, as we we go on increasing the number of constituent have shown in the supplementary file, [4] that parameters, the numerical result of the ESI might global optima is unaffected under additional input go on decreasing: there is no guarantee of stability. The reason for this is that each constituent term TABLE I: Differences between ESI and CDHS in the ESI is a number between 0 and 1 – now as at a glance we add more terms to this, the value of the score S. ESI CDHS tends to zero. This is not the case with our model, No. Derived for four input parame- CD-HPF, as we have shown in the supplementary On the contrary, the CDHS is ters only. It is unclear how the solidly grounded in optimiza- file of [4] that global optima is unaffected un- ESI will behave if additional 1. tion theory as we have shown der additional input parameters (finitely many). parameters such as eccentric- (in the supplementary file of ity, flux, radial velocity, etc. are However this throws a computational challenge [4]). added. of local oscillations which has been tackled by The exponents α, β, γ and δ The exponents of each term in Stochastic Gradient Ascent [4]. are not predetermined; com- 2. the CDHS, w /n is predeter- i puting them is a part of the d. It is easy to misconstrue CDHS with ESI as being mined. optimization problem. one since the parameters of the CD-HPF are In all likelihood, the inclusion the same as that of the ESI, and the functional of additional input parameters This does not happen with the will diminish the ESI since all form in either metric is multiplicative in nature. 3. CDHS even if we include addi- input parameters are scaled be- tional parameters. However, that is not the case. We have laid a little tween 0 and 1, and the weights emphasis on how our metric is similar to ESI in are fixed. saying that the ESI is a special case of the CD- HPF. The crux of the philosophy behind the CD- HPF is essentially that of adaptive modeling, i.e., between the CDHS and the ESI, the ESI would have the score generated is based on the best combi- no guarantee of having a maxima. The CD-HPF is nation of the factors and not by static weights as based on an adaptive modeling, that is, when the followed by ESI. CDHS is computed, the response is a maximum, • That, essentially, the ESI score gives which is based on the functional form of the CD- non-dynamic weights to all the different HPF. We reported the proof in order to substantiate planetary (with no trade-off between the the fact that a maximum does exist of the functional weights) observables or calculated features form that we have used. The highlights of the CD- considered, which in practice may not be HPF are: the best approach, or at least, the only 1) The exponents (or, the elasticities) of each way of indicating habitability. It might observable in the function, R, D, Ts, Ve are be reasonable to say that for different denoted using α, β, γ, and δ respectively. The exoplanets, the various planetary observables constraints on the permissible values of the may weigh each other out to create a unique elasticities are: – kind of favorable condition. For instance, in α + β + γ + δ ≤ 1 one planet, the mass may be optimal, but the temperature may be higher than the average and of the Earth, but still within permissible α, β, γ, δ ≥ 0 limits; in another planet, the temperature may be similar to that of the Earth, but the Thus, the computation of the CDHS of a planet mass may be lower. By discovering the best is essentially a constrained optimization problem. combination of the weights (or, as we call it, 2) In [4], we have also shown that the number