Dark Matter: A Result of nonadditive gravitational forces
Jesús Arturo Robles-Gutiérrez, Ernesto Lacomba Zamora, Jesús Martiniano Arturo Robles-Domínguez
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Universidad Autónoma Metropolitana México DF, México
Cinna Lomnitz
Instituto de Geofísica, Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México DF, México
María Eugenia Robles-Gutiérrez
Universidad Panamericana México DF, México
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Three <span class="elsevierStyleItalic">L/d</span> ratios and five ρ<span class="elsevierStyleInf">2</span> /p, ratios are considered. The unity departure time is <span class="elsevierStyleItalic">ρ</span><span class="elsevierStyleInf">1</span> indicated</p>" ] ] ] "autores" => array:1 [ 0 => array:2 [ "autoresLista" => "Carlos Flores, José M. Romo, Mario Vega" "autores" => array:3 [ 0 => array:2 [ "nombre" => "Carlos" "apellidos" => "Flores" ] 1 => array:2 [ "nombre" => "José M." "apellidos" => "Romo" ] 2 => array:2 [ "nombre" => "Mario" "apellidos" => "Vega" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716913714703?idApp=UINPBA00004N" "url" => "/00167169/0000005200000002/v2_201505081345/S0016716913714703/v2_201505081345/en/main.assets" ] "itemAnterior" => array:18 [ "pii" => "S0016716913714685" "issn" => "00167169" "doi" => "10.1016/S0016-7169(13)71468-5" "estado" => "S300" "fechaPublicacion" => "2013-04-01" "aid" => "71468" "copyright" => "Universidad Nacional Autónoma de México" "documento" => "article" "licencia" => "http://creativecommons.org/licenses/by-nc-nd/4.0/" "subdocumento" => "fla" "cita" => "Geofisica Internacional. 2013;52:135-52" "abierto" => array:3 [ "ES" => true "ES2" => true "LATM" => true ] "gratuito" => true "lecturas" => array:2 [ "total" => 978 "formatos" => array:3 [ "EPUB" => 25 "HTML" => 708 "PDF" => 245 ] ] "en" => array:11 [ "idiomaDefecto" => true "titulo" => "Effect of galvanic distortions on the series and parallel magnetotelluric impedances and comparison with other responses" "tienePdf" => "en" "tieneTextoCompleto" => "en" "tieneResumen" => array:2 [ 0 => "es" 1 => "en" ] "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "135" "paginaFinal" => "152" ] ] "contieneResumen" => array:2 [ "es" => true "en" => true ] "contieneTextoCompleto" => array:1 [ "en" => true ] "contienePdf" => array:1 [ "en" => true ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 1080 "Ancho" => 1282 "Tamanyo" => 102650 ] ] "descripcion" => array:1 [ "en" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall"><span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span> correspond to the <span class="elsevierStyleSmallCaps">te</span> and <span class="elsevierStyleSmallCaps">tm</span> modes of sounding # 11 of the synthetic data set <span class="elsevierStyleSmallCaps">coprod2s1</span> made available in <span class="elsevierStyleSmallCaps">mtn</span>et.dias.ie by <a class="elsevierStyleCrossRef" href="#bib0080">Varentsov (1998)</a>. When <span class="elsevierStyleItalic">θ</span> ≠ 0 in <a class="elsevierStyleCrossRef" href="#eq0025">equation (5)</a> the elements of the resulting impedance all are mixtures of the phases of <span class="elsevierStyleItalic">A</span> and <span class="elsevierStyleItalic">B</span>. Calculations were made using <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>= 1.97, <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">2</span> = −−0.77, c<span class="elsevierStyleInf">3</span> = −0.35 and <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">4</span> = 0.64. The graphs correspond to the phase of the impedance.</p>" ] ] ] "autores" => array:1 [ 0 => array:2 [ "autoresLista" => "Enrique Gómez-Treviño, Francisco Javier Esparza Hernández, José Manuel Romo Jones" "autores" => array:3 [ 0 => array:2 [ "nombre" => "Enrique" "apellidos" => "Gómez-Treviño" ] 1 => array:2 [ "nombre" => "Francisco Javier" "apellidos" => "Esparza Hernández" ] 2 => array:2 [ "nombre" => "José Manuel" "apellidos" => "Romo Jones" ] ] ] ] ] "idiomaDefecto" => "en" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716913714685?idApp=UINPBA00004N" "url" => "/00167169/0000005200000002/v2_201505081345/S0016716913714685/v2_201505081345/en/main.assets" ] "en" => array:18 [ "idiomaDefecto" => true "titulo" => "Dark Matter: A Result of nonadditive gravitational forces" "tieneTextoCompleto" => true "paginas" => array:1 [ 0 => array:2 [ "paginaInicial" => "153" "paginaFinal" => "157" ] ] "autores" => array:3 [ 0 => array:4 [ "autoresLista" => "Jesús Arturo Robles-Gutiérrez, Ernesto Lacomba Zamora, Jesús Martiniano Arturo Robles-Domínguez" "autores" => array:3 [ 0 => array:2 [ "nombre" => "Jesús Arturo" "apellidos" => "Robles-Gutiérrez" ] 1 => array:2 [ "nombre" => "Ernesto" "apellidos" => "Lacomba Zamora" ] 2 => array:4 [ "nombre" => "Jesús Martiniano Arturo" "apellidos" => "Robles-Domínguez" "email" => array:1 [ 0 => "rodj@xanum.uam.mx" ] "referencia" => array:1 [ 0 => array:2 [ "etiqueta" => "*" "identificador" => "cor0005" ] ] ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Universidad Autónoma Metropolitana México DF, México" "identificador" => "aff0005" ] ] "correspondencia" => array:1 [ 0 => array:3 [ "identificador" => "cor0005" "etiqueta" => "*" "correspondencia" => "Corresponding author" ] ] ] 1 => array:3 [ "autoresLista" => "Cinna Lomnitz" "autores" => array:1 [ 0 => array:2 [ "nombre" => "Cinna" "apellidos" => "Lomnitz" ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Instituto de Geofísica, Universidad Nacional Autónoma de México Ciudad Universitaria Delegación Coyoacán, 04510 México DF, México" "identificador" => "aff0010" ] ] ] 2 => array:3 [ "autoresLista" => "María Eugenia Robles-Gutiérrez" "autores" => array:1 [ 0 => array:2 [ "nombre" => "María Eugenia" "apellidos" => "Robles-Gutiérrez" ] ] "afiliaciones" => array:1 [ 0 => array:2 [ "entidad" => "Universidad Panamericana México DF, México" "identificador" => "aff0015" ] ] ] ] "resumenGrafico" => array:2 [ "original" => 0 "multimedia" => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 730 "Ancho" => 818 "Tamanyo" => 54388 ] ] "descripcion" => array:1 [ "en" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall">Observations of NGC 3198 (From <a class="elsevierStyleCrossRef" href="#bib0005">Begeman, 1987</a>).</p>" ] ] ] "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0025"><a name="p154"></a>Introduction</span><p id="par0005" class="elsevierStylePara elsevierViewall">Zwicky (1933) interpreted some observations of the Coma Cumulus and concluded that its gravitational mass is several times the luminous mass. He called the difference the <span class="elsevierStyleItalic">dark mass</span>. This work is regarded as the starting point of the Dark Matter problem. <a class="elsevierStyleCrossRef" href="#bib0030">Sciama (1993)</a> discussed several theories intended to explain this interesting phenomenon, including the existence of new particles which do not emit electromagnetic radiation but contribute to the gravitational mass, but without experimental confirmation. Other solutions involve a mathematical modification of Newton’s laws but insufficient physical reasons in support of such modification have been presented.</p><p id="par0010" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0020">Robles-Domínguez (2007)</a> and <a class="elsevierStyleCrossRef" href="#bib0025">Robles-Gutiérrez (2010)</a> proposed new intermolecular forces in field theory that arise from interactions between three and more molecules. We showed that such interactions have not been previously considered, and that such nonadditive electromagnetic forces are needed in order to account for the experimental data on critical points in fluids, triple states, and liquid-solid phase transitions. We call these forces <span class="elsevierStyleItalic">nonadditive</span> because their mathematical form differs from that of binary, or additive, forces. Nonadditive forces are important within relatively short distances. In gases the mean intermolecular distances are large and nonadditive forces are not significant as compared with additive forces. In liquids and solids, however, the density is large and the mean intermolecular distance is very short. In this case the nonadditive forces may overcome the additive forces. We show that the existence of liquid and solid phases is experimental proof for the existence of nonadditive forces.</p><p id="par0015" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a> of this paper we provide a short derivation of the equation of state in fluids from statistical mechanics as provided in greater detail elsewhere (<a class="elsevierStyleCrossRef" href="#bib0020">Robles-Domínguez, 2007</a>; <a class="elsevierStyleCrossRef" href="#bib0025">Robles-Gutiérrez, 2010</a>). This derivation includes nonadditive forces. In Section 3 we apply the equivalent equation from <a class="elsevierStyleCrossRef" href="#sec0010">section 2</a> to the gravitational field using data from <a class="elsevierStyleCrossRef" href="#bib0005">Begeman (1987)</a> on galaxy NGC 3198, and we show how dark mass will arise.