Timing of young radio pulsars II. Braking indices and their interpretation
A. Parthasarathy, S. Johnston, R. M. Shannon, L. Lentati, M. Bailes, S. Dai, M. Kerr, R. N. Manchester, S. Osłowski, C. Sobey. W. van Straten, P. Weltevrede
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 $\sim$ 10 years 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.
Braking indices are used to describe the evolution of pulsars rotation, and can offer insights into the braking mechanism that dominates the slow down. Here we discuss the main difficulties associated with measuring braking indices and the complexity of interpreting these measurements. Considering recent braking index measurements on pulsars with large and regular glitches, we comment on the significant effects that the loosely coupled superfluid inside pulsars might have on their spin evolution.
The analysis of some braking mechanisms for neutron stars was carried out to determine the sign of the second derivative of the pulsar period. This quantity is the important parameter for calculations of the braking index n. It is shown that this derivative can be positive and lead to decreasing of n. It is necessary to correct the methods of calculations of n used this moment because they are based as a rule on the suggestion on the constancy of pulsar parameters (magnetic fields, angles between some axes and so on). The estimations of corrections to braking indices are obtained. It is shown …
Eight pulsars have low braking indices, which challenge the magnetic dipole braking of pulsars. 222 pulsars and 15 magnetars have abnormal distribution of frequency second derivatives, which also make contradiction with classical understanding. How neutron star magnetospheric activities affect these two phenomena are investigated by using the wind braking model of pulsars. It is based on the observational evidence that pulsar timing is correlated with emission and both aspects reflect the magnetospheric activities. Fluctuations are unavoidable for a physical neutron star magnetosphere. Young p…
Recently, Parthsarathy et al. analysed long-term timing observations of 85 young radio pulsars. They found that 11 objects have braking indices ranging $\sim 10-100$, far from the classical value $n=3$. They also noted a mild correlation between measured value of $n$ and characteristic age of a radio pulsar. In this article we systematically analyse possible physical origin of large braking indices. We find that a small fraction of these measurements could be caused by gravitational acceleration from an unseen ultra-wide companion of a pulsar or by precession. Remaining braking indices cannot …
PSR B0540-69 has braking index measurement in its persistent state: n=2.129 \pm 0.012. Recently, it is reported to have spin-down state changes: a suddenly 36 % increase in the spin-down rate. Combining the persistent state braking index measurement and different spin-down states, PSR B0540-69 is more powerful than intermittent pulsars in constraining pulsar spin-down models. The pulsar wind model is applied to explain the variable timing behavior of PSR B0540-69. The persistent state braking index of PSR B0540-69 is the combined effect of magnetic dipole radiation and particle wind. The parti…
Due to the lack of long term pulsed emission in quiescence and the strong timing noise, it is impossible to directly measure the braking index $n$ of a magnetar. 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 they cluster in a range of $1\sim$42. Five magnetars have smaller mean braking indices of $1<n<3$, and we interpret them within a combination of magneto-dipole radiation and wind aided braking, while the larger mean braking indices of $n>3$ for other three…
Braking indices of pulsars present a scientific challenge as their theoretical calculation is still an open problem. In this paper we report results of a study regarding such calculation which adapts the canonical model (which admits that pulsars are rotating magnetic dipoles) basically by introducing a compensating component in the energy conservation equation of the system. This component would correspond to an effective force that varies with the first power of the tangential velocity of the pulsar's crust. We test the proposed model using data available and predict braking indices values f…
The ~800 yr-old pulsar J1846-0258 is a unique transition object between rotation-powered pulsars and magnetars: though behaving like a rotation-powered pulsar most of the time, in 2006 it exhibited a distinctly magnetar-like outburst accompanied by a large glitch. Here we present X-ray timing observations taken with the Rossi X-ray Timing Explorer over a 2.2-yr period after the X-ray outburst and glitch had recovered. We observe that the braking index of the pulsar, previously measured to be n=2.65+/-0.01, is now n=2.16+/-0.13, a decrease of 18+/-5%. We also note a persistent increase in the t…
We present a phase-coherent timing solution for PSR J1640-4631, a young 206 ms pulsar using X-ray timing observations taken with NuSTAR. Over this timing campaign, we have measured the braking index of PSR J1640-4631 to be n = 3.15+/-0.03. Using a series of simulations, we argue that this unusually high braking index is not due to timing noise, but is intrinsic to the pulsar's spin-down. We cannot, however, rule out contamination due to an unseen glitch recovery, although the recovery timescale would have to be longer than most yet observed. If this braking index is eventually proven to be sta…
