Birth Spin Period Research Articles

Fallback onto kicked neutron stars and its effect on spin-kick alignment

ABSTRACT Fallback in core-collapse supernova explosions is potentially of significant importance for the birth spins of neutron stars and black holes. It has recently been pointed out that the angular momentum imparted onto a compact remnant by fallback material is subtly intertwined with its kick because fallback onto a moving neutron star or black hole will preferentially come for a conical region around its direction of travel. We show that contrary to earlier expectations such one-sided fallback accretion onto a neutron star will tend to produce spin-kick misalignment. Since the baroclinic driving term in the vorticity equation is perpendicular to the nearly radial pressure gradient, convective eddies in the progenitor as well as Rayleigh–Taylor plumes growing during the explosion primarily carry angular momentum perpendicular to the radial direction. Fallback material from the accretion volume of a moving neutron star therefore carries substantial angular momentum perpendicular to the kick velocity. We estimate the seed angular momentum fluctuations from convective motions in core-collapse supernova progenitors and argue that accreted fallback material will almost invariably be accreted with the maximum permissible specific angular momentum for reaching the Alfvén radius. This imposes a limit of ${\sim }10^{-2}\, \mathrm{M}_\odot$ of fallback accretion for fast-spinning young neutron stars with periods of ${\sim }20\, \mathrm{ms}$ and less for longer birth spin periods.

Differential rotation in a 3D simulation of oxygen shell burning

ABSTRACTWe study differential rotation in late-stage shell convection in a 3D hydrodynamic simulation of a rapidly rotating $16\, {\rm M}_{\odot }$ helium star with a particular focus on the convective oxygen shell. We find that the oxygen shell develops a quasi-stationary pattern of differential rotation that is described neither by uniform angular velocity as assumed in current stellar evolution models of supernova progenitors, nor by uniform specific angular momentum. Instead, the oxygen shell develops a positive angular velocity gradient with faster rotation at the equator than at the pole by tens of per cent. We show that the angular momentum transport inside the convection zone is not adequately captured by a diffusive mixing-length flux proportional to the angular velocity or angular momentum gradient. Zonal flow averages reveal stable large-scale meridional flow and an entropy deficit near the equator that mirrors the patterns in the angular velocity. The structure of the flow is reminiscent of simulations of stellar surface convection zones and the differential rotation of the Sun, suggesting that similar effects are involved; future simulations will need to address in more detail how the interplay of buoyancy, inertial forces, and turbulent stresses shapes differential rotation during late-stage convection in massive stars. If convective regions develop positive angular velocity gradients, angular momentum could be shuffled out of the core region more efficiently, potentially making the formation of millisecond magnetars more difficult. Our findings have implications for neutron star birth spin periods and supernova explosion scenarios that involve rapid core rotation.

Stochastic core spin-up in massive stars – implications of 3D simulations of oxygen shell burning

ABSTRACT It has been suggested based on analytic theory that even in non-rotating supernova progenitors stochastic spin-up by internal gravity waves (IGWs) during the late burning stages can impart enough angular momentum to the core to result in neutron star birth spin periods below $100\, \mathrm{ms}$, and a relatively firm upper limit of $500\, \mathrm{ms}$ for the spin period. We here investigate this process using a 3D simulation of oxygen shell burning in a 3 M⊙ He star. Our model indicates that stochastic spin-up by IGWs is less efficient than previously thought. We find that the stochastic angular momentum flux carried by waves excited at the shell boundary is significantly smaller for a given convective luminosity and turnover time than would be expected from simple dimensional analysis. This can be explained by noting that the waves launched by overshooting convective plumes contain modes of opposite angular wavenumber with similar amplitudes, so that the net angular momentum of excited wave packets almost cancels. We find that the wave-mediated angular momentum flux from the oxygen shell follows a random walk, but again dimensional analysis overestimates the random walk amplitudes since the correlation time is only a fraction of the convective turnover time. Extrapolating our findings over the entire lifetime of the last burning stages prior to collapse, we predict that the core angular momentum from stochastic spin-up would translate into long birth spin periods of several seconds for low-mass progenitors and no less than $100\, \mathrm{ms}$ even for high-mass progenitors.

