Difference between revisions of "User:Tohline/Appendix/Equation templates"

Revision as of 20:13, 24 January 2010

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{{ User:Tohline/Math/Template_Name }}

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: §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: §8.5
(8.53)

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

I: §10.2
(10.1)
Note

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

§II.1
(1)
Note

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_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.

[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).

@@ Line 166: / Line 166: @@
 * [LL75] '''Laundau, L. D. &amp; Lifshitz, E. M.''' 1975 (originally, 1959), Fluid Mechanics (New York: Pergamon Press)
+** <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.
 * [BT87] '''Binney, J. &amp; Tremaine, S.''' 1987, Galactic Dynamics (Princeton, NJ: Princeton University Press)
 * [KW94] '''Kippenhahn, R. &amp; Weigert, A.''' 1994, Stellar Structure and Evolution (New York: Springer-Verlag)