  Spherical hydrodynamic models with a polytropic equation of state (EoS) for
forming protostars are revisited in order to investigate the so-called
luminosity conundrum highlighted by observations. For a molecular cloud (MC)
core with such an EoS with polytropic index $\gamma$ >1, the central mass
accretion rate (MAR) decreases with increasing time as a protostar emerges,
offering a sensible solution to this luminosity problem. As the MAR decreases,
the protostellar luminosity also decreases, meaning that it is invalid to infer
the star formation time from the currently observed luminosity using an
isothermal model. Furthermore, observations of radial density profiles and the
radio continua of numerous MC cores evolving towards protostars also suggest
that polytropic dynamic spheres of $\gamma$ > 1 should be used in physical
models.
