$R$-mode frequencies of rapidly and differentially rotating relativistic neutron stars

$R$-modes of neutron stars could be a source of gravitational waves for ground based detectors. If the precise $r$-mode frequency $\sigma$ in an inertial frame is known, guided gravitational wave searches with enhanced detectability are possible. Because of its physical importance, many authors have calculated the $r$-mode frequency. For the dominant mode, the associated gravitational wave frequency is $4/3$ times the angular velocity of the star $\mathrm{\Omega }$, subject to various corrections of which relativistic and rotational corrections are the most important. This has led several authors to investigate the dependence of the $r$-mode frequency on factors such as the relativistic compactness parameter $M/R$ and the angular velocity of stars with different equations of state. The results found so far, however, are almost independent of the equation of state. Here, we investigate the effect of rapid rotation and differential rotation on $\sigma$. We work with a series of rotating neutron stars with polytropic equations of state, and we perform the time evolution of the linear perturbation equations in the relativistic Cowling approximation with a finite-difference code. We find that rotational effects in the $r$-mode frequency can be larger than relativistic effects for rapidly spinning stars with low compactness, reducing the observed frequency. Differential rotation also acts to decrease $\sigma$, but its effect increases with the compactness. The results presented here are relevant to the design of gravitational-wave and electromagnetic $r$-mode searches.

Figure 1

$\kappa$ vs $M/R$ for sequences of stars with different values of $f$. From top to bottom, $f=700$, 600, 500, 400, 300, 100 Hz. Note that $\kappa$ decreases (and $\sigma$ increases) for more compact stars, but $\kappa$ increases (and $\sigma$ decreases) for more rapidly rotating stars.

Figure 2

$\kappa$ vs $f$ for sequences of stars with different values of $M/R$, which were obtained by interpolation from the data in Fig. 1 (dashed lines). From top to bottom, $M/R=0.12$, 0.14, 0.16, 0.18, 0.20. The solid lines represent quadratic fits of the form $\kappa =a+b{f}^2$, where both $a$ and $b$ decrease with the compactness.

Figure 3

${\kappa }_0$ vs $M/R$. The dashed line represents Eq. (2), and the solid line is given by a quadratic fit of the parameter $a$ in Fig. 2 that gives ${\kappa }_0=0.64-0.35(M/R)-2.0(M/R{)}^2$. The difference between the two curves shows the accuracy of the Cowling approximation.

Figure 4

${\kappa }_2$ vs $M/R$. The solid lines shows a quadratic fit of ${\kappa }_2$ obtained from the parameter $b$ in Fig. 2 that gives ${\kappa }_2=0.68-4.1(M/R)+9.8(M/R{)}^2$. The rotational correction ${\kappa }_2$ is more significant for less compact stars.

Figure 5

The rotation profiles as a function of the radial coordinate scaled by the equatorial radius $R_{\text{eq}}$ of six differentially rotating stars with ${\widehat{A}}^{-1}=0.5$ that have the same total angular momentum as the corresponding uniformly rotating models that have the same compactness and $f=600\mathrm{Hz}$. We see that, although the differential rotation parameter across the models is set constant, stars with higher compactness have slightly more physical differential rotation. $M/R=0.11,0.13,\dots ,0.21$ as indicated in the figure.

Figure 6

$\kappa$ vs $M/R$ for sequences of stars with different values of the differential rotation parameter ${\widehat{A}}^{-1}$. The upper curve shows $\kappa$ for the models presented in Fig. 5. The other curves are similar, but with ${\widehat{A}}^{-1}=0.4$, 0.3 and 0.0 (from top to bottom). Note that $\kappa$ increases (and $\sigma$ decreases) for stars with stronger differential rotation.

Figure 7

${\kappa }_3$ vs ${\widehat{A}}^{-1}$ for sequences of stars with different values of $M/R$, which were obtained by interpolation from the data in Fig. 6. The solid lines represent fits of the form ${\kappa }_3=c{\widehat{A}}^{-1}+d{\widehat{A}}^{-3}$. From top to bottom, $M/R=0.2,0.18,\dots ,0.12$. The differential rotation correction ${\kappa }_3$ is more significant for more compact stars.

Figure 8

${\kappa }_3$ vs ${\widehat{A}}^{-1}$ for sequences of stars with different values of $f$ ranging from 200–600 Hz and two different values of compactness: $M/R=0.20$ (upper curves) and $M/R=0.14$ (lower curves). In this range, ${\kappa }_3$ is almost independent of the total angular momentum of the star and depends much more strongly on the compactness. (The color code is $\{\text{blue},\text{yellow},\text{green},\text{red},\text{violet}\}$ for $f=600,500,400,300,200\mathrm{Hz}$, respectively).

Figure 9

$c$ vs $M/R$ as defined by Eq. (6) for sequences of stars with different values of $f$ ranging from 200–600 Hz. The color code is the same as in Fig. 8.

Figure 10

$d$ vs $M/R$ as defined by Eq. (6) for sequences of stars with different values of $f$ ranging from 200–600 Hz. The color code is the same as in Fig. 8.