What the Timing of Millisecond Pulsars Can Teach us about Their Interior

The cores of compact stars reach the highest densities in nature and therefore could consist of novel phases of matter. We demonstrate via a detailed analysis of pulsar evolution that precise pulsar timing data can constrain the star’s composition, through unstable global oscillations ($r$ modes) whose damping is determined by microscopic properties of the interior. If not efficiently damped, these modes emit gravitational waves that quickly spin down a millisecond pulsar. As a first application of this general method, we find that ungapped interacting quark matter is consistent with both the observed radio and x-ray data, whereas for ordinary nuclear matter some additional enhanced damping mechanism is required.

Figure 1

Boundaries of the $r$-mode instability regions for different star compositions compared to pulsar data. Left: Standard static instability boundary compared to x-ray data [15, 22] with error estimates from different envelope models [16, 23]. Right: Dynamic instability boundary in timing parameter space compared to radio data [6] (all data points are upper limits for the $r$-mode component of the spin-down). The curves represent $1.4M_{\odot }$ neutron star with standard viscous damping [18, 20] (long-dashed line) and with additional boundary layer rubbing [14] at a rigid crust (dotted line), as well as $1.4M_{\odot }$ strange star [19, 24] (short-dashed line) and same with long-ranged NFL interactions causing enhanced damping [11] (using the strong coupling ${\alpha }_s=1$) (solid line)—more massive stars are not qualitatively different. The thin curves show for the neutron star exemplarily the analytic approximation for the individual segments. The encircled points denote the only millisecond-radio pulsar $\mathrm{J}0437-4715$ with a temperature estimate and the only LMXB IGR $\mathrm{J}00291+5934$ that has been observed to spin down during quiescence.

Figure 2

The thermal steady-state spin-down curves, along which the star evolves, for several $T$- and $\mathrm{\Omega }$-independent saturation amplitudes. Also shown are the boundaries of the instability regions for neutron stars with different damping sources as discussed in Fig. 1. Left: Static instability boundary compared to x-ray data [15, 22]. Evolution curves are shown for a $1.4M_{\odot }$ neutron star with modified Urca cooling. The vertical line gives the temperature below which photon emission replaces neutrino emission as the dominant cooling mechanism. Right: Dynamic instability boundary in timing parameter space compared to radio data [6].

Figure 3

The $r$-mode temperatures as observed at infinity of radio pulsars with known timing data as well as the corresponding temperatures of LMXBs (dots) compared to $r$-mode instability boundaries. $r$-mode temperatures are given for sources with $f\ge 200\mathrm{Hz}$ assuming neutron stars with modified Urca (stars) and direct Urca cooling (diamonds).