Neutron star bulk viscosity, ‘spin-flip’ and GW emission of newly born magnetars

The viscosity-driven "spin-flip" instability in newborn magnetars with interior toroidal magnetic fields is re-examined. We calculate the bulk viscosity coefficient ($\zeta$) of cold, $npe \mu$ matter in neutron stars (NS), for selected values of the nuclear symmetry energy and in the regime where $\beta$-equilibration is slower than characteristic oscillation periods. We show that: i) $\zeta$ is larger than previously assumed and the instability timescale correspondingly shorter; ii) for a magnetically-induced ellipticity $\epsilon_B \lesssim 4 \times 10^{-3}$, typically expected in newborn magnetars, spin-flip occurs for initial spin periods $\lesssim 2-3$ ms, with some dependence on the NS equation of state (EoS). We then calculate the detectability of GW signals emitted by newborn magnetars subject to "spin-flip", by accounting also for the reduction in range resulting from realistic signal searches. For an optimal range of $\epsilon_B \sim (1-5) \times 10^{-3}$, and birth spin period $\lesssim 2$ ms, we estimate an horizon of $\gtrsim 4$ Mpc, and $\gtrsim 30$ Mpc, for Advanced and third generation interferometers at design sensitivity, respectively. A supernova (or a kilonova) is expected as the electromagnetic counterpart of such GW events. Outside of the optimal range for GW emission, EM torques are more efficient in extracting the NS spin energy, which may power even brighter EM transients.

Effects of Fallback Accretion on Protomagnetar Outflows in Gamma-Ray Bursts and Superluminous Supernovae

Abstract Rapidly spinning, strongly magnetized protoneutron stars (“millisecond protomagnetars”) are candidate central engines of long-duration gamma-ray bursts (GRBs), superluminous supernovae (SLSNe), and binary neutron star mergers. Magnetar birth may be accompanied by the fallback of stellar debris, lasting for seconds or longer following the explosion. Accretion alters the magnetar evolution by (1) providing an additional source of rotational energy (or a potential sink, if the propeller mechanism operates), (2) enhancing the spin-down luminosity above the dipole rate by compressing the magnetosphere and expanding the polar cap region of open magnetic field lines, and (3) supplying an additional accretion-powered neutrino luminosity that sustains the wind baryon loading, even after the magnetar’s internal neutrino luminosity has subsided. The more complex evolution of the jet power and magnetization of an accreting magnetar more readily accounts for the high 56Ni yields of GRB SNe and the irregular time evolution of some GRB light curves (e.g., bursts with precursors followed by a long quiescent interval before the main emission episode). Additional baryon loading from accretion-powered neutrino irradiation of the polar cap lengthens the time frame over which the jet magnetization is in the requisite range σ ≲ 103 for efficient gamma-ray emission, thereby accommodating GRBs with ultralong durations. Though accretion does not significantly raise the maximum energy budget from the limit of ≲ few × 1052 erg for an isolated magnetar, it greatly expands the range of magnetic field strengths and birth spin periods capable of powering GRB jets, reducing the differences between the magnetar properties normally invoked to explain GRBs versus SLSNe.

