Estimation of the Pulsar Braking Index Using the Evolution of the Rotational Kinetic Energy Loss Rate. Testing on the Crab Pulsar

Abstract

The slowdown mechanism of rotating neutron stars (NSs) has long been an important topic in pulsar astronomy, which is still an open question of detailed debate. It is widely understood that it arises from the loss of the rotational kinetic energy ($E$) through the magnetic dipole radiation (MDR). This simple model would lead to a relation $\dot {\Omega } \propto {{\Omega }^{3}}$, where $\Omega$ and $\dot {\Omega }$ are the spin angular velocity and its time derivative. However, a sole MDR torque may be oversimple and incorrect, and the above relation could be written in a more general form $\dot {\Omega } = - K{{\Omega }^{n}}$, where $K$ is a constant, and $n$ is known as the braking index, being 3 for the MDR model. The $n$ value of a pulsar could in principle be directly measured through its $\ddot {\Omega }$ observation. However, it is difficult to measure the values of $\ddot {\Omega }$ due to the effects of the timing noise or glitches, making the sample of pulsars with reliable $n$ observations is very tiny (the number ~10). Therefore, this study aims to estimate a pulsar’s $n$ value without relying on the observation of $\ddot {\Omega }$, by employing a simple and convenient $\dot {E}$ evolution model. In which, the information about the real age ${{t}_{o}}$ of the target pulsar is required as input. Here, we applied this $\dot {E}$ model to the Crab pulsar since it is the only NS source with a known age. Our results inferred its $n$ value to be $2.3 < n < 2.7$, and also constrained several of its initial parameters, e.g., its initial spin period and energy loss rate were limited to be $0.0180 < {{P}_{i}}$ [s] $< 0.0197$ and $2.56 \times {{10}^{{39}}} < {{\dot {E}}_{i}}$ [erg/s] $< 4.38 \times {{10}^{{39}}}$, respectively. Our above estimated $n$ value for the Crab pulsar is consistent with the observation based on its $\ddot {\Omega }$ measurement, as ${{n}_{{{\text{obs}}}}} = 2.51 \pm 0.01$. The departure of its $n$ value from the canonical 3 may be interpreted by a combination of a particle wind flow torque and an increase of the magnetic inclination angle, along with the classical MDR model. Our method may be applied to approximately evaluate the $n$ values for more Crab-like pulsars in the future if their true ages can be accurately measured through a new methodology.