elasticities) to maximize the resultant score, of components in the CDHS can be countably to the different planetary observables, we are infinite. Of course, the components being con- creating a score which presents the best case sidered for each planet should be the same and scenario for the habitability of a planet. the scoring system becomes different, but the possibility still exists that n number of observ- ables and/or input parameters can be accom- Key Differences between ESI and CDHS: modated into the function, while keeping the In the case of the ESI, there is no evidence of a constraints on the elasticities intact. Moreover, rigorous functional analysis. Schulze-Makuch et al. we have proved a very important theorem that have not reported the existence of extrema for the even if n number of observables and/or input ESI. We tried to find the maximum for the ESI in parameters make the functional form extremely a manner similar to that of CD-HPF and we were complicated, a global optima is still guaranteed. unable to do so with the appropriate mathematical The reason why CDHS computation is a challeng- tools. This means that if we were to draw parallels ing problem (and ESI is not!) is Fig. 1: ESI vs CDHS(CRS) Fig. 3: ESI vs CDHS(DRS)

Fig. 2: ESI vs CDHS (CRS) – a closeup Fig. 4: ESI vs CDHS (CRS) – a closeup

• Local oscillations about the optima [3], [6], [8] two fundamentally different approaches (one of are difficult to mitigate even though we have which is CDHS) that lead to a similar conclusion shown there exists a theoretical guarantee for about an exoplanet. While the CDHS provides the same. a numerical indicator (in fact the existence • Extrema doesn’t occur at the corner points and of one global optima shouldn’t be a concern is therefore the location of such is difficult to at all, rather a vantage point of the model predict. that this eliminates the possibility of computing • We have emphasized enough on how the CD- scores arbitrarily), the machine classification HPF reconciles with the machine learning meth- bolsters our proposition by telling us automat- ods that we have used to automatically clas- ically which class of habitability an exoplanet sify exoplanets. It is not easy two to propose belongs to. The performance of machine clas- sification is evaluated by class-wise accuracy. the new metric holds for IRS condition only (neither The accuracies achieved are remarkably high, maxima nor minima). The new ESI input structure, and at the same time, we see that the values of despite a functional form very similar to CDHS is not the CDHS for the sample of potentially habit- able to reproduce the dynamic and flexible behavior able exoplanets which we have considered are of the Earth Similarity Score reported by CDHS. The also close to 1. Therefore, the computational details are documented in Appendix A. approaches map earth similarity to habitability. This is remarkable and non-trivial. This degree II. Discussion of computational difficulty is non-existent in While each of these analyses is conducted indepen- ESI. dently, the results reconcile in a way that is sensible. • NOTE: One should not miss the crucial point The earth similarity indices, usually a solution that of computation of CDHS which is only a part of doesn’t address the habitability classification prob- the exercise. The greater challenge is the vindi- lem has been reconciled with machine classification cation of the CDHS metric in the classification approach. In the process of doing so, we developed of habitability of exoplanets by reconciliation of CDHS which is more representative (as opposed to the