Estimating the Injection Duration of 20 MeV Protons in Large Western Solar Energetic Particle Events

An ad hoc analytical calculation is presented to infer the duration of injection of 20 MeV protons in 21 selected western solar energetic particle (SEP) events. We convolve the solution of diffusion equation with a "triangle" source to model the time-intensity profiles over the onset and the peaking phase. The effects of "corotating" flux tubes and of solar wind convection are neglected. To accommodate these simplifications, only western events whose associated flares erupted between W15 and W90 are selected. The time-intensity profiles of these events are reconstructed from the timescales presented in Kahler (2005) and Kahler (2013) using the modified Weibull function. From the linear relation between the logarithm of the peak intensity and the logarithm of the fluence of 27-37 MeV protons presented in Kahler & Ling, we derive an optimal radial mean free path (lambda (mfp)) of 0.08 au and adopt this value to fit all selected events. The inferred duration of injection for the selected events, which in general increases with the initial speed of the associated coronal mass ejection (CME) (V (cme)), is less than 1 hr for V (cme) < 1000 km s(-1) and varies from a few to similar to 10 hr for 1000 km s(-1) < V (cme) < 2000 km s(-1). We then estimate the distance that the associated CMEs have traveled over the duration of injection. Most CMEs in selected events have traveled to less than 60 solar radii by the time the majority of accelerated particles have been injected into the interplanetary space.