We have investigated the long-term evolutions of the high-magnetic field radio pulsars (HBRPs) with measured braking indices in the same model that was applied earlier to individual anomalous X-ray pulsars (AXPs), soft gamma repeaters (SGRs) and dim isolated neutron stars (XDINs). We have shown that the rotational properties (period, period derivative and braking index) and the X-ray luminosity of individual HBRPs can be acquired simultaneously by the neutron stars evolving with fallback discs. The model sources reach the observed properties of HBRPs in the propeller phases, when pulsed radio …
In this paper, we attribute high braking indices $n>3$ of two magnetars SGR 0501$+$4516 and 1E 2259$+$586 to the decrease in their inclination angles using the double magnetic-dipole model proposed by Hamil et al.(2016). In this model, there are two magnetic moments inside a neutron star, one is generated by the rotation effect of a charged sphere, $M_{1}$, and the other is generated by the magnetization of ferromagnetically ordered material, $M_{2}$. Our calculations indicate that the magnetic moment $M_{2}$ would evolve towards alignment with the spin axis of the two magnetars, and cause the…
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…
Braking index measurements offer the opportunity to explore the processes affecting the long-term spin evolution of pulsars and possible evolutionary connections between the various pulsar populations. For young pulsars the long-term trends are generally obscured by short term phenomena such as timing noise and the recoveries form large glitches. Here we present a new method to overcome the latter and report on braking index measurements for the Vela-like pulsars PSR B1800-21 and PSR B1823-13, an updated measurement for Vela and new estimates for four more glitching pulsars observed at Jodrell…
Pulsars are stars that emit electromagnetic radiation in well-defined time intervals. The frequency of such pulses decays with time as is quantified by the {\it braking index} ($n$). In the canonical model $n = 3$ for all pulsars, but observational data show that $n \neq 3$, indicating a limitation of the model. In this work we present a new approach to study the frequency decay of the rotation of a pulsar, based on an adaptation of the canonical one. We consider the pulsar a star that rotates in vacuum and has a strong magnetic field but, differently from the canonical model, we assume that i…
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 th…
A nascent neutron star may be exposed to fallback accretion soon after the proto-neutron star stage. This high accretion episode can submerge the magnetic field deep in the crust. The diffusion of the magnetic field back to the surface will take hundreds to millions of years depending on the amount of mass accreted and the consequent depth the field is buried. Neutron stars with large kick velocities will accrete less amount of fallback material leading to shallower submergence of their fields and shorter time-scales for the growth of their fields. We obtain the relation $τ_{\rm Ohm} \propto v…
We study the role of time dependent magnetic ellipticities ($ε_{B}$) on the calculation of the braking index of pulsars. Moreover, we study the consequences of such a $ε_{B}$ on the amplitude of gravitational waves (GWs) generated by pulsars with measured braking indices. We show that, since the ellipticity generated by the magnetic dipole is extremely small, the corresponding amplitude of GWs is much smaller than the amplitude obtained via the spindown limit.
We report on the timing properties of the `Crab twin' pulsar PSR~B0540$-$69 measured with X-ray data taken with the Swift telescope over a period of 1100\,days. The braking index of the pulsar was estimated to be $n = 0.03\pm0.013$ in a previous study performed in 2015 with 500-day Swift data. This small value of $n$ is unusual for pulsars, and a comparison to old measurements of $n\approx2.1$ for the same target determined $\sim$10 years ago suggests a dramatic change in the braking index. To confirm the small value and large change of $n$, we used 1100-day Swift observations including the da…
The magnetic fields of the nascent neutron stars could be submerged in the crust by rapid fallback accretion and could diffuse to the surface later in life. According to this field burial scenario young pulsars may have growing magnetic fields, a process known to result with less-than-three braking indices; larger braking indices implying longer field-growth time-scales. A nascent neutron star with a larger kick velocity would accrete less amount of matter leading to a shallower burial of its field and a more rapid field growth. Such an inverse relation between the field growth time-scale infe…
PSR J1846-0258 is an object which straddles the boundary between magnetars and rotation powered pulsars. Though behaving for many years as a rotation-powered pulsar, in 2006, it exhibited distinctly magnetar-like behavior - emitting several short hard X-ray bursts, and a flux increase. Here we report on 7 years of post-outburst timing observations of PSR J1846-0258 using the Rossi X-ray Timing Explorer and the Swift X-ray Telescope. We measure the braking index over the post-magnetar outburst period to be $n=2.19\pm0.03$. This represents a change of $Δn=-0.46\pm0.03$ or a 14.5$\;σ$ difference …
