earth mover distance - Githubissues

/usr/lib/jvm/java-8-oracle/bin/java -ea -Didea.launcher.port=7532 -Didea.launcher.bin.path=/home/felix/dev/idea-IU-163.7743.44/bin -Dfile.encoding=UTF-8 -classpath /home/felix/dev/idea-IU-163.7743.44/lib/idea_rt.jar:/home/felix/dev/idea-IU-163.7743.44/plugins/junit/lib/junit-rt.jar:/usr/lib/jvm/java-8-oracle/jre/lib/charsets.jar:/usr/lib/jvm/java-8-oracle/jre/lib/deploy.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/cldrdata.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/dnsns.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/jaccess.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/jfxrt.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/localedata.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/nashorn.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/sunec.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/sunjce_provider.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/sunpkcs11.jar:/usr/lib/jvm/java-8-oracle/jre/lib/ext/zipfs.jar:/usr/lib/jvm/java-8-oracle/jre/lib/javaws.jar:/usr/lib/jvm/java-8-oracle/jre/lib/jce.jar:/usr/lib/jvm/java-8-oracle/jre/lib/jfr.jar:/usr/lib/jvm/java-8-oracle/jre/lib/jfxswt.jar:/usr/lib/jvm/java-8-oracle/jre/lib/jsse.jar:/usr/lib/jvm/java-8-oracle/jre/lib/management-agent.jar:/usr/lib/jvm/java-8-oracle/jre/lib/plugin.jar:/usr/lib/jvm/java-8-oracle/jre/lib/resources.jar:/usr/lib/jvm/java-8-oracle/jre/lib/rt.jar:/home/felix/IdeaProjects/mathosphere/mathosphere-core/target/test-classes:/home/felix/IdeaProjects/mathosphere/mathosphere-core/target/classes:/home/felix/.m2/repository/junit/junit/4.11/junit-4.11.jar:/home/felix/.m2/repository/org/hamcrest/hamcrest-core/1.3/hamcrest-core-1.3.jar:/home/felix/.m2/repository/com/google/guava/guava/17.0/guava-17.0.jar:/home/felix/.m2/repository/net/sf/saxon/Saxon-HE/9.5.1-6/Saxon-HE-9.5.1-6.jar:/home/felix/IdeaProjects/mathosphere/lib/MathMLQueryGenerator/target/classes:/home/felix/IdeaProjects/mathosphere/lib/MathMLCan/target/classes:/home/felix/.m2/repository/org/jdom/jdom2/2.0.5/jdom2-2.0.5.jar:/home/felix/.m2/repository/commons-cli/commons-cli/1.2/commons-cli-1.2.jar:/home/felix/.m2/repository/xmlunit/xmlunit/1.5/xmlunit-1.5.jar:/home/felix/.m2/repository/org/apache/commons/commons-lang3/3.3.2/commons-lang3-3.3.2.jar:/home/felix/.m2/repository/org/fusesource/wikitext/wikitext-core/1.3/wikitext-core-1.3.jar:/home/felix/.m2/repository/org/fusesource/wikitext/mediawiki-core/1.3/mediawiki-core-1.3.jar:/home/felix/.m2/repository/com/beust/jcommander/1.48/jcommander-1.48.jar:/home/felix/.m2/repository/org/sweble/wikitext/swc-engine/2.0.0/swc-engine-2.0.0.jar:/home/felix/.m2/repository/com/sun/xml/bind/jaxb-impl/2.2.5/jaxb-impl-2.2.5.jar:/home/felix/.m2/repository/org/jsoup/jsoup/1.6.3/jsoup-1.6.3.jar:/home/felix/.m2/repository/log4j/log4j/1.2.14/log4j-1.2.14.jar:/home/felix/.m2/repository/org/sweble/wikitext/swc-parser-lazy/2.0.0/swc-parser-lazy-2.0.0.jar:/home/felix/.m2/repository/de/fau/cs/osr/ptk/ptk-common/2.0.0/ptk-common-2.0.0.jar:/home/felix/.m2/repository/commons-jxpath/commons-jxpath/1.3/commons-jxpath-1.3.jar:/home/felix/.m2/repository/xtc/rats-runtime/1.15.0/rats-runtime-1.15.0.jar:/home/felix/.m2/repository/de/fau/cs/osr/utils/utils/2.0.0/utils-2.0.0.jar:/home/felix/.m2/repository/net/sf/jopt-simple/jopt-simple/4.3/jopt-simple-4.3.jar:/home/felix/.m2/repository/org/apache/flink/flink-clients/0.10.1/flink-clients-0.10.1.jar:/home/felix/.m2/repository/org/apache/flink/flink-core/0.10.1/flink-core-0.10.1.jar:/home/felix/.m2/repository/org/apache/flink/flink-runtime/0.10.1/flink-runtime-0.10.1.jar:/home/felix/.m2/repository/io/netty/netty-all/4.0.31.Final/netty-all-4.0.31.Final.jar:/home/felix/.m2/repository/org/javassist/javassist/3.18.2-GA/javassist-3.18.2-GA.jar:/home/felix/.m2/repository/org/codehaus/jettison/jettison/1.1/jettison-1.1.jar:/home/felix/.m2/repository/stax/stax-api/1.0.1/stax-api-1.0.1.jar:/home/felix/.m2/repository/org/scala-lang/scala-library/2.10.4/scala-library-2.10.4.jar:/home/felix/.m2/repository/com/typesafe/akka/akka-actor_2.10/2.3.7/akka-actor_2.10-2.3.7.jar:/home/felix/.m2/repository/com/typesafe/config/1.2.1/config-1.2.1.jar:/home/felix/.m2/repository/com/typesafe/akka/akka-remote_2.10/2.3.7/akka-remote_2.10-2.3.7.jar:/home/felix/.m2/repository/io/netty/netty/3.8.0.Final/netty-3.8.0.Final.jar:/home/felix/.m2/repository/com/google/protobuf/protobuf-java/2.5.0/protobuf-java-2.5.0.jar:/home/felix/.m2/repository/org/uncommons/maths/uncommons-maths/1.2.2a/uncommons-maths-1.2.2a.jar:/home/felix/.m2/repository/com/typesafe/akka/akka-slf4j_2.10/2.3.7/akka-slf4j_2.10-2.3.7.jar:/home/felix/.m2/repository/org/clapper/grizzled-slf4j_2.10/1.0.2/grizzled-slf4j_2.10-1.0.2.jar:/home/felix/.m2/repository/com/github/scopt/scopt_2.10/3.2.0/scopt_2.10-3.2.0.jar:/home/felix/.m2/repository/io/dropwizard/metrics/metrics-core/3.1.0/metrics-core-3.1.0.jar:/home/felix/.m2/repository/io/dropwizard/metrics/metrics-jvm/3.1.0/metrics-jvm-3.1.0.jar:/home/felix/.m2/repository/io/dropwizard/metrics/metrics-json/3.1.0/metrics-json-3.1.0.jar:/home/felix/.m2/repository/org/apache/zookeeper/zookeeper/3.4.6/zookeeper-3.4.6.jar:/home/felix/.m2/repository/jline/jline/0.9.94/jline-0.9.94.jar:/home/felix/.m2/repository/org/apache/flink/flink-optimizer/0.10.1/fl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com.intellij.rt.execution.application.AppMain com.intellij.rt.execution.junit.JUnitStarter -ideVersion5 com.formulasearchengine.mathosphere.mathpd.FlinkPdTest
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<title>Radial oscillations of relativistic stars</title>
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<div class="ltx_document ltx_authors_1line">
<h1 class="ltx_title ltx_title_document">Radial oscillations of relativistic stars</h1>
<div class="ltx_authors">
<span class="ltx_creator ltx_role_author">
<span class="ltx_personname">K. D. Kokkotas  
</span><span class="ltx_author_notes"><span>Department of Physics, Aristotle University of
Thessaloniki,
Thessaloniki 54006, Greece
<br class="ltx_break"/>email: kokkotas@astro.auth.gr
<br class="ltx_break"/>ruoff@astro.auth.gr
<br class="ltx_break"/></span></span></span>
<span class="ltx_author_before">  </span><span class="ltx_creator ltx_role_author">
<span class="ltx_personname">J. Ruoff
</span><span class="ltx_author_notes"><span>Department of Physics, Aristotle University of
Thessaloniki,
Thessaloniki 54006, Greece
<br class="ltx_break"/>email: kokkotas@astro.auth.gr
<br class="ltx_break"/>ruoff@astro.auth.gr
<br class="ltx_break"/>Institut für Astronomie und Astrophysik, Universität
Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany </span></span></span></div>

<div class="ltx_keywords">
<h6 class="ltx_title ltx_title_keywords">Key Words.:</h6>neutron stars – oscillations of stars –
equation of state
</div><span class="ltx_note ltx_role_offprints"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">offprints: </span>K. D. Kokkotas</span></span></span>
<div class="ltx_abstract">
<p class="ltx_p">We present a new survey of the radial oscillation modes of
neutron stars. This study complements and corrects earlier studies
of radial oscillations. We present an extensive list of frequencies
for the most common equations of state and some more recent ones.
In order to check the accuracy, we use two different numerical
schemes which yield the same results. The stimulation for this work
comes from the need of the groups that evolve the full nonlinear
Einstein equation to have reliable results from perturbation theory
for comparison.</p>
</div>
<div id="S1" class="ltx_section">
<h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2>

<div id="S1.p1" class="ltx_para">
<p class="ltx_p">As they are the simplest oscillation modes of neutron stars, radial
modes have been the first under investigation, more than 35 years ago
(Chandrasekhar 1964a, 1964b). More important, they can give
information about the stability of the stellar model under
consideration. Since radial oscillations do not couple to
gravitational waves, the appropriate equations are quite simple, and
it is relatively easy to numerically solve the eigenvalue problem that
leads to the discrete set of oscillation frequencies of a neutron
star. In the absence of any dissipative processes, the oscillation
spectrum of a stable stellar model forms a complete set; it is
therefore possible to describe any arbitrary periodic radial motion of
a neutron star as a superposition of its various eigenmodes.</p>
</div>
<div id="S1.p2" class="ltx_para">
<p class="ltx_p">The radial modes of neutron stars have been thoroughly investigated by
various authors mostly for zero temperature equations of state (EOS)
(e.g. <cite class="ltx_cite ltx_citemacro_cite">Harrison et al. (<a href="#bib.bib20" title="" class="ltx_ref">1965</a>)</cite>, <cite class="ltx_cite ltx_citemacro_cite">Chanmugam (<a href="#bib.bib11" title="" class="ltx_ref">1977</a>)</cite>, <cite class="ltx_cite ltx_citemacro_cite">Glass &amp; Lindblom (<a href="#bib.bib15" title="" class="ltx_ref">1983</a>)</cite>, <cite class="ltx_cite ltx_citemacro_cite">Väth &amp; Chanmugam (<a href="#bib.bib29" title="" class="ltx_ref">1992</a>)</cite> and
references therein). But also protoneutron stars with a finite
temperature EOS (<cite class="ltx_cite ltx_citemacro_cite">Gondek et al. (<a href="#bib.bib17" title="" class="ltx_ref">1997</a>)</cite>) and strange stars were studied
(<cite class="ltx_cite ltx_citemacro_cite">Benvenuto &amp; Horvath (<a href="#bib.bib6" title="" class="ltx_ref">1991</a>)</cite>, <cite class="ltx_cite ltx_citemacro_cite">Väth &amp; Chanmugam (<a href="#bib.bib29" title="" class="ltx_ref">1992</a>)</cite>, <cite class="ltx_cite ltx_citemacro_cite">Gondek &amp; Zdunik (<a href="#bib.bib18" title="" class="ltx_ref">1999</a>)</cite>).</p>
</div>
<div id="S1.p3" class="ltx_para">
<p class="ltx_p">The first exhaustive compilation of radial modes for various zero
temperature EOS was presented by <cite class="ltx_cite ltx_citemacro_cite">Glass &amp; Lindblom (<a href="#bib.bib15" title="" class="ltx_ref">1983</a>)</cite> (hereafter GL). However,
as was later pointed out by <cite class="ltx_cite ltx_citemacro_cite">Väth &amp; Chanmugam (<a href="#bib.bib29" title="" class="ltx_ref">1992</a>)</cite> (hereafter VC), their
numerical values for the oscillation frequencies seemed to be flawed
although their equations were correct. VC computed the radial
frequencies for 6 equations of state of dense matter and corroborated
their own results using the argument (<cite class="ltx_cite ltx_citemacro_cite">Harrison et al. (<a href="#bib.bib20" title="" class="ltx_ref">1965</a>)</cite>) that for the
numerical code to be correct, it must yield a zero-frequency mode at
exactly that central density for which the neutron star reaches its
maximal mass. This is the point where the stellar model becomes
unstable with respect to radial collapse if the central density is
further increased. Yet, this is not the case for the results of GL, as
was noticed by VC. However, the above mentioned test can be used only
in the case when both in the stellar model and in the perturbation
equations the equilibrium adiabatic index is used. In general,
different adiabatic indices can be used depending on the physical
conditions inside a star (<cite class="ltx_cite ltx_citemacro_cite">Gondek et al. (<a href="#bib.bib17" title="" class="ltx_ref">1997</a>)</cite>). For example, if the slowness
of weak interaction processes are taken into account, the regions of
configurations stable with respect to radial perturbations extend
beyond the central density of the star with the minimum mass
(e.g. <cite class="ltx_cite ltx_citemacro_cite">Chanmugam (<a href="#bib.bib11" title="" class="ltx_ref">1977</a>)</cite>) and of the star with maximum mass (<cite class="ltx_cite ltx_citemacro_cite">Gourgoulhon et al. (<a href="#bib.bib19" title="" class="ltx_ref">1995</a>)</cite>).</p>
</div>
<div id="S1.p4" class="ltx_para">
<p class="ltx_p">In this paper, we repeat the numerical calculation of the radial
oscillation modes of neutron stars for various zero temperature
equations of state using the equilibrium adiabatic index. To verify
the results, we use two different formulations of the equations
together with two different numerical methods to solve the eigenvalue
problem. We find that in all cases we obtain matching values for the
eigenfrequencies. In addition we have verified that the codes yield
zero frequency modes not only at the maxima but also at the minima of
the mass curves.</p>
</div>
<div id="S1.p5" class="ltx_para">
<p class="ltx_p">We give corrected values for the equations of state used by GL, and we
add some new equations of state. It is not clear to us what went wrong
in their calculations, since for certain EOS our values agree with
theirs (EOS C, E, O), for others they differ only slightly (EOS F, L,
N), but for some EOS the discrepancy is quite large (EOS A, B, D, G,
I).</p>
</div>
<div id="S1.p6" class="ltx_para">
<p class="ltx_p">Additionally we include six more recent equations of state: Two models
of <cite class="ltx_cite ltx_citemacro_cite">Glendenning (<a href="#bib.bib16" title="" class="ltx_ref">1985</a>)</cite>, one of the model of <cite class="ltx_cite ltx_citemacro_cite">Wiringa et al. (<a href="#bib.bib30" title="" class="ltx_ref">1988</a>)</cite>, the EOS MPA of
<cite class="ltx_cite ltx_citemacro_cite">Wu et al. (<a href="#bib.bib31" title="" class="ltx_ref">1991</a>)</cite>, and two EOS of <cite class="ltx_cite ltx_citemacro_cite">Akmal et al. (<a href="#bib.bib1" title="" class="ltx_ref">1998</a>)</cite>. Finally, we include three
more tables for polytropic equations of state with different
polytropic indices. The form we use is given by</p>
<table id="S1.E1" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.E1.m1" class="ltx_Math" alttext="p=\kappa\rho^{1+1/n}\&gt;." display="block"><apply><eq/><ci>𝑝</ci><apply><times/><ci>𝜅</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜌</ci><apply><plus/><cn type="integer">1</cn><apply><divide/><cn type="integer">1</cn><ci>𝑛</ci></apply></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td></tr>
</table>
<p class="ltx_p">In particular, we present for the following values of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m1" class="ltx_Math" alttext="\kappa" display="inline"><ci>𝜅</ci></math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m2" class="ltx_Math" alttext="n" display="inline"><ci>𝑛</ci></math>: (<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m3" class="ltx_Math" alttext="n=1,\kappa=100\," display="inline"><apply><csymbol cd="ambiguous">formulae-sequence</csymbol><apply><eq/><ci>𝑛</ci><cn type="integer">1</cn></apply><apply><eq/><ci>𝜅</ci><cn type="integer">100</cn></apply></apply></math>km<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m4" class="ltx_Math" alttext="{}^{2}" display="inline"><apply><cn type="integer">2</cn></apply></math>), <math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m5" class="ltx_Math" alttext="(n=0.8,\kappa=700\," display="inline"><cerror><csymbol cd="ambiguous">fragments</csymbol><ci>(</ci><csymbol cd="unknown">n</csymbol><eq/><cn type="float">0.8</cn><ci>,</ci><csymbol cd="unknown">κ</csymbol><eq/><cn type="integer">700</cn></cerror></math>km<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m6" class="ltx_Math" alttext="{}^{2.5}" display="inline"><apply><cn type="float">2.5</cn></apply></math>),<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m7" class="ltx_Math" alttext="(n=0.5,\kappa=2\cdot 10^{5}\," display="inline"><cerror><csymbol cd="ambiguous">fragments</csymbol><ci>(</ci><csymbol cd="unknown">n</csymbol><eq/><cn type="float">0.5</cn><ci>,</ci><csymbol cd="unknown">κ</csymbol><eq/><cn type="integer">2</cn><ci>⋅</ci><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">5</cn></apply></cerror></math>km<math xmlns="http://www.w3.org/1998/Math/MathML" id="S1.p6.m8" class="ltx_Math" alttext="{}^{4})" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>)</ci><cn type="integer">4</cn></apply></math>.</p>
</div>
<div id="S1.p7" class="ltx_para">
<p class="ltx_p">Another interesting feature is the occurrence of avoided mode
crossings for realistic EOS. This phenomenon has been thoroughly
studied by <cite class="ltx_cite ltx_citemacro_cite">Gondek &amp; Zdunik (<a href="#bib.bib18" title="" class="ltx_ref">1999</a>)</cite> for a realistic nucleon EOS and an EOS
representing a strange star model. We find that it occurs for all
considered realistic EOS, for some it is quite strongly pronounced,
for others it is less obvious.</p>
</div>
</div>
<div id="S2" class="ltx_section">
<h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">2 </span>Equations and numerical methods</h2>

