Surface modelling of non-radial pulsators: alternative formalisms within the linear approximation

The problem of modelling the surface of stars undergoing non-radial pulsation is reviewed. Linear-approximation expressions for the surface radius, temperature, velocity and geometry of a pulsating star are derived and discussed using both the Lagrangian (fixed-element) and Eulerian (fixed-position) formalisms. In each case, small numerical discrepancies are found between the perturbed states predicted by these alternative approaches. These discrepancies are shown to scale quadratically with the pulsation amplitude, and are therefore attributed to a transgression of the linear-approximation limits. Singled out for particular attention are the expressions for the surface geometry perturbations predicted by each formalism. Marked differences are apparent between these expressions: terms containing the horizontal fluid displacement appear explicitly in the Lagrangian result, but are absent from the corresponding Eulerian one. By examining the physical origin of these terms, it is demonstrated that the two formalisms are, in fact, perfectly consistent with regard to the geometry perturbations, and - as with all other perturbations - simply furnish alternative representations of the same physical processes. The conclusion is that either formalism is an appropriate choice when modelling the surface of a pulsating star.