Grunsky's theorem: Difference between revisions

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[[File:Terrestrial planets size comparison.png|thumb|510px|Though differing in size and temperature, terrestrial planets of the Solar System tend to have high Earth Similarity Indexes - [[Mercury (planet)|Mercury]] (0.39), [[Venus]] (0.78), [[Earth]] (1.00) and [[Mars]] (0.64) in true colors, sizes to scale.<ref name="UPR-Catalog">http://phl.upr.edu/projects/habitable-exoplanets-catalog/data</ref> ]]
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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 [[planet]]s, however the formula can also be applied to large [[natural satellite]]s and other objects. The ESI is a function of the planet's [[radius]], [[density]], [[escape velocity]], and [[surface temperature]].<ref name=esi>{{cite web|title=Earth Similarity Index (ESI)|url=http://phl.upr.edu/projects/earth-similarity-index-esi|publisher=Planetary Habitability Laboratory}}</ref><ref>Schulze-Makuch, D., Méndez, A., Fairén, A. G., von Paris, P., Turse, C., Boyer, G., Davila, A. F., Resendes de Sousa António, M., Irwin, L. N., and Catling, D. (2011) [http://online.liebertpub.com/doi/abs/10.1089/ast.2010.0592 A Two-Tiered Approach to Assess the Habitability of Exoplanets.] Astrobiology 11(10): 1041-1052.</ref> 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 planet|terrestrial]] rocky composition.
 
ESI is not a measure of [[planetary habitability|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 [[exoplanet]]s have been found with values in this range. [[Kepler-62e]] has the highest Earth Similarity of [[Exoplanet#Confirmed discoveries|confirmed]] exoplanets at 0.83, while [[Kepler Object of Interest|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.<ref name="UPR-Catalog" />
 
On November 4, 2013, astronomers reported, based on [[Kepler (spacecraft)|''Kepler'' space mission]] data, that there could be as many as 40 billion [[Terrestrial planet|Earth-sized]] [[extrasolar planets|planets]] orbiting in the [[habitable zone]]s of [[sun-like|sun-like stars]] and [[red dwarf stars]] within the [[Milky Way Galaxy]].<ref name="NYT-20131104">{{cite news |last=Overbye |first=Dennis|title=Far-Off Planets Like the Earth Dot the Galaxy|url=http://www.nytimes.com/2013/11/05/science/cosmic-census-finds-billions-of-planets-that-could-be-like-earth.html |date=November 4, 2013 |work=[[New York Times]] |accessdate=November 5, 2013 }}</ref><ref name="PNAS-20131031">{{cite journal |last1=Petigura |first1=Eric A.|last2=Howard |first2=Andrew W. |last3=Marcy |first3=Geoffrey W. |title=Prevalence of Earth-size planets orbiting Sun-like stars|url=http://www.pnas.org/content/early/2013/10/31/1319909110 |date=October 31, 2013 |journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|doi=10.1073/pnas.1319909110 |accessdate=November 5, 2013 }}</ref> 11 billion of these estimated planets may be orbiting sun-like stars.<ref name="LATimes-20131104">{{cite news |last=Khan |first=Amina |title=Milky Way may host billions of Earth-size planets |url=http://www.latimes.com/science/la-sci-earth-like-planets-20131105,0,2673237.story |date=November 4, 2013 |work=[[Los Angeles Times]] |accessdate=November 5, 2013 }}</ref> The nearest such planet may be 12 light-years away, according to the scientists.<ref name="NYT-20131104" /><ref name="PNAS-20131031" />
 
== Formulation ==
The ESI is defined by the expression
 
: <math> ESI = \prod_{i=1}^n \left(1 - \left| \frac{x_i - x_{i_0}}{x_i + x_{i_0}} \right| \right)^\frac{w_i}{n} </math>
 
where <math> x_i </math> is one of the planetary properties (e.g. surface temperature), <math> x_{i_0} </math> is the corresponding terrestrial reference value (e.g. 288 K) for the property, <math> w_i </math> 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.
 
{| class="wikitable"
! Property
! Reference Value
! Weight Exponent
|-
| [[Mean]] [[radius]]    ||  1.0 [[Image:Earth symbol.svg|12x19px|Earth]] || 0.57
|-
| [[Bulk density]]        ||  1.0 [[Image:Earth symbol.svg|12x19px|Earth]] || 1.07
|-
| [[Escape velocity]]    ||  1.0 [[Image:Earth symbol.svg|12x19px|Earth]] || 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==
{{main|List of potential habitable exoplanets}}
[[File:Relative sizes of all of the habitable-zone planets discovered to date alongside Earth.jpg|thumb|Comparison of the sizes of planets [[Kepler-69]]c, [[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:
 
{| class="wikitable"
! Planet
! Global ESI
! Status/Note/References
|-
| [[KOI-3284.01]] || 0.90 || Unconfirmed<ref name="UPR-Catalog"/>
|-
| [[KOI-1686.01]] || 0.89 || Unconfirmed<ref name="UPR-Catalog"/>
|-
| [[KOI-3010.01]] || 0.87 || Unconfirmed<ref name="UPR-Catalog"/>
|-
| [[KOI-4742.01]] || 0.83 || Unconfirmed<ref name="UPR-Catalog"/>
|-
| [[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==
[[File:Io Earth Moon Comparison.png|thumb|right|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 moon]]s and other non-planets to have high ESIs.]]
The ESI can be applied to objects other than planets, including [[natural satellite]]s, [[dwarf planet]]s and [[asteroid]]s 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.<ref>pg 143. [http://www.statslab.cam.ac.uk/~pat/misc.pdf Multivariate and other worksheets for R (or S-Plus): a miscellany] P.M.E.Altham, Statistical Laboratory, University of Cambridge. January 10, 2013</ref>
 
The following non-planetary objects have relatively high global ESIs:
 
{| class="wikitable"
! Object
! Classification
! Global ESI
|-
| [[HD 222582 b m]] || Exomoon (unconfirmed) || 0.86
|-
| [[Moon]] || Natural satellite || 0.56
|-
| [[Io (moon)|Io]] || Natural satellite || 0.36
|-
| [[Callisto (moon)|Callisto]] || Natural satellite || 0.34
|-
| [[Ganymede (moon)|Ganymede]] || Natural satellite || 0.29
|-
| [[Ceres (dwarf planet)|Ceres]] || Dwarf planet/Asteroid || 0.27
|-
| [[Europa (moon)|Europa]] || Natural satellite || 0.26
|-
| [[4 Vesta]] || Asteroid || 0.256
|-
| [[Titan (moon)|Titan]] || Natural satellite || 0.24
|-
| [[2 Pallas]] || Asteroid || 0.222
|-
| [[Titania (moon)|Titania]] || Natural satellite || 0.10
|-
| [[Enceladus (moon)|Enceladus]] || Natural satellite || 0.094
|-
| [[Pluto]] || Dwarf planet || 0.075
|-
| [[Triton (moon)|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==
{{reflist|30em}}
 
==External links==
*[http://phl.upr.edu/projects/habitable-exoplanets-catalog/data HEC: Data of Potential Habitable Exoplanets and Exomoons]
 
{{Extraterrestrial life}}
{{Exoplanet}}
 
[[Category:Planetary habitability]]
[[Category:Search for extraterrestrial intelligence]]

Latest revision as of 03:49, 8 December 2014

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