{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:19:31Z","timestamp":1760242771594,"version":"build-2065373602"},"reference-count":67,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2016,5,19]],"date-time":"2016-05-19T00:00:00Z","timestamp":1463616000000},"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>Rotation of galaxies is examined by the general principle of least action. This law of nature describes a system in its surroundings, here specifically a galaxy in the surrounding Universe. According to this holistic theory the gravitational potential due to all matter in the expanding Universe relates to the universal curvature which, in turn, manifests itself as the universal acceleration. Then the orbital velocities from the central bulge to distant perimeters are understood to balance both the galactic and universal acceleration. Since the galactic acceleration decreases with distance from the galaxy\u2019s center to its luminous edge, the orbital velocities of ever more distant stars and gas clouds tend toward a value that tallies the universal acceleration. This tiny term has been acknowledged earlier by including it as a parameter in the modified gravitational law, but here the tiny acceleration is understood to result from the gravitational potential that spans across the expanding Universe. This resolution of the galaxy rotation problem is compared with observations and contrasted with models of dark matter. Also, other astronomical observations that have been interpreted as evidence for dark matter are discussed in light of the least-action principle.<\/jats:p>","DOI":"10.3390\/e18050191","type":"journal-article","created":{"date-parts":[[2016,5,19]],"date-time":"2016-05-19T20:43:57Z","timestamp":1463690637000},"page":"191","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Rotation of Galaxies within Gravity of the Universe"],"prefix":"10.3390","volume":"18","author":[{"given":"Arto","family":"Annila","sequence":"first","affiliation":[{"name":"Department of Physics, University of Helsinki, Helsinki FI-00014, Finland"}]}],"member":"1968","published-online":{"date-parts":[[2016,5,19]]},"reference":[{"key":"ref_1","unstructured":"De Maupertuis, P.-L.M. (1744). Accord de diff\u00e9rentes lois de la nature qui avaient jusqu\u2019ici paru incompatibles. M\u00e9m. l'Acad. R. Sci. Paris, 417\u2013426. (In French)."},{"key":"ref_2","unstructured":"De Maupertuis, P.-L.M. (1746). Les loix du mouvement et du repos d\u00e9duites d\u2019un principe metaphysique. Hist. l'Acad. R. Sci. B.-Lett. Berlin, 267\u2013294. (In French)."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.3390\/e12112333","article-title":"All in action","volume":"12","author":"Annila","year":"2010","journal-title":"Entropy"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2944","DOI":"10.1111\/j.1365-2966.2011.19242.x","article-title":"Least-time paths of light","volume":"416","author":"Annila","year":"2011","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1111\/j.1365-2966.2011.19364.x","article-title":"Least-action perihelion precession","volume":"417","author":"Koskela","year":"2011","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1973","DOI":"10.1111\/j.1365-2966.2012.21022.x","article-title":"Probing Mach\u2019s principle","volume":"423","author":"Annila","year":"2012","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"135025","DOI":"10.1155\/2015\/135025","article-title":"Cosmic rays report from the structure of space","volume":"2015","author":"Annila","year":"2015","journal-title":"Adv. Astron."},{"key":"ref_8","first-page":"411","article-title":"Aperture Synthesis Study of Neutral Hydrogen in NGC 2403 and NGC 4236. II. Discussion","volume":"24","author":"Shostak","year":"1973","journal-title":"Astron. Astrophys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1086\/153889","article-title":"The rotation curve and geometry of M31 at large galactocentric distances","volume":"201","author":"Roberts","year":"1975","journal-title":"Astrophys. J."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"L107","DOI":"10.1086\/182804","article-title":"Extended rotation curves of high-luminosity spiral galaxies. IV\u2013Systematic dynamical properties, SA through SC","volume":"225","author":"Rubin","year":"1978","journal-title":"Astrophys. J."},{"key":"ref_11","first-page":"661","article-title":"A new method of determining distances to galaxies","volume":"54","author":"Tully","year":"1977","journal-title":"Astron. Astrophys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"L99","DOI":"10.1086\/312628","article-title":"The Baryonic Tully\u2013Fisher Relation","volume":"533","author":"McGaugh","year":"2000","journal-title":"Astrophys. J."},{"key":"ref_13","unstructured":"Einstein, A. (1979). Autobiographical Notes, Open Court Publishing."},{"key":"ref_14","unstructured":"Koschmieder, E.L. (1993). B\u00e9nard Cells and Taylor Vortices, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Choudhuri, A.R. (1998). The Physics of Fluids and Plasmas: An Introduction for Astrophysicists, Cambridge University Press.","DOI":"10.1017\/CBO9781139171069"},{"key":"ref_16","unstructured":"Hoffmann, A.C., and Stein, L.E. (2007). Gas Cyclones and Swirl Tubes: Principles, Design, and Operation, Springer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1038\/156463a0","article-title":"Lognormal distributions","volume":"156","author":"Gaddum","year":"1945","journal-title":"Nature"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1641\/0006-3568(2001)051[0341:LNDATS]2.0.CO;2","article-title":"Log-normal distributions across the sciences: Keys and clues","volume":"51","author":"Limpert","year":"2001","journal-title":"Bioscience"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Baryshev, Y., and Teerikorpi, P. (2002). Discovery of Cosmic Fractals, World Scientific.","DOI":"10.1142\/9789812388780"},{"key":"ref_20","first-page":"41","article-title":"Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy","volume":"6","year":"1963","journal-title":"Bol. Asoc. Argent. Astron."