{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T15:55:47Z","timestamp":1768319747434,"version":"3.49.0"},"reference-count":60,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,13]],"date-time":"2020-02-13T00:00:00Z","timestamp":1581552000000},"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 velocities of space plasma particles, often follow kappa distribution functions. The kappa index, which labels and governs these distributions, is an important parameter in understanding the plasma dynamics. Space science missions often carry plasma instruments on board which observe the plasma particles and construct their velocity distribution functions. A proper analysis of the velocity distribution functions derives the plasma bulk parameters, such as the plasma density, speed, temperature, and kappa index. Commonly, the plasma bulk density, velocity, and temperature are determined from the velocity moments of the observed distribution function. Interestingly, recent studies demonstrated the calculation of the kappa index from the speed (kinetic energy) moments of the distribution function. Such a novel calculation could be very useful in future analyses and applications. This study examines the accuracy of the specific method using synthetic plasma proton observations by a typical electrostatic analyzer. We analyze the modeled observations in order to derive the plasma bulk parameters, which we compare with the parameters we used to model the observations in the first place. Through this comparison, we quantify the systematic and statistical errors in the derived moments, and we discuss their possible sources.<\/jats:p>","DOI":"10.3390\/e22020212","type":"journal-article","created":{"date-parts":[[2020,2,14]],"date-time":"2020-02-14T10:09:57Z","timestamp":1581674997000},"page":"212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["On the Determination of Kappa Distribution Functions from Space Plasma Observations"],"prefix":"10.3390","volume":"22","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":[[2020,2,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/s10509-007-9679-1","article-title":"Suprathermal particle distributions in space physics: Kappa distributions and entropy","volume":"312","author":"Shizgal","year":"2007","journal-title":"Astrophys. Space Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"A11105","DOI":"10.1029\/2009JA014352","article-title":"Beyond kappa distributions: Exploiting Tsalis statistical mechanics in space plasmas","volume":"114","author":"Livadiotis","year":"2009","journal-title":"J. Geophys. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s11214-013-9982-9","article-title":"Understanding kappa distributions: A toolbox for space science and astrophysics","volume":"175","author":"Livadiotis","year":"2013","journal-title":"Space Sci. Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s11207-010-9640-2","article-title":"Kappa distributions: Theory and applications in space plasmas","volume":"267","author":"Pierrard","year":"2010","journal-title":"Sol. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1002\/2014JA020825","article-title":"Introduction to special section on origins and properties of kappa distributions: Statistical Background and properties of kappa distributions in space plasmas","volume":"120","author":"Livadiotis","year":"2015","journal-title":"J. Geophys. Res."},{"key":"ref_6","unstructured":"Livadiotis, G. (2017). Kappa Distributions: Theory and Applications in Plasmas, Elsavier."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s41116-019-0021-0","article-title":"The multi-scale nature of the solar wind","volume":"16","author":"Verscharen","year":"2019","journal-title":"Living Rev. Sol. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1029\/97GL00992","article-title":"Ulysses electron distributions fitted with kappa functions","volume":"24","author":"Maksimovic","year":"1997","journal-title":"Geophys. Res. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"A09104","DOI":"10.1029\/2005JA011119","article-title":"Radial evolution of the electron distribution functions in the fast solar wind between 0.3 and 1.5 AU","volume":"110","author":"Maksimovic","year":"2005","journal-title":"J. Geophys. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"17021","DOI":"10.1029\/1999JA900169","article-title":"Electron velocity distribution functions from the solar wind to the corona","volume":"104","author":"Pierrard","year":"1999","journal-title":"J. Geophys. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.12942\/lrsp-2006-1","article-title":"Kinetic Physics of the Solar Wind","volume":"3","author":"Marsch","year":"2006","journal-title":"Living Rev. Sol. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1086\/382866","article-title":"A transonic collisionless model of the solar wind","volume":"606","author":"Zouganelis","year":"2004","journal-title":"Astrophys. J."},{"key":"ref_13","first-page":"A05104","article-title":"Radial evolution of nonthermal electron populations in the low-latitude solar wind: Helios, Cluster, and Ulysses Observations","volume":"114","author":"Maksimovic","year":"2009","journal-title":"J. Geophys. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7074","DOI":"10.1002\/2014JA020353","article-title":"Electron kappa distribution and quasi-thermal noise","volume":"119","author":"Yoon","year":"2014","journal-title":"J. Geophys. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1516","DOI":"10.1002\/2014JA020636","article-title":"\u03ba-distributed protons in the solar wind and their charge-exchange coupling to energetic hydrogen","volume":"120","author":"Heerikhuisen","year":"2015","journal-title":"J. Geophys. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/0019-1035(87)90040-6","article-title":"A comparison of the Mercury and Earth magnetospheres: Electron measurements and substorm time scales","volume":"71","author":"Christon","year":"1987","journal-title":"Icarus"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"A09S12","DOI":"10.1029\/2003JA010270","article-title":"Energetic ion characteristics and neutral gas interactions in Jupiter\u2019s magnetosphere","volume":"109","author":"Mauk","year":"2004","journal-title":"J. Geophys. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"A01212","DOI":"10.1029\/2008JA013761","article-title":"Energetic ion soectral characteristics in the Saturnian magnetosphere using Cassini\/MIMI measurements","volume":"114","author":"Dialynas","year":"2009","journal-title":"J. Geophys. Res."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3126","DOI":"10.1002\/jgra.50353","article-title":"Characterizing the dayside magnetosheath using energetic neutral atoms: IBEX and THEMIS observations","volume":"118","author":"Ogasawara","year":"2013","journal-title":"J. Geophys. Res."},{"key":"ref_20","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_21","doi-asserted-by":"crossref","first-page":"3475","DOI":"10.1002\/2017GL072715","article-title":"Properties of suprathermal electrons associated with discrete auroral arcs","volume":"44","author":"Ogasawara","year":"2017","journal-title":"Geophys. Res. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7407","DOI":"10.1002\/2016JA022972","article-title":"Characterizing cometary electrons with kappa distributions","volume":"121","author":"Broiles","year":"2016","journal-title":"J. Geophys. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1016\/S0273-1177(03)00356-9","article-title":"Voyager observations of low-energy ions during solar cycle 23","volume":"32","author":"Decker","year":"2003","journal-title":"Adv. Space Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1092","DOI":"10.1088\/0004-637X\/708\/2\/1092","article-title":"Microstructure of the heliospheric termination shock: Implications for energetic neutral atom observations","volume":"708","author":"Zank","year":"2010","journal-title":"Astrophys. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/0004-637X\/734\/1\/1","article-title":"First Sky map of the inner heliosheath temperature using IBEX spectra","volume":"734","author":"Livadiotis","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1088\/0004-637X\/751\/1\/64","article-title":"Pick-up ion distributions and their influence on energetic neutral atom spectral curvature","volume":"751","author":"Livadiotis","year":"2012","journal-title":"Astrophys. J."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1088\/0004-637X\/738\/1\/64","article-title":"The influence of pick-up ions on space plasma distributions","volume":"738","author":"Livadiotis","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_29","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_30","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_31","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics: Approaching A Complex World, Springer."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/S0378-4371(98)00437-3","article-title":"The role of constraints within generalized nonextensive statistics","volume":"261","author":"Tsallis","year":"1998","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"50001","DOI":"10.1209\/0295-5075\/122\/50001","article-title":"Thermodynamic origin of kappa distributions","volume":"122","author":"Livadiotis","year":"2018","journal-title":"Eur. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s10509-016-2949-z","article-title":"Misestimation of temperature when applying Maxwellian distributions to space plasmas described by kappa distributions","volume":"361","author":"Nicolaou","year":"2016","journal-title":"Astrophys. Space Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3","DOI":"10.3847\/1538-4357\/aad45d","article-title":"Determining the kappa distributions of space plasmas from observations in a limited energy range","volume":"864","author":"Nicolaou","year":"2018","journal-title":"Astrophys. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"101","DOI":"10.3847\/1538-4357\/ab48e3","article-title":"The impact of turbulent solar wind fluctuations on Solar Orbiter plasma proton measurements","volume":"886","author":"Nicolaou","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Nicolaou, G., Wicks, R.T., Livadiotis, G., Verscharen, D., Owen, C.J., and Kataria, D.O. (2020). Determining the bulk parameters of plasma electrons from pitch-angle distribution measurements. Entropy, 22.","DOI":"10.3390\/e22010103"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"142","DOI":"10.3847\/1538-4357\/aaa713","article-title":"Generation of Kappa Distributions in Solar Wind at 1au","volume":"853","author":"Livadiotis","year":"2018","journal-title":"Astrophys. J."