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Three-phase Interstellar Medium in Galaxies Resolving Evolution with Star Formation and Supernova Feedback (TIGRESS): Algorithms, Fiducial Model, and Convergence

We introduce TIGRESS, a novel framework for multi-physics numerical simulations of the star-forming interstellar medium (ISM) implemented in the Athena MHD code. The algorithms of TIGRESS are designed to spatially and temporally resolve key physical features, including: (1) the gravitational collapse and ongoing accretion of gas that leads to star formation in clusters; (2) the explosions of supernovae (SNe), both near their progenitor birth sites and from runaway OB stars, with time delays relative to star formation determined by population synthesis; (3) explicit evolution of SN remnants prior to the onset of cooling, which leads to the creation of the hot ISM; (4) photoelectric heating of the warm and cold phases of the ISM that tracks the time-dependent ambient FUV field from the young cluster population; (5) large-scale galactic differential rotation, which leads to epicyclic motion and shears out overdense structures, limiting large-scale gravitational collapse; (6) accurate evolution of magnetic fields, which can be important for vertical support of the ISM disk as well as angular momentum transport. We present tests of the newly implemented physics modules, and demonstrate application of TIGRESS in a fiducial model representing the solar neighborhood environment. We use a resolution study to demonstrate convergence and evaluate the minimum resolution Δx required to correctly recover several ISM properties, including the star formation rate, wind mass-loss rate, disk scale height, turbulent and Alfvénic velocity dispersions, and volume fractions of warm and hot phases. For the solar neighborhood model, all these ISM properties are converged at Δx⩽8 pc.

79 ASTRONOMY AND ASTROPHYSICS↗

Mixing in magnetized turbulent media

Turbulent motions are essential to the mixing of entrained fluids and are also capable of amplifying weak initial magnetic fields by small-scale dynamo action. Here we perform a systematic study of turbulent mixing in magnetized media, using three-dimensional magnetohydrodynamic simulations that include a scalar concentration field. We focus on how mixing depends on the magnetic Prandtl number, Pm, from 1 to 4 and the Mach number, M, from 0.3 to 2.4. For all subsonic flows, we find that the velocity power spectrum has a k {sup –5/3} slope in the early kinematic phase, but steepens due to magnetic back reactions as the field saturates. The scalar power spectrum, on the other hand, flattens compared to k {sup –5/3} at late times, consistent with the Obukohov-Corrsin picture of mixing as a cascade process. At higher Mach numbers, the velocity power spectrum also steepens due to the presence of shocks, and the scalar power spectrum again flattens accordingly. Scalar structures are more intermittent than velocity structures in subsonic turbulence, whereas for supersonic turbulence, velocity structures appear more intermittent than the scalars only in the kinematic phase. Independent of the Mach number of the flow, scalar structures are arranged in sheets in both the kinematic and saturated phases of the magnetic field evolution. For subsonic turbulence, scalar dissipation is hindered in the strong magnetic field regions, probably due to Lorentz forces suppressing the buildup of scalar gradients, whereas for supersonic turbulence, scalar dissipation increases monotonically with increasing magnetic field strength. At all Mach numbers, mixing is significantly slowed by the presence of dynamically important small-scale magnetic fields, implying that mixing in the interstellar medium and in galaxy clusters is less efficient than modeled in hydrodynamic simulations.

79 ASTRONOMY AND ASTROPHYSICS↗

ON THE PARALLEL AND PERPENDICULAR PROPAGATING MOTIONS VISIBLE IN POLAR PLUMES: AN INCUBATOR FOR (FAST) SOLAR WIND ACCELERATION?

We combine observations of the Coronal Multi-channel Polarimeter and the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory to study the characteristic properties of (propagating) Alfvénic motions and quasi-periodic intensity disturbances in polar plumes. This unique combination of instruments highlights the physical richness of the processes taking place at the base of the (fast) solar wind. The (parallel) intensity perturbations with intensity enhancements around 1% have an apparent speed of 120 km s{sup −1} (in both the 171 and 193 Å passbands) and a periodicity of 15 minutes, while the (perpendicular) Alfvénic wave motions have a velocity amplitude of 0.5 km s{sup −1}, a phase speed of 830 km s{sup −1}, and a shorter period of 5 minutes on the same structures. These observations illustrate a scenario where the excited Alfvénic motions are propagating along an inhomogeneously loaded magnetic field structure such that the combination could be a potential progenitor of the magnetohydrodynamic turbulence required to accelerate the fast solar wind.

