Length and time scales relevant to fragmentation as a function of f_cyl, in units of the FWHM (left axis) and , the sound crossing time of r₀ (right), respectively. The shortest wavelength growing mode is λ_cr (dashed-multiple dotted lines). The fastest growing mode is λ_m (dashed-single dotted lines) which, in the linear regime, develops on a time scale τ_m (long-dashed lines). For comparison, λ_cr,sph is the length along the filament required to encompass a mass equal to that of a critical stable sphere for the same p_ext and K (solid line). The black broken lines correspond to non-magnetized cylinders. The related grey lines show the increasing effect of a magnetic field and correspond to p² = 1%, 10%, and 100% of equipartition. The values are based on the work of Nagasawa (1987), see Appendix E. The vertical dotted line marks the mass ratio f_cyl = 0.75.