Self-gravity at the scale of the polar cell

We present the exact calculus of the gravitational potential and acceleration along the symmetry axis of a plane, homogeneous, polar cell as a function of mean radius a, radial extension e, and opening angle f. Accurate approximations are derived in the limit of high numerical resolution at the geometrical mean <a> of the inner and outer radii (a key-position in current FFT-based Poisson solvers). Our results are the full extension of the approximate formula given in the textbook of Binney & Tremaine to all resolutions. We also clarify definitely the question about the existence (or not) of self-forces in polar cells. We find that there is always a self-force at radius <a> except if the shape factor a.f/e reaches ~ 3.531, asymptotically. Such cells are therefore well suited to build a polar mesh for high resolution simulations of self-gravitating media in two dimensions. A by-product of this study is a newly discovered indefinite integral involving complete elliptic integral of the first kind over modulus.