  Calcium-rich (SN 2005E-like) explosions are very faint (typical -15.5, type I
supernovae (SNe) showing strong Ca-lines, mostly observed in old stellar
environments. Several models for such SNe had been explored and debated, but
non were able to consistently reproduce the observed properties of Ca-rich SNe,
nor their rates and host-galaxy distributions. Here we show that the
disruptions of low-mass carbon-oxygen (CO) white dwarfs (WDs) by hybrid
Helium-CO (HeCO) WDs during their merger could explain the origin and
properties of such SNe. We make use of detailed multi-dimensional
hydrodynamical-thermonuclear (FLASH) simulations models of HeCO-CO WD-WD
mergers to characterize such explosions. We find that the accretion of CO
material onto a HeCO-WD heats its He-shell and eventually leads to its "weak"
detonation and ejection and the production of a sub-energetic $\sim10^{49}$erg
Ca-rich SN, while leaving the CO core of the HeCO-WD intact as a hot remnant
white dwarf, possibly giving rise to X-ray emission as it cools down. We model
the detailed light-curve and spectra of such explosions to find an excellent
agreement with observations of Ia/c Ca-rich SNe. In these models most of the He
is burned, producing Ia/c Ca-rich SNe but higher He-enrichment levels might
potentially also explain the Ib Ca-rich SNe. We thereby provide a viable,
consistent model for the origins of Ca-rich. These findings can shed new light
on the role of Ca-rich in the chemical evolution of galaxies and the
intra-cluster medium, and their contribution to the observed 511 kev signal in
the Galaxy originating from positrons produced from ${}^{44}\textrm{Ti}$ decay.
Finally, the origins of such SNe points to the key-role of HeCO WDs as SN
progenitors, and their potential role as progenitors of other thermonuclear SNe
including normal type Ia SNe.