<div id="S2.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">2.1 </span>The radial equations</h3>

<div id="S2.SS1.p1" class="ltx_para">
<p class="ltx_p">The static and spherically symmetric metric which describes an
equilibrium stellar model is given by the following line element:</p>
<table id="S2.E2" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E2.m1" class="ltx_Math" alttext="ds^{2}=-e^{2\nu}dt^{2}+e^{2\lambda}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d%&#10;\phi^{2})\;." display="block"><apply><eq/><apply><times/><ci>𝑑</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑠</ci><cn type="integer">2</cn></apply></apply><apply><plus/><apply><minus/><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><cn type="integer">2</cn><ci>𝜈</ci></apply></apply><ci>𝑑</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑡</ci><cn type="integer">2</cn></apply></apply></apply><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply></apply><ci>𝑑</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply></apply><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply><apply><plus/><apply><times/><ci>𝑑</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜃</ci><cn type="integer">2</cn></apply></apply><apply><apply><csymbol cd="ambiguous">superscript</csymbol><sin/><cn type="integer">2</cn></apply><apply><times/><ci>𝜃</ci><ci>𝑑</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>italic-ϕ</ci><cn type="integer">2</cn></apply></apply></apply></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td></tr>
</table>
<p class="ltx_p">Together with the energy-momentum tensor for a perfect fluid</p>
<table id="S2.E3" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E3.m1" class="ltx_Math" alttext="T_{\mu\nu}=(\rho+p)u_{\mu}u_{\mu}+p\,g_{\mu\nu}\;," display="block"><apply><eq/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑇</ci><apply><times/><ci>𝜇</ci><ci>𝜈</ci></apply></apply><apply><plus/><apply><times/><apply><plus/><ci>𝜌</ci><ci>𝑝</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑢</ci><ci>𝜇</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑢</ci><ci>𝜇</ci></apply></apply><apply><times/><ci>𝑝</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑔</ci><apply><times/><ci>𝜇</ci><ci>𝜈</ci></apply></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td></tr>
</table>
<p class="ltx_p">Einstein’s field equations yield three independent ordinary
differential equations for the four unknowns <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p1.m1" class="ltx_Math" alttext="\nu,\mu,\rho" display="inline"><list><ci>𝜈</ci><ci>𝜇</ci><ci>𝜌</ci></list></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p1.m2" class="ltx_Math" alttext="p" display="inline"><ci>𝑝</ci></math>.
To complete the set of equations, an equation of state</p>
<table id="S2.E4" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E4.m1" class="ltx_Math" alttext="p=p(\rho)" display="block"><apply><eq/><ci>𝑝</ci><apply><times/><ci>𝑝</ci><ci>𝜌</ci></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td></tr>
</table>
<p class="ltx_p">must be supplemented. For a given central density, those equations
then yield a unique stellar model with radius <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p1.m3" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math> and mass <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p1.m4" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math>.
Usually one introduces the mass function <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p1.m5" class="ltx_Math" alttext="m" display="inline"><ci>𝑚</ci></math> via</p>
<table id="S2.E5" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E5.m1" class="ltx_Math" alttext="e^{-2\lambda}=1-\frac{2m(r)}{r}" display="block"><apply><eq/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><minus/><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply></apply></apply><apply><minus/><cn type="integer">1</cn><apply><divide/><apply><times/><cn type="integer">2</cn><ci>𝑚</ci><ci>𝑟</ci></apply><ci>𝑟</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td></tr>
</table>
<p class="ltx_p">in order to replace the metric function <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p1.m6" class="ltx_Math" alttext="\lambda" display="inline"><ci>𝜆</ci></math>.</p>
</div>
<div id="S2.SS1.p2" class="ltx_para">
<p class="ltx_p">To obtain the equations that govern the radial oscillations, both
fluid and spacetime variables are perturbed in such a way that the
spherical symmetry of the background body is not violated. If we
define as <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p2.m1" class="ltx_Math" alttext="\delta r(r,t)" display="inline"><apply><times/><ci>𝛿</ci><ci>𝑟</ci><interval closure="open"><ci>𝑟</ci><ci>𝑡</ci></interval></apply></math> the time dependent radial displacement of a
fluid element located at the position <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p2.m2" class="ltx_Math" alttext="r" display="inline"><ci>𝑟</ci></math> in the unperturbed model and
assume a harmonic time dependence</p>
<table id="S2.E6" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E6.m1" class="ltx_Math" alttext="\delta r(r,t)=X(r)e^{i\omega t}\;," display="block"><apply><eq/><apply><times/><ci>𝛿</ci><ci>𝑟</ci><interval closure="open"><ci>𝑟</ci><ci>𝑡</ci></interval></apply><apply><times/><ci>𝑋</ci><ci>𝑟</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><ci>𝑖</ci><ci>𝜔</ci><ci>𝑡</ci></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td></tr>
</table>
<p class="ltx_p">we obtain the following equation describing the radial
oscillations</p>
<table id="A1.EGx1" class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table">

<tr id="S2.E7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"/>
<td class="ltx_td ltx_align_center ltx_eqn_cell"/>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.Ex1.m3" class="ltx_Math" alttext="\displaystyle C_{s}^{2}X^{\prime\prime}+\left((C_{s}^{2})^{\prime}-Z+4\pi r%&#10;\gamma pe^{2\lambda}-\nu^{\prime}\right)X^{\prime}" display="inline"><apply><plus/><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐶</ci><ci>𝑠</ci></apply><cn type="integer">2</cn></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑋</ci><ci>′′</ci></apply></apply><apply><times/><apply><minus/><apply><plus/><apply><minus/><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐶</ci><ci>𝑠</ci></apply><cn type="integer">2</cn></apply><ci>′</ci></apply><ci>𝑍</ci></apply><apply><times/><cn type="integer">4</cn><ci>𝜋</ci><ci>𝑟</ci><ci>𝛾</ci><ci>𝑝</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply></apply></apply></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜈</ci><ci>′</ci></apply></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑋</ci><ci>′</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td></tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"/>
<td class="ltx_td ltx_align_center ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.Ex2.m2" class="ltx_Math" alttext="\displaystyle+" display="inline"><plus/></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.Ex2.m3" class="ltx_Math" alttext="\displaystyle\left[2(\nu^{\prime})^{2}+{2m\over r^{3}}e^{2\lambda}-Z^{\prime}-%&#10;4\pi(\rho+p)Zre^{2\lambda}+\omega^{2}e^{2\lambda-2\nu}\right]X" display="inline"><apply><times/><apply><csymbol cd="latexml">delimited-[]</csymbol><apply><plus/><apply><minus/><apply><plus/><apply><times/><cn type="integer">2</cn><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜈</ci><ci>′</ci></apply><cn type="integer">2</cn></apply></apply><apply><times/><apply><divide/><apply><times/><cn type="integer">2</cn><ci>𝑚</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">3</cn></apply></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply></apply></apply></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑍</ci><ci>′</ci></apply><apply><times/><cn type="integer">4</cn><ci>𝜋</ci><apply><plus/><ci>𝜌</ci><ci>𝑝</ci></apply><ci>𝑍</ci><ci>𝑟</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply></apply></apply></apply><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><minus/><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply><apply><times/><cn type="integer">2</cn><ci>𝜈</ci></apply></apply></apply></apply></apply></apply><ci>𝑋</ci></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
<tr class="ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"/>
<td class="ltx_td ltx_align_center ltx_eqn_cell"/>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E7.m3" class="ltx_Math" alttext="\displaystyle=0\;," display="inline"><apply><eq/><csymbol cd="latexml">absent</csymbol><cn type="integer">0</cn></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/></tr>
</table>
<p class="ltx_p">where <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p2.m3" class="ltx_Math" alttext="C_{s}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐶</ci><ci>𝑠</ci></apply></math> is the sound speed, which is calculated from the
unperturbed background for a specific equation of state</p>
<table id="S2.E8" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E8.m1" class="ltx_Math" alttext="C_{s}^{2}={dp\over d\rho}\;," display="block"><apply><eq/><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐶</ci><ci>𝑠</ci></apply><cn type="integer">2</cn></apply><apply><divide/><apply><times/><ci>𝑑</ci><ci>𝑝</ci></apply><apply><times/><ci>𝑑</ci><ci>𝜌</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td></tr>
</table>
<p class="ltx_p">and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p2.m4" class="ltx_Math" alttext="\gamma" display="inline"><ci>𝛾</ci></math> is the adiabatic index, which, for adiabatic oscillations,
is related to the sound speed through</p>
<table id="S2.E9" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E9.m1" class="ltx_Math" alttext="\gamma={\rho+p\over p}{dp\over d\rho}\;." display="block"><apply><eq/><ci>𝛾</ci><apply><times/><apply><divide/><apply><plus/><ci>𝜌</ci><ci>𝑝</ci></apply><ci>𝑝</ci></apply><apply><divide/><apply><times/><ci>𝑑</ci><ci>𝑝</ci></apply><apply><times/><ci>𝑑</ci><ci>𝜌</ci></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td></tr>
</table>
<p class="ltx_p">Finally</p>
<table id="S2.E10" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E10.m1" class="ltx_Math" alttext="Z(r)=C_{s}^{2}\left(\nu^{\prime}-{2\over r}\right)\;." display="block"><apply><eq/><apply><times/><ci>𝑍</ci><ci>𝑟</ci></apply><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐶</ci><ci>𝑠</ci></apply><cn type="integer">2</cn></apply><apply><minus/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜈</ci><ci>′</ci></apply><apply><divide/><cn type="integer">2</cn><ci>𝑟</ci></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td></tr>
</table>
<p class="ltx_p">The boundary condition at the center is that</p>
<table id="S2.E11" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E11.m1" class="ltx_Math" alttext="\delta r(r=0)=0\;," display="block"><cerror><csymbol cd="ambiguous">fragments</csymbol><csymbol cd="unknown">δ</csymbol><csymbol cd="unknown">r</csymbol><cerror><csymbol cd="ambiguous">fragments</csymbol><ci>(</ci><csymbol cd="unknown">r</csymbol><eq/><cn type="integer">0</cn><ci>)</ci></cerror><eq/><cn type="integer">0</cn><ci>,</ci></cerror></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td></tr>
</table>
<p class="ltx_p">while at the surface, the Lagrangian variation of the pressure
should vanish, i.e.
</p>
<table id="S2.E12" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E12.m1" class="ltx_Math" alttext="\Delta p=0\;." display="block"><apply><eq/><apply><times/><ci>Δ</ci><ci>𝑝</ci></apply><cn type="integer">0</cn></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td></tr>
</table>
<p class="ltx_p">This leads to the condition</p>
<table id="S2.E13" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E13.m1" class="ltx_Math" alttext="\gamma p\zeta(r)^{\prime}=0\;,\quad\mbox{\rm where}\quad\zeta=r^{2}e^{-\nu}X\;." display="block"><apply><csymbol cd="ambiguous">formulae-sequence</csymbol><apply><eq/><apply><times/><ci>𝛾</ci><ci>𝑝</ci><ci>𝜁</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><ci>′</ci></apply></apply><list><cn type="integer">0</cn><mtext>where</mtext></list></apply><apply><eq/><ci>𝜁</ci><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><minus/><ci>𝜈</ci></apply></apply><ci>𝑋</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td></tr>
</table>
<p class="ltx_p">Equation (<a href="#S2.E7" title="(7) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>) together with the boundary conditions
(<a href="#S2.E11" title="(11) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">11</span></a>) and (<a href="#S2.E13" title="(13) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">13</span></a>) form a self-adjoint
boundary value problem for <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p2.m5" class="ltx_Math" alttext="\omega^{2}" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply></math>.</p>
</div>
<div id="S2.SS1.p3" class="ltx_para">
<p class="ltx_p">As an alternative, the master equation can be written in the variable
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p3.m1" class="ltx_Math" alttext="\zeta" display="inline"><ci>𝜁</ci></math> to yield Equation (26.6) of <cite class="ltx_cite ltx_citemacro_cite">Misner, Thorne &amp; Wheeler (<a href="#bib.bib24" title="" class="ltx_ref">1973</a>)</cite>, which explicitely
shows its self-adjoint nature:</p>
<table id="S2.E14" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E14.m1" class="ltx_Math" alttext="0=\frac{d}{dr}\left(P\frac{d\zeta}{dr}\right)+\left(Q+\omega^{2}W\right)\zeta\;," display="block"><apply><eq/><cn type="integer">0</cn><apply><plus/><apply><times/><apply><divide/><ci>𝑑</ci><apply><times/><ci>𝑑</ci><ci>𝑟</ci></apply></apply><apply><times/><ci>𝑃</ci><apply><divide/><apply><times/><ci>𝑑</ci><ci>𝜁</ci></apply><apply><times/><ci>𝑑</ci><ci>𝑟</ci></apply></apply></apply></apply><apply><times/><apply><plus/><ci>𝑄</ci><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><ci>𝑊</ci></apply></apply><ci>𝜁</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td></tr>
</table>
<p class="ltx_p">with</p>
<table id="A1.EGx2" class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table">

<tr id="S2.E15" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E15.m1" class="ltx_Math" alttext="\displaystyle r^{2}W" display="inline"><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply><ci>𝑊</ci></apply></math></td>
<td class="ltx_td ltx_align_center ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E15.m2" class="ltx_Math" alttext="\displaystyle=" display="inline"><eq/></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E15.m3" class="ltx_Math" alttext="\displaystyle\left(\rho+p\right)e^{3\lambda+\nu}" display="inline"><apply><times/><apply><plus/><ci>𝜌</ci><ci>𝑝</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><plus/><apply><times/><cn type="integer">3</cn><ci>𝜆</ci></apply><ci>𝜈</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td></tr>
<tr id="S2.E16" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E16.m1" class="ltx_Math" alttext="\displaystyle r^{2}P" display="inline"><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply><ci>𝑃</ci></apply></math></td>
<td class="ltx_td ltx_align_center ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E16.m2" class="ltx_Math" alttext="\displaystyle=" display="inline"><eq/></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E16.m3" class="ltx_Math" alttext="\displaystyle\gamma p\,e^{\lambda+3\nu}" display="inline"><apply><times/><ci>𝛾</ci><ci>𝑝</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><plus/><ci>𝜆</ci><apply><times/><cn type="integer">3</cn><ci>𝜈</ci></apply></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td></tr>
<tr id="S2.E17" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E17.m1" class="ltx_Math" alttext="\displaystyle r^{2}Q" display="inline"><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply><ci>𝑄</ci></apply></math></td>
<td class="ltx_td ltx_align_center ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E17.m2" class="ltx_Math" alttext="\displaystyle=" display="inline"><eq/></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E17.m3" class="ltx_Math" alttext="\displaystyle e^{\lambda+3\nu}\left(\rho+p\right)\left((\nu^{\prime})^{2}+4%&#10;\frac{\nu^{\prime}}{r}-8\pi e^{2\lambda}p\right)\;." display="inline"><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><plus/><ci>𝜆</ci><apply><times/><cn type="integer">3</cn><ci>𝜈</ci></apply></apply></apply><apply><plus/><ci>𝜌</ci><ci>𝑝</ci></apply><apply><minus/><apply><plus/><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜈</ci><ci>′</ci></apply><cn type="integer">2</cn></apply><apply><times/><cn type="integer">4</cn><apply><divide/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜈</ci><ci>′</ci></apply><ci>𝑟</ci></apply></apply></apply><apply><times/><cn type="integer">8</cn><ci>𝜋</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑒</ci><apply><times/><cn type="integer">2</cn><ci>𝜆</ci></apply></apply><ci>𝑝</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td></tr>
</table>
<p class="ltx_p">At the origin, we have <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p3.m2" class="ltx_Math" alttext="\zeta(r=0)=0" display="inline"><cerror><csymbol cd="ambiguous">fragments</csymbol><csymbol cd="unknown">ζ</csymbol><cerror><csymbol cd="ambiguous">fragments</csymbol><ci>(</ci><csymbol cd="unknown">r</csymbol><eq/><cn type="integer">0</cn><ci>)</ci></cerror><eq/><cn type="integer">0</cn></cerror></math>, and at the surface, the boundary
condition is also given by Eq. (<a href="#S2.E13" title="(13) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">13</span></a>).</p>
</div>
<div id="S2.SS1.p4" class="ltx_para">
<p class="ltx_p">Since in both the general relativistic and in Newtonian theory, the
oscillation problem is described by a Sturm-Liouville boundary value
problem, the mathematical features that are known for the Newtonian
problem (see <cite class="ltx_cite ltx_citemacro_cite">Ledoux &amp; Walraven (<a href="#bib.bib21" title="" class="ltx_ref">1958</a>)</cite>) also apply to the general relativistic case,
i.e. the frequency spectrum is discrete, there are <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p4.m1" class="ltx_Math" alttext="n" display="inline"><ci>𝑛</ci></math> nodes between
the center and the surface of the eigenfunction of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p4.m2" class="ltx_Math" alttext="n" display="inline"><ci>𝑛</ci></math>th mode, and
the eigenfunctions are orthogonal.</p>
</div>
<div id="S2.SS1.p5" class="ltx_para">
<p class="ltx_p">Since <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m1" class="ltx_Math" alttext="\omega" display="inline"><ci>𝜔</ci></math> is real for <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m2" class="ltx_Math" alttext="\omega^{2}&gt;0" display="inline"><apply><gt/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><cn type="integer">0</cn></apply></math>, the solution is purely
oscillatory. However for <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m3" class="ltx_Math" alttext="\omega^{2}&lt;0" display="inline"><apply><lt/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><cn type="integer">0</cn></apply></math>, the frequency <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m4" class="ltx_Math" alttext="\omega" display="inline"><ci>𝜔</ci></math> is
imaginary, which corresponds to an exponentially growing solution.
This means that for negative values of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m5" class="ltx_Math" alttext="\omega^{2}" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply></math>, we have unstable
radial oscillations. For neutron stars, it is the fundamental mode
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m6" class="ltx_Math" alttext="\omega_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜔</ci><cn type="integer">0</cn></apply></math> which becomes imaginary at central densities <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m7" class="ltx_Math" alttext="\rho_{c}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply></math>
larger than the critical density <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m8" class="ltx_Math" alttext="\rho_{crit}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><apply><times/><ci>𝑐</ci><ci>𝑟</ci><ci>𝑖</ci><ci>𝑡</ci></apply></apply></math> for which the total
stellar mass <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m9" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m10" class="ltx_Math" alttext="\rho_{c}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply></math> is maximal. In this case,
the star will ultimately collapse to a black hole. For
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m11" class="ltx_Math" alttext="\rho_{c}=\rho_{crit}" display="inline"><apply><eq/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><apply><times/><ci>𝑐</ci><ci>𝑟</ci><ci>𝑖</ci><ci>𝑡</ci></apply></apply></apply></math>, there frequency of the fundamental mode
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS1.p5.m12" class="ltx_Math" alttext="\omega_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜔</ci><cn type="integer">0</cn></apply></math> must vanish. The higher modes become unstable for higher
densities than for maximum mass models. For realistic equations of
state, there are several regions in the mass-central-density curve,
which are unstable. On the neutron star branch, there is another
instability point on the low density side, where the star can become
unstable with respect to explosion. This point limits the minimal mass
of a neutron star.</p>
</div>
</div>
<div id="S2.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection"><span class="ltx_tag ltx_tag_subsection">2.2 </span>The numerical methods</h3>