Recent timing observation constrained the braking index of the X-ray pulsar PSR J1640-4631 to be $n=3.15\pm0.03$, which is the highest value of all pulsars with measured braking indices so far. In this Letter, we investigate whether pulsar braking by combined between the magnetic dipole emission and the gravitational radiation might have a braking index greater than three. For conventional neutron star and low mass quark star candidates, the inferred ellipticities derived by the observed braking index are obviously much larger than the theoretical estimated maximum value. If PSR J1640-4631 is …
The magnetic dipole radiation (MDR) model is currently the best approach we have to explain pulsar radiation. However a most characteristic parameter of the observed radiation, the braking index n$_{\rm obs}$ shows deviations for all the eight best studied isolated pulsars, from the simple model prediction n$_{\rm dip}$ = 3. The index depends upon the rotational frequency and its first and second time derivatives, but also on the assumption of that the magnetic dipole moment and inclination angle, and the moment of inertia of the pulsar are constant in time. In a recent paper [Phys. Rev. D 91,…
Recent timing observation reported that the radio pulsar PSR J1734 - 3333 with a rotating period $P=1.17~\rm s$ is slowing down with a period derivative $\dot{P}=2.28\times 10^{-12}\rm s\,s^{-1}$. Its derived braking index $n=0.9 \pm 0.2$ is the lowest value among young radio pulsars with the measured braking indices. In this Letter, we attempt to investigate the influence of the braking torque caused by the interaction between the fall-back disk and the strong magnetic field of the pulsar on the spin evolution of PSR J1734 - 3333. Analytical result show that this braking torque is obviously f…
We make the first measurement of the braking index $n$ of two putative millisecond magnetars born in short gamma-ray bursts. We measure $n=2.9\pm0.1$ and $n=2.6\pm0.1$ for millisecond magnetars born in GRB 130603B and GRB 140903A respectively. The neutron star born in GRB 130603B has the only known braking index consistent with the fiducial $n=3$ value. This value is ruled out with 99.95\% confidence for GRB 140903A. We discuss possible causes of $n<3$ braking indices in millisecond magnetars, showing that several models can account for the measurement of the braking index in GRB 140903A, whil…
The binary neutron star merger GW170817/GRB170817A confirmed that at least some neutron star mergers are the progenitors of short gamma-ray bursts. Many short gamma-ray bursts have long-term x-ray afterglows that have been interpreted in terms of post-merger millisecond magnetars---rapidly rotating, highly magnetised, massive neutron stars. We review our current understanding of millisecond magnetars born in short gamma-ray bursts, focusing particularly three main topics. First, whether millisecond magnetars really do provide the most plausible explain for the x-ray plateau. Second, determinin…
We report the discovery and timing measurements of PSR J1208-6238, a young and highly magnetized gamma-ray pulsar, with a spin period of 440 ms. The pulsar was discovered in gamma-ray photon data from the Fermi Large Area Telescope (LAT) during a blind-search survey of unidentified LAT sources, running on the distributed volunteer computing system Einstein@Home. No radio pulsations were detected in dedicated follow-up searches with the Parkes radio telescope, with a flux density upper limit at 1369 MHz of 30 $μ$Jy. By timing this pulsar's gamma-ray pulsations, we measure its braking index over…
Pulsars are good clocks in the universe. One fundamental question is that why they are good clocks? This is related to the braking mechanism of pulsars. Nowadays pulsar timing is done with unprecedented accuracy. More pulsars have braking indices measured. The period derivative of intermittent pulsars and magnetars can vary by a factor of several. However, during pulsar studies, the magnetic dipole braking in vacuum is still often assumed. It is shown that the fundamental assumption of magnetic dipole braking (vacuum condition) does not exist and it is not consistent with the observations. The…
The wind braking model and its applications of magnetars are discussed. The decreasing torque of magnetars during outbursts, anti-glitch, and anti-correlations between radiation and timing are understandable in the wind braking model. Recent timing observations of magnetars are also consistent with the previous modeling. A magnetism-powered wind nebula and a braking index smaller than three are the two predictions. Besides isolated magnetars, there may also be accreting magnetars in binary systems and magnetars accreting from fallback disks. Observationally, ultra-luminous X-ray pulsars may be…
The recently discovered rotationally powered pulsar PSR J1640-4631 is the first to have a braking index measured, with high enough precision, that is greater than three. An inclined magnetic rotator in vacuum or plasma would be subject not only to spin-down but also to an alignment torque. The vacuum model can address the braking index only for an almost orthogonal rotator that is incompatible with the single peaked pulse profile. The magnetic dipole model with the corotating plasma predicts braking indices between $3-3.25$. We find that the braking index of $3.15$ is consistent with two diffe…
