<?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>S0185-11012007000100005</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Tidal Heating: A Restricted three Body Problem Investigation of the spitzer and Weinberg Approximations]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Cruz]]></surname>
<given-names><![CDATA[Fidel]]></given-names>
</name>
<xref ref-type="aff" rid="A01"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Aguilar]]></surname>
<given-names><![CDATA[Luis A.]]></given-names>
</name>
<xref ref-type="aff" rid="A01"/>
</contrib>
</contrib-group>
<aff id="A01">
<institution><![CDATA[,Universidad Nacional Autónoma de México Instituto de Astronomía ]]></institution>
<addr-line><![CDATA[Ensenada B. C.]]></addr-line>
<country>México</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>00</month>
<year>2007</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>00</month>
<year>2007</year>
</pub-date>
<volume>43</volume>
<numero>1</numero>
<fpage>89</fpage>
<lpage>94</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S0185-11012007000100005&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S0185-11012007000100005&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S0185-11012007000100005&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Estudiamos el calentamiento por fuerzas de marea de cúmulos globulares utilizando el problema restringido de tres cuerpos. Nuestros resultados concuerdan más con la aproximación de Spitzer en el caso adiabático que con la de Gnedin y Ostriker para la aproximación de Weinberg. De estos resultados concluimos que la aproximación de Spitzer es válida para cúmulos en órbitas de alta excentricidad. Encontramos que un escalamiento lineal del calentamiento después de un choque de marea subestima el efecto que se mide después de varios choques.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[We study the tidal heating of globular clusters using the restricted three body approximation. Our results are more consistent with the Spitzer correction for adiabaticity than with the Gnedin and Ostriker formula for the Weinberg correction. From these results we conclude that the Spitzer approximation is valid for clusters in orbits with high eccentricity. We find that a simple linear scaling of the effect after one tidal shock greatly underestimates the actual effect after repeated shocks.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[CELESTIAL MECHANICS]]></kwd>
<kwd lng="en"><![CDATA[GLOBULAR CLUSTERS: GENERAL]]></kwd>
<kwd lng="en"><![CDATA[METHODS: NUMERICAL]]></kwd>
<kwd lng="en"><![CDATA[STELLAR DYNAMICS]]></kwd>
</kwd-group>
</article-meta>
</front><body><![CDATA[ <p align="center"><font face="verdana" size="4"><b>TIDAL HEATING: A RESTRICTED THREE BODY PROBLEM INVESTIGATION OF THE SPITZER AND WEINBERG APPROXIMATIONS</b></font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="center"><font face="verdana" size="2"><b>Fidel Cruz and Luis A. Aguilar</b></font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><i>Instituto de Astronom&iacute;a, Universidad Nacional Aut&oacute;noma de M&eacute;xico, Apdo. Postal 877, 22860 Ensenada, B. C., M&eacute;xico (<a href="mailto:aguilar@astrosen.unam.mx">aguilar<i>@astrosen.unam.mx</i></a>, <a href="mailto:fidel@astrosen.unam.mx">fidel@astrosen.unam.mx</a>)</i></font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2">Received 2005 September 13    <br> Accepted 2006 October 10</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>RESUMEN</b></font></p>     ]]></body>
<body><![CDATA[<p align="justify"><font face="verdana" size="2">Estudiamos el calentamiento por fuerzas de marea de c&uacute;mulos globulares utilizando el problema restringido de tres cuerpos. Nuestros resultados concuerdan m&aacute;s con la aproximaci&oacute;n de Spitzer en el caso adiab&aacute;tico que con la de Gnedin y Ostriker para la aproximaci&oacute;n de Weinberg. De estos resultados concluimos que la aproximaci&oacute;n de Spitzer es v&aacute;lida para c&uacute;mulos en &oacute;rbitas de alta excentricidad. Encontramos que un escalamiento lineal del calentamiento despu&eacute;s de un choque de marea subestima el efecto que se mide despu&eacute;s de varios choques.</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>ABSTRACT</b></font></p>     <p align="justify"><font face="verdana" size="2">We study the tidal heating of globular clusters using the restricted three body approximation. Our results are more consistent with the Spitzer correction for adiabaticity than with the Gnedin and Ostriker formula for the Weinberg correction. From these results we conclude that the Spitzer approximation is valid for clusters in orbits with high eccentricity. We find that a simple linear scaling of the effect after one tidal shock greatly underestimates the actual effect after repeated shocks.</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b><i>Key Words: </i></b>CELESTIAL   MECHANICS &#151; GLOBULAR  CLUSTERS: GENERAL&nbsp;&#151; METHODS:   NUMERICAL &#151; STELLAR DYNAMICS</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><a href="/pdf/rmaa/v43n1/v43n1a5.pdf">DESCARGAR ART&Iacute;CULO EN FORMATO PDF</a></font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>REFERENCES</b></font></p>     ]]></body>
