Abstract:
The theoretical determination of braking indices of pulsars is still an open problem. In this paper we report results of a study concerning such determination based on a modification of the canonical model, which admits that pulsars are rotating magnetic dipoles, and on data from the seven pulsars with known braking indices. In order to test the modified model we predict ranges for the braking indices of other pulsars.

Abstract:
Braking index measurements of young radio pulsars are all smaller than the value expected for spin down by magnetic dipole braking. We investigate magnetic field evolution in the neutron star crust due to Hall drift as an explanation for observed braking indices. Using numerical simulations and a semi-analytic model, we show that a $\approx 10^{14}\ {\rm G}$ quadrupolar toroidal field in the neutron star crust at birth leads to growth of the dipole moment at a rate large enough to agree with measured braking indices. A key factor is the density at which the crust yields to magnetic stresses that build up during the evolution, which sets a characteristic minimum Hall timescale. The observed braking indices of pulsars with inferred dipole fields of $\lesssim 10^{13}\ {\rm G}$ can be explained in this picture, although with a significant octupole component needed in some cases. For the stronger field pulsars, those with $B_d\gtrsim 10^{13}\ {\rm G}$, we find that the magnetic stresses in the crust exceed the maximum shear stress before the pulsar reaches its current age, likely quenching the Hall effect. This may have implications for the magnetar activity seen in the high magnetic field radio pulsar PSR~J1846-0258. Observations of braking indices may therefore be a new piece of evidence that neutron stars contain subsurface toroidal fields that are significantly stronger than the dipole field, and may indicate that the Hall effect is important in a wider range of neutron stars than previously thought.

Abstract:
Isolated pulsars are rotating neutron stars with accurately measured angular velocities $\Omega$, and their time derivatives which show unambiguously that the pulsars are slowing down. Although the exact mechanism of the spin-down is a question of debate in detail, 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 $\Omega$. This relation leads to the power law $\dot{\Omega}$ = -K $\Omega^{\rm n}$ 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 as yet. We employ four physically realistic equations of state, and two computational codes, to model the dynamical effects of rotation on the braking index in the MDR model. In addition to this we simulate an effect on moment of inertia where we assume a certain amount of superfluid matter has manifest between the crust and core region of the star thus essentially eliminating momentum transfer between the two regions. We find that the effects of rotation on braking index are significant at high frequencies, but have little effect at frequencies consistent with the most accurately measured pulsars to-date.

Abstract:
Eight pulsars have low braking indices value which challenge the traditional model of dipole magnetic braking of pulsars. 222 pulsars and 15 magnetars have abnormal distribution of $\ddot{\nu}$ that also make contradiction with classical theory. How neutron star magnetospheric activities affect both these two phenomenons by using the updated wind braking model are investigated. It bases on the observational evidence that pulsar timing is related to radiation and both these two aspects can reflect magnetospheric activities. The new formulate of $n$ and $\ddot{\nu}$ are proposed to understand a more general situation. As neutron star magnetospheric fluctuation is unavoidable so it must be considered. Young pulsars have meaningful braking indices, while old pulsars' and magnetars' fluctuation item of $\ddot{\nu}$ dominates them and it reflects the timing noise. It can explain both the two questions above uniformly. The steady braking indices of eight young pulsars can be seen as small fluctuation amplitude. On the other hand, the abnormal distribution of $\ddot{\nu}$ can be seen as the case of timing noise for pulsar spin down. An equation like Langevin equation for Brownian motion was derived for pulsar spin-down. The fluctuation in magnetosphere can be either periodic or random, which result in timing noise and they have similar results. The magnetospheric activities of magnetars are always stronger than those of normal pulsars.

Abstract:
Isolated pulsars are rotating neutron stars with accurately measured angular velocities $\Omega$, and their time derivatives which show unambiguously that the pulsars are slowing down. Although the exact mechanism of the spin-down is a question of debate, the commonly accepted view is that it arises either through emission of magnetic dipole radiation (MDR) from a rotating magnetized body, through emission of a relativistic particle wind, or via higher order magnetic multipole or gravitational quadrupole radiation. The calculated energy loss by a rotating pulsar is model dependent and leads to the power law $\dot{\Omega}$ = -K $\Omega^{\rm n}$ where $n$ is called the braking index. The theoretical value for braking index is $n = 1, 3, 5$ for wind, MDR, quadrupole radiation respectively. The accepted view is that pulsar braking is strongly dominated by MDR. Highly precise observations of isolated pulsars yield braking index values in the range $1 < n < 2.8$ which are consistently less than the value predicted from the MDR model. We discuss possible ways to bring theory closer to observation for the MDR, and also consider how the other mechanisms may play a role in future study of the braking index problem.

