# Similarity indexing of exoplanets in search for potential habitability:
application to Mars-like worlds.

J. M. Kashyap11affiliation<EMAIL_ADDRESS>S. B.
Gudennavar22affiliation<EMAIL_ADDRESS>Department
of Physics, Christ University, Bengaluru 560 029, India  Urmi
Doshi33affiliation<EMAIL_ADDRESS>M. Safonova44affiliation:
<EMAIL_ADDRESS>M. P. Birla Institute of Fundamental Research,
Bangalore 560 001, India

###### Abstract

Study of exoplanets is one of the main goals of present research in planetary
sciences and astrobiology. Analysis of huge planetary data from space missions
such as CoRoT and Kepler is directed ultimately at finding a planet similar to
Earth – the Earth’s twin, and answering the question of potential exo-
habitability. The Earth Similarity Index (ESI) is a first step in this quest,
ranging from 1 (Earth) to 0 (totally dissimilar to Earth). It was defined for
the four physical parameters of a planet: radius, density, escape velocity and
surface temperature. The ESI is further sub-divided into interior ESI
(geometrical mean of radius and density) and surface ESI (geometrical mean of
escape velocity and surface temperature). The challenge here is to determine
which exoplanet parameter(s) is important in finding this similarity; how
exactly the individual parameters entering the interior ESI and surface ESI
are contributing to the global ESI. Since the surface temperature entering
surface ESI is a non-observable quantity, it is difficult to determine its
value. Using the known data for the Solar System objects, we established the
calibration relation between surface and equilibrium temperatures to devise an
effective way to estimate the value of the surface temperature of exoplanets
for further analysis with our graphic methodology.

ESI is a first step in determining potential exo-habitability that may not be
very similar to a terrestrial life. A new approach, called Mars Similarity
Index (MSI), is introduced to identify planets that may be habitable to the
extreme forms of life. MSI is defined in the range between 1 (present Mars)
and 0 (dissimilar to present Mars) and uses the same physical parameters as
ESI. We are interested in Mars-like planets to search for planets that may
host the extreme life forms, such as the ones living in extreme environments
on Earth; for example, methane on Mars may be a product of the methane-
specific extremophile life form metabolism.

Earth Similarity Index (ESI), Mars Similarity Index (MSI), potential
habitability, surface temperature, exoplanets

##  1 Introduction

The search for life elsewhere outside the Earth has been a very fascinating
area in recent years. A lot of efforts have been channeled in this direction
in the form of space missions looking for potential habitable planets (e.g.
Seager, 2010). Analysis of huge volume of collected planetary data from space
missions such as CoRoT and Kepler is directed at finding a planet similar to
Earth – the Earth’s twin, and answering the question of exo-habitability. A
full assessment of the habitability of any planet requires very detailed
information about it, which we still cannot achieve. Currently, the best we
can do is to compare the properties we can measure (or infer), in other words
determine the potential habitability. Here, we consider habitability in the
Earth way, or in some modified but still recognizable by us way (e.g. mars, or
Titan, or Europa/Enciladus). The key parameters that are typically available
from observations are stellar flux that can help to judge the temperature of
the planet, and the planet’s size, derived either from transit observation –
radius, or from RV measurements – low limit on mass. In the latter case, from
the statistical point of view the true mass differs from the low limit by only
a factor of $4/\pi\sim 1.2$ (e.g. Maruyana et al. 2013). With the developed
theoretical models, it is already possible to exclude planets from the
potential habitability list based on mass/radius values. Thus, since we prefer
to start the search only on the rocky worlds, only planets of up to
$1.6\,R_{\rm E}$ or $2\,M_{\rm E}$ can be rocky (e.g. Grosset, Mocqut&Sotin,
2007; Chen & Kipping, 2016). With the basic criteria and data already
available, we can start gauging the potential habitability of exoplanets. The
challenge here is to determine which exoplanet parameter(s) are important in
finding this similarity. To address this challenge, the Earth Similarity Index
(ESI), a parametric index to analyze the exoplanets’ data, was introduced to
assess the Earth-likeness of exoplanets (Schulze-Makuch et al., 2011a;
Mascaro, 2011). This multi-parameter ESI scale depends on radius, density,
escape velocity and surface temperature of a planet. The ESI index is used by
the Planetary Habitability Laboratory (PHL), University of Puerto Rico,
Arecibo, to estimate the potential habitability of all discovered to date
exoplanets. Here, the total ESI ranges from 0 (totally dissimilar to Earth) to
1 (identical to Earth). A planet with ESI $\geq 0.8$ is considered an Earth-
like, but even planets with ESI $\geq 0.73$ (e.g. Mars) are optimistically
called potentially habitable planets (PHPs). As of now (January 2017), there
are 44 assumed PHPs111http://phl.upr.edu/projects/habitable-exoplanets-
catalog. We focus on indexing the exoplanet data taken from the online
Exoplanets Catalog (PHL-EC)222http://phl.upr.edu/projects/habitable-
exoplanets-catalog/data/database maintained by the PHL. The ESI is subdivided
into the interior ESI — an estimate of probability of a planet to have a rocky
interior (calculated from radius and density), and exterior ESI – a surface
similarity, an estimate of probability of a surface temperature to be within a
habitable range (calculated from surface temperature and escape velocity); the
total (global) ESI is their geometric mean. This way, the scale is useful for
the overall concept of planet similarity for both interior and surface
properties. One of our objectives here is to establish how the individual
parameters, entering the interior and surface ESI, are contributing to the
global ESI, using graphic analysis approach.

