Neutron Star Birth Research Articles

Isolated Pulsar Population Synthesis with Simulation-based Inference

We combine pulsar population synthesis with simulation-based inference (SBI) to constrain the magnetorotational properties of isolated Galactic radio pulsars. We first develop a framework to model neutron star birth properties and their dynamical and magnetorotational evolution. We specifically sample initial magnetic field strengths, B, and spin periods, P, from lognormal distributions and capture the late-time magnetic field decay with a power law. Each lognormal is described by a mean, μlogB,μlogP , and standard deviation, σlogB,σlogP , while the power law is characterized by the index, a late. We subsequently model the stars’ radio emission and observational biases to mimic detections with three radio surveys, and we produce a large database of synthetic P– Ṗ diagrams by varying our five magnetorotational input parameters. We then follow an SBI approach that focuses on neural posterior estimation and train deep neural networks to infer the parameters’ posterior distributions. After successfully validating these individual neural density estimators on simulated data, we use an ensemble of networks to infer the posterior distributions for the observed pulsar population. We obtain μlogB=13.10−0.10+0.08 , σlogB=0.45−0.05+0.05 and μlogP=−1.00−0.21+0.26 , σlogP=0.38−0.18+0.33 for the lognormal distributions and alate=−1.80−0.61+0.65 for the power law at the 95% credible interval. We contrast our results with previous studies and highlight uncertainties of the inferred a late value. Our approach represents a crucial step toward robust statistical inference for complex population synthesis frameworks and forms the basis for future multiwavelength analyses of Galactic pulsars.

TRAPUM search for pulsars in supernova remnants and pulsar wind nebulae − I. Survey description and initial discoveries

ABSTRACT We present the description and initial results of the TRAPUM (TRAnsients And PUlsars with MeerKAT) search for pulsars associated with supernova remnants (SNRs), pulsar wind nebulae, and unidentified TeV emission. The list of sources to be targeted includes a large number of well-known candidate pulsar locations but also new candidate SNRs identified using a range of criteria. Using the 64-dish MeerKAT radio telescope, we use an interferometric beamforming technique to tile the potential pulsar locations with coherent beams which we search for radio pulsations, above a signal-to-noise of 9, down to an average flux density upper limit of 30 μJy. This limit is target-dependent due to the contribution of the sky and nebula to the system temperature. Coherent beams are arranged to overlap at their 50 per cent power radius, so the sensitivity to pulsars is not degraded by more than this amount, though realistically averages around 65 per cent if every location in the beam is considered. We report the discovery of two new pulsars; PSR J1831−0941 is an adolescent pulsar likely to be the plerionic engine of the candidate PWN G20.0+0.0, and PSR J1818−1502 appears to be an old and faint pulsar that we serendipitously discovered near the centre of a SNR already hosting a compact central object. The survey holds importance for better understanding of neutron star birth rates and the energetics of young pulsars.

Investigating the Lower Mass Gap with Low-mass X-Ray Binary Population Synthesis

Mass measurements from low-mass black hole X-ray binaries (LMXBs) and radio pulsars have been used to identify a gap between the most massive neutron stars (NSs) and the least massive black holes (BHs). BH mass measurements in LMXBs are typically only possible for transient systems: outburst periods enable detection via all-sky X-ray monitors, while quiescent periods enable radial velocity measurements of the low-mass donor. We quantitatively study selection biases due to the requirement of transient behavior for BH mass measurements. Using rapid population synthesis simulations (COSMIC), detailed binary stellar-evolution models (MESA), and the disk instability model of transient behavior, we demonstrate that transient LMXB selection effects introduce observational biases, and can suppress mass-gap BHs in the observed sample. However, we find a population of transient LMXBs with mass-gap BHs form through accretion-induced collapse of an NS during the LMXB phase, which is inconsistent with observations. These results are robust against variations of binary evolution prescriptions. The significance of this accretion-induced collapse population depends upon the maximum NS birth mass . To reflect the observed dearth of low-mass BHs, COSMIC and MESA models favor . In the absence of further observational biases against LMXBs with mass-gap BHs, our results indicate the need for additional physics connected to the modeling of LMXB formation and evolution.