</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2</span><span class="elsevierStyleSectionTitle" id="sect0030">Nonadditive forces in fluids</span><p id="par0020" class="elsevierStylePara elsevierViewall">Molecules in a fluid interact mainly through the electromagnetic field: the gravitational, weak and strong nuclear fields are irrelevant. We consider a monocomponent fluid that contains <span class="elsevierStyleItalic">N</span> molecules at an absolute temperature <span class="elsevierStyleItalic">T</span>, inside a volume <span class="elsevierStyleItalic">V</span>, which is described in Statistical Mechanics by the canonical partition function Z:<elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><p id="par0025" class="elsevierStylePara elsevierViewall">where β=1kBT, <span class="elsevierStyleItalic">k<span class="elsevierStyleInf">B</span></span> is Boltzmann’s cons-tant, <span class="elsevierStyleItalic">E</span> is</p><p id="par0030" class="elsevierStylePara elsevierViewall">the Energy of the system, Γ=Γri_;,pi_;,i=1…N is the phase space of the system, <span class="elsevierStyleItalic">r<span class="elsevierStyleInf"><span class="elsevierStyleBold">i</span></span></span> and pi_; are the vector position and momentum of the <span class="elsevierStyleItalic">i</span>th particle, <span class="elsevierStyleItalic">N</span> is the total number of molecules, <span class="elsevierStyleItalic">V</span> is the volume of the system, and <span class="elsevierStyleItalic">T</span> is the absolute temperature. We may write<elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0035" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">K</span> is the total kinetic energy of the system and Φ is the total potential energy; and<elsevierMultimedia ident="eq0015"></elsevierMultimedia></p><p id="par0040" class="elsevierStylePara elsevierViewall">as we are considering a single-component fluid.</p><p id="par0045" class="elsevierStylePara elsevierViewall">Integration of <a class="elsevierStyleCrossRef" href="#eq0005">(1)</a> over the momenta is well known (<a class="elsevierStyleCrossRef" href="#bib0015">Reichl, 1998</a>) and the result is<elsevierMultimedia ident="eq0020"></elsevierMultimedia></p><p id="par0050" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">λ</span> = <span class="elsevierStyleItalic">h</span>(2π<span class="elsevierStyleItalic">mk<span class="elsevierStyleInf">B</span>T</span>)<span class="elsevierStyleSup">−½</span> and <span class="elsevierStyleItalic">h</span> is Planck’s Constant.</p><span id="sec0015" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2.1</span><span class="elsevierStyleSectionTitle" id="sect0035">van der Waals equation of state</span><p id="par0055" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0020">Equation (4)</a> considers only the positions of all <span class="elsevierStyleItalic">N</span> molecules in the system. Ornstein and van Kampen (in <a class="elsevierStyleCrossRef" href="#bib0015">Reichl, 1998</a>), consider than the molecules interact with the next binary additive potential energy<elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0060" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">hc</span> stands for “hard core”. The subscript <span class="elsevierStyleItalic">2</span> denotes binary additive interaction; <span class="elsevierStyleItalic">r<span class="elsevierStyleInf">i</span></span> and <span class="elsevierStyleItalic">r<span class="elsevierStyleInf">j</span></span> are position of molecules <span class="elsevierStyleItalic">i</span> and <span class="elsevierStyleItalic">j</span> respectively, and it is assumed that the molecules have a hard core of radius <span class="elsevierStyleItalic">a</span> and a smooth attractive binary-additive interaction Φ<span class="elsevierStyleInf">2</span> with a very long range. Ornstein and van Kampen assume that the density is high enough and the range of the attractive interaction is wide enough so that many molecules will interact simultaneously. Volume <span class="elsevierStyleItalic">V</span> may be divided in cells of volume <span class="elsevierStyleBold">∆</span> large enough to contain