ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: selected from an ensemble of similar objects. Light curves were constructed by taking one of the observable parameters from the formal solutions and computing its response to the driving in depth from equation ( 8 ). In detail, this was carried out by fitting a cubic natural spline through the parameter values at the known depths. 4.4 Variable background radiation The operations for variation of the background radiation follow a broadly similar approach to those used in Section 4.3 . However, there are important differences. First, we apply only the default sinusoid and function D 6 as driving functions, since only D 6 is based on a model where the background, rather than the pump, is varied. The process is easier in the sense that the parameter that is varying in time is simply a parameter in equation ( 4 ), and no complicated justification is required for its use. A significant complication in the variable background case, however, is that the driver in the background analogue of equation ( 8 ) now demands a response from a number of different models, each computed with a different value of i BG . Therefore, modelling flares from background variation is computationally considerably more expensive. 5 VARIABLE PUMPING RESULTS We plot, in Fig. 3 , the flare light curves, over one period, that result from driving with pump functions D 0 – D 5 for a prolate cloud with Γ = 0.6, with the observer placed in the optimum position (viewing the cloud along the z -axis close to the long axis of the cloud). The cloud style is uniform in unsaturated inversion. The parameters of equation ( 8 ) in this case are τ min = 10.0 and Δτ = 5.0, corresponding to a regime that is already characteristic of quite strong saturation. Throughout Section 5 , the y -axes of graphs are shown in flux density, based on a background specific intensity that is scaled to the saturation intensity of the maser transition. Flux density is likely the best parameter to plot in view of the prominent role of single-dish monitoring in repetitive flare studies. An explanation of the flux-density scale appears in Appendix A . Owing to the nature of the saturation intensity, and particularly the fact that this quantity may itself depend on the energy density of pumping radiation, a conversion formula to units of Jy at a typical source distance, and for a particular maser transition, for example, is not trivial. Such a conversion is also discussed in detail in Appendix A . The variability index of the maser light curve, defined as in the Introduction as F pk / F qui is 1.74 in Fig. 3 . The same parameter for the driving functions, replacing flux densities with values of the depth multiplier, is 1.5. Although the variability index of the maser is significantly larger here, the light curve in Fig. 3 demonstrates the smoothing effect of saturation: as exponential growth gives way to a functional form that is closer to linear at high flux density, the maser responds more weakly to the driving