THE SPIN RATE OF PRE-COLLAPSE STELLAR CORES: WAVE-DRIVEN ANGULAR MOMENTUM TRANSPORT IN MASSIVE STARS

The core rotation rates of massive stars have a substantial impact on the nature of core-collapse supernovae and their compact remnants. We demonstrate that internal gravity waves (IGW), excited via envelope convection during a red supergiant phase or during vigorous late time burning phases, can have a significant impact on the rotation rate of the pre-SN core. In typical ($10 \, M_\odot \lesssim M \lesssim 20 \, M_\odot$) supernova progenitors, IGW may substantially spin down the core, leading to iron core rotation periods $P_{\rm min,Fe} \gtrsim 30 \, {\rm s}$. Angular momentum (AM) conservation during the supernova would entail minimum NS rotation periods of $P_{\rm min,NS} \gtrsim 3 \, {\rm ms}$. In most cases, the combined effects of magnetic torques and IGW AM transport likely lead to substantially longer rotation periods. However, the stochastic influx of AM delivered by IGW during shell burning phases inevitably spin up a slowly rotating stellar core, leading to a maximum possible core rotation period. We estimate maximum iron core rotation periods of $P_{\rm max,Fe} \lesssim 5 \times 10^3 \, {\rm s}$ in typical core-collapse supernova progenitors, and a corresponding spin period of $P_{\rm max, NS} \lesssim 500 \, {\rm ms}$ for newborn neutron stars. This is comparable to the typical birth spin periods of most radio pulsars. Stochastic spin-up via IGW during shell O/Si burning may thus determine the initial rotation rate of most neutron stars. For a given progenitor, this theory predicts a Maxwellian distribution in pre-collapse core rotation frequency that is uncorrelated with the spin of the overlying envelope.

Population synthesis of isolated neutron stars with magneto-rotational evolution

We revisit the population synthesis of isolated radio-pulsars incorporating recent advances on the evolution of the magnetic field and the angle between the magnetic and rotational axes from new simulations of the magneto-thermal evolution and magnetosphere models, respectively. An interesting novelty in our approach is that we do not assume the existence of a death line. We discuss regions in parameter space that are more consistent with the observational data. In particular, we find that any broad distribution of birth spin periods with $P_0\lesssim 0.5$ s can fit the data, and that if the alignment angle is allowed to vary consistently with the torque model, realistic magnetospheric models are favoured compared to models with classical magneto-dipolar radiation losses. Assuming that the initial magnetic field is given by a lognormal distribution, our optimal model has mean strength $\langle\log B_0{\rm [G]}\rangle \approx 13.0-13.2$ with width $\sigma (\log B_0) = 0.6-0.7$. However, there are strong correlations between parameters. This degeneracy in the parameter space can be broken by an independent estimate of the pulsar birth rate or by future studies correlating this information with the population in other observational bands (X-rays and $\gamma$-rays).

Angular momentum redistribution by SASI spiral modes and consequences for neutron star spins

In the collapsing core of massive stars, the standing accretion shock instability (SASI) can drive spiral modes that efficiently redistribute angular momentum. This process can impart a spin to the forming neutron star even when the progenitor star is non-rotating. Here we develop the first analytical description of the angular momentum redistribution driven by a spiral mode of the SASI. Our analysis, valid in the limit of small mode amplitude, shows that the angular momentum separation is driven by the Reynolds stress generated by the spiral mode. The resulting solutions compare favorably with previous three-dimensional hydrodynamic simulations of the SASI in the linear and weakly non-linear phases. Reasonable agreement is also found when extrapolating the solutions into the fully non-linear phase. A Reynolds-decomposition of the flow is performed in the saturated state of these simulations, showing that outward angular momentum transport by the Reynolds stress and the fluctuating component of the mass flux balance inward transport by advection. We derive an approximate analytic expression for the maximum angular momentum deposited in the neutron star as a function of the mass accretion rate, shock radius, shock compression ratio, and amplitude of the spiral mode at the time of explosion. Implications for the birth spin periods of neutron stars are discussed.

