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 _ { kick } \propto { } f _ { H 05 } { } { } m _ { ej } { } { } m _ { rem } ^ { -1 } , where f _ { H 05 } is a normalization factor , drawn from a Maxwellian distribution with one-dimensional root-mean-square velocity \sigma { } = 265 km s ^ { -1 } , m _ { ej } is the mass of the supernova ( SN ) ejecta and m _ { 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 _ { BNS } ) , BH–NS binaries ( R _ { BHNS } ) and binary BHs ( R _ { BBH } ) , based on the cosmic star formation rate density and metallicity evolution .
In our fiducial model , we find R _ { BNS } \sim { } 600 Gpc ^ { -3 } yr ^ { -1 } , R _ { BHNS } \sim { } 10 Gpc ^ { -3 } yr ^ { -1 } and R _ { BBH } \sim { } 50 Gpc ^ { -3 } yr ^ { -1 } , fairly consistent with the numbers inferred from the LIGO-Virgo collaboration .