modeling approach with the machine classi- ESI) of the mapping between these two completely fication approach (where explicit scores were not different approaches. used to classify habitability of exoplanets but A different perspective of the inference sought the outcome validates the metric). The greatest from the results of the ML algorithms lies in strength of CDHS is its flexibility in functional the values of the Cobb-Douglas Habitability Scores form helping forge the unification paradigm. (CDHS). We see that the values of CDHS of To be specific, even under the constrained op- Proxima-b and TRAPPIST-1 c, d, and e (on which timization problem, the search space is countably the probability of life is deemed to be more prob- 2 infinite making the problem of finding the global able), [4] are closer to the CDHS of Earth than optima computationally intractable until some in- those of the remaining planetary samples which we tervention is made. We have mitigated the problem have specifically explored in our study. by employing stochastic gradient ascent. If it’s a The exposition of the efficacy of our methods are simplex problem, we know the optima is at one concurrently being made by astrophysicists. In a of the corner points (theoretically) and therefore recent article in Astrobiology, it has been said that don’t bother looking for it in the entire search space two planets in the TRAPPIST-1 system are likely to (please note even in that case, the computational be habitable: http://astrobiology.com/2018/01/two- complexity is hard to ignore and people develop trappist-1-system-planets-are-potentially- different kinds of approximation algorithms to im- habitable.html. TRAPPIST-1 b and c are likely to prove upon the complexity). The functional form in have molten-core mantles and rocky surfaces, and our case is non-linear, convex and is bounded by hence, moderate surface temperatures and modest constraints (which is typically a geometric region, amounts of tidal heating. The expected favorable contour/curved) making the search space complex conditions are also reflected in the CDHS values of enough to find the optima. The only positive thing in these planets as reported in the 2018 paper. our favor is the theoretical existence of global optima The computational aspect is also worth mention- which ensures, once our algorithm finds the optima, ing here so as to provide the reader a thorough it terminates! There is the other issue of handling understanding of the method we have explored, oscillation around the optima, coupled with finding their advantages over existing methods, and notwith- ”the optima”. standing, their limitations. It is, indeed the question of finding the optima The essence of the CD-HPF, and consequently, efficiently that distinguishes CDHS from ESI. As an that of the CDHS is indeed orthogonal to the essence exploratory investigation, we rendered ESI a func- of the ESI or BCI. We argue not in favor of the tional form, similar to CDHS with the express objec- superiority of our metric, but for the new approach tive of finding elasticity (dynamic weights to each of that is developed. We believe that there should actu- the parameters) that may compute optimum Earth ally be various metrics arising from different schools similarity of any exoplanet. What we found was such of thought so that the habitability of an exoplanet theoretical guarantee is non-existent for ESI and it may be collectively determined from all these. Such a may ensure global optima only in the case of IRS, 2CDHS for MARS = 0.583 for all the variation of S.Temp; which is infeasible (Please refer to [1]). Therefore, our CDHS for MOON = 0.336, ESI of MARS=0.64, ESI of attempt to render theoretical credence to ESI fails as MOON=0.56 kind of adaptive modeling has not been used in the context of planetary habitability prior to the CD-  2R α−2 1 = α. (α − 1) . . . (α 4 (7) HPF. While not in agreement with the structure of Re + R (Re + R) ESI, CDHS does complement ESI. − 1).4R2 − 8RR