Young rotation-powered pulsars as ultraluminous X-ray sources

The aim of this paper is to investigate a possible contribution of the rotation-powered pulsars and pulsar wind nebulae to the population of ultraluminous X-ray sources (ULXs). We first develop an analytical model for the evolution of the distribution function of pulsars over the spin period and find both the steady-state and the time-dependent solutions. Using the recent results on the X-ray efficiency dependence on pulsar characteristic age, we then compute the X-ray luminosity function (XLF) of rotation-powered pulsars. In a general case, it has a broken power-law shape with a high-luminosity cutoff, which depends on the distributions of the birth spin period and the magnetic field. Using the observed XLF of sources in the nearby galaxies and the condition that the pulsar XLF does not exceed that, we find the allowed region for the parameters describing the birth period distribution. We find that the mean pulsar period should be greater than 10–40 ms. These results are consistent with the constraints obtained from the X-ray luminosity of core-collapse supernovae. We estimate that the contribution of the rotation-powered pulsars to the ULX population is at a level exceeding 3 per cent. For a wide birth period distribution, this fraction grows with luminosity and above 1040 erg s−1 pulsars can dominate the ULX population.

Can magnetar spin-down power extended emission in some short GRBs?

Extended emission gamma-ray bursts are a subset of the ‘short’ class of burst which exhibit an early time rebrightening of gamma emission in their light curves. This extended emission arises just after the initial emission spike, and can persist for up to hundreds of seconds after trigger. When their light curves are overlaid, our sample of 14 extended emission bursts show a remarkable uniformity in their evolution, strongly suggesting a common central engine powering the emission. One potential central engine capable of this is a highly magnetized, rapidly rotating neutron star, known as a magnetar. Magnetars can be formed by two compact objects coalescing, a scenario which is one of the leading progenitor models for short bursts in general. Assuming that a magnetar is formed, we gain a value for the magnetic field and late-time spin period for nine of the extended emission bursts by fitting the magnetic dipole spin-down model of Zhang and Mészáros. Assuming that the magnetic field is constant, and the observed energy release during extended emission is entirely due to the spin-down of this magnetar, we then derive the spin period at birth for the sample. We find that all birth spin periods are in good agreement with those predicted for a newly born magnetar.

SXP 1062, a young Be X-ray binary pulsar with long spin period

(shortened) The SMC is ideally suited to investigating the recent star formation history from X-ray source population studies. It harbours a large number of Be/X-ray binaries, and the supernova remnants can be easily resolved with imaging X-ray instruments. We search for new supernova remnants in the SMC and in particular for composite remnants with a central X-ray source. We study the morphology of newly found candidate supernova remnants using radio, optical and X-ray images and investigate their X-ray spectra. Here we report on the discovery of the new supernova remnant around the recently discovered Be/X-ray binary pulsar SXP 1062 in radio and X-ray images. The Be/X-ray binary system is found near the centre of the supernova remnant, which is located at the outer edge of the eastern wing of the SMC. The remnant is oxygen-rich, indicating that it developed from a type Ib event. From XMM-Newton observations we find that the neutron star with a spin period of 1062 s shows a very high average spin-down rate of 0.26 s per day over the observing period of 18 days. From the currently accepted models, our estimated age of around 10000-25000 years for the supernova remnant is not long enough to spin down the neutron star from a few 10 ms to its current value. Assuming an upper limit of 25000 years for the age of the neutron star and the extreme case that the neutron star was spun down by the accretion torque that we have measured during the XMM-Newton observations since its birth, a lower limit of 0.5 s for the birth spin period is inferred. For more realistic, smaller long-term average accretion torques our results suggest that the neutron star was born with a correspondingly longer spin period. This implies that neutron stars in Be/X-ray binaries with long spin periods can be much younger than currently anticipated.

