Pre-collapse Core Research Articles

The Jittering Jets Explosion Mechanism in Electron Capture Supernovae

We conduct one-dimensional stellar-evolution simulations of stars with zero-age main-sequence (ZAMS) masses of M ZAMS = 8.8 − 9.45 M ⊙ toward core collapse by electron capture and find that the convective zone of the precollapse core can supply the required stochastic angular momentum fluctuations to set a jet-driven electron capture supernova explosion in the frame of the jittering jets explosion mechanism. By our assumed criteria of a minimum convective specific angular momentum and an accreted mass during jet launching of M acc ≃ 0.001−0.01 M ⊙, the layer in the convective zone that when accreted launches the exploding jittering jets resides in the helium-rich zone. Depending on the model, this exploding layer is accreted at about a minute to a few hours after core collapse occurs, much shorter than the time the exploding shock crosses the star. The final (gravitational) mass of the neutron star (NS) remnant is in the range of M NS = 1.25−1.43 M ⊙.

Supernova 1987A’s Keyhole: A Long-lived Jet-pair in the Final Explosion Phase of Core-collapse Supernovae

I further study the manner by which a pair of opposite jets shape the “keyhole” morphological structure of the core-collapse supernova (CCSN) SN 1997A, now the CCSN remnant (CCSNR) 1987A. By doing so, I strengthen the claim that the jittering-jet explosion mechanism accounts for most, likely all, CCSNe. The “keyhole” structure comprises a northern low-intensity zone closed with a bright rim on its front and an elongated low-intensity nozzle in the south. This rim-nozzle asymmetry is observed in some cooling flow clusters and planetary nebulae that are observed to be shaped by jets. I build a toy model that uses the planar jittering jets pattern, where consecutive pairs of jets tend to jitter in a common plane, implying that the accreted gas onto the newly born neutron star at the late explosion phase flows perpendicular to that plane. This allows for a long-lived jet-launching episode. This long-lasting jet-launching episode launches more mass into the jets that can inflate larger pairs of ears or bubbles, forming the main jets’ axis of the CCSNR that is not necessarily related to a possible pre-collapse core rotation. I discuss the relation of the main jets’ axis to the neutron star’s natal kick velocity.

Classifying Core Collapse Supernova Remnants by Their Morphology as Shaped by the Last Exploding Jets

Under the assumption that jets explode all core collapse supernovae (CCSNe), I classify 14 CCSN remnants (CCSNRs) into five groups according to their morphology as shaped by jets, and attribute the classes to the specific angular momentum of the pre-collapse core. Point-symmetry (one CCSNR): According to the jittering jets explosion mechanism (JJEM) when the pre-collapse core rotates very slowly, the newly born neutron star (NS) launches tens of jet-pairs in all directions. The last several jet-pairs might leave an imprint of several pairs of “ears,” i.e., a point-symmetric morphology. One pair of ears (eight CCSNRs): More rapidly rotating cores might force the last pair of jets to be long-lived and shape one pair of jet-inflated ears that dominates the morphology. S-shaped (one CCSNR): The accretion disk might precess, leading to an S-shaped morphology. Barrel-shaped (three CCSNRs): Even more rapidly rotating pre-collapse cores might result in a final energetic pair of jets that clear the region along the axis of the pre-collapse core rotation and form a barrel-shaped morphology. Elongated (one CCSNR): A very rapidly rotating pre-collapse core forces all jets to be along the same axis such that the jets are inefficient in expelling mass from the equatorial plane and the long-lasting accretion process turns the NS into a black hole. The two new results of this study are the classification of CCSNRs into five classes based on jet-shaped morphological features, and the attribution of the morphological classes mainly to the pre-collapse core rotation in the frame of the JJEM.

