  The structural stability of a lightsail under the laser flux necessary for
interstellar flight is studied analytically and numerically. A sinusoidal
perturbation is introduced into a two-dimensional thin-film sail to determine
if the sail remains stable or if the perturbations grow in amplitude. A
reflective material that gives specular reflection of the laser illumination is
assumed in determining the resulting loading on the sail, although other
reflection models can be incorporated as well. The quasi-static solution of the
critical point between shape stability and instability is found by equating the
bending moments induced on the sail due to radiation pressure with the
restoring moments caused by the strength of the sail and the tension applied at
the edges of the sail, permitting analytical expressions for the elastic
modulus and boundary tension magnitude to be found as a function of sail
properties and the amplitude and wave number of the initial perturbation. These
expressions are also derived from a formal variational energy approach. A
numerical model of the sail dynamics is developed by discretizing the sail into
finite elements. By introducing torsional and rectilinear springs between the
elements into the numerical model, a hierarchy of models is produced that can
incorporate the effects of bending and applied tension. The numerical models
permit the transient dynamics of a perturbed sail to be compared to the
analytic results of the quasi-static analysis. The analytic expressions can
correctly predict the stability boundary found in the numerical simulations.
The stiffness required to make a sail stable against uncontrolled perturbation
growth is shown to be unfeasible for known materials. A modest tensioning of
the sail (e.g., via an inflatable structure or spinning of the sail) can
maintain a stable sail shape under all wavelengths and amplitudes of
perturbations.
