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A user with 3,474 edits. Account created on 10 December 2023.
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22 June 2024

21 June 2024

  • 20:3820:38, 21 June 2024 diff hist 0 N File:MovieWongN4b.gif No edit summary current
  • 20:3720:37, 21 June 2024 diff hist 0 N File:Montage04.png No edit summary current
  • 20:3620:36, 21 June 2024 diff hist 0 N File:MovieWongN2.gif No edit summary current
  • 20:3520:35, 21 June 2024 diff hist 0 N File:Wong73OurFig6b.png No edit summary current
  • 20:3320:33, 21 June 2024 diff hist 0 N File:Wong111N4.png No edit summary current
  • 20:2620:26, 21 June 2024 diff hist +169,163 N Apps/Wong1973Potential Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Wong's (1973) Analytic Potential= ==Technical Background Review== Drawing principally from the published works of {{ Wong73full }} and {{ CT99full }}, in an accompanying discussion we have detailed how the gravitational potential of any finite mass distribution can be determined using toroidal coordinates, rathe..." current
  • 20:2420:24, 21 June 2024 diff hist 0 N File:Wong73VaryingRoverd.gif No edit summary current
  • 20:2320:23, 21 June 2024 diff hist 0 N File:MovieWongComposite.gif No edit summary current
  • 20:2020:20, 21 June 2024 diff hist 0 N File:FlatColorContoursCropped.png No edit summary current
  • 20:0920:09, 21 June 2024 diff hist +68,786 N 2DStructure/ToroidalGreenFunction Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Using Toroidal Coordinates to Determine the Gravitational Potential= NOTE:   An earlier version of this chapter has been shifted to our "Ramblings" Appendix. Here we build upon our accompanying review of..." current
  • 20:0520:05, 21 June 2024 diff hist +139,984 N 2DStructure/ToroidalCoordinates Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Using Toroidal Coordinates to Determine the Gravitational Potential= ==Preface== <table border="1" width="90%" align="center" cellpadding="15"> <tr><td align="left"> <table border="0" cellpadding="8" align="center" width="60%"> <tr><td align="center"> <font size="+1"><b>Abstract</b></font><br />Joel E. Tohline (September, 2017) </td></tr> </table> <table border="0" cellpadding="3" align="center" width="60%"> <tr><td..." current
  • 19:5719:57, 21 June 2024 diff hist +136,010 N AxisymmetricConfigurations/PoissonEq Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Solving the (Multi-dimensional) Poisson Equation Numerically= ==Overview== The set of Principal Governing Equations 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 ''Euler'' equation) that includes an acceleration due to local gradients in the (Newtonian) gravitational p..."
  • 19:5419:54, 21 June 2024 diff hist +106,923 N Apps/DysonWongTori Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Self-Gravitating, Incompressible (Dyson-Wong) Tori= <!--Much of the introductory material of this chapter has been drawn from the paper by [http://adsabs.harvard.edu/abs/1990ApJ...361..394T Tohline & Hachisu (1990)] titled, ''The Breakup of Self-Gravitating Rings, Tori, and Accretion Disks.''--> <table border="1" cellpadding="8" align="center"> <tr> <th algin="center" colspan="5"> Schematics Illustrating ''Anch..." current
  • 19:4919:49, 21 June 2024 diff hist +8,317 N Template:LSU CT99CommonTheme3B Created page with "Beginning with ''his'' integral expression for <math>\Phi(\eta,\theta)|_\mathrm{axisym}</math>, {{ Wong73 }} was able to complete the integrals in ''both'' coordinate directions and obtain an analytic expression for the potential both inside and outside of a uniformly charged (equivalently, uniform-density), circular torus. <font color="orange">This is a remarkable result that has been largely unnoticed and unappreciated by the astrophysics community.</font> We detail..." current
  • 19:4819:48, 21 June 2024 diff hist +4,051 N Template:LSU CT99CommonTheme3A Created page with "Referencing [<b><font color="red">MF53</font></b>], {{ Wong73full }} developed an integral expression for the Coulomb (equivalently, gravitational) potential whose foundation is the single summation, Green's function expression for the reciprocal distance between two points in toroidal coordinates. For axisymmetric mass distributions<sup>‡</sup>, Wong obtained, <table border="0" cellpadding="5" align="center"> <tr> <td align="righ..." current
  • 19:4719:47, 21 June 2024 diff hist +2,551 N Template:LSU CT99CommonTheme3 Created page with "Now, beginning with Version 2 of our expression for the ''Gravitational Potential of Axisymmetric Mass Distributions'', let's also map the (unprimed) cylindrical coordinate pair, <math>~(\varpi, z)</math>, to the same (but, unprimed) toroidal coordinate system, <math>~(\eta,\theta)</math>, and place the toroidal coordinate system's ''anchor ring'' in the equatorial plane of the cylindrical-coordinate system such that, <math>~(\varpi_a,z_a) = (a,0)</math>. This gives, wh..." current
  • 19:4619:46, 21 June 2024 diff hist +3,585 N Template:LSU CT99CommonTheme2A Created page with "Referencing (equivalently, Version 1 of) the above-identified integral expression for the ''Gravitational Potential of an Axisymmetric Mass Distribution,'' [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] offer the following assessment in §6 of their paper: <table border="0" cellpadding="3" align="center" width="60%"> <tr><td align="left"> <font color="darkgreen"> "The important question we have tried to clarify..." current
  • 19:4519:45, 21 June 2024 diff hist +2,795 N Template:LSU CT99CommonTheme2 Created page with "Suppose we rewrite (Version 1 of) the above-highlighted Key integral expression such that the (primed) coordinate location of each mass element is mapped from cylindrical coordinates <math>~(\varpi^', z^')</math> to a toroidal-coordinate system <math>~(\eta^',\theta^')</math> whose ''anchor ring'' cuts through the meridional plane at the cylindrical-coordinate location, <math>~(\varpi_a,z_a)</math>. This desired mapping is handled via the pair of relations,..." current