many molecules but small enough so that the attraction<a name="p155"></a> between molecules within a given cell is constant regardless of separation in the cell. If the number of molecules in cell α is <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf">α</span> we may write, from <a class="elsevierStyleCrossRef" href="#eq0020">(4)</a> and <a class="elsevierStyleCrossRef" href="#eq0025">(5)</a>:<elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0065" class="elsevierStylePara elsevierViewall">where<elsevierMultimedia ident="eq0035"></elsevierMultimedia></p><p id="par0070" class="elsevierStylePara elsevierViewall">and δ is the volume of the hard core of one molecule.</p><p id="par0075" class="elsevierStylePara elsevierViewall">The most probable distribution of molecules in <span class="elsevierStyleItalic">V</span> at thermodynamic equilibrium is the uniform distribution<elsevierMultimedia ident="eq0040"></elsevierMultimedia></p><p id="par0080" class="elsevierStylePara elsevierViewall">for all α, and defining<elsevierMultimedia ident="eq0045"></elsevierMultimedia></p><p id="par0085" class="elsevierStylePara elsevierViewall">for all α.</p><p id="par0090" class="elsevierStylePara elsevierViewall">We use mean field theory and the definition of Helmholtz energy <span class="elsevierStyleItalic">A</span>:<elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0095" class="elsevierStylePara elsevierViewall">The thermodynamics of the systems is deduced from Helmholtz energy <span class="elsevierStyleItalic">A</span>, ie,: fluids. Pressure <span class="elsevierStyleItalic">p</span> is defined as<elsevierMultimedia ident="eq0055"></elsevierMultimedia></p><p id="par0100" class="elsevierStylePara elsevierViewall">Using <a class="elsevierStyleCrossRef" href="#eq0025">equations (5)</a> through <a class="elsevierStyleCrossRef" href="#eq0055">(11)</a> we obtain van der Waals equation of state<elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0105" class="elsevierStylePara elsevierViewall">Let us express this equation in terms of fluid density:<elsevierMultimedia ident="eq0065"></elsevierMultimedia></p><p id="par0110" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">m</span> is the mass of one molecule; and we find<elsevierMultimedia ident="eq0070"></elsevierMultimedia></p><p id="par0115" class="elsevierStylePara elsevierViewall">where<elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0120" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#eq0070">equation (14)</a> the first term of the right-hand side contains the hard core contribution and the second contains all binary-additive interactions. This equation can approximately represent the pressure of the gaseous phase but cannot describe the pressures of the liquid and solid phases in the fluid (<a class="elsevierStyleCrossRef" href="#bib0015">Reichl, 1998</a>).</p><p id="par0125" class="elsevierStylePara elsevierViewall">In the van der Waals equation of state <a class="elsevierStyleCrossRef" href="#eq0070">(14)</a>, we note three important properties:<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005"><span class="elsevierStyleLabel">1.</span><p id="par0130" class="elsevierStylePara elsevierViewall">The second term on the right-hand side contains <span class="elsevierStyleItalic">all</span> binary-additive intermolecular interactions;</p></li><li class="elsevierStyleListItem" id="lsti0010"><span class="elsevierStyleLabel">2.</span><p id="par0135" class="elsevierStylePara elsevierViewall">The two terms on the right-hand side are linearly independent in a functional-analysis sense; and</p></li><li class="elsevierStyleListItem" id="lsti0015"><span class="elsevierStyleLabel">3.</span><p id="par0140" class="elsevierStylePara elsevierViewall">This equation cannot reproduce the experimental data in fluids.