79 ASTRONOMY AND ASTROPHYSICS↗

Derivation of the anisotropy profile, constraints on the local velocity dispersion, and implications for direct detection

We study the implications of a pseudo-phase-space density power-law for the anisotropy profile of a Milky Way-like dark matter halo. Requiring that the anisotropy parameter does not take non-physical values within the virial radius places a maximum value on the local radial velocity dispersion. For a plausible range of halo parameters, it is possible to take a local total velocity dispersion of up to about 300 km/s. Making this choice uniquely specifies the anisotropy and dispersion profiles. We introduce a way to model the local velocity distribution that incorporates this anisotropy and study the impact on direct detection.

79 ASTRONOMY AND ASTROPHYSICS↗

Conservative velocity mappings for discontinuous Galerkin kinetics

Continuum computational kinetic plasma models evolve the distribution function of a plasma species f s on a phase-space grid over time. In many problems of interest the distribution function has limited extent in velocity space; hence, using a uniform, highly refined mesh would be costly and slow. Nonuniform velocity grids can reduce the computational cost by placing more degrees of freedom where f s is appreciable and fewer where it is not. In this work we introduce a first-of-its kind discontinuous Galerkin approach to nonuniform velocity-space discretization using mapped velocity coordinates. This new method is presented in the context of a gyrokinetic model used to study magnetized plasmas. We create discretizations of collisionless and collisional terms using mappings in a way that exactly conserves particles and energy. Numerical tests of such properties are presented, and we show that this new discretization can reproduce earlier gyrokinetic simulations using grids with up to 6–60 times fewer cells and 22X-60X speed-ups depending on dimensionality, geometry and plasma parameters.

Discontinuous Galerkin↗

A SEARCH FOR NEW CANDIDATE SUPER-CHANDRASEKHAR-MASS TYPE Ia SUPERNOVAE IN THE NEARBY SUPERNOVA FACTORY DATA SET

We present optical photometry and spectroscopy of five Type Ia supernovae discovered by the Nearby Supernova Factory selected to be spectroscopic analogs of the candidate super-Chandrasekhar-mass events SN 2003fg and SN 2007if. Their spectra are characterized by hot, highly ionized photospheres near maximum light, for which SN 1991T supplies the best phase coverage among available close spectral templates. Like SN 2007if, these supernovae are overluminous (-19.5 < M{sub V} < -20) and the velocity of the Si II {lambda}6355 absorption minimum is consistent with being constant in time from phases as early as a week before, and up to two weeks after, B-band maximum light. We interpret the velocity plateaus as evidence for a reverse-shock shell in the ejecta formed by interaction at early times with a compact envelope of surrounding material, as might be expected for SNe resulting from the mergers of two white dwarfs. We use the bolometric light curves and line velocity evolution of these SNe to estimate important parameters of the progenitor systems, including {sup 56}Ni mass, total progenitor mass, and masses of shells and surrounding carbon/oxygen envelopes. We find that the reconstructed total progenitor mass distribution of the events (including SN 2007if) is bounded from below by the Chandrasekhar mass, with SN 2007if being the most massive. We discuss the relationship of these events to the emerging class of super-Chandrasekhar-mass SNe Ia, estimate the relative rates, compare the mass distribution to that expected for double-degenerate SN Ia progenitors from population synthesis, and consider implications for future cosmological Hubble diagrams.