<body><![CDATA[<!-- ref --><p align="justify"><font face="verdana" size="2">Aguilar, L., Hut, P., &amp; Ostriker, J. P. 1988, ApJ, 335, 720 (AHO) </font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420518&pid=S0185-1101200700010000500001&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Chernoff, D. F., Kochanek, C. S., &amp; Shapiro, S. L. 1986, ApJ, 309, 183</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420519&pid=S0185-1101200700010000500002&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Gnedin, O. Y., Lee, H. M., &amp; Ostriker, J. P. 1999, ApJ, 522, 935</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420520&pid=S0185-1101200700010000500003&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Gnedin, O. Y., &amp; Ostriker, J. P. 1997, ApJ, 474, 223 (GO)</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420521&pid=S0185-1101200700010000500004&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">_________. 1999, ApJ, 513, 626</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420522&pid=S0185-1101200700010000500005&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">H&eacute;non, M. 1970, A &amp; A, 9, 24</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420523&pid=S0185-1101200700010000500006&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Innanen, K. A. 1979, AJ, 84, 960</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420524&pid=S0185-1101200700010000500007&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Innanen, K. A., Harris, W. E., &amp; Webbink, R. F. 1983, AJ, 88, 338</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420525&pid=S0185-1101200700010000500008&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Jefferys, W. H. 1976, AJ, 81, 983</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420526&pid=S0185-1101200700010000500009&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Odenkirchen, M., et al. 2003, AJ, 126, 2385</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420527&pid=S0185-1101200700010000500010&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Ostriker, J. P., Spitzer, L., &amp; Chevalier, R. A. 1972, ApJ, 176, L51</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420528&pid=S0185-1101200700010000500011&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Press, W. H., Flannery, B. P., Teukolsky, S. A, &amp; Vettering, W. T. 1992, Numerical Recipes in Fortran 77: Art of Scientific Computing (Cambridge: Cambridge Univ. Press)</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420529&pid=S0185-1101200700010000500012&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Spitzer, L. 1958, ApJ, 127, 17</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420530&pid=S0185-1101200700010000500013&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">_________. 1987, Dynamical Evolution of Globular Clusters (Princeton: Princeton Univ. Press)</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420531&pid=S0185-1101200700010000500014&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">Weinberg, M. D. 1994, AJ, 108, 1398</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=7420532&pid=S0185-1101200700010000500015&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --> ]]></body><back>
<ref-list>
<ref id="B1">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Aguilar]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Hut]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
<name>
<surname><![CDATA[Ostriker]]></surname>
<given-names><![CDATA[J. P.]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>335</volume>
<page-range>720</page-range></nlm-citation>
</ref>
<ref id="B2">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Chernoff]]></surname>
<given-names><![CDATA[D. F.]]></given-names>
</name>
<name>
<surname><![CDATA[Kochanek]]></surname>
<given-names><![CDATA[C. S.]]></given-names>
</name>
<name>
<surname><![CDATA[Shapiro]]></surname>
<given-names><![CDATA[S. L.]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>309</volume>