</p></li></ul></p><p id="par0145" class="elsevierStylePara elsevierViewall">Fourier theory (<a class="elsevierStyleCrossRef" href="#bib0010">Kolmogorov and Fomin, 1961</a>) may be used to describe a function as a Fourier series. Properties 2 and 3 above mean that the van der Waals equation contains only two terms of the infinite series which we need to reproduce the experimental data of fluids. Property 1 means that the last right-hand terms must be expressed in terms of new forces. Thus in terms of linearly independent nonadditive forces, the third term will contain all third-order nonadditive interactions, the fourth term all fourth-order nonadditive interactions and so on. The total pressure may be written as follows:<elsevierMultimedia ident="eq0080"></elsevierMultimedia></p><p id="par0150" class="elsevierStylePara elsevierViewall">where <span class="elsevierStyleItalic">p<span class="elsevierStyleInf">hc</span></span> contains the hard core interactions, <span class="elsevierStyleItalic">p<span class="elsevierStyleInf">2</span></span></p><p id="par0155" class="elsevierStylePara elsevierViewall">the total scalar additive binary potential energy, <span class="elsevierStyleItalic">p<span class="elsevierStyleInf">3</span></span> the total scalar third-order or triadic nonadditive potential energy, <span class="elsevierStyleItalic">p4</span> the total scalar fourth-order<a name="p156"></a> (or tetradic) nonadditive potential energy, and so on. Thus<elsevierMultimedia ident="eq0085"></elsevierMultimedia><elsevierMultimedia ident="eq0090"></elsevierMultimedia><elsevierMultimedia ident="eq0095"></elsevierMultimedia><elsevierMultimedia ident="eq0100"></elsevierMultimedia></p><p id="par0160" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#eq0085">Equations (17)</a>, <a class="elsevierStyleCrossRef" href="#eq0090">(18)</a>, etc. constitute an infinite Fourier series of independent linear functions.</p><p id="par0165" class="elsevierStylePara elsevierViewall">The hard core cannot allow the fluid volume to vanish and the density cannot be infinite: thus the Fourier series can converge.</p></span></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0040">Nonadditive forces in the gravitational field</span><p id="par0170" class="elsevierStylePara elsevierViewall">In <a class="elsevierStyleCrossRef" href="#bib0025">Robles-Gutiérrez <span class="elsevierStyleItalic">et al</span>. (2010)</a>, we postulated the existence of nonadditive gravitational forces in the gravitational field. Before we extend the theory to galaxies let us describe the astronomical observations of galaxy NGC 3198 (<a class="elsevierStyleCrossRef" href="#bib0005">Begeman, 1987</a>). <a class="elsevierStyleCrossRef" href="#fig0005">Figure 1</a> shows Begeman’s graph of rotational velocity vs. radius in minutes of arc between the vector from Earth to the object and the vector from Earth to the center of NGC 3198. The distance from Earth to NGC 3198 is <span class="elsevierStyleItalic">9.2 Mpc</span>. The maximum distances of masses from the center of the galaxy should provide information on the total mass of the galaxy.</p><elsevierMultimedia ident="fig0005"></elsevierMultimedia><p id="par0175" class="elsevierStylePara elsevierViewall">The galaxy may be likened to a gas inside a jar. The jar exerts a pressure on the gas. In a state of equilibrium the pressure should be constant everywhere in the gas. In the case of a galaxy the peripherical stars plus gas are confined to a galactic volume by a pressure called galactic pressure, which equals gravity per unit area. It is equal to the centripetal force per unit area, and it is not identical at all points of the galaxy.</p><p id="par0180" class="elsevierStylePara elsevierViewall">The centripetal force on an object <span class="elsevierStyleItalic">l</span> is equal to the gravitational force on this object, which is the sum of all gravitational interactions, additive and nonadditive:<elsevierMultimedia ident="eq0105"></elsevierMultimedia></p><p id="par0185" class="elsevierStylePara elsevierViewall">where subscript <span class="elsevierStyleItalic">l</span> refers to the peripheral stars or gas clouds observed by Begeman in NGC 3198 galaxy and subscript <span class="elsevierStyleItalic">j</span> refers to the forces. Let <span class="elsevierStyleItalic">j</span> = 2 refer to binary additive forces, <span class="elsevierStyleItalic">j</span> = 3 to ternary nonadditive forces and so on. We omit the hardcore term because <span class="elsevierStyleItalic">m</span>, is far away from the galaxy.