79 ASTRONOMY AND ASTROPHYSICS↗

INTERFERENCE OF THE RUNNING WAVES AT LIGHT BRIDGES OF A SUNSPOT

The observations of chromospheric oscillations of two umbral light bridges (LBs) within a sunspot from NOAA Active Region 12127 are presented. It was found that the running umbral waves with periods of 2.2–2.6 minutes underwent very fast damping before approaching umbral boundaries, while those with higher periods (>2.6 minutes) could propagate outside umbrae. On two sides of each LB adjacent to umbrae, the cross-wavelet spectra displayed that the oscillations on them had a common significant power region with dominant frequencies of 2–6 minutes and phase differences of ∼90°. A counterstream of two running umbral waves in the 2–6 minute frequency range propagated toward the LBs, where they encountered each other and gave rise to constructive or even destructive interference on the LBs. In addition, the velocity and density perturbations on the LBs were found in opposite phases suggesting that the perturbations were caused by the downward propagating waves.

79 ASTRONOMY AND ASTROPHYSICS↗

Sound velocities in highly oriented pyrolytic graphite shocked to 18 GPa: Orientational order dependence and elastic instability

Previous reports of rapid phase transformation above 18 GPa [Erskine and Nellis, Nature 349, 317 (1991)] and large elastic waves below 18 GPa [Lucas et al., J. Appl. Phys. 114, 093515 (2013)] for shock-compressed ZYB-grade highly-oriented pyrolytic graphite (HOPG), but not for less oriented ZYH-grade HOPG, indicated a link between the orientational order dependence of the HOPG response above and below the phase transformation stress. To gain insight into this link and into the mechanical response of HOPG shocked to peak stresses approaching the phase transformation onset, the compressibility of ZYB- and ZYH-grade HOPG in the shocked state was examined using front surface impact experiments. Particle velocity histories and sound velocities were measured for peak stresses reaching 18 GPa. Although the locus of the measured peak stress-particle velocity states is indistinguishable for the two grades of HOPG, the measured sound velocities in the peak state reveal significant differences between the two grades. Specifically, 1) The measured sound velocities are somewhat higher for ZYH-grade HOPG, compared to ZYB-grade HOPG. 2) The measured sound velocities for ZYH-grade HOPG increase smoothly with compression, whereas those for ZYB-2 grade HOPG exhibit a significant reduction in the compression dependence from 12 GPa to 17 GPa and an abrupt increase from 17 GPa to 18 GPa. 3) The longitudinal moduli, determined from the measured sound velocities, are smaller than the calculated bulk moduli for ZYB-grade HOPG shocked to peak stresses above 15 GPa, indicating the onset of an elastic instability. The present findings demonstrate that the softening of the longitudinal modulus (or elastic instability) presented here is linked to the large elastic waves and the rapid phase transformation reported previously – all observed only for shocked ZYB-grade HOPG. The elastic instability in shocked ZYB-grade HOPG is likely a precursor to the rapid phase transformation observed above 18 GPa for this HOPG grade.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

ROTATING STARS AND THE FORMATION OF BIPOLAR PLANETARY NEBULAE. II. TIDAL SPIN-UP

We present new binary stellar evolution models that include the effects of tidal forces, rotation, and magnetic torques with the goal of testing planetary nebulae (PNs) shaping via binary interaction. We explore whether tidal interaction with a companion can spin-up the asymptotic giant brach (AGB) envelope. To do so, we have selected binary systems with main-sequence masses of 2.5 M {sub ⊙} and 0.8 M {sub ⊙} and evolve them allowing initial separations of 5, 6, 7, and 8 au. The binary stellar evolution models have been computed all the way to the PNs formation phase or until Roche lobe overflow (RLOF) is reached, whatever happens first. We show that with initial separations of 7 and 8 au, the binary avoids entering into RLOF, and the AGB star reaches moderate rotational velocities at the surface (∼3.5 and ∼2 km s{sup −1}, respectively) during the inter-pulse phases, but after the thermal pulses it drops to a final rotational velocity of only ∼0.03 km s{sup −1}. For the closest binary separations explored, 5 and 6 au, the AGB star reaches rotational velocities of ∼6 and ∼4 km s{sup −1}, respectively, when the RLOF is initiated. We conclude that the detached binary models that avoid entering the RLOF phase during the AGB will not shape bipolar PNs, since the acquired angular momentum is lost via the wind during the last two thermal pulses. This study rules out tidal spin-up in non-contact binaries as a sufficient condition to form bipolar PNs.

