Star formation triggered by SN explosions: an application to the stellar   association of $\beta$ Pictoris
  In the present study, considering the physical conditions that are relevant in interactions between supernova remnants (SNRs) and dense molecular clouds for triggering star formation we have built a diagram of SNR radius versus cloud density in which the constraints above delineate a shaded zone where star formation is allowed. We have also performed fully 3-D radiatively cooling numerical simulations of the impact between SNRs and clouds under different initial conditions in order to follow the initial steps of these interactions. We determine the conditions that may lead either to cloud collapse and star formation or to complete cloud destruction and find that the numerical results are consistent with those of the SNR-cloud density diagram. Finally, we have applied the results above to the $\beta-$Pictoris stellar association which is composed of low mass Post-T Tauri stars with an age of 11 Myr. It has been recently suggested that its formation could have been triggered by the shock wave produced by a SN explosion localized at a distance of about 62 pc that may have occurred either in the Lower Centaurus Crux (LCC) or in the Upper Centaurus Lupus (UCL) which are both nearby older subgroups of that association (Ortega and co-workers). Using the results of the analysis above we have shown that the suggested origin for the young association at the proposed distance is plausible only for a very restricted range of initial conditions for the parent molecular cloud, i.e., a cloud with a radius of the order of 10 pc and density of the order of 20 cm$^{-3}$ and a temperature of the order of 50$-$100 K. 
C. Melioli, E.M. de Gouveia Dal Pino, R. de la Reza, A. Raga