[Paper Review] Universal Relations for the Increase in the Mass and Radius of a Rotating Neutron Star

Andreas Konstantinou, Sharon M. Morsink|arXiv (Cornell University)|Jun 24, 2022

Pulsars and Gravitational Waves ResearchPhysics and Astronomy54 references37 citations

TL;DR

This paper proposes universal, equation-of-state-independent correction factors for the mass and radius of rotating neutron stars, derived from non-rotating Tolman-Oppenheimer-Volkoff (TOV) solutions. Using constant-central-density sequences and a diverse set of randomly generated equations of state, the authors show that fractional increases in mass and radius depend only on the non-rotating star's mass and radius and the spin frequency, enabling efficient inclusion of rotation in equation-of-state inference. The key contribution is a computationally inexpensive mapping from rotating to non-rotating neutron stars, allowing reconstruction of the zero-spin mass-radius relation from observed rotating systems.

ABSTRACT

Rotation causes an increase in a neutron star's mass and equatorial radius. The mass and radius depend sensitively on the unknown equation of state (EOS) of cold, dense matter. However, the increases in mass and radius due to rotation are almost independent of the EOS. The EOS independence leads to the idea of neutron star universality. In this paper, we compute sequences of rotating neutron stars with constant central density. We use a collection of randomly generated EOS to construct simple correction factors to the mass and radius computed from the equations of hydrostatic equilibrium for non-rotating neutron stars. The correction factors depend only on the non-rotating star's mass and radius and are almost independent of the EOS. This makes it computationally inexpensive to include observations of rotating neutron stars in EOS inference codes. We also construct a mapping from the measured mass and radius of a rotating neutron star to a corresponding non-rotating star. The mapping makes it possible to construct a zero-spin mass-radius curve if the masses and radii of many neutron stars with different spins are measured. We show that the changes in polar and equatorial radii are symmetric, in that the polar radius shrinks at the same rate that the equatorial radius grows. This symmetry is related to the observation that the equatorial compactness (the ratio of mass to radius) is almost constant on one of the constant-density sequences.

Motivation & Objective

• To develop a computationally efficient method to correct non-rotating neutron star models for rotation effects in equation-of-state (EOS) inference.
• To investigate whether the increases in mass and radius due to rotation are approximately independent of the equation of state (EOS), enabling universal relations.
• To construct a mapping from observed rotating neutron stars to equivalent non-rotating stars with the same central density, facilitating reconstruction of the zero-spin mass-radius relation.
• To assess the practical significance of rotational corrections for current and future X-ray missions like NICER, Strobe-X, and eXTP.

Proposed method

• Computed sequences of rapidly rotating neutron stars with constant central density using the rns code and a library of 100 randomly generated equations of state (EOS), including piecewise polytropes and SLy-type models.
• Derived empirical correction factors for mass and radius as functions of the non-rotating star's mass and radius and the normalized spin frequency $\Omega_n^2 = \Omega / \Omega_K^2$, where $\Omega_K^2$ is the Kepler frequency.
• Introduced a normalized spin parameter $\Omega_n^2$ to enhance universality and reduce scatter compared to using $\chi / \chi_K$.
• Constructed an inverse mapping from rotating star properties (mass, equatorial radius, spin frequency) to equivalent non-rotating star properties via Equations (9)–(12), using coefficients from Table 1.
• Validated the method using the PP0 EOS and applied it to NICER observations of PSR J0030+0451, showing small corrections relative to measurement uncertainties.
• Explored the symmetry between polar radius shrinkage and equatorial radius expansion, linking it to the near-constancy of equatorial compactness $C_e = M/R_e$ along constant-density sequences.

Experimental results

Research questions

• RQ1 Can the increase in mass and equatorial radius due to rotation be described by a universal relation independent of the equation of state?
• RQ2 To what extent do the fractional changes in mass and radius of rotating neutron stars depend only on the non-rotating star’s mass and radius and the spin frequency?
• RQ3 Can a reliable mapping be constructed from observed rotating neutron stars to equivalent non-rotating stars with the same central density?
• RQ4 How significant are rotational corrections compared to current and future measurement uncertainties in mass and radius?
• RQ5 Why is the equatorial compactness nearly constant along constant-central-density sequences, and what physical symmetry underlies this?

Key findings

• The fractional increase in mass and radius due to rotation is well described by empirical formulae that depend only on the non-rotating star’s mass and radius and the normalized spin frequency $\Omega_n^2$, with minimal dependence on the equation of state.
• For PSR J0030+0451 (ν = 205 Hz), the rotational correction reduces the radius by about 0.08 km and the mass by 0.01 M⊙, which is an order of magnitude smaller than the 1σ measurement uncertainties of ~1.2 km and ~0.15 M⊙.
• The equatorial compactness $C_e = M/R_e$ remains nearly constant along constant-central-density sequences, suggesting an underlying symmetry that balances polar contraction and equatorial expansion.
• The inverse mapping from rotating to non-rotating stars, based on Equations (9)–(12), successfully reconstructs the zero-spin mass-radius relation with a small spread in the mapped points, much smaller than typical observational uncertainties.
• The correction factors are small for slow rotators (e.g., 200 Hz) but will become increasingly important for future missions like Strobe-X and eXTP, which aim for 1% radius precision and target faster-spinning millisecond pulsars.
• The use of $\Omega_n^2 = \Omega / \Omega_K^2$ as a normalized spin parameter yields better universality than using $\chi / \chi_K$, reducing scatter in the relations.