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Whitworth's (1981) Isothermal Free-Energy Surface
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Sometimes I explore some ideas to a sufficient depth that it seems worthwhile for me to archive the technical derivations even if the idea itself does not immediately produce a publishable result. This page, which has a simple outline layout, provides links to these various pages of technical notes.

  1. Orthogonal Curvilinear Coordinate Systems
    1. Direction Cosines
    2. (Confocal) Elliptic Cylinder Coordinates plus Concentric Analog (T5 Coordinates)
    3. Concentric Ellipsoidal (T6) Coordinates
    4. Concentric Ellipsoidal (T8) Coordinates
    5. Concentric Ellipsoidal (T12) Coordinates
    6. Daring Attack
  2. Relationship between HNM82 models and T1 coordinates
  3. Playing with the Spherical Wave Equation
  4. Analyzing Azimuthal Distortions
    1. Summary for Hadley & Imamura
    2. Detailed Notes   🎦
    3. Supplementary database generated by the Hadley & Imamura collaboration
    4. Large supplementary dataset accumulated by the Hadley & Imamura collaboration
    5. YouTube videos that supplement simulations of J. W. Woodward, J. E. Tohline, & I. Hachisu (1994)
    6. Stability Analyses of PP Tori
    7. Stability Analyses of PP Tori (Part 2)
  5. Integrals of Motion
    1. Old discussion
    2. T3 Coordinates
      1. Special (quadratic) case: Joel's Derivation vs. Jay's Derivation
    3. Killing Vector Approach; Jay Call's related Talk page
    4. Characteristic Vector for T3 Coordinates
    5. T4 Coordinates (Abandoned by Joel 7/6/2010 because non-orthogonal)
  6. Marcello's Radiation-Hydro Simulations
    1. Determining Code Units
    2. Summary of Scalings
    3. Initial Temperature Distributions
  7. Photosphere of Stably Accreting DWD
  8. Binary Polytropes
  9. A* Scheme
    1. Initial Effort to Explain Jay Call's Hybrid Scheme in the Context of Zach Byerly's Dissertation
    2. Implications of Hybrid Scheme
  10. Exploring the Properties of Radial Oscillations in Pressure-Truncated n = 5 Polytropes
  11. Instabilities Associated with Equilibrium Sequence Turning Points
  12. Derivations Related to Ledoux's Variational Principle
  13. More on Zero-Zero Bipolytropes
    1. Pt 1: Radial Oscillations of a Zero-Zero-Bipolytrope (Early Flawed Summary)
    2. Pt 2: Details
    3. Pt 3: Searching for Additional Eigenvectors
    4. Pt 4: Good Summary
    5. Numerically Determined Eigenvectors
  14. Analyzing Five-One Bipolytropes
    1. Assessing the Stability of Spherical, BiPolytropic Configurations
    2. Searching for Analytic EigenVector for (5,1) Bipolytropes
    3. Discussing Patrick Motl's 2019 Simulations
    4. Continue Search
  15. On the Origin of Planetary Nebulae (Investigation Resulting from a July, 2013 Discussion with Kundan Kadam)
  16. Looking outward, from Inside a Black Hole
  17. Radial Dependence of the Strong Nuclear Force
  18. Dyson (1893a) Part I:  Some Details
  19. Radiation-Hydrodynamics
  20. Saturn
  21. Doctoral students Tohline has advised over the years
  22. For Richard H. Durisen
  23. For Shangli Ou
  24. For Paul Fisher
  25. For PJ in April 2021
  26. Riemann Meets COLLADA and Oculus Rift S: Example (b/a, c/a) = (0.41, 0.385)
    1. Virtual Reality and 3D Printing
    2. Success Importing Animated Scene into Oculus Rift S
    3. Carefully (Re)Build Riemann Type S Ellipsoids Inside Oculus Rift Environment
    4. Other Example S-type Riemann Ellipsoids:
      1. (b/a, c/a) = (0.90, 0.333)
      2. (b/a, c/a) = (0.74, 0.692)
      3. (b/a, c/a) = (0.28, 0.256)
  27. Challenges Constructing Ellipsoidal-Like Configurations
    1. Riemann Type 1 Ellipsoids
    2. Construction Challenges (Pt. 1)
    3. Construction Challenges (Pt. 2)
    4. Construction Challenges (Pt. 3)
    5. Construction Challenges (Pt. 4)
    6. Construction Challenges (Pt. 5)
    7. Related discussions of models viewed from a rotating reference frame:
      1. PGE
      2. NOTE to Eric Hirschmann & David Neilsen... I have moved the earlier contents of this page to a new Wiki location called Compressible Riemann Ellipsoids.
  28. Bordeaux University
    1. External Gravitational Potential of Toroids
    2. Spheroid-Ring Sequences
    3. Discussions Following Dissertation Defense
  29. Copyright Issues