Presently, one of the ESI parameters, the mean surface temperature, is
estimated for the rocky planets in PHL-EC by following a correction factor of
30-33 K based on the Earth’s greenhouse effect. Another objective of our work
in calculations of the global ESI is to try to introduce a better estimate of
the surface temperature of rocky exoplanets, because it may be a crucial
element in the search for habitable planets (Carone et al. 2016). From graphic
analysis of the known data for some of the Solar System objects, we establish
the calibrated relation between surface and equilibrium temperatures. This
relation is extended to estimate the surface temperatures of exoplanets from
their equilibrium temperatures.

Though Earth is currently the only place where life as we know it exists,
there is a good probability that life could have existed on Mars in the past
(Abramov & Mojzsis, 2016). Indeed, Mars is technically inside the Sun’s
habitable zone (HZ), though its ESI is only 0.73 (PHL). However, Mars-like
conditions may be suitable for the extremophile life forms, such as the ones
living in extreme environments on Earth (numerous experiments have proven
several organisms to survive the simulated martian conditions, e.g. Onofri et
al. 2015). Reports on the presence of methane in the martian atmosphere
(Webster et al. 2015) raise the possibility of the existence of a methane-
specific extremophile life forms; in fact, on Earth methanogens thrive in
conditions similar to martian environments: in dry desert soils and in 3-km
deep glacial ice (see Hu et al. 2016, and references therein). Therefore, we
are interested in Mars similarity to look for planets potentially habitable by
extremophiles, say extreme PHPs. For that, we introduce the new indexing
formulation, called Mars Similarity Index (MSI), where the range is 0 (no
similarity to present Mars) and 1 (present Mars). Mars here represents a sort
of test-bed that can indicate the potential habitability of small rocky
planets. A planet with MSI $>0.63$ is recognized to be Mars-like, e.g. Earth,
which is a habitable planet. We find from our data analysis six Mars-like
planets (Mars, Earth, Kepler-186 f, Kepler-442 b, Kepler-438 b, Kepler-62 f,
GJ 667C c, GJ 667C f), which are also included in the PHP list.

The structure of this paper is as follows. In Section 2, we give the detailed
derivation of the ESI and MSI scales. Data analysis is presented in Section 3,
which includes ESI analysis, surface temperature analysis, and MSI analysis as
subsections. Finally, discussion and conclusions are given in Section 4.

##  2 Mathematical formulation of the ESI and MSI

Distance/similarity measurements are widely used in, for example,
classification of objects, clustering and retrieval problems, or to compute
the overlaps between quantitative data. Here, the distance $d$ is represented
as dissimilarity, and proximity is equivalent to similarity $s$.
Mathematically, the concept of distance is a metric one – a measure of a true
distance in Euclidean space ${\bf R}^{n}$. This problem can be addressed by
using Minkowski’s space of $L_{p}$ form (Sung-Hyuk Cha, 2007), in which
$p$-norm stands for finite $n$-dimensional vector space,

$d=\left(\sum_{i=1}^{n}|p_{i}-q_{i}|^{p}\right)^{1/p}\,,$ (1)

where the city-block, or Manhattan, $L_{1}$ distance

$d_{\rm CB}=\sum_{i=1}^{n}|p_{i}-q_{i}|\,,$ (2)

and Euclidean $L_{2}$ distance

$d_{\rm Euc}=\left(\sum_{i=1}^{n}(p_{i}-q_{i})^{2}\right)^{1/2}\,,$ (3)

are the special cases. Here, $p_{i}$ and $q_{i}$ are the coordinates of $p$
and $q$ in dimension $i$, and $i=1,2,3,\ldots,n$. $L_{1}$ form has an
advantage that it can be decomposed into contributions made by each variable
being the sum of absolute differences (e.g. for the $L_{2}$ form, it would be
the decomposition of the squared distance). However, in comparing multivariate
data sets, some distance measures can be applied that do not conform to the
usual definition of a metric (i.e. metric axioms, e.g. Deza and Deza 2016),
but are still very useful as a measure of difference (or similarity) between
samples. Here, we are interested in finding similarities between different
planets based on their various characteristics, in other words, a multivariate
data sets. In such cases, especially abundant in ecological and environmental
studies to quantify the differences between samples collected at different
sampling locations (e.g Looman, 1960), the Bray-Curtis distance is the most
used scale (Bray and Curtis, 1957); it is also sometimes called an ecological
distance (Kindt and Coe, 2005). The advantages in using the ecological
distance is that differences between datasets can be expressed by a single
statistic. Bray-Curtis is a modified Manhattan distance333Bray-Curtis distance
becomes Manhattan when it is applied to relative counts, as opposed to the
absolute abundances (Greenacre and Primicerio, 2013)., where the summed
differences between the variables are standardized by the summed variables of
the objects,

$d_{\rm
BC}={\sum_{i=1}^{n}|p_{i}-q_{i}|\over\sum_{i=1}^{n}\left(p_{i}+q_{i}\right)}\,.$
(4)

Here $p_{i}$ and $q_{i}$ are two different precisely measurable quantities
between which the distance is to be measured, and $n$ is the total number of
variables. The assumption in the Bray-Curtis scale is that samples are taken
from same physical measure, e.g. mass, or volume. It is because the distance
is found from the raw counts, so that if there is a higher abundance in one
sample comparing to the other, it is a part of the difference between the two
samples. The advantage of the Bray-Curtis scale is the simplicity in
interpretation: $0$ means the samples are exactly the same, and $1$ means they
are completely disjoint. It shall be kept in mind that Bray-Curtis distance is
not the true metric distance since it violates the triangular inequality – one
of the axioms of a true metric. Though it is still called a semi-metric in
some earlier works, in modern mathematics it is defined as a non-metric (Deza
and Deza, 2007).

