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NASA NTRS1980

Supercritical time-dependent accretion onto compact objects. I - Neutron stars

The time-dependent flow resulting from spherically symmetric, supercritical accretion onto a nonmagnetic neutron star is computed for accretion rates between 3 and 30. In addition, the fully time-dependent self-consistent set of coupled radiation-hydrodynamic equations governing such flow has been solved. Effects that have not previously been treated are taken into account, including separate energy equations for ions and electrons allowing for nonequilibrium, variable Eddington factors in the radiation transfer to account for anisotropy of the radiation field due to highly spherically extended envelopes, and convection of ions, electrons and photons in the accretion envelope. It is found that supercritical flow, for material falling from a radius of less than 10 to the 10th cm, is characterized by stable accretion with no evidence of stellar wind outflow at any phase of the evolution.

Klein, R. I.↗

DOE OSTI2023

Tidal Stripping of a White Dwarf by an Intermediate-mass Black Hole

During the inspiralling of a white dwarf (WD) into an intermediate-mass black hole (~10 2-5 M ⊙ ), both gravitational waves (GWs) and electromagnetic (EM) radiation are emitted. Once the eccentric orbit's pericenter radius approaches the tidal radius, the WD would be tidally stripped upon each pericenter passage. The accretion of this stripped mass would produce EM radiation. It is suspected that the recently discovered new types of transients, namely the quasiperiodic eruptions and the fast ultraluminous x-ray bursts, might originate from such systems. Modeling these flares requires a prediction of the amount of stripped mass from the WD and the details of the mass supply to the accretion disk. We run hydrodynamical simulations to study the orbital parameter dependence of the stripped mass. We find that our results match the analytical estimate that the stripped mass is proportional to z 5/2 , where z is the excess depth by which the WD overfills its instantaneous Roche lobe at the pericenter. The corresponding fallback rate of the stripped mass is calculated, which may be useful in interpreting the individual flaring light curve in candidate EM sources. We further calculate the long-term mass-loss evolution of a WD during its inspiral and the detectability of the GW and EM signals. The EM signal from the mass-loss stage can be easily detected: the limiting distance is ~320(M h /10 4 M ⊙ ) Mpc for the Einstein Probe. The GW signal, for space-borne detectors such as Laser Interferometer Space Antenna or TianQin, can be detected only within the Local Supercluster (~33 Mpc).

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1992

Semiempirical limits on the thermal conductivity of intracluster gas

A semiempirical method for establishing lower limits on the thermal conductivity of hot gas in clusters of galaxies is described. The method is based on the observation that the X-ray imaging data (e.g., Einstein IPC) for clusters are well described by the hydrostatic-isothermal beta model, even for cooling flow clusters beyond about one core radius. In addition, there are strong indications that noncooling flow clusters (like the Coma Cluster) have a large central region (up to several core radii) of nearly constant gas temperature. This suggests that thermal conduction is an effective means of transporting and redistributing the thermal energy of the gas. This in turn has implications for the extent to which magnetic fields in the cluster are effective in reducing the thermal conductivity of the gas. Time-dependent hydrodynamic simulations for the gas in the Coma Cluster under two separate evolutionary scenarios are presented. One scenario assumes that the cluster potential is static and that the gas has an initial adiabatic distribution. The second scenario uses an evolving cluster potential. These models along with analytic results show that the thermal conductivity of the gas in the Coma Cluster cannot be less than 0.1 of full Spitzer conductivity. These models also show that high gas conductivity assists rather than hinders the development of radiative cooling in the central regions of clusters.