The Neutron Star to Black Hole Mass Gap in the Frame of the Jittering Jets Explosion Mechanism (JJEM)

I build a toy model in the frame of the jittering jets explosion mechanism (JJEM) of core collapse supernovae that incorporates both the stochastically varying angular momentum component of the material that the newly born neutron star (NS) accretes and the constant angular momentum component, and show that the JJEM can account for the ≃2.5–5M ⊙ mass gap between NSs and black holes (BHs). The random component of the angular momentum results from pre-collapse core convection fluctuations that are amplified by post-collapse instabilities. The fixed angular momentum component results from pre-collapse core rotation. For slowly rotating pre-collapse cores the stochastic angular momentum fluctuations form intermittent accretion disks (or belts) around the NS with varying angular momentum axes in all directions. The intermittent accretion disk/belt launches jets in all directions that expel the core material in all directions early on, hence leaving an NS remnant. Rapidly rotating pre-collapse cores form an accretion disk with angular momentum axis that is about the same as the pre-collapse core rotation. The NS launches jets along this axis and hence the jets avoid the equatorial plane region. Inflowing core material continues to feed the central object from the equatorial plane increasing the NS mass to form a BH. The narrow transition from slow to rapid pre-collapse core rotation, i.e., from an efficient to inefficient jet feedback mechanism, accounts for the sparsely populated mass gap.

Rapid decline in the lightcurves of luminous supernovae by jet-driven bipolar explosions

ABSTRACT We calculate the lightcurves of jet-driven bipolar core collapse supernova (CCSN) explosions into a bipolar circumstellar matter (CSM) and show that an equatorial observer finds the lightcurves to possess a rapid, and even an abrupt drop. The scenario that might lead to such an explosive morphology is a common envelope evolution (CEE), where shortly before the CCSN explosion the red-supergiant progenitor interacts with a more compact companion that spirals-in and spins-up the core. The companion can be a main sequence star, a neutron star, or a black hole. The binary interaction ejects a shell through an intensive wind and the CEE ejects a denser gas in the equatorial plane. We assume that the companion accretes mass and launches jets. We conduct three-dimensional (3D) hydrodynamical simulations where we launch weak jets, the shaping jets, into the dense shell and show that the interaction forms a bipolar CSM. As a result of the rapid pre-collapse core rotation jets drive the CCSN explosion. We simulate the interaction of the jets with the bipolar CSM and use a simple scheme to calculate the lightcurves. We show that the abrupt drop in the lightcurve of an observer not too close to the polar directions can account for the lightcurve of the hydrogen poor luminous supernova (LSN) SN 2018don.

Boosting Jittering Jets by Neutrino Heating in Core Collapse Supernovae

I estimate the energy that neutrino heating adds to the outflow that jets induce in the collapsing core material in core collapse supernovae (CCSNe), and find that this energy crudely doubles the energy that the jets deposit into the outer core. I consider the jittering jets explosion mechanism where there are several stochastic jet-launching episodes, each lasting for about 0.01–0.1 s. The collapsing core material passes through the stalled shock at about 100 km and then slowly flows onto the proto-neutron star (NS). I assume that the proto-NS launches jittering jets, and that the jets break out from the stalled shock. I examine the boosting process by which the high-pressure gas inside the stalled shock, the gain region material, expands alongside the jets and does work on the material that the jets shock, the cocoon. This work is crudely equal to the energy that the original jets carry. I argue that the coupling between instabilities, stochastic rotation, magnetic fields, and jittering jets leads to most CCSN explosions. In other cases, the pre-collapse core is rapidly rotating and therefore ordered rotation replaces stochastic rotation and fixed jets replace jittering jets.

Remnant masses of core collapse supernovae in the jittering jets explosion mechanism

ABSTRACT We conduct one-dimensional (1D) stellar evolution simulations of non-rotating stars with initial masses in the range of $11\!-\!48 \, \mathrm{M}_{\odot }$ to the time of core collapse and, using a criterion on the specific angular momentum fluctuations in the inner convective zones, estimate the masses of the neutron star (NS) remnants according to the jittering jets explosion mechanism. From the 1D simulations, we find that several convective zones with specific angular momentum fluctuations of $j_{\rm {conv}} \gtrsim 2.5 \times 10^{15} {\, \rm cm}^2 {\, \rm s}^{-1}$ develop near the edge of the iron core in all models. For this condition for explosion, we find the NS remnant masses to be in the range of $1.3\!-\!1.8 \, \mathrm{M}_\odot$, while if we require twice as large values, i.e. $j_{\rm {conv}} \gtrsim 5 \times 10^{15} {\, \rm cm}^2 {\, \rm s}^{-1}$, we find the NS remnant masses to be in the range of $1.4\!-\!2.8 \, \mathrm{M}_\odot$ (the upper values here might form black holes). Note that in general, the formation of black holes in the jittering jets explosion mechanism requires a rapidly rotating pre-collapse core, while we simulate non-rotating stars.

