Four-hair relations for differentially rotating neutron stars in the weak-field limit

Abstract

The opportunity to study physics at supra-nuclear densities through X-ray observations of neutron stars has led to in-depth investigations of certain approximately universal relations that can remove degeneracies in pulse profile models. One such set of relations determines all of the multipole moments of a neutron star just from the first three (the mass monopole, the current dipole and the mass quadrupole moment) approximately independently of the equation of state. These three-hair relations were found to hold in neutron stars that rotate rigidly, as is the case in old pulsars, but neutron stars can also rotate differentially, as is the case for proto-neutron stars and hypermassive transient remnants of binary mergers. We here extend the three-hair relations to differentially rotating stars for the first time with a generic rotation law using two approximations: a weak-field scheme (an expansion in powers of the neutron star compactness) and a perturbative differential rotation scheme (an expansion about rigid rotation). These approximations allow us to analytically derive approximately universal relations that allow us to determine all of the multipole moments of a (perturbative) differentially rotating star in terms of only the first four moments. These new fourhair relations for differentially rotating neutron stars are found to be approximately independent of the equation of state to a higher degree than the three-hair relations for uniformly rotating stars. Our results can be instrumental in the development of four-hair relations for rapidly differentially rotating stars in full General Relativity using numerical simulations.

Approximate universal relations for neutron stars and quark stars

“…Completely analytic three-hair relations can also be obtained if one studies stars with equations of state that are perturbations from n = 0 polytropes [174,176]. Doing so, one finds that [174,175]…”

“…The first (and only) study of the no-hair relations in differentially-rotating stars is that of Bretz et al [175], which restricts attention to the Newtonian limit and the small differential-rotation limit. Although the actual profile of rotation depends on the astrophysical situation one considers, the rotation law in the Newtonian limit can be parametrized in a generic way as [189] Ω Ω c =…”

An entropy-area law for neutron stars near the black hole threshold

“…Another idea one could explore in the future is whether the universality we have found is robust to the inclusion of strong magnetic fields or rapid rotation. In the I-Love-Q case, the universality was found to be robust to the latter for a fixed dimensionless spin angular momentum [5,7,32], but not to the former [33], thus breaking in proto-neutron stars (see [34,35] for related work) and in magnetars (but not in millisecond pulsars). One may expect the same to be true for the entropy of neutron stars, but this should be verified through numerical calculations.…”

I–Love–Q relations for realistic white dwarfs

“…Although the focus of our paper is for WDs, we here comment on how differential rotation, finite temperatures and deviations from chemical equilibrium affect the I-Love-Q relations for NSs. The effect of differential rotation on universal relations 2 has been studied in (Bretz et al 2015) within the Newtonian limit and small differential-rotation approximation. The authors showed that the fractional difference in the relations from the uniformly-rotating case is comparable to the fractional amount of differential rotation over uniform rotation.…”

“…The universal relations studied in(Bretz et al 2015) were those among stellar multipole moments rather than the I-Love-Q relations.…”