<div id="S2.SS2.p1" class="ltx_para">
<p class="ltx_p">Since the radial perturbation problem is an old one, various
methods have been used to estimate the radial mode frequencies for
a given equations of state. A numerical integration scheme, which
is similar to what we will describe here, has widely been used,
while a Rayleigh-Ritz variational technique has also been used in
the early times, see <cite class="ltx_cite ltx_citemacro_cite">Bardeen et al. (<a href="#bib.bib4" title="" class="ltx_ref">1966</a>)</cite> for details.</p>
</div>
<div id="S2.SS2.p2" class="ltx_para">
<p class="ltx_p">Given the discrepancies existing in the literature, we have
derived the results via two different numerical methods.</p>
</div>
<div id="S2.SS2.p3" class="ltx_para">
<p class="ltx_p">The <span class="ltx_text ltx_font_italic">first method</span> is called in numerical analysis the <span class="ltx_text ltx_font_italic">shooting method</span>. In this case, one starts the integration for a
trial value of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p3.m1" class="ltx_Math" alttext="\omega^{2}" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply></math> and a given set of initial values of
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p3.m2" class="ltx_Math" alttext="X(r=0)" display="inline"><cerror><csymbol cd="ambiguous">fragments</csymbol><csymbol cd="unknown">X</csymbol><cerror><csymbol cd="ambiguous">fragments</csymbol><ci>(</ci><csymbol cd="unknown">r</csymbol><eq/><cn type="integer">0</cn><ci>)</ci></cerror></cerror></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p3.m3" class="ltx_Math" alttext="X^{\prime}(r=0)" display="inline"><cerror><csymbol cd="ambiguous">fragments</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑋</ci><ci>′</ci></apply><cerror><csymbol cd="ambiguous">fragments</csymbol><ci>(</ci><csymbol cd="unknown">r</csymbol><eq/><cn type="integer">0</cn><ci>)</ci></cerror></cerror></math> which satisfy at the center the boundary
condition (<a href="#S2.E11" title="(11) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">11</span></a>) and integrates towards the surface. The
discrete values of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p3.m4" class="ltx_Math" alttext="\omega^{2}" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply></math> for which the boundary condition
(<a href="#S2.E13" title="(13) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">13</span></a>) is satisfied are the eigenfrequencies of the radial
perturbations.</p>
</div>
<div id="S2.SS2.p4" class="ltx_para">
<p class="ltx_p">We will apply this method to equation (<a href="#S2.E14" title="(14) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">14</span></a>), but we first
transform it into two first order differential equations. By
introducing</p>
<table id="S2.E18" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E18.m1" class="ltx_Math" alttext="\eta=P\zeta^{\prime}\;," display="block"><apply><eq/><ci>𝜂</ci><apply><times/><ci>𝑃</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜁</ci><ci>′</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td></tr>
</table>
<p class="ltx_p">we obtain</p>
<table id="A1.EGx3" class="ltx_equationgroup ltx_eqn_eqnarray ltx_eqn_table">

<tr id="S2.E19" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E19.m1" class="ltx_Math" alttext="\displaystyle\frac{d\zeta}{dr}" display="inline"><apply><divide/><apply><times/><ci>𝑑</ci><ci>𝜁</ci></apply><apply><times/><ci>𝑑</ci><ci>𝑟</ci></apply></apply></math></td>
<td class="ltx_td ltx_align_center ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E19.m2" class="ltx_Math" alttext="\displaystyle=" display="inline"><eq/></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E19.m3" class="ltx_Math" alttext="\displaystyle\frac{\eta}{P}" display="inline"><apply><divide/><ci>𝜂</ci><ci>𝑃</ci></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td></tr>
<tr id="S2.E20" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E20.m1" class="ltx_Math" alttext="\displaystyle\frac{d\eta}{dr}" display="inline"><apply><divide/><apply><times/><ci>𝑑</ci><ci>𝜂</ci></apply><apply><times/><ci>𝑑</ci><ci>𝑟</ci></apply></apply></math></td>
<td class="ltx_td ltx_align_center ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E20.m2" class="ltx_Math" alttext="\displaystyle=" display="inline"><eq/></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E20.m3" class="ltx_Math" alttext="\displaystyle-\left(\omega^{2}W+Q\right)\zeta\;." display="inline"><apply><minus/><apply><times/><apply><plus/><apply><times/><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><ci>𝑊</ci></apply><ci>𝑄</ci></apply><ci>𝜁</ci></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td></tr>
</table>
<p class="ltx_p">Through Taylor expansion, we find that close to the origin we have
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p4.m1" class="ltx_Math" alttext="\zeta(r)=\zeta_{0}\,r^{3}+{\cal O}(r^{5})" display="inline"><apply><eq/><apply><times/><ci>𝜁</ci><ci>𝑟</ci></apply><apply><plus/><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜁</ci><cn type="integer">0</cn></apply><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">3</cn></apply></apply><apply><times/><ci>𝒪</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">5</cn></apply></apply></apply></apply></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p4.m2" class="ltx_Math" alttext="\eta(r)=\eta_{0}+{\cal O}(r^{2})" display="inline"><apply><eq/><apply><times/><ci>𝜂</ci><ci>𝑟</ci></apply><apply><plus/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜂</ci><cn type="integer">0</cn></apply><apply><times/><ci>𝒪</ci><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝑟</ci><cn type="integer">2</cn></apply></apply></apply></apply></math>. From equation (<a href="#S2.E19" title="(19) ‣ 2.2 The numerical methods ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">19</span></a>) it then follows that the
leading order coefficients are related by <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p4.m3" class="ltx_Math" alttext="3\zeta_{0}=\eta_{0}/P(0)" display="inline"><apply><eq/><apply><times/><cn type="integer">3</cn><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜁</ci><cn type="integer">0</cn></apply></apply><apply><times/><apply><divide/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜂</ci><cn type="integer">0</cn></apply><ci>𝑃</ci></apply><cn type="integer">0</cn></apply></apply></math>.
Choosing <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p4.m4" class="ltx_Math" alttext="\eta_{0}=1" display="inline"><apply><eq/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜂</ci><cn type="integer">0</cn></apply><cn type="integer">1</cn></apply></math>, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p4.m5" class="ltx_Math" alttext="\zeta_{0}=1/(3P(0))" display="inline"><apply><eq/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜁</ci><cn type="integer">0</cn></apply><apply><divide/><cn type="integer">1</cn><apply><times/><cn type="integer">3</cn><ci>𝑃</ci><cn type="integer">0</cn></apply></apply></apply></math>, which gives us
the initial values for the integration.</p>
</div>
<div id="S2.SS2.p5" class="ltx_para">
<p class="ltx_p">The <span class="ltx_text ltx_font_italic">second method</span> is based on finite differencing the radial
perturbation equation (<a href="#S2.E7" title="(7) ‣ 2.1 The radial equations ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>) using second order accurate
schemes for the spatial derivatives. The coefficients of the equation
are calculated for a certain number of, say, <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m1" class="ltx_Math" alttext="N" display="inline"><ci>𝑁</ci></math> grid points. In this
way a matrix equation of the form</p>
<table id="S2.E21" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E21.m1" class="ltx_Math" alttext="\left(A-\omega^{2}_{n}\mbox{I}\right)y=0\;,\qquad 0\leq n\leq N" display="block"><apply><csymbol cd="ambiguous">formulae-sequence</csymbol><apply><eq/><apply><times/><apply><minus/><ci>𝐴</ci><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><ci>𝑛</ci></apply><mtext>I</mtext></apply></apply><ci>𝑦</ci></apply><cn type="integer">0</cn></apply><apply><and/><apply><leq/><cn type="integer">0</cn><ci>𝑛</ci></apply><apply><leq/><share href="#S2.E21.m1.2.sh1"/><ci>𝑁</ci></apply></apply></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td></tr>
</table>
<p class="ltx_p">is constructed. <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m2" class="ltx_Math" alttext="A" display="inline"><ci>𝐴</ci></math> is the tridiagonal matrix of the coefficients, I
is the identity matrix, <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m3" class="ltx_Math" alttext="\omega^{2}_{n}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><ci>𝑛</ci></apply></math> is the squared frequency of the
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m4" class="ltx_Math" alttext="n" display="inline"><ci>𝑛</ci></math>th mode, and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m5" class="ltx_Math" alttext="y" display="inline"><ci>𝑦</ci></math> is the vector with the unknown values of the
eigenfunction of the specific mode at the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m6" class="ltx_Math" alttext="N" display="inline"><ci>𝑁</ci></math> grid points. The
homogeneous linear equation (<a href="#S2.E21" title="(21) ‣ 2.2 The numerical methods ‣ 2 Equations and numerical methods ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">21</span></a>) has a nontrivial solution
only if the determinant of the coefficient matrix is equal to zero, i.e.</p>
<table id="S2.E22" class="ltx_equation ltx_eqn_table">

<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"/>
<td class="ltx_eqn_cell ltx_align_center"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.E22.m1" class="ltx_Math" alttext="\det|A-\omega^{2}_{n}\mbox{I}|=0\;." display="block"><apply><eq/><apply><determinant/><apply><abs/><apply><minus/><ci>𝐴</ci><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><ci>𝑛</ci></apply><mtext>I</mtext></apply></apply></apply></apply><cn type="integer">0</cn></apply></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"/>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td></tr>
</table>
<p class="ltx_p">This means that <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m7" class="ltx_Math" alttext="\omega^{2}_{n}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜔</ci><cn type="integer">2</cn></apply><ci>𝑛</ci></apply></math> are the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m8" class="ltx_Math" alttext="N" display="inline"><ci>𝑁</ci></math> eigenvalues of the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m9" class="ltx_Math" alttext="N\times N" display="inline"><apply><times/><ci>𝑁</ci><ci>𝑁</ci></apply></math>-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" id="S2.SS2.p5.m10" class="ltx_Math" alttext="A" display="inline"><ci>𝐴</ci></math>. Their numerical evaluation has been achieved using the
routines F01AKF and F02APF of the Nag library.</p>
</div>
<div id="S2.SS2.p6" class="ltx_para">
<p class="ltx_p">In this way one can calculate hundreds of radial eigenvalues for a
specific stellar model in a single run. This method is more time
consuming, but one avoids to search for each eigenvalue separately. In
numerical analysis, this method is referred to as <span class="ltx_text ltx_font_italic">Numerov method</span>.</p>
</div>
<div id="S2.SS2.p7" class="ltx_para">
<p class="ltx_p">Using both methods, we have calculated for each stellar model a large
number of eigenvalues, though in the tables of the Appendix, we list
only the three lowest ones. A further check of consistency is that for
each EOS, the maximum mass model must yield zero frequency for the
first mode. This is indeed the case as is it not for the results of
GL.</p>
</div>
</div>
</div>
<div id="S3" class="ltx_section">
<h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">3 </span>Results</h2>