  • 19:4419:44, 21 June 2024 diff hist +3,185 N Template:LSU CT99CommonTheme1B Created page with "[http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] and, separately, [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] have argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — with <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring e..." current
  • 19:4319:43, 21 June 2024 diff hist +1,393 N Template:LSU CT99CommonTheme1Awhite Created page with "In §102 of a book titled, [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential ''The Theory of the Potential'', MacMillan (1958; originally, 1930)] derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>. Throughout our related discussions, we generally will refer to this addition..." current
  • 19:4119:41, 21 June 2024 diff hist +1,512 N Template:LSU CT99CommonTheme1 Created page with "The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the following integral expression that we will refer to as the, <div align="center"> <font color="#770000">'''Gravitational Potential of an Axisymmetric Mass Distribution (Version 1)'''</font> {{ Math/EQ_CT99Axisymmetric }} </div> and, <math>~K(\mu)</math> is the complete elliptic integral of the first kind. This Key Equation<sup>†</sup> may be strai..." current
  • 19:3819:38, 21 June 2024 diff hist +5,166 N AxisymmetricConfigurations/SolvingPE Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions= You have arrived at this page from our ''Tiled Menu'' by clicking on the chapter title that is also identified, below, by the light-blue-colored icon. You may proceed directly to that chapter by clicking (again) on that highlighted chapter icon. However, we have brought y..." current
  • 19:3719:37, 21 June 2024 diff hist +18,905 N AxisymmetricConfigurations/HSCF Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Introduction= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:white;" |<b>Hachisu Self-<br />Consistent-Field<br />[HSCF]<br />Technique</b> |} Text colored dark green has been extracted ''verbatim'' from..." current
  • 19:3419:34, 21 June 2024 diff hist +11,748 N AxisymmetricConfigurations/Equilibria Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Axisymmetric Configurations (Steady-State Structures)= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:lightgreen;" |<b>Constructing<br />Steady-State<br />Axisymmetric<br />Configurations</b> |} Equilibr..." current
  • 19:3319:33, 21 June 2024 diff hist +29,571 N AxisymmetricConfigurations/PGE Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Axisymmetric Configurations (Governing Equations)= {| class="AxisymmetricConfigurationsPGE" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:lightgreen;" |<b>PGEs<br />for<br />Axisymmetric<br />Systems</b> |} If the self-gravitating configura..." current
  • 19:3219:32, 21 June 2024 diff hist +5,195 N AxisymmetricConfigurations/Storyline Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =(Initially) Axisymmetric Configurations= {| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:black;" |<font color="white">Storyline</font> |} <table border="0" cellpadding="3" align="center" width="60%"> <t..." current
  • 19:3119:31, 21 June 2024 diff hist +55,705 N Apps/GoldreichWeber80 Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Homologously Collapsing Stellar Cores= {| class="n3Dynamics" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffff99;;" |<b>Collapse of<br />an Isolated<br />n = 3</br>Polytrope</b> |} ==Review of Goldreich and Weber (1980)== This is principally a review of the..." current
  • 19:3019:30, 21 June 2024 diff hist +55,422 N SSC/Dynamics/IsothermalSimilaritySolution Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Similarity Solution= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffff99;;" |<b>Similarity<br />Solution</b> |} Much of the material in this chapter has been drawn from §4.1 of a [http://adsabs.harvard.edu/abs/1982FCPh......." current
  • 19:2819:28, 21 June 2024 diff hist +14,380 N SSC/Dynamics/IsothermalCollapse Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Collapse of Isothermal Spheres= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 150px; width: 150px; background-color:#ffff99;;" |<b>Collapse of<br />Isothermal<br />Spheres</b> |}  <br />  <br />  <br />  <br />  <br />  <br />  <br /..." current
  • 19:2319:23, 21 June 2024 diff hist +106,322 N SSC/Dynamics/FreeFall Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Free-Fall= {| class="PolytropeStability" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:#ffff99;;" |<b>Free-Fall<br />Collapse</b> |} In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of phys..." current
  • 19:1919:19, 21 June 2024 diff hist +57,840 N SSC/StabilityConjecture/Bipolytrope51 Created page with "__FORCETOC__ =Examine B-KB74 Conjecture in the Context of <math>(n_c, n_e) = (5 , 1)</math> Bipolytropes= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" |<font size="-1"><b>B-KB74<br />Conjecture<br /> RE: Bipolytrope</b><br />(n<sub>c</sub>, n<sub>e</sub>) = (5, 1)</font> |} In §6 of their paper, [..." current
  • 19:1519:15, 21 June 2024 diff hist +21,588 N SSC/Stability/Yabushita75 Created page with "__FORCETOC__ <!-- __NOTOC__ will force TOC off --> =Stability of a BiPolytrope with an Isothermal Core= This analysis pulls largely from {{ Yabushita75full }}; the focus is on bipolytropes having <math>(n_c, n_e) = (\infty, \tfrac{3}{2})</math>. In an accompanying discussion, we summarize the steps that Yabushita took — from 1968 and 1974, to 1975 — that led up to his discovery of an analytic description of the..." current

23 January 2024

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