<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0185-1101</journal-id>
<journal-title><![CDATA[Revista mexicana de astronomía y astrofísica]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. mex. astron. astrofis]]></abbrev-journal-title>
<issn>0185-1101</issn>
<publisher>
<publisher-name><![CDATA[Universidad Nacional Autónoma de México, Instituto de Astronomía]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0185-11012021000200407</article-id>
<article-id pub-id-type="doi">10.22201/ia.01851101p.2021.57.02.13</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Stability analysis of relativistic polytropes]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Saad]]></surname>
<given-names><![CDATA[Abdel-Naby S.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
<xref ref-type="aff" rid="Aaf"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Nouh]]></surname>
<given-names><![CDATA[Mohamed I.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Shaker]]></surname>
<given-names><![CDATA[Ashraf A.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Kamel]]></surname>
<given-names><![CDATA[Tarek M.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Qassim University Deanship of Educational Services Department of Mathematics]]></institution>
<addr-line><![CDATA[Buraidah ]]></addr-line>
<country>Saudi Arabia</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,National Research Institute of Astronomy and Geophysics Astronomy Department ]]></institution>
<addr-line><![CDATA[Cairo ]]></addr-line>
<country>Egypt</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>00</month>
<year>2021</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>00</month>
<year>2021</year>
</pub-date>
<volume>57</volume>
<numero>2</numero>
<fpage>407</fpage>
<lpage>418</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S0185-11012021000200407&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S0185-11012021000200407&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S0185-11012021000200407&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[ABSTRACT We study the relativistic self-gravitating, hydrostatic spheres with a polytropic equation of state, considering structures with the polytropic indices n = 1(0.5)3 and illustrate the results for the relativistic parameters &#963; = 0 &#8722; 0.75. We determine the critical relativistic parameter at which the mass of the polytrope has a maximum value and represents the first mode of radial instability. For n = 1(0.5)2.5, stable relativistic polytropes occur for &#963; less than the critical values 0.42,0.20,0.10, and 0.04, respectively, while unstable relativistic polytropes are obtained when &#963; is greater than the same values. When n = 3.0 and &#963; &gt; 0.5, energetically unstable solutions occur. The results of critical values are in full agreement with those evaluated by several authors. Comparisons between analytical and numerical solutions of the given relativistic functions provide a maximum relative error of order 10&#8722;3.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[RESUMEN Estudiamos esferas hidrostáticas, autogravitantes y relativistas, con una ecuación de estado politrópica, considerando estructuras con índices politrópicos n = 1(0.5)3, e ilustramos los resultados para parámetros relativistas &#963; = 0 &#8722; 0.75. Determinamos el parámetro relativista crítico, para el cual la masa del politropo alcanza un valor máximo y representa el primer modo de inestabilidad radial. Con n = 1(0.5)2.5 encontramos politropos relativistas estables para &#963; menor que los valores críticos 0.42,0.20,0.10, y 0.04, respectivamente. Se obtienen politropos relativistas inestables para valores mayores de &#963;. Cuando n = 3.0 y &#963; &gt; 0.5 encontramos soluciones energéticamente inestables. Los resutados sobre los valores críticos concuerdan muy bien con los de otros autores. Al comparar las soluciones analíticas y numéricas de las funciones relativistas estudiadas se encuentran errores relativos máximos del orden de 10&#8722;3.]]></p></abstract>
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