</p><p id="par0190" class="elsevierStylePara elsevierViewall">If we divide <a class="elsevierStyleCrossRef" href="#eq0105">(21)</a> by the unit of area we obtain the equation of the pressures, as in a fluid:<elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0195" class="elsevierStylePara elsevierViewall">But a similar expression should obtain for the centripetal and gravitational accelerations, after dividing <a class="elsevierStyleCrossRef" href="#eq0105">equation (21)</a> by <span class="elsevierStyleItalic">m</span> which is necessary because the mass <span class="elsevierStyleItalic">m</span> is unknown. Thus the centripetal acceleration equals the gravitational acceleration <span class="elsevierStyleItalic">g<span class="elsevierStyleInf">l</span></span> generated by the total galaxy mass including the dark mass, over <span class="elsevierStyleItalic">m</span>, which is the sum of all binary-additive contributions <span class="elsevierStyleItalic">g<span class="elsevierStyleInf">12</span></span> plus all third-order nonadditive accelerations g<span class="elsevierStyleInf">13</span>, and so on, over <span class="elsevierStyleItalic">m<span class="elsevierStyleInf">l</span></span>:<elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0200" class="elsevierStylePara elsevierViewall">Here the first term of the summand is Newton’s acceleration of gravity:<elsevierMultimedia ident="eq0120"></elsevierMultimedia></p><p id="par0205" class="elsevierStylePara elsevierViewall">Written out as<elsevierMultimedia ident="eq0125"></elsevierMultimedia></p><p id="par0210" class="elsevierStylePara elsevierViewall">By analogy with the case of a fluid we may write <span class="elsevierStyleItalic">g<span class="elsevierStyleInf">l</span></span>, in terms of an infinite density series:<a name="p157"></a><elsevierMultimedia ident="eq0130"></elsevierMultimedia></p><p id="par0215" class="elsevierStylePara elsevierViewall">where the density ρ<span class="elsevierStyleItalic"><span class="elsevierStyleInf">l</span></span> equals the luminous mass <span class="elsevierStyleItalic">M<span class="elsevierStyleInf">llum</span></span> within a sphere of radius <span class="elsevierStyleItalic">r<span class="elsevierStyleInf">l</span></span> from the center of the galaxy to <span class="elsevierStyleItalic">m<span class="elsevierStyleInf">l</span></span> over the volume of the sphere. <a class="elsevierStyleCrossRef" href="#eq0130">Equation (26)</a> may be written<elsevierMultimedia ident="eq0135"></elsevierMultimedia></p><p id="par0220" class="elsevierStylePara elsevierViewall">or<elsevierMultimedia ident="eq0140"></elsevierMultimedia></p><p id="par0225" class="elsevierStylePara elsevierViewall">If<elsevierMultimedia ident="eq0145"></elsevierMultimedia></p><p id="par0230" class="elsevierStylePara elsevierViewall">this term will contribute to dark matter.</p><p id="par0235" class="elsevierStylePara elsevierViewall">The series of <a class="elsevierStyleCrossRef" href="#eq0140">equation (28)</a> converges because the density is finite. We may consider only the first and second terms of the series and neglect the small higher terms. Thus<elsevierMultimedia ident="eq0150"></elsevierMultimedia></p><p id="par0240" class="elsevierStylePara elsevierViewall">from which<elsevierMultimedia ident="eq0155"></elsevierMultimedia></p><p id="par0245" class="elsevierStylePara elsevierViewall">and the terms inside the square brackets are the real mass.</p><p id="par0250" class="elsevierStylePara elsevierViewall"><a class="elsevierStyleCrossRef" href="#bib0005">Begeman (1987)</a> obtained the following values of <span class="elsevierStyleItalic">r<span class="elsevierStyleInf">l</span></span> =30 <span class="elsevierStyleItalic">kcp</span>, <span class="elsevierStyleItalic">V<span class="elsevierStyleInf">c</span></span>=150 km/s for this galaxy:<elsevierMultimedia ident="eq0160"></elsevierMultimedia></p><p id="par0255" class="elsevierStylePara elsevierViewall">From <a class="elsevierStyleCrossRef" href="#eq0150">Equations (30)</a>, <a class="elsevierStyleCrossRef" href="#eq0155">(31)</a> and <a class="elsevierStyleCrossRef" href="#eq0160">(32)</a> we find<elsevierMultimedia ident="eq0165"></elsevierMultimedia></p><p id="par0260" class="elsevierStylePara elsevierViewall">and by introducing this value in <a class="elsevierStyleCrossRef" href="#eq0155">(31)</a> we obtain, for <span class="elsevierStyleItalic">l</span>=30 <span class="elsevierStyleItalic">kcp</span>,<elsevierMultimedia ident="eq0170"></elsevierMultimedia></p><p id="par0265" class="elsevierStylePara elsevierViewall">As this value is positive, we fnd that<elsevierMultimedia ident="eq0175"></elsevierMultimedia></p><p id="par0270" class="elsevierStylePara elsevierViewall">This result proves that the dark mass must be due to the presence of nonadditive forces.