The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. The Deformed Figures of the Maclaurin Spheroids

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The post-Newtonian figure is obtained as a deformation of the Newtonian Maclaurin spheroid (with semi-axes and , say) by a Lagrangian displacement proportional to (2) =a'12( 2Xl 2x2, - ), where w denotes the distance from the axis of rotation and x1 and x2 are the Cartesian coordinates in the equatorial plane. It is shown that the equation defining the boundary of the post-Newtonian configuration is of the form 2 2 R 5 4 2+-2-t-2S2t(e)--4-22 =0, ai a3 a1 a1 a1 a3 (e) is a determinate function of the eccentricity e of the Maclaurin spheroid and R5 (= 2CM/c2) is the Schwarzschild radius. The function S t(e) is tabulated in the paper. Further, the angular velocity of rotation of the post-Newtonian configuration differs from that of the Maclaurin spheroid by an amount which is also tabulated. The solution of the post-Newtonian equations exhibits a singularity at a certain eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin of this singularity is that at e* the Maclaurin spheroid allows an infinitesimal neutral deformation by a displacement proportional to (2); and the Newtonian instability of the Maclaurin spheroid at e* is excited by the post-Newtonian effects of general relativity.">  <meta property="og:type" content="article"> <meta property="og:title" content="The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. The Deformed Figures of the Maclaurin Spheroids"> <meta property="og:site_name" content="NASA/ADS"> <meta property="og:description" content="The equations of post-Newtonian hydrodynamics are solved appropriately for a uniformly rotating homogeneous mass with symmetry about the axis of rotation. The post-Newtonian figure is obtained as a deformation of the Newtonian Maclaurin spheroid (with semi-axes and , say) by a Lagrangian displacement proportional to (2) =a'12( 2Xl 2x2, - ), where w denotes the distance from the axis of rotation and x1 and x2 are the Cartesian coordinates in the equatorial plane. It is shown that the equation defining the boundary of the post-Newtonian configuration is of the form 2 2 R 5 4 2+-2-t-2S2t(e)--4-22 =0, ai a3 a1 a1 a1 a3 (e) is a determinate function of the eccentricity e of the Maclaurin spheroid and R5 (= 2CM/c2) is the Schwarzschild radius. The function S t(e) is tabulated in the paper. Further, the angular velocity of rotation of the post-Newtonian configuration differs from that of the Maclaurin spheroid by an amount which is also tabulated. The solution of the post-Newtonian equations exhibits a singularity at a certain eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin of this singularity is that at e* the Maclaurin spheroid allows an infinitesimal neutral deformation by a displacement proportional to (2); and the Newtonian instability of the Maclaurin spheroid at e* is excited by the post-Newtonian effects of general relativity."> <meta property="og:url" content="https://ui.adsabs.harvard.edu/abs/1967ApJ...147..334C/abstract"> <meta property="og:image" content="https://ui.adsabs.harvard.edu/styles/img/transparent_logo.svg"> <meta property="article:published_time" content="01/1967"> <meta property="article:author" content="Chandrasekhar, S.">  <meta name="citation_journal_title" content="The Astrophysical Journal"> <meta name="citation_authors" content="Chandrasekhar, S."> <meta name="citation_title" content="The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. 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The function S t(e) is tabulated in the paper. Further, the angular velocity of rotation of the post-Newtonian configuration differs from that of the Maclaurin spheroid by an amount which is also tabulated. The solution of the post-Newtonian equations exhibits a singularity at a certain eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin of this singularity is that at e* the Maclaurin spheroid allows an infinitesimal neutral deformation by a displacement proportional to (2); and the Newtonian instability of the Maclaurin spheroid at e* is excited by the post-Newtonian effects of general relativity."/> <meta name="twitter:title" content="The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. 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II. The Deformed Figures of the Maclaurin Spheroids <a href=""></a> </h2> <div id="authors-and-aff" class="s-authors-and-aff"> <ul class="list-inline"> <li class="author"><a href="/search/?q=author%3A%22Chandrasekhar%2C+S.%22">Chandrasekhar, S.