In this work, we interpreted the high braking index of PSR J1640$-$4631 with a combination of the magneto-dipole radiation and dipole magnetic field decay models. By introducing a mean rotation energy conversion coefficient $\overlineζ$, the ratio of the total high-energy photon energy to the total rotation energy loss in the whole life of the pulsar, and combining the pulsar's high-energy and timing observations with reliable nuclear equation of state, we estimate the pulsar's initial spin period, $P_{0}\sim (17-44)$ ms, corresponding to the moment of inertia $I\sim (0.8-2.1)\times 10^{45}$ g…
Spontaneous wandering of the magnetic axis in pulsars: 3D magneto-thermal simulations and the imprint on braking indices
Departament de Física, Universitat d’Alacant（阿利坎特大学物理系）；Institut de Ciències de I’Espai (ICE-CSIC)（空间科学研究所（CSIC））；Institut d’Estudis Espacials de Catalunya (IEEC)（加泰罗尼亚空间研究学院）
We report the results of a 16-month monitoring campaign using the Swift satellite of PSR B0540-69, a young pulsar in the Large Magellanic Cloud. Phase connection was maintained throughout the campaign so that a reliable ephemeris could be determined, and the length of the campaign is adequate to accurately determine the spin frequency and its first and second derivatives. The braking index is 0.031 +/- 0.013 (90% confidence), a value much lower than previously reported for B0540-69 and almost all other young pulsars. We use data from the extensive monitoring campaign with RXTE to show that tim…
The spin evolution of isolated neutron stars (NSs) is dominatd by their magnetic fields. The measured braking indices of young NSs show that the spin-down mechanism due to magnetic dipole radiation with constant magnetic fields is inadequate. Assuming that the NS magnetic field is buried by supernova fallback matter and re-emerges after accretion stops, we carry out Monte-Carlo simulation of the evolution of young NSs, and show that most of the pulsars have the braking indices ranging from -1 to 3. The results are compatible with the observational data of NSs associated with supernova remnants…
The inter-glitch timing of the Vela pulsar is characterized by a constant second derivative of the rotation rate. This takes over after the post-glitch exponential relaxation, and is completed at about the time of the next glitch. The vortex creep model explains the second derivatives in terms of non-linear response to the glitch. We present inter-glitch timing fits to the present sample covering 16 large glitches, taking into account the possibility that in some glitches part of the step in spin-down rate may involve a "persistent shift", as observed in the Crab pulsar. Modifying the expressi…
Though pulsars spin regularly, the differences between the observed and predicted ToA (time of arrival), known as "timing noise", can still reach a few milliseconds or more. We try to understand the noise in this paper. As proposed by Xu & Qiao in 2001, both dipole radiation and particle emission would result in pulsar braking. Accordingly, possible fluctuation of particle current flow is suggested here to contribute significant ToA variation of pulsars. We find that the particle emission fluctuation could lead to timing noise which can't be eliminated in timing process, and that a longer peri…
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 $n=3.15\pm0.03$. The decrease of the inclination angle between 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 et al (2016) to explain the decrease of the magnetic inclination angle of PSR J1640-4631. The rotation effect of a ch…
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 $α$ 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 J11…
We study the possibility that the long term red timing-noise in pulsars originates from the evolution of the magnetic inclination angle $χ$. The braking torque under consideration is a combination of the dipole radiation and the current loss. We find that the evolution of $χ$ can give rise to extra cubic and fourth-order polynomial terms in the timing residuals. These two terms are determined by the efficiency of the dipole radiation, the relative electric-current density in the pulsar tube and $χ$. The following observation facts can be explained with this model: a) young pulsars have positiv…
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 {\em 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…
Radio timing observations of the high-magnetic-field pulsar PSR J1718-3718 have shown that it suffered a large glitch with Dnu_g/nu = (33.25 +/- 0.01) x 10^{-6} between 2007 September (MJD 54336) and 2009 January (MJD 54855). This is the largest pulsar glitch ever observed. As is common, there was a small increase in braking torque at the time of the glitch but, unlike all other pulsars, the braking torque has continued to increase over the two years since the glitch. Polarization observations show that the mean pulse profile has about 30% linear polarization with a smooth change of position a…
