Braking Index Research Articles

Results of 12 yr of Pulsar Timing at Nanshan. I.

Abstract We have used the Nanshan 25 m Radio Telescope at Xinjiang Astronomical Observatory to obtain timing observations of 87 pulsars from 2002 July to 2014 March. Using the “Cholesky” timing analysis method we have determined positions and proper motions for 48 pulsars, 24 of which are improved positions compared to previously published values. We also present the first published proper motions for nine pulsars and improved proper motions for 21 pulsars using the pulsar timing and position-comparison method. The pulsar rotation parameters are derived and are more accurate than previously published values for 36 pulsars. Glitches are detected in three pulsars: PSRs J1722−3632, J1852−0635, and J1957+2831. For the first two, the glitches are large, with Δν g /ν > 10−6, and they are the first detected glitches in these pulsars. PSR J1722−3632 is the second oldest pulsar, with a large glitch. For the middle-age pulsars (τ c > 105 yr), the calculated braking indices, , are strongly correlated with τ c and the numbers of positive and negative values of n are almost equal. For young pulsars (τ c < 105 yr), there is no correlation between and τ c and most have n > 0.

Could acceleration of a pulsar affect braking index?

In the coming era of multi-messenger astrophysics, pulsars might be one of the most possible electromagnetic counterparts of the gravitational wave. The braking indices, which are related closely to the electromagnetic radiation of pulsars, are shown to be larger for the pulsars with companion. It motivates us to set up a modified spin-down equation for accelerated pulsars. In this model, we attempt to figure out whether acceleration of a pulsar can cause a larger braking index.

Measurement of the Rate Distribution of the Population of Repeating Fast Radio Bursts: Implications for Progenitor Models

Abstract The discovery of many repeating fast radio bursts (FRBs) by the Canadian Hydrogen Intensity Mapping Experiment, the high rate of individual bursts, and the observation of repeat bursts from a bright FRB initially detected by the Australian Square Kilometre Array Pathfinder (ASKAP) suggest a significant population of repeating FRBs. Here, we analyze the rate distribution of this population using results from follow-up observations of the Commensal Real-time ASKAP Fast Transients Survey. Characterizing a repeating FRB by its burst rate R 0 above an energy of 1038 erg, we consider a volumetric density Φ(R 0) with rate distribution . We use maximum-likelihood methods to constrain ζ < −1.94 at 90% confidence. We discuss how this constraint can be used to limit different classes of progenitor models for repeating FRBs: it excludes FRBs with burst rate proportional to neutron star spin-down power with braking index n ≤ 14, i.e., magnetic dipole radiation (n = 3); and it excludes some scenarios for FRB emission from magnetars, and the magnetic field interactions of compact binaries.

Timing of young radio pulsars – II. Braking indices and their interpretation

ABSTRACT In Paper I of this series, we detected a significant value of the braking index (n) for 19 young, high-$\dot{E}$ radio pulsars using ∼10 yr of timing observations from the 64-m Parkes radio telescope. Here, we investigate this result in more detail using a Bayesian pulsar timing framework to model timing noise and to perform selection to distinguish between models containing exponential glitch recovery and braking index signatures. We show that consistent values of n are maintained with the addition of substantial archival data, even in the presence of glitches. We provide strong arguments that our measurements are unlikely due to exponential recovery signals from unseen glitches even though glitches play a key role in the evolution of a pulsar’s spin frequency. We conclude that, at least over decadal time-scales, the value of n can be significantly larger than the canonical 3 and discuss the implications for the evolution of pulsars.

The braking index of PSR B0540−69 and the associated pulsar wind nebula emission after spin-down rate transition

ABSTRACT In 2011 December, PSR B054−69 experienced a spin-down rate transition (SRT), after which the spin-down power of the pulsar increased by $\sim 36{{\ \rm per\ cent}}$. About 1000 d after the SRT, the X-ray luminosity of the associated pulsar wind nebula (PWN) was found to brighten by $32\pm 8{{\ \rm per\ cent}}$. After the SRT, the braking index n of PSR B0540−69 changes from n = 2.12 to 0.03 and then keeps this value for about five years before rising to n = 0.9 in the following years. We find that most of the current models have difficulties in explaining the measured braking index. One exceptive model of the braking index evolution is the increasing dipole magnetic field of PSR B0540−69. We suggest that the field increase may result from some instabilities within the pulsar core that enhance the poloidal component at the price of toroidal component of the magnetic field. The increasing dipole magnetic field will result in the X-ray brightening of the PWN. We fit the PWN X-ray light curve by two models: one assumes a constant magnetic field within the PWN during the brightening and the other assumes an enhanced magnetic field proportional to the energy density of the PWN. It appears that the two models fit the data well, though the later model seems to fit the data a bit better. This provides marginal observational evidence that magnetic field in the PWN is generated by the termination shock. Future high-quality and high-cadence data are required to draw a solid conclusion.