<div id="S3.p1" class="ltx_para">
<p class="ltx_p">Although in principle we could compute the eigenfrequencies up to
arbitrary precision, this would make no sense, since the overall
accuracy of the frequencies is not limited by the machine precision,
but by the number of tabulated values of the equation of state. For
the construction of the stellar background model, one therefore has to
interpolate between the given points. As it turns out, different
interpolation scheme can yield different mode frequencies. Even though
the bulk parameters of the stellar models are not very sensitive to
the actual interpolation scheme, it is the profile of the sound speed,
or equivalently, the adiabatic index which enters into the oscillation
equations, and this quantity is highly sensitive to the interpolation
scheme, especially in the regions where the EOS changes quite
abruptly, such as, for instance, at the neutron drip point. Since
this region lies in the low pressure regime and is therefore located
close to the surface of the neutron star, it has a quite large
influence on the modes because their amplitudes peak at the surface.
From trying different interpolation schemes, such as linear
logarithmic interpolation or spline interpolation, we find that the
frequencies may vary up to about three per cent. We therefore tabulate
our value with only two significant digits. Only for the polytropic
equations of state, we include three significant digits, since in this
case, the equation of state is analytic, and we do not have to rely on
interpolation.</p>
</div>
<div id="S3.p2" class="ltx_para">
<p class="ltx_p">VC have given tabulated values for EOS D and EOS N. For EOS D, our
results agree with theirs, however, for EOS N we find a
quite significant discrepancy for the stellar parameters like mass and
radius, especially in the high density regime. For instance, for
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m1" class="ltx_Math" alttext="\rho_{c}=2\times 10^{15}\;" display="inline"><apply><eq/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><times/><cn type="integer">2</cn><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></apply></math>g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m2" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math>, they find a mass of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m3" class="ltx_Math" alttext="M=2.563M_{\odot}" display="inline"><apply><eq/><ci>𝑀</ci><apply><times/><cn type="float">2.563</cn><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑀</ci><csymbol cd="latexml">direct-product</csymbol></apply></apply></apply></math>, whereas we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m4" class="ltx_Math" alttext="M=2.621M_{\odot}" display="inline"><apply><eq/><ci>𝑀</ci><apply><times/><cn type="float">2.621</cn><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑀</ci><csymbol cd="latexml">direct-product</csymbol></apply></apply></apply></math>. Since also GL
find the former value, it seems that both GL and VC have used the
tabulated version of the EOS N provided by <cite class="ltx_cite ltx_citemacro_cite">Lindblom &amp; Detweiler (<a href="#bib.bib22" title="" class="ltx_ref">1983</a>)</cite>. If we also
use this table, we, again, agree with VC, both
in the stellar parameters and the radial oscillation frequencies.
However, we have access to a table with a larger number of values
(about twice a many in the density range from <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m5" class="ltx_Math" alttext="10^{14}\;" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">14</cn></apply></math>g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m6" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math> to
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m7" class="ltx_Math" alttext="10^{16}\;" display="inline"><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">16</cn></apply></math>g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p2.m8" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math>), which yields the latter value. In Table <a href="#A1.T10" title="Table 10 ‣ Appendix A Results for various equations of state ‣ Radial oscillations of relativistic stars" class="ltx_ref"><span class="ltx_text ltx_ref_tag">10</span></a>,
we give the frequencies obtained with the more refined EOS, which,
especially for the fundamental mode, are quite different from the
values of VC. These discrepancies show very drastically that the results
are quite sensitive to the number of tabulated values of a given EOS.</p>
</div>
<div id="S3.F1" class="ltx_figure"><img src="x1.png" id="S3.F1.g1" class="ltx_graphics ltx_centering" width="390" height="512" alt=""/>
<div class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>We show the first five radial modes as a function of
the central energy density. The frequency of the fundamental
mode goes to zero at a density of about <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.F1.m3" class="ltx_Math" alttext="2.35\times 10^{15}\," display="inline"><apply><times/><cn type="float">2.35</cn><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math>g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.F1.m4" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math>, which indicates the onset of radial
instability with respect to collapse to a black hole. The arrows
indicate the avoided crossings between the different modes.</div>
</div>
<div id="S3.p3" class="ltx_para">
<p class="ltx_p">In Fig. 1, we show the five radial modes as a function of the central
density for the quite recent EOS APR1 (<cite class="ltx_cite ltx_citemacro_cite">Akmal et al. (<a href="#bib.bib1" title="" class="ltx_ref">1998</a>)</cite>). It is clearly
discernible that the fundamental mode becomes unstable at central
densities above <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p3.m1" class="ltx_Math" alttext="2.35\times 10^{15}\," display="inline"><apply><times/><cn type="float">2.35</cn><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math>g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p3.m2" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math>. The instability point
corresponds to a stellar model with the maximal allowed mass of <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p3.m3" class="ltx_Math" alttext="2.38M_{\odot}" display="inline"><apply><times/><cn type="float">2.38</cn><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑀</ci><csymbol cd="latexml">direct-product</csymbol></apply></apply></math> and a radius of 10.77 km.</p>
</div>
<div id="S3.p4" class="ltx_para">
<p class="ltx_p">Another prominent feature is the occurrence of a series of avoided
crossings between the various modes. This peculiarity has been
observed in previous calculations (<cite class="ltx_cite ltx_citemacro_cite">Gondek et al. (<a href="#bib.bib17" title="" class="ltx_ref">1997</a>)</cite>), and has been
extensively discussed by <cite class="ltx_cite ltx_citemacro_cite">Gondek &amp; Zdunik (<a href="#bib.bib18" title="" class="ltx_ref">1999</a>)</cite>. It should be noted that those
avoided crossings do not appear when a polytropic EOS is used (in this
case, one also does not have the second instability point at the low
density region), but it is a characteristic of realistic equations of
state.</p>
</div>
<div id="S3.p5" class="ltx_para">
<p class="ltx_p">The phenomenon of avoided crossings is also known to appear for other
types of oscillations. Depending on the stellar models, there can be
avoided crossings between <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p5.m1" class="ltx_Math" alttext="g" display="inline"><ci>𝑔</ci></math>-modes and <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p5.m2" class="ltx_Math" alttext="p" display="inline"><ci>𝑝</ci></math>-modes. Furthermore,
<cite class="ltx_cite ltx_citemacro_cite">Anderson et al. (<a href="#bib.bib2" title="" class="ltx_ref">1996</a>)</cite> have reported it to occur between the <math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p5.m3" class="ltx_Math" alttext="f" display="inline"><ci>𝑓</ci></math>-mode and the
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S3.p5.m4" class="ltx_Math" alttext="w" display="inline"><ci>𝑤</ci></math>-modes. Also in rotating stars a similar phenomenon shows up for the
quasi-radial modes when their oscillation frequencies are plotted as a
function of the rotational frequency (<cite class="ltx_cite ltx_citemacro_cite">Clement (<a href="#bib.bib12" title="" class="ltx_ref">1986</a>)</cite>, <cite class="ltx_cite ltx_citemacro_cite">Yoshida &amp; Eriguchi (<a href="#bib.bib32" title="" class="ltx_ref">1999</a>)</cite>).</p>
</div>
<div id="S3.p6" class="ltx_para">
<p class="ltx_p">All these cases have in common that there usually exist two or more
families of modes, which arise from different physical origins.
However, since they are described by a common set of equations, a
particular frequency can only correspond to one single mode.
Therefore, if the frequencies of two modes belonging to different
families start to approach each other, they eventually have to repel
each other before they come too close. This goes along with the modes
exchanging their “family membership”.</p>
</div>
<div id="S3.p7" class="ltx_para">
<p class="ltx_p">The radial oscillation modes, too, can be divided into two more or
less independent families. According to <cite class="ltx_cite ltx_citemacro_cite">Gondek &amp; Zdunik (<a href="#bib.bib18" title="" class="ltx_ref">1999</a>)</cite>, one family lives
predominately in the high density core of the neutron star and the
other in the low density envelope. The two regions are divided by a
“wall” in the adiabatic index which results from the abrupt change
in the stiffness of the matter at the neutron drip point (c.f. Fig.2
of <cite class="ltx_cite ltx_citemacro_cite">Gondek &amp; Zdunik (<a href="#bib.bib18" title="" class="ltx_ref">1999</a>)</cite>). This wall effect is present for any realistic EOS,
since it is associated with the neutron drip point, which belongs
to the low pressure regime and is the same for all EOS.</p>
</div>
<div id="S3.p8" class="ltx_para">
<p class="ltx_p">In a model problem, <cite class="ltx_cite ltx_citemacro_cite">Gondek &amp; Zdunik (<a href="#bib.bib18" title="" class="ltx_ref">1999</a>)</cite> have decoupled both families, and in
this case, both spectra show real crossings, when plotted on top over
each other. When the coupling is brought back, the crossings vanish
and the usual avoided crossing picture reemerges.</p>
</div>
</div>
<div id="S4" class="ltx_section">
<h2 class="ltx_title ltx_title_section"><span class="ltx_tag ltx_tag_section">4 </span>Summary</h2>

<div id="S4.p1" class="ltx_para">
<p class="ltx_p">We have presented updated results for radial oscillations of neutron
stars using a quite exhaustive list of currently available equations
of state, including some very recent ones. For most equations of state
we significantly disagree with the values given by <cite class="ltx_cite ltx_citemacro_cite">Glass &amp; Lindblom (<a href="#bib.bib15" title="" class="ltx_ref">1983</a>)</cite>. We
have obtained our results by means of two different numerical methods,
which agree up to arbitrary precision.</p>
</div>
<div id="S4.p2" class="ltx_para">
<p class="ltx_p">Furthermore, we have checked that our numerical codes yield a zero
frequency mode located exactly at both instability points, which are
characterized by the local extrema in the mass-density curve. Here, we
also obtain full agreement. The overall accuracy, however, is limited
by the number of tabulated points for a given equation of state. Here,
different numerical interpolation schemes may yield variations in the
frequencies up to about three per cent.</p>
</div>
<div id="S4.p3" class="ltx_para">
<p class="ltx_p">Our results agree with the previous results of <cite class="ltx_cite ltx_citemacro_cite">Väth &amp; Chanmugam (<a href="#bib.bib29" title="" class="ltx_ref">1992</a>)</cite>. However,
we use a more complete table for the EOS N (<cite class="ltx_cite ltx_citemacro_cite">Serot (<a href="#bib.bib28" title="" class="ltx_ref">1979</a>)</cite>), which
significantly alters the values one obtains when the table provided by
<cite class="ltx_cite ltx_citemacro_cite">Lindblom &amp; Detweiler (<a href="#bib.bib22" title="" class="ltx_ref">1983</a>)</cite> is used.</p>
</div>
<div id="S4.p4" class="ltx_para">
<p class="ltx_p">We have repeated the calculations for the equations of state already
used by GL, and we have corrected their given values. In
addition, we have included a large number of more recent equations of
state. Since most of the present non-linear evolution codes use
polytropic equations of state, we also have tabulated the mode
frequencies for three different values of the polytropic index <math xmlns="http://www.w3.org/1998/Math/MathML" id="S4.p4.m1" class="ltx_Math" alttext="n" display="inline"><ci>𝑛</ci></math>.</p>
</div>
</div>
<div class="ltx_acknowledgements">
<h6 class="ltx_title ltx_title_acknowledgements">Acknowledgements.</h6>
J.R. was supported by the Deutsche Forschungsgemeinschaft through
SFB 382 and the Marie Curie Fellowship No. HPMF-CT-1999-00364.

</div>
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Wu X., Müther H., Soffel M., Herold H., Ruder H., 1991, A&amp;A 246, 411

</span></li>
<li id="bib.bib32" class="ltx_bibitem"><span class="ltx_bibtag ltx_role_refnum">Yoshida &amp; Eriguchi  (1999)</span>
<span class="ltx_bibblock">
Yoshida S., Eriguchi Y., astro-ph/9908359

</span></li>
</ul>
</div>
<div id="A1" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix"><span class="ltx_tag ltx_tag_appendix">Appendix A </span>Results for various equations of state</h2>

