  This study introduces the exocomet curve of growth, a new method to analyse
the variable absorptions observed in $\beta$ Pictoris spectrum and link them to
the physical properties of the transiting cometary tails. We show that the
absorption depth of a comet in a set of lines arising from similar excitation
levels of a given chemical species follows a simple curve as a function of the
gf-values of the lines. This curve is the analogue of the curve of growth for
interstellar absorption lines, where equivalent widths are replaced by
absorption depths. To fit this exocomet curve of growth, we introduce a model
where the cometary absorption is produced by a homogeneous cloud, covering a
limited fraction of the stellar disc. This model is defined by two parameters:
$\alpha$, the covering factor of the cloud, and $\beta$, related to its typical
the optical depth. This model is tested on two comets observed with the Hubble
Space Telescope in December 1997 and October 2018, in a set of Fe II lines at
275 nm. The measured absorption depths are found to satisfactory match the
two-parameter curve of growth model, indicating that both comets cover roughly
40 % of the stellar disc ($\alpha=0.4$) and have optical thicknesses close to
unity. Then, we show that if we consider a set of lines arising from a wider
range of energy levels, the absorbing species seems to be populated at
thermodynamical equilibrium, causing the cometary absorption to follow a curve
of growth as a function of $gf \cdot e^{-E_l/k_B T}$ (where T is the
temperature of the absorbing medium). For the comet observed on December 6,
1997, we derive a temperature of $10500\pm500$ K and a total Fe II column
density of $(1.11\pm0.09)\times10^{15}$ cm$^{-2}$. By probing the population of
the highest excited energy levels ($E_l\sim25000$ cm$^{-1}$), we also estimate
an electronic density of $(3\pm1)\times10^{7}$ cm$^{-3}$.
