On the maximum compactness of neutron stars

The stellar compactness, that is, the dimensionless ratio between the mass and radius of a compact star, mathcal{C} := M/R, plays a fundamental role in characterising the gravitational and nuclear-physics aspects of neutron stars. Yet, because the compactness depends sensitively on the unknown equation of state (EOS) of nuclear matter, the simple question: "how compact can a neutron star be?" remains unanswered. To address this question, we adopt a statistical approach and consider a large number of parameterised EOSs that satisfy all known constraints from nuclear theory, perturbative Quantum Chromodynamics (QCD), and astrophysical observations. Next, we conjecture that, for any given EOS, the maximum compactness is attained by the star with the maximum mass of the sequence of nonrotating configurations. While we can prove this conjecture for a rather large class of solutions, its general proof is still lacking. However, the evidence from all of the EOSs considered strongly indicates that it is true in general. Exploiting the conjecture, we can concentrate on the compactness of the maximum-mass stars and show that an upper limit appears for the maximum compactness and is given by mathcal{C}_{rm max} = 1/3. Importantly, this upper limit is essentially independent of the stellar mass and a direct consequence of perturbative-QCD constraints.