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On the time dependence of the $h$-index

The time dependence of the $h$-index is analyzed by considering the average behaviour of $h$ as a function of the academic age $A_A$ for about 1400 Italian physicists, with career lengths spanning from 3 to 46 years. The individual $h$-index is strongly correlated with the square root of the total citations $N_C$: $h \approx 0.53 \sqrt{N_C}$. For academic ages ranging from 12 to 24 years, the distribution of the time scaled index $h/\sqrt{A_A}$ is approximately time-independent and it is well described by the Gompertz function. The time scaled index $h/\sqrt{A_A}$ has an average approximately equal to 3.8 and a standard deviation approximately equal to 1.6. Finally, the time scaled index $h/\sqrt{A_A}$ appears to be strongly correlated with the contemporary $h$-index $h_c$.