The orbit will close upon itself provided that Δφ equals a rational fraction of 2 π, i.e., Δ Δ φ φ = 2 π π m n where m and n are integers. In that case, the radius oscillates exactly m times while the azimuthal angle φ makes exactly n revolutions. In general, however, Δφ/2π will not be such a rational number, and thus the orbit will not be closed. In that case, the particle will eventually pass arbitrarily close to every point within the annulus. Two types of central force always produce closed orbits: F (r) = α r (a linear force) and F (r) = α/ r (an inverse-square law). As shown by Bertrand, these two central forces are the only ones that guarantee closed orbits.