Explodability fluctuations of massive stellar cores enable asymmetric compact object mergers such as GW190814

The first three observing runs with Advanced LIGO and Virgo have resulted in the detection of binary black hole (BBH) mergers with highly unequal mass components, which are difficult to reconcile with standard formation paradigms. The most representative of these is GW190814, a highly asymmetric merger between a 23 M⊙ black hole (BH) and a 2.6 M⊙ compact object. Here, we explore recent results, suggesting that a sizable fraction of stars with pre-collapse carbon-oxygen core masses above 10 M⊙, and extending up to at least 30 M⊙, may produce objects inside the so-called lower mass gap that bridges the division between massive pulsars and BHs in Galactic X-ray binaries. We demonstrate that such an explosion landscape would naturally cause a fraction of massive binaries to produce GW190814-like systems instead of symmetric-mass BBHs. We present examples of specific evolutionary channels leading to the formation of GW190814 and GW200210, a 24 + 2.8 M⊙ merger discovered during the O3b observing run. We estimate the merger-rate density of these events in our scenario to be 𝒪(5%) of the total BBH merger rate. Finally, we discuss the broader implications of this formation channel for compact object populations, and its possible relevance to less asymmetric merger events such as GW200105 and GW200115.

Determining the Structure of Rotating Massive Stellar Cores with Gravitational Waves

Abstract The gravitational wave (GW) signal resulting from stellar core collapse encodes a wealth of information about the physical parameters of the progenitor star and the resulting core-collapse supernova (CCSN). We present a novel approach to constrain CCSN progenitor properties at collapse using two of the most detectable parts of the GW signal: the core-bounce signal and evolution of the dominant frequency mode from the protoneutron star. We focus on the period after core bounce but before explosion and investigate the predictive power of GWs from rotating CCSNe to constrain properties of the progenitor star. We analyze 34 2D and four 3D neutrino-radiation-hydrodynamic simulations of stellar core collapse in progenitors of varied initial mass and rotation rate. Extending previous work, we verify the compactness of the progenitor at collapse to correlate with the early ramp-up slope, and in rotating cases, also with the core angular momentum. Combining this information with the bounce signal, we present a new analysis method to constrain the pre-collapse core compactness of the progenitor. Because these GW features occur less than a second after core bounce, this analysis could allow astronomers to predict electromagnetic properties of a resulting CCSN even before shock breakout.

The Role of Core-collapse Physics in the Observability of Black Hole Neutron Star Mergers as Multimessenger Sources

Abstract Recent 1D core-collapse simulations indicate a nonmonotonicity of the explodability of massive stars with respect to their precollapse core masses, which is in contrast to commonly used prescriptions. In this work, we explore the implications of these results on the formation of coalescing black hole (BH)–neutron star (NS) binaries. Furthermore, we investigate the effects of natal kicks and the NS’s radius on the synthesis of such systems and potential electromagnetic counterparts (EMCs) linked to them. Models based on 1D core-collapse simulations result in a BH–NS merger detection rate ( ∼ 2.3 yr−1), 5–10 times larger than the predictions of “standard” prescriptions. This is primarily due to the formation of low-mass BHs via direct collapse, and hence no natal kicks, favored by the 1D simulations. The fraction of observed systems that will produce an EMC, with the supernova engine from 1D simulations, ranges from 2% to 25%, depending on the NS equation of state. Notably, in most merging systems with EMCs, the NS is the first-born compact object, as long as the NS’s radius is ≲ 12 km. Furthermore, models with negligible kicks for low-mass BHs increase the detection rate of GW190426_152155-like events to ∼ 0.6 yr−1, with an associated probability of EMC ≤10% for all supernova engines. Finally, models based on 1D core-collapse simulations predict a ratio of BH–NSs to binary BHs’ merger rate density that is at least twice as high as other prescriptions, but at the same time overpredicting the measured local merger density rate of binary black holes.

