User:Tohline/Appendix/Equation templates

Key Equations

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Parallel References (Section & Eq. #)

Template_Name

Resulting Equation

C67

LL75

BT87

KW94

P00

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

§I.1
(1.2)
Note

I: §8.5
(8.45)

<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

§I.2
(2.1)
Note

I: §8.5
(8.48)

<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)

<math>\nabla^2 \Phi = 4\pi G \rho</math>

§I.3
(3.5)
Note

I: §10.2
(10.1)
Note

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

§II.1
(1)
Note

§IX.80
(80.8)

I: §5.6
(5.91)

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

[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_Continuity01 — LL75 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
- 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_FirstLaw01 — 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.
- 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).