The rate at which exoplanets are being discovered e e is rapidly increasing. Given the current situation and hence final second order derivative is- how technology is rapidly improving in astronomy, 2  α−2 we firmly believe that eventually, problems such as ∂ Y 2R 1 2
2 = α. (α − 1) . . 4 . (α − 1).4Re − 8RRe this in observational astronomy will require a thor- ∂R Re + R (Re + R) ough dealing of data science to be handled efficiently. (8) As the number of confirmed exoplanets grow, it for concavity eq.3 must be greater than 0 so, might be wise to use an indexing and classification α−2  2R  1 method which can do the job automatically, with α. (α − 1) . . . (α − 1).4R2 − 8RR
> 0 R + R (R + R)4 e e little need for human intervention. This is where e e both ESI and CDHS shall play useful roles, we believe. (9) Appendix A: Constraint Conditions for elas- on solving above inequality we will get , ticity; α, β, γ and δ: ESI with dynamic input elasticity fails to be an optimizer. R α − 1 > 2 (10) In the function, Re  R − Rα  D − D β similarly it can be proved for other 3 elasticity Y = k. 1 − e . 1 − e . R + R D + D constants which gives us constrained conditions e e (2)  T − T γ  V − V δ D 1 − e . 1 − e β − 1 > 2 , (11) Te + T Ve + V De T where Re, De,Te and Ve are the radius, density, γ − 1 > 2 , (12) surface temperature and escape velocity of Earth Te V and are constant terms. R,D.T and V are radius , δ − 1 > 2 (13) density ,surface temperature and escape velocity for Ve the planet under study.k is also a constant parameter Summing up equations (10),(11,(12) and (13), Differentiating eq.1 above partially w.r.t. R  R D T V   α−1 α + β + γ + δ − 4 > 2 + + + ∂Y 2R 2Re Re De Te Ve = α. . 2 (3) ∂R Re + R (Re + R) (14) finding second derivative from eq.2  R D T V  ⇒ α + β + γ + δ > 2 + + + + 4 ∂2Y Re De Te Ve = α. (α (4) (15) ∂R2  α−2     2R 2Re 2Re Above Equation (10) shows that sum of the four −1) . . 2 . 2 Re + R (Re + R) (Re + R) elasticity constants cannot be less than or equal to  α−1   1 (in fact cannot be less than 1). This is the case of 2R 4Re(Re + R) − α. . IRS (increasing return to scale) in CDHPF function, R + R (R + R)4 e e which means that function is neither concave nor convex. The new metric holds for IRS condition only  α−2 2 2R 4Re which doesn’t ensure a global maxima implying lack = α. (α − 1) . . 4 Re + R (Re + R) (5) of theoretical foundation for the ESI input structure.  α−1  2  Appendix B: Optimization proof using La- 2R 4Re + 4Re.R) − α. . 4 grangian Multiplier. Re + R (Re + R) Here we provide the analytical proof of the claim made that this is the case of IRS (increasing return  α−2  2R 1 to scale) in CDHPF function, which means that = α. (α − 1) . . 4 . (α Re + R (Re + R) (6) function is neither concave nor convex. The new  2 2R 2 metric holds for IRS condition only, which doesn’t − 1).4Re − .(4Re + 4RRe) Re + R ensure a global maxima. The function Y is given as ∂L  2R   2D  specified in eq.(2) in Appendix A, reproduced here = −(w + w + ∂λ 1 R + R 2 D + D for ease- e e (23)  2T   2V   α  β w + w − m) = 0 Re − R De − D 3 4 Y = k. 1 − . 1 − . Te + T Ve + V Re + R De + D (16) From Eq. (19) and (20),  T − T γ  V − V δ 1 − e . 1 − e T + T V + V  α−1.  β  α  β−1 e e α. 2R . 2D β. 2R . 