We studied the emergence process of 42 active region (ARs) by analyzing the time derivative, R(t), of the total unsigned flux. Line-of-sight magnetograms acquired by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) were used. A continuous piecewise linear fitting to the R(t)-profile was applied to detect an interval, dt_2, of nearly-constant R(t) covering one or several local maxima. The averaged over dt_2 magnitude of R(t) was accepted as an estimate of the maximal value of the flux growth rate, R_MAX, which varies in a range of (0.5-5)x10^20 Mx hour^-1 for active regions with the maximal total unsigned flux of (0.5-3)x10^22 Mx. The normalized flux growth rate, R_N, was defined under an assumption that the saturated total unsigned flux, F_MAX, equals unity. Out of 42 ARs in our initial list, 36 event were successfully fitted and they form two subsets (with a small overlap of 8 events): the ARs with a short (<13 hours) interval dt_2 and a high (>0.024 hour^-1) normalized flux emergence rate, R_N, form the rapid emergence event subset. The second subset consists of gradual emergence events and it is characterized by a long (>13 hours) interval dt_2 and a low R_N (<0.024 hour^-1). In diagrams of R_MAX plotted versus F_MAX, the events from different subsets are not overlapped and each subset displays an individual power law. The power law index derived from the entire ensemble of 36 events is 0.69+-0.10. The rapid emergence is consistent with a two-step emergence process of a single twisted flux tube. The gradual emergence is possibly related to a consecutive rising of several flux tubes emerging at nearly the same location in the photosphere.
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