The intersection between two distributions is a more widely used form of
similarity (as opposed to distance). Most similarity measures for intersection
can be transformed from the distance measure by the transformation technique
(see Bloom 1981, but not exclusively),

$s_{\rm BC}=1-d_{\rm
BC}=1-{\sum_{i=1}^{n}|p_{i}-q_{i}|\over\sum_{i=1}^{n}\left(p_{i}+q_{i}\right)}\,.$
(5)

Here, the value of $0$ means complete absence of relationships, and the value
of $1$ shows a complete matching of the two data records in the
$n$-dimensional space (Schulz, 2007).

Distances/similarities based on heterogeneous data can be found after a
process of standardization — balancing of the contribution of different types
of variables in an equitable way (Greenacre and Primicerio, 2013). One way to
do that is to calculate the similarity for each set of homogeneous variables
and then combine them using various methods (see Sec. 3.2). Higher values in
one set may impact the result of the Bray-Curtis similarity more dominantly
and imply that these variables are more likely to discriminate between sets.
Therefore, user-defined weighting is a convenient (though subjective) method
for down-weighing the differences for a set of variables.

In our case, we would like to compare sets of different variables of one
planet with that of the reference value, for example, Earth, to find planets
that are similar to Earth. We rewrite Eq. (5) as

$s=\left[1-\frac{|x-x_{0}|}{(x+x_{0})}\right]^{w_{x}}\,,$ (6)

where $x$ is the physical property of the exoplanet, $w_{x}$ is the weight for
this property, $x_{0}$ is the reference value, and the dimension $n=1$, since
we are constructing the index separately for each physical property. We find
the weights by defining the threshold value ($V$) in the similarity scale for
each quantity,

$V=\left[1-\Big{|}\frac{x_{0}-x}{x_{0}+x}\Big{|}\right]^{w_{x}}\,.$ (7)

Traditionally, the similarity indices are subdivided into equal 0.2 intervals
(Bloom, 1981), defining very low, low, moderate, high and very high similarity
regions. Therefore, the threshold can be defined on this grounds, for example
considering only very high similarity region with the threshold $V=0.8$.
Defining the physical limits $x_{a}$ and $x_{b}$ of the permissible variation
of a variable with respect to $x_{0}$ (i.e. $x_{a}<x_{0}<x_{b}$), we calculate
the weight exponents for the lower $w_{a}$ and upper $w_{a}$ limits,

$w_{a}=\frac{\ln{V}}{\ln\left[1-\left|\frac{x_{0}-x_{a}}{x_{0}+x_{a}}\right|\right]}\,,\quad
w_{b}=\frac{\ln{V}}{\ln\left[1-\left|\frac{x_{b}-x_{0}}{x_{b}+x_{0}}\right|\right]}\,,\quad$
(8)

The average weight is found by the geometric mean,

$w_{x}=\sqrt{{w_{a}}\times{w_{b}}}\,.$ (9)

In this paper, we use Eq. 5 to define the Earth similarity index

$ESI_{x}={\left[1-\Big{|}\frac{x-x_{0}}{x+x_{0}}\Big{|}\right]^{w_{x}}}\,,$
(10)

and Mars similarity index

$MSI_{x}={\left[1-\Big{|}\frac{x-x_{0}}{x+x_{0}}\Big{|}\right]^{w_{x}}}\,,$
(11)

where $x$ is the physical property of the exoplanet (for example, radius or
density), and $x_{0}$ is the reference to Earth in ESI, and to Mars in MSI.

We focus on indexing the exoplanet data taken from the online PHL Exoplanets
Catalog (PHL-EC)444Maintained by the PHL,
http://phl.upr.edu/projects/habitable-exoplanets-catalog/data/database. The
catalog contains more than 60 observed and derived stellar and planetary
parameters for all currently confirmed exoplanets. The input parameters for
similarity scales are radius $R$, density $\rho$, surface temperature $T_{S}$
and escape velocity ${V_{e}}$. These parameters, except the surface
temperature, are expressed in Earth Units (EU) in the ESI calculations. In
addition, we normalized the mean radius, bulk density and escape velocity to
Mars Units (MU) in the calculations of the Mars Similarity Index. The
corresponding weight exponents for both ESI and MSI scales were found using
the threshold value $V=0.8$, indicating very high similarity region. The
weight exponents for the upper and lower limits of parameters were calculated
for the Earth-like parameter range (Schulze-Makuch et al., 2011a): radius 0.5
to 1.9 EU, mass 0.1 to 10 EU, density 0.7 to 1.5 EU, surface temperature 273
to 323 K, and escape velocity 0.4 to 1.4 EU, through Eqs. (8) and (9).
Similarly, the weight exponents for the lower and upper limits of parameters
were defined for the Mars-like conditions: radius range 0.72 to 1.88 MU, mass
range 0.514 to 9.30 MU, density range 0.89 to 1.402 MU, surface temperature
range 233 to 418 K, and escape velocity range 0.85 to 2.23 MU. Here, MU are
Mars Units, where radius is 3390 km, density is 3.93 g/cm3, escape velocity is
5.03 km/s, and the mean surface temperature 240 K (Barlow, 2014). The reason
behind the limits definitions were to have a rocky planet, with lower limit in
comparison to Mars (mass and radius are chosen as for Mercury, the smallest
planet in our Solar System, and density as for Io), and with Earth as the
upper limit. The temperature range is chosen on the basis of the range known
to be suitable for extremophile life forms on our planet, between
$-40^{\circ}$ and $+145^{\circ}$C (Tung et al., 2007). The corresponding
weight exponents were calculated using the same method as for the ESI, and are
given in Table 1 along with Earth and Mars unit parameters.

