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NQG III -- Two-Centre Problems, Whirlpool Galaxy and Toy Neutron Stars

Richard Durran, Aubrey Truman

TL;DR

The work develops a rigorous semi-classical quantisation of the Euler two-centre problem and its extension with a central restoring force, yielding elliptic-spiral stationary states expressible through elliptic-function machinery. A quantum Liouville condition enables tractable, elliptic-integral formulations and a clean connection between quantum states and classical ellipses, with stochastic-mechanics perspectives linking to Burgers–Zeldovich fluids. The authors apply these constructs to a Schrödinger–heat framework that motivates a Burgers-type model of spiral galaxy formation, deriving explicit spiral-arm geometry and potential implications for dark-matter phenomena in galaxies like the Whirlpool (M51). The results fuse classical Liouville integrability, Weierstrass elliptic-function representations, and quantum–stochastic formalisms to provide a new lens on structure formation in astrophysical contexts, including barred and spiral galaxies.

Abstract

In the hunt for WIMPish dark matter and testing our new theory, we extend the results obtained for the Kepler problem in NQG I and NQG II to the Euler two-centre problem and to other classical Hamiltonian systems with planar periodic orbits. In the first case our results lead to quantum elliptical spirals converging to elliptical orbits where stars and other celestial bodies can form as the corresponding WIMP/molecular clouds condense. The examples inevitably involve elliptic integrals as was the case in our earlier work on equatorial orbits of toy neutron stars (see Ref. [27]). Hence this is the example on which we focus in this work on quantisation. The main part of our analysis which leans heavily on Hamilton-Jacobi theory is applicable to any KLMN integrable planar periodic orbits for Hamiltonian systems. The most useful results on Weierstrass elliptic functions needed in these two works we have summarised with complete proofs in the appendix. This has been one of the most enjoyable parts of this research understanding in more detail the genius of Weierstrass and Jacobi. However we have to say that the beautiful simplicity of the Euler two-centre results herein transcend even this as far as we are concerned. At the end of the paper we see how the Burgers-Zeldovich fluid model relates to our set-up through Nelson's stochastic mechanics.

NQG III -- Two-Centre Problems, Whirlpool Galaxy and Toy Neutron Stars

TL;DR

The work develops a rigorous semi-classical quantisation of the Euler two-centre problem and its extension with a central restoring force, yielding elliptic-spiral stationary states expressible through elliptic-function machinery. A quantum Liouville condition enables tractable, elliptic-integral formulations and a clean connection between quantum states and classical ellipses, with stochastic-mechanics perspectives linking to Burgers–Zeldovich fluids. The authors apply these constructs to a Schrödinger–heat framework that motivates a Burgers-type model of spiral galaxy formation, deriving explicit spiral-arm geometry and potential implications for dark-matter phenomena in galaxies like the Whirlpool (M51). The results fuse classical Liouville integrability, Weierstrass elliptic-function representations, and quantum–stochastic formalisms to provide a new lens on structure formation in astrophysical contexts, including barred and spiral galaxies.

Abstract

In the hunt for WIMPish dark matter and testing our new theory, we extend the results obtained for the Kepler problem in NQG I and NQG II to the Euler two-centre problem and to other classical Hamiltonian systems with planar periodic orbits. In the first case our results lead to quantum elliptical spirals converging to elliptical orbits where stars and other celestial bodies can form as the corresponding WIMP/molecular clouds condense. The examples inevitably involve elliptic integrals as was the case in our earlier work on equatorial orbits of toy neutron stars (see Ref. [27]). Hence this is the example on which we focus in this work on quantisation. The main part of our analysis which leans heavily on Hamilton-Jacobi theory is applicable to any KLMN integrable planar periodic orbits for Hamiltonian systems. The most useful results on Weierstrass elliptic functions needed in these two works we have summarised with complete proofs in the appendix. This has been one of the most enjoyable parts of this research understanding in more detail the genius of Weierstrass and Jacobi. However we have to say that the beautiful simplicity of the Euler two-centre results herein transcend even this as far as we are concerned. At the end of the paper we see how the Burgers-Zeldovich fluid model relates to our set-up through Nelson's stochastic mechanics.

Paper Structure

This paper contains 18 sections, 22 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. On the Semi-Classical Mechanics of Euler's Two-Centre Problem and a Quantum Elliptical Spiral Connecting Two Focal Centres
  3. Semi-Classical Analysis of Two Centres
  4. Underlying Classical Mechanics of Euler's Two Centre Problem
  5. Semi-Classical Analysis of Two Gravitational Centres with a Central Linear Restoring Force
  6. The Effective Potential, Stability of Circular Orbits and Potential Wells
  7. Stability of Circular Orbits
  8. $\lambda$-Analysis
  9. Graphical Analysis for small $|\textbf{B}|$
  10. On the $\mathbf{r=r_{0}(\theta)}$ equation for KLMN problem for small $\mathbf{|B|}$, $\mathbf{C^6>\frac{729}{4}\mu^2B^2}$ and $\mathbf{{|}\frac{B}{C}{|}\sim0}$
  11. Semi-Classical Wavefunctions for Stationary States Corresponding to Periodic Orbits
  12. Solving the $R_{0}$ equation
  13. Quantum Mechanics, Elementary Formula and Burgers-Zeldovich Fluids
  14. Stochastic Mechanics, Nelson Newton Law and Burgers Equation
  15. Classical Mechanics as a Limiting Case of Schrödinger Quantum Mechanics
  16. ...and 3 more sections

Key Result

Theorem 1.1

If $z=z(u)=\int\limits_{u_{0}}^{u}\{f(u)\}^{-\frac{1}{2}}du$, $u>u_{0}$, where $f(u)$ is a quartic polynomial with no repeated factors, then where $\wp(z)=\wp(z;g_{2},g_{3})$ is the Weierstrass function formed with the invariants $g_{2}$ and $g_{3}$ of the quartic $f$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • proof
  • Theorem 4.5
  • ...and 23 more