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Non-Linear Stability of Spherical Systems in MOND

Joachim Frenkler

Abstract

We prove the non-linear stability of a large class of spherically symmetric equilibrium solutions of both the collisonless Boltzmann equation and of the Euler equations in MOND. This is the first such stability result that is proven with mathematical rigour in MOND. While we strive to prove our stability theorems, we develop new, genuinely Mondian ideas how arising mathematical difficulties can be solved. At some points it was necessary to restrict our analysis to spherical symmetry. We discuss every point where this extra assumption was necessary and outline which efforts must be undertaken to get along without it in future works. In the end we show on the example of a polytropic model how our stability result can be applied.

Non-Linear Stability of Spherical Systems in MOND

Abstract

We prove the non-linear stability of a large class of spherically symmetric equilibrium solutions of both the collisonless Boltzmann equation and of the Euler equations in MOND. This is the first such stability result that is proven with mathematical rigour in MOND. While we strive to prove our stability theorems, we develop new, genuinely Mondian ideas how arising mathematical difficulties can be solved. At some points it was necessary to restrict our analysis to spherical symmetry. We discuss every point where this extra assumption was necessary and outline which efforts must be undertaken to get along without it in future works. In the end we show on the example of a polytropic model how our stability result can be applied.
Paper Structure (17 sections, 28 theorems, 305 equations, 2 figures)

This paper contains 17 sections, 28 theorems, 305 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Principle results and basic ideas - Collisionless models
  3. Principle results and basic ideas - Fluid models
  4. Potential theory
  5. Newtonian potentials
  6. Mondian potentials
  7. Potential energy
  8. Finding minimizers
  9. Finding a minimizer in the fluid situation
  10. Euler-Lagrange equation for a minimizer in the fluid situation
  11. Constructing a minimizer in the collisionless situation
  12. Stability
  13. Stability in the collisionless situation
  14. Stability in the fluid situation
  15. Discussion about spherical symmetry
  16. ...and 2 more sections

Key Result

Theorem 1.1

A minimizer $f_0$ of the variational problem variational problem exists if we limit the variational problem to spherically symmetric distribution functions $f$. This minimizer is of the form ansatz where $\phi=(\Phi')^{-1}$. If this minimizer is unique, then every spherically symmetric solution $f(t

Figures (2)

  • Figure 1: We calculated numerically the solution $\rho_s$ of \ref{['equ ode for rho']} with central value $\rho_s(0)=s$. In the above figure we plotted the mass $M_s$ of $\rho_s$ against the central value $s\in[0.01,1]$. We see that in the depicted range for every given mass $M>0$ there is only one $s$ such that $\rho_s$ has mass $M$. The behaviour of the graph of $M_s$ for $s$ very small (the deep MOND limit) and for $s$ very large (the Newtonian limit) is given in Figure \ref{['figure-mass-of-minimizer-limits']}.
  • Figure 2: The two diagrams show the same information as Figure \ref{['figure-mass-of-minimizer']} but with different ranges. In the left diagram we zoomed in on the range $s\in[0.0001,0.01]$. This is the deep MOND limit. In this range the graph of $M_s$ is approximately a parabola. For comparison we plotted (dashed line) also the parabola $1.06\,s^2$. In the right diagram we zoomed out on the range $s\in[10,1000]$. This is the Newtonian limit. There the graph of $M_s$ becomes a straight line. Putting the informations from Figure \ref{['figure-mass-of-minimizer']} and Figure \ref{['figure-mass-of-minimizer-limits']} together, we conclude that for every given mass $M>0$ there is exactly one $s>0$ such that $\rho_s$ has mass $M$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Regularity of Newtonian potentials
  • Lemma 2.2: Newtons shell theorem
  • Lemma 2.3
  • Lemma 2.4: Regularity of interpolation function $\lambda$
  • Lemma 2.5: Regularity of Mondian potentials
  • Lemma 2.6: Regularity of Mondian potential in spherical symmetry
  • proof
  • Definition 2.7: Mondian potential energy
  • ...and 45 more