Low-energy core-collapse supernovae in the frame of the jittering jets explosion mechanism

ABSTRACT We relate the pre-explosion binding energy of the ejecta of core-collapse supernovae (CCSNe) of stars with masses in the lower range of CCSNe and the location of the convection zones in the pre-collapse core of these stars, to explosion properties in the frame of the jittering jets explosion mechanism. Our main conclusion is that in the frame of the jittering jets explosion mechanism the remnant of a pulsar in these low-energy CCSNe has some significance, in that the launching of jets by the newly born neutron star (NS) spins-up the NS and create a pulsar. We crudely estimated the period of the pulsars to be tens of milliseconds in these cases. The convective zones seed perturbations that lead to accretion of stochastic angular momentum that in turn is assumed to launch jittering jets in this explosion mechanism. We calculate the binding energy and the location of the convective zones with the stellar evolution code mesa. For the lowest stellar masses, we study, MZAMS ≃ 8.5–11 M⊙, the binding energy above the convective zones is low, and so is the expected explosion energy in the jittering jets explosion mechanism that works in a negative feedback cycle. The expected mass of the NS remnant is MNS ≈ 1.25–1.6 M⊙, even for these low-energy CCSNe.

The strongly interacting binary scenarios of the enigmatic supernova iPTF14hls

Abstract We argue that some scenarios for the enigmatic supernova (SN) iPTF14hls and its progenitor require a strong binary interaction. We examine scenarios that attribute the extra power of iPTF14hls to a magnetar, to a late fallback on to the neutron star (NS) that launches jets, to an interaction of the ejecta with a circumstellar matter (CSM), or to a common envelope jets SN (CEJSN). For each of these four scenarios, we study the crucial process that supplies the extra energy and conclude that a binary companion to the progenitor must be present. For the magnetar scenario and late jets we claim that a companion should spin-up the pre-collapse core, in the ejecta-CSM scenario we find that the formation of the equatorial CSM requires a companion, and in the CEJSN where a NS spirals-in inside the giant envelope of the progenitor and launches jets the strong binary interaction is built-in. We argue that these types of strong binary interactions make the scenarios rare and explain the enigmatic nature of iPTF14hls. We further study processes that might accompany the binary interaction, in particular, the launching of jets before, during and after the explosion and their observational consequences. We do not consider the difficulties of the different scenarios and neither do we determine the best scenario for iPTF14hls. We rather focus on the binary nature of these scenarios that might as well explain other rare types of SNe.

Neutron Star Kicks by the Gravitational Tug-boat Mechanism in Asymmetric Supernova Explosions: Progenitor and Explosion Dependence

Abstract Asymmetric mass ejection in the early phase of supernova (SN) explosions can impart a kick velocity to the new-born neutron star (NS). For neutrino-driven explosions the NS acceleration has been shown to be mainly caused by the gravitational attraction of the anisotropically expelled inner ejecta, while hydrodynamic forces contribute on a subdominant level, and asymmetric neutrino emission plays only a secondary role. Two- and three-dimensional hydrodynamic simulations have demonstrated that this gravitational tug-boat mechanism can explain the observed space velocities of young NSs up to more than 1000 km s−1. Here, we discuss how the NS kick depends on the energy, ejecta mass, and asymmetry of the SN explosion, and what role the compactness of the pre-collapse stellar core plays for the momentum transfer to the NS. We also provide simple analytic expressions for the NS velocity in terms of these quantities. Referring to results of hydrodynamic simulations in the literature, we argue why, within the discussed scenario of NS acceleration, electron-capture SNe, low-mass Fe-core SNe, and ultra-stripped SNe can be expected to have considerably lower intrinsic NS kicks than core-collapse SNe of massive stellar cores. Our basic arguments also remain valid if progenitor stars possess large-scale asymmetries in their convective silicon and oxygen burning layers. Possible scenarios for spin-kick alignment are sketched. Much of our discussion stays on a conceptual and qualitative level, and more work is necessary on the numerical modeling side to determine the dependences of involved parameters, whose prescriptions will be needed for recipes that can be used to better describe NS kicks in binary evolution and population synthesis studies.