},{"key":"ref_39","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_40","doi-asserted-by":"crossref","first-page":"10","DOI":"10.3847\/1538-4357\/ab05b7","article-title":"On the Origin of Polytropic Behavior in Space and Astrophysical Plasmas","volume":"874","author":"Livadiotis","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_41","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_42","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_43","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_44","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_45","doi-asserted-by":"crossref","first-page":"3","DOI":"10.3847\/1538-4357\/ab487a","article-title":"Rankine\u2013Hugoniot Shock Conditions for Space and Astrophysical Plasmas Described by Kappa Distributions","volume":"886","author":"Livadiotis","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"106","DOI":"10.3847\/1538-4357\/aaef77","article-title":"The Fluid-like and Kinetic Behavior of Kinetic Alfv\u00e9n Turbulence in Space Plasma","volume":"870","author":"Wu","year":"2019","journal-title":"Astrophys. J."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"105307","DOI":"10.1063\/1.5125714","article-title":"Collision frequency and mean free path for plasmas described by kappa distributions","volume":"9","author":"Livadiotis","year":"2019","journal-title":"AIP Adv."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"105009","DOI":"10.1088\/1402-4896\/ab12ab","article-title":"Theoretical aspects of Hamiltonian kappa distributions","volume":"94","author":"Livadiotis","year":"2019","journal-title":"Phys. Scr."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1088\/0004-637X\/741\/2\/88","article-title":"Invariant kappa distribution in space plasmas out of equilibrium","volume":"741","author":"Livadiotis","year":"2011","journal-title":"Astrophys. J."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"19","DOI":"10.3847\/0067-0049\/223\/2\/19","article-title":"The New Horizons Solar Wind Around Pluto (SWAP) observations of the solar wind from 11\u201333 au","volume":"223","author":"Elliott","year":"2016","journal-title":"Astrophys. J. Supp. Ser."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Vaivads, A., Retin\u00f2, A., Soucek, J., Khotyaintsev, Y.V., Valentini, F., Escoubet, C.P., Alexandrova, O., Andr\u00e9, M., Bale, S.D., and Balikhin, M. (2016). Turbulence heating ObserveR-satellite mission proposal. J. Plasma Phys., 82.","DOI":"10.1017\/S0022377816000775"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1439","DOI":"10.1002\/2016JA023269","article-title":"Electrostatic analyzer design for solar wind proton measurements with high temporal, energy, and angular resolutions","volume":"122","author":"Cara","year":"2017","journal-title":"J. Geophys. Res."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.pss.2015.03.020","article-title":"Jupiter\u2019s deep magnetotail boundary layer","volume":"111","author":"Nicolaou","year":"2015","journal-title":"Planet. Space Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.pss.2015.10.001","article-title":"Plasma Properties in the deep jovian magnetotail","volume":"119","author":"Nicolaou","year":"2015","journal-title":"Planet. Space Sci."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"7256","DOI":"10.1002\/2017JA024117","article-title":"Survey of thermal ions in Saturn\u2019s magnetosphere utilizing a forward model","volume":"122","author":"Wilson","year":"2017","journal-title":"J. Geophys. Res."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"e2018JA026169","DOI":"10.1029\/2018JA026169","article-title":"Method to derive ion properties from Juno JADE including abundance estimates for O+ and S2+","volume":"125","author":"Kim","year":"2020","journal-title":"J. Geophys. Res."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1029\/JA082i010p01493","article-title":"Properties of the solar wind at 0.3 AU inferred from measurements at 1 AU","volume":"82","author":"Barouch","year":"1977","journal-title":"J. Geophys. Res."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1029\/GL015i001p00088","article-title":"Estimates of solar wind heating inside 0.3 AU","volume":"15","author":"Freeman","year":"1988","journal-title":"Geophys. Res. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"033302","DOI":"10.1063\/1.4866850","article-title":"Angular scattering of 1\u201350 keV ions through graphene and thin carbon foils: Potential applications for space plasma instrumentation","volume":"85","author":"Ebert","year":"2014","journal-title":"Rev. Sci. Instrum."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.nimb.2015.07.098","article-title":"Semi-empirical relationships for the energy loss and straggling of 1\u201350 keV hydrogen ions passing through thin carbon foils","volume":"359","author":"Allegrini","year":"2015","journal-title":"Nuclear Instrum. Methods Phys. Res. Sect. B"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:57:30Z","timestamp":1760173050000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,13]]},"references-count":60,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["e22020212"],"URL":"https:\/\/doi.org\/10.3390\/e22020212","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,13]]}}}