Second Braking Index of Intermittent Pulsar and Nulling Pulsar**This work was funded by Xin Jiang Autonomous Region Science and Technology Department Natural Science Fund Youth Project under No. 2017D01B02.

The braking index parameter of intermittent pulsar would provide important insight into the pulsar slowdown and emission mechanism. In previous work, we presented the first braking index of the intermittent pulsar B1931+24, J1841−0500, J1832+0029 and nulling pulsar B0823+26, by using their observed spin frequency first derivatives in “on” and “off” states. In this paper, to improve and justify our method for studying the braking mechanism of intermittent pulsars, we continue to formulate a method in which the second braking indices, , of these pulsars can also be given with only the observed spin frequency first derivatives and . This paper will discuss, according to the emission and spin-down property of intermittent pulsars, three different magnetospheric configuration changes in the polar cap region. We find our approach produces feasible values of the second braking index of intermittent pulsar. Our calculations of the second braking indices are in close agreement with predicted second braking indices m = n(2n − 1), which are given by simple spin-down law of pulsar spin evolution, resulting in indirect observational support to our model.

The First Day in the Life of a Magnetar: Evolution of the Inclination Angle, Magnetic Dipole Moment, and Braking Index of Millisecond Magnetars during Gamma-Ray Burst Afterglows

Abstract The afterglow emission of some gamma-ray bursts (GRBs) shows a shallow decay (plateau) phase implying continuous injection of energy. The source of this energy is very commonly attributed to the spin-down power of a nascent millisecond magnetar. The magnetic dipole radiation torque is considered to be the mechanism causing the spin-down of the neutron star. This torque has a component working for the alignment of the angle between rotation and the magnetic axis, i.e., the inclination angle, which has been neglected in modeling GRB afterglow light curves. Here, we demonstrate the evolution of the inclination angle and magnetic dipole moment of nascent magnetars associated with GRBs. We constrain the initial inclination angle, magnetic dipole moment, and rotation period of seven magnetars by modeling the seven long-GRB afterglow light curves. We find that, in its first day, the inclination angle of a magnetar decreases rapidly. The rapid alignment of the magnetic and rotation axis may address the lack of persistent radio emission from mature magnetars. We also find that in three cases the magnetic dipole moments of magnetars decrease exponentially to a value a few times smaller than the initial value. The braking index of nascent magnetars, as a result of the alignment and magnetic dipole moment decline, is variable during the afterglow phase and always greater than three.

The evolution of magnetic field and spin‐down of young pulsars

AbstractHere, we summarize our recent work on the evolution of magnetic field and spin‐down of young pulsars including high‐B pulsars and magnetars. Our work includes the following three parts: (a) Based on the estimated ages of their potentially associated supernova remnants (SNRs), we estimate the values of the mean braking indices of eight magnetars with SNRs and find that five magnetars have smaller mean braking indices of 1 < n < 3, and we interpret them within a combination of magnetodipole radiation and wind‐aided braking. (b) The low braking index of n = 0.9(2) for PSR J1734‐3333 could undergo a supercritical accretion soon after its formation in a supernova explosion. The buried multipole magnetic fields will merge into a dipole magnetic field. Retaining the current field growth index ϵ = 1.34, this pulsar will become a magnetar with dBp∼6.2(2) × 1014 G and dBp∼1.06(4) × 1015 G after 50 kyrs and 100 kys, respectively. (c) By introducing a mean rotation energy conversion coefficient and combining the pulsar's high‐energy and timing observations with a reliable nuclear equation of state, we estimate the initial spin period, initial dipole magnetic field, and true age of PSR J1640‐4631. For this source, its measured high braking index n = 3.15(3) is attributed to a long‐term dipole magnetic field decay, and its soft X‐ray emission is attributed to anisotropic heating.

