Shadowing High-Dimensional Hamiltonian Systems: The Gravitational $N$-body Problem

A shadow is an exact solution to a chaotic system of equations that remains close to a numerically computed solution for a long time. Using a variable-order, variable–time-step integrator, we numerically compute solutions to a gravitational $N$-body problem in which many particles move and interact in a fixed potential. We then search for shadows of these solutions with the longest possible duration. We find that in “softened” potentials, shadow durations are sufficiently long for significant evolution to occur. However, in unsoftened potentials, shadow durations are typically very short.

Figure 1

Histogram of shadow durations of 1000 unsoftened $M$-body systems. Each system has 99 fixed particles and one moving particle (i.e., $M=1$). Noisy orbits have local error of about ${10}^{-5}$ per crossing time; numerical shadows were required to have a maximum local error no bigger than ${10}^{-14}$. The horizontal axis is in crossing times; the vertical axis is the measured probability density. The distribution fits an exponential curve with a mean glitch rate of about 0.07 per crossing time, indicating that the moving particle encounters glitches as a Poisson process in an unsoftened system.

Figure 2

Average shadow durations in crossing times of an unsoftened gravitational $M$-body system in which there are 100 particles, $M$ of which move, as a function of $M$. The noisy orbits have local error of ${10}^{-5}$ per crossing time; each numerical shadow was required to have a maximum local error no bigger than ${10}^{-12}$. The dots represent sample shadow durations, 30 samples each for $M=1,2,\dots ,19,20,25$ and 10 samples each for $M=30,35,\dots ,50$. The “coupled average” line joins their averages, while $55/M^{0.9}$ and $55/M^{1.1}$ are plotted for comparison. The “coupled average” line is statistically indistinguishable from the “uncoupled average” one, in which the gravitational interaction between moving particles is deleted. This suggests that even coupled particles encounter glitches independently of one another.

Figure 3

Histogram of shadow durations of 1000 systems identical to those in Fig. 1 except with softening $\epsilon =0.1$. Note that, in contrast to Fig. 1, the $x$ axis is logarithmic and extends to 1000 crossing times, and that the histogram height is zero near a shadow duration of zero, meaning that no particles undergo glitches until several tens of crossing times have occurred; in fact, of the 1000 systems sampled, the shortest shadow lasts 33 crossing times.

Figure 4

Average shadow durations in crossing times of a system with softening parameter $\epsilon =0.1$ as a function of $M$. The points represent sample shadow durations, 9 samples each for $M=1–5,10,20,25$, while the “coupled average” line joins their averages. Following the assumption that particles encounter glitches independently of one another, the “predicted average” is artificially constructed for each $M=1,\dots ,25$ by superimposing $M$ samples chosen at random from Fig. 3 and taking the minimum shadow duration of those samples. The resulting graph resembles the “coupled average” graph reasonably well, except for the surprising effect of underestimating shadow duration of the real system by about $(20\pm 10)\%$.