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert","year":"2002","journal-title":"Rev. Modern Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1016\/j.plrev.2010.10.001","article-title":"Natural patterns of energy dispersal","volume":"7","author":"Annila","year":"2010","journal-title":"Phys. Life Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1126\/science.84.2188.506","article-title":"Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field","volume":"84","author":"Einstein","year":"1936","journal-title":"Science"},{"key":"ref_24","unstructured":"Berry, M. (2001). Principles of Cosmology and Gravitation, Cambridge University Press."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1088\/0067-0049\/180\/2\/330","article-title":"Five-Year Wilkinson Microwave Anisotropy Probe Observations: Cosmological Interpretation","volume":"180","author":"Komatsu","year":"2009","journal-title":"Astrophys. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1086\/161130","article-title":"A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis","volume":"270","author":"Milgrom","year":"1983","journal-title":"Astrophys. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1086\/161131","article-title":"A modification of the Newtonian dynamics\u2014Implications for galaxies","volume":"270","author":"Milgrom","year":"1983","journal-title":"Astrophys. J."},{"key":"ref_28","unstructured":"Milgrom, M. (2008). The MOND Paradigm."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Milgrom, M. (2011). MD or DM? Modified dynamics at low accelerations vs. dark matter. Proc. Sci.","DOI":"10.22323\/1.109.0033"},{"key":"ref_30","unstructured":"Liddle, A.R. (2007). An Introduction to Modern Cosmology, Wiley."},{"key":"ref_31","unstructured":"Feynman, R.P., Morinigo, F.B., Wagner, W.G., and Hatfield, B. (1995). Feynman Lectures on Gravitation, Addison-Wesley."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"67","DOI":"10.5923\/j.ijtmp.20120204.03","article-title":"The meaning of mass","volume":"2","author":"Annila","year":"2012","journal-title":"Int. J. Theor. Math. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"208","DOI":"10.4006\/0836-1398-28.2.208","article-title":"The substance of gravity","volume":"28","author":"Annila","year":"2015","journal-title":"Phys. Essays"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"061102","DOI":"10.1103\/PhysRevLett.116.061102","article-title":"Observation of Gravitational Waves from a Binary Black Hole Merger","volume":"116","author":"Abbott","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1086\/164387","article-title":"The redshift-distance relation. IX\u2013Perturbation of the very nearby velocity field by the mass of the Local Group","volume":"307","author":"Sandage","year":"1986","journal-title":"Astrophys. J."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s001590050019","article-title":"The local group of galaxies","volume":"9","year":"1999","journal-title":"Astron. Astrophys. Rev."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1051\/0004-6361:20078894","article-title":"Dark energy in the environments of the Local Group, the M 81 group, and the CenA group: The normalized Hubble diagram","volume":"483","author":"Teerikorpi","year":"2008","journal-title":"Astron. Astrophys."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bennett, C.L., Larson, D., Weiland, J.L., Jarosik, N., Hinshaw, G., Odegard, N., Smith, K.M., Hill, R.S., Gold, B., and Halpern, M. (2013). Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results. Astrophys. J., 208.","DOI":"10.1088\/0067-0049\/208\/2\/20"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1093\/mnras\/249.3.523","article-title":"Extended rotation curves of spiral galaxies\u2013Dark haloes and modified dynamics","volume":"249","author":"Begeman","year":"1991","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2446","DOI":"10.1111\/j.1365-2966.2011.18564.x","article-title":"Mass models of the Milky Way","volume":"414","author":"McMillan","year":"2011","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Tamm, A., Tempel, E., Tenjes, P., Tihhonova, O., and Tuvikene, T. (2012). Stellar mass map and dark matter distribution in M31. Astron. Astrophys., 546.","DOI":"10.1051\/0004-6361\/201220065"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1111\/j.1365-2966.2007.11964.x","article-title":"The RAVE survey: Constraining the local Galactic escape speed","volume":"379","author":"Smith","year":"2007","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1086\/313274","article-title":"A Compilation of Redshifts and Velocity Dispersions for ACO Clusters","volume":"125","author":"Struble","year":"1999","journal-title":"Astrophys. J."