View source for User:Tohline/Appendix/Equation templates

{{LSU_HBook_header}}

==<font size="+2" color="darkblue">Key Equations</font>==

Each of the equations displayed in the Table, below, encapsulates a physical concept that is fundamental to our understanding of &#8212; and, hence our discussion of &#8212; the '''structure, stability, and dynamics of self-gravitating systems.'''  The pervasiveness of these physical concepts throughout astrophysics is reflected in the fact that the same equations &#8212; perhaps written in slightly different forms &#8212; 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.  The Table offers a guide to some parallel discussions that have appeared in published texts over the past 5+ decades in connection with a selected set of key physical relations. 
   

<div align="center">
<table border=3 cellpadding=5 cellspacing=1 width="95%" bordercolor="darkblue">
<tr>
  <td colspan=2>
To insert a given equation into any Wiki document, type ...<br /><center>
&#123;&#123; User:Tohline/Math/<i><font color="red">Template_Name</font></i> &#125;&#125;</center>
  </td>
  <td colspan=5 align="center">
<font color="red">Parallel References</font> <br />&sect; no. and (Eq. no.) 
  </td>
</tr>
<tr>
  <th width="20%">
<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">BT87</font>
  </th>
  <th colspan=1>
<font color="red">KW94</font>
  </th>
  <th colspan=1>
<font color="red">P00</font>
  </th>
</tr>


<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_Continuity01 EQ_Continuity01]
  </td>
  <td align="center">
Continuity Equation:<br />
{{User:Tohline/Math/EQ_Continuity01}}
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;I.1<br /> (1.2)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#LL75note_Continuity01  Note]
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;2.5<br /> (2.22)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#Kw94note_Continuity01  Note]
  </td>
  <td colspan=1 align="center">
I: &#167;8.5<br /> (8.45)
  </td>
</tr>


<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_Euler01 EQ_Euler01]
  </td>
  <td align="center">
Euler Equation:<br />
{{User:Tohline/Math/EQ_Euler01}}
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;I.2<br /> (2.1)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#LL75note_Euler01  Note]
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;2.5<br /> (2.20)
  </td>
  <td colspan=1 align="center">
I: &#167;8.5<br /> (8.48)
  </td>
</tr>


<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_FirstLaw01 EQ_FirstLaw01]
  </td>
  <td align="center">
1<sup>st</sup> Law of Thermodynamics:<br />
{{User:Tohline/Math/EQ_FirstLaw01}}
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;I.2<br /> (2.5)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#LL75note_FirstLaw01  Note]
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;4.1<br /> (4.1)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#KW94note_FirstLaw01  Note]
  </td>
  <td colspan=1 align="center">
I: &#167;8.5<br /> (8.53)
  </td>
</tr>


<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_Poisson01 EQ_Poisson01]
  </td>
  <td align="center">
Poisson Equation:<br />
{{User:Tohline/Math/EQ_Poisson01}}
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;I.3<br /> (3.5)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#LL75note_Poisson01  Note]
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;1.3<br /> (1.9)
  </td>
  <td colspan=1 align="center">
I: &#167;10.2<br /> (10.1)<br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#P00note_Poisson01  Note]
  </td>
</tr>


<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_EOSideal00 EQ_EOSideal00]
  </td>
  <td align="center">
{{User:Tohline/Math/EQ_EOSideal00}}
  </td>
  <td colspan=5>
&nbsp;
  </td>
</tr>


<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_EOSideal0A EQ_EOSideal0A]
  </td>
  <td align="center">
Ideal Gas Equation of State:<br />
{{User:Tohline/Math/EQ_EOSideal0A}}
  </td>
  <td colspan=1 align="center">
&#167;II.1<br /> (1) <br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#C67note_EOSideal0A  Note]
  </td>
  <td colspan=1 align="center">
&#167;IX.80<br /> (80.8) <br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#LL75note_EOSideal0A  Note]
  </td>
  <td colspan=1 align="center">
&nbsp;
  </td>
  <td colspan=1 align="center">
&#167;13.0<br /> (13.1) <br />[http://www.vistrails.org/index.php/User:Tohline/Appendix/Equation_templates#KW94note_EOSideal0A  Note]
  </td>
  <td colspan=1 align="center">
I: &#167;5.6<br /> (5.91)
  </td>
</tr>