Table 1: ESI and MSI Parametric Table

Planetary Property | Ref. Value | Ref. Value | Weight Exponents | Weight Exponents
---|---|---|---|---
| for ESI | for MSI | for ESI | for MSI
Mean Radius | 1EU | 1MU | 0.57 | 0.86
Bulk Density | 1EU | 1MU | 1.07 | 2.10
Escape Velocity | 1EU | 1MU | 0.70 | 1.09
Surface Temperature | 288K | 240K | 5.58 | 3.23

EU = Earth Units, with Earth radius 6371 km, density 5.51 g/cm3, escape
velocity 11.19 km/s.
MU = Mars Units, with Mars radius 3390 km, density 3.93 g/cm3, escape velocity
5.03 km/s.

##  3 Data analysis

The PHL-EC contains 3635 confirmed exoplanets (as of January 2017). However,
some of them do not have all the required input parameters for calculating the
global ESI. Wherever we could, we supplemented the missing data after the
additional search through the following catalogs: Habitable Zone Gallery, Open
Exoplanet Catalogue, Extrasolar Planets Encyclopaedia, Exoplanets Data
Explorer and Nasa Exoplanet Archive. In addition, we discarded the entries
with unrealistic values. For example, some of the planets had the
equilibrium/surface temperatures of less than 3.2 K, and some had the density
values of around 500 EU. In such cases, we have done extensive search through
all available exoplanet catalogs and discovery papers, and supplemented those
values that we could find. As an example, there is a lot of confusion with the
available data on Kepler-53c, Kepler-57c and Kepler-59b planets. Their
densities are listed in the PHL-EC as 162, 573.18 and 492 Earth Units,
respectively. In such cases, the density of, say, Kepler 57c, becomes 21 times
the density of the Sun’s core. There is obviously a mistake in the retrieved
data. We have searched through the catalogs and found that for Kepler 53c, the
mass of 5007.56 EU used in calculating that density was, in fact, an upper
limit from the stability analysis (Steffen et al. 2013), and it was
subsequently updated (Haden et al. 2014) to $35.4^{+19.}_{-14.8}$ EU with
nearly the same value for the radius. Using this number, the density becomes
1.169 EU. Similarly, for Kepler 57c, the density of 573.18 EU was obtained
using the upper limit on mass of 2208.83 EU which, after updating the mass to
$7.4^{+9.4}_{-6.3}$ EU (Haden et al. 2014) with essentially the same radius,
fell within the normal range, 1.139 EU. Correspondingly, we have corrected the
data for these planets in our catalog. Some of the entries had to be removed
owing to the absence of available data (or very conflicting values), which
left us with 3566 exoplanets for our analysis (1650 of them are rocky).

###  3.1 SURFACE TEMPERATURE ANALYSIS

Surface temperature of the planet enters the exterior ESI index – a measure of
surface similarity. Usually, the extrasolar planets temperatures are estimated
from the calculated temperature of the parent star and other observational
data (e.g. distance to the star, etc.) (Fressin et al., 2012), in the
assumption that the planet does not have an atmosphere e.g. (e.g. Williams,
2014), which gives a so-called radiative equilibrium temperature. It is
essentially an effective temperature attained by an isothermal planet that is
in complete radiation equilibrium with the parent star. If a planet has no
atmosphere, this temperature equals the surface temperature of the planet.
Actual temperatures of the gas giants are usually higher because there is an
internal heat source. If the planet has a substantial atmosphere, then this is
the temperature near the tropopause of the planet – approximately the level
where the radiation from the planet is emitted to space. Terrestrial planets
with atmospheres, such as Earth or Venus, for example, have higher
temperatures due to the greenhouse effect. So, though equilibrium temperature
may be different from the actual surface temperature of the planet, it is a
useful parameter in comparing planets. It has been used to estimate the
boundaries of Habitable Zones around stars (e.g. Kaltenegger and Sasselov
2011; Schulze-Makuch et al. 2011), and is an essential quantity for exoplanets
when it is not possible to measure (or calculate) the surface T.

The albedo entering the surface temperature equation is generally not known
and has to be assumed. The albedo depends on many factors, some of which are
geometry, composition, and atmospheric properties. In addition, there is also
the emissivity index which represents atmospheric effects, and depends on the
type of surface and many climate models. In calculations of the surface
temperatures of rocky planets by the PHL, a correction factor of $30-33$ K is
usually used, based on the Earth’s green-house effect (Schulze-Makuch et al.,
2011a; Volokin and ReLlez, 2014). However, we shall not ignore the good
quality data available for other planets in our Solar System, because unlike
with the exoplanets, many parameters are actually been measured. We have used
the known data of our Solar System objects (only rocky bodies), using several
available sources, such as NASA planetary fact sheet (Williams, 2014), and
extrapolated the resulting relation to obtain the surface temperatures of
rocky exoplanets in our dataset. The relation between the surface temperature
$(T_{s})$ and the equilibrium temperature $(T_{e})$ (Fig. 1) is

$T_{s}=9.65+1.0956\times T_{e}\,,$ (12)

Figure 1: Calibration of surface temperature. Venus is masked as red dot due
its very high surface temperature.

