{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T18:43:57Z","timestamp":1772477037027,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,12]],"date-time":"2019-10-12T00:00:00Z","timestamp":1570838400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The polytropic index of space plasmas is typically determined from the relationship between the measured plasma density and temperature. In this study, we quantify the errors in the determination of the polytropic index, due to uncertainty in the analyzed measurements. We model the plasma density and temperature measurements for a certain polytropic index, and then, we apply the standard analysis to derive the polytropic index. We explore the accuracy of the derived polytropic index for a range of uncertainties in the modeled density and temperature and repeat for various polytropic indices. Our analysis shows that the uncertainties in the plasma density introduce a systematic error in the determination of the polytropic index which can lead to artificial isothermal relations, while the uncertainties in the plasma temperature increase the statistical error of the calculated polytropic index value. We analyze Wind spacecraft observations of the solar wind protons and we derive the polytropic index in selected intervals over 2002. The derived polytropic index is affected by the plasma measurement uncertainties, in a similar way as predicted by our model. Finally, we suggest a new data-analysis approach, based on a physical constraint, that reduces the amount of erroneous derivations.<\/jats:p>","DOI":"10.3390\/e21100997","type":"journal-article","created":{"date-parts":[[2019,10,14]],"date-time":"2019-10-14T03:54:13Z","timestamp":1571025253000},"page":"997","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["On the Calculation of the Effective Polytropic Index in Space Plasmas"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3623-4928","authenticated-orcid":false,"given":"Georgios","family":"Nicolaou","sequence":"first","affiliation":[{"name":"Department of Space and Climate Physics, Mullard Space Science Laboratory, University College London, Dorking, Surrey RH5 6NT, UK"}]},{"given":"George","family":"Livadiotis","sequence":"additional","affiliation":[{"name":"Southwest Research Institute, San Antonio, TX 78238, USA"}]},{"given":"Robert T.","family":"Wicks","sequence":"additional","affiliation":[{"name":"Department of Space and Climate Physics, Mullard Space Science Laboratory, University College London, Dorking, Surrey RH5 6NT, UK"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,12]]},"reference":[{"key":"ref_1","unstructured":"Parker, E.N. (1963). Interplanetary Dynamical Processes, Wiley-Interscience."},{"key":"ref_2","unstructured":"Chandrasekhar, S. (1967). An Introduction to the Study of Stellar Structure, Dover Publications."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3847\/0067-0049\/223\/1\/13","article-title":"Superposition of polytropic in the inner heliosheath","volume":"223","author":"Livadiotis","year":"2016","journal-title":"Astrophys. J. Suppl. Ser."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"7","DOI":"10.3847\/1538-4357\/aa61ff","article-title":"Modeling the plasma flow in the inner heliosheath with a spatially varying compression ratio","volume":"838","author":"Nicolaou","year":"2017","journal-title":"Astrophys. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1029\/94JA02420","article-title":"An empirical determination of the polytropic index for the free-streaming solar wind using Helios 1 data","volume":"100","author":"Totten","year":"1995","journal-title":"J. Geophys. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1029\/1998GL900306","article-title":"Multi-tube model for interplanetary magnetic clouds","volume":"26","author":"Osherovich","year":"1999","journal-title":"Geophys. Res. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"19851","DOI":"10.1029\/1999JA900254","article-title":"On the determination of electron polytrope indices within coronal mass ejections in the solar wind","volume":"104","author":"Gosling","year":"1999","journal-title":"J. Geophys. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0273-1177(94)90042-6","article-title":"Gasdynamic and magnetohydrodynamic modeling of the magnetosheath: A tutorial","volume":"14","author":"Spreiter","year":"1994","journal-title":"Adv. Space Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1086\/147124","article-title":"The stellar-wind regions","volume":"134","author":"Parker","year":"1961","journal-title":"Astrophys. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1088\/0004-637X\/809\/2\/111","article-title":"Shock strength in space and astrophysical plasmas","volume":"809","author":"Livadiotis","year":"2015","journal-title":"Astrophys. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"38","DOI":"10.3847\/1538-4357\/833\/1\/38","article-title":"Generalized multi-polytropic Rankine-Hugoniot relations and the entropy conditions","volume":"833","author":"Scherer","year":"2016","journal-title":"Astrophys. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"L5","DOI":"10.3847\/2041-8213\/ab0f43","article-title":"First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_13","first-page":"1","article-title":"The solar wind as a turbulence laboratory","volume":"4","author":"Bruno","year":"2013","journal-title":"Living Rev. Sol. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.1002\/2017JA024978","article-title":"Using kappa distributions to identify the potential energy","volume":"123","author":"Livadiotis","year":"2018","journal-title":"J. Geophys. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"10","DOI":"10.3847\/1538-4357\/ab05b7","article-title":"On the polytropic behavior in space and astrophysical plasma","volume":"874","author":"Livadiotis","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"52","DOI":"10.3847\/1538-4357\/ab31ad","article-title":"Long term correlations of polytropic indices with kappa distributions in solar wind plasma near 1 au","volume":"884","author":"Nicolaou","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1431","DOI":"10.1029\/97GL01204","article-title":"Solar wind polytropic index in the vicinity of stream interactions","volume":"24","author":"Newbury","year":"1997","journal-title":"Geophys. Res. Lett."