User:Tohline/Appendix/Equation templates

Key Equations

To insert a given equation into any Wiki document, type ... {{ User:Tohline/Math/Template_Name }}		Parallel References (Section & Eq. #)
Template_Name	Resulting Equation	C67	LL75	BT87	KW94	P00
EQ_Continuity01	<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>		§I.1 (1.2) Note			I: §8.5 (8.45)
EQ_Euler01	<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>		§I.2 (2.1) Note			I: §8.5 (8.48)
EQ_FirstLaw01	<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>		§I.2 (2.5) Note			I: §8.5 (8.53)
EQ_Poisson01	<math>\nabla^2 \Phi = 4\pi G \rho</math>		§I.3 (3.5) Note			I: §10.2 (10.1) Note
EQ_EOSideal00	<math>~P = n_g k T</math>
EQ_EOSideal0A	<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>	§II.1 (1) Note				I: §5.6 (5.91)
EQ_EOSideal02	<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

 

Key Parallel Printed Reference Texts (spanning 5+ decades)

• [C67] Chandrasekhar, S. 1967 (originally, 1939), An Introduction to the Study of Stellar Structure (New York: Dover)
  • EQ_EOSideal0A — In C67, the ideal gas equation of state is initially written in terms of the specific volume <math>~V</math>, instead of the mass density <math>~\rho</math>; also, it is initially assumed that <math>~\bar{\mu}</math> = 1. Both <math>~\rho</math> and <math>~\bar{\mu}</math> are introduced in §III.1, Eq.(5).

• [LL75] Laundau, L. D. & Lifshitz, E. M. 1975 (originally, 1959), Fluid Mechanics (New York: Pergamon Press)
  • EQ_Euler01 — In the Euler equation, LL75 do not initially include a source term to account for a gradient in the Newtonian gravitational potential, <math>~\Phi</math>; 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.
  • EQ_Poisson01 — In LL75, the symbol <math>\Delta</math>, rather than <math>\nabla^2</math>, is used to represent the Laplacian spatial operator.

• [BT87] Binney, J. & Tremaine, S. 1987, Galactic Dynamics (Princeton, NJ: Princeton University Press)
• [KW94] Kippenhahn, R. & Weigert, A. 1994, Stellar Structure and Evolution (New York: Springer-Verlag)
• [P00] 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)
  • EQ_Poisson01 — See also Vol.I: §10.4, Eq.(10.58).

 

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