2D Re+R De+D Re+R De+D It can be rewritten as- = w1 w2  2R α  2D β (24) Y = k. . . Re + R De + D or γ δ (17)  2T   2V  2D 2R β w . = . . 1 (25) Te + T Ve + V De + D Re + R α w2 The Lagrangian function for the optimization prob- Similarly, we get lem is hence given as- 2T 2R γ w = . . 3 (26) T + T R + R α w  2R   2D  e e 2 L = Y − λ(w1 + w2 and Re + R De + D     (18) 2T 2V 2V 2R δ w4 + w3 + w4 − m) = . . (27) Te + T Ve + V Ve + D Re + R α w2 Putting the values from eq. (25),(26) and (27) in  2R α  2D β = k. . . eq.(19) and solving it further we get, Re + R De + D 1 2R   1−(α+β+γ+δ)  2T γ  2V δ  2R  = λ.k.α1−(β+γ+δ).ββ.γγ .δδ . − λ(w1 + Re + R Te + T Ve + V Re + R 1 β+γ+δ−1 −β −γ −δ
1−(α+β+γ+δ)  2D   2T  × w1 w2 w3 w4 w2 + w3 + (28) De + D Te + T  2V  Similarly, we get following w4 − m) Ve + V 1 2D   1−(α+β+γ+δ) = λ.k.β1−(α+γ+δ).αα.γγ .δδ The first order conditions are- De + D α−1 β 1 ∂L  2R   2D  × w α+γ+δ−1w −αw −γ w −δ
1−(α+β+γ+δ) = kα 2 1 3 4 ∂R R + R D + D (29) e e (19)  γ  δ 2T 2V 1 − w1λ = 0 2T  1−(α+β+δ) α β δ 1−(α+β+γ+δ) Te + T Ve + V = λ.k.γ .α .β .δ Te + T α β−1 1 ∂L  2R   2D  α+β+δ−1 −α −β −δ
1−(α+β+γ+δ) = kα × w3 w1 w2 w4 ∂D R + R D + D (30) e e (20)  γ  δ 2T 2V 1 − w2λ = 0 2V  1−(α+β+γ) α β γ  1−(α+β+γ+δ) Te + T Ve + V = λ.k.δ .α .β .γ Ve + V  α  β 1 ∂L 2R 2D α+β+γ−1 −α −β −γ
1−(α+β+γ+δ) = kα × w4 w1 w2 w3 ∂T R + R D + D e e (21) (31)  2T γ−1  2V δ − w3λ = 0 Therefore, Lagrangian function is given as Te + T Ve + V  2R   2D  α β ∂L  2R   2D  L = w1 + w2 = kα Re + R De + D ∂V Re + R De + D     (22) 2T 2V 0  γ  δ−1 + w3 + w4 − λ (f(R,D,T,V ) − y ) 2T 2V Te + T Ve + V − w4λ = 0 Te + T Ve + V (32) 1 2T −1 α+β+δ −α −β −δ
α+β+γ+δ For above function , 1st order conditions can be given = k .γ α β δ Te + T as- 1 (40)  −α−β−δ β  α+β+γ+δ    α−1 × w .wα.w .wδ.y0 ∂L 2Re 2R 3 1 2 4 = w1 2 − λkα ∂R (Re + R) Re + R 1 2V −1 α+β+γ −α −β −γ
α+β+γ+δ  β  γ  δ = k .δ α β γ 2D 2T 2V V + V = 0 e (41) D + D T + T V + V 1 e e e  −α−β−γ α β γ 0 α+β+γ+δ (33) × w4 .w1 .w2 .w3 .y    α ∂L 2De 2R Cost for getting y’ in optimized way is say C where = w2 2 − λkβ ∂D (De + D) Re + R     (34) 2R 2D  β−1  γ  δ C = w1 + w2 2D 2T 2V Re + R De + D = 0     (42) De + D Te + T Ve + V 2T 2V +w3 + w4    α Te + T Ve + V ∂L 2Te 2R = w3 2 − λkγ ∂T (Te + T ) Re + R So, analytically C can be written as- (35)  β  γ−1  δ 1 2D 2T 2V  β γ  α+β+γ+δ 0 1 = 0 α δ α+β+γ+δ (43) C = Q. w1 w2 w3 w4. .y De + D Te + T Ve + V ∂L  2V   2R α where Q is given as- = w e − λkδ 4 2 1 ∂V (Ve + V ) Re + R β+γ+δ α+γ+δ α+β+γ+δ −1  α β  β γ δ−1 (36) α+β+γ+δ  2D   2T   2V  Q = k β γ δ + α γ δ + = 0 β + γ + δ α + γ + δ D + D T + T V + V 1 e e e α+β+δ α+β+γ α+β+γ+δ −1  γ δ  k α+β+γ+δ + ∂L  2R α  2D β  2T γ αα + ββ + δδ αα + ββ + γγ = k ∂λ Re + R De + D Te + T (44)  2V δ with C as − y0 = 0 avg Ve + V 1 C h α β γ δi α+β+γ+δ 0 1 −1 (37) C = = Q w w w w .y ( α+β+γ+δ ) avg y0 1 2 3 4 Putting the values from eq. (25),(26) and (27) in (45) eq.(37) we get, Deriving conditions for optimization-  2R α  β w  2R β y0 = k. 1  2R α−1  2D β  2T γ Re + R α w2 Re + R λαk Re + R De + D Te + T  γ w  2R γ  δ w  2R δ (46) 1 1  2V δ α w3 Re + R α w4 Re + R = w1 Ve + V  2R α+β+γ+δ y0 = k α−β−γ−δββγγ δδ  2R α  2D β−1  2T γ Re + R λβk β+γ+δ −β −γ −δ Re + R De + D Te + T w1 w2 w3 w4 (47)  2V δ On solving for radius, we get = w2 Ve + V 1 2R −1 β+γ+δ −β −γ −δ
α+β+γ+δ = k .α β γ δ  2R α  2D β  2T γ−1 Re + R λγk 1 (38)   α+β+γ+δ Re + R De + D Te + T × w−β−γ−δ.wβ.wγ .wδ.y0 (48) 1 2 3 4  2V δ = w3 Similarly, we have following calculations for other 3 Ve + V parameters-  2R α  2D β  2T γ 2D 1 λδk = k−1.βα+γ+δα−αγ−γ δ−δ