Origin and evolution of magnetars

Abstract We present a population synthesis study of the observed properties of the magnetars investigating the hypothesis that they are drawn from a population of progenitors that are more massive than those of the normal radio pulsars. We assume that the anomalous X-ray emission is caused by the decay of a toroidal or tangled up field that does not take part in the spin-down of the star. Our model assumes that the magnetic flux of the neutron star is distributed as a Gaussian in the logarithm about a mean value that is described by a power law , where Mp is the mass of the progenitor. We find that we can explain the observed properties of the magnetars for a model with Φ0 = 2 × 1025 G cm2 and γ= 5 if we suitably parametrize the time evolution of the anomalous X-ray luminosity as an exponentially decaying function of time. Our modelling suggests that magnetars arise from stars in the high-mass end (20 M⊙≤Mp≤ 45 M⊙) of this distribution. The lower mass progenitors are assumed to give rise to the radio pulsars. The high value of γ can be interpreted in one of two ways. It may indicate that the magnetic flux distribution on the main sequence is a strong function of mass and that this is reflected in the magnetic fluxes of the neutron stars that form from this mass range (the fossil field hypothesis). The recent evidence for magnetic fluxes similar to those of the magnetars in a high fraction (∼25 per cent) of massive O-type stars lends support to such a hypothesis. Another possibility is that the spin of the neutron star is a strong function of the progenitor mass, and it is only for stars that are more massive than ∼20 M⊙ that magnetar-type fields can be generated by the α−ω dynamo mechanism (the convective dynamo hypothesis). In either interpretation, it has to be assumed that all or a subset of stars in the mass range ∼20–45 M⊙, which on standard stellar evolution models lead to black holes via the formation of a fall-back disc, must give rise to magnetars. Unlike with the radio pulsars, the magnetars only weakly constrain the birth spin period, due to their rapid spin-down. Our model predicts a birthrate of ∼1.5–3 × 10−3 yr−1 for the magnetars.

The birth properties of Galactic millisecond radio pulsars

We model the population characteristics of the sample of millisecond pulsars (MSPs) within a distance of 1.5 kpc. We find that for a braking index n = 3, the birth magnetic field distribution of the neutron stars as they switch on as radio-emitting MSPs can be represented by a Gaussian in the logarithm with mean log B(G) = 8.1 and σ log B = 0.4 and their birth spin period by a Gaussian with mean P 0 = 4 ms and σ P 0 = 1.3 ms. We assume no field decay during the lifetime of MSPs. Our study, which takes into consideration acceleration effects on the observed spin-down rate, shows that most MSPs are born with periods that are close to the currently observed values and with average characteristic ages that are typically larger by a factor of ∼1.5 compared to the true age. The Galactic birth rate of the MSPs is deduced to be ≥3.2 x 10 -6 yr -1 near the upper end of previous estimates and larger than the semi-empirical birth rate ∼10 -7 yr -1 of the low-mass X-ray binaries (LMXBs), the currently favoured progenitors. The mean birth spin period deduced by us for the radio MSPs is a factor of ∼2 higher than the mean spin period observed for the accretion and nuclear powered X-ray pulsars, although this discrepancy can be resolved if we use a braking index n = 5, the value appropriate to spin-down caused by angular momentum losses by gravitational radiation or magnetic multipolar radiation. We discuss the arguments for and against the hypothesis that accretion-induced collapse (AIC) may constitute the main route to the formation of the MSPs, pointing out that on the AIC scenario the low magnetic fields of the MSPs may simply reflect the field distribution in isolated magnetic white dwarfs which has recently been shown to be bi-modal with a dominant component that is likely to peak at fields below 10 3 G which would scale to neutron star fields below 10 9 G, under magnetic flux conservation.

Birth and Evolution of Isolated Radio Pulsars

We investigate the birth and evolution of Galactic isolated radio pulsars. We begin by estimating their birth space velocity distribution from proper motion measurements of Brisken et al. (2002, 2003). We find no evidence for multimodality of the distribution and favor one in which the absolute one-dimensional velocity components are exponentially distributed and with a three-dimensional mean velocity of 380^{+40}_{-60} km s^-1. We then proceed with a Monte Carlo-based population synthesis, modelling the birth properties of the pulsars, their time evolution, and their detection in the Parkes and Swinburne Multibeam surveys. We present a population model that appears generally consistent with the observations. Our results suggest that pulsars are born in the spiral arms, with a Galactocentric radial distribution that is well described by the functional form proposed by Yusifov & Kucuk (2004), in which the pulsar surface density peaks at radius ~3 kpc. The birth spin period distribution extends to several hundred milliseconds, with no evidence of multimodality. Models which assume the radio luminosities of pulsars to be independent of the spin periods and period derivatives are inadequate, as they lead to the detection of too many old simulated pulsars in our simulations. Dithered radio luminosities proportional to the square root of the spin-down luminosity accommodate the observations well and provide a natural mechanism for the pulsars to dim uniformly as they approach the death line, avoiding an observed pile-up on the latter. There is no evidence for significant torque decay (due to magnetic field decay or otherwise) over the lifetime of the pulsars as radio sources (~100 Myr). Finally, we estimate the pulsar birthrate and total number of pulsars in the Galaxy.

