  An intrinsic correlation has been identified between the luminosity and
duration of plateaus in the X-ray afterglows of Gamma-Ray Bursts (GRBs;
Dainotti et al. 2008), suggesting a central engine origin. The magnetar central
engine model predicts an observable plateau phase, with plateau durations and
luminosities being determined by the magnetic fields and spin periods of the
newly formed magnetar. This paper analytically shows that the magnetar central
engine model can explain, within the 1$\sigma$ uncertainties, the correlation
between plateau luminosity and duration. The observed scatter in the
correlation most likely originates in the spread of initial spin periods of the
newly formed magnetar and provides an estimate of the maximum spin period of
~35 ms (assuming a constant mass, efficiency and beaming across the GRB
sample). Additionally, by combining the observed data and simulations, we show
that the magnetar emission is most likely narrowly beamed and has $\lesssim$20%
efficiency in conversion of rotational energy from the magnetar into the
observed plateau luminosity. The beaming angles and efficiencies obtained by
this method are fully consistent with both predicted and observed values. We
find that Short GRBs and Short GRBs with Extended Emission lie on the same
correlation but are statistically inconsistent with being drawn from the same
distribution as Long GRBs, this is consistent with them having a wider beaming
angle than Long GRBs.