</a> </li> </ul> </div> <div class="s-abstract-text"> <h4 class="sr-only">Abstract</h4> <p> The equations of post-Newtonian hydrodynamics are solved appropriately for a uniformly rotating homogeneous mass with symmetry about the axis of rotation. The post-Newtonian figure is obtained as a deformation of the Newtonian Maclaurin spheroid (with semi-axes and , say) by a Lagrangian displacement proportional to (2) =a'12( 2Xl 2x2, - ), where w denotes the distance from the axis of rotation and x1 and x2 are the Cartesian coordinates in the equatorial plane. It is shown that the equation defining the boundary of the post-Newtonian configuration is of the form 2 2 R 5 4 2+-2-t-2S2t(e)--4-22 =0, ai a3 a1 a1 a1 a3 (e) is a determinate function of the eccentricity e of the Maclaurin spheroid and R5 (= 2CM/c2) is the Schwarzschild radius. The function S t(e) is tabulated in the paper. Further, the angular velocity of rotation of the post-Newtonian configuration differs from that of the Maclaurin spheroid by an amount which is also tabulated. The solution of the post-Newtonian equations exhibits a singularity at a certain eccentricity e*(= 0.985226) of the Maclaurin spheroid. The origin of this singularity is that at e* the Maclaurin spheroid allows an infinitesimal neutral deformation by a displacement proportional to (2); and the Newtonian instability of the Maclaurin spheroid at e* is excited by the post-Newtonian effects of general relativity. </p> </div> <br> <dl class="s-abstract-dl-horizontal"> <dt>Publication:</dt> <dd> <div id="article-publication">The Astrophysical Journal</div> </dd> <dt>Pub Date:</dt> <dd>January 1967</dd> <dt>DOI:</dt> <dd> <span> <a href="/link_gateway/1967ApJ...147..334C/doi:10.1086/149003" target="_blank" rel="noopener">10.1086/149003</a> <i class="fa fa-external-link"></i> </span> </dd> <dt>Bibcode:</dt> <dd> <a href="/abs/1967ApJ...147..334C/abstract"> 1967ApJ...147..334C </a> <i class="icon-help" title="The bibcode is assigned by the ADS as a unique identifier for the paper."></i> </dd> </dl> </article> </div> <div data-widget="ShowCitations"></div> <div data-widget="ShowReferences"></div> <div data-widget="ShowCoreads"></div> <div data-widget="ShowSimilar"></div> <div data-widget="ShowTableofcontents"></div> <div data-widget="ShowGraphics"></div> <div data-widget="ShowExportcitation" data-origin="abstract"></div> <div data-widget="ShowMetrics" data-allow-redirect="false"></div> <div data-widget="MetaTagsWidget"></div> </div> </div> </div> <div class="s-right-col-container col-xs-12 col-sm-12 col-md-3 col-lg-2 s-right-column" id="right-col-container" > <div data-widget="ShowResources"> <div data-reactroot="" class="s-right-col-widget-container" style="padding: 10px" > <div> <div class="resources__container"> <div class="resources__full__list"> <div class="resources__header__row"> <i class="fa fa-file-text-o" aria-hidden="true"> </i> <div class="resources__header__title">full text sources</div> </div> <div class="resources__content"> <div class="resources__content__title">ADS</div> <div class="resources__content__links"> <span> <a href="/link_gateway/1967ApJ...147..334C/ADS_PDF" rel="noopener" class="resources__content__link unlock" > <i class="fa fa-file-pdf-o" aria-hidden="true"> </i> </a> <div class="resources__content__link__separator">|</div> </span> <span> <a href="/link_gateway/1967ApJ...147..334C/ADS_SCAN" rel="noopener" class="resources__content__link unlock" > <i class="fa fa-file-image-o" aria-hidden="true"> </i> </a> </span> </div> </div> </div> </div> <div data-widget="ShowAssociated"> <div data-reactroot="" class="resources__container"> <div class="resources__header__row"> <i class="fa fa-folder-open"></i> <div class="resources__header__title"> Related Materials (5) </div> </div> <div> <div style=" font-size: 1em; 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// Add class "autocomplete-active": x[currentFocus].classList.add("autocomplete-active"); } function removeActive(x) { // Remove the "active" class from all autocomplete items: for (var i = 0; i < x.length; i++) { x[i].classList.remove("autocomplete-active"); } } function closeAllLists(elmnt) { // Close all autocomplete lists in the document, except the one passed as an argument: var x = document.getElementsByClassName("autocomplete-items"); for (var i = 0; i < x.length; i++) { if (elmnt != x[i] && elmnt != searchBox) { x[i].parentNode.removeChild(x[i]); } } } // Any other clicks in the document: document.addEventListener("click", function (e) { closeAllLists(e.target); }); } var autoList = [ { value: 'author:""', label: 'Author', match: 'author:"' }, { value: 'author:"^"', label: 'First Author', match: 'first author' }, { value: 'author:"^"', label: 'First Author', match: 'author:"^' }, { value: 'bibcode:""', label: 'Bibcode', desc: 'e.g. bibcode:1989ApJ...342L..71R', match: 'bibcode:"' }, { value: 'bibstem:""', label: 'Publication', desc: 'e.g. bibstem:ApJ', match: 'bibstem:"' }, { value: 