David, Laurence P.↗

NASA NTRS2026

The Role of Disruptive Impacts on Ocean Generation and Longevity in Icy Moons

Among icy moons of the outer Solar System, subsurface water oceans and large collisions both seem to be common, but the impact of the latter on the presence and persistence of the former is unclear and has rarely been investigated. Here we interface a smoothed-particle hydrodynamics model to simulate collisions with a thermal-structural evolution model to simulate the evolution of moons pre-collision, post-collision and without a collision. Overall, even such large-scale collisions affect only the ocean thickness or longevity, and the presence or absence of an ocean is affected for only a part of a moon’s history. In reaccreted moons, the ocean survives the impact and becomes much thicker inside larger moons with radius near 1,000 km, whereas an ocean that would otherwise arise inside moons with radius near 500 km is absent because the collision promotes ice–rock differentiation. Our simulations have not yielded an ocean developed post-impact—whether directly via collisional or reaccretional heating or indirectly through tidal heating due to collision-induced orbital changes—in a moon that would otherwise have remained frozen. The ocean-enhancing effect is pronounced only for late disruptive impacts onto large, 1,000-km-class targets, which are unlikely in recent solar system history.

Marc Neveu↗

NASA NTRS1993

Study of bubble behavior in weightlessness (effects of thermal gradient and acoustic stationary wave) (M-16)

The aim of this experiment is to understand how bubbles behave in a thermal gradient and acoustic stationary wave under microgravity. In microgravity, bubble or bubbles in a liquid will not rise upward as they do on Earth but will rest where they are formed because there exists no gravity-induced buoyancy. We are interested in how bubbles move and in the mechanisms which support the movement. We will try two ways to make bubbles migrate. The first experiment concerns behavior of bubbles in a thermal gradient. It is well known than an effect of surface tension which is masked by gravity on the ground becomes dominant in microgravity. The surface tension on the side of the bubble at a lower temperature is stronger than at a higher temperature. The bubble migrates toward the higher temperature side due to the surface tension difference. The migration speed depends on the so-called Marangoni number, which is a function of the temperature difference, the bubble diameter, liquid viscosity, and thermal diffusivity. At present, some experimental data about migration speeds in liquids with very small Marangoni numbers were obtained in space experiments, but cases of large Marangoni number are rarely obtained. In our experiment a couple of bubbles are to be injected into a cell filled with silicon oil, and the temperature gradient is to be made gradually in the cell by a heater and a cooler. We will be able to determine migration speeds in a very wide range of Marangoni numbers, as well as study interactions between the bubbles. We will observe bubble movements affected by hydrodynamical and thermal interactions, the two kinds of interactions which occur simultaneously. These observation data will be useful for analyzing the interactions as well as understanding the behavior of particles or drops in materials processing. The second experiment concerns bubble movement in an acoustic stationary wave. It is known that a bubble in a stationary wave moves toward the node or the loop according to whether its diameter is larger or smaller than that of the main resonant radius. In our experiment fine bubbles will be observed to move according to an acoustic field formed in a cylindrical cell. The existence of bubbles varies the acoustic speed, and the interactive force between bubbles will make the bubble behavior collective and complicated. This experiment will be very useful to development of bubble removable technology as well as to the understanding of bubble behavior.

Azuma, H.↗

NASA NTRS1995

Hydrodynamics of binary coalescence. 2: Polytropes with gamma = 5/3

We present a new numerical study of the equilibrium and stability properties of close binary systems. We use the smoothed-particle hydrodynamics (SPH) technique both to construct accurate equilibrium configurations in three dimensions and to follow their hydrodynamic evolution. We adopt a simple polytropic equation of state p = K(sub rho)(exp gamma) with gamma = 5/3 and K = constant within each star, applicable to low-mass degenerate dwarfs as well as low-mass main-sequence stars. For degenerate configurations, we set the two polytropic constants equal, K = K prime, independent of the mass ratio. For main-sequence stars, we adjust K and K prime so as to obtain a simple mass-radius relation of the form R/R prime = M/M prime, where R prime and M prime are the radius and mass of the secondary. Along a sequence of binary equilibrium configurations for two identical stars, we demonstrate the existence of both secular and dynamical instabilities, confirming directly the results of recent analytic work. We use the SPH method to calculate the nonlinear development of the dynamical instability and to determine the final fate of the system. We find that the two stars merge together into a single, rapidly rotating object in just a few orbital periods. Equilibrium sequences are also constructed for systems containing two nonidentical stars. These sequences terminate at a Roche limit, which we can determine very accurately using SPH. For two low-mass main-sequence stars with mass ratio q approximately less than 0.4 we find that the (synchronized) Roche limit configuration is secularly unstable. We discuss the implications of our results for the evolution of double white-dwarf systems and W Ursae Majoris binaries.