The Galactic population of canonical pulsars

Context. Current wisdom suggests that the observed population of neutron stars are manifestations of their birth scenarios and their thermal and magnetic field evolution. Neutron stars can be observed at various wavebands as pulsars, and radio pulsars represent by far the largest population of neutron stars. Aims. In this paper, we aim to constrain the observed population of the canonical neutron star period, its magnetic field, and its spatial distribution at birth in order to understand the radio and high-energy emission processes in a pulsar magnetosphere. For this purpose we design a population synthesis method, self-consistently taking into account the secular evolution of a force-free magnetosphere and the magnetic field decay. Methods. We generated a population of pulsars and evolved them from their birth to the present time, using the force-free approximation. We assumed a given initial distribution for the spin period, surface magnetic field, and spatial Galactic location. Radio emission properties were accounted for by the polar cap geometry, whereas the gamma-ray emission was assumed to be produced within the striped wind model. Results. We find that a decaying magnetic field gives better agreement with observations compared to a constant magnetic field model. Starting from an initial mean magnetic field strength of B = 2.5 × 108 T with a characteristic decay timescale of 4.6 × 105 yr, a neutron star birth rate of 1/70 yr and a mean initial spin period of 60 ms, we find that the force-free model satisfactorily reproduces the distribution of pulsars in the P−Ṗ diagram with simulated populations of radio-loud, radio-only, and radio quiet gamma-ray pulsars similar to the observed populations.

The SPAN512 mid-latitude pulsar survey at the Nançay Radio Telescope

Context. The large number of ongoing surveys for pulsars and transients at various radio observatories is motivated by the science obtained from these sources. Timing and polarisation analysis of relativistic binaries can place strong constraints on theories of gravity. The observation of a growing number of millisecond pulsars (MSPs) spread over the celestial sphere may allow the detection of a stochastic gravitational wave background arising from supermassive black hole binaries. A more complete sample of young pulsars improves our knowledge of neutron star birth and evolution. Transients such as fast radio bursts can serve to probe the intergalactic medium. Aims. The SPAN512 pulsar survey covers intermediate Galactic latitudes using the L-band receiver of the Nançay Radio Telescope (NRT). The survey covers 224 sq. deg. of the sky for a total exposure time of 2200 h. Population syntheses predict the discovery of 3 to 19 new normal pulsars and a few MSPs. Methods. We present detailed modelling of the NRT beam with its L-band receiver and its sensitivity which we used to precisely assess the expected survey yield. We used the flexible Pulsar Arecibo L-band Feed Array data processing pipeline to search the 47 TB of SPAN512 data for pulsars and transients. Results. The SPAN512 survey discovered two new MSPs and one new middle-aged pulsar. We focus on the analysis of the 2.4-ms spin period pulsar J2205+6012 for which we also report the detection of gamma-ray pulsations. Its narrow pulse width (35µs at an observing frequency of 2.55 GHz) allows for sub-microsecond timing precision over 8 yr, with exciting prospects for pulsar timing array programs.

X-Raying the Birth of Binary Neutron Stars and Neutron Star–Black Hole Binaries

We consider fallback accretion after an ultrastripped supernova (USSN) that accompanies formation of a binary neutron star (BNS) or a neutron star–black hole binary (NS–BH). The fallback matter initially accretes directly to the nascent NS, while it starts to accrete to the circumbinary disk, typically 0.1–1 day after the onset of the USSN explosion. The circumbinary disk mass further accretes, forming mini disks around each compact object, with a super-Eddington rate up to a few years. We show that such a system constitutes a binary ultraluminous X-ray source, and a fraction of the X-rays can emerge through the USSN ejecta. We encourage follow-up observations of USSNe within ≲100 Mpc and ∼100–1000 days after the explosion using Chandra, XMM Newton, and NuSTAR, which could detect the X-ray counterpart with time variations representing the properties of the nascent compact binary, e.g., the orbital motion of the binary, the spin of the NS, and/or the quasiperiodic oscillation of the mini disks.