79 ASTRONOMY AND ASTROPHYSICS↗

Two-phase air-water stratified flow measurement using ultrasonic techniques

In this paper, a time resolved ultrasound system was developed for investigating two-phase air-water stratified flow. The hardware of the system includes a pulsed wave transducer, a pulser/receiver, and a digital oscilloscope. The time domain cross correlation method is used to calculate the velocity profile along ultrasonic beam. The system is able to provide velocities with spatial resolution of around 1mm and the temporal resolution of 200μs. Experiments were carried out on single phase water flow and two-phase air-water stratified flow. For single phase water flow, the flow rates from ultrasound system were compared with those from electromagnetic flow (EM) meter, which showed good agreement. Then, the experiments were conducted on two-phase air-water stratified flow and the results were given. Compared with liquid height measurement from conductance probe, it indicated that the measured velocities were explainable.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Self-propagating reactions in Al/Zr multilayers: Anomalous dependence of reaction velocity on bilayer thickness

High temperature, self-propagating reactions are observed in vapor-deposited Al/Zr multilayered foils of overall atomic ratios 3 Al:1 Zr and 2 Al:1 Zr and nanoscale layer thicknesses; however, the reaction velocities do not exhibit the inverse dependence on bilayer thickness that is expected based on changes in the average diffusion distance. Instead, for bilayer thicknesses of 20-30 nm, the velocity is essentially constant at ∼7.7 m/s. We explore several possible explanations for this anomalous behavior, including microstructural factors, changes in the phase evolution, and phase transformations in the reactant layers, but find no conclusive explanations. We determine that the phase evolution during self-propagating reactions in foils with a 3 Al:1 Zr stoichiometry is a rapid transformation from Al/Zr multilayers to the equilibrium intermetallic Al{sub 3}Zr compound with no intermediate crystalline phases. This phase evolution is the same for foils of 90 nm bilayer thicknesses and foils of bilayer thicknesses in the range of 27 nm to 35 nm. Further, for foils with a bilayer thickness of 90 nm and a 3 Al:1 Zr overall chemistry, the propagation front is planar and steady, in contrast to unsteady reaction fronts in foils with 1 Al:1 Zr overall chemistry and similar bilayer thicknesses.

36 MATERIALS SCIENCE↗

An Advanced N -body Model for Interacting Multiple Stellar Systems

We construct an advanced model for interacting multiple stellar systems in which we compute all trajectories with a numerical N -body integrator, namely the Bulirsch–Stoer from the SWIFT package. We can then derive various observables: astrometric positions, radial velocities, minima timings (TTVs), eclipse durations, interferometric visibilities, closure phases, synthetic spectra, spectral energy distribution, and even complete light curves. We use a modified version of the Wilson–Devinney code for the latter, in which the instantaneous true phase and inclination of the eclipsing binary are governed by the N -body integration. If all of these types of observations are at one’s disposal, a joint χ {sup 2} metric and an optimization algorithm (a simplex or simulated annealing) allow one to search for a global minimum and construct very robust models of stellar systems. At the same time, our N -body model is free from artifacts that may arise if mutual gravitational interactions among all components are not self-consistently accounted for. Finally, we present a number of examples showing dynamical effects that can be studied with our code and we discuss how systematic errors may affect the results (and how to prevent this from happening).