Abstract:
In the standard scenario for spin evolution of isolated neutron stars, a young pulsar slows down with a surface magnetic field B that does not change. Thus the pulsar follows a constant B trajectory in the phase space of spin period and spin period time derivative. Such an evolution predicts a braking index n = 3 while the field is constant and n > 3 when the field decays. This contrasts with all nine observed values being n < 3. Here we consider a magnetic field that is buried soon after birth and diffuses to the surface. We use a model of a growing surface magnetic field to fit observations of the three pulsars with lowest n: PSR J0537-6910 with n = -1.5, PSR B0833-45 (Vela) with n = 1.4, and PSR J1734-3333 with n = 0.9. By matching the age of each pulsar, we determine their magnetic field and spin period at birth and confirm the magnetar-strength field of PSR J1734-3333. Our results indicate that all three pulsars formed in a similar way to central compact objects (CCOs), with differences due to the amount of accreted mass. We suggest that magnetic field emergence may play a role in the distinctive glitch behaviour of low braking index pulsars, and we propose glitch behaviour and characteristic age as possible criteria in searches for CCO descendants.

Abstract:
We present a systematic study of the evolution of intermediate- and low-mass X-ray binaries consisting of an accreting neutron star of mass $1.0-1.8 M_{\odot}$ and a donor star of mass $1.0-6.0 M_{\odot}$. In our calculations we take into account physical processes such as unstable disk accretion, radio ejection, bump-induced detachment, and outflow from the $L_{2}$ point. Comparing the calculated results with the observations of binary radio pulsars, we report the following results. (1) The allowed parameter space for forming binary pulsars in the initial orbital period - donor mass plane increases with increasing neutron star mass. This may help explain why some MSPs with orbital periods longer than $\sim 60$ days seem to have less massive white dwarfs than expected. Alternatively, some of these wide binary pulsars may be formed through mass transfer driven by planet/brown dwarf-involved common envelope evolution. (2) Some of the pulsars in compact binaries might have evolved from intermediate-mass X-ray binaries with anomalous magnetic braking. (3) The equilibrium spin periods of neutron stars in low-mass X-ray binaries are in general shorter than the observed spin periods of binary pulsars by more than one order of magnitude, suggesting that either the simple equilibrium spin model does not apply, or there are other mechanisms/processes spinning down the neutron stars.

Abstract:
Here we report that the observed braking indices of the 366 pulsars in the sample of Hobbs et al. range from about $-10^8$ to about $+10^8$ and are significantly correlated with their characteristic ages. Using the model of magnetic field evolution we developed previously based on the same data, we derived an analytical expression for the braking index, which agrees with all the observed statistical properties of the braking indices of the pulsars in the sample of Hobbs et al. Our model is, however, incompatible with the previous interpretation that magnetic field growth is responsible for the small values of braking indices ($<3$) observed for "baby" pulsars with characteristic ages of less than $2\times 10^3$ yr. We find that the "instantaneous" braking index of a pulsar may be different from the "averaged" braking index obtained from fitting the data over a certain time span. The close match between our model-predicted "instantaneous" braking indices and the observed "averaged" braking indices suggests that the time spans used previously are usually smaller than or comparable to their magnetic field oscillation periods. Our model can be tested with the existing data, by calculating the braking index as a function of the time span for each pulsar. In doing so, one can obtain for each pulsar all the parameters in our magnetic field evolution model, and may be able to improve the sensitivity of using pulsars to detect gravitational waves.

Abstract:
Ap/Bp stars are magnetic chemically peculiar early A and late B type stars of the main sequence. They exhibit peculiar surface abundance anomalies that are thought to be the result of gravitational settling and radiative levitation. The physics of diffusion in these stars are reviewed briefly and some model predictions are discussed. While models reproduce some observations reasonably well, more work is needed before the behavior of diffusing elements in a complex magnetic field is fully understood.

Abstract:
The old question of rotational braking of Ap Si stars is revisited on the empirical side, taking advantage of the recent Hipparcos results. Field stars with various evolutionary states are considered, and it is shown that the loose correlation between their rotational period and their surface gravity is entirely compatible with conservation of angular momentum. No evidence is found for any loss of angular momentum on the Main Sequence, which confirms earlier results based on less reliable estimates of surface gravity. The importance of reliable, fundamental Teff determinations of Bp and Ap stars is emphasized.