Rasio, Frederic A.↗

DOE OSTI2023

Universal scaling laws and density slopes for dark matter haloes

Smalls scale challenges suggest some missing pieces in our current understanding of dark matter. In this work, a cascade theory for dark matter is proposed to provide extra insights, similar to the cascade phenomenon in hydrodynamic turbulence. The kinetic energy is cascaded in dark matter from small to large scales involves a constant rate $ε_u$ (≈–4.6×10 –7 m 2 / s 3 ). Confirmed by N-body simulations, the energy cascade leads to a two-thirds law for kinetic energy $v^2_r$ on scale $\textit{r}$ such that $v^2_r∝(ε_ur)^{2/3}$. Equivalently, a four-thirds law can be established for mean halo density $ρ_s$ enclosed in the scale radius $r_s$ such that $ρ_s ∝ε^{2/3}_uG^{–1}r^{–4/3}_s$, which was confirmed by galaxy rotation curves. Critical properties of dark matter might be obtained by identifying key constants on relevant scales. First, the largest halo scale $r_l$ can be determined by $–u^3_0/ε_u$, where u0 is the velocity dispersion. Second, the smallest scale rη is dependent on the nature of dark matter. For collisionless dark matter, $r_η∝(–Gℏ/ε_u)^{1/3}$ ≈ 10 –13 m , where ℏ is the Planck constant. An uncertainty principle for momentum and acceleration fluctuations is also postulated. For self-interacting dark matter, $r_η∝ε^2_uG^{–3}(σ/m)^3$, where σ/m is the cross-section of interaction. On halo scale, the energy cascade leads to an asymptotic density slope γ = –4/3 for fully virialized haloes with a vanishing radial flow, which might explain the nearly universal halo density. Based on the continuity equation, halo density is analytically shown to be closely dependent on the radial flow and mass accretion, such that simulated haloes can have different limiting slopes. A modified Einasto density profile is proposed accordingly.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Anisotropic Flow in Fixed-Target 208 Pb + 20 Ne Collisions as a Probe of Quark-Gluon Plasma

The System for Measuring Overlap with Gas (SMOG2) at the LHCb detector enables the study of fixed-target ion-ion collisions at relativistic energies ($\sqrt{𝑠_{NN}}$ ∼ 100 GeV in the center of mass). Here, with input from ab initio calculations of the structure of 16 O and 20 Ne , we compute 3+1⁢D hydrodynamic predictions for the anisotropic flow of Pb+Ne and Pb+O collisions to be tested with upcoming LHCb data. This will allow the detailed study of quark-gluon plasma formation as well as experimental tests of the predicted nuclear shapes. Elliptic flow (𝑣 2 ) in Pb + Ne collisions is greatly enhanced compared to the Pb + O baseline due to the shape of 20 Ne , which is deformed in a bowling-pin geometry. Owing to the large 208 Pb radius, this effect is seen in a broad centrality range, a unique feature of this collision configuration. Larger elliptic flow further enhances the quadrangular flow (𝑣 4 ) of Pb + Ne collisions via nonlinear coupling, and impacts the sign of the kurtosis of the elliptic flow vector distribution (𝑐 2 ⁡{4}). Exploiting the shape of 20 Ne proves thus an ideal method to investigate the formation of quark-gluon plasma in fixed-target experiments at LHCb, and demonstrates the power of System for Measuring Overlap with Gas as a tool to image nuclear ground states.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS2013

Accretion Disks Around Binary Black Holes of Unequal Mass: GRMHD Simulations Near Decoupling