},{"key":"ref_18","first-page":"A10107","article-title":"Solar wind polytropic index estimates based on single spacecraft plasma and interplanetary field measurements","volume":"111","author":"Kartalev","year":"2006","journal-title":"J. Geophys. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"15331","DOI":"10.1029\/93JA01012","article-title":"Polytropic relation in interplanetary clouds","volume":"98","author":"Osherovich","year":"1993","journal-title":"J. Geophys. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4736","DOI":"10.1002\/2014JA020855","article-title":"Polytropic index of central plasma sheet ions based on MHD Bernoulli integral","volume":"120","author":"Pang","year":"2015","journal-title":"J. Geophys. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1866","DOI":"10.1029\/2018JA026108","article-title":"Dependence of thermodynamic processes on upstream interplanetary magnetic field conditions for magnetosheath ions","volume":"124","author":"Park","year":"2019","journal-title":"J. Geophys. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1086\/378878","article-title":"A three-dimensional model of the solar wind incorporating solar magnetogram observations","volume":"595","author":"Roussev","year":"2003","journal-title":"Astrophys. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1002\/ctpp.201000071","article-title":"Effective polytropic indices of anisotropic magnetosheath plasmas with magnetoacoustic waves","volume":"51","author":"Meister","year":"2011","journal-title":"Contrib. Plasma Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1371","DOI":"10.1007\/s11207-013-0401-x","article-title":"Long-term variability of the polytropic index of solar wind protons at 1 AU","volume":"289","author":"Nicolaou","year":"2014","journal-title":"Sol. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"88","DOI":"10.3847\/0004-637X\/829\/2\/88","article-title":"Plasma-field coupling at small length scales in solar wind near 1 au","volume":"829","author":"Livadiotis","year":"2016","journal-title":"Astrophys. J."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Livadiotis, G. (2018). Long-term independence of solar wind polytropic index on plasma flow speed. Entropy, 20.","DOI":"10.3390\/e20100799"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Schwenn, R. (1990). Large-Scale Structure of the Interplanetary Medium, in Physics of the Inner Heliosphere, Springer.","DOI":"10.1007\/978-3-642-75361-9"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"8066","DOI":"10.1029\/2018JA025820","article-title":"Energetic ion moments and polytropic index in Saturn\u2019s magnetosphere using Cassini\/MIMI measurements: A simple model based on \u03ba-distribution functions","volume":"123","author":"Dialynas","year":"2018","journal-title":"J. Geophys. Res."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2032","DOI":"10.1016\/j.pss.2009.09.007","article-title":"Plasma electrons in Saturn\u2019s magnetotail: Structure, distribution and energisation","volume":"57","author":"Arridge","year":"2009","journal-title":"Planet. Space Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3463","DOI":"10.1002\/2013JA019665","article-title":"Properties of plasma ions in the distant Jovian magnetosheath using Solar Wind Around Pluto data on New Horizons","volume":"119","author":"Nicolaou","year":"2014","journal-title":"J. Geophys. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1088\/0004-637X\/762\/2\/134","article-title":"Pressure of the proton plasma in the inner heliosheath","volume":"762","author":"Livadiotis","year":"2013","journal-title":"Astrophys. J."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1088\/0004-637X\/749\/1\/11","article-title":"Non-equilibrium thermodynamic processes: Space plasmas and the inner heliosheath","volume":"749","author":"Livadiotis","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2863","DOI":"10.1002\/jgra.50304","article-title":"Fitting method based on correlation maximization: Applications in Astrophysics","volume":"118","author":"Livadiotis","year":"2013","journal-title":"J. Geophys. Res"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Nicolaou, G., Verscharen, D., Wicks, R.T., and Owen, C.J. (2019). The Impact of Turbulent Solar Wind Fluctuations on Solar Orbiter Plasma Proton Measurements. Astrophys. J., in press.","DOI":"10.3847\/1538-4357\/ab48e3"},{"key":"ref_35","first-page":"A03105","article-title":"Physics-based tests to identify the accuracy of solar wind ion measurements: A case study with the Wind Faraday Cups","volume":"111","author":"Kasper","year":"2006","journal-title":"J. Geophys. Res."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1134\/S0010952510020012","article-title":"Algebra and statistics of the solar wind","volume":"48","author":"Veselovsky","year":"2010","journal-title":"Cosmic Res."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"A36","DOI":"10.1051\/0004-6361\/201731831","article-title":"Solar-wind predictions for the Parker Solar Probe orbit: Near-Sun extrapolations derived from an empirical solar-wind model based on Helios and OMNI observations","volume":"611","author":"Venzmer","year":"2018","journal-title":"Astron. Astrophys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/BF00751326","article-title":"SWE, a comprehensive plasma instrument for the WIND spacecraft","volume":"71","author":"Ogilvie","year":"1995","journal-title":"Space Sci. Rev."},{"key":"ref_39","unstructured":"Kasper, J. (2003). Solar Wind Plasma: Kinetic Properties and Micro-Instabilities. [Ph.D. Thesis, Massachusetts Institute of Technology]."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/BF00751330","article-title":"The WIND magnetic field investigation","volume":"71","author":"Lepping","year":"1995","journal-title":"Space Sci. Rev."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"416","DOI":"10.3390\/stats2040028","article-title":"Linear Regression with Optimal Rotation","volume":"2","author":"Livadiotis","year":"2019","journal-title":"Stats"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/10\/997\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:25:39Z","timestamp":1760189139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/10\/997"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,12]]},"references-count":41,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,10]]}},"alternative-id":["e21100997"],"URL":"https:\/\/doi.org\/10.3390\/e21100997","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,12]]}}}