In Table 2, we present the equilibrium and surface temperatures of several
Solar System objects, and the sample of our results for a few potentially
habitable exoplanets.

Table 2: Equilibrium and surface temperatures Planet | Equilibrium Temp | Surface Temp | $T_{s}=9.65+1.0966\,T_{e}$
---|---|---|---
| (K) | (K) | (K)
Earth | 255 | 288 | 289.28
Mars | 217 | 240 | 247.61
Moon | 157 | 197 | 181.81
Venus | 227 | 730 | 258.57
Titan | 82 | 94 | 99.57
Triton | 34.2 | 38 | 47.15
GJ 667Cc | 246.5 | 277.4 | 279.96
Kepler-442 b | 233 | unknown | 265.16
Kepler-438 b | 276 | unknown | 312.31
GJ 667 C f | 220.7 | unknown | 251.67

Surface temperature of the Moon varies from Farside 120 K to the 350 K of a
subsolar point, hence we adopted the average surface temperature for the Moon
of 197 K as the most physically robust estimate (Volokin and ReLlez, 2014).

This empirical correlation is, of course, very simplified. However, such
regression analysis was already successfully used to predict the mean global
temperatures of rocky planets in Volokin and ReLlez (2015) (see their Figs. 1
– 4). The objective of our analysis was to evaluate how well the resulting new
relation may predict the observed mean surface temperatures; in Table 2 we
show the measured and calculated by Eq. 12 surface temperatures, and observe
that calculated values are pretty close to the measured ones.

###  3.2 ESI CALCULATION

First we converted the input parameters to Earth Units (EU), except the
surface temperature, which is left in Kelvin. Some of the planets in the PHL-
EC do not have estimates for the surface temperature. To mitigate this, we
have calculated the surface temperature of all rocky planets using Eq. (12).
The corresponding Earth Similarity Index for each parameter was calculated
using Eq. (10). Then these indices were separately combined to form an
interior similarity and surface similarity. The interior ESI is thus

$ESI_{I}=\sqrt{{ESI_{R}}\times{ESI_{\rho}}}\,,$ (13)

and surface ESI is:

$ESI_{S}=\sqrt{ESI_{T}\times ESI_{V_{e}}}\,,$ (14)

where $ESI_{R}$, $ESI_{\rho}$, $ESI_{T}$ & $ESI_{V_{e}}$ are Earth Similarity
Indices, calculated for radius, density, surface temperature and escape
velocity, respectively. The global ESI is their geometric mean:

$ESI=\sqrt{{ESI_{I}}\times{ESI_{S}}}\,.$ (15)

The results of the ESI calculations for all 3566 currently confirmed
exoplanets are presented in Table 3 (only few planets are shown as an
example). The full dataset is available online (Kashyap et al. 2017). The
sample calculation of ESI for Mars is given in the Appendix.

Table 3: A sample of calculated ESI Names | Radius | Density | Temp | E. Vel | $ESI_{S}$ | $ESI_{I}$ | ESI
---|---|---|---|---|---|---|---
| (EU) | (EU) | (K) | (EU) |  |  |
Earth | 1.00 | 1.00 | 288 | 1.00 | 1.00 | 1.00 | 1.00
Mars | 0.53 | 0.73 | 240 | 0.45 | 0.65 | 0.82 | 0.73
Kepler-438 b | 1.12 | 0.90 | 312 | 1.06 | 0.88 | 0.95 | 0.91
Proxima Cen b | 1.12 | 0.90 | 259 | 1.06 | 0.85 | 0.95 | 0.90
GJ 667C c | 1.54 | 1.05 | 280 | 1.57 | 0.88 | 0.92 | 0.90
Kepler-296 e | 1.48 | 1.03 | 302 | 1.5 | 0.86 | 0.93 | 0.89

###  3.3 ESI ANALYSIS

Results of Section 3.1 are presented as a histogram of global ESI (Fig. 2).
According to the PHL project, surface ESI is dominating the interior ESI,
because the surface temperature weight exponent value is much higher than that
of the interior parameters. We found, however, that this is only true for the
giant planets. For the rocky planets, we found that the interior ESI is a
predominant factor in the global ESI, where the real values of interior and
surface ESI play a larger role than the weight exponent. The 3-D histogram
(Fig. 3) is the result of overplotting interior and surface ESI for all the
rocky exoplanets.

Figure 2: Histogram of the global ESI values of 3566 exoplanets Figure 3: 3-D
histogram of interior and surface ESI for 1650 rocky planets.

In Fig. 4, we present a scatter plot of interior ESI versus surface ESI. Blue
dots are the giant planets, red dots are the rocky planets, and cyan circles
are the Solar System objects. The dashed curves are the isolines of constant
global ESI, with values shown in the plot. Planets above ESI=0.8 are
conservatively considered Earth-like, and planets with ESI $\gtrsim 0.73$ are
optimistically potentially habitable planets (PHL). We see that there are 29
Earth-like planets (ESI ¿ 0.8) in 3566 planets that we have considered. In
this plot we also see the predominant nature of the interior ESI. However, due
to the geometrical mean nature of the global ESI formula, we need to consider
all the four parameters to check the habitability of the planet. We also find
from the plot that there seem to be a definite division between gaseous and
rocky planets, at approximately $ESI_{I}=0.67$ (interior ESI of the Moon). It
is interesting to note that this division separates Moon and Io, rocky
satellites, (especially Io, which is closer in bulk composition to the
terrestrial planets) and, say Pluto and Europa, which are composed of water
ice–rock.