The Character of Three-Dimensional Core-Collapse Simulation Results

We here in graphical form highlight some of the early phases in the explosion of core-collapse supernovae and the birth of neutron stars. The graphics presented summarizes the results of the almost fifty three-dimensional radiation/hydrodynamic simulations the Princeton group has simulated during the last three years using its state-of-the-art (though still evolving and improving) code Fornax.

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.

Evidence for three-dimensional spin–velocity alignment in a pulsar

More than 50 years after the discovery of pulsars1 and confirmation of their association with supernova explosions2–4, the origin of the initial spin and velocity of pulsars remains largely a mystery. The typical space velocities of several hundred km s−1 have been attributed to ‘kicks’ resulting from asymmetries either in the supernova ejecta or in the neutrino emission5–7. Observations have shown a strong tendency for alignment between the pulsar space velocity and the spin axis in young pulsars, but until now these comparisons have been restricted to two dimensions. Here, we report evidence for three-dimensional alignment between the spin and velocity vectors, largely based on observations made with the Five-hundred-meter Aperture Spherical Radio Telescope (FAST) of the pulsar PSR J0538+2817 and its associated supernova remnant S147. Analysis of these and related observations has enabled us to determine the location of the pulsar within the supernova remnant and hence its radial velocity. Current simulations of supernova explosions have difficulty producing such three-dimensional alignment7–9. Our results, which depend on the unprecedented sensitivity of the observations, add another dimension to the intriguing correlation between pulsar spin-axis and birth-kick directions, thereby deepening the mysteries surrounding the birth of neutron stars. Fast-moving pulsars and neutron stars in general may have received a kinetic ‘kick’ from an asymmetric element in the supernova explosion that formed them. Here, the spin axis of a pulsar is determined to lie along the three-dimensional direction of the pulsar’s motion, providing a challenging constraint on supernova explosion modelling.

Lower bound for the neutrino magnetic moment from kick velocities induced at the birth of neutron stars

We show that the neutrino chirality flip, which can take place in the core of a neutron star at birth, is an efficient process to allow neutrinos to anisotropically escape, thus providing a to induce the neutron star kick velocities. The process is not subject to the no-go theorem since although the flip from left- to right-handed neutrinos happens at equilibrium, the reverse process does not take place given that right-handed neutrinos do not interact with matter and therefore detailed balance is lost. For simplicity, we model the neutron star core as being made of strange quark matter. We find that the process is efficient when the neutrino magnetic moment is not smaller than [Formula: see text], where [Formula: see text] is the Bohr magneton. When this lower bound is combined with the most stringent upper bound, which uses the luminosity data obtained from the analysis of SN 1987A, our results set a range for the neutrino magnetic moment given by [Formula: see text]. The obtained kick velocities for natal conditions are consistent with the observed ones and span the correct range of radii for typical magnetic field intensities.

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.

Double Neutron Star Mergers from Hierarchical Triple-star Systems

Abstract The isolated binary evolution model for merging neutron stars (NSs) involves processes such as mass transfer, common-envelope evolution, and natal kicks, all of which are poorly understood. Also, the predicted NS–NS merger rates are typically lower than the rates inferred from the LIGO GW170817 event. Here, we investigate merger rates of NS and black hole–NS binaries in hierarchical triple-star systems. In such systems, the tertiary can induce Lidov–Kozai (LK) oscillations in the inner binary, accelerating its coalescence and potentially enhancing compact object merger rates. However, because compact objects originate from massive stars, the prior evolution should also be taken into account. Natal kicks, in particular, could significantly reduce the rates by unbinding the tertiary before it can affect the inner binary through LK evolution. We carry out simulations of massive triples, taking into account stellar evolution starting from the main sequence, secular and tidal evolution, and the effects of supernovae. For large NS birth kicks ( ), we find that the triple NS–NS merger rate (several hundred ) is lower by a factor of ∼2–3 than the binary rate, but for no kicks ( ), the triple rate (several thousand ) is comparable to the binary rate. Our results indicate that a significant fraction of NS–NS mergers could originate from triples if a substantial portion of the NS population is born with low kick velocities, as indicated by other work. However, uncertainties and open questions remain because of our simplifying assumption of dynamical decoupling after inner binary interaction has been triggered.

