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=[[File:LSUkey.png|50px]]<font size="+2" color="black">Key Equations</font>= Each of the equations displayed in the Tables, below, encapsulates a physical concept that is fundamental to our understanding of — and, hence our discussion of — the '''structure, stability, and dynamics of self-gravitating fluids.''' The pervasiveness of these physical concepts throughout astrophysics is reflected in the fact that the same equations — perhaps written in slightly different forms — appear in numerous published books and research papers. When attempting to understand the physical concept that is associated with any one of these mathematical relations, it can be helpful to read how and in what context different authors have introduced the expression in their own work. These Tables offer guides to some parallel discussions that have appeared in published texts over the past 5+ decades in connection with selected sets of key physical relations. <font color="darkgreen">EXAMPLE:</font> Suppose you want to gain a better understanding of the origin of the ideal gas equation of state, the definition of the gas constant {{ Template:Math/C_GasConstant }}, or how to determine the value of the mean molecular weight {{Template:Math/MP_MeanMolecularWeight}} of a gas. According to the Table entitled ''Equations of State'', you will find a discussion of the ideal gas equation of state: near Eq. (1) in §II.1 of Chandrasekhar (1967); near Eq. (80.8) in §IX.80 of Landau & Lifshitz (1975); near Eq. (5.91) in Vol. I, §5.6 of Padmanabhan (2000); etc. A "note" (linked to a comment further down on this page) appears along with a table entry if the relevant equation in the cited reference contains notations or symbol names that differ significantly from the equation as displayed here. ==Principal Governing Equations== <span id="PGE"> <div align="center"> <table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue"> <tr> <th colspan=9 align="center"> <font size="+1" color="darkblue">Principal Governing Equations</font> </th> </tr> <tr> <td colspan=2> To insert a given equation into any Wiki document, type ...<br /><center> {{ Math/<i><font color="red">Template_Name</font></i> }}</center> </td> <td colspan=7 align="center"> <font color="red">Parallel References</font> <br />§ no. and (Eq. no.) </td> </tr> <tr> <th width="15%"> <font color="red">Template_Name</font> </th> <th width="40%"> <font color="red">Resulting Equation</font> </th> <th colspan=1> <font color="red">C67</font> </th> <th colspan=1> <font color="red">LL75</font> </th> <th colspan=1> <font color="red">H87</font> </th> <th colspan=1> <font color="red">ST83</font> </th> <th colspan=1> <font color="red">KW94</font> </th> <th colspan=1> <font color="red">P00</font> </th> <th colspan=1> <font color="red">BLRY07</font> </th> </tr> <tr> <td> [[Template:Math/EQ_Continuity01|EQ_Continuity01]] </td> <td align="center"> Continuity Equation:<br /> {{ Math/EQ_Continuity01 }} </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §I.1<br /> (1.2)<br />[[#LL75note_Continuity01|Note]] </td> <td colspan=1 align="center"> §5.4<br /> (5.37)<br />[[#H87note_Continuity01|Note]] </td> <td colspan=1 align="center"> §6.1<br /> (6.1.1)<br />[[#ST83note_Continuity01|Note]] </td> <td colspan=1 align="center"> §2.5<br /> (2.22)<br />[[#KW94note_Continuity01|Note]] </td> <td colspan=1 align="center"> I: §8.5<br /> (8.45) </td> <td colspan=1 align="center"> §1.4<br /> (1.53) </td> </tr> <tr> <td> [[Template:Math/EQ_Euler01|EQ_Euler01]] </td> <td align="center"> Euler Equation:<br /> {{ Math/EQ_Euler01 }} </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §I.2<br /> (2.1)<br />[[#LL75note_Euler01|Note]] </td> <td colspan=1 align="center"> §5.4<br /> (5.38)<br />[[#H87note_Euler01|Note]] </td> <td colspan=1 align="center"> §6.1<br /> (6.1.2)<br /> </td> <td colspan=1 align="center"> §2.5<br /> (2.20) </td> <td colspan=1 align="center"> I: §8.5<br /> (8.48) </td> <td colspan=1 align="center"> §1.4<br /> (1.55) </td> </tr> <tr> <td> [[Template:Math/EQ_FirstLaw01|EQ_FirstLaw01]] </td> <td align="center"> 1<sup>st</sup> Law of Thermodynamics:<br /> {{ Math/EQ_FirstLaw01 }} </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §I.2<br /> (2.5)<br />[[#LL75note_FirstLaw01|Note]] </td> <td colspan=1 align="center"> §4.2<br /> (4.31)<br />[[#H87note_FirstLaw01|Note]] </td> <td colspan=1 align="center"> §6.1<br /> (6.1.8)<br /> </td> <td colspan=1 align="center"> §4.1<br /> (4.1)<br />[[#KW94note_FirstLaw01|Note]] </td> <td colspan=1 align="center"> I: §8.5<br /> (8.53) </td> <td colspan=1 align="center"> </td> </tr> <tr> <td> [[Template:Math/EQ_Poisson01|EQ_Poisson01]] </td> <td align="center"> Poisson Equation:<br /> {{Template:Math/EQ_Poisson01}} </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §I.3<br /> (3.5)<br />[[#LL75note_Poisson01|Note]] </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §6.1<br /> (6.1.4)<br /> </td> <td colspan=1 align="center"> §1.3<br /> (1.9) </td> <td colspan=1 align="center"> I: §10.2<br /> (10.1)<br />[[#P00note_Poisson01|Note]] </td> <td colspan=1 align="center"> Chap. 7 </td> </tr> </table> </div> </span> ==Equations of State== <span id="EOS"> <div align="center"> <table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue"> <tr> <th colspan=9 align="center"> <font size="+1" color="darkblue">Equations of State</font> </th> </tr> <tr> <td colspan=2> To insert a given equation into any Wiki document, type ...<br /><center> {{ Math/<i><font color="red">Template_Name</font></i> }}</center> </td> <td colspan=7 align="center"> <font color="red">Parallel References</font> <br />§ no. and (Eq. no.) </td> </tr> <tr> <th width="15%"> <font color="red">Template_Name</font> </th> <th width="40%"> <font color="red">Resulting Equation</font> </th> <th colspan=1> <font color="red">C67</font> </th> <th colspan=1> <font color="red">LL75</font> </th> <th colspan=1> <font color="red">H87</font> </th> <th colspan=1> <font color="red">ST83</font> </th> <th colspan=1> <font color="red">KW94</font> </th> <th colspan=1> <font color="red">P00</font> </th> <th colspan=1> <font color="red">BLRY07</font> </th> </tr> <tr> <td> [[Template:Math/EQ_EOSideal0A|EQ_EOSideal0A]] </td> <td align="center"> Ideal Gas Equation of State:<br /> {{ Math/EQ_EOSideal0A }} </td> <td colspan=1 align="center"> §II.1<br /> (1) <br />[[#C67note_EOSideal0A|Note]] </td> <td colspan=1 align="center"> §IX.80<br /> (80.8) <br />[[#LL75note_EOSideal0A|Note]] </td> <td colspan=1 align="center"> §1.1<br /> ("n") </td> <td colspan=1 align="center"> §2.3<br /> (2.3.32)<br /> or <br /> (3.2.12) </td> <td colspan=1 align="center"> §13.0<br /> (13.1) <br />[[#KW94note_EOSideal0A|Note]] </td> <td colspan=1 align="center"> I: §5.6<br /> (5.91) </td> <td colspan=1 align="center"> §5.4<br /> (5.34) </td> </tr> <tr> <td> [[Template:Math/EQ_ZTFG01|EQ_ZTFG01]] </td> <td align="center"> Degenerate Electron Pressure:<br /> {{ Math/EQ_ZTFG01 }} ——— <font color="darkgreen">NOTE:</font> ———<br /> <math> F(\chi) = \frac{8}{5}\chi^5 - \frac{4}{7}\chi^7 + \cdots ~~~~~~(\mathrm{for}~~ \chi\ll 1) </math> <math> F(\chi) = 2\chi^4 - 2\chi^2 + \cdots ~~~~~~~(\mathrm{for}~~ \chi\gg 