We report on simulations in general relativity of magnetized disks onto black hole binaries. We vary the binary mass ratio from 1:1 to 1:10 and evolve the systems when they orbit near the binary disk decoupling radius. We compare (surface) density profiles, accretion rates (relative to a single, non-spinning black hole), variability, effective alpha-stress levels and luminosities as functions of the mass ratio. We treat the disks in two limiting regimes: rapid radiative cooling and no radiative cooling. The magnetic field lines clearly reveal jets emerging from both black hole horizons and merging into one common jet at large distances. The magnetic fields give rise to much stronger shock heating than the pure hydrodynamic flows, completely alter the disk structure, and boost accretion rates and luminosities. Accretion streams near the horizons are among the densest structures; in fact, the 1:10 no-cooling evolution results in a refilling of the cavity. The typical effective temperature in the bulk of the disk is approx. 10(exp5) (M / 10(exp 8)M solar mass (exp -1/4(L/L(sub edd) (exp 1/4K) yielding characteristic thermal frequencies approx. 10 (exp 15) (M /10(exp 8)M solar mass) (exp -1/4(L/L (sub edd) (1+z) (exp -1)Hz. These systems are thus promising targets for many extragalactic optical surveys, such as LSST, WFIRST, and PanSTARRS.

Supermassive↗

DOE OSTI2020

High-speed imaging of transition from fluid breakup to phase explosion in electric explosion of tungsten wires in air

High-speed visible imaging of sub-microsecond electric explosion of wires at the low specific energy deposition threshold reveals three distinct modes of wire failure as capacitor charge voltage and energy deposition are increased. For 100 micron diameter gold-plated tungsten wires of 2 cm length, deposited energies of 1.9 eV/atom produces a liquid column that undergoes hydrodynamic breakup into droplets with radii of order of wire diameter on timescales of 200 μs. Instability growth, column breakup, and droplet coalescence follow classic Rayleigh-Plateau predictions for instability of viscous fluid column. Above 3.2 eV/atom of deposited energy, wires are seen to abruptly transition to an expanding mixture of micron scale liquid-droplets and vapor within one frame (less than 3.33 μs), which has been termed ‘phase explosion’ in literature. Between these two limits, at 2.5 eV/atom of deposited energy, wire radius is unchanged for the first 10 μs before the onset of a rapid expansion and disintegration that resembles homogenous nucleation of mechanically unstable bubbles. Here, thermodynamic calculations are presented that separate cases by temperature obtained during heating: below boiling point, near boiling point, and exceeding boiling point.

42 ENGINEERING↗

DOE OSTI2020

First study of Hohlraum x-ray preheat asymmetry inside an ICF capsule

In indirect drive inertial confinement fusion (ICF), laser induced Hohlraum preheat radiation (so-called M-band, >1.8 keV) asymmetry will lead to asymmetric ablation front and ablator–fuel interface hydrodynamic instability growth in an imploding capsule. First experiments to infer the M-band asymmetries at the capsule were performed on the National Ignition Facility for high density carbon (HDC) ICF capsules in low density fill (0.3 mg/cc 4 He) Au Hohlraums by time resolved imaging of 2.3 keV fluorescence emission of a smaller Mo sphere placed inside the capsule. In this study, the measured Mo emission is pole hot (P 2 > 0) since M-band is generated mainly by the outer laser beams as their irradiance at the Hohlraum wall is 5× higher than for the inner beams. P 2 has a greater negative than positive swing vs time [Δ(P 2 /P 0 )/Δt ~ 0.2/ns], giving insight into laser heated Hohlraum dynamics. P 4 asymmetry is small at the sphere due to efficient geometric smoothing of Hohlraum asymmetries at large Hohlraum -to-capsule ratios. The M-band P 2 history is qualitatively reproduced by radiation hydrodynamic HYDRA simulations. The smaller P 2 than that calculated earlier suggests either less outer beam spot motion and/or preheat emission. At late times, the observed P 2 swing is larger and P 4 is more negative than simulated, which could be due to inner beams being stopped more in the outer beams wall plasma bubble than simulated. Asymmetry at the HDC capsule inner surface (“ice–ablator interface”) is also inferred from the Mo emission asymmetry by an analytic viewfactor model, accounting for the Mo/HDC radius difference and HDC capsule opacity.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