Long-term simulations of multi-dimensional core-collapse supernovae: Implications for neutron star kicks

Abstract Core-collapse supernovae (CCSNe) are the final stage of massive stars, marking the birth of neutron stars (NSs). The aspherical mass ejection drives a natal kick of the forming NS. In this work we study the properties of the NS kick based on our long-term hydrodynamics CCSN simulations. We perform two-dimensional (2D) simulations for ten progenitors from a 10.8 to 20$\, M_{\odot }$ star covering a wide range of the progenitor’s compactness parameter, and two three-dimensional (3D) simulations for an 11.2$\, M_{\odot }$ star. Our 2D models present a variety of explosion energies between ∼1.3 × 1050 erg and ∼1.2 × 1051 erg, and NS kick velocities between ∼100 km s−1 and ∼1500 km s−1. For the 2D exploding models, we find that the kick velocities tend to become higher with the progenitor’s compactness. This is because the high progenitor compactness results in high neutrino luminosity from the proto-neutron star (PNS), leading to more energetic explosions. Since high-compactness progenitors produce massive PNSs, we point out that the NS masses and the kick velocities can be correlated, which is moderately supported by observation. Comparing 2D and 3D models of the 11.2$\, M_{\odot }$ star, the diagnostic explosion energy in 3D is, as previously identified, higher than that in 2D, whereas the 3D model results in a smaller asymmetry in the ejecta distribution and a smaller kick velocity than in 2D. Our results confirm the importance of self-consistent CCSN modeling covering a long-term post-bounce evolution in 3D for a quantitative prediction of the NS kicks.

Three-dimensional simulations of neutrino-driven core-collapse supernovae from low-mass single and binary star progenitors

We present a suite of seven 3D supernova simulations of non-rotating low-mass progenitors using multi-group neutrino transport. Our simulations cover single star progenitors with zero-age main sequence masses between $9.6 M_\odot$ and $12.5 M_\odot$ and (ultra)stripped-envelope progenitors with initial helium core masses between $2.8 M_\odot$ and $3.5 M_\odot$. We find explosion energies between $0.1\,\mathrm{Bethe}$ and $0.4\,\mathrm{Bethe}$, which are still rising by the end of the simulations. Although less energetic than typical events, our models are compatible with observations of less energetic explosions of low-mass progenitors. In six of our models, the mass outflow rate already exceeds the accretion rate onto the proto-neutron star, and the mass and angular momentum of the compact remnant have closely approached their final value, barring the possibility of later fallback. While the proto-neutron star is still accelerated by the gravitational tug of the asymmetric ejecta, the acceleration can be extrapolated to obtain estimates for the final kick velocity. We obtain gravitational neutron star masses between $1.22 M_\odot$ and $1.44 M_\odot$, kick velocities between $11\, \mathrm{km}\, \mathrm{s}^{-1}$ and $695\, \mathrm{km}\, \mathrm{s}^{-1}$, and spin periods from $20\, \mathrm{ms}$ to $2.7\,\mathrm{s}$, which suggests that typical neutron star birth properties can be naturally obtained in the neutrino-driven paradigm. We find a loose correlation between the explosion energy and the kick velocity. There is no indication of spin-kick alignment, but a correlation between the kick velocity and the neutron star angular momentum, which needs to be investigated further as a potential point of tension between models and observations.