<div id="A1.p1" class="ltx_para">
<p class="ltx_p">This appendix provides the numerical data for the radial mode
frequencies of 17 realistic and 3 polytropic EOS. We present the data
in the form of one table for each EOS. In each table we list the
central density <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.p1.m1" class="ltx_Math" alttext="\rho_{c}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply></math>, the radius <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.p1.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math>, and the mass <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.p1.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math> of the
stellar model, and the frequencies <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.p1.m4" class="ltx_Math" alttext="\nu_{n}=\omega_{n}/(2\pi)" display="inline"><apply><eq/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><ci>𝑛</ci></apply><apply><divide/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜔</ci><ci>𝑛</ci></apply><apply><times/><cn type="integer">2</cn><ci>𝜋</ci></apply></apply></apply></math> of the
first three radial modes. We also include one stellar model above the
stability limit. For this case, we give the <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.p1.m5" class="ltx_Math" alttext="e" display="inline"><ci>𝑒</ci></math>-folding time in <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.p1.m6" class="ltx_Math" alttext="ms" display="inline"><apply><times/><ci>𝑚</ci><ci>𝑠</ci></apply></math>
for the fundamental mode, which is marked by an asterisk.</p>
</div>
<div id="A1.T1" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span>Data for the EOS A (<cite class="ltx_cite ltx_citemacro_cite">Pandharipande (<a href="#bib.bib26" title="" class="ltx_ref">1971</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T1.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.200</td>
<td class="ltx_td ltx_align_center">8.335</td>
<td class="ltx_td ltx_align_center">1.654</td>
<td class="ltx_td ltx_align_center">0.34*</td>
<td class="ltx_td ltx_align_center">7.55</td>
<td class="ltx_td ltx_align_center">11.91</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.100</td>
<td class="ltx_td ltx_align_center">8.373</td>
<td class="ltx_td ltx_align_center">1.654</td>
<td class="ltx_td ltx_align_center">0.28</td>
<td class="ltx_td ltx_align_center">7.58</td>
<td class="ltx_td ltx_align_center">11.95</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.980</td>
<td class="ltx_td ltx_align_center">8.419</td>
<td class="ltx_td ltx_align_center">1.654</td>
<td class="ltx_td ltx_align_center">0.66</td>
<td class="ltx_td ltx_align_center">7.63</td>
<td class="ltx_td ltx_align_center">12.00</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">8.874</td>
<td class="ltx_td ltx_align_center">1.621</td>
<td class="ltx_td ltx_align_center">1.97</td>
<td class="ltx_td ltx_align_center">7.98</td>
<td class="ltx_td ltx_align_center">12.33</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.344</td>
<td class="ltx_td ltx_align_center">9.256</td>
<td class="ltx_td ltx_align_center">1.536</td>
<td class="ltx_td ltx_align_center">2.62</td>
<td class="ltx_td ltx_align_center">8.29</td>
<td class="ltx_td ltx_align_center">12.27</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.995</td>
<td class="ltx_td ltx_align_center">9.479</td>
<td class="ltx_td ltx_align_center">1.447</td>
<td class="ltx_td ltx_align_center">2.94</td>
<td class="ltx_td ltx_align_center">8.46</td>
<td class="ltx_td ltx_align_center">11.88</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.698</td>
<td class="ltx_td ltx_align_center">9.667</td>
<td class="ltx_td ltx_align_center">1.329</td>
<td class="ltx_td ltx_align_center">3.23</td>
<td class="ltx_td ltx_align_center">8.57</td>
<td class="ltx_td ltx_align_center">11.31</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.259</td>
<td class="ltx_td ltx_align_center">9.890</td>
<td class="ltx_td ltx_align_center">1.050</td>
<td class="ltx_td ltx_align_center">3.67</td>
<td class="ltx_td ltx_align_center">8.04</td>
<td class="ltx_td ltx_align_center">10.67</td></tr>
</tbody>
</table>
</div>
<div id="A1.T2" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span>Data for the EOS B (<cite class="ltx_cite ltx_citemacro_cite">Pandharipande (<a href="#bib.bib26" title="" class="ltx_ref">1971</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T2.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">6.100</td>
<td class="ltx_td ltx_align_center">7.024</td>
<td class="ltx_td ltx_align_center">1.413</td>
<td class="ltx_td ltx_align_center">0.32*</td>
<td class="ltx_td ltx_align_center">8.75</td>
<td class="ltx_td ltx_align_center">13.18</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">6.000</td>
<td class="ltx_td ltx_align_center">7.048</td>
<td class="ltx_td ltx_align_center">1.413</td>
<td class="ltx_td ltx_align_center">0.25</td>
<td class="ltx_td ltx_align_center">8.76</td>
<td class="ltx_td ltx_align_center">13.20</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.900</td>
<td class="ltx_td ltx_align_center">7.072</td>
<td class="ltx_td ltx_align_center">1.413</td>
<td class="ltx_td ltx_align_center">0.61</td>
<td class="ltx_td ltx_align_center">8.78</td>
<td class="ltx_td ltx_align_center">13.21</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.500</td>
<td class="ltx_td ltx_align_center">7.175</td>
<td class="ltx_td ltx_align_center">1.411</td>
<td class="ltx_td ltx_align_center">1.30</td>
<td class="ltx_td ltx_align_center">8.86</td>
<td class="ltx_td ltx_align_center">13.32</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.012</td>
<td class="ltx_td ltx_align_center">7.316</td>
<td class="ltx_td ltx_align_center">1.404</td>
<td class="ltx_td ltx_align_center">1.82</td>
<td class="ltx_td ltx_align_center">8.91</td>
<td class="ltx_td ltx_align_center">13.45</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.981</td>
<td class="ltx_td ltx_align_center">7.686</td>
<td class="ltx_td ltx_align_center">1.361</td>
<td class="ltx_td ltx_align_center">2.69</td>
<td class="ltx_td ltx_align_center">8.94</td>
<td class="ltx_td ltx_align_center">13.71</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.388</td>
<td class="ltx_td ltx_align_center">7.953</td>
<td class="ltx_td ltx_align_center">1.304</td>
<td class="ltx_td ltx_align_center">3.10</td>
<td class="ltx_td ltx_align_center">8.86</td>
<td class="ltx_td ltx_align_center">13.71</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">8.145</td>
<td class="ltx_td ltx_align_center">1.247</td>
<td class="ltx_td ltx_align_center">3.34</td>
<td class="ltx_td ltx_align_center">8.81</td>
<td class="ltx_td ltx_align_center">13.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.995</td>
<td class="ltx_td ltx_align_center">8.766</td>
<td class="ltx_td ltx_align_center">0.971</td>
<td class="ltx_td ltx_align_center">3.58</td>
<td class="ltx_td ltx_align_center">8.65</td>
<td class="ltx_td ltx_align_center">11.28</td></tr>
</tbody>
</table>
</div>
<div id="A1.T3" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 3: </span>Data for the EOS C (<cite class="ltx_cite ltx_citemacro_cite">Bethe &amp; Johnson (<a href="#bib.bib5" title="" class="ltx_ref">1974</a>)</cite>, model I)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T3.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.100</td>
<td class="ltx_td ltx_align_center">9.884</td>
<td class="ltx_td ltx_align_center">1.852</td>
<td class="ltx_td ltx_align_center">0.39* 6.23</td>
<td class="ltx_td ltx_align_center">9.55</td>
<td class="ltx_td"/></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">9.952</td>
<td class="ltx_td ltx_align_center">1.852</td>
<td class="ltx_td ltx_align_center">0.30</td>
<td class="ltx_td ltx_align_center">6.23</td>
<td class="ltx_td ltx_align_center">9.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.800</td>
<td class="ltx_td ltx_align_center">10.095</td>
<td class="ltx_td ltx_align_center">1.850</td>
<td class="ltx_td ltx_align_center">0.79</td>
<td class="ltx_td ltx_align_center">6.25</td>
<td class="ltx_td ltx_align_center">9.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.500</td>
<td class="ltx_td ltx_align_center">10.326</td>
<td class="ltx_td ltx_align_center">1.840</td>
<td class="ltx_td ltx_align_center">1.24</td>
<td class="ltx_td ltx_align_center">6.28</td>
<td class="ltx_td ltx_align_center">9.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.995</td>
<td class="ltx_td ltx_align_center">10.779</td>
<td class="ltx_td ltx_align_center">1.790</td>
<td class="ltx_td ltx_align_center">1.80</td>
<td class="ltx_td ltx_align_center">6.34</td>
<td class="ltx_td ltx_align_center">9.60</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.778</td>
<td class="ltx_td ltx_align_center">11.010</td>
<td class="ltx_td ltx_align_center">1.746</td>
<td class="ltx_td ltx_align_center">2.02</td>
<td class="ltx_td ltx_align_center">6.34</td>
<td class="ltx_td ltx_align_center">9.58</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.413</td>
<td class="ltx_td ltx_align_center">11.443</td>
<td class="ltx_td ltx_align_center">1.619</td>
<td class="ltx_td ltx_align_center">2.36</td>
<td class="ltx_td ltx_align_center">6.33</td>
<td class="ltx_td ltx_align_center">9.47</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.122</td>
<td class="ltx_td ltx_align_center">11.834</td>
<td class="ltx_td ltx_align_center">1.436</td>
<td class="ltx_td ltx_align_center">2.56</td>
<td class="ltx_td ltx_align_center">6.25</td>
<td class="ltx_td ltx_align_center">9.23</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">12.018</td>
<td class="ltx_td ltx_align_center">1.323</td>
<td class="ltx_td ltx_align_center">2.59</td>
<td class="ltx_td ltx_align_center">6.13</td>
<td class="ltx_td ltx_align_center">8.90</td></tr>
</tbody>
</table>
</div>
<div id="A1.T4" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 4: </span>Data for the EOS D (<cite class="ltx_cite ltx_citemacro_cite">Bethe &amp; Johnson (<a href="#bib.bib5" title="" class="ltx_ref">1974</a>)</cite>, model V)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T4.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.370</td>
<td class="ltx_td ltx_align_center">9.360</td>
<td class="ltx_td ltx_align_center">1.651</td>
<td class="ltx_td ltx_align_center">0.71*</td>
<td class="ltx_td ltx_align_center">6.99</td>
<td class="ltx_td ltx_align_center">10.27</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.300</td>
<td class="ltx_td ltx_align_center">9.403</td>
<td class="ltx_td ltx_align_center">1.651</td>
<td class="ltx_td ltx_align_center">0.30</td>
<td class="ltx_td ltx_align_center">6.99</td>
<td class="ltx_td ltx_align_center">10.27</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">9.598</td>
<td class="ltx_td ltx_align_center">1.648</td>
<td class="ltx_td ltx_align_center">0.84</td>
<td class="ltx_td ltx_align_center">6.96</td>
<td class="ltx_td ltx_align_center">10.32</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.512</td>
<td class="ltx_td ltx_align_center">9.944</td>
<td class="ltx_td ltx_align_center">1.631</td>
<td class="ltx_td ltx_align_center">1.36</td>
<td class="ltx_td ltx_align_center">6.81</td>
<td class="ltx_td ltx_align_center">10.46</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.778</td>
<td class="ltx_td ltx_align_center">10.447</td>
<td class="ltx_td ltx_align_center">1.547</td>
<td class="ltx_td ltx_align_center">2.43</td>
<td class="ltx_td ltx_align_center">6.71</td>
<td class="ltx_td ltx_align_center">9.88</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.413</td>
<td class="ltx_td ltx_align_center">10.678</td>
<td class="ltx_td ltx_align_center">1.424</td>
<td class="ltx_td ltx_align_center">2.96</td>
<td class="ltx_td ltx_align_center">7.25</td>
<td class="ltx_td ltx_align_center">10.37</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.122</td>
<td class="ltx_td ltx_align_center">10.965</td>
<td class="ltx_td ltx_align_center">1.186</td>
<td class="ltx_td ltx_align_center">3.09</td>
<td class="ltx_td ltx_align_center">6.93</td>
<td class="ltx_td ltx_align_center">9.71</td></tr>
</tbody>
</table>
</div>
<div id="A1.T5" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 5: </span>Data for the EOS E (<cite class="ltx_cite ltx_citemacro_cite">Moszkowski (<a href="#bib.bib25" title="" class="ltx_ref">1974</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T5.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">9.061</td>
<td class="ltx_td ltx_align_center">1.726</td>
<td class="ltx_td ltx_align_center">1.78</td>
<td class="ltx_td ltx_align_center">7.62</td>
<td class="ltx_td ltx_align_center">11.56</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.818</td>
<td class="ltx_td ltx_align_center">9.171</td>
<td class="ltx_td ltx_align_center">1.711</td>
<td class="ltx_td ltx_align_center">1.98</td>
<td class="ltx_td ltx_align_center">7.68</td>
<td class="ltx_td ltx_align_center">11.60</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.239</td>
<td class="ltx_td ltx_align_center">9.562</td>
<td class="ltx_td ltx_align_center">1.624</td>
<td class="ltx_td ltx_align_center">2.58</td>
<td class="ltx_td ltx_align_center">7.84</td>
<td class="ltx_td ltx_align_center">11.65</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.778</td>
<td class="ltx_td ltx_align_center">9.915</td>
<td class="ltx_td ltx_align_center">1.474</td>
<td class="ltx_td ltx_align_center">3.02</td>
<td class="ltx_td ltx_align_center">7.90</td>
<td class="ltx_td ltx_align_center">11.44</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.585</td>
<td class="ltx_td ltx_align_center">10.068</td>
<td class="ltx_td ltx_align_center">1.376</td>
<td class="ltx_td ltx_align_center">3.18</td>
<td class="ltx_td ltx_align_center">7.86</td>
<td class="ltx_td ltx_align_center">11.19</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.259</td>
<td class="ltx_td ltx_align_center">10.314</td>
<td class="ltx_td ltx_align_center">1.144</td>
<td class="ltx_td ltx_align_center">3.39</td>
<td class="ltx_td ltx_align_center">7.59</td>
<td class="ltx_td ltx_align_center">10.36</td></tr>
</tbody>
</table>
</div>
<div id="A1.T6" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 6: </span>Data for the EOS F (<cite class="ltx_cite ltx_citemacro_cite">Arponen (<a href="#bib.bib3" title="" class="ltx_ref">1972</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T6.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.200</td>
<td class="ltx_td ltx_align_center">7.881</td>
<td class="ltx_td ltx_align_center">1.463</td>
<td class="ltx_td ltx_align_center">0.40*</td>
<td class="ltx_td ltx_align_center">7.40</td>
<td class="ltx_td ltx_align_center">11.88</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.100</td>
<td class="ltx_td ltx_align_center">7.922</td>
<td class="ltx_td ltx_align_center">1.463</td>
<td class="ltx_td ltx_align_center">0.20</td>
<td class="ltx_td ltx_align_center">7.41</td>
<td class="ltx_td ltx_align_center">11.86</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.012</td>
<td class="ltx_td ltx_align_center">7.961</td>
<td class="ltx_td ltx_align_center">1.463</td>
<td class="ltx_td ltx_align_center">0.46</td>
<td class="ltx_td ltx_align_center">7.42</td>
<td class="ltx_td ltx_align_center">11.83</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.500</td>
<td class="ltx_td ltx_align_center">8.204</td>
<td class="ltx_td ltx_align_center">1.459</td>
<td class="ltx_td ltx_align_center">1.09</td>
<td class="ltx_td ltx_align_center">7.48</td>
<td class="ltx_td ltx_align_center">11.66</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.981</td>
<td class="ltx_td ltx_align_center">8.490</td>
<td class="ltx_td ltx_align_center">1.449</td>
<td class="ltx_td ltx_align_center">1.42</td>
<td class="ltx_td ltx_align_center">7.54</td>
<td class="ltx_td ltx_align_center">11.49</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.162</td>
<td class="ltx_td ltx_align_center">9.088</td>
<td class="ltx_td ltx_align_center">1.412</td>
<td class="ltx_td ltx_align_center">1.63</td>
<td class="ltx_td ltx_align_center">7.40</td>
<td class="ltx_td ltx_align_center">11.15</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.239</td>
<td class="ltx_td ltx_align_center">9.923</td>
<td class="ltx_td ltx_align_center">1.333</td>
<td class="ltx_td ltx_align_center">1.84</td>
<td class="ltx_td ltx_align_center">6.76</td>
<td class="ltx_td ltx_align_center">10.23</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.585</td>
<td class="ltx_td ltx_align_center">10.465</td>
<td class="ltx_td ltx_align_center">1.222</td>
<td class="ltx_td ltx_align_center">2.36</td>
<td class="ltx_td ltx_align_center">6.62</td>
<td class="ltx_td ltx_align_center">9.74</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.122</td>
<td class="ltx_td ltx_align_center">10.889</td>
<td class="ltx_td ltx_align_center">1.032</td>
<td class="ltx_td ltx_align_center">2.75</td>
<td class="ltx_td ltx_align_center">6.57</td>
<td class="ltx_td ltx_align_center">9.03</td></tr>
</tbody>
</table>
</div>
<div id="A1.T7" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 7: </span>Data for the EOS G (<cite class="ltx_cite ltx_citemacro_cite">Canuto &amp; Chitre (<a href="#bib.bib8" title="" class="ltx_ref">1974</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T7.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">6.300</td>
<td class="ltx_td ltx_align_center">6.945</td>
<td class="ltx_td ltx_align_center">1.357</td>
<td class="ltx_td ltx_align_center">0.70*</td>
<td class="ltx_td ltx_align_center">8.77</td>
<td class="ltx_td ltx_align_center">13.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">6.200</td>
<td class="ltx_td ltx_align_center">6.970</td>
<td class="ltx_td ltx_align_center">1.357</td>
<td class="ltx_td ltx_align_center">0.48</td>
<td class="ltx_td ltx_align_center">8.77</td>
<td class="ltx_td ltx_align_center">13.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">6.100</td>
<td class="ltx_td ltx_align_center">6.996</td>
<td class="ltx_td ltx_align_center">1.357</td>
<td class="ltx_td ltx_align_center">0.72</td>
<td class="ltx_td ltx_align_center">8.77</td>
<td class="ltx_td ltx_align_center">13.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.800</td>
<td class="ltx_td ltx_align_center">7.075</td>
<td class="ltx_td ltx_align_center">1.356</td>
<td class="ltx_td ltx_align_center">1.18</td>
<td class="ltx_td ltx_align_center">8.79</td>
<td class="ltx_td ltx_align_center">13.53</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.500</td>
<td class="ltx_td ltx_align_center">7.159</td>
<td class="ltx_td ltx_align_center">1.353</td>
<td class="ltx_td ltx_align_center">1.53</td>
<td class="ltx_td ltx_align_center">8.81</td>
<td class="ltx_td ltx_align_center">13.52</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.000</td>
<td class="ltx_td ltx_align_center">7.308</td>
<td class="ltx_td ltx_align_center">1.344</td>
<td class="ltx_td ltx_align_center">2.00</td>
<td class="ltx_td ltx_align_center">8.87</td>
<td class="ltx_td ltx_align_center">13.51</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.503</td>
<td class="ltx_td ltx_align_center">7.472</td>
<td class="ltx_td ltx_align_center">1.327</td>
<td class="ltx_td ltx_align_center">2.40</td>
<td class="ltx_td ltx_align_center">8.95</td>
<td class="ltx_td ltx_align_center">13.53</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.498</td>
<td class="ltx_td ltx_align_center">7.899</td>
<td class="ltx_td ltx_align_center">1.253</td>
<td class="ltx_td ltx_align_center">2.98</td>
<td class="ltx_td ltx_align_center">8.97</td>
<td class="ltx_td ltx_align_center">13.61</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.631</td>
<td class="ltx_td ltx_align_center">8.399</td>
<td class="ltx_td ltx_align_center">1.114</td>
<td class="ltx_td ltx_align_center">3.25</td>
<td class="ltx_td ltx_align_center">8.48</td>
<td class="ltx_td ltx_align_center">12.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.376</td>
<td class="ltx_td ltx_align_center">8.557</td>
<td class="ltx_td ltx_align_center">1.057</td>
<td class="ltx_td ltx_align_center">3.35</td>
<td class="ltx_td ltx_align_center">8.34</td>
<td class="ltx_td ltx_align_center">11.86</td></tr>
</tbody>
</table>
</div>
<div id="A1.T8" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 8: </span>Data for the EOS I (<cite class="ltx_cite ltx_citemacro_cite">Cohen et al. (<a href="#bib.bib13" title="" class="ltx_ref">1970</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T8.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.100</td>
<td class="ltx_td ltx_align_center">11.795</td>
<td class="ltx_td ltx_align_center">2.446</td>
<td class="ltx_td ltx_align_center">0.33*</td>
<td class="ltx_td ltx_align_center">5.27</td>
<td class="ltx_td ltx_align_center">8.15</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">11.900</td>
<td class="ltx_td ltx_align_center">2.447</td>
<td class="ltx_td ltx_align_center">0.24</td>
<td class="ltx_td ltx_align_center">5.32</td>
<td class="ltx_td ltx_align_center">8.23</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">12.161</td>
<td class="ltx_td ltx_align_center">2.441</td>
<td class="ltx_td ltx_align_center">0.84</td>
<td class="ltx_td ltx_align_center">5.40</td>
<td class="ltx_td ltx_align_center">8.29</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.585</td>
<td class="ltx_td ltx_align_center">12.470</td>
<td class="ltx_td ltx_align_center">2.418</td>
<td class="ltx_td ltx_align_center">1.22</td>
<td class="ltx_td ltx_align_center">5.50</td>
<td class="ltx_td ltx_align_center">8.41</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.259</td>
<td class="ltx_td ltx_align_center">13.025</td>
<td class="ltx_td ltx_align_center">2.324</td>
<td class="ltx_td ltx_align_center">1.69</td>
<td class="ltx_td ltx_align_center">5.60</td>
<td class="ltx_td ltx_align_center">8.52</td></tr>
java.io.IOException: Server returned HTTP response code: 429 for URL: http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd
    at sun.net.www.protocol.http.HttpURLConnection.getInputStream0(HttpURLConnection.java:1876)
    at sun.net.www.protocol.http.HttpURLConnection.getInputStream(HttpURLConnection.java:1474)
    at org.apache.xerces.impl.XMLEntityManager.setupCurrentEntity(Unknown Source)
    at org.apache.xerces.impl.XMLEntityManager.startEntity(Unknown Source)
    at org.apache.xerces.impl.XMLEntityManager.startDTDEntity(Unknown Source)
    at org.apache.xerces.impl.XMLDTDScannerImpl.setInputSource(Unknown Source)
    at org.apache.xerces.impl.XMLDocumentScannerImpl$DTDDispatcher.dispatch(Unknown Source)
    at org.apache.xerces.impl.XMLDocumentFragmentScannerImpl.scanDocument(Unknown Source)
    at org.apache.xerces.parsers.XML11Configuration.parse(Unknown Source)
    at org.apache.xerces.parsers.XML11Configuration.parse(Unknown Source)
    at org.apache.xerces.parsers.XMLParser.parse(Unknown Source)
    at org.apache.xerces.parsers.DOMParser.parse(Unknown Source)
    at org.apache.xerces.jaxp.DocumentBuilderImpl.parse(Unknown Source)
    at com.formulasearchengine.mathmlquerygenerator.xmlhelper.XMLHelper.String2Doc(XMLHelper.java:203)
    at com.formulasearchengine.mathosphere.mathpd.pojos.ArxivDocument.getDoc(ArxivDocument.java:30)
    at com.formulasearchengine.mathosphere.mathpd.pojos.ArxivDocument.getMathTags(ArxivDocument.java:34)
    at com.formulasearchengine.mathosphere.mathpd.Distances.getDocumentHistogram(Distances.java:100)
    at com.formulasearchengine.mathosphere.mathpd.Distances.testdist(Distances.java:137)
    at com.formulasearchengine.mathosphere.mathpd.FlinkPd$1.reduce(FlinkPd.java:45)
    at org.apache.flink.runtime.operators.AllGroupReduceDriver.run(AllGroupReduceDriver.java:148)
    at org.apache.flink.runtime.operators.BatchTask.run(BatchTask.java:489)
    at org.apache.flink.runtime.operators.BatchTask.invoke(BatchTask.java:354)
    at org.apache.flink.runtime.taskmanager.Task.run(Task.java:584)
    at java.lang.Thread.run(Thread.java:745)
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">13.499</td>
<td class="ltx_td ltx_align_center">2.154</td>
<td class="ltx_td ltx_align_center">2.05</td>
<td class="ltx_td ltx_align_center">5.74</td>
<td class="ltx_td ltx_align_center">8.61</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.794</td>
<td class="ltx_td ltx_align_center">13.883</td>
<td class="ltx_td ltx_align_center">1.883</td>
<td class="ltx_td ltx_align_center">2.31</td>
<td class="ltx_td ltx_align_center">5.77</td>
<td class="ltx_td ltx_align_center">8.51</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.631</td>
<td class="ltx_td ltx_align_center">14.127</td>
<td class="ltx_td ltx_align_center">1.561</td>
<td class="ltx_td ltx_align_center">2.46</td>
<td class="ltx_td ltx_align_center">5.67</td>
<td class="ltx_td ltx_align_center">7.95</td></tr>
</tbody>
</table>
</div>