Modelling of isolated radio pulsars and magnetars on the fossil field hypothesis

We explore the hypothesis that the magnetic fields of neutron stars are of fossil origin. For parametrised models of the distribution of magnetic flux on the Main Sequence and of the birth spin period of the neutron stars, we calculate the expected properties of isolated radio pulsars in the Galaxy using as our starting point the initial mass function and star formation rate as a function of galacto-centric radius. We then use the 1374 MHz Parkes Multi-Beam Survey of isolated radio pulsars to constrain the parameters in our model and to deduce the required distribution of magnetic fields on the main sequence. We find agreement with observations for a model with a star formation rate that corresponds to a supernova rate of 2 per century in the Galaxy from stars with masses in the range 8 - 45 solar masses and predict 447,000 active pulsars in the Galaxy with luminosities greater than 0.19 mJy kpc^2. The progenitor OB stars have a field distribution which peaks near 46 Gauss with about 8 percent of stars having fields in excess of 1,000 Gauss. The higher field progenitors yield a population of 24 neutron stars with fields in excess of 10^14 Gauss, periods ranging from 5 to 12 seconds, and ages of up to 100,000 years, which we identify as the dominant component of the magnetars. We also predict that high field neutron stars (log B>13.5) originate preferentially from higher mass progenitors and have a mean mass of 1.6 solar masses, which is significantly above the mean mass of 1.4 solar masses calculated for the overall population of radio pulsars.

Phase-coherent timing of PSR J1811-1925: the pulsar at the heart of G11.2-0.3

The X-ray Pulsar PSR J1811-1925 in the historic supernova remnant G11.2-0.3 has a characteristic age over 10 times the age of the remnant. This likely implies that its current spin period, 65 ms, is close to its birth spin period. Alternatively, the pulsar may have an unusually high braking index. We report here on regular Rossi X-ray Timing Explorer/Proportional Counting Array (RXTE/PCA) timing observations of the pulsar that were designed to measure its braking index. We provide a preliminary phase-coherent timing solution which includes a significant ν ̈ . The braking index we measure is ≫3, likely a result of conventional timing noise. We also report on a preliminary analysis of the pulsar's hard spectrum: we determine a photon index of Γ=1.78±0.74, for the pulsed component of the spectrum, consistent within the uncertainties with previous ASCA and Chandra observations. The pulsed emission of PSR J1811-1925 is seen beyond 30 keV, and the pulsations remain sinusoidal up to and beyond this energy.

The Pulsar Wind Nebula Torus of PSR J0538+2817 and the Origin of Pulsar Velocities

We find evidence for a faint wind nebula surrounding PSR J0538+2817 in Chandra X-Ray Observatory ACIS imaging. This object is particularly interesting, as the pulsar spin-down age is largest for any such X-ray pulsar wind nebula (PWN). If interpreted as an equatorial torus, the PWN supports the claimed association with the S147 supernova remnant and implies good alignment between the pulsar spin and space velocity. Comparison of the supernova remnant age, X-ray cooling age, and characteristic age suggests a birth spin period of ≳130 ms. In turn, if we accept as causal the alignment of the linear and angular momenta, this places strong constraints on the origin of the "kick" at the neutron star birth.