Infinite gravitating particle lattice

An infinite gravitating particle lattice with spacing $h$ in the azimuthal direction is a shearing-sheet equilibrium when all particles sit at $x_n=0,y_n=nh$. Opposite-neighbor accelerations cancel absolutely. The linear gravitational response to pair displacement differences has transverse and longitudinal coefficients $1$ and $-2$, divided by $h^3|j-n{|}^3$.

For $q=GmF(hk)/(h^3{\Omega }^2)$, the squared dimensionless growth rates are $s^2/{\Omega }^2=(q-1\pm \sqrt{9q^2-26q+1})/2$. Exponential growth begins for $q>(13-4\sqrt{10})/9$. Since the gravitating lattice Fourier kernel is largest at $hk=\pi$, the full infinite lattice is exponentially unstable when $Gm/(h^3{\Omega }^2)>(13-4\sqrt{10})/[9F(\pi )]$. Marginal repeated imaginary roots and zero-frequency secular modes require separate treatment.

The discrete normal modes of an infinite gravitating particle lattice involve $F(\xi )=2{\sum }_{l=1}^{\infty }(1-\cos (l\xi ))/l^3$. It is even, nonnegative and $2\pi$-periodic. Its derivative is a positive reciprocal-square sine series for $0<\xi <\pi$, so its maxima occur exactly at odd multiples of $\pi$, with $F(\pi )=(7/2)\zeta (3)$.