<div id="A1.T9" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 9: </span>Data for the EOS L (<cite class="ltx_cite ltx_citemacro_cite">Pandharipande et al. (<a href="#bib.bib27" title="" class="ltx_ref">1976</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T9.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.500</td>
<td class="ltx_td ltx_align_center">13.618</td>
<td class="ltx_td ltx_align_center">2.662</td>
<td class="ltx_td ltx_align_center">0.68*</td>
<td class="ltx_td ltx_align_center">4.66</td>
<td class="ltx_td ltx_align_center">7.39</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">13.747</td>
<td class="ltx_td ltx_align_center">2.660</td>
<td class="ltx_td ltx_align_center">0.56</td>
<td class="ltx_td ltx_align_center">4.70</td>
<td class="ltx_td ltx_align_center">7.47</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.259</td>
<td class="ltx_td ltx_align_center">13.936</td>
<td class="ltx_td ltx_align_center">2.649</td>
<td class="ltx_td ltx_align_center">0.97</td>
<td class="ltx_td ltx_align_center">4.79</td>
<td class="ltx_td ltx_align_center">7.58</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.150</td>
<td class="ltx_td ltx_align_center">14.087</td>
<td class="ltx_td ltx_align_center">2.630</td>
<td class="ltx_td ltx_align_center">1.25</td>
<td class="ltx_td ltx_align_center">4.92</td>
<td class="ltx_td ltx_align_center">7.69</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">14.299</td>
<td class="ltx_td ltx_align_center">2.579</td>
<td class="ltx_td ltx_align_center">1.59</td>
<td class="ltx_td ltx_align_center">5.21</td>
<td class="ltx_td ltx_align_center">8.05</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.794</td>
<td class="ltx_td ltx_align_center">14.681</td>
<td class="ltx_td ltx_align_center">2.391</td>
<td class="ltx_td ltx_align_center">2.03</td>
<td class="ltx_td ltx_align_center">5.66</td>
<td class="ltx_td ltx_align_center">8.35</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.631</td>
<td class="ltx_td ltx_align_center">14.989</td>
<td class="ltx_td ltx_align_center">2.044</td>
<td class="ltx_td ltx_align_center">2.27</td>
<td class="ltx_td ltx_align_center">5.71</td>
<td class="ltx_td ltx_align_center">8.20</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">15.025</td>
<td class="ltx_td ltx_align_center">1.959</td>
<td class="ltx_td ltx_align_center">2.32</td>
<td class="ltx_td ltx_align_center">5.69</td>
<td class="ltx_td ltx_align_center">8.15</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.500</td>
<td class="ltx_td ltx_align_center">15.056</td>
<td class="ltx_td ltx_align_center">1.636</td>
<td class="ltx_td ltx_align_center">2.53</td>
<td class="ltx_td ltx_align_center">5.58</td>
<td class="ltx_td ltx_align_center">7.52</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.398</td>
<td class="ltx_td ltx_align_center">14.889</td>
<td class="ltx_td ltx_align_center">1.214</td>
<td class="ltx_td ltx_align_center">2.77</td>
<td class="ltx_td ltx_align_center">5.47</td>
<td class="ltx_td ltx_align_center">6.09</td></tr>
</tbody>
</table>
</div>
<div id="A1.T10" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 10: </span>Data for the EOS N (<cite class="ltx_cite ltx_citemacro_cite">Serot (<a href="#bib.bib28" title="" class="ltx_ref">1979</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T10.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.700</td>
<td class="ltx_td ltx_align_center">12.740</td>
<td class="ltx_td ltx_align_center">2.634</td>
<td class="ltx_td ltx_align_center">0.47*</td>
<td class="ltx_td ltx_align_center">5.10</td>
<td class="ltx_td ltx_align_center">7.97</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.600</td>
<td class="ltx_td ltx_align_center">12.852</td>
<td class="ltx_td ltx_align_center">2.633</td>
<td class="ltx_td ltx_align_center">0.49</td>
<td class="ltx_td ltx_align_center">5.19</td>
<td class="ltx_td ltx_align_center">8.09</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">13.107</td>
<td class="ltx_td ltx_align_center">2.619</td>
<td class="ltx_td ltx_align_center">1.03</td>
<td class="ltx_td ltx_align_center">5.36</td>
<td class="ltx_td ltx_align_center">8.30</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.200</td>
<td class="ltx_td ltx_align_center">13.385</td>
<td class="ltx_td ltx_align_center">2.575</td>
<td class="ltx_td ltx_align_center">1.47</td>
<td class="ltx_td ltx_align_center">5.62</td>
<td class="ltx_td ltx_align_center">8.57</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">13.686</td>
<td class="ltx_td ltx_align_center">2.468</td>
<td class="ltx_td ltx_align_center">1.90</td>
<td class="ltx_td ltx_align_center">5.90</td>
<td class="ltx_td ltx_align_center">8.88</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">13.951</td>
<td class="ltx_td ltx_align_center">2.233</td>
<td class="ltx_td ltx_align_center">2.39</td>
<td class="ltx_td ltx_align_center">6.27</td>
<td class="ltx_td ltx_align_center">9.19</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">13.980</td>
<td class="ltx_td ltx_align_center">1.729</td>
<td class="ltx_td ltx_align_center">2.96</td>
<td class="ltx_td ltx_align_center">6.61</td>
<td class="ltx_td ltx_align_center">8.79</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.500</td>
<td class="ltx_td ltx_align_center">13.757</td>
<td class="ltx_td ltx_align_center">1.313</td>
<td class="ltx_td ltx_align_center">3.20</td>
<td class="ltx_td ltx_align_center">6.38</td>
<td class="ltx_td ltx_align_center">7.56</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.400</td>
<td class="ltx_td ltx_align_center">13.349</td>
<td class="ltx_td ltx_align_center">0.836</td>
<td class="ltx_td ltx_align_center">3.26</td>
<td class="ltx_td ltx_align_center">5.02</td>
<td class="ltx_td ltx_align_center">5.68</td></tr>
</tbody>
</table>
</div>
<div id="A1.T11" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 11: </span>Data for the EOS O (<cite class="ltx_cite ltx_citemacro_cite">Bowers et al. (<a href="#bib.bib7" title="" class="ltx_ref">1975</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T11.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.100</td>
<td class="ltx_td ltx_align_center">11.502</td>
<td class="ltx_td ltx_align_center">2.379</td>
<td class="ltx_td ltx_align_center">0.55*</td>
<td class="ltx_td ltx_align_center">5.56</td>
<td class="ltx_td ltx_align_center">8.64</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">11.587</td>
<td class="ltx_td ltx_align_center">2.378</td>
<td class="ltx_td ltx_align_center">0.53</td>
<td class="ltx_td ltx_align_center">5.63</td>
<td class="ltx_td ltx_align_center">8.72</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">11.765</td>
<td class="ltx_td ltx_align_center">2.370</td>
<td class="ltx_td ltx_align_center">1.05</td>
<td class="ltx_td ltx_align_center">5.80</td>
<td class="ltx_td ltx_align_center">8.99</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.600</td>
<td class="ltx_td ltx_align_center">11.974</td>
<td class="ltx_td ltx_align_center">2.346</td>
<td class="ltx_td ltx_align_center">1.43</td>
<td class="ltx_td ltx_align_center">5.94</td>
<td class="ltx_td ltx_align_center">9.23</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">12.201</td>
<td class="ltx_td ltx_align_center">2.296</td>
<td class="ltx_td ltx_align_center">1.81</td>
<td class="ltx_td ltx_align_center">6.14</td>
<td class="ltx_td ltx_align_center">9.51</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.200</td>
<td class="ltx_td ltx_align_center">12.442</td>
<td class="ltx_td ltx_align_center">2.199</td>
<td class="ltx_td ltx_align_center">2.18</td>
<td class="ltx_td ltx_align_center">6.38</td>
<td class="ltx_td ltx_align_center">9.93</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">12.672</td>
<td class="ltx_td ltx_align_center">2.019</td>
<td class="ltx_td ltx_align_center">2.56</td>
<td class="ltx_td ltx_align_center">6.75</td>
<td class="ltx_td ltx_align_center">10.21</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">12.832</td>
<td class="ltx_td ltx_align_center">1.682</td>
<td class="ltx_td ltx_align_center">2.86</td>
<td class="ltx_td ltx_align_center">7.09</td>
<td class="ltx_td ltx_align_center">9.43</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">12.760</td>
<td class="ltx_td ltx_align_center">1.173</td>
<td class="ltx_td ltx_align_center">3.09</td>
<td class="ltx_td ltx_align_center">6.58</td>
<td class="ltx_td ltx_align_center">7.92</td></tr>
</tbody>
</table>
</div>
<div id="A1.T12" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 12: </span>Data for the EOS G<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m2" class="ltx_Math" alttext="{}_{240}" display="inline"><apply><cn type="integer">240</cn></apply></math> (<cite class="ltx_cite ltx_citemacro_cite">Glendenning (<a href="#bib.bib16" title="" class="ltx_ref">1985</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m3" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m4" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m5" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m6" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m7" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m8" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m9" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T12.m10" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.600</td>
<td class="ltx_td ltx_align_center">10.850</td>
<td class="ltx_td ltx_align_center">1.553</td>
<td class="ltx_td ltx_align_center">0.66*</td>
<td class="ltx_td ltx_align_center">5.39</td>
<td class="ltx_td ltx_align_center">8.59</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.500</td>
<td class="ltx_td ltx_align_center">10.928</td>
<td class="ltx_td ltx_align_center">1.553</td>
<td class="ltx_td ltx_align_center">0.33</td>
<td class="ltx_td ltx_align_center">5.40</td>
<td class="ltx_td ltx_align_center">8.58</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.200</td>
<td class="ltx_td ltx_align_center">11.209</td>
<td class="ltx_td ltx_align_center">1.549</td>
<td class="ltx_td ltx_align_center">0.77</td>
<td class="ltx_td ltx_align_center">5.42</td>
<td class="ltx_td ltx_align_center">8.48</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">11.647</td>
<td class="ltx_td ltx_align_center">1.529</td>
<td class="ltx_td ltx_align_center">1.11</td>
<td class="ltx_td ltx_align_center">5.43</td>
<td class="ltx_td ltx_align_center">8.37</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">12.200</td>
<td class="ltx_td ltx_align_center">1.481</td>
<td class="ltx_td ltx_align_center">1.37</td>
<td class="ltx_td ltx_align_center">5.34</td>
<td class="ltx_td ltx_align_center">8.10</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">12.798</td>
<td class="ltx_td ltx_align_center">1.374</td>
<td class="ltx_td ltx_align_center">1.70</td>
<td class="ltx_td ltx_align_center">5.28</td>
<td class="ltx_td ltx_align_center">7.98</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">13.110</td>
<td class="ltx_td ltx_align_center">1.267</td>
<td class="ltx_td ltx_align_center">1.91</td>
<td class="ltx_td ltx_align_center">5.17</td>
<td class="ltx_td ltx_align_center">7.96</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">13.351</td>
<td class="ltx_td ltx_align_center">1.095</td>
<td class="ltx_td ltx_align_center">2.27</td>
<td class="ltx_td ltx_align_center">5.27</td>
<td class="ltx_td ltx_align_center">6.86</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.400</td>
<td class="ltx_td ltx_align_center">13.482</td>
<td class="ltx_td ltx_align_center">0.711</td>
<td class="ltx_td ltx_align_center">2.36</td>
<td class="ltx_td ltx_align_center">4.54</td>
<td class="ltx_td ltx_align_center">5.23</td></tr>
</tbody>
</table>
</div>
<div id="A1.T13" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 13: </span>Data for the EOS G<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m2" class="ltx_Math" alttext="{}_{300}" display="inline"><apply><cn type="integer">300</cn></apply></math> (<cite class="ltx_cite ltx_citemacro_cite">Glendenning (<a href="#bib.bib16" title="" class="ltx_ref">1985</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m3" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m4" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m5" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m6" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m7" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m8" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m9" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T13.m10" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.200</td>
<td class="ltx_td ltx_align_center">11.687</td>
<td class="ltx_td ltx_align_center">1.788</td>
<td class="ltx_td ltx_align_center">0.33*</td>
<td class="ltx_td ltx_align_center">5.02</td>
<td class="ltx_td ltx_align_center">8.06</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.100</td>
<td class="ltx_td ltx_align_center">11.762</td>
<td class="ltx_td ltx_align_center">1.788</td>
<td class="ltx_td ltx_align_center">0.12</td>
<td class="ltx_td ltx_align_center">5.08</td>
<td class="ltx_td ltx_align_center">8.10</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">11.842</td>
<td class="ltx_td ltx_align_center">1.787</td>
<td class="ltx_td ltx_align_center">0.51</td>
<td class="ltx_td ltx_align_center">5.15</td>
<td class="ltx_td ltx_align_center">8.15</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">12.027</td>
<td class="ltx_td ltx_align_center">1.782</td>
<td class="ltx_td ltx_align_center">0.86</td>
<td class="ltx_td ltx_align_center">5.25</td>
<td class="ltx_td ltx_align_center">8.23</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">12.542</td>
<td class="ltx_td ltx_align_center">1.742</td>
<td class="ltx_td ltx_align_center">1.28</td>
<td class="ltx_td ltx_align_center">5.37</td>
<td class="ltx_td ltx_align_center">8.22</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">13.182</td>
<td class="ltx_td ltx_align_center">1.624</td>
<td class="ltx_td ltx_align_center">1.66</td>
<td class="ltx_td ltx_align_center">5.36</td>
<td class="ltx_td ltx_align_center">7.99</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">13.482</td>
<td class="ltx_td ltx_align_center">1.506</td>
<td class="ltx_td ltx_align_center">1.92</td>
<td class="ltx_td ltx_align_center">5.42</td>
<td class="ltx_td ltx_align_center">8.12</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">13.718</td>
<td class="ltx_td ltx_align_center">1.286</td>
<td class="ltx_td ltx_align_center">2.27</td>
<td class="ltx_td ltx_align_center">5.33</td>
<td class="ltx_td ltx_align_center">7.50</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.500</td>
<td class="ltx_td ltx_align_center">13.733</td>
<td class="ltx_td ltx_align_center">1.119</td>
<td class="ltx_td ltx_align_center">2.54</td>
<td class="ltx_td ltx_align_center">5.61</td>
<td class="ltx_td ltx_align_center">6.60</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.400</td>
<td class="ltx_td ltx_align_center">13.630</td>
<td class="ltx_td ltx_align_center">0.825</td>
<td class="ltx_td ltx_align_center">2.58</td>
<td class="ltx_td ltx_align_center">5.04</td>
<td class="ltx_td ltx_align_center">5.59</td></tr>
</tbody>
</table>
</div>
<div id="A1.T14" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 14: </span>Data for the EOS WFF (<cite class="ltx_cite ltx_citemacro_cite">Wiringa et al. (<a href="#bib.bib30" title="" class="ltx_ref">1988</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T14.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.200</td>
<td class="ltx_td ltx_align_center">9.510</td>
<td class="ltx_td ltx_align_center">1.840</td>
<td class="ltx_td ltx_align_center">0.47*</td>
<td class="ltx_td ltx_align_center">6.66</td>
<td class="ltx_td ltx_align_center">10.38</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.100</td>
<td class="ltx_td ltx_align_center">9.558</td>
<td class="ltx_td ltx_align_center">1.840</td>
<td class="ltx_td ltx_align_center">0.42</td>
<td class="ltx_td ltx_align_center">6.70</td>
<td class="ltx_td ltx_align_center">10.44</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">9.612</td>
<td class="ltx_td ltx_align_center">1.840</td>
<td class="ltx_td ltx_align_center">0.69</td>
<td class="ltx_td ltx_align_center">6.73</td>
<td class="ltx_td ltx_align_center">10.45</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.800</td>
<td class="ltx_td ltx_align_center">9.729</td>
<td class="ltx_td ltx_align_center">1.836</td>
<td class="ltx_td ltx_align_center">1.08</td>
<td class="ltx_td ltx_align_center">6.80</td>
<td class="ltx_td ltx_align_center">10.49</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.600</td>
<td class="ltx_td ltx_align_center">9.850</td>
<td class="ltx_td ltx_align_center">1.828</td>
<td class="ltx_td ltx_align_center">1.38</td>
<td class="ltx_td ltx_align_center">6.90</td>
<td class="ltx_td ltx_align_center">10.55</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">10.278</td>
<td class="ltx_td ltx_align_center">1.759</td>
<td class="ltx_td ltx_align_center">2.13</td>
<td class="ltx_td ltx_align_center">7.20</td>
<td class="ltx_td ltx_align_center">10.89</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">10.441</td>
<td class="ltx_td ltx_align_center">1.710</td>
<td class="ltx_td ltx_align_center">2.37</td>
<td class="ltx_td ltx_align_center">7.30</td>
<td class="ltx_td ltx_align_center">10.99</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">10.774</td>
<td class="ltx_td ltx_align_center">1.538</td>
<td class="ltx_td ltx_align_center">2.87</td>
<td class="ltx_td ltx_align_center">7.47</td>
<td class="ltx_td ltx_align_center">11.12</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.200</td>
<td class="ltx_td ltx_align_center">10.925</td>
<td class="ltx_td ltx_align_center">1.389</td>
<td class="ltx_td ltx_align_center">3.12</td>
<td class="ltx_td ltx_align_center">7.52</td>
<td class="ltx_td ltx_align_center">10.78</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">11.038</td>
<td class="ltx_td ltx_align_center">1.178</td>
<td class="ltx_td ltx_align_center">3.34</td>
<td class="ltx_td ltx_align_center">7.54</td>
<td class="ltx_td ltx_align_center">9.42</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.900</td>
<td class="ltx_td ltx_align_center">11.075</td>
<td class="ltx_td ltx_align_center">1.044</td>
<td class="ltx_td ltx_align_center">3.41</td>
<td class="ltx_td ltx_align_center">7.46</td>
<td class="ltx_td ltx_align_center">8.51</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">11.104</td>
<td class="ltx_td ltx_align_center">0.889</td>
<td class="ltx_td ltx_align_center">3.42</td>
<td class="ltx_td ltx_align_center">6.99</td>
<td class="ltx_td ltx_align_center">7.68</td></tr>
</tbody>
</table>
</div>
<div id="A1.T15" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 15: </span>Data for the EOS MPA (<cite class="ltx_cite ltx_citemacro_cite">Wu et al. (<a href="#bib.bib31" title="" class="ltx_ref">1991</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T15.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.800</td>
<td class="ltx_td ltx_align_center">7.899</td>
<td class="ltx_td ltx_align_center">1.560</td>
<td class="ltx_td ltx_align_center">0.39*</td>
<td class="ltx_td ltx_align_center">7.82</td>
<td class="ltx_td ltx_align_center">11.96</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.700</td>
<td class="ltx_td ltx_align_center">7.930</td>
<td class="ltx_td ltx_align_center">1.560</td>
<td class="ltx_td ltx_align_center">0.37</td>
<td class="ltx_td ltx_align_center">7.85</td>
<td class="ltx_td ltx_align_center">11.99</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.600</td>
<td class="ltx_td ltx_align_center">7.963</td>
<td class="ltx_td ltx_align_center">1.559</td>
<td class="ltx_td ltx_align_center">0.67</td>
<td class="ltx_td ltx_align_center">7.87</td>
<td class="ltx_td ltx_align_center">12.02</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.500</td>
<td class="ltx_td ltx_align_center">7.996</td>
<td class="ltx_td ltx_align_center">1.559</td>
<td class="ltx_td ltx_align_center">0.87</td>
<td class="ltx_td ltx_align_center">7.90</td>
<td class="ltx_td ltx_align_center">12.06</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.400</td>
<td class="ltx_td ltx_align_center">8.030</td>
<td class="ltx_td ltx_align_center">1.558</td>
<td class="ltx_td ltx_align_center">1.04</td>
<td class="ltx_td ltx_align_center">7.92</td>
<td class="ltx_td ltx_align_center">12.10</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.200</td>
<td class="ltx_td ltx_align_center">8.104</td>
<td class="ltx_td ltx_align_center">1.556</td>
<td class="ltx_td ltx_align_center">1.31</td>
<td class="ltx_td ltx_align_center">7.96</td>
<td class="ltx_td ltx_align_center">12.14</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.000</td>
<td class="ltx_td ltx_align_center">8.186</td>
<td class="ltx_td ltx_align_center">1.551</td>
<td class="ltx_td ltx_align_center">1.55</td>
<td class="ltx_td ltx_align_center">7.99</td>
<td class="ltx_td ltx_align_center">12.16</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.500</td>
<td class="ltx_td ltx_align_center">8.407</td>
<td class="ltx_td ltx_align_center">1.531</td>
<td class="ltx_td ltx_align_center">2.07</td>
<td class="ltx_td ltx_align_center">8.08</td>
<td class="ltx_td ltx_align_center">12.26</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">8.669</td>
<td class="ltx_td ltx_align_center">1.489</td>
<td class="ltx_td ltx_align_center">2.51</td>
<td class="ltx_td ltx_align_center">8.13</td>
<td class="ltx_td ltx_align_center">12.24</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.500</td>
<td class="ltx_td ltx_align_center">8.973</td>
<td class="ltx_td ltx_align_center">1.410</td>
<td class="ltx_td ltx_align_center">2.90</td>
<td class="ltx_td ltx_align_center">8.16</td>
<td class="ltx_td ltx_align_center">12.10</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">9.328</td>
<td class="ltx_td ltx_align_center">1.269</td>
<td class="ltx_td ltx_align_center">3.19</td>
<td class="ltx_td ltx_align_center">8.08</td>
<td class="ltx_td ltx_align_center">11.61</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.500</td>
<td class="ltx_td ltx_align_center">9.747</td>
<td class="ltx_td ltx_align_center">1.033</td>
<td class="ltx_td ltx_align_center">3.29</td>
<td class="ltx_td ltx_align_center">7.58</td>
<td class="ltx_td ltx_align_center">10.48</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.200</td>
<td class="ltx_td ltx_align_center">10.031</td>
<td class="ltx_td ltx_align_center">0.844</td>
<td class="ltx_td ltx_align_center">3.27</td>
<td class="ltx_td ltx_align_center">6.95</td>
<td class="ltx_td ltx_align_center">9.00</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">10.251</td>
<td class="ltx_td ltx_align_center">0.698</td>
<td class="ltx_td ltx_align_center">3.22</td>
<td class="ltx_td ltx_align_center">6.36</td>
<td class="ltx_td ltx_align_center">7.47</td></tr>
</tbody>
</table>
</div>
<div class="ltx_pagination ltx_role_newpage"/>
<div id="A1.T16" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 16: </span>Data for the EOS APR1 (<cite class="ltx_cite ltx_citemacro_cite">Akmal et al. (<a href="#bib.bib1" title="" class="ltx_ref">1998</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T16.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.400</td>
<td class="ltx_td ltx_align_center">10.746</td>
<td class="ltx_td ltx_align_center">2.379</td>
<td class="ltx_td ltx_align_center">0.40*</td>
<td class="ltx_td ltx_align_center">6.01</td>
<td class="ltx_td ltx_align_center">9.14</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.300</td>
<td class="ltx_td ltx_align_center">10.822</td>
<td class="ltx_td ltx_align_center">2.379</td>
<td class="ltx_td ltx_align_center">0.47</td>
<td class="ltx_td ltx_align_center">6.08</td>
<td class="ltx_td ltx_align_center">9.21</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.200</td>
<td class="ltx_td ltx_align_center">10.904</td>
<td class="ltx_td ltx_align_center">2.377</td>
<td class="ltx_td ltx_align_center">0.80</td>
<td class="ltx_td ltx_align_center">6.16</td>
<td class="ltx_td ltx_align_center">9.29</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.100</td>
<td class="ltx_td ltx_align_center">10.990</td>
<td class="ltx_td ltx_align_center">2.373</td>
<td class="ltx_td ltx_align_center">1.04</td>
<td class="ltx_td ltx_align_center">6.24</td>
<td class="ltx_td ltx_align_center">9.37</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">11.080</td>
<td class="ltx_td ltx_align_center">2.366</td>
<td class="ltx_td ltx_align_center">1.25</td>
<td class="ltx_td ltx_align_center">6.32</td>
<td class="ltx_td ltx_align_center">9.45</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">11.277</td>
<td class="ltx_td ltx_align_center">2.340</td>
<td class="ltx_td ltx_align_center">1.64</td>
<td class="ltx_td ltx_align_center">6.50</td>
<td class="ltx_td ltx_align_center">9.62</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.500</td>
<td class="ltx_td ltx_align_center">11.611</td>
<td class="ltx_td ltx_align_center">2.250</td>
<td class="ltx_td ltx_align_center">2.19</td>
