https://selfgravitatingfluids.education/JETohline/index.php?action=history&feed=atom&title=AxisymmetricConfigurations%2FPoissonEq 2026-04-29T17:47:33Z Revision history for this page on the wiki MediaWiki 1.43.1 https://selfgravitatingfluids.education/JETohline/index.php?title=AxisymmetricConfigurations/PoissonEq&diff=1673&oldid=prev 2024-06-30T22:01:34Z

<p><span class="autocomment">Stahler (1983)</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:01, 30 June 2024</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1218">Line 1,218:</td> <td colspan="2" class="diff-lineno">Line 1,218:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;/div&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;/div&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>--&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>--&gt;</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In describing [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)] method, we will first draw upon our "[[Appendix/<del style="font-weight: bold; text-decoration: none;">Equation_templates</del>#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In describing [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)] method, we will first draw upon our "[[Appendix/<ins style="font-weight: bold; text-decoration: none;">EquationTemplates</ins>#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;table border=&quot;0&quot; align=&quot;center&quot; cellpadding=&quot;10&quot;&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;table border=&quot;0&quot; align=&quot;center&quot; cellpadding=&quot;10&quot;&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ Math/EQ TRApproximation }}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ Math/EQ TRApproximation }}</div></td></tr> </table>

Joel2 https://selfgravitatingfluids.education/JETohline/index.php?title=AxisymmetricConfigurations/PoissonEq&diff=1672&oldid=prev 2024-06-30T22:00:13Z

<p><span class="autocomment">Comparison With Other Related Derivations</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:00, 30 June 2024</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l857">Line 857:</td> <td colspan="2" class="diff-lineno">Line 857:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;/table&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;/table&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Notice that the first of these two rewritten expressions aligns perfectly with our "[[Appendix/<del style="font-weight: bold; text-decoration: none;">Equation_templates</del>#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Notice that the first of these two rewritten expressions aligns perfectly with our "[[Appendix/<ins style="font-weight: bold; text-decoration: none;">EquationTemplates</ins>#Other_Equations_with_Assigned_Templates|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;table border=&quot;0&quot; align=&quot;center&quot; cellpadding=&quot;10&quot;&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>&lt;table border=&quot;0&quot; align=&quot;center&quot; cellpadding=&quot;10&quot;&gt;&lt;tr&gt;&lt;td align=&quot;center&quot;&gt;</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ Math/EQ TRApproximation }}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{ Math/EQ TRApproximation }}</div></td></tr> </table>

Joel2 https://selfgravitatingfluids.education/JETohline/index.php?title=AxisymmetricConfigurations/PoissonEq&diff=1403&oldid=prev 2024-06-21T23:57:51Z

<p>Created page with &quot;__FORCETOC__ &lt;!-- __NOTOC__ will force TOC off --&gt; =Solving the (Multi-dimensional) Poisson Equation Numerically= ==Overview== The set of <a href="/JETohline/index.php/PGE#Principal_Governing_Equations" title="PGE">Principal Governing Equations</a> that serves as the foundation of our study of the structure, stability, and dynamical evolution of self-gravitating fluids contains an equation of motion (the &#039;&#039;Euler&#039;&#039; equation) that includes an acceleration due to local gradients in the (Newtonian) gravitational p...&quot;</p> <a href="https://selfgravitatingfluids.education/JETohline/index.php?title=AxisymmetricConfigurations/PoissonEq&amp;diff=1403">Show changes</a>

Joel2