EXPLAINING THE MOST ENERGETIC SUPERNOVAE WITH AN INEFFICIENT JET-FEEDBACK MECHANISM

ABSTRACT We suggest that the energetic radiation from core-collapse super-energetic supernovae (SESNe) is due to a long-lasting accretion process onto the newly born neutron star (NS), resulting from an inefficient operation of the jet-feedback mechanism (JFM). The jets that are launched by the accreting NS or black hole maintain their axis due to a rapidly rotating pre-collapse core and do not manage to eject core material from near the equatorial plane. The jets are able to eject material from the core along the polar directions and reduce the gravity near the equatorial plane. The equatorial gas expands, and part of it falls back over a timescale of minutes to days to prolong the jet-launching episode. According to the model for SESNe proposed in the present paper, the principal parameter that distinguishes between the different cases of core-collapse supernova (CCSN) explosions, such as between normal CCSNe and SESNe, is the efficiency of the JFM. This efficiency, in turn, depends on the pre-collapse core mass, envelope mass, core convection, and, most of all, the angular momentum profile in the core. One prediction of the inefficient JFM for SESNe is the formation of a slow equatorial outflow in the explosion. The typical velocity and mass of this outflow are estimated to be v eq ≈ 1000 km s−1 and M eq ≳ 1 M ⊙, respectively, though quantitative values will have to be checked in future hydrodynamic simulations.

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.

GRAVITATIONAL WAVE SIGNATURES OF HYPERACCRETING COLLAPSAR DISKS

By performing two-dimensional special relativistic (SR) magnetohydrodynamic simulations, we study possible signatures of gravitational waves (GWs) in the context of the collapsar model for long-duration gamma-ray bursts. In our SR simulations, the central black hole is treated as an absorbing boundary. By doing so, we focus on the GWs generated by asphericities in neutrino emission and matter motions in the vicinity of the hyperaccreting disks. We compute nine models by adding initial angular momenta and magnetic fields parametrically to a precollapse core of a $35 M_{\odot}$ progenitor star. As for the microphysics, a realistic equation of state is employed and the neutrino cooling is taken into account via a multiflavor neutrino leakage scheme. To accurately estimate GWs produced by anisotropic neutrino emission, we perform a ray-tracing analysis in general relativity by a post-processing procedure. By employing a stress formula that includes contributions both from magnetic fields and special relativistic corrections, we study also the effects of magnetic fields on the gravitational waveforms. We find that the GW amplitudes from anisotropic neutrino emission show a monotonic increase with time, whose amplitudes are much larger than those from matter motions of the accreting material. We show that the increasing trend of the neutrino GWs stems from the excess of neutrino emission in the direction near parallel to the spin axis illuminated from the hyperaccreting disks. We point out that a recently proposed future space-based interferometer like Fabry-Perot type DECIGO would permit the detection of these GW signals within $\approx$ 100 Mpc.

Core-collapse supernovae: Reflections and directions

Core-collapse supernovae are among the most fascinating phenomena in astrophysics and provide a formidable challenge for theoretical investigation. They mark the spectacular end of the lives of massive stars and, in an explosive eruption, release as much energy as the sun produces during its whole life. A better understanding of the astrophysical role of supernovae as birth sites of neutron stars, black holes, and heavy chemical elements, and more reliable predictions of the observable signals from stellar death events are tightly linked to the solution of the longstanding puzzle of how collapsing stars achieve explosion. In this article our current knowledge of the processes that contribute to the success of the explosion mechanism is concisely reviewed. After a short overview of the sequence of stages of stellar core-collapse events, the general properties of progenitor-dependent neutrino emission will be briefly described. Applying sophisticated neutrino transport in axisymmetric (2D) simulations with general relativity as well as in simulations with an approximate treatment of relativistic effects, we can find successful neutrinodriven explosions for a growing set of progenitor stars. The first results of three-dimensional (3D) models have been obtained, and magnetohydrodynamic simulations demonstrate that strong initial magnetic fields in the pre-collapse core can foster the onset of neutrino-powered supernova explosions even in nonrotating stars. These results are discussed in the context of the present controversy about the value of 2D simulations for exploring the supernova mechanism in realistic 3D environments, and they are interpreted against the background of the current disagreement on the question of whether the standing accretion shock instability (SASI) or neutrino-driven convection is the crucial agency that supports the onset of the explosion.

