Exactly solved model of a one dimensional self gravitating system

TL;DR

The paper constructs a one-dimensional, collisionless self-gravitating system in which all particles share the same energy, implemented via a delta-function energy distribution and analyzed in unit choices $G=1/4\pi$, $f_0=1/2$, $E=1/2$. It derives a compact parametric solution for the orbit and potential, with $A(T)=-T$, $V(T)=1-\tfrac{T^2}{2}$, $X(T)=T-\tfrac{T^3}{6}$ and $\Phi(T)=\tfrac{1}{2}-\tfrac{1}{2}(1-\tfrac{T^2}{2})^2$, where $X$ and $V$ are two-term Taylor truncations of sine and cosine and the solution is valid for $-\sqrt{2}<T<\sqrt{2}$ in the first half-cycle. The real-space density is $\rho(x)=2f_0/\sqrt{2(E-\phi(x))}$ and develops an integrable inverse-square-root singularity at turning points, characteristic of fold caustics, with the time derivative of the acceleration experiencing a finite discontinuity at those caustics. Numerical experiments on a cold 1D sheet system indicate the configuration is unstable, evolving toward a broader energy spectrum, which supports the interpretation that caustic features imprint weak, localized singular behavior and may persist primarily under smooth, cold initial conditions. Overall, the work provides a pedagogical, analytically tractable example that illuminates caustic phenomena in collisionless self-gravitating systems and their possible relevance to cold dark matter simulations.

Abstract

A model one-dimensional self consistent steady state collisionless self-gravitating system in which all the particles have the same energy is presented. This has the remarkable property that the position and velocity of the particles orbiting in their own self consistent potential are given exactly, in terms of time, by the truncations of sine and cosine functions to the first two terms in their respective Taylor series. The potential and density also have simple analytic expressions in terms of time as parameter. It is not being claimed that this system has any direct astronomical application. However, it does motivate a conjecture about the behaviour of the density, potential, and orbits near caustics in simulations of cold collisionless dark matter. It is a rather surprising result which might interest practitioners of stellar dynamics and serve as an elementary example in teaching the subject.

Figures (5)

• Figure 1: The position and velocity vs. time are plotted for the one dimensional motion of the particles in the model, over a quarter cycle, For comparison, the same quantities for the reference harmomic oscillator are plotted as well
• Figure 2: Phase trajectory of the motion in the X-V plane for a quarter cycle. The trajectory for the reference harmonic oscillator is shown for comparison
• Figure 3: The self consistent density and potential as functions of position. The potential has been multiplied by 4, its actual value at the ends is 0.5. The reference harmonic oscillator potential also scaled by 4, is plotted as well The density plot has been truncated, since it rises to infinity at the two extreme values of x
• Figure 4: The initial phase space distribution of the model discretized using 30 sheets
• Figure 5: The phase space distribution afer 60 crossing times