Application of a two-dipole model to PSR J1640–4631, a pulsar with an anomalous braking index

Abstract Recent timing observation provides an intriguing result for the braking index of the X-ray pulsar PSR$\:$J1640$-$4631, which has a measured braking index $n=3.15\pm 0.03$. The decrease of the inclination angle between the spin axis and the magnetic axis can be responsible for such a high braking index. However, the physical mechanisms causing the change of the magnetic inclination angle have not been fully understood. In this Letter, we apply a two-dipole model given by Hamil, Stone, and Stone (2016, Phys. Rev. D., 94, 063012) to explain the decrease of the magnetic inclination angle of PSR$\:$J1640$-$4631. The rotation effect of a charged sphere and the magnetization of ferromagnetically ordered material produce magnetic moments $M_{\,1}$ and $M_{\,2}$, respectively. There exists a minimum of the potential energy for the magnetic moment $M_{\,2}$ in the magnetic field of $M_{\,1}$, hence the $M_{\,2}$ will freely rotate around the minimum energy position (i.e., equilibrium position), similar to a simple pendulum. Our calculation indicates that the magnetic moment $M_{\,2}$ would evolve towards alignment with the spin axis for PSR$\:$J1640$-$4631, and cause the magnetic inclination angle to decrease. The single peak in the pulse profile favors a relatively low change rate for the magnetic inclination angle.

Inferring the Presence of Tides in Detached White Dwarf Binaries

Abstract Tidal interactions can play an important role as compact white dwarf (WD) binaries are driven together by gravitational waves (GWs). This will modify the strain evolution measured by future space-based GW detectors and impact the potential outcome of the mergers. Surveys now and in the near future will generate an unprecedented population of detached WD binaries to constrain tidal interactions. Motivated by this, I summarize the deviations between a binary evolving under the influence of only GW emission and a binary that is also experiencing some degree of tidal locking. I present analytic relations for the first and second derivative of the orbital period and braking index. Measurements of these quantities will allow the inference of tidal interactions, even when the masses of the component WDs are not well constrained. Finally, I discuss tidal heating and how it can provide complimentary information.

Timing of young radio pulsars – I. Timing noise, periodic modulation, and proper motion

ABSTRACT The smooth spin-down of young pulsars is perturbed by two non-deterministic phenomenon, glitches, and timing noise. Although the timing noise provides insights into nuclear and plasma physics at extreme densities, it acts as a barrier to high-precision pulsar timing experiments. An improved methodology based on the Bayesian inference is developed to simultaneously model the stochastic and deterministic parameters for a sample of 85 high-$\dot{E}$ radio pulsars observed for ∼10 yr with the 64-m Parkes radio telescope. Timing noise is known to be a red process and we develop a parametrization based on the red-noise amplitude (Ared) and spectral index (β). We measure the median Ared to be $-10.4^{+1.8}_{-1.7}$ yr3/2 and β to be $-5.2^{+3.0}_{-3.8}$ and show that the strength of timing noise scales proportionally to $\nu ^{1}|\dot{\nu }|^{-0.6\pm 0.1}$, where ν is the spin frequency of the pulsar and $\dot{\nu }$ is its spin-down rate. Finally, we measure significant braking indices for 19 pulsars and proper motions for 2 pulsars, and discuss the presence of periodic modulation in the arrival times of 5 pulsars.

Using pulsar’s braking indices to estimate changes in their moments of inertia with age-related considerations

Pulsars are modeled as neutron stars originated from the collapse of a progenitor one. In the canonical model they are described by spherical magnetized dipoles that rotate with the magnetic axis usually misaligned relative to the rotation axis, and such misalignment would explain the observation of radiation emitted in pulses in a certain direction rendering the typical observational characteristic of this kind of star. The frequency of such pulses decays with time and it can be quantified by the braking index (n). In the canonical model n = 3 for all pulsars but observational data show that n [negationslash] = 3. In this work we present a model for the understanding of the frequency decay of the rotation of a pulsar adapting the canonical one. We consider the pulsar a star that rotates in vacuum and has a strong magnetic field but, in contrast to the canonical model, we assume that its moment of inertia changes in time due to a uniform variation of a displacement parameter in time. We found that the old pulsars that present high values of the braking index tend to present smaller internal displacements of mass, in particular the superfluid neutron matter in the core. We relate this trend to neutron vortices’ creep in rotating superfluids, indicating a possible reason for this coincidence.

The influence of superfluid core cooling in the braking index of Pulsars.

Pulsars are stars from which electromagnetic radiation is observed to pulsate in well-defined time intervals as the star rotates and the emission of eletromagnetic signal is located in a place different from the rotation center. The frequencies of the pulses decay with time, quantified by the braking index (n). In the canonical model n = 3, in general, for all pulsars, but observational data shows that n is lower than 3. In this work a new model is presented, based on a modification of the canonical one incorporating the influence of neutron and proton density that appear in the superfluid core and, as the star cools down, the density of the superfluid core increases making the star to shrink with time and temperature, making the inertia moment to decrease. The difference ot this model from the canonical one is that the star moment of inertia decreases with time (what would accelerate the rotation of the star) what makes the star to not slow down as fast as it should without this process.

