{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,6]],"date-time":"2026-09-06T05:16:00Z","timestamp":1788671760027,"version":"build-2803163510"},"reference-count":29,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T00:00:00Z","timestamp":1788480000000},"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 transformation of a diffuse molecular cloud into a star necessarily increases the entropy of the universe, chiefly through the radiation emitted as gravitational binding energy is released. We present a compact, fully closed-form thermodynamic model of this process: the Sackur\u2013Tetrode equation gives the entropy of the initial cloud and, generously, of the stellar material itself, while the released gravitational potential energy is converted into a radiation-entropy term Srad=\u0394Epot\/(2Teff), the factor of one-half following from the virial theorem for a self-gravitating star in hydrostatic equilibrium. For a solar-type star we obtain \u0394S\u22431.9\u00d71037JK\u22121, consistent with independent literature estimates of stellar and interstellar entropy. Extending the calculation across the main sequence (O through M) gives \u0394S\u221dM0.71, rising from 1.2\u00d71037JK\u22121 for a 0.3M\u2299 M dwarf to 2.0\u00d71038JK\u22121 for a 20M\u2299 O star. We then map the full (M,R,Teff) parameter space to locate the locus of \u0394S=0\u2014the formal boundary of thermodynamic feasibility for a single monolithic collapse\u2014and show that every real main-sequence star lies deep in the entropy-producing region, with the boundary itself displaced to radii and masses far outside the stellar regime. Applying the same closed-form model to representative red giants, supergiants, white dwarfs and neutron stars (not as a model of their true formation, but as a diagnostic of how compactness controls radiative entropy production) shows that \u0394S is set primarily by the compactness GM2\/(RTeff) of the final configuration, so that degenerate remnants\u2014if they were assembled by a single collapse from a diffuse cloud\u2014would be substantially larger entropy sources than main-sequence stars, while extended giants are comparatively modest ones. The same closed-form machinery gives direct access to a full thermodynamic feasibility map, something that would otherwise require a large grid of numerical simulations to reconstruct, and we compare our results throughout with the current literature on stellar and cosmic entropy rather than with ad hoc benchmarks.<\/jats:p>","DOI":"10.3390\/e28090991","type":"journal-article","created":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T17:25:56Z","timestamp":1788542756000},"page":"991","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Entropy Production During Star Formation: An Analytic Thermodynamic Framework from the Main Sequence to Compact Remnants"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6479-2236","authenticated-orcid":false,"given":"Javier","family":"Mart\u00edn-Torres","sequence":"first","affiliation":[{"name":"Earth-Life Science Institute (ELSI), Institute of Science Tokyo, Meguro-ku, Tokyo 152-8550, Japan"},{"name":"Instituto Andaluz de Ciencias de la Tierra (CSIC-UGR), 18100 Armilla, Spain"},{"name":"School of Engineering, Physics and Mathematics, Northumbria University, Newcastle-upon-Tyne NE1 8ST, UK"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"resources","vocabulary":"credit"},{"role":"software","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4492-9650","authenticated-orcid":false,"given":"Mar\u00eda-Paz","family":"Zorzano","sequence":"additional","affiliation":[{"name":"Centro de Astrobiolog\u00eda (CSIC-INTA), Torrej\u00f3n de Ardoz, 28850 Madrid, Spain"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"writing-review-editing","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1146\/annurev.aa.25.090187.000323","article-title":"Star formation in molecular clouds: Observation and theory","volume":"25","author":"Shu","year":"1987","journal-title":"Annu. 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