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Mathematical Geosciences

, Volume 49, Issue 2, pp 179–194 | Cite as

Relative Abundances of Mineral Species: A Statistical Measure to Characterize Earth-like Planets Based on Earth’s Mineralogy

  • Grethe HystadEmail author
  • Robert T. Downs
  • Robert M. Hazen
  • Joshua J. Golden
Article

Abstract

The mineral frequency distribution of Earth’s crust provides a mineralogy-based statistical measure for characterizing Earth-like planets. It has previously been shown that this distribution conforms to a generalized inverse Gauss–Poisson large number of rare events model. However, there is no known analytic expression for the probability distribution of this model; therefore, the population probabilities do not exist in closed forms. Consequently, in this paper, the population probabilities are calculated numerically for all mineral species in Earth’s crust, including the predicted undiscovered species. These population probabilities provide an estimate of the occurrence probabilities of species in a random sample of N mineral species–locality pairs. These estimates are used to characterize Earth in terms of its mineralogy. The study demonstrates that Earth is mineralogically unique in the cosmos. In spite of this uniqueness, the frequency distribution of minerals from Earth can be used to quantify the extent to which another planet is Earth-like. Quantitative criteria for characterizing Earth-like planets are given. An example, involving mineral species found on Mars by the CheMin instrument during the Mars Science Laboratory mission suggests that Mars is mineralogically similar to an Earth-like planet.

Keywords

Statistical mineralogy Mineral frequency distribution Large number of rare events distribution Mineral ecology Earth-like planets Mars mineralogy 

Notes

Acknowledgements

Two anonymous reviewers provided detailed and valuable suggestions for improving the manuscript. We received critical advice and data from Edward Grew. We gratefully acknowledge support from NASA Mars Science Laboratory Mission NNX11AP82A, as well as support from the Alfred P. Sloan Foundation (Grant Number 2013-10-01), the W. M. Keck Foundation (Grant Number 140002372), the Deep Carbon Observatory, the Carnegie Institution for Science, and an anonymous private foundation.