<page-range>183</page-range></nlm-citation>
</ref>
<ref id="B3">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Gnedin]]></surname>
<given-names><![CDATA[O. Y.]]></given-names>
</name>
<name>
<surname><![CDATA[Lee]]></surname>
<given-names><![CDATA[H. M.]]></given-names>
</name>
<name>
<surname><![CDATA[Ostriker]]></surname>
<given-names><![CDATA[J. P.]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>522</volume>
<page-range>935</page-range></nlm-citation>
</ref>
<ref id="B4">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Gnedin]]></surname>
<given-names><![CDATA[O. Y.]]></given-names>
</name>
<name>
<surname><![CDATA[Ostriker]]></surname>
<given-names><![CDATA[J. P.]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>474</volume>
<page-range>223</page-range></nlm-citation>
</ref>
<ref id="B5">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Gnedin]]></surname>
<given-names><![CDATA[O. Y.]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>513</volume>
<page-range>626</page-range></nlm-citation>
</ref>
<ref id="B6">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Hénon]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[A & A]]></source>
<year></year>
<volume>9</volume>
<page-range>24</page-range></nlm-citation>
</ref>
<ref id="B7">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Innanen]]></surname>
<given-names><![CDATA[K. A.]]></given-names>
</name>
</person-group>
<source><![CDATA[AJ]]></source>
<year></year>
<volume>84</volume>
<page-range>960</page-range></nlm-citation>
</ref>
<ref id="B8">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Innanen]]></surname>
<given-names><![CDATA[K. A.]]></given-names>
</name>
<name>
<surname><![CDATA[Harris]]></surname>
<given-names><![CDATA[W. E.]]></given-names>
</name>
<name>
<surname><![CDATA[Webbink]]></surname>
<given-names><![CDATA[R. F.]]></given-names>
</name>
</person-group>
<source><![CDATA[AJ]]></source>
<year></year>
<volume>88</volume>
<page-range>338</page-range></nlm-citation>
</ref>
<ref id="B9">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Jefferys]]></surname>
<given-names><![CDATA[W. H.]]></given-names>
</name>
</person-group>
<source><![CDATA[AJ]]></source>
<year></year>
<volume>81</volume>
<page-range>983</page-range></nlm-citation>
</ref>
<ref id="B10">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Odenkirchen]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[AJ]]></source>
<year></year>
<volume>126</volume>
<page-range>2385</page-range></nlm-citation>
</ref>
<ref id="B11">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ostriker]]></surname>
<given-names><![CDATA[J. P.]]></given-names>
</name>
<name>
<surname><![CDATA[Spitzer]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Chevalier]]></surname>
<given-names><![CDATA[R. A.]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>176</volume>
<page-range>L51</page-range></nlm-citation>
</ref>
<ref id="B12">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Press]]></surname>
<given-names><![CDATA[W. H.]]></given-names>
</name>
<name>
<surname><![CDATA[Flannery]]></surname>
<given-names><![CDATA[B. P.]]></given-names>
</name>
<name>
<surname><![CDATA[Teukolsky]]></surname>
<given-names><![CDATA[S. A]]></given-names>
</name>
<name>
<surname><![CDATA[Vettering]]></surname>
<given-names><![CDATA[W. T.]]></given-names>
</name>
</person-group>
<source><![CDATA[Numerical Recipes in Fortran 77: Art of Scientific Computing]]></source>
<year>1992</year>
<publisher-loc><![CDATA[Cambridge ]]></publisher-loc>
<publisher-name><![CDATA[Cambridge Univ. Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B13">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Spitzer]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<source><![CDATA[ApJ]]></source>
<year></year>
<volume>127</volume>
<page-range>17</page-range></nlm-citation>
</ref>
<ref id="B14">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Spitzer]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
</person-group>
<person-group person-group-type="editor">
<name>
</name>
</person-group>
<source><![CDATA[Dynamical Evolution of Globular Clusters]]></source>
<year>1987</year>
<publisher-loc><![CDATA[Princeton ]]></publisher-loc>
<publisher-name><![CDATA[Princeton Univ. Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B15">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Weinberg]]></surname>
<given-names><![CDATA[M. D.]]></given-names>
</name>
</person-group>
<source><![CDATA[AJ]]></source>
<year></year>
<volume>108</volume>
<page-range>1398</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
