Appendix/EquationTemplates: Difference between revisions

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Revision as of 14:39, 30 June 2021


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:

 

§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:

 

§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:

 

§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:

 

§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

Eulerian (and Conservative) form of the continuity equation.

EQ_Euler02

Eulerian form of the Euler equation.

EQ_Euler03

Conservative form of the Euler equation.

EQ_Euler04

Euler equation in terms of vorticity.

EQ_FirstLaw02

Adiabatic form of the 1st Law of Thermodynamics.

EQ_Polytrope01

Polytropic equation of state.

EQ_Polytrope02

Enthalpy in a polytrope.

EQ_Polytrope03

User:Tohline/Math/EQ Polytrope03

Density in terms of enthalpy for polytrope.

EQ_EOSideal00

User:Tohline/Math/EQ EOSideal00

Alternate form of the ideal gas equation of state.

EQ_EOSideal02

Alternate form of the ideal gas equation of state.

EQ_TRApproximation

User:Tohline/Math/EQ TRApproximation

Gravitational potential exterior to an axisymmetric torus,
in the Thin Ring (TR) Approximation.

EQ_CT99Axisymmetric

User:Tohline/Math/EQ CT99Axisymmetric

Gravitational potential of any axisymmetric mass distribution.


 

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