References

  1. Baayen RH (2001) Word frequency distributions, text, speech and language technology, vol 18. Kluwer Academic Publishers, DordrechtCrossRefGoogle Scholar
  2. Borucki W, Koch D, Batalha N, Caldwell D, Christensen-Dalsgaard J, Cochran WD, Dunham E, Gautier TN, Geary J, Gilliland R, Jenkins J, Kjeldsen H, Lissauer JJ, Rowe J (2008) Kepler: search for Earth-size planets in the habitable zone. Proc Int Astron Union 4:289–299CrossRefGoogle Scholar
  3. Borucki WJ, Koch DG, Basri GB, Caldwell DA, Caldwell JF, Cochran WD, DeVore E, Dunham EW, Geary JC, Gilliland RL, Gould A, Jenkins JM, Kondo Y, Latham DW, Lissauer JJ (2003) The Kepler mission: finding the sizes, orbits and frequencies of Earth-size and larger extrasolar planets. In: Deming D, Seager S (eds) Scientific frontiers in research on extrasolar planets, ASP conference series, vol 294, pp 427–440Google Scholar
  4. Donahoe FJ (1966) On the abundance of Earth-like planets. Icarus 5:303–304CrossRefGoogle Scholar
  5. Downs RT (2015) Determining mineralogy on Mars with the CheMin X-ray diffractometer. Elements 11:45–50CrossRefGoogle Scholar
  6. Evert S (2004) A simple LNRE model for random character sequences. In: Proceedings of the 7èmes Journées Internationales d’Analyse Statistique des Données Textuelles, Louvain-la-Neuve, Belgium, pp 411–422Google Scholar
  7. Evert S, Baroni M (2007) zipfR: word frequency distributions in R. In: Proceedings of the 45th annual meeting of the association for computational linguistics, Prague, Czech Republic, Posters and Demonstrations Session, pp 29–32Google Scholar
  8. Evert S, Baroni M (2008) Statistical models for word frequency distributions, package zipfR. http://zipfr.r-forge.r-project.org/materials/zipfR_0.6-5.pdf. Accessed 1 June 2015
  9. Grew ES, Hazen RM (2014) Beryllium mineral evolution. Am Miner 99:999–1021CrossRefGoogle Scholar
  10. Hazen RM, Papineau D, Bleeker W, Downs RT, Ferry JM, McCoy TJ, Sverjensky DA, Yang H (2008) Mineral evolution. Am Miner 93:1693–1720CrossRefGoogle Scholar
  11. Hazen RM, Bekker A, Bish DL, Bleeker W, Downs RT, Farquhar J, Ferry JM, Grew ES, Knoll AH, Papineau D, Ralph JP, Sverjensky DA, Valley JW (2011) Needs and opportunities in mineral evolution research. Am Miner 96:953–963CrossRefGoogle Scholar
  12. Hazen RM, Grew ES, Downs RT, Golden JJ, Hystad G (2015a) Mineral ecology: chance and necessity in the mineral diversity of terrestrial planets. Can Miner 00:1–29Google Scholar
  13. Hazen RM, Hystad G, Downs RT, Golden JJ, Pires AJ, Grew ES (2015b) Earth’s ‘missing’ minerals. Am Miner 100:2344–2347CrossRefGoogle Scholar
  14. Hystad G, Downs RT, Grew ES, Hazen RM (2015a) Statistical analysis of mineral diversity and distribution: Earth’s mineralogy is unique. Earth Planet Sci Lett 426:154–157CrossRefGoogle Scholar
  15. Hystad G, Downs RT, Hazen RM (2015b) Mineral species fequency distribution conforms to a large number of rare events model: prediction of Earth’s missing minerals. Math Geosci 47:647–661CrossRefGoogle Scholar
  16. Johnson RA, Wichern DW (2007) Applied multivariate statistical analysis, 6th edn. Pearson, Upper Saddle RiverGoogle Scholar
  17. Ryaben’kii VS, Tsynkov SV (2006) A theoretical introduction to numerical analysis. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  18. Seager S (2003) The search for extrasolar Earth-like planets. Earth Planet Sci Lett 208:113–124CrossRefGoogle Scholar
  19. Shen TJ, Chao A, Lin CF (2003) Predicting the number of new species in further taxonomic sampling. Ecology 84(3):798–804CrossRefGoogle Scholar
  20. Sichel HS (1971) On a family of discrete distributions particularly suited to represent long-tailed frequency data. In: Proceedings of the third symposium on mathematical statistics, Pretoria, South Africa, pp 51–97Google Scholar
  21. Sichel HS (1975) On a distribution law for word frequencies. J Am Stat Assoc 70:542–547Google Scholar
  22. Sichel HS (1986) Word frequency distributions and type-token characteristics. Math Sci 11:45–72Google Scholar
  23. Ward PD, Brownlee D (2003) Rare Earth: why complex life is uncommon in the universe. Copernicus, New YorkGoogle Scholar
  24. Williams RE, Blacker B, Dickinson M, Dixon WVD, Ferguson HC, Fruchter AS, Giavalisco M, Gilliland RL, Heyer I, Katsanis R, Levay Z, Lucas RA, McElroy DB, Petro L, Postman M, Adorf HM, Hook R (1996) The Hubble deep field: observations, data reduction, and galaxy photometry. Astron J 112:1335CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2016

Authors and Affiliations

  • Grethe Hystad
    • 1
    Email author
  • Robert T. Downs
    • 2
  • Robert M. Hazen
    • 3
  • Joshua J. Golden
    • 2
  1. 1.Mathematics, Statistics, and Computer SciencePurdue University NorthwestHammondUSA
  2. 2.Department of GeosciencesUniversity of ArizonaTucsonUSA
  3. 3.Geophysical LaboratoryCarnegie Institution for ScienceWashingtonUSA

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