Appendix/EquationTemplates

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LSUkey.pngKey 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 , or how to determine the value of the mean molecular weight 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 ...
{{ Math/Template_Name }}

Parallel References
§ no. and (Eq. no.)

Template_Name

Resulting Equation

C67

LL75

H87

ST83

KW94

P00

BLRY07

EQ_Continuity01

Continuity Equation:

LSUkey.png

 

§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:

LSUkey.png

 

§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:

LSUkey.png

 

§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:

LSUkey.png

 

§I.3
(3.5)
Note

 

§6.1
(6.1.4)

§1.3
(1.9)

I: §10.2
(10.1)
Note

Chap. 7

Equations of State

Equations of State

To insert a given equation into any Wiki document, type ...
{{ Math/Template_Name }}

Parallel References
§ no. and (Eq. no.)

Template_Name

Resulting Equation

C67

LL75

H87

ST83

KW94

P00

BLRY07

EQ_EOSideal0A

Ideal Gas Equation of State:

LSUkey.png

§II.1
(1)
Note

§IX.80
(80.8)
Note

§1.1
("n")

§2.3
(2.3.32)
or
(3.2.12)

§13.0
(13.1)
Note

I: §5.6
(5.91)

§5.4
(5.34)

EQ_ZTFG01

Degenerate Electron Pressure:

LSUkey.png

where: 

and:   

———   NOTE:   ———

§X.1
(19)
+
(20)

 

§11.2
(11.41)

§2.3
(2.3.5)
+
(2.3.6)

§15.0
(15.13)
+
(15.14)

I: §5.9.2
(5.156)
+
(5.158)

§5.6.1
(5.86)
+
(5.87)
+
(5.88)

EQ_EOSradiation01

Radiation Pressure:

LSUkey.png

 

 

§12.1
(12.12)
+
(12.15)

 

 

 

§5.6.1
(5.85)

EQ_PressureTotal01

Normalized Total Pressure:

LSUkey.png

 

 

 

 

 

 

 

Traditional Equations of (Spherical) Stellar Structure

Traditional Equations of (Spherical) Stellar Structure

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{{ Math/Template_Name }}

Parallel References
§ no. and (Eq. no.)

Template_Name

Resulting Equation

C67

LL75

H87

ST83

KW94

P00

BLRY07

EQ_SSmassConservation01

Mass Conservation:

LSUkey.png

§IV.2
(6)

 

 

§3.2
(3.2.1)

§2.1
(2.4)

II: §2.2
(2.2)

§5.1
(5.2)

EQ_SShydrostaticBalance01

Hydrostatic Balance:

LSUkey.png

§IV.2
(6)

 

 

§3.2
(3.2.2)

§1.1
(1.2)
Note

II: §2.2
(2.1)

§5.1
(5.1)

EQ_SSLaneEmden01

Polytropic Lane-Emden Equation:

LSUkey.png


§IV.2
(11)
Note

 

 

§3.3
(3.3.6)

§19.2
(19.10)

I: §10.3
(10.4)

 

EQ_SSLaneEmden02

Isothermal Lane-Emden Equation:

LSUkey.png


§IV.22
(374)

 

 

 

§19.8
(19.35)

I: §10.3.3
(10.23)

 

EQ_SSradiationTransport01

Radiation Transport:

LSUkey.png


§IV.22
(374)

 

 

 

§5.1.2
(5.11)

II: §2.2
(2.8)

 

EQ_SSenergyConservation01

Energy Conservation:

LSUkey.png


§IV.22
(374)

 

 

 

§4.2
(4.22)

II: §2.2
(2.18)

 

Stability: Radial Pulsation

Stability: Radial Pulsation

To insert a given equation into any Wiki document, type ...
{{ Math/Template_Name }}

Parallel References
§ no. and (Eq. no.)

Template_Name

Resulting Equation

C67

ST83

KW94

HK94

P00

EQ_RadialPulsation01

LAWE:   Linear Adiabatic Wave (or Radial Pulsation) Equation

LSUkey.png


 

§6.5
(6.5.6)

§38.1
(38.8)

§10.1.1
(10.16)

II: §3.7.1
(3.144)

EQ_RadialPulsation04

Δ-Highlighted LAWE:

LSUkey.png

where:      


 

 

 

 

 

EQ_RadialPulsation02

Polytropic LAWE:

LSUkey.png

where:           and,    


 

 

 

 

 

EQ_RadialPulsation03

Isothermal LAWE:

LSUkey.png

where:        and,    


 

 

 

 

 

Key Parallel References (printed 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 , instead of the mass density ; also, it is initially assumed that = 1. Both and are introduced in §III.1, Eq.(5).
    • EQ_SSLaneEmden01 — At the end of his Chapter IV, C67 writes an extensive history of the earliest work on stellar structure pointing especially the origins of the so-called Lane-Emden equation. He points out, for example, that Ritter (1880) actually published this governing differential equation prior to Emden.


  • [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, ; a term representing acceleration due to gravity, , 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 , rather than , is used to represent the Laplacian spatial operator.
    • EQ_EOSideal0A — In LL75, the ideal gas equation of state is written in terms of the specific volume , as well as in terms of the mass density .


  • [ST83] Shapiro, S. L. & Teukolsky, S. A. 1983, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (New York: John Wiley & Sons); republished in 2004 by WILEY-VCH Verlag GmbH & Co. KGaA
    • EQ_Continuity01 — ST83 present the Eulerian, rather than the Lagrangian form of the Continuity equation.


  • [H87] Huang, K. 1987 (originally 1963), Statistical Mechanics (New York: John Wiley & Sons)
    • EQ_Continuity01 — H87 presents the Eulerian, rather than the Lagrangian form of the Continuity equation, and the variable is used instead of to represent the velocity.
    • EQ_Euler01 — H87 presents the Eulerian, rather than the Lagrangian form of the Euler equation, and the variable is used instead of to represent the velocity. Furthermore, to match the source term in our version of the Euler equation, we must set H87's applied acceleration, .
    • EQ_FirstLaw01 — H87 begins a discussion of the 1st Law of Thermodynamics in the first section of the first chapter, but it does not appear in the form we present (relevant for a "dilute gas") until Eq.(4.31).


  • [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)
    • EQ_Continuity01 — KW94 present the Eulerian, rather than the Lagrangian form of the Continuity equation.
    • EQ_FirstLaw01 — In KW94, the symbol instead of is used to represent the specific internal energy.
    • EQ_EOSideal0A — In KW94, the ideal gas equation of state is actually first introduced in §2.2, Eq.(27), but it is seriously discussed in Chapter 13. KW94 provide a particularly nice explanation of how to calculate the model parameter, .
    • EQ_SShydrostaticBalance01 — In KW94, the hydrostatic balance equation is expressed in terms of instead of ; and the second term on the right-hand-side allows for a net radial acceleration.


  • [HK94] Hansen, C. J. & Kawaler, S. D. 1994, Stellar Interiors: Physical Principles, Structure, and Evolution (New York: Springer)


  • [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).


  • [BLRY07] Bodenheimer, P., Laughlin, G. P., Różyczka, M. & Yorke, H. W. 2007, Numerical Methods in Astrophysics An Introduction (New York: Taylor & Francis)

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

Density in terms of enthalpy for polytrope.

EQ_EOSideal00

Alternate form of the ideal gas equation of state.

EQ_EOSideal02

Alternate form of the ideal gas equation of state.

EQ_TRApproximation

LSUkey.png

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

EQ_CT99Axisymmetric

LSUkey.png

Gravitational potential of any axisymmetric mass distribution.


 

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