# Main Page

## Preamble

Much of the astrophysics community's present understanding of the structure, stability, and dynamical evolution of individual stars, short-period binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) are derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations — also heavily used to model continuum flows in terrestrial environments — are thought to govern the underlying physics of the vast majority of macroscopic fluid configurations in astronomy. Although relatively simple in form, they prove to be very rich in nature.

The literature on this subject is enormous, as serious discussions of the structure and dynamical properties of stars and galaxies date back well over a century. The primary purpose of this work is two-fold:

1. To document in an electronically accessible format many of the key physical principles that underlie modern discussions of the structure, stability, and dynamics of self-gravitating (astrophysical) fluid systems.
2. To take advantage of the added dimensions offered by the hypertext medium — such as color, text/equation linkages, animation, and virtual reality environments — to effectively illustrate many of these physical principles.

We have adopted MediaWiki as the hosting environment of choice most significantly because, after incorporating the proper set of extensions, it facilitates the insertion of complex, LaTeX-formulated mathematical expressions into the text.

## Highlights

• Introductory discussions of the Principal Governing Equations.
• Roughly 50 chapters that examine the structure, stability, and dynamical evolution of (1D) spherically symmetric configurations.
• Approximately 30 chapters that focus on the properties and behavior of (2D) axisymmetric configurations.
• Approximately 15 chapters that review what is presently understood about the structure and dynamical evolution of fully 3D configurations.

April 2022:  Presently our Tiled Menu provides links to roughly 100 separate chapter discussions; these chapters, in turn, contain links to at least a hundred additional pages of supporting material. These numbers will steadily increase as we continue to examine the behavior of a wider variety of astrophysical fluid systems.

### Classic Works

1. Maclaurin's (1742) Original Text & Analysis
2. Bernhard Riemann's (1861) collected works
3. J. H. Lane (1870)
4. Eddington's (1926) Derivation of the LAWE
5. Chandrasekhar Limiting (White Dwarf) Mass (1935)
6. Schönberg - Chandrasekhar Mass (1942)
7. Bonnor - Ebert Isothermal Spheres (1955 - 56)
8. S. Chandrasekhar's (1969) Ellipsoidal Figures of Equilibrium
9. Papaloizou - Pringle Tori (1984)

### Under-Appreciated Works

1. Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids
2. Srivastava's (1968) analytic (F-type) solution to the Lane-Emden equation of index, ${\displaystyle n=5}$ — hereinafter referred to as ${\displaystyle \theta _{5F}(\xi )}$.
3. Wong's (1973) Analytic Potential for a Uniform-Density Torus
4. Yabushita's (1974) Analytic Eigenvector for Marginally Unstable, Pressure-Truncated Isothermal Spheres
5. Hayashi, Narita, & Miyama's (1982) Analytic Description of Rotating Isothermal Configurations with Flat Rotation Curves
6. Murphy's (1985) Analytic Prescription of the Equilibrium Structure of ${\displaystyle (n_{c},n_{e})=(1,5)}$ Bipolytropes
7. Eggleton, Faulkner & Cannon's (1998) Analytic Prescription of the Equilibrium Structure of ${\displaystyle (n_{c},n_{e})=(5,1)}$ Bipolytropes

### Our (Tohline's) Recent Contributions

1. The maximum of Srivastava's ${\displaystyle \theta _{5F}(\xi )}$ function occurs precisely when the function argument, ${\displaystyle \xi =\xi _{\mathrm {crit} }\equiv e^{2\tan ^{-1}(1+2^{1/3})}.}$
2. Analytic Determination of the Eigenvector Associated with Marginally Unstable, Pressure-Truncated Polytropic Spheres
3. The task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having any surface shape has been reduced to a problem of carrying out a single, line integration.
4. Exact demonstration of the validity of the B-KB74 conjecture — see 📚 Bisnovatyi-Kogan & Blinnikov (1974) — in the context of spherically symmetric, pressure-truncated, ${\displaystyle n=5}$ polytropes.
5. Analytic Prescription of the Trajectories of Lagrangian Fluid Elements in Riemann Type I Ellipsoids
6. Virtual Reality:  Riemann meets COLLADA & Oculus Rift S; see, for example, our Table of Accessible COLLADA Models