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1111\/j.1365-2966.2005.09590.x","article-title":"New constraints on modified Newtonian dynamics from galaxy clusters","volume":"364","author":"Pointecouteau","year":"2005","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1111\/j.1365-2966.2007.11696.x","article-title":"The Universal Rotation Curve of Spiral Galaxies","volume":"378","author":"Salucci","year":"2007","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Ibata, R., Nipoti, C., Sollima, A., Bellazzini, M., Chapman, S., and Dalessandro, E. (2012). Do globular clusters possess Dark Matter halos? A case study in NGC 2419. Mon. Not. R. Astron. Sci.","DOI":"10.1093\/mnras\/sts302"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"838","DOI":"10.1086\/338222","article-title":"Are Dwarf Spheroidal Galaxies Dark Matter Dominated or Remnants of Disrupted Larger Satellite Galaxies? A Possible Test","volume":"566","author":"Klessen","year":"2002","journal-title":"Astrophys. J."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1086\/521816","article-title":"The Kinematics of the Ultra-faint Milky Way Satellites: Solving the Missing Satellite Problem","volume":"670","author":"Simon","year":"2007","journal-title":"Astrophys. J."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"L21","DOI":"10.1088\/2041-8205\/804\/1\/L21","article-title":"Small Scatter and Nearly Isothermal Mass Profiles to Four Half-light Radii from Two-dimensional Stellar Dynamics of Early-type Galaxies","volume":"804","author":"Cappellari","year":"2015","journal-title":"Astrophys. J. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1086\/154215","article-title":"Velocity dispersions and mass-to-light ratios for elliptical galaxies","volume":"204","author":"Faber","year":"1976","journal-title":"Astrophys. J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"221102","DOI":"10.1103\/PhysRevLett.112.221102","article-title":"Clear and Measurable Signature of Modified Gravity in the Galaxy Velocity Field","volume":"112","author":"Hellwing","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1098\/rspa.2008.0494","article-title":"In the light of time","volume":"465","author":"Tuisku","year":"2009","journal-title":"Proc. R. Soc. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1016\/j.physa.2015.10.105","article-title":"Natural thermodynamics","volume":"444","author":"Annila","year":"2016","journal-title":"Phys. A"},{"key":"ref_54","first-page":"21","article-title":"Binary and Millisecond Pulsars","volume":"11","author":"Lorimer","year":"2008","journal-title":"Living Rev. Relat."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1086\/432968","article-title":"Baryonic Tully\u2013Fisher Relation","volume":"632","author":"McGaugh","year":"2005","journal-title":"Astrophys. J."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1080\/00107510701244055","article-title":"The modified Newtonian dynamics\u2014MOND and its implications for new physics","volume":"47","author":"Bekenstein","year":"2006","journal-title":"Contemp. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.mbs.2007.07.004","article-title":"Natural distribution","volume":"210","author":"Annila","year":"2007","journal-title":"Math. Biosci."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1093\/mnras\/265.4.1013","article-title":"On the Shape of the Light Profiles of Early Type Galaxies","volume":"265","author":"Caon","year":"1993","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_59","first-page":"99","article-title":"Stellar systems following the R exp 1\/m luminosity law","volume":"249","author":"Ciotti","year":"1991","journal-title":"Astron. Astrophys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"L11","DOI":"10.1093\/mnras\/268.1.L11","article-title":"A New Extragalactic Distance Indicator Based on the Surface Brightness Profiles of Dwarf Elliptical Galaxies","volume":"268","author":"Young","year":"1994","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_61","unstructured":"Mihalas, D., and Routly, P.M. (1968). Galactic Astronomy, Freeman."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1088\/0004-637X\/717\/2\/825","article-title":"The Fermi haze: A gamma-ray counterpart to the microwave haze","volume":"717","author":"Dobler","year":"2010","journal-title":"Astrophys. J."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1111\/j.1365-2966.2009.14938.x","article-title":"How flat can you get? A model comparison perspective on the curvature of the Universe","volume":"397","author":"Vardanyan","year":"2009","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_64","first-page":"281","article-title":"A gravitational and electromagnetic analogy, Part I","volume":"31","author":"Heaviside","year":"1893","journal-title":"Electrician"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1093\/mnras\/113.1.34","article-title":"On the origin of inertia","volume":"113","author":"Sciama","year":"1953","journal-title":"Mon. Not. R. Astron. Sci."},{"key":"ref_66","unstructured":"Assis, A.K.T. (2014). Relational Mechanics and Implementation of Mach\u2019s Principle with Weber\u2019s Gravitational Force, Aperion."},{"key":"ref_67","unstructured":"Mandl, F., and Shaw, G. (2002). Quantum Field Theory, John Wiley & Sons."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/5\/191\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:24:09Z","timestamp":1760210649000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/5\/191"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,19]]},"references-count":67,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2016,5]]}},"alternative-id":["e18050191"],"URL":"https:\/\/doi.org\/10.3390\/e18050191","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,5,19]]}}}