1) </math> </td> <td colspan=1 align="center"> §X.1<br /> (19) <br /> + <br /> (20) </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §11.2<br /> (11.41) </td> <td colspan=1 align="center"> §2.3<br /> (2.3.5)<br /> + <br /> (2.3.6) </td> <td colspan=1 align="center"> §15.0<br /> (15.13)<br /> + <br /> (15.14) </td> <td colspan=1 align="center"> I: §5.9.2<br /> (5.156)<br /> + <br /> (5.158) </td> <td colspan=1 align="center"> §5.6.1<br /> (5.86)<br /> + <br /> (5.87)<br /> + <br /> (5.88) </td> </tr> <tr> <td> [[Template:Math/EQ_EOSradiation01|EQ_EOSradiation01]] </td> <td align="center"> Radiation Pressure:<br /> {{ Math/EQ_EOSradiation01 }} </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §12.1<br /> (12.12)<br /> +<br /> (12.15) </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §5.6.1<br /> (5.85) </td> </tr> <tr> <td> [[Template:Math/EQ_PressureTotal01|EQ_PressureTotal01]] </td> <td align="center"> Normalized Total Pressure:<br /> {{ Math/EQ_PressureTotal01 }} </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> </tr> </table> </div> </span> ==Traditional Equations of (Spherical) Stellar Structure== <span id="SS"> <div align="center"> <table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue"> <tr> <th colspan=9 align="center"> <font size="+1" color="darkblue">Traditional Equations of (Spherical) Stellar Structure</font> </th> </tr> <tr> <td colspan=2> To insert a given equation into any Wiki document, type ...<br /><center> {{ Math/<i><font color="red">Template_Name</font></i> }}</center> </td> <td colspan=7 align="center"> <font color="red">Parallel References</font> <br />§ no. and (Eq. no.) </td> </tr> <tr> <th width="15%"> <font color="red">Template_Name</font> </th> <th width="40%"> <font color="red">Resulting Equation</font> </th> <th colspan=1> <font color="red">C67</font> </th> <th colspan=1> <font color="red">LL75</font> </th> <th colspan=1> <font color="red">H87</font> </th> <th colspan=1> <font color="red">ST83</font> </th> <th colspan=1> <font color="red">KW94</font> </th> <th colspan=1> <font color="red">P00</font> </th> <th colspan=1> <font color="red">BLRY07</font> </th> </tr> <tr> <td> [[Template:Math/EQ_SSmassConservation01|EQ_SSmassConservation01]] </td> <td align="center"> Mass Conservation:<br /> {{ Math/EQ_SSmassConservation01 }} </td> <td colspan=1 align="center"> §IV.2<br /> (6) </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §3.2<br /> (3.2.1) </td> <td colspan=1 align="center"> §2.1<br /> (2.4) </td> <td colspan=1 align="center"> II: §2.2<br /> (2.2) </td> <td colspan=1 align="center"> §5.1<br /> (5.2) </td> </tr> <tr> <td> [[Template:Math/EQ_SShydrostaticBalance01|EQ_SShydrostaticBalance01]] </td> <td align="center"> Hydrostatic Balance:<br /> {{ Math/EQ_SShydrostaticBalance01 }} </td> <td colspan=1 align="center"> §IV.2<br /> (6) </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §3.2<br /> (3.2.2) </td> <td colspan=1 align="center"> §1.1<br /> (1.2) <br />[[#KW94note_SShydrostaticBalance|Note]] </td> <td colspan=1 align="center"> II: §2.2<br /> (2.1) </td> <td colspan=1 align="center"> §5.1<br /> (5.1) </td> </tr> <tr> <td> [[Template:Math/EQ_SSLaneEmden01|EQ_SSLaneEmden01]] </td> <td align="center"> Polytropic Lane-Emden Equation:<br /> {{ Math/EQ_SSLaneEmden01 }}<br /> <!