</p></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleSectionTitle" id="sect0045">Conclusions</span><p id="par0275" class="elsevierStylePara elsevierViewall">Whenever nonadditive interactions, that are multi-body terms, are taken into account, Newton’s law of universal gravitation is sufficient to explain the astronomical observations of a “dark mass”. The example of Galaxy NGC 3198 (where substantial amounts of dark matter had been detected) shows that nonadditive terms in <a class="elsevierStyleCrossRef" href="#eq0125">Equation (25)</a>, a generalization of Newton’s law of gravitation, can provide a satisfying explanation of the difference between luminous and gravitational matter.</p></span></span>" "textoCompletoSecciones" => array:1 [ "secciones" => array:10 [ 0 => array:3 [ "identificador" => "xres497961" "titulo" => "Resumen" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec519509" "titulo" => "Palabras clave" ] 2 => array:3 [ "identificador" => "xres497962" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0010" ] ] ] 3 => array:2 [ "identificador" => "xpalclavsec519508" "titulo" => "Key words" ] 4 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 5 => array:3 [ "identificador" => "sec0010" "titulo" => "Nonadditive forces in fluids" "secciones" => array:1 [ 0 => array:2 [ "identificador" => "sec0015" "titulo" => "van der Waals equation of state" ] ] ] 6 => array:2 [ "identificador" => "sec0020" "titulo" => "Nonadditive forces in the gravitational field" ] 7 => array:2 [ "identificador" => "sec0025" "titulo" => "Conclusions" ] 8 => array:2 [ "identificador" => "xack161001" "titulo" => "Acknowledgments" ] 9 => array:1 [ "titulo" => "Bibliography" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "fechaRecibido" => "2012-04-22" "fechaAceptado" => "2012-10-02" "PalabrasClave" => array:2 [ "es" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Palabras clave" "identificador" => "xpalclavsec519509" "palabras" => array:2 [ 0 => "materia oscura" 1 => "fuerzas no aditivas" ] ] ] "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Key words" "identificador" => "xpalclavsec519508" "palabras" => array:2 [ 0 => "Dark Matter" 1 => "nonadditive forces" ] ] ] ] "tieneResumen" => true "resumen" => array:2 [ "es" => array:2 [ "titulo" => "Resumen" "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0005" class="elsevierStyleSimplePara elsevierViewall">Basándonos en los datos experimentales en fluidos encontramos, en referencias <a class="elsevierStyleCrossRef" href="#bib0020">Robles-Domínguez et al. (2007)</a> y <a class="elsevierStyleCrossRef" href="#bib0025">Robles-Gutiérrez <span class="elsevierStyleItalic">et al</span>. (2010)</a>, que en el Campo Electromagnético existen realmente nuevas fuerzas no-aditivas entre 3 o más moléculas; postulamos que también existen nuevas fuerzas no-aditivas en el Campo Gravitacional y al agregarlas a la Ley de Gravitación Universal de Newton éstas dan lugar a la Masa Obscura.</p></span>" ] "en" => array:2 [ "titulo" => "Abstract" "resumen" => "<span id="abst0010" class="elsevierStyleSection elsevierViewall"><p id="spar0010" class="elsevierStyleSimplePara elsevierViewall">Experimental data in fluids suggest that nonadditive electromagnetic forces between 3 or more molecules account for the existence of critical points, triple states and phase transitions (<a class="elsevierStyleCrossRef" href="#bib0020">Robles-Domínguez et al., 2007</a>; <a class="elsevierStyleCrossRef" href="#bib0025">Robles-Gutiérrez <span class="elsevierStyleItalic">et al</span>., 2010</a>). Similar nonadditive forces between 3 or more molecules in the gravitational field incorporated into Newton’s universal gravitational law may also explain the existence of dark matter.</p></span>" ] ] "multimedia" => array:36 [ 0 => array:7 [ "identificador" => "fig0005" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 730 "Ancho" => 818 "Tamanyo" => 54388 ] ] "descripcion" => array:1 [ "en" => "<p id="spar0015" class="elsevierStyleSimplePara elsevierViewall">Observations of NGC 3198 (From <a class="elsevierStyleCrossRef" href="#bib0005">Begeman, 1987</a>).