  Calcium-rich supernovae (Ca-rich SNe) are faint, rapidly evolving transients
whose progenitor system is yet to be determined. We derive the $\gamma$-ray
deposition histories of five Ca-rich SNe from the literature in order to place
constraints on possible progenitor systems. We find that the $ \gamma $-ray
escape time, $ t_0 $, of the Ca-rich SNe sample is $\approx35$-$65 \,\rm{d}$,
within the unoccupied region between Type Ia SNe and stripped envelope
supernovae (SESNe). The $ t_0$-$M_\mathrm{Ni56} $ distribution of these SNe,
where $M_\mathrm{Ni56}$ is the synthesised $^{56}$Ni mass in the explosion,
creates a continuum between the Type Ia and SESNe $ t_0$-$M_\mathrm{Ni56} $
distribution, hinting at a possible connection between all the events. By
comparing our results to models from the literature, we were able to determine
that helium shell detonation models and core-collapse models of ultra-stripped
stars are unlikely to explain Ca-rich SNe, since the gamma-ray escape time in
these models is smaller than the observed values. Models that agree with the
observed $ t_0$-$M_\mathrm{Ni56} $ distribution are explosions of low mass,
$M\approx0.75$-$0.8\,M_\odot $, white dwarfs and core-collapse models of
stripped stars with an ejecta mass of $M\approx1$-$3\,M_{\odot}$.