'bibstem:""', label: 'Publication', desc: 'e.g. bibstem:ApJ', match: 'publication (bibstem)' }, { value: 'arXiv:', label: 'arXiv ID', match: 'arxiv:' }, { value: 'doi:', label: 'DOI', match: 'doi:' }, { value: 'full:""', label: 'Full text search', desc: 'title, abstract, and body', match: 'full:' }, { value: 'full:""', label: 'Full text search', desc: 'title, abstract, and body', match: 'fulltext' }, { value: 'full:""', label: 'Full text search', desc: 'title, abstract, and body', match: 'text' }, { value: 'year:', label: 'Year', match: 'year' }, { value: 'year:1999-2005', label: 'Year Range', desc: 'e.g. 1999-2005', match: 'year range' }, { value: 'aff:""', label: 'Affiliation', match: 'aff:' }, { value: 'abs:""', label: 'Search abstract + title + keywords', match: 'abs:' }, { value: 'database:astronomy', label: 'Limit to papers in the astronomy database', match: 'database:astronomy' }, { value: 'database:physics', label: 'Limit to papers in the physics database', match: 'database:physics' }, { value: 'title:""', label: 'Title', match: 'title:"' }, { value: 'orcid:', label: 'ORCiD identifier', match: 'orcid:' }, { value: 'object:', label: 'SIMBAD object (e.g. object:LMC)', match: 'object:' }, { value: 'property:refereed', label: 'Limit to refereed', desc: '(property:refereed)', match: 'refereed' }, { value: 'property:refereed', label: 'Limit to refereed', desc: '(property:refereed)', match: 'property:refereed' }, { value: 'property:notrefereed', label: 'Limit to non-refereed', desc: '(property:notrefereed)', match: 'property:notrefereed' }, { value: 'property:notrefereed', label: 'Limit to non-refereed', desc: '(property:notrefereed)', match: 'notrefereed' }, { value: 'property:eprint', label: 'Limit to eprints', desc: '(property:eprint)', match: 'eprint' }, { value: 'property:eprint', label: 'Limit to eprints', desc: '(property:eprint)', match: 'property:eprint' }, { value: 'property:openaccess', label: 'Limit to open access', desc: '(property:openaccess)', match: 'property:openaccess' }, { value: 'property:openaccess', label: 'Limit to open access', desc: '(property:openaccess)', match: 'openaccess' }, { value: 'doctype:software', label: 'Limit to software', desc: '(doctype:software)', match: 'software' }, { value: 'doctype:software', label: 'Limit to software', desc: '(doctype:software)', match: 'doctype:software' }, { value: 'property:inproceedings', label: 'Limit to papers in conference proceedings', desc: '(property:inproceedings)', match: 'proceedings' }, { value: 'property:inproceedings', label: 'Limit to papers in conference proceedings', desc: '(property:inproceedings)', match: 'property:inproceedings' }, { value: 'citations()', label: 'Citations', desc: 'Get papers citing your search result set', match: 'citations(' }, { value: 'references()', label: 'References', desc: 'Get papers referenced by your search result set', match: 'references(' }, { value: 'trending()', label: 'Trending', desc: 'Get papers most read by users who recently read your search result set', match: 'trending(' }, { value: 'reviews()', label: 'Review Articles', desc: 'Get most relevant papers that cite your search result set', match: 'reviews(' }, { value: 'useful()', label: 'Useful', desc: 'Get papers most frequently cited by your search result set', match: 'useful(' }, { value: 'similar()', label: 'Similar', desc: 'Get papers that have similar full text to your search result set', match: 'similar(' }, ]; // initiate the autocomplete function on the "q" element, and pass along the operators array as possible autocomplete values: inputBox = document.getElementById("q") if (inputBox) { inputBox.focus() // autofucs inputBox.setSelectionRange(inputBox.value.length, inputBox.value.length); // bring cursor to the end autocomplete(inputBox, autoList); } </script> <script> (function() { // turn off no-js if we have javascript document.documentElement.className = document.documentElement.className.replace("no-js", "js"); function getCookie(cname) { var name = cname + "="; var decodedCookie = decodeURIComponent(document.cookie); var ca = decodedCookie.split(';'); for (var i = 0; i < ca.length; i++) { var c = ca[i]; while (c.charAt(0) == ' ') { c = c.substring(1); } if (c.indexOf(name) == 0) { return c.substring(name.length, c.length); } } return ""; } (function() { // looks for the cookie, and sets true if its 'always' const coreCookie = getCookie('core') === 'always'; // only load bumblebee if we detect the core cookie and we are on abstract page if (coreCookie || (!(/^\/abs\//.test(document.location.pathname)) && !coreCookie)) { return; 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The Post-Newtonian Effects of General Relativity on the Equilibrium of Uniformly Rotating Bodies. II. The Deformed Figures of the Maclaurin Spheroids - NASA/ADS