EXPLOSIVE NUCLEOSYNTHESIS IN THE NEUTRINO-DRIVEN ASPHERICAL SUPERNOVA EXPLOSION OF A NON-ROTATING 15M☉STAR WITH SOLAR METALLICITY

We investigate explosive nucleosynthesis in a non-rotating 15$M_\odot$ star with solar metallicity that explodes by a neutrino-heating supernova (SN) mechanism aided by both standing accretion shock instability (SASI) and convection. To trigger explosions in our two-dimensional hydrodynamic simulations, we approximate the neutrino transport with a simple light-bulb scheme and systematically change the neutrino fluxes emitted from the protoneutron star. By a post-processing calculation, we evaluate abundances and masses of the SN ejecta for nuclei with the mass number $\le 70$ employing a large nuclear reaction network. Aspherical abundance distributions, which are observed in nearby core-collapse SN remnants, are obtained for the non-rotating spherically-symmetric progenitor, due to the growth of low-mode SASI. Abundance pattern of the supernova ejecta is similar to that of the solar system for models whose masses ranges $(0.4-0.5) \Ms$ of the ejecta from the inner region ($\le 10,000\km$) of the precollapse core. For the models, the explosion energies and the \nuc{Ni}{56} masses are $ \simeq 10^{51} \rm erg$ and $(0.05-0.06) \Ms$, respectively; their estimated baryonic masses of the neutron star are comparable to the ones observed in neutron-star binaries. These findings may have little uncertainty because most of the ejecta is composed by matter that is heated via the shock wave and has relatively definite abundances. The abundance ratios for Ne, Mg, Si and Fe observed in Cygnus loop are well reproduced with the SN ejecta from an inner region of the $15\Ms$ progenitor.

Gravitational Waves from Stellar Collapse: Correlations to Explosion Asymmetries

The collapse of massive stars not only produces observable outbursts across the entire electromagnetic spectrum but, for Galactic (or near-Galactic) supernovae, detectable signals for ground-based neutrino and gravitational wave detectors. Gravitational waves and neutrinos provide the only means to study the actual engine behind the optical outbursts: the collapsed stellar core. While the neutrinos are most sensitive to details of the equation of state, gravitational waves provide a means to study the mass asymmetries in this central core. We present gravitational wave signals from a series of 3-dimensional core-collapse simulations with asymmetries derived from initial perturbations caused by pre-collapse convection, core rotation, and low-mode convection in the explosion engine itself. A Galactic supernovae will allow us to differentiate these different sources of asymmetry. Combining this signal with other observations of the supernova, from neutrinos to gamma-rays to the compact remnant, dramatically increases the predictive power of the gravitational wave signal. We conclude with a discussion of the gravitational wave signal arising from collapsars, the leading engine for long-duration gamma-ray bursts.

Collapse of Magnetized Singular Isothermal Toroids. II. Rotation and Magnetic Braking

We study numerically the collapse of rotating magnetized molecular cloud cores, focusing on rotation and magnetic braking during the main accretion phase of isolated star formation. Motivated by previous numerical work and analytic considerations, we idealize the precollapse core as a magnetized singular isothermal toroid, with a constant rotational speed everywhere. The collapse starts from the center and propagates outward in an inside-out fashion, satisfying exact self-similarity in space and time. For rotation rates and field strengths typical of dense low-mass cores, the main feature remains the flattening of the mass distribution along field lines—the formation of a pseudodisk, as in the nonrotating cases. The density distribution of the pseudodisk is little affected by rotation. On the other hand, the rotation rate is strongly modified by pseudodisk formation. Most of the centrally accreted material reaches the vicinity of the protostar through the pseudodisk. The specific angular momentum can be greatly reduced on the way, by an order of magnitude or more, even when the precollapse field strength is substantially below the critical value for dominant cloud support. The efficient magnetic braking is due to the pinched geometry of the magnetic field in the pseudodisk, which strengthens the magnetic field and lengthens the level arm for braking. Both effects enhance the magnetic transport of angular momentum from inside to outside. The excess angular momentum is carried away in a low-speed outflow that has, despite claims made by other workers, little in common with observed bipolar molecular outflows. We discuss the implications of our calculations for the formation of true disks that are supported against gravity by rotation.