</p>" ] ] 1 => array:6 [ "identificador" => "eq0005" "etiqueta" => "(1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Z(T,V,N)=∫Γexp(−βE)dΓ," "Fichero" => "si1.jpeg" "Tamanyo" => 2088 "Alto" => 19 "Ancho" => 198 ] ] 2 => array:6 [ "identificador" => "eq0010" "etiqueta" => "(2)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "E=K+Φ" "Fichero" => "si5.jpeg" "Tamanyo" => 780 "Alto" => 12 "Ancho" => 79 ] ] 3 => array:6 [ "identificador" => "eq0015" "etiqueta" => "(3)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "dΓ=πi−1Ndri_;dpi_;" "Fichero" => "si6.jpeg" "Tamanyo" => 1227 "Alto" => 31 "Ancho" => 104 ] ] 4 => array:6 [ "identificador" => "eq0020" "etiqueta" => "(4)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Z(T,V,N)=(N!λ3N)−1∫...∫dr1_;...drN_;exp(−βΦ)" "Fichero" => "si7.jpeg" "Tamanyo" => 3472 "Alto" => 24 "Ancho" => 370 ] ] 5 => array:6 [ "identificador" => "eq0025" "etiqueta" => "(5)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Φri_;−rj_;=+∞=Φhc,ri_;−rj_;<aΦ2ri_;−rj_;<0,ri_;−rj_;>a" "Fichero" => "si8.jpeg" "Tamanyo" => 4917 "Alto" => 64 "Ancho" => 380 ] ] 6 => array:6 [ "identificador" => "eq0030" "etiqueta" => "(6)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Z(T,V,N)=(N!λ3N)−1∑{Nα}'N!ΠαNα![Παγ(Nα)]exp12β∑α,α'Φα,α'NαNα'≈1λT3N∑{Nα}'exp∑αNαlnΔ−NαδNα=Nα+12β∑α'Φα,α'NαNα'" "Fichero" => "si9.jpeg" "Tamanyo" => 10275 "Alto" => 58 "Ancho" => 811 ] ] 7 => array:6 [ "identificador" => "eq0035" "etiqueta" => "(7)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "γ(Nα)=(Δ−Nαδ)Nα" "Fichero" => "si10.jpeg" "Tamanyo" => 1441 "Alto" => 18 "Ancho" => 148 ] ] 8 => array:6 [ "identificador" => "eq0040" "etiqueta" => "(8)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Nα=NVΔ" "Fichero" => "si11.jpeg" "Tamanyo" => 787 "Alto" => 20 "Ancho" => 68 ] ] 9 => array:6 [ "identificador" => "eq0045" "etiqueta" => "(9)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "W0Δ=∑α'Φα,α'" "Fichero" => "si12.jpeg" "Tamanyo" => 1222 "Alto" => 36 "Ancho" => 97 ] ] 10 => array:6 [ "identificador" => "eq0050" "etiqueta" => "(10)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A(T,V,N)=−1βlnZ(T,V,N)." 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"Fichero" => "si38.jpeg" "Tamanyo" => 1009 "Alto" => 13 "Ancho" => 110 ] ] ] "bibliografia" => array:2 [ "titulo" => "Bibliography" "seccion" => array:1 [ 0 => array:2 [ "identificador" => "bibs0005" "bibliografiaReferencia" => array:6 [ 0 => array:3 [ "identificador" => "bib0005" "etiqueta" => "Begeman, 1987" "referencia" => array:1 [ 0 => array:2 [ "contribucion" => array:1 [ 0 => array:1 [ "autores" => array:1 [ 0 => array:2 [ "etal" => false "autores" => array:1 [ 0 => "Begeman K.H." ] ] ] ] ] "host" => array:1 [ 0 => array:1 [ "Libro" => array:3 [ "titulo" => "Rotation Curves of Spiral Galaxies" "fecha" => "1987" "editorial" => "Ph. D. 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Sylvia Robles Gutiérrez is gratefully acknowledged.</p>" "vista" => "all" ] ] ] "idiomaDefecto" => "en" "url" => "/00167169/0000005200000002/v2_201505081345/S0016716913714697/v2_201505081345/en/main.assets" "Apartado" => array:4 [ "identificador" => "36047" "tipo" => "SECCION" "es" => array:2 [ "titulo" => "Original paper" "idiomaDefecto" => true ] "idiomaDefecto" => "es" ] "PDF" => "https://static.elsevier.es/multimedia/00167169/0000005200000002/v2_201505081345/S0016716913714697/v2_201505081345/en/main.pdf?idApp=UINPBA00004N&text.app=https://www.elsevier.es/" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S0016716913714697?idApp=UINPBA00004N" ]
| Year/Month | Html | Total | |
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| 2020 March | 12 | 3 | 15 |
| 2020 February | 10 | 0 | 10 |
| 2020 January | 5 | 4 | 9 |
| 2019 December | 15 | 8 | 23 |
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| 2018 September | 1 | 3 | 4 |
| 2018 August | 5 | 1 | 6 |
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| 2018 June | 4 | 0 | 4 |
| 2018 May | 3 | 1 | 4 |
| 2018 April | 7 | 1 | 8 |
| 2018 February | 3 | 3 | 6 |
| 2017 November | 1 | 0 | 1 |
| 2017 October | 6 | 0 | 6 |
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| 2017 July | 2 | 0 | 2 |
| 2017 June | 7 | 20 | 27 |
| 2017 May | 11 | 8 | 19 |
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| 2016 October | 17 | 3 | 20 |
| 2016 September | 11 | 2 | 13 |
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| 2016 March | 7 | 3 | 10 |
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| 2015 December | 10 | 4 | 14 |
| 2015 November | 13 | 3 | 16 |
| 2015 October | 17 | 2 | 19 |
| 2015 September | 8 | 3 | 11 |
| 2015 August | 2 | 4 | 6 |
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| 2015 June | 1 | 0 | 1 |
| 2015 May | 1 | 1 | 2 |
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