In continuation of our earlier work on the long-term evolution of the so-called high--B radio pulsars (HBRPs) with measured braking indices, we have investigated the long-term evolution of the remaining five HBRPs for which braking indices have not been measured yet. This completes our source-by-source analyses of HBRPs in the fallback disc model that was also applied earlier to anomalous X-ray pulsars (AXPs), soft gamma repeaters (SGRs), and dim isolated neutron stars (XDINs). Our results show that the X-ray luminosities and the rotational properties of these rather different neutron star pop…
The pulsar timing is observed to be different from predicted by a simple magnetic dipole radiation. We choose eight pulsars whose braking index was reliably determined. Assuming the smaller values of braking index are dominated by the secular evolution of the magnetic inclination, we calculate the increasing rate of the magnetic inclination for each pulsar. We find a possible relation between the rotation frequency of each pulsar and the inferred evolution of the magnetic inclination. Due to the model-dependent fit of the magnetic inclination and other effects, more observational indicators fo…
We discuss some aspects of de Araujo, Coelho & Costa (2016, 2017) concerning the role of a time dependent magnetic ellipticity on the pulsars' braking indices and on the putative gravitational waves (GWs) these objects can emit. Since only nine of more than 2000 known pulsars have accurately measured braking indices, it is of interest to extend this study to all known pulsars, in particular as regards GW generation. In summary, our results show a pessimistic prospect for the detection of GWs generated by these pulsars, even for advanced detectors such as aLIGO and AdVirgo, and the planned Eins…
PSR J0537$-$6910 is the most active glitching pulsar with a glitch rate $\sim0.3$ yr $^{-1}$. We have reanalysed 45 glitches of PSR J0537$-$6910 published in the literature and have done post-glitch timing fits within the vortex creep model. Moment of inertia fractions of the superfluid regions participating in glitches are obtained for each event and the model predictions for the inter-glitch time are confronted with the observed time-scales. Similarities and differences with the glitching behaviours of the well studied Crab and Vela pulsars are highlighted. From superfluid recoupling time-sc…
We investigate the impact of noise processes on high-precision pulsar timing. Our analysis focuses on the measurability of the second spin frequency derivative $\ddotν$. This $\ddotν$ can be induced by several factors including the radial velocity of a pulsar. We use Bayesian methods to model the pulsar times-of-arrival in the presence of red timing noise and dispersion measure variations, modelling the noise processes as power laws. Using simulated times-of-arrival that both include red noise, dispersion measure variations and non-zero $\ddotν$ values, we find that we are able to recover the …
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 pulsar timing and position comparison method. The pulsar rotation parameters are derived and are more accurate than previously published value…
Highly precise pulsar timing is very important for understanding the nature of neutron stars and the implied physics, even being used to detect directly gravitational waves. Unfortunately, the accuracy of the pulsar timing is seriously affected by the spin-down irregularities, such as spin fluctuations with a manifestation of low frequency structures (the so-called red noise processes), and various activities of magnetospheres. Except from the random timing noise, significant periodic or quasi-periodic components could also be found in some timing residuals of pulsars, indicating the presence …
PSR J0537-6910, also known as the Big Glitcher, is the most prolific glitching pulsar known, and its spin-induced pulsations are only detectable in X-ray. We present results from analysis of 2.7 years of NICER timing observations, from 2017 August to 2020 April. We obtain a rotation phase-connected timing model for the entire timespan, which overlaps with the third observing run of LIGO/Virgo, thus enabling the most sensitive gravitational wave searches of this potentially strong gravitational wave-emitting pulsar. We find that the short-term braking index between glitches decreases towards a …
It has long been thought that the pulsed X-ray properties of rotation-powered pulsars are stable on long time scales. However, long-term, systematic studies of individual sources have been lacking. Furthermore, dramatic X-ray variability has now been observed from two pulsars having inferred sub-critical dipole magnetic fields. Here we present an analysis of the long-term pulsed X-ray properties of the young, energetic pulsar PSR B1509-58 using data from the Rossi X-ray Timing Explorer. We measured the 2-50 keV pulsed flux for 14.7 yr of X-ray observations and found that it is consistent with …
Measuring a pulsar's rotational evolution is crucial to understanding the nature of the pulsar. Here we provide updated timing models for the rotational evolution of six pulsars, five of which are rotation phase-connected using primarily NICER X-ray data. For the newly-discovered fast energetic young pulsar, PSR J0058-7218, we increase the baseline of its timing model from 1.4 days to 8 months and not only measure more precisely its spin-down rate nudot = (-6.2324+/-0.0001)x10^-11 Hz s^-1 but also for the first time the second time derivative of spin rate nuddot = (4.2+/-0.2)x10^-21 Hz s^-2. F…
