We address the general problem of the luminosity-specific planetary nebula ( PN ) number , better known as the “ \alpha ” ratio , given by \alpha = { N _ { PN } / L _ { gal } } , and its relationship with age and metallicity of the parent stellar population .
Our analysis relies on population synthesis models , that account for simple stellar populations ( SSPs ) , and more elaborated galaxy models covering the full star-formation range of the different Hubble morphological types .
This theoretical framework is compared with the updated census of the PN population in Local Group galaxies and external ellipticals in the Leo group , and the Virgo and Fornax clusters .

The main conclusions of our study can be summarized as follows :

i ) according to the Post-AGB stellar core mass , PN lifetime in a SSP is constrained by three relevant regimes , driven by the nuclear ( M _ { core } \gtrsim 0.57 ~ { } M _ { \odot } ) , dynamical ( 0.57 ~ { } M _ { \odot } \gtrsim M _ { core } \gtrsim 0.55 ~ { } M _ { \odot } ) and transition ( 0.55 ~ { } M _ { \odot } \gtrsim M _ { core } \gtrsim 0.52 ~ { } M _ { \odot } ) timescales .
The lower limit for M _ { core } also sets the minimum mass for stars to reach the AGB thermal-pulsing phase and experience the PN event ;

ii ) mass loss is the crucial mechanism to constrain the value of \alpha , through the definition of the initial-to-final mass relation ( IFMR ) .
The Reimers mass-loss parameterization , calibrated on Pop II stars of Galactic globular clusters , poorly reproduces the observed value of \alpha in late-type galaxies , while a better fit is obtained using the empirical IFMR derived from white-dwarf observations in the Galaxy open clusters ;

iii ) the inferred PN lifetime for Local Group spirals and irregulars exceeds 10 000 yr , which suggests that M _ { core } \lesssim 0.65 ~ { } M _ { \odot } cores dominate , throughout ;

iv ) the relative PN deficiency in elliptical galaxies , and the observed trend of \alpha with galaxy optical colors support the presence of a prevailing fraction of low-mass cores ( M _ { core } \lesssim 0.55 ~ { } M _ { \odot } ) in the PN distribution , and a reduced visibility timescale for the nebulae as a consequence of the increased AGB transition time .
The stellar component with M _ { core } \lesssim 0.52 ~ { } M _ { \odot } , which overrides the PN phase , could provide an enhanced contribution to hotter HB and Post-HB evolution , as directly observed in M 32 and the bulge of M 31 .
This implies that the most UV-enhanced ellipticals should also display the lowest values of \alpha , as confirmed by the Virgo cluster early-type galaxy population ;

v ) any blue-straggler population , invoked as progenitor of the M _ { core } \gtrsim 0.7 ~ { } M _ { \odot } PNe in order to preserve the constancy of the bright luminosity-function cut-off magnitude in ellipticals , must be confined to a small fraction ( few percents at most ) of the whole galaxy PN population .