Revising natal kick prescriptions in population synthesis simulations

Natal kicks are matter of debate and significantly affect the merger rate density of compact objects. Here, we present a new simple formalism for natal kicks of neutron stars (NSs) and black holes (BHs). We describe the magnitude of the kick as $v_{\rm kick}\propto{}f_{\rm H05}\,{}\,{}m_{\rm ej}\,{}\,{}m_{\rm rem}^{-1}$, where $f_{\rm H05}$ is a normalization factor, drawn from a Maxwellian distribution with one-dimensional root-mean-square velocity $\sigma{}=265$~km~s$^{-1}$, $m_{\rm ej}$ is the mass of the supernova (SN) ejecta and $m_{\rm rem}$ is the mass of the compact object. This formalism matches the proper motions of young Galactic pulsars and can naturally account for the differences between core-collapse SNe of single stars, electron-capture SNe and ultra-stripped SNe occurring in interacting binaries. Finally, we use our new kick formalism to estimate the local merger rate density of binary NSs ($R_{\rm BNS}$), BH--NS binaries ($R_{\rm BHNS}$) and binary BHs ($R_{\rm BBH}$), based on the cosmic star formation rate density and metallicity evolution. In our fiducial model, we find $R_{\rm BNS}\sim{}600$~Gpc$^{-3}$~yr$^{-1}$, $R_{\rm BHNS}\sim{}10$~Gpc$^{-3}$~yr$^{-1}$ and $R_{\rm BBH}\sim{}50$~Gpc$^{-3}$~yr$^{-1}$, fairly consistent with the numbers inferred from the LIGO-Virgo collaboration.