<tr>
  <td>
[http://www.vistrails.org/index.php/User:Tohline/Math/EQ_EOSideal02 EQ_EOSideal02]
  </td>
  <td align="center">
{{User:Tohline/Math/EQ_EOSideal02}}
  </td>
  <td colspan=5>
&nbsp;
  </td>
</tr>


</table>
</div>

&nbsp;<br />

<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 {{User:Tohline/Math/C_GasConstant}}, or how to determine the value of the mean molecular weight {{User:Tohline/Math/MP_MeanMolecularWeight}} of a gas.  According to the Table, you will find a discussion of the ideal gas equation of state: near Eq. (1) in &sect;II.1 of Chandrasekhar (1967); near Eq. (80.8) in &sect;IX.80 of Landau &amp; Lifshitz (1975); near Eq. (5.91) in Vol. I, &sect;5.6 of Padmanabhan (2000); etc. A "note" (linked to a comment at the bottom of this page) appears along with a table entry if the relevant equation in the cited reference contains notations or symbol names that differ from the equation as displayed here.

==<font size="+2" color="darkblue">Key Parallel (Printed Text) References (spanning 5+ decades)</font>==


* [<b><font color="red">C67</font></b>] '''Chandrasekhar, S.''' 1967 (originally, 1939), An Introduction to the Study of Stellar Structure (New York: Dover)
** <span id="C67note_EOSideal0A">EQ_EOSideal0A</span> &#8212; In C67, the ideal gas equation of state is initially written in terms of the specific volume {{User:Tohline/Math/VAR_SpecificVolume01}}, instead of the mass density {{User:Tohline/Math/VAR_Density01}}; also, it is initially assumed that {{User:Tohline/Math/MP_MeanMolecularWeight}} = 1.  Both {{User:Tohline/Math/VAR_Density01}} and {{User:Tohline/Math/MP_MeanMolecularWeight}} are introduced in &#167;III.1, Eq.(5). 

* [<b><font color="red">LL75</font></b>] '''Laundau, L. D. &amp; Lifshitz, E. M.''' 1975 (originally, 1959), Fluid Mechanics (New York: Pergamon Press)
** <span id="LL75note_Continuity01">EQ_Continuity01</span> &#8212; LL75 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
** <span id="LL75note_Euler01">EQ_Euler01</span> &#8212; In the Euler equation, LL75 do not initially include a source term to account for a gradient in the Newtonian gravitational potential, {{User:Tohline/Math/VAR_NewtonianPotential01}}; a term representing acceleration due to gravity, <math>\vec{g} = -\nabla\Phi</math>, is introduced in Eq.(2.4), but in LL75 this is intended primarily to describe gravity at the surface of the Earth.
** <span id="LL75note_FirstLaw01">EQ_FirstLaw01</span> &#8212; LL75's Eq.(2.5) must be combined with their discussion of what they refer to as ''the familiar thermodynamic relation'' (between LL75 Eqs. 2.8 and 2.9) in order to appreciate the similarity with our expression.
** <span id="LL75note_Poisson01">EQ_Poisson01</span> &#8212; In LL75, the symbol <math>\Delta</math>, rather than <math>\nabla^2</math>, is used to represent the Laplacian spatial operator.
** <span id="LL75note_EOSideal0A">EQ_EOSideal0A</span> &#8212; In LL75, the ideal gas equation of state is written in terms of the specific volume {{User:Tohline/Math/VAR_SpecificVolume01}}, as well as in terms of the mass density {{User:Tohline/Math/VAR_Density01}}.

* [<b><font color="red">BT87</font></b>] '''Binney, J. &amp; Tremaine, S.''' 1987, Galactic Dynamics (Princeton, NJ: Princeton University Press)
* [<b><font color="red">KW94</font></b>] '''Kippenhahn, R. &amp; Weigert, A.''' 1994, Stellar Structure and Evolution (New York: Springer-Verlag)
* [<b><font color="red">P00</font></b>] '''Padmanabhan, T.''' 2000, Theoretical Astrophysics.  Volume I: Astrophysical Processes (Cambridge: Cambridge University Press); and Padmanabhan, T. 2001, Theoretical Astrophysics.  Volume II: Stars and Stellar Systems (Cambridge: Cambridge University Press)
** <span id="P00note_Poisson01">EQ_Poisson01</span> &#8212; See also Vol.I: &#167;10.4, Eq.(10.58). 



&nbsp;<br />
{{LSU_HBook_footer}}