-- [[File:EmdenGaskugeln1907.jpg|200px|thumb|left|Emden (1907)]] [[File:RitterEquation1880.jpg|90px|thumb|right|Ritter (1880)]] --> </td> <td colspan=1 align="center"> §IV.2<br /> (11) <br />[[#C67note_SSLaneEmden01|Note]] </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §3.3<br /> (3.3.6) </td> <td colspan=1 align="center"> §19.2<br /> (19.10) </td> <td colspan=1 align="center"> I: §10.3<br /> (10.4) </td> <td colspan=1 align="center"> </td> </tr> <tr> <td> [[Template:Math/EQ_SSLaneEmden02|EQ_SSLaneEmden02]] </td> <td align="center"> Isothermal Lane-Emden Equation:<br /> {{ Math/EQ_SSLaneEmden02 }}<br /> </td> <td colspan=1 align="center"> §IV.22<br /> (374) <br /> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §19.8<br /> (19.35) </td> <td colspan=1 align="center"> I: §10.3.3<br /> (10.23) </td> <td colspan=1 align="center"> </td> </tr> <tr> <td> [[Template:Math/EQ_SSradiationTransport01|EQ_SSradiationTransport01]] </td> <td align="center"> Radiation Transport:<br /> {{ Math/EQ_SSradiationTransport01 }}<br /> </td> <td colspan=1 align="center"> §IV.22<br /> (374) <br /> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §5.1.2<br /> (5.11) </td> <td colspan=1 align="center"> II: §2.2<br /> (2.8) </td> <td colspan=1 align="center"> </td> </tr> <tr> <td> [[Template:Math/EQ_SSenergyConservation01|EQ_SSenergyConservation01]] </td> <td align="center"> Energy Conservation:<br /> {{ Math/EQ_SSenergyConservation01 }}<br /> </td> <td colspan=1 align="center"> §IV.22<br /> (374) <br /> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §4.2<br /> (4.22) </td> <td colspan=1 align="center"> II: §2.2<br /> (2.18) </td> <td colspan=1 align="center"> </td> </tr> </table> </div> </span> ==Stability: Radial Pulsation== <span id="RadialStability"> <div align="center"> <table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue"> <tr> <th colspan="7" align="center"> <font size="+1" color="darkblue">Stability: Radial Pulsation</font> </th> </tr> <tr> <td colspan=2> To insert a given equation into any Wiki document, type ...<br /><center> {{ Math/<i><font color="red">Template_Name</font></i> }}</center> </td> <td colspan=5 align="center"> <font color="red">Parallel References</font> <br />§ no. and (Eq. no.) </td> </tr> <tr> <th width="15%"> <font color="red">Template_Name</font> </th> <th width="40%"> <font color="red">Resulting Equation</font> </th> <th colspan=1> <font color="red">C67</font> </th> <th colspan=1> <font color="red">ST83</font> </th> <th colspan=1> <font color="red">KW94</font> </th> <th colspan=1> <font color="red">HK94</font> </th> <th colspan=1> <font color="red">P00</font> </th> </tr> <tr> <td> [[Template:Math/EQ_RadialPulsation01|EQ_RadialPulsation01]] </td> <td align="center"> LAWE: Linear Adiabatic Wave (or Radial Pulsation) Equation<br /> {{ Math/EQ_RadialPulsation01 }}<br /> <!-- [[File:Eddington1930Cover.png|100px|thumb|right|Eddington (1926)]] --> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> §6.5<br /> (6.5.6) </td> <td colspan=1 align="center"> §38.1<br /> (38.8) </td> <td colspan=1 align="center"> §10.1.1<br /> (10.16) </td> <td colspan=1 align="center"> II: §3.7.1<br /> (3.144) </td> </tr> <tr> <td> [[Template:Math/EQ_RadialPulsation04|EQ_RadialPulsation04]] </td> <td align="center"> Δ-Highlighted LAWE:<br /> {{ Math/EQ_RadialPulsation04 }}<br /> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> </tr> <tr> <td> [[Template:Math/EQ_RadialPulsation02|EQ_RadialPulsation02]] </td> <td align="center"> Polytropic LAWE:<br /> {{ Math/EQ_RadialPulsation02 }}<br /> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> </tr> <tr> <td> [[Template:Math/EQ_RadialPulsation03|EQ_RadialPulsation03]] </td> <td align="center"> Isothermal LAWE:<br /> {{ Math/EQ_RadialPulsation03 }}<br /> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> <td colspan=1 align="center"> </td> </tr> </table> </div> </span>
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