Figure 4: Plot of interior ESI versus surface ESI. Blue dots are the giant
planets, red dots are the rocky planets, and cyan circles are the Solar System
objects (Table 3). The dashed curves are the isolines of constant global ESI,
with values shown in the plot. The giant planets above 0.67 dotted lines are
of water-gas composition, and the planets to the right of 0.67 dotted line are
rocky planets. Planets with ESI$\gtrsim 0.8$ are considered Earth-like.
However, the optimistic limit is $\sim 0.67$.

When the ESI index was proposed, it was accepted that even planets with ESI
between 0.6 and 0.8 could be potentially habitable, or at least similar to
Earth. Thus, we propose to extend the optimistic limit from 0.73 to 0.67.
There are now we about 100 rocky exoplanets with ESI $>0.67$. For example, the
ESI of Kepler-445 d is 0.76. It is located in the HZ, and has an estimated
surface temperature of 305 K, which would make it suitable for life.

###  3.4 MSI CALCULATION

In our Solar System, we have discovered many planetary bodies with conditions
similar to some of the terrestrial environments where we know that Earth
extremophiles live. Particularly notable here is Mars. It is now believed that
Mars had a much wetter and warmer environment in its early history (Grotzinger
et al. 2015; Wray et al. 2016). The discovery of a desert varnish on Mars
(Krinsley et al. 2012), believed to be the product of the specific bacteria on
Earth (Dorn and Oberlander 1981), has gotten us even further down the road of
whether life existed/exists on Mars with its super-extreme conditions for
habitation. We introduce here the Mars Similarity Index (MSI), calculated
using Eq. (11), to look for Mars-like planets as potential planets to host
extremophile life forms. We are interested in this study to compare extreme
environments similar to Mars. Interior MSI is

$MSI_{I}=\sqrt{{MSI_{R}}\times{MSI_{\rho}}}\,,$ (16)

and the surface MSI is

$MSI_{S}=\sqrt{MSI_{T_{S}}\times MSI_{V_{e}}}\,,$ (17)

where $MSI_{R}$, $MSI_{\rho}$, $MSI_{T_{S}}$ & $MSI_{V_{e}}$ are Mars
similarity indices calculated for radius, density, surface temperature and
escape velocity, respectively. The global MSI is given by:

$MSI=\sqrt{{MSI_{I}}\times{MSI_{S}}}$ (18)

A result of MSI calculations is presented in Table 4 (only few entries are
shown). The full dataset is available online (Kashyap et al. 2017). For the
graphic analysis part we have used all 3566 confirmed exoplanets as in the ESI
case.

Table 4: A sample of determined MSI Names | Radius | Density | Temp | E. Vel | $MSI_{S}$ | $MSI_{I}$ | MSI
---|---|---|---|---|---|---|---
| (MU) | (MU) | (K) | (MU) |  |  |
Mars | 1.00 | 1.00 | 240 | 1.00 | 1.00 | 1.00 | 1.00
Earth | 1.88 | 1.407 | 288 | 2.23 | 0.659 | 0.70 | 0.68
Moon | 0.513 | 0.853 | 197 | 0.48 | 0.66 | 0.77 | 0.71
Kepler-42 d | 1.07 | 1.05 | 503 | 1.11 | 0.58 | 0.95 | 0.74
Kepler-378 c | 1.29 | 1.08 | 462 | 1.37 | 0.59 | 0.90 | 0.73
Kepler-438 b | 2.10 | 1.23 | 312 | 2.35 | 0.70 | 0.73 | 0.72
Proxima Cen b | 2.10 | 1.23 | 259 | 2.35 | 0.69 | 0.73 | 0.71

###  3.5 ANALYSIS OF THE MSI

Results of Section 3.3 are represented as a histogram of the global MSI (Fig.
5) for 3566 confirmed exoplanets. In Fig. 6, we show the 3-D histogram of the
interior and surface MSI. As with the ESI, we can see that interior MSI is
more dominant factor than surface MSI for the rocky exoplanets in the global
MSI.

Figure 5: Histogram of the global MSI values for 3566 confirmed exoplanets.
Figure 6: Interior and Surface MSI for 1650 rocky planets.

In Fig. 7, we present a scatter plot of interior MSI versus surface MSI for
3566 confirmed exoplanets. The dashed curves are the isolines of constant
global MSI, with values shown in the plot (the value for the Earth is 0.68).
Planets above MSI $\sim 0.63$ are considered Mars-like. For example,
Kepler-186 f has MSI$\gtrsim 0.69$ is potentially habitable for extremophile
life forms. There is a noticeable difference of similarity to Mars planets
(29) compared with the Earth-like planets, where we find 99 of the Earth-like
PHPs. The reason is that, probably due to the selection bias, smaller planets
are under-represented in the catalog of detected planets. In Fig. 8, we
present mass-radius plots for small size planets. Left plot is in the Earth
units for planets of $\leq 20$ Earth masses, and the right plot in Mars units
for planets $\leq 2$ Earth masses, or $\sim 20$ Mars masses. The regular
features seen on this plot, as well as on Figs. 4 and 7, are due to the use of
modelled values, inferring mass from radius or radius from mass. We have
plotted the model curves for the mass-radius relation adopted by the PHL on
both plots. It is obvious that data for the most of the rocky planets was
estimated from these relations. Planets that are noticeably not following the
relations are those whose mass and radius were estimated independently, such
as, e.g. Kepler-138 b (Jontof-Hutter et al. 2015).

Figure 7: Plot of interior MSI versus surface MSI. Blue dots are the giant
planets, red dots are the rocky planets, and cyan circles are the Solar System
objects. The dashed curves are the isolines of constant global MSI, with
values shown in the plot. Planets with MSI$\gtrsim 0.63$ are assumed Mars-
like; we obtain 29 planets.