High-entropy ejections from magnetized proto-neutron star winds: implications for heavy element nucleosynthesis

Although initially thought to be promising for production of the r-process nuclei, standard models of neutrino-heated winds from proto-neutron stars (PNSs) do not reach the requisite neutron-to-seed ratio for production of the lanthanides and actinides. However, the abundance distribution created by the r-, rp-, or νp-processes in PNS winds depends sensitively on the entropy and dynamical expansion time-scale of the flow, which may be strongly affected by high magnetic fields. Here, we present results from magnetohydrodynamic simulations of non-rotating neutrino-heated PNS winds with strong dipole magnetic fields from 1014 to 1016 G, and assess their role in altering the conditions for nucleosynthesis. The strong field forms a closed zone and helmet streamer configuration at the equator, with episodic dynamical mass ejections in toroidal plasmoids. We find dramatically enhanced entropy in these regions and conditions favourable for third-peak r-process nucleosynthesis if the wind is neutron-rich. If instead the wind is proton-rich, the conditions will affect the abundances from the νp-process. We quantify the distribution of ejected matter in entropy and dynamical expansion time-scale, and the critical magnetic field strength required to affect the entropy. For B ≳1015 G, we find that ≳10−6 M⊙ and up to ∼10−5 M⊙ of high-entropy material is ejected per highly magnetized neutron star birth in the wind phase, providing a mechanism for prompt heavy element enrichment of the universe. Former binary companions identified within (magnetar-hosting) supernova remnants, the remnants themselves, and runaway stars may exhibit overabundances. We provide a comparison with a semi-analytic model of plasmoid eruption and discuss implications and extensions.

On the Progenitor of Binary Neutron Star Merger GW170817

Abstract On 2017 August 17 the merger of two compact objects with masses consistent with two neutron stars was discovered through gravitational-wave (GW170817), gamma-ray (GRB 170817A), and optical (SSS17a/AT 2017gfo) observations. The optical source was associated with the early-type galaxy NGC 4993 at a distance of just ∼40 Mpc, consistent with the gravitational-wave measurement, and the merger was localized to be at a projected distance of ∼2 kpc away from the galaxy’s center. We use this minimal set of facts and the mass posteriors of the two neutron stars to derive the first constraints on the progenitor of GW170817 at the time of the second supernova (SN). We generate simulated progenitor populations and follow the three-dimensional kinematic evolution from binary neutron star (BNS) birth to the merger time, accounting for pre-SN galactic motion, for considerably different input distributions of the progenitor mass, pre-SN semimajor axis, and SN-kick velocity. Though not considerably tight, we find these constraints to be comparable to those for Galactic BNS progenitors. The derived constraints are very strongly influenced by the requirement of keeping the binary bound after the second SN and having the merger occur relatively close to the center of the galaxy. These constraints are insensitive to the galaxy’s star formation history, provided the stellar populations are older than 1 Gyr.

Rapidly Rising Optical Transients from the Birth of Binary Neutron Stars

Abstract We study optical counterparts of a new-born pulsar in a double neutron star system like PSR J0737-3039A/B. This system is believed to have ejected a small amount of mass of  ( 0.1 M ⊙ ) at the second core-collapse supernova. We argue that the initial spin of the new-born pulsar can be determined by the orbital period at the time when the second supernova occurs. The spin angular momentum of the progenitor is expected to be similar to that of the He-burning core, which is tidally synchronized with the orbital motion, and then the second remnant may be born as a millisecond pulsar. If the dipole magnetic field strength of the nascent pulsar is comparable with that inferred from the current spin-down rate of PSR J0737-3039B, the initial spin-down luminosity is comparable to the luminosity of super-luminous supernovae. We consider thermal emission arising from the supernova ejecta driven by the relativistic wind from such a new-born pulsar. The resulting optical light curves have a rise time of ∼10 days and a peak luminosity of ∼1044 erg s−1. The optical emission may last for a month to several months, due to the reprocessing of X-rays and UV photons via photoelectric absorption. These features are broadly consistent with those of the rapidly rising optical transients. The high spin-down luminosity and small ejecta mass are favorable for the progenitor of the repeating fast radio burst, FRB 121102. We discuss a possible connection between new-born double pulsars and fast radio bursts.