<td class="ltx_td ltx_align_center">6.76</td>
<td class="ltx_td ltx_align_center">9.89</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.200</td>
<td class="ltx_td ltx_align_center">11.966</td>
<td class="ltx_td ltx_align_center">2.040</td>
<td class="ltx_td ltx_align_center">2.75</td>
<td class="ltx_td ltx_align_center">6.96</td>
<td class="ltx_td ltx_align_center">10.16</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">12.171</td>
<td class="ltx_td ltx_align_center">1.774</td>
<td class="ltx_td ltx_align_center">3.10</td>
<td class="ltx_td ltx_align_center">6.94</td>
<td class="ltx_td ltx_align_center">10.27</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">12.294</td>
<td class="ltx_td ltx_align_center">1.365</td>
<td class="ltx_td ltx_align_center">3.28</td>
<td class="ltx_td ltx_align_center">6.75</td>
<td class="ltx_td ltx_align_center">8.82</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.700</td>
<td class="ltx_td ltx_align_center">12.336</td>
<td class="ltx_td ltx_align_center">1.109</td>
<td class="ltx_td ltx_align_center">3.21</td>
<td class="ltx_td ltx_align_center">6.53</td>
<td class="ltx_td ltx_align_center">7.39</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">12.435</td>
<td class="ltx_td ltx_align_center">0.841</td>
<td class="ltx_td ltx_align_center">2.95</td>
<td class="ltx_td ltx_align_center">5.52</td>
<td class="ltx_td ltx_align_center">6.37</td></tr>
</tbody>
</table>
</div>
<div id="A1.T17" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 17: </span>Data for the EOS APR2 (<cite class="ltx_cite ltx_citemacro_cite">Akmal et al. (<a href="#bib.bib1" title="" class="ltx_ref">1998</a>)</cite>)</div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m1" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m2" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m3" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m4" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m5" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m6" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m7" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T17.m8" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.800</td>
<td class="ltx_td ltx_align_center">9.998</td>
<td class="ltx_td ltx_align_center">2.201</td>
<td class="ltx_td ltx_align_center">0.39*</td>
<td class="ltx_td ltx_align_center">6.43</td>
<td class="ltx_td ltx_align_center">9.71</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.700</td>
<td class="ltx_td ltx_align_center">10.059</td>
<td class="ltx_td ltx_align_center">2.201</td>
<td class="ltx_td ltx_align_center">0.45</td>
<td class="ltx_td ltx_align_center">6.50</td>
<td class="ltx_td ltx_align_center">9.79</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.600</td>
<td class="ltx_td ltx_align_center">10.122</td>
<td class="ltx_td ltx_align_center">2.199</td>
<td class="ltx_td ltx_align_center">0.77</td>
<td class="ltx_td ltx_align_center">6.57</td>
<td class="ltx_td ltx_align_center">9.88</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.500</td>
<td class="ltx_td ltx_align_center">10.193</td>
<td class="ltx_td ltx_align_center">2.197</td>
<td class="ltx_td ltx_align_center">1.01</td>
<td class="ltx_td ltx_align_center">6.63</td>
<td class="ltx_td ltx_align_center">9.92</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.400</td>
<td class="ltx_td ltx_align_center">10.269</td>
<td class="ltx_td ltx_align_center">2.192</td>
<td class="ltx_td ltx_align_center">1.21</td>
<td class="ltx_td ltx_align_center">6.69</td>
<td class="ltx_td ltx_align_center">9.98</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.200</td>
<td class="ltx_td ltx_align_center">10.428</td>
<td class="ltx_td ltx_align_center">2.176</td>
<td class="ltx_td ltx_align_center">1.58</td>
<td class="ltx_td ltx_align_center">6.83</td>
<td class="ltx_td ltx_align_center">10.08</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">10.598</td>
<td class="ltx_td ltx_align_center">2.148</td>
<td class="ltx_td ltx_align_center">1.92</td>
<td class="ltx_td ltx_align_center">6.98</td>
<td class="ltx_td ltx_align_center">10.22</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">10.789</td>
<td class="ltx_td ltx_align_center">2.098</td>
<td class="ltx_td ltx_align_center">2.25</td>
<td class="ltx_td ltx_align_center">7.12</td>
<td class="ltx_td ltx_align_center">10.35</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">11.203</td>
<td class="ltx_td ltx_align_center">1.890</td>
<td class="ltx_td ltx_align_center">2.89</td>
<td class="ltx_td ltx_align_center">7.26</td>
<td class="ltx_td ltx_align_center">10.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">11.572</td>
<td class="ltx_td ltx_align_center">1.410</td>
<td class="ltx_td ltx_align_center">3.37</td>
<td class="ltx_td ltx_align_center">7.01</td>
<td class="ltx_td ltx_align_center">10.07</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">11.737</td>
<td class="ltx_td ltx_align_center">1.032</td>
<td class="ltx_td ltx_align_center">3.25</td>
<td class="ltx_td ltx_align_center">6.59</td>
<td class="ltx_td ltx_align_center">7.58</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.700</td>
<td class="ltx_td ltx_align_center">11.884</td>
<td class="ltx_td ltx_align_center">0.826</td>
<td class="ltx_td ltx_align_center">3.01</td>
<td class="ltx_td ltx_align_center">5.88</td>
<td class="ltx_td ltx_align_center">6.54</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.600</td>
<td class="ltx_td ltx_align_center">12.189</td>
<td class="ltx_td ltx_align_center">0.632</td>
<td class="ltx_td ltx_align_center">2.63</td>
<td class="ltx_td ltx_align_center">4.50</td>
<td class="ltx_td ltx_align_center">5.76</td></tr>
</tbody>
</table>
</div>
<div id="A1.T18" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 18: </span>Data for the polytropic EOS with <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m4" class="ltx_Math" alttext="n=1" display="inline"><apply><eq/><ci>𝑛</ci><cn type="integer">1</cn></apply></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m5" class="ltx_Math" alttext="\kappa=100\," display="inline"><apply><eq/><ci>𝜅</ci><cn type="integer">100</cn></apply></math>km<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m6" class="ltx_Math" alttext="{}^{2}" display="inline"><apply><cn type="integer">2</cn></apply></math></div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m7" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m8" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m9" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m10" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m11" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m12" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m13" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T18.m14" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.700</td>
<td class="ltx_td ltx_align_center">7.518</td>
<td class="ltx_td ltx_align_center">1.351</td>
<td class="ltx_td ltx_align_center">0.618*</td>
<td class="ltx_td ltx_align_center">7.582</td>
<td class="ltx_td ltx_align_center">11.569</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.650</td>
<td class="ltx_td ltx_align_center">7.535</td>
<td class="ltx_td ltx_align_center">1.351</td>
<td class="ltx_td ltx_align_center">0.180</td>
<td class="ltx_td ltx_align_center">7.576</td>
<td class="ltx_td ltx_align_center">11.556</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.600</td>
<td class="ltx_td ltx_align_center">7.554</td>
<td class="ltx_td ltx_align_center">1.351</td>
<td class="ltx_td ltx_align_center">0.358</td>
<td class="ltx_td ltx_align_center">7.569</td>
<td class="ltx_td ltx_align_center">11.542</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.500</td>
<td class="ltx_td ltx_align_center">7.590</td>
<td class="ltx_td ltx_align_center">1.351</td>
<td class="ltx_td ltx_align_center">0.569</td>
<td class="ltx_td ltx_align_center">7.557</td>
<td class="ltx_td ltx_align_center">11.520</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.300</td>
<td class="ltx_td ltx_align_center">7.667</td>
<td class="ltx_td ltx_align_center">1.350</td>
<td class="ltx_td ltx_align_center">0.838</td>
<td class="ltx_td ltx_align_center">7.524</td>
<td class="ltx_td ltx_align_center">11.457</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">5.000</td>
<td class="ltx_td ltx_align_center">7.787</td>
<td class="ltx_td ltx_align_center">1.348</td>
<td class="ltx_td ltx_align_center">1.129</td>
<td class="ltx_td ltx_align_center">7.475</td>
<td class="ltx_td ltx_align_center">11.365</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.000</td>
<td class="ltx_td ltx_align_center">8.256</td>
<td class="ltx_td ltx_align_center">1.326</td>
<td class="ltx_td ltx_align_center">1.755</td>
<td class="ltx_td ltx_align_center">7.244</td>
<td class="ltx_td ltx_align_center">10.950</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">8.862</td>
<td class="ltx_td ltx_align_center">1.266</td>
<td class="ltx_td ltx_align_center">2.141</td>
<td class="ltx_td ltx_align_center">6.871</td>
<td class="ltx_td ltx_align_center">10.319</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">9.673</td>
<td class="ltx_td ltx_align_center">1.126</td>
<td class="ltx_td ltx_align_center">2.323</td>
<td class="ltx_td ltx_align_center">6.237</td>
<td class="ltx_td ltx_align_center">9.295</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.500</td>
<td class="ltx_td ltx_align_center">10.19</td>
<td class="ltx_td ltx_align_center">0.998</td>
<td class="ltx_td ltx_align_center">2.302</td>
<td class="ltx_td ltx_align_center">5.737</td>
<td class="ltx_td ltx_align_center">8.513</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">10.81</td>
<td class="ltx_td ltx_align_center">0.802</td>
<td class="ltx_td ltx_align_center">2.150</td>
<td class="ltx_td ltx_align_center">5.007</td>
<td class="ltx_td ltx_align_center">7.394</td></tr>
</tbody>
</table>
</div>
<div id="A1.T19" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 19: </span>Data for the polytropic EOS with <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m4" class="ltx_Math" alttext="n=0.8" display="inline"><apply><eq/><ci>𝑛</ci><cn type="float">0.8</cn></apply></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m5" class="ltx_Math" alttext="\kappa=700\," display="inline"><apply><eq/><ci>𝜅</ci><cn type="integer">700</cn></apply></math>km<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m6" class="ltx_Math" alttext="{}^{2.5}" display="inline"><apply><cn type="float">2.5</cn></apply></math></div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m7" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m8" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m9" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m10" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m11" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m12" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m13" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T19.m14" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.800</td>
<td class="ltx_td ltx_align_center">7.832</td>
<td class="ltx_td ltx_align_center">1.609</td>
<td class="ltx_td ltx_align_center">0.630*</td>
<td class="ltx_td ltx_align_center">7.601</td>
<td class="ltx_td ltx_align_center">11.626</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.750</td>
<td class="ltx_td ltx_align_center">7.853</td>
<td class="ltx_td ltx_align_center">1.609</td>
<td class="ltx_td ltx_align_center">0.281</td>
<td class="ltx_td ltx_align_center">7.602</td>
<td class="ltx_td ltx_align_center">11.624</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.700</td>
<td class="ltx_td ltx_align_center">7.874</td>
<td class="ltx_td ltx_align_center">1.609</td>
<td class="ltx_td ltx_align_center">0.459</td>
<td class="ltx_td ltx_align_center">7.604</td>
<td class="ltx_td ltx_align_center">11.623</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.600</td>
<td class="ltx_td ltx_align_center">7.917</td>
<td class="ltx_td ltx_align_center">1.609</td>
<td class="ltx_td ltx_align_center">0.690</td>
<td class="ltx_td ltx_align_center">7.606</td>
<td class="ltx_td ltx_align_center">11.618</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.500</td>
<td class="ltx_td ltx_align_center">7.961</td>
<td class="ltx_td ltx_align_center">1.608</td>
<td class="ltx_td ltx_align_center">0.862</td>
<td class="ltx_td ltx_align_center">7.608</td>
<td class="ltx_td ltx_align_center">11.612</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.300</td>
<td class="ltx_td ltx_align_center">8.053</td>
<td class="ltx_td ltx_align_center">1.606</td>
<td class="ltx_td ltx_align_center">1.132</td>
<td class="ltx_td ltx_align_center">7.609</td>
<td class="ltx_td ltx_align_center">11.596</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">4.000</td>
<td class="ltx_td ltx_align_center">8.199</td>
<td class="ltx_td ltx_align_center">1.600</td>
<td class="ltx_td ltx_align_center">1.455</td>
<td class="ltx_td ltx_align_center">7.610</td>
<td class="ltx_td ltx_align_center">11.573</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.500</td>
<td class="ltx_td ltx_align_center">8.470</td>
<td class="ltx_td ltx_align_center">1.579</td>
<td class="ltx_td ltx_align_center">1.868</td>
<td class="ltx_td ltx_align_center">7.577</td>
<td class="ltx_td ltx_align_center">11.477</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">8.778</td>
<td class="ltx_td ltx_align_center">1.539</td>
<td class="ltx_td ltx_align_center">2.199</td>
<td class="ltx_td ltx_align_center">7.501</td>
<td class="ltx_td ltx_align_center">11.314</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.500</td>
<td class="ltx_td ltx_align_center">9.126</td>
<td class="ltx_td ltx_align_center">1.468</td>
<td class="ltx_td ltx_align_center">2.464</td>
<td class="ltx_td ltx_align_center">7.361</td>
<td class="ltx_td ltx_align_center">11.051</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.000</td>
<td class="ltx_td ltx_align_center">9.509</td>
<td class="ltx_td ltx_align_center">1.351</td>
<td class="ltx_td ltx_align_center">2.656</td>
<td class="ltx_td ltx_align_center">7.121</td>
<td class="ltx_td ltx_align_center">10.636</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.500</td>
<td class="ltx_td ltx_align_center">9.908</td>
<td class="ltx_td ltx_align_center">1.161</td>
<td class="ltx_td ltx_align_center">2.741</td>
<td class="ltx_td ltx_align_center">6.711</td>
<td class="ltx_td ltx_align_center">9.969</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">10.251</td>
<td class="ltx_td ltx_align_center">0.865</td>
<td class="ltx_td ltx_align_center">2.649</td>
<td class="ltx_td ltx_align_center">5.998</td>
<td class="ltx_td ltx_align_center">8.856</td></tr>
</tbody>
</table>
</div>
<div id="A1.T20" class="ltx_table">
<div class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 20: </span>Data for the polytropic EOS with <math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m4" class="ltx_Math" alttext="n=0.5" display="inline"><apply><eq/><ci>𝑛</ci><cn type="float">0.5</cn></apply></math> and
<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m5" class="ltx_Math" alttext="\kappa=2\cdot 10^{5}\," display="inline"><apply><eq/><ci>𝜅</ci><apply><ci>⋅</ci><cn type="integer">2</cn><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">5</cn></apply></apply></apply></math>km<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m6" class="ltx_Math" alttext="{}^{4}" display="inline"><apply><cn type="integer">4</cn></apply></math></div>
<table class="ltx_tabular ltx_guessed_headers ltx_align_middle">
<thead class="ltx_thead">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m7" class="ltx_Math" alttext="\rho_{c}\times 10^{15}" display="inline"><apply><times/><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜌</ci><ci>𝑐</ci></apply><apply><csymbol cd="ambiguous">superscript</csymbol><cn type="integer">10</cn><cn type="integer">15</cn></apply></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m8" class="ltx_Math" alttext="R" display="inline"><ci>𝑅</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m9" class="ltx_Math" alttext="M" display="inline"><ci>𝑀</ci></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m10" class="ltx_Math" alttext="\nu_{0}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">0</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m11" class="ltx_Math" alttext="\nu_{1}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">1</cn></apply></math></th>
<th class="ltx_td ltx_align_center ltx_th ltx_th_column"><math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m12" class="ltx_Math" alttext="\nu_{2}" display="inline"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜈</ci><cn type="integer">2</cn></apply></math></th></tr>
</thead>
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">g/cm<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m13" class="ltx_Math" alttext="{}^{3}" display="inline"><apply><cn type="integer">3</cn></apply></math></td>
<td class="ltx_td ltx_align_center">km</td>
<td class="ltx_td ltx_align_center">M<math xmlns="http://www.w3.org/1998/Math/MathML" id="A1.T20.m14" class="ltx_Math" alttext="{}_{\odot}" display="inline"><apply><csymbol cd="latexml">direct-product</csymbol></apply></math></td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td>
<td class="ltx_td ltx_align_center">kHz</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.500</td>
<td class="ltx_td ltx_align_center">8.604</td>
<td class="ltx_td ltx_align_center">2.120</td>
<td class="ltx_td ltx_align_center">0.322*</td>
<td class="ltx_td ltx_align_center">7.308</td>
<td class="ltx_td ltx_align_center">11.220</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.450</td>
<td class="ltx_td ltx_align_center">8.629</td>
<td class="ltx_td ltx_align_center">2.120</td>
<td class="ltx_td ltx_align_center">0.244</td>
<td class="ltx_td ltx_align_center">7.344</td>
<td class="ltx_td ltx_align_center">11.268</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.400</td>
<td class="ltx_td ltx_align_center">8.655</td>
<td class="ltx_td ltx_align_center">2.120</td>
<td class="ltx_td ltx_align_center">0.620</td>
<td class="ltx_td ltx_align_center">7.402</td>
<td class="ltx_td ltx_align_center">11.347</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.300</td>
<td class="ltx_td ltx_align_center">8.708</td>
<td class="ltx_td ltx_align_center">2.120</td>
<td class="ltx_td ltx_align_center">0.848</td>
<td class="ltx_td ltx_align_center">7.452</td>
<td class="ltx_td ltx_align_center">11.412</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.200</td>
<td class="ltx_td ltx_align_center">8.763</td>
<td class="ltx_td ltx_align_center">2.118</td>
<td class="ltx_td ltx_align_center">1.140</td>
<td class="ltx_td ltx_align_center">7.548</td>
<td class="ltx_td ltx_align_center">11.542</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">3.000</td>
<td class="ltx_td ltx_align_center">8.881</td>
<td class="ltx_td ltx_align_center">2.111</td>
<td class="ltx_td ltx_align_center">1.519</td>
<td class="ltx_td ltx_align_center">7.698</td>
<td class="ltx_td ltx_align_center">11.740</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.600</td>
<td class="ltx_td ltx_align_center">9.140</td>
<td class="ltx_td ltx_align_center">2.075</td>
<td class="ltx_td ltx_align_center">2.151</td>
<td class="ltx_td ltx_align_center">8.005</td>
<td class="ltx_td ltx_align_center">12.139</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">2.200</td>
<td class="ltx_td ltx_align_center">9.419</td>
<td class="ltx_td ltx_align_center">1.988</td>
<td class="ltx_td ltx_align_center">2.716</td>
<td class="ltx_td ltx_align_center">8.305</td>
<td class="ltx_td ltx_align_center">12.516</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.800</td>
<td class="ltx_td ltx_align_center">9.672</td>
<td class="ltx_td ltx_align_center">1.809</td>
<td class="ltx_td ltx_align_center">3.235</td>
<td class="ltx_td ltx_align_center">8.555</td>
<td class="ltx_td ltx_align_center">12.804</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.400</td>
<td class="ltx_td ltx_align_center">9.784</td>
<td class="ltx_td ltx_align_center">1.484</td>
<td class="ltx_td ltx_align_center">3.665</td>
<td class="ltx_td ltx_align_center">8.651</td>
<td class="ltx_td ltx_align_center">12.850</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.200</td>
<td class="ltx_td ltx_align_center">9.713</td>
<td class="ltx_td ltx_align_center">1.252</td>
<td class="ltx_td ltx_align_center">3.792</td>
<td class="ltx_td ltx_align_center">8.551</td>
<td class="ltx_td ltx_align_center">12.653</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">1.000</td>
<td class="ltx_td ltx_align_center">9.491</td>
<td class="ltx_td ltx_align_center">0.977</td>
<td class="ltx_td ltx_align_center">3.870</td>
<td class="ltx_td ltx_align_center">8.369</td>
<td class="ltx_td ltx_align_center">12.334</td></tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_center">0.800</td>
<td class="ltx_td ltx_align_center">9.045</td>
<td class="ltx_td ltx_align_center">0.678</td>
<td class="ltx_td ltx_align_center">3.810</td>
<td class="ltx_td ltx_align_center">7.962</td>
<td class="ltx_td ltx_align_center">11.690</td></tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div class="ltx_page_footer">
<div class="ltx_page_logo">Generated  on Thu Oct 27 17:25:13 2016 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img src="data:image/png;base64,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" alt="[LOGO]"/></a></div></div>
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12/05/2016 19:10:13 GroupReduce (GroupReduce at run(FlinkPd.java:40))(1/1) switched to FAILED 
; Line#: 1; Column#: -1
net.sf.saxon.trans.XPathException: Finding root of tree: the context item is absent
    at net.sf.saxon.expr.Expression.dynamicError(Expression.java:1147)
    at net.sf.saxon.expr.RootExpression.getNode(RootExpression.java:93)
    at net.sf.saxon.expr.SingleNodeExpression.evaluateItem(SingleNodeExpression.java:149)
    at net.sf.saxon.expr.SingleNodeExpression.evaluateItem(SingleNodeExpression.java:25)
    at net.sf.saxon.expr.SimpleStepExpression.iterate(SimpleStepExpression.java:86)
    at net.sf.saxon.xpath.XPathExpressionImpl.rawIterator(XPathExpressionImpl.java:213)
    at net.sf.saxon.xpath.XPathExpressionImpl.evaluate(XPathExpressionImpl.java:370)
    at com.formulasearchengine.mathmlquerygenerator.xmlhelper.XMLHelper.getElementsB(XMLHelper.java:184)
    at com.formulasearchengine.mathosphere.mathpd.pojos.ArxivDocument.getMathTags(ArxivDocument.java:34)
    at com.formulasearchengine.mathosphere.mathpd.Distances.getDocumentHistogram(Distances.java:100)
    at com.formulasearchengine.mathosphere.mathpd.Distances.testdist(Distances.java:137)
    at com.formulasearchengine.mathosphere.mathpd.FlinkPd$1.reduce(FlinkPd.java:45)
    at org.apache.flink.runtime.operators.AllGroupReduceDriver.run(AllGroupReduceDriver.java:148)
    at org.apache.flink.runtime.operators.BatchTask.run(BatchTask.java:489)
    at org.apache.flink.runtime.operators.BatchTask.invoke(BatchTask.java:354)
    at org.apache.flink.runtime.taskmanager.Task.run(Task.java:584)
    at java.lang.Thread.run(Thread.java:745)
--------------- linked to ------------------
javax.xml.xpath.XPathExpressionException: net.sf.saxon.trans.XPathException: Finding root of tree: the context item is absent
    at net.sf.saxon.xpath.XPathExpressionImpl.evaluate(XPathExpressionImpl.java:378)
    at com.formulasearchengine.mathmlquerygenerator.xmlhelper.XMLHelper.getElementsB(XMLHelper.java:184)
    at com.formulasearchengine.mathosphere.mathpd.pojos.ArxivDocument.getMathTags(ArxivDocument.java:34)
    at com.formulasearchengine.mathosphere.mathpd.Distances.getDocumentHistogram(Distances.java:100)
    at com.formulasearchengine.mathosphere.mathpd.Distances.testdist(Distances.java:137)
    at com.formulasearchengine.mathosphere.mathpd.FlinkPd$1.reduce(FlinkPd.java:45)
    at org.apache.flink.runtime.operators.AllGroupReduceDriver.run(AllGroupReduceDriver.java:148)
    at org.apache.flink.runtime.operators.BatchTask.run(BatchTask.java:489)
    at org.apache.flink.runtime.operators.BatchTask.invoke(BatchTask.java:354)
    at org.apache.flink.runtime.taskmanager.Task.run(Task.java:584)
    at java.lang.Thread.run(Thread.java:745)
Caused by: net.sf.saxon.trans.XPathException: Finding root of tree: the context item is absent
    at net.sf.saxon.expr.Expression.dynamicError(Expression.java:1147)
    at net.sf.saxon.expr.RootExpression.getNode(RootExpression.java:93)
    at net.sf.saxon.expr.SingleNodeExpression.evaluateItem(SingleNodeExpression.java:149)
    at net.sf.saxon.expr.SingleNodeExpression.evaluateItem(SingleNodeExpression.java:25)
    at net.sf.saxon.expr.SimpleStepExpression.iterate(SimpleStepExpression.java:86)
    at net.sf.saxon.xpath.XPathExpressionImpl.rawIterator(XPathExpressionImpl.java:213)
    at net.sf.saxon.xpath.XPathExpressionImpl.evaluate(XPathExpressionImpl.java:370)
    ... 10 more