Figure 8: Left: Mass-radius diagram for exoplanets with measured masses less
than 20 EU along with model curves for different mass-radius relation: black
line is $R=M^{0.3}$ for $M_{\rm E}<1$; blue-dotted line is $R=M^{0.5}$ for
$1<M_{\rm E}<200$. Red crosses indicate rocky planets, blue crosses are gas
giants, cyan squares are our Solar System objects. In 2016, the data in the
catalog suggested only two rocky exoplanets smaller than Earth. In the present
data, there are many more smaller planets. Right: Blow-up of the previous plot
for small-size planets, in terms of the Mars units. Line of same mass-radius
relation are marked on the plot, along with the isolines of constant density.
Some interesting planets are marked by names.

##  4 Discussion and Conclusion

The current definition of the habitable planets as those that are to be found
in habitable zones (HZ) of the hosts has a caveat. A habitable planet may not
necessarily harbour life (e.g. Venus), although from a distance its
biosignatures may show positive signs; while Earth-like planets (similar
atmosphere, size, mass, at a similar distance and orbiting a solar-type star)
with high probability may actually have life on them (e.g. Jheeta 2013). Other
factors can come in next, such as, for example, the age of the planet (e.g.
Safonova et al., 2016) since life requires time to change its environment to
become noticeable.

The search for habitable exoplanets has essentially two goals, both of which
(if fulfilled) will have profound implications to our civilization. One is to
seek life elsewhere outside the Earth. Another one is to seek a twin-Earth,
preferably nearby. The second goal, in principle, is to have a planet
habitable for our kind of life, but uninhabited, so that we can shift there in
the far away future, if needed. Another aspect of the second goal is that it
is easier to search for the biosignatures of life as we know it on a planet
which looks just like Earth. It is estimated that one in five solar-type stars
and approximately half of all M-dwarf stars may host an Earth-like planet in
the HZ. Extrapolation of Kepler’s data shows that in our Galaxy alone there
could be as many as 40 billion such planets (e.g. Borucki et al. 2010; Batalha
et al. 2013; Petigura et al. 2013). And it is quite possible that quite soon
we may actually observe most of them. With the ultimate goal of a discovery of
life, astronomers do not have millennia to quietly sit and sift through more
information than even pentabytes of data. In addition, obtaining the spectra
of an Earth-like planet around a Sun-like star is difficult, and would require
a large-scale expensive space mission (such as e.g. JWST), which still may be
able to observe only about a hundred stars over its lifetime (e.g., Turnbull
et al. 2012). Thus, it is necessary to prioritise the planets to look at, to
introduce a quick estimate of whether a planet can be habitable from the
measured properties of the star and the planet. The Earth Similarity Index
(ESI), a parametric index to analyze the exoplanets data, was introduced to
access precisely that; to evaluate the potential habitability (Earth-likeness)
of all discovered to date exoplanets. Since our search for habitable
exoplanets (aka Earth-like life, which is clearly favoured by Earth-like
conditions) is by necessity anthropocentric, as all we know for sure is only
the Earth-based habitability, any such indexing has to be centred around
finding Earth-like life, at least initially. Here, we have shown how the ESI
can be derived from the initial mathematical concept of similarity.

Out of the four parameters, entering the global ESI, only one (radius) is a
direct observable, while the remaining three parameters, surface temperature,
escape velocity and density, are generally calculated. According to the PHL
project, surface ESI is dominating the interior ESI, because the weight
exponent value for the surface temperature is much higher than that of the
interior parameters. We found, however, that the interior ESI is a predominant
factor in the global ESI for the rocky exoplanets, where the real values of
interior and surface ESI play a larger role than the weight exponents (Fig.
2). However, even though evaluation of only radius and density parameters may
be enough to suggest a rocky nature of an exoplanet, due to the geometrical
mean nature of the ESI formulation, we need to consider the surface
temperature to verify the Earth-likeness. For example, if we consider surface
temperature values as 10 K, 100 K and 2500 K, and keep interior ESI the same
as for the Earth, the corresponding global ESI values will be 0.02, 0.40 and
0.11, respectively; clearly not habitable. Thus, the surface temperature and
plays a key role in balancing the global ESI equation. However, there is
always an observational difficulty in estimating the surface temperature value
of the exoplanet. We introduced the calibration technique in Section 3.2 to
try to mitigate this difficulty for the case of rocky planets. We have used
the existing observational data from Solar System rocky objects and performed
the linear regression to extrapolate to all rocky exoplanets.