The Atacama Cosmology Telescope: Probing the baryon content of SDSS DR15 galaxies with the thermal and kinematic Sunyaev-Zel’dovich effects

We present high signal-to-noise measurements (up to 12$\sigma$) of the average thermal Sunyaev Zel'dovich (tSZ) effect from optically selected galaxy groups and clusters and estimate their baryon content within a 2.1$^\prime$ radius aperture. Sources from the Sloan Digital Sky Survey (SDSS) Baryon Oscillation Spectroscopic Survey (BOSS) DR15 catalog overlap with 3,700 sq. deg. of sky observed by the Atacama Cosmology Telescope (ACT) from 2008 to 2018 at 150 and 98 GHz (ACT DR5), and 2,089 sq. deg. of internal linear combination component-separated maps combining ACT and $\it{Planck}$ data (ACT DR4). The corresponding optical depths, $\bar{\tau}$, which depend on the baryon content of the halos, are estimated using results from cosmological hydrodynamic simulations assuming an AGN feedback radiative cooling model. We estimate the mean mass of the halos in multiple luminosity bins, and compare the tSZ-based $\bar{\tau}$ estimates to theoretical predictions of the baryon content for a Navarro-Frenk-White profile. We do the same for $\bar{\tau}$ estimates extracted from fits to pairwise baryon momentum measurements of the kinematic Sunyaev-Zel'dovich effect (kSZ) for the same data set obtained in a companion paper. We find that the $\bar{\tau}$ estimates from the tSZ measurements in this work and the kSZ measurements in the companion paper agree within $1\sigma$ for two out of the three disjoint luminosity bins studied, while they differ by 2-3$\sigma$ in the highest luminosity bin. The optical depth estimates account for one third to all of the theoretically predicted baryon content in the halos across luminosity bins. Potential systematic uncertainties are discussed. The tSZ and kSZ measurements provide a step towards empirical Compton-$\bar{y}$-$\bar{\tau}$ relationships to provide new tests of cluster formation and evolution models.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1988

A numerical study of viscous vortex rings using a spectral method

Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass.

Stanaway, S. K.↗

NASA NTRS1995

Hydrodynamics of coalescing binary neutron stars: Ellipsoidal treatment

We employ an approximate treatment of dissipative hydrodynamics in three dimensions to study the coalescence of binary neutron stars driven by the emission of gravitational waves. The stars are modeled as compressible ellipsoids obeying a polytropic equation of state; all internal fluid velocities are assumed to be linear functions of the coordinates. The hydrodynamics equations then reduce to a set of coupled ordinary differential equations for the evolution of the principal axes of the ellipsoids, the internal velocity parameters, and the binary orbital parameters. Gravitational radiation reaction and viscous dissipation are both incorporated. We set up exact initial binary equilibrium configurations and follow the transition from the quasi-static, secular decay of the orbit at large separation to the rapid dynamical evolution of the configurations just prior to contact. A hydrodynamical instability resulting from tidal interactions significantly accelerates the coalescence at small separation, leading to appreciable radial infall velocity and tidal lag angles near contact. This behavior is reflected in the gravitational waveforms and may be observable by gravitational wave detectors under construction. In cases where the neutron stars have spins which are not aligned with the orbital angular momentum, the spin-induced quadrupole moment can lead to precession of the orbital plane and therefore modulation of the gravitational wave amplitude even at large orbital radius. However, the amplitude of the modulation is small for typical neutron star binaries with spins much smaller than the orbital angular momentum.