Determining the Efficiency of Converting Magnetar Spindown Energy into Gamma-Ray Burst X-Ray Afterglow Emission and Its Possible Implications

Abstract Plateaus are common in X-ray afterglows of gamma-ray bursts. Among the few scenarios regarding their origin, the leading one is that there exists a magnetar inside and it persistently injects its spindown energy into an afterglow. In previous studies, the radiation efficiency of this process is assumed to be a constant at ≳0.1, which is quite simple and strong. In this work we obtain the efficiency from a physical point of view and find that this efficiency strongly depends on the injected luminosity. One implication of this result is that those X-ray afterglow light curves that show steeper temporal decay than t −2 after the plateau phase can be naturally understood now. Also, the braking indexes deduced from afterglow fitting are found to be larger than those in previous studies, which are more reasonable for newborn magnetars.

What can PSR J1640-4631 tell us about the internal physics of this neutron star?

Gravitational wave emissions (GWEs) of pulsars could not only make them promising targets for continuous gravitational wave searches but also leave imprints in their timing data. We interpret the measured braking index of PSR J1640-4631 with a model involving both the GWE and dipole magnetic field decay. Combining the timing data of PSR J1640-4631 and the theory of magnetic field decay, we propose a new approach of constraining the number of precession cycles, $\xi$, which is highly uncertain currently but can be tightly related to the interior physics of a neutron star and its GWE. We suggest that future observation of the tilt angle $\chi$ of PSR J1640-4631 would not merely help to constrain $\xi$ but also possibly provide information about the internal magnetic field configuration of this pulsar. We find that $\xi$ would be larger than previous estimates unless a tiny angle $\chi\lesssim5^\circ$ is observed. Furthermore, a measured angle $\chi\gtrsim12^\circ$ would indicate $\xi\gtrsim 10^6$, which is at least ten times larger than that suggested previously.

Investigations of the Ohmic Decay and the Soft X-Ray Emission of the High-braking-index Pulsar PSR J1640−4631

The young pulsar PSR J1640−4631 is the first pulsar reported with a high braking index, n = 3.15 ± 0.03, and has attracted extensive attention. In this work, by inputting the equation of state, we obtain two typical and feasible ranges of the electrical conductivity in the crust of PSR J1640−4631. Then, by considering the effects of general relativity, we formulate the induction equation and use it to investigate the decay of crustal magnetic fields within two scenarios: pure Ohmic decay, and the combination of Ohmic decay and Hall drift. In the former scenario, we calculate the current magnetic field decay rates as dBp/dt ∼ (0.78–1.00) × 107 G yr−1 and dBt/dt ∼ (0.82–1.05) × 108 G yr−1 for the poloidal and toroidal parts, respectively, and estimate the current magnetic energy decay rate as LB ∼ (0.95–1.22) × 1032 erg s−1. In the latter scenario, the combination of Ohmic decay and Hall drift accelerates the decay of magnetic fields, and we calculate the current magnetic field decay rates as dBp/dt ∼ (1.26–1.47) × 107 G yr−1, and dBt/dt ∼ (1.32–1.54) × 108 G yr−1 for the poloidal and toroidal parts, respectively, and estimate the current magnetic energy decay rate as LB ∼ (1.86–2.18) × 1032 erg s−1. It is found that the recalculated value of the braking index n of the star is lower than its observed value, which implies that other factors influence the spin-down evolution. The current magnetic energy decay rates calculated in both scenarios are far less than the isotropic soft X-ray luminosity obtained from Chandra+NuStar telescopes. The observed soft X-ray flux in the (2–10) keV band is FX ∼ 1.8(4) × 10−13 erg cm−2 s−1, which corresponds to a high surface temperature, T ∼ 1.54(6) × 106 K. The observed X-ray flux of the star could be caused by the decay of a multipolar magnetic field near the pole, which is strong enough to activate the slot-gap mechanism. The high surface temperature of the star is attributed either to magnetic spot formation or thermoplastic wave heating due to the decay of the toroidal field near the pole. Our results could be useful in interpreting the high soft X-ray flux from high magnetic field pulsars.