  1. Roots of Cubic Equation
    1. In the context of T2 Coordinates, when <math>~q^2 = (a_1/a_3)^2=3</math>.
    2. PP Tori — Also includes cube root of a complex number
    3. Srivastava's F-Type solution for <math>~n=5</math> polytropes.
    4. Murphy & Fiedler's Bipolytrope with <math>~(n_c, n_e) = (1,5)</math>
    5. Analytic Eigenfunctions for Bipolytropes with <math>~(n_c, n_e) = (0, 0)</math> — also involves cube root of a complex number
  2. Roots of Quartic Equation
    1. Analytic Eigenfunction for Bipolytropes with <math>~(n_c, n_e) = (0, 0)</math>
    2. Determine temperature from total pressure
  3. Singular Sturm-Liouville (eigenvalue) Problem
    1. Oscillations of PP Tori in the slim torus limit
    2. Characteristics of unstable eigenvectors in self-gravitating tori
  4. Approximate Power-Series Expressions
  5. Fourier Series
  6. Special Functions & Other Broadly Used Representations
    1. Spherical Harmonics and Associated Legendre Functions
    2. Multipole Expansions
    3. Familiar Expression for the Cylindrical Green's Function Expansion
    4. Toroidal Functions
  7. Green's Function in terms of Toroidal Functions
    1. Compact Cylindrical Green Function
    2. Toroidal configurations & related coordinate systems — Includes EUREKA! moment; also uses wikitable overflow (scrolling) box
    3. Toroidal Coordinate Integration Limits <math>~\Leftarrow ~~</math> Includes Table of Example K(k) and E(k) Function Values; see a separate set of K(k) and E(k) evaluations in the context of Our Attempt to Replicate Dyson's results.
    4. Using Toroidal Coordinates to Determine the Gravitational Potential (Initial Presentation)
    5. Using Toroidal Coordinates to Determine the Gravitational Potential (Improved Presentation)   <math>~\Leftarrow</math>    includes series expansions for K(k) and E(k)
    6. Relationships between Toroidal Functions <math>~\Leftarrow ~~</math> 5 plots of [MF53] data included here
    7. Confusion Regarding Whipple Formulae
    8. Pulling It All Together <math>~\Leftarrow ~~</math> 2 additional plots of [MF53] data included here
  8. Scale Factors for Orthogonal Curvilinear Coordinate Systems

Computer-Generated Holography

Computer Generated Holgram (Fall 2004)
in collaboration with Richard Muffoletto
and others from as cited

Muffoletto's CGH
  1. Lead in …
    1. Original Table of Contents
    2. Preface
  2. Apertures that are Parallel to the Image Screen:
    1. One-dimensional Aperture
      1. Initial Ideas
      2. Consolidate Expressions
      3. T. Kreis, P. Aswendt, & R. Höfling (2001), Optical Engineering, vol. 40, no. 6, 926 - 933:   Hologram reconstruction using a digital micromirror device
    2. Two-dimensional, Rectangular Aperture
    3. Relevance to Holograms
    4. Caution and Words of Wisdom
  3. Apertures that are Tilted with Respect to the Image Screen:
  4. Building Holograms from VRML Files:
  5. ZebraImaging and Southwestern Medical Center
  6. Embracing COLLADA (2020)
    1. Principal Illustration
    2. Demonstration Steps
  7. Quantum Mechanics
  8. On 4/15/2021, Google brought the following article to my attention:  S. Igarashi, T. Nakamura, K. Matsushima, & M. Yamaguchi (2018), Optics Express, Vol. 26, Issue 8, pp.10773-10786, Efficient tiled calculation of over-10-gigapixel holograms using ray-wavefront conversion. It heavily references [22] the 2007 (Opt. Express, 15(9), 5631-5640, Shifted Fresnel diffraction for computational holography) work that I published in collaboration with R. Muffoletto and John Tyler.

Computer Algorithms

  1. Directory …/fortran/FreeEnergy/EFE: README
  2. Directory …/numRecipes/EllipticIntegrals/Riemann


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) publication,