It is now believed that in its early history, Mars had a much wetter and
warmer environment, just at the time when life on Earth is now known to have
originated (this date was recently moved back to 4.1 Ga (Gigayears ago) (Bell
et al. 2015). Curiosity data indicates early ($\sim 3.8$ Ga) Martian climate
with stable water lakes on the surface for thousands to millions of years at a
time (Grotzinger et al. 2015), and a recently discovered evidence of
carbonate-rich ($\sim 3.8$ Ga) bedrock (Wray et al. 2016) suggested the
habitable warm environment. It is possible that after the presumed
catastrophic impact-caused loss of most of the atmosphere (e.g., Melosh et
al., 1989; Webster et al. 2013), only the toughest life forms had survived,
the ones we call here on Earth as extremophiles. They would have adapted to
the currently existing conditions and just like the terrestrial extremophiles
would need such conditions for the survival; for example, terrestrial
methanogens have developed biological mechanism that allows them to repair DNA
and protein damage to survive at temperatures from $-40^{\circ}$C to
$145^{\circ}$C (Tung et al., 2007). The usual conditions for habitability
would be different for such life forms. Carbon and water have the dominant
role as the backbone molecule and a solvent of biochemistry for Earth life.
However, the abundance of carbon may not be a useful indication of the
habitability of an exoplanet. The Earth is actually significantly depleted in
carbon compared with the outer Solar System. Here, on Earth, we have examples
of life, both microbial and animal, that do not require large amounts of water
either. For example, both bacteria and archaea are found thriving in the hot
asphalt lakes (Schulze-Makuch et al. 2011b) with no oxygen and virtually no
water present. They respire with the aid of metals, perhaps iron or manganese,
and create their own water by breaking down hydrocarbons, just like E. coli
gut bacteria that can generate most of their own water from light hydrocarbons
(Kreuzer-Martin et al. 2005). We have introduced the Mars similarity index to
study the Mars-like planets as potential planets to host extremophile life
forms. In this scale, Moon has the MSI of $\sim 0.75$, Earth has the MSI of
$\sim 0.68$, and the next closest exoplanet is Kepler-186 f (MSI=0.69), which
is listed as potentially habitable planet in the HEC. Mars-like planets can
tell us about the habitability of small worlds rather than planets that are
far from their star. For example, Earth at Mars distance would most probably
still be habitable (Wordsworth, 2016). Given constant exchange of impact
ejecta between the planets, it is possible that biota from the Earth reached
and survived on Mars, which thus could have been ‘extremophile’-habitable
throughout all its history. It is interesting to note that when we started
this work, only two small (less than Earth in size or mass) exoplanets were
known. In this year, many new small planets were discovered, with resulting 29
planets that we can call Mars-like.

So can such similarity scale be useful? It actually might be developed into a
sort of same scale as the stellar types in astronomy, such as classifying
stars based on information about size, temperature, and brightness. It can be
used as a quick tool of screening planets in important characteristics in
Earth-likeness. Different ranking scales for evaluating habitability
perspectives for follow-up targets have been already proposed (e.g.
habitability index for transiting exoplanets (HITE, Barnes et al. 2015), or
Cobb-Douglas Habitability Index (CDHI, Bora et al. 2016). We conclude that it
is necessary to arrive at the a multiparameter calculator that, though based
on a current similarity scale – ESI, may include more input parameters (e.g.
orbital properties, temperature, escape velocity, radius, density, activation
energy and so on), and extended to other planet-likeness, such as similarity
to Mars – MSI. If we find habitable possibilities on Titan for example, the
scale can be modified for the Titan-like habitability. We would like to call
this future calculator a Life Information Score (LIS), which shall be used as
an overall calculator to detect life itself. The LIS is almost similar to the
anthropic selection, which basically deals with the preconditions for the
emergence of life and, ultimately, intelligent observers (Waltham, 2011). But
the expected outcome of this LIS is to accurately measure the possibility of a
planet to host any form of life using only the parametric data.

## Calculation of Mars ESI as an example

$ESI_{x}$ calculations for Mars are performed using Eq. (10), with weight
exponents from Table 1, by using the following values for the input
parameters,

$\displaystyle R=0.53\times 6371\,km=3376.63\,km\,,$
$\displaystyle\rho=0.71\times 5.51\,g/cm^{3}=3.9121\,g/cm^{3}\,,$
$\displaystyle V_{e}=0.45\times 11.19\,km/s=5.0355\,km/s\,,$ $\displaystyle
T_{s}=240\,K\,.$

The ESI for each parameter are, accordingly,

$\displaystyle
ESI_{R}=\left(1-\left|3376.63\,km-6371\,km\right|\Big{/}\left|3376.63\,km+6371\,km\right|\right)^{0.57}=0.8124\,,$
$\displaystyle
ESI_{\rho}=\left(1-|3.9121\,g/cm^{3}-5.51\,g/cm^{3}|\Big{/}|3.9121\,g/cm^{3}+5.51g/cm^{3}|\right)^{1.07}=0.8218\,,$
$\displaystyle
ESI_{v_{e}}=\left(1-|5.0355\,km/s-11.19\,km/s|\Big{/}|5.0355\,km/s+11.19\,km/s|\right)^{0.7}=0.7162\,,$
$\displaystyle
ESI_{T_{s}}=\left(1-|240\,K-288\,K|\Big{/}|240\,K+288\,K|\right)^{5.58}=0.5875\,.$

Interior ESI from Eq. (13) is:

$ESI_{I}=\sqrt{0.8124\times 0.8218}\approx 0.8171$.

Surface ESI from Eq. (14) is:

$ESI_{S}=\sqrt{0.7162\times 0.5875}\approx 0.6487$.

And the global ESI for Mars (Eq. 15) is:

$ESI=\sqrt{0.8171\times 0.6487}\approx 0.728$.

## Acknowledgments

We would like to thank Jayant Murthy (Indian Institute of Astrophysics,
Bangalore) and Yuri Shchekinov (Lebedev Institute, Moscow) for the fruitful
discussions. This research has made use of the Extrasolar Planets
Encyclopaedia at http://www.exoplanet.eu, Exoplanets Data Explorer at
http://exoplanets.org, the Habitable Zone Gallery at
http://www.hzgallery.org/, the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program
at http://exoplanetarchive.ipac.caltech.edu/ and NASA Exoplanet Archive at
http://exoplanetarchive.ipac.caltech.edu and NASA Astrophysics Data System
Abstract Service.

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