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
$N_C$: $h \approx 0.53 N_C$. For academic ages ranging from 12 to 24
years, the distribution of the time scaled index $h/A_A$ is
approximately time-independent and it is well described by the Gompertz
function. The time scaled index $h/A_A$ has an average approximately
equal to 3.8 and a standard deviation approximately equal to 1.6. Finally, the
time scaled index $h/A_A$ appears to be strongly correlated with the
contemporary $h$-index $h_c$.
