Universal relations for the Keplerian sequence of rotating neutron stars

We investigate the Keplerian (mass-shedding) sequence of rotating neutron stars. Twelve different equations of state are used to describe the nuclear structure. We find four fitting relations which connect the rotating frequency, mass and radius of stars in the mass-shedding limit to the mass and radius of stars in the static sequence. We show the breakdown of approximate relation for the Keplerian frequency derived by Lattimer and Prakash [Science 304, 536 (2004)] and then we present a new, equation of state (EOS)-independent and more accurate relation. This relation fits the Keplerian frequency of rotating neutron stars to about 2% for a large range of the compactness $M_S/R_S$ of the reference nonrotating neutron star, namely the static star with the same central density as the rotating one. The performance of the fitting formula is close to 4% for $M_S/R_S\le 0.05M_{\odot }/\mathrm{km}$ ($f_K\le 350\mathrm{Hz}$). We present additional EOS-independent relations for the Keplerian sequence including relations for $M_K{f}_K$ and $R_K{f}_K$ in terms of $M_S{f}_S$ and $R_S{f}_S$, respectively, one of $M_K/R_K$ as a function of $f_K/f_S$ and $M_S/R_S$, and a relation between the $M_K$, $R_K$ and $f_K$. These new fitting relations are approximately EOS independent with an error in the worst case of 8%. The universality of the Keplerian sequence properties presented here add to the set of other neutron star universal relations in the literature such as the $I$-Love-$Q$ relation, the gravitational binding energy and the energy, angular momentum and radius of the last circular orbit of a test particle around rotating neutron stars. This set of universal, analytic formulas facilitates the inclusion of general relativistic effects in the description of relativistic astrophysical systems involving fast rotating neutron stars.

Figure 1

Mass versus equatorial radius along constant high-frequency sequences for the GM1 EOS. It can be seen how above some frequency value (here $\approx 1200\mathrm{Hz}$) the mass at which the constant frequency sequence cuts the Keplerian sequence does not represent a minimum mass bound.

Figure 2

Mass versus equatorial radius of the configurations at the Keplerian sequence for the selected EOS used in this work.

Figure 3

$f_K/f_S$ ratio versus $M_S$ for each selected EOS.

Figure 4

The fitting curve given by Eq. (2) (black solid curve) and $f_K/f_S$ ratio versus $M_S/R_S$ for each selected EOS (upper panel). The relative error between the calculated data and Eq. (2) (lower panel).

Figure 5

Comparison of the new relation, Eq. (2), and the L&P relation, Eq. (1). We here use the fastest observed pulsar up to now, PSR J1748-2446ad [54] with $f=716\mathrm{Hz}$ and a frequency of $f=1122\mathrm{Hz}$, the rotation rate claimed of the neutron star in the XTE J1739-285 [55], but not yet confirmed. The light gray dot-dashed and dotted curves give the relation (1) from Ref. [10]. The dark gray dot-dashed and dotted curves are obtained from our Eq. (2). We also show the nonrotating mass-radius relation obtained with all the EOSs used in this work.

Figure 6

$M_K{f}_K$ versus $M_S{f}_S$ for each selected EOS. The black solid curve indicates Eq. (3) (upper panel). The relative error between the curve and data (lower panel).

Figure 7

$R_K{f}_K$ versus $R_S{f}_S$ for each selected EOS. The black solid curve indicates Eq. (4) (upper panel). The relative error between the curve and data (lower panel).

Figure 8

The dimensionless parameter, compactness, versus $(M_S/R_S)(f_K/f_S)$ for each selected EOS. The black solid curve indicates Eq. (5) (upper panel). The relative error between the curve and data (lower panel).

Figure 9

$R_K{f}_K$ versus $M_K{f}_K$ for each selected EOS. The black solid curve indicates Eq. (6) (upper panel). The relative error between the curve and data (lower panel).