A Flaring Magnetar in FRB 121102?

Abstract The persistent radio counterpart of fast radio burst (FRB) 121102 is estimated to have particles, energy erg, and size cm. The source can be nebula inflated and heated by an intermittent outflow from a magnetar—a neutron star powered by its magnetic (rather than rotational) energy. The object is young and frequently liberating energy in magnetic flares driven by accelerated ambipolar diffusion in the neutron star core, feeding the nebula and producing bright millisecond bursts. The particle number in the nebula is consistent with ion ejecta from giant flares. The nebula may also contain the freeze-out of electron–positron pairs created months after the neutron star birth; the same mechanism offers an explanation for in the Crab Nebula. The persistent source around FRB 121102 is likely heated by magnetic dissipation and internal waves excited by the magnetar ejecta. The volumetric heating by waves explains the nebula’s enormous efficiency in producing radio emission. The repeating radio bursts are suggested to occur much closer to the magnetar, as a result of ultrarelativistic internal shocks in the magnetar wind, which are launched by the magnetospheric flares. The shocks are mediated by Larmor rotation, which forms a GHz maser with the observed ms duration. Furthermore, the flare ejecta can become charge-starved and then convert to electromagnetic waves.

Neutron stars are gold mines

Neutron stars are not only mines for clues to dense matter physics but may also be the auspicious sources of half of all nuclei heavier than [Formula: see text] in the universe, including the auric isotopes. Although the cold dense matter above the nuclear saturation density cannot be directly explored in the laboratory, gilded constraints on the properties of matter from 1 to 10 times higher density can now be panned from neutron star observations. We show how upcoming observations, such as gravitational wave from mergers, precision timing of pulsars, neutrinos from neutron star birth and X-rays from bursts and thermal emissions, will provide the bullion from which further advances can be smelted.

Merger of a white dwarf–neutron star binary to 1029carat diamonds: origin of the pulsar planets

We show that the merger and tidal disruption of a C/O white dwarf (WD) by a neutron star (NS) binary companion provides a natural formation scenario for the PSR B1257+12 planetary system. Starting with initial conditions for the debris disk produced of the disrupted WD, we model its long term viscous evolution, including for the first time the effects of mass and angular momentum loss during the early radiatively inefficient accretion flow (RIAF) phase and accounting for the unusual C/O composition on the disk opacity. For plausible values of the disk viscosity $\alpha \sim 10^{-3}-10^{-2}$ and the RIAF mass loss efficiency, we find that the disk mass remaining near the planet formation radius at the time of solid condensation is sufficient to explain the pulsar planets. Rapid rocky planet formation via gravitational instability of the solid carbon-dominated disk is facilitated by the suppression of vertical shear instabilities due to the high solid-to-gas ratio. Additional evidence supporting a WD-NS merger scenario includes (1) the low observed occurrence rate of pulsar planets ($\lesssim 1\%$ of NS birth), comparable to the expected WD-NS merger rate; (2) accretion by the NS during the RIAF phase is sufficient to spin PSR B1257+12 up to its observed 6 ms period; (3) similar models of `low angular momentum' disks, such as those produced from supernova fallback, find insufficient mass reaching the planet formation radius. The unusually high space velocity of PSR B1257+12 of $\gtrsim 326\,{\rm km\,s}^{-1}$ suggests a possible connection to the Calcium-rich transients, dim supernovae which occur in the outskirts of their host galaxies and were proposed to result from mergers of WD-NS binaries receiving SN kicks. The C/O disk composition implied by our model likely results in carbon-rich planets with diamond interiors.