Unifying neutron star sub-populations in the supernova fallback accretion model

We employ the supernova fallback disk model to simulate the spin evolution of isolated young neutron stars (NSs). We consider the submergence of the NS magnetic fields during the supercritical accretion stage and its succeeding reemergence. It is shown that the evolution of the spin periods and the magnetic fields in this model is able to account for the relatively weak magnetic fields of central compact objects and the measured braking indices of young pulsars. For a range of initial parameters, evolutionary links can be established among various kinds of NS sub-populations including magnetars, central compact objects and young pulsars. Thus, the diversity of young NSs could be unified in the framework of the supernova fallback accretion model.

Constraining the Braking Index and Energy Partition of Magnetar Spindown with Swift/XRT Data

Abstract The long-lasting X-ray plateau emission in long gamma-ray bursts (GRBs) shows observational evidence for ongoing energy injection, which may be from magnetar spindown due to energy released via either magnetic dipole (MD) or gravitational wave (GW) radiation. In this paper, by systematically analyzing the Swift/XRT light curves detected before 2018 July, we find 45 light curves with a measured redshift that monotonically decay as a smooth broken power law. By assuming that the central engines of these GRBs are newly born magnetars, we measure the braking index n of putative millisecond magnetars, due to MD and GW radiations. The inferred braking indices are not close to 3 or 5, but range between them with a normal distribution ( ). We define a dimensionless parameter , which is the ratio between the MD and GW components, and find that the energy released via magnetar spindown in most GRBs of our sample is dominated by GW radiation for ms and ϵ = 0.005 and 0.01. On the other hand, we find that and the braking index n seem to be anticorrelated within a large systematic error at t = 0, but depend on the values of the parameters P 0 and ϵ. These results suggest that the contribution of GW radiation cannot be ignored, and that a larger braking index leads to GWs dominating the energy released during magnetar spindown if indeed magnetars are operating in some long GRBs.

Braking index jumps in young pulsars

The departure of all measured pulsar braking indexes from the "canonical" dipole value 3 has been attributed to several causes in the past. Careful monitoring of the Crab pulsar has revealed permanent changes in the spin-down rate which are most likely the accumulation of small jumps in the angle $\alpha$ between the magnetic and spin axis. Recently, a large permanent change in the braking index of the in the "Crab twin" pulsar B0540-69 has been reported, and an analogous phenomenon seen in the high-field pulsar PSR 1846-0258 has been seen following a glitch, while another similar event (in PSR J119-6127) needs to be confirmed. We argue in this work that a common physical origin of all these observations can be attributed to the counter-alignment of the axis without serious violations of the observed features and with very modest inferred values of the hypothesized jump in the $\alpha$ angle. In addition, detected increases of the X-ray luminosities after the events are an additional ingredient for this interpretation. We argue that a component of a time-dependent torque has been identified, being an important ingredient towards a full solution of observed pulsar timing behavior which is in search of a consistent modeling.

On the Magnetar Origin of the GRBs Presenting X-Ray Afterglow Plateaus

Abstract The X-ray afterglow plateau emission observed in many gamma-ray bursts (GRBs) has been interpreted as being fueled either by fallback onto a newly formed black hole or by the spin-down luminosity of an ultra-magnetized millisecond neutron star. If the latter model is assumed, GRB X-ray afterglow light curves can be reproduced analytically. We fit a sample of GRB X-ray plateaus, interestingly yielding a distribution in the diagram of magnetic field versus spin period (B–P) consistent with B ∝ P 7/6, which is consistent with GRB expectations of the well-established physics of the spin-up line for accreting Galactic X-ray pulsars. The normalization of the relation that we obtain perfectly matches spin-up line predictions for typical neutron star masses (∼1 M ⊙) and radii (∼10 km), and for mass accretion rates typically expected in GRBs, . Short GRBs with extended emission (SEEs) appear toward the long-period end of the distribution, and long GRBs (LGRBs) toward the short-period end. This result is consistent with expectations from the spin-up limit, where the total accreted mass determines the position of the neutron star in the B–P diagram. The B–P distributions for LGRBs and SEEs are statistically different, further supporting the idea that the fundamental plane relation—a tri-dimensional correlation between the X-ray luminosity at the end of the plateau, the end time of the plateau, and the 1 s peak luminosity in the prompt emission—is a powerful discriminant among those populations. Our conclusions are robust against suppositions regarding the collimation angle of the GRB and the magnetar braking index, which shift the resulting properties of the magnetar parallel to the spin-up line, and strongly support a magnetar origin for GRBs presenting X-ray plateaus.