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Two low mass neutron stars, J0737-3039B and the companion to J1756-2251, showstrong evidence of being formed from the collapse of an ONeMg core in anelectron capture supernova (ECSN) or in an ultra-stripped iron core collapsesupernova (FeCCSN). Using three different systematically generated sets ofequations of state we explore the relationship between the moment of inertia ofJ0737-3039A and the binding energy of the two low mass neutron stars. We findthis relationship, a less strict variant of the recently discovered I-Love-Qrelations, is nevertheless more robust than a previously explored correlationbetween the binding energy and the slope of the nuclear symmetry energy L. Wefind that, if either J0737-3039B or the J1756-2251 companion were formed in anECSN, no more than 0.06 solar masses could have been lost from the progenitorcore, more than four times the mass loss predicted by current supernovamodeling. A measurement of the moment of inertia of J0737-3039A to within 10%accuracy from pulsar timing, possible within a decade, can discriminate betweenformation scenarios such as ECSN or ultra-stripped FeCCSN and, given currentconstraints on the predicted core mass loss, potentially rule them out. Usingthe I-Love-Q relations we find that an Advanced LIGO can potentially measurethe neutron star tidal polarizability to equivalent accuracy in a neutronstar-neutron star merger at a distance of 200 Mpc, thus obtaining similarconstraints on the formation scenarios. Such information on the occurrence ofECSNe is important for population synthesis calculations, especially forestimating the rate of binary neutron star mergers and resultingelectromagnetic and gravitational wave signals. Further progress needs to bemade modeling the core collapse process that leads to low-mass neutron stars,particularly in making robust predictions for the mass loss from the progenitorcore.

Testing the formation scenarios of binary neutron star systems with measurements of the neutron star moment of inertia