Grunsky's theorem

Though differing in size and temperature, terrestrial planets of the Solar System tend to have high Earth Similarity Indexes - Mercury (0.39), Venus (0.78), Earth (1.00) and Mars (0.64) in true colors, sizes to scale.

The Earth Similarity Index, ESI or "easy scale" is a measure of how physically similar a planetary mass object is to Earth. It is a scale from zero to one, with Earth having a value of one. The ESI was designed to measure planets, however the formula can also be applied to large natural satellites and other objects. The ESI is a function of the planet's radius, density, escape velocity, and surface temperature. These parameters are often estimated based on one or more known variables. Such variables depend greatly on the method of observation used. For example, surface temperatures is influenced by a variety of factors including irradiance, tidal heating, albedo, insolation and greenhouse warming. Where these are not known, planetary equilibrium temperature is frequently used, or the variable is inferred from other known attributes.

A planet with a high ESI (values in the range from 0.8 and 1.0) is likely to be of terrestrial rocky composition.

ESI is not a measure of habitability, though given the point of reference being Earth, some of its functions match closely to those used by habitability measures. The ESI and habitable zone share in common the use of surface temperature as a primary function (and the terrestrial point of reference).

According to this measure there are no other Earth-like planets or moons in the Solar System (second-ranked Venus is 0.78), though a number of exoplanets have been found with values in this range. Kepler-62e has the highest Earth Similarity of confirmed exoplanets at 0.83, while Kepler candidate KOI-1686.01, if confirmed, would likely have an ESI of 0.89. Further, the candidate exomoon HD 222582 b m of a confirmed exoplanet, and several candidate exomoons (KOI 375.01 m, KOI-2933.01 m, KOI-422.01 m) of unconfirmed exoplanets, all have an ESI of 0.86.

On November 4, 2013, astronomers reported, based on Kepler space mission data, that there could be as many as 40 billion Earth-sized planets orbiting in the habitable zones of sun-like stars and red dwarf stars within the Milky Way Galaxy. 11 billion of these estimated planets may be orbiting sun-like stars. The nearest such planet may be 12 light-years away, according to the scientists.

Formulation

The ESI is defined by the expression

$ESI=\prod _{i=1}^n{(1-\left|\frac{x_i-x_{i_0}}{x_i+x_{i_0}}\right|)}^{\frac{w_i}{n}}$

where $x_i$ is one of the planetary properties (e.g. surface temperature), $x_{i_0}$ is the corresponding terrestrial reference value (e.g. 288 K) for the property, $w_i$ is a weight exponent for the property, and n is the total number of planetary properties. The weight exponents adjust the sensitivity of the scale and equalize their meanings across the different properties. The set of properties, their reference values, and their weight exponents are found in the following table.

Property	Reference Value	Weight Exponent
Mean radius	1.0	0.57
Bulk density	1.0	1.07
Escape velocity	1.0	0.70
Surface temperature	288 K	5.58

ESI has been split into two components to measure different aspects of physical similarity - the Interior ESI and the Surface ESI. The Mean radius and bulk density comprise the Interior ESI, while the escape velocity and surface temperature comprise the Surface ESI. Global ESI is typically cited as the global measure.

Planets with relatively high ESI

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Comparison of the sizes of planets Kepler-69c, Kepler-62e (0.82), Kepler-62f (0.69), and the Earth. All planets except the Earth are artists' conceptions.

Extrasolar planets dominate the list of known Earth-like objects. However the classification is made difficult in that of the methods of extrasolar planet detection, many methods leave ESI parameters unquantified. For example, with the transit method, one of the more successful methods, measurement of radius can be highly accurate, however mass and density is often estimated. Likewise with radial velocity methods, which provide accurate measurements of mass but are less successful measuring radius. Planets observed via a number of different methods therefore have the most accurate measures of ESI, though this is not possible in many situations.

The following planets have been determined to have higher ESI than Venus or Mars:

Planet	Global ESI	Status/Note/References
KOI-3284.01	0.90	Unconfirmed
KOI-1686.01	0.89	Unconfirmed
KOI-3010.01	0.87	Unconfirmed
KOI-4742.01	0.83	Unconfirmed
Kepler-62e	0.83
Gliese 667C c	0.82
Gliese 581 g	0.82	Unconfirmed
KOI-1422.04	0.82	Unconfirmed

ESIs of non-planets

The Moon, Io and Earth shown to scale. Although significantly smaller, some of the Solar System's moons and dwarf planets share similarities to Earth's density and temperature resulting in relatively high ESIs. It is theoretically possible for Earth-sized extrasolar moons and other non-planets to have high ESIs.

The ESI can be applied to objects other than planets, including natural satellites, dwarf planets and asteroids though comparisons typically draw lower global ESI due to the lower average density and temperature of these objects, at least for those known in the Solar System.

The following non-planetary objects have relatively high global ESIs:

Object	Classification	Global ESI
HD 222582 b m	Exomoon (unconfirmed)	0.86
Moon	Natural satellite	0.56
Io	Natural satellite	0.36
Callisto	Natural satellite	0.34
Ganymede	Natural satellite	0.29
Ceres	Dwarf planet/Asteroid	0.27
Europa	Natural satellite	0.26
4 Vesta	Asteroid	0.256
Titan	Natural satellite	0.24
2 Pallas	Asteroid	0.222
Titania	Natural satellite	0.10
Enceladus	Natural satellite	0.094
Pluto	Dwarf planet	0.075
Triton	Natural satellite	0.074

Note that of these, only Titan is known to hold onto a significant atmosphere despite an overall lower size and density.

See also

• Earth analog
• List of potential habitable exoplanets
• Planetary Habitability Index

References

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External links

• HEC: Data of Potential Habitable Exoplanets and Exomoons