12/05/2016 19:10:13 Job execution switched to status FAILING.
12/05/2016 19:10:13 DataSink (TextOutputFormat (/tmp/temp8703750503609043885201498954750741) - UTF-8)(1/1) switched to CANCELED 
12/05/2016 19:10:13 Job execution switched to status FAILED.

org.apache.flink.runtime.client.JobExecutionException: Job execution failed.

    at org.apache.flink.runtime.jobmanager.JobManager$$anonfun$handleMessage$1$$anonfun$applyOrElse$5.apply$mcV$sp(JobManager.scala:563)
    at org.apache.flink.runtime.jobmanager.JobManager$$anonfun$handleMessage$1$$anonfun$applyOrElse$5.apply(JobManager.scala:509)
    at org.apache.flink.runtime.jobmanager.JobManager$$anonfun$handleMessage$1$$anonfun$applyOrElse$5.apply(JobManager.scala:509)
    at scala.concurrent.impl.Future$PromiseCompletingRunnable.liftedTree1$1(Future.scala:24)
    at scala.concurrent.impl.Future$PromiseCompletingRunnable.run(Future.scala:24)
    at akka.dispatch.TaskInvocation.run(AbstractDispatcher.scala:41)
    at akka.dispatch.ForkJoinExecutorConfigurator$AkkaForkJoinTask.exec(AbstractDispatcher.scala:401)
    at scala.concurrent.forkjoin.ForkJoinTask.doExec(ForkJoinTask.java:260)
    at scala.concurrent.forkjoin.ForkJoinPool$WorkQueue.pollAndExecAll(ForkJoinPool.java:1253)
    at scala.concurrent.forkjoin.ForkJoinPool$WorkQueue.runTask(ForkJoinPool.java:1346)
    at scala.concurrent.forkjoin.ForkJoinPool.runWorker(ForkJoinPool.java:1979)
    at scala.concurrent.forkjoin.ForkJoinWorkerThread.run(ForkJoinWorkerThread.java:107)
Caused by: javax.xml.xpath.XPathExpressionException: net.sf.saxon.trans.XPathException: Finding root of tree: the context item is absent
    at net.sf.saxon.xpath.XPathExpressionImpl.evaluate(XPathExpressionImpl.java:378)
    at com.formulasearchengine.mathmlquerygenerator.xmlhelper.XMLHelper.getElementsB(XMLHelper.java:184)
    at com.formulasearchengine.mathosphere.mathpd.pojos.ArxivDocument.getMathTags(ArxivDocument.java:34)
    at com.formulasearchengine.mathosphere.mathpd.Distances.getDocumentHistogram(Distances.java:100)
    at com.formulasearchengine.mathosphere.mathpd.Distances.testdist(Distances.java:137)
    at com.formulasearchengine.mathosphere.mathpd.FlinkPd$1.reduce(FlinkPd.java:45)
    at org.apache.flink.runtime.operators.AllGroupReduceDriver.run(AllGroupReduceDriver.java:148)
    at org.apache.flink.runtime.operators.BatchTask.run(BatchTask.java:489)
    at org.apache.flink.runtime.operators.BatchTask.invoke(BatchTask.java:354)
    at org.apache.flink.runtime.taskmanager.Task.run(Task.java:584)
    at java.lang.Thread.run(Thread.java:745)
Caused by: net.sf.saxon.trans.XPathException: Finding root of tree: the context item is absent
    at net.sf.saxon.expr.Expression.dynamicError(Expression.java:1147)
    at net.sf.saxon.expr.RootExpression.getNode(RootExpression.java:93)
    at net.sf.saxon.expr.SingleNodeExpression.evaluateItem(SingleNodeExpression.java:149)
    at net.sf.saxon.expr.SingleNodeExpression.evaluateItem(SingleNodeExpression.java:25)
    at net.sf.saxon.expr.SimpleStepExpression.iterate(SimpleStepExpression.java:86)
    at net.sf.saxon.xpath.XPathExpressionImpl.rawIterator(XPathExpressionImpl.java:213)
    at net.sf.saxon.xpath.XPathExpressionImpl.evaluate(XPathExpressionImpl.java:370)
    ... 10 more

Process finished with exit code 255
gipplab / mathosphere

earth mover distance #126
