Braking index of isolated pulsars

Isolated pulsars are rotating neutron stars with accurately measured angular velocities $\mathrm{\Omega }$, and their time derivatives that show unambiguously that the pulsars are slowing down. Although the exact mechanism of the spin-down is a question of detailed debate, the commonly accepted view is that it arises through emission of magnetic dipole radiation (MDR) from a rotating magnetized body. Other processes, including the emission of gravitational radiation, and of relativistic particles (pulsar wind), are also being considered. The calculated energy loss by a rotating pulsar with a constant moment of inertia is assumed proportional to a model dependent power of $\mathrm{\Omega }$. This relation leads to the power law $\dot{\mathrm{\Omega }}=-K{\mathrm{\Omega }}^n$ where $n$ is called the braking index. The MDR model predicts $n$ exactly equal to 3. Selected observations of isolated pulsars provide rather precise values of $n$, individually accurate to a few percent or better, in the range $1<n<2.8$, which is consistently less than the predictions of the MDR model. In spite of an extensive investigation of various modifications of the MDR model, no satisfactory explanation of observation has been found yet. The aim of this work is to determine the deviation of the value of $n$ from the canonical $n=3$ for a star with a frequency dependent moment of inertia in the region of frequencies from zero (static spherical star) to the Kepler velocity (onset of mass shedding by a rotating deformed star), in the macroscopic MDR model. For the first time, we use microscopic realistic equations of state (EoS) of the star to determine its behavior and structure. In addition, we examine the effects of the baryonic mass $M_B$ of the star, and possible core superfluidity, on the value of the braking index within the MDR model. Four microscopic equations of state are employed as input to two different computational codes that solve Einstein’s equations numerically, either exactly or using the perturbative Hartle-Thorne method, to calculate the moment of inertia and other macroscopic properties of rotating neutron stars. The calculations are performed for fixed values of $M_B$ (as masses of isolated pulsars are not known) ranging from $1.0–2.2M_{\odot }$, and fixed magnetic dipole moment and inclination angle between the rotational and magnetic field axes. The results are used to solve for the value of the braking index as a function of frequency, and find the effect of the choice of the EoS, $M_B$. The density profile of a star with a given $M_B$ is calculated to determine the transition between the crust and the core and used in estimation of the effect of core superfluidity on the braking index. Our results show conclusively that, within the model used in this work, any significant deviation of the braking index away from the value $n=3$ occurs at frequencies higher than about ten times the frequency of the slow rotating isolated pulsars most accurately measured to date. The rate of change of $n$ with frequency is related to the softness of the EoS and the $M_B$ of the star as this controls the degree of departure from sphericity. Change in the moment of inertia in the MDR model alone, even with the more realistic features considered here, cannot explain the observational data on the braking index and other mechanisms have to be sought.

Figure 1

MoI as a function of frequency for a pulsar with $M_B=2.0M_{\odot }$ as calculated with both RPN and PRNS9 numerical codes.

Figure 2

Pressure vs energy density $\epsilon$ (in units of the energy density of symmetric nuclear matter at saturation ${\epsilon }_0$) as predicted by the four EoS used in this work.

Figure 3

Braking index as a function of frequency calculated for a pulsar with $M_B=2.0M_{\odot }$ with all EoS adopted in this work.

Figure 4

Braking index as a function of frequency calculated of pulsars with $M_B=1.0–2.2{\mathrm{M}}_{\odot }$.

Figure 5

Pressure as a function central density of a pulsar with $M_B=2.0{\mathrm{M}}_{\odot }$, rotating with frequencies decreasing from the Kepler limit to zero as predicted by the four EoS used in this work. Relation between the central density and frequency is given in Table 3. For more explanation see the text.

Figure 6

Percentage of the core in a static star and a star rotating at 600 Hz as a function of $M_B$. For more explanation see the text.

Figure 7

Total (crust and core) and the crust-only MoI as a function of frequency, calculated for a pulsar with $M_B=1.0{\mathrm{M}}_{\odot }$.

Figure 8

$\mathrm{\Delta }$ n represent the difference in braking index as a function of frequency between stars with (see Fig. 4) and without core contribution to the MoI. Each curve is displayed up to the Kepler frequency of the star.

Figure 9

Lower and upper limits on values of the braking index as a function of frequency, including results from both numerical codes, all EoS, $M_B$, and the superfluid condition. The (yellow) shaded area between the two lines defines the location of all results within the limits. The pulsar with baryonic $M_B=2.2{\mathrm{M}}_{\odot }$ and the KDE0v1 EoS has the highest Kepler frequency (see Table 2) and defines the frequency limit in this work.

Figure 10

The lower limit of the braking index (see Fig. 9) as a function of frequency (solid line) compared with the ratio between polar ($R_p$) and equatorial radii ($R_{\text{eq}}$), normalized to three, which determines deformation of the star. The difference between the two lines represents a correlation between deviations of the braking index from $n=3$ and deformation for a $1.0{\mathrm{M}}_{\odot }$ pulsar rotating at frequencies below 160 Hz (notice the expanded y scale). It is seen that the shape deformation, even for this most deformable star, is small at these frequencies and quite unable to reproduce the observed range of braking indices.