Star formation triggering is about when, where, and how star formation begins. The global laws of star formation, such as the Kennicutt (1998) or Bigiel et al. (2008) relations, show a correlation between the star formation rate and the supply of gas of various types. The Kennicutt (1998) study shows a star formation rate per unit area that scales with the 1.4 power of the total gas column density in the main disks and central regions of spiral galaxies, and the Bigiel et al. (2008) and Leroy et al. (2008) studies show a linear relation between the star formation rate per unit area and the CO emission per unit area. A linear relation for dense gas traced by HCN was first shown by Gao & Solomon (2004) and Wu et al. (2005).
In all cases, star formation is assumed to occur only in cold dark gas, which is traced by CO at low density and HCN and other molecules at high density. The empirical laws suggest that the star formation rate is independent of how the gas is assembled, and therefore independent of the triggering mechanisms. This is particularly true if the average pressure and radiation field determine the general molecular state of the gas, and the pressures which make shells and rings do not change this state much, they just push the gas around. Triggering will also not influence the empirical laws much if the various types of triggering are always present to the same degree. For example, if every initial incidence of star formation is followed by a certain proportion of additional stars that form by radiative implosion or in shell and ring collapse, then the conversion efficiency from ambient gas to young stars will contain all of these effects combined. Each triggering process will not stand out separately in the total star formation rate.
According to Gnedin & Kravtsov (2011) and others, CO and star formation are predictable from the average pressure, self-shielding, radiation field, and other quantities. The pressure that determines molecular self-shielding does not vary much on the scale of shells and rings once they are old enough for star formation to begin. Regions with radiative implosion have high pressure, but this may be a relatively minor star formation event. The pressure varies in spiral arms, but this should only put scatter in the empirical relations. Thus the signatures of localized triggering may be imperceptible in the empirical star formation laws.