Lai, Dong↗

NASA NTRS1993

Collisions and close encounters between massive main-sequence stars

Smooth-particle hydrodynamics is used to investigate collisions and close encounters between two massive main-sequence stars. Approximate scaling relations and fitting formulas are given for the amount of mass loss and for the critical impact parameters for capture or merging. The more massive stars, which have smaller ratios of specific binding energy to the square of escape velocity, are more easily disrupted in collisions. As the relative velocity increases, the orbital energy of the colliding stars can only be dissipated by shock heating, and the critical capture radius decreases much faster than predicted by linear theory. The average fractional mass loss per collision in a cluster is found not to depend sensitively on the stellar velocity dispersion.

Lai, Dong↗

NASA NTRS1997

Modeling of Water Injection into a Vacuum

A loosely coupled two-phase vacuum water plume model has been developed. This model consists of a source flow model to describe the expansion of water vapor, and the Lagrangian equations of motion for particle trajectories. Gas/Particle interaction is modeled through the drag force induced by the relative velocities. Particles are assumed traveling along streamlines. The equations of motion are integrated to obtain particle velocity along the streamline. This model has been used to predict the mass flux in a 5 meter radius hemispherical domain resulting from the burst of a water jet of 1.5 mm in diameter, mass flow rate of 24.2 g/s, and stagnation pressure of 21.0 psia, which is the nominal Orbiter water dump condition. The result is compared with an empirical water plume model deduced from a video image of the STS-29 water dump. To further improve the model, work has begun to numerically simulate the bubble formation and bursting present in a liquid stream injected into a vacuum. The technique of smoothed particle hydrodynamics was used to formulate this simulation. A status and results of the on-going effort are presented and compared to results from the literature.

Alred, John W.↗

DOE OSTI2020

Ring patterns generated by an expanding colloidal meniscus

The drop-and-dry is a common technique allowing for creation of periodic nanoparticle (NP) structures for sensing, photonics, catalysis, etc. However, the reproducibility and scalability of this approach for fabrication of NP-based structures faces serious challenges due to the complexity of the simple, at first glance, evaporation process. In this work we study the effect of the spatial confinement on the NP self-assembly under slow solvent evaporation, when the air-liquid-substrate contact line (CL) expands from the center towards the walls of a cylindrical cell, forming a toroid. Using in situ video monitoring of the stick-slip CL motion, we find regular hydrodynamic perturbations in the meniscus, and reveal fine details of the formation of quasiperiodic rings of close packed NP layers. We report that drying of the toroidal NP droplet has a number of important differences from drying of the classical hemispherical colloidal drops. In toroidal drops we observe linear-in-time average meniscus motion, in contrast to the hemispherical drops where the meniscus moves as a square root of time. While both droplet geometries produce NP ring patterns, the ring width for the toroidal drop decreases with increasing ring radius, while it decreases with decreasing the radius of the hemispherical drop. We suggest that free ligands are the main cause of the Marangoni instabilities driving the periodic vorticity in the meniscus. Finally, we show that the usually ignored contact line tension may yield a considerable contribution to the CL pinning causing the CL slip-stick motion and the ring formation.

36 MATERIALS SCIENCE↗

NASA NTRS2020

On the Maximum Stellar Rotation to Form a Black Hole without an Accompanying Luminous Transient

The collapse of a massive star with low angular momentum content is commonly argued to result in the formation of a black hole without an accompanying bright transient. Our goal in this Letter is to understand the flow in and around a newly-formed black hole, involving accretion and rotation, via general relativistic hydrodynamics simulations aimed at studying the conditions under which infalling material can accrete without forming a centrifugally supported structure and, as a result, generate no effective feedback. If the feedback from the black hole is, on the other hand, significant, the collapse would be halted and we suggest that the event is likely to be followed by a bright transient. We find that feedback is only efficient if the specific angular momentum of the infalling material at the innermost stable circular orbit exceeds that of geodesic circular flow at that radius by at least≈20%. We use the results of our simulations to constrain the maximal stellar rotation rates of the disappearing massive progenitors PHL293B-LBV and N6946-BH1, and to provide an estimate of the overall rate of disappearing massive stars. We find that about a few percent of single O-type stars with measured rotational velocities are expected to spin below the critical value before collapse and are thus predicted to vanish without a trace.

Scott C Noble↗