Appendix/EquationTemplates

Key Equations

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.

EXAMPLE: 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 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

Principal Governing Equations
To insert a given equation into any Wiki document, type ... {{ Template:Math/Template_Name }}		Parallel References § no. and (Eq. no.)
Template_Name	Resulting Equation	C67	LL75	H87	ST83	KW94	P00	BLRY07
EQ_Continuity01	Continuity Equation: $\frac{d\rho }{dt}+\rho \mathrm{\nabla }\cdot \overrightarrow{v}=0$		§I.1 (1.2) Note	§5.4 (5.37) Note	§6.1 (6.1.1) Note	§2.5 (2.22) Note	I: §8.5 (8.45)	§1.4 (1.53)
EQ_Euler01	Euler Equation: $\frac{d\overrightarrow{v}}{dt}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }$		§I.2 (2.1) Note	§5.4 (5.38) Note	§6.1 (6.1.2)	§2.5 (2.20)	I: §8.5 (8.48)	§1.4 (1.55)
EQ_FirstLaw01	1st Law of Thermodynamics: $T\frac{ds}{dt}=\frac{dϵ}{dt}+P\frac{d}{dt}\left(\frac{1}{\rho }\right)$		§I.2 (2.5) Note	§4.2 (4.31) Note	§6.1 (6.1.8)	§4.1 (4.1) Note	I: §8.5 (8.53)
EQ_Poisson01	Poisson Equation: ${\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho$		§I.3 (3.5) Note		§6.1 (6.1.4)	§1.3 (1.9)	I: §10.2 (10.1) Note	Chap. 7

Other Equations with Assigned Templates

To insert a given equation into any Wiki document, type ... {{ Template:Math/Template_Name }}
Template_Name	Resulting Equation	Description
EQ_Continuity02	$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \overrightarrow{v})=0$	Eulerian (and Conservative) form of the continuity equation.
EQ_Euler02	$\frac{\partial \overrightarrow{v}}{\partial t}+(\overrightarrow{v}\cdot \mathrm{\nabla })\overrightarrow{v}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }\mathrm{\Phi }$	Eulerian form of the Euler equation.
EQ_Euler03	$\frac{\partial (\rho \overrightarrow{v})}{\partial t}+\mathrm{\nabla }\cdot [(\rho \overrightarrow{v})\overrightarrow{v}]=-\mathrm{\nabla }P-\rho \mathrm{\nabla }\mathrm{\Phi }$	Conservative form of the Euler equation.
EQ_Euler04	$\frac{\partial \overrightarrow{v}}{\partial t}+\overrightarrow{\zeta }\times \overrightarrow{v}=-\frac{1}{\rho }\mathrm{\nabla }P-\mathrm{\nabla }[\mathrm{\Phi }+\frac{1}{2}{v}^2]$	Euler equation in terms of vorticity.
EQ_FirstLaw02	$\frac{dϵ}{dt}+P\frac{d}{dt}\left(\frac{1}{\rho }\right)=0$	Adiabatic form of the 1st Law of Thermodynamics.
EQ_Polytrope01	$P=K_{\mathrm{n}}{\rho }^{1+1/n}$	Polytropic equation of state.
EQ_Polytrope02	$H=(n+1)K_{\mathrm{n}}{\rho }^{1/n}$	Enthalpy in a polytrope.
EQ_Polytrope03	$\rho =[\frac{H}{(n+1)K_{\mathrm{n}}}{]}^n$	Density in terms of enthalpy for polytrope.
EQ_EOSideal00	$P=n_{g}kT$	Alternate form of the ideal gas equation of state.
EQ_EOSideal02	$P=({\gamma }_{\mathrm{g}}-1)ϵ\rho$	Alternate form of the ideal gas equation of state.
EQ_TRApproximation		Gravitational potential exterior to an axisymmetric torus, in the Thin Ring (TR) Approximation.
EQ_CT99Axisymmetric		Gravitational potential of any axisymmetric mass distribution.