A semi-analytic model for the temporal evolution of the episodic
  disc-to-star accretion rate during star formation
  We develop a semi-analytic formalism for the determination of the evolution
of the stellar mass accretion rate for specified density and velocity profiles
that emerge from the runaway collapse of a prestellar cloud core. In the early
phase, when the infall of matter from the surrounding envelope is substantial,
the star accumulates mass primarily because of envelope-induced gravitational
instability in a protostellar disc. In this phase, we model the envelope mass
accretion rate from the isothermal free-fall collapse of a molecular cloud
core. The disc gains mass from the envelope, and also transports matter to the
star via a disc accretion mechanism that includes episodic gravitational
instability and mass accretion bursts according to the Toomre $Q$-criterion. In
the early phase the envelope accretion is dominant, whereas in the late phase
the disc accretion is dominant. In the disc accretion phase, mass is accreted
on to the star due to gravitational torques within the spiral structures in the
disc, in a manner that analytic theory suggests has a mass accretion rate
$\propto t^{-6/5}$. Our model provides a self-consistent evolution of the mass
accretion rate by joining the spherical envelope accretion with the disc
accretion and accounts for the presence of episodic accretion bursts at
appropriate times. We show using a simple example that the burst mode is
essential to explain the long-standing 'luminosity problem' of young stellar
objects. The bursts are needed to provide a good match to the observed
distribution of bolometric luminosities. In contrast, a smoothly time-dependent
mass accretion rate, whether monotonically increasing or decreasing, is unable
to do so. Our framework reproduces key elements of detailed numerical
simulations of disc accretion and can aid in developing intuition about the
basic physics as well as in comparing theory with observations.