α+β+γ+δ R + R D + D T + T D + D e e e e δ−1 (49) 1 (39)     α+β+γ+δ 2V × w−α−γ−δ.wα.wγ .wδ.y0 = w4 2 1 3 4 Ve + V 1 2D   1−(α+β+γ+δ) Multiplying eq (46),(47),(48) and (49) by = λ.kααβ1−(α+γ+δ)γγ δδ  2R   2D   2T   2V  De + D R +R , D +D , T +T , V +V respectively, 1 e e e e  −α α+γ+δ−1 −γ −δ 1−(α+β+γ+δ) × w1 w2 w3 w4  2R α  2D β  2T γ λαk (63) Re + R De + D Te + T (50) 1 δ 2T   1−(α+β+γ+δ)  2V   2R  = λkααββγ1−(α+β+δ)δδ = w1 Te + T Ve + V Re + R 1 (64)  −α −β α+β+δ−1 −δ 1−(α+β+γ+δ)  2R α  2D β  2T γ × w1 w2 w3 w4 λβk Re + R De + D Te + T 1 (51) 2V  α β γ 1−(α+β+γ) 1−(α+β+γ+δ)  δ   = λkα β γ δ 2V 2D Ve + V = w2 1 Ve + V De + D  −α −β −γ α+β+γ−1 1−(α+β+γ+δ) × w1 w2 w3 w4  2T α  2D β  2T γ λγk (65) Te + T De + D Te + T (52) These above obtained expressions are to be maxi-  2V δ  2T  = w3 mized. Substituting these values in function Y (given Ve + V Te + T as), α β γ  2V   2D   2T   α  β λγk Re − R De − D Ve + V De + D Te + T Y = 1 − . 1 − . (53) Re + R De + D δ (66)  2V   2V   γ  δ = w4 Te − T Ve − V Ve + V Ve + V 1 − . 1 − Te + T Ve + V From eq(17), the above equations can be expressed we obtain, as- 1  2R  Y = kpα+β+γ+δααββγγ δδ
1−(α+β+γ+δ) λα.Y = w1 (54) 1 (67) Re + R   1−(α+β+γ+δ) × w−αw−βw−γ w−δ  2D  1 2 3 4 λβ.Y = w2 (55) De + D If α + β + γ + δ is 1, the model is undefined, which is  2T  the case of constant return to scale (CRS).In this λγ.Y = w3 (56) case the sum of these elasticities must be greater T + T e than 1 for Y to be maximized (IRS) case. This   2V is infeasible from earlier observations implying the λδ.Y = w4 (57) Ve + V CDHS with ESI input structure doesn’t guarantee  2R  a global optima, certainly a drawback that CDHS λα.Y = w1 (58) doesn’t have. Re + R From above equations, it can be concluded References

2D β w1 2R [1] Bora, K., Saha, S., Agrawal, S., Safonova, M., Routh, S., = (59) Narasimhamurthy, A., 2016. CD-HPF: New Habitability De + D α w2 Re + R Score Via Data Analytic Modeling. Astronomy and Com- 2T γ w1 2R puting, 17, 129-143 = (60) [2] Schulze-Makuch, D., M´endez,A., Fair´en,A. G., et al., Te + T α w3 Re + R 2011. A Two-Tiered Approach to Assessing the Habit- 2V δ w1 2R ability of Exoplanets. Astrobiology, 11, 1041. = (61) [3] Saha S., Sarkar J., Dwivedi A., Dwivedi N., Anand M. Ve + V α w4 Re + R N., Roy R., 2016. A novel revenue optimization model Putting the values from eq.(59),(60),(61) into to address the operation and maintenance cost of a data center. Journal of Cloud Computing, 5:1, 1-23. eq.(46),(47),(48) and (49) and solving them we get [4] Saha, S., Basak, S., Agrawal, S., Safonova, M., Bora, K., following- Sarkar, P., Murthy, J. 2018. Theoretical Validation of 1 Potential Habitability via Analytical and Boosted Tree 2R   1−(α+β+γ+δ) = λ.kα1−(β+γ+δ)ββγγ δδ Methods: An Optimistic Study on Recently Discovered Exoplanets. Astronomy and Computing, Arxiv, arXiv.org Re + R 1 ¿ astro-ph ¿ arXiv:1712.01040   1−(α+β+γ+δ) × wβ+γ+δ−1w−βw−γ w−δ [5] Cobb, C. W. and Douglas, P. H., 1928. A Theory of Pro- 1 2 3 4 duction. American Economic Review, 18 (Supplement), (62) 139. [6] Ginde, G., Saha, S., Mathur, A., Venkatagiri, S., Vadakkepat, S., Narasimhamurthy, A., B.S. Daya Sagar., 2016. ScientoBASE: A Framework and Model for Com- puting Scholastic Indicators of Non-Local Influence of Journals via Native Data Acquisition Algorithms. J. Sci- entometrics, 107:1, 1-51 [7] Wolf, E. T., 2017. Assessing the Habitability of the TRAPPIST-1 System Using a 3D Climate Model. Astro- phys. J., 839:L1. [8] B. Goswami, J. Sarkar, S. Saha and S. Kar., CD-SFA: Stochastic Frontier Analysis Approach to Revenue Mod- eling in Cloud Data Centers, International Journal of Computer Networks and Distributed Systems, Inder- science (Forthcoming)