79 ASTRONOMY AND ASTROPHYSICS↗

DG-IMEX method for a two-moment model for radiation transport in the $\mathscr{O}$($v$/$c$) limit

Here, we consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to $\mathscr{O}$($v$/$c$), expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct $\mathscr{O}$($v$/$c$) Eulerian-frame number density and is consistent, to $\mathscr{O}$($v$/$c$), with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical uxes; (iii) a realizability-preserving implicit collision update; and(iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a "spectral redistribution" scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Analysis of toroidal phasing of resonant magnetic perturbation effects on edge transport in the DIII-D tokamak

Resonant Magnetic Perturbation (RMP) fields produced by external control coils are considered a viable option for the suppression of Edge Localized Modes in present and future tokamaks. In DIII-D, the RMPs are generated by six pairs of I-coils, each spanning 60° in toroidal angle, with the currents flowing in opposite directions in adjacent pairs of I-coils. Reversal of the currents in all I-coils, which produces a 60° toroidal shift in the RMP field configuration, generates uniquely different edge pedestal profiles of the density, temperature, and rotation velocities, implying different effects on the related edge transport phenomena caused by the difference in toroidal phase of the I-coil currents. The diffusive and non-diffusive transport effects of this RMP toroidal phase reversal are analyzed by comparing the ion and electron heat diffusivities, angular momentum transport frequencies, ion diffusion coefficients, and the particle pinch velocities interpreted from the measured profiles for the two phases of the I-coil currents.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Substructure in the stellar halo near the Sun: I. Data-driven clustering in integrals-of-motion space

Context. Merger debris is expected to populate the stellar haloes of galaxies. In the case of the Milky Way, this debris should be apparent as clumps in a space defined by the orbital integrals of motion of the stars. Aims. Our aim is to develop a data-driven and statistics-based method for finding these clumps in integrals-of-motion space for nearby halo stars and to evaluate their significance robustly. Methods. We used data from Gaia EDR3, extended with radial velocities from ground-based spectroscopic surveys, to construct a sample of halo stars within 2.5 kpc from the Sun. We applied a hierarchical clustering method that makes exhaustive use of the single linkage algorithm in three-dimensional space defined by the commonly used integrals of motion energy E, together with two components of the angular momentum, L z and L ⊥ . To evaluate the statistical significance of the clusters, we compared the density within an ellipsoidal region centred on the cluster to that of random sets with similar global dynamical properties. By selecting the signal at the location of their maximum statistical significance in the hierarchical tree, we extracted a set of significant unique clusters. By describing these clusters with ellipsoids, we estimated the proximity of a star to the cluster centre using the Mahalanobis distance. Additionally, we applied the HDBSCAN clustering algorithm in velocity space to each cluster to extract subgroups representing debris with different orbital phases. Results. Our procedure identifies 67 highly significant clusters (> 3σ), containing 12% of the sources in our halo set, and 232 subgroups or individual streams in velocity space. In total, 13.8% of the stars in our data set can be confidently associated with a significant cluster based on their Mahalanobis distance. Inspection of the hierarchical tree describing our data set reveals a complex web of relations between the significant clusters, suggesting that they can be tentatively grouped into at least six main large structures, many of which can be associated with previously identified halo substructures, and a number of independent substructures. This preliminary conclusion is further explored in a companion paper, in which we also characterise the substructures in terms of their stellar populations. Conclusions. Our method allows us to systematically detect kinematic substructures in the Galactic stellar halo with a data-driven and interpretable algorithm. The list of the clusters and the associated star catalogue are provided in two tables available at the CDS.

79 ASTRONOMY AND ASTROPHYSICS↗

Effect of H2O on Upper Mantle Phase Transitions in MgSiO3: is the Seismic X-discontinuity an Indicator of Mantle Water Content

The mantle X-discontinuity, usually assigned to positive seismic velocity reflectors in the 260-330 km depth range, has proved difficult to explain in terms of a single mineralogical phase transformation in part because of its depth variability. The coesite to stishovite transition of SiO{sub 2} matches deeper X-discontinuity depths but requires 5-10% free silica in the mantle to match observed impedance contrast. The orthoenstatite (OEn) to high-pressure clinoenstatite (HPCen) transformation of MgSiO{sub 3} also broadly coincides with depths of the X but requires chemically depleted and orthoenstatite-rich lithology at 300 km depth in order to match observed seismic impedance contrast. On the basis of high-pressure infrared spectroscopy, X-ray diffraction, and Raman spectroscopy, we show that 1300 ppm variation of H{sub 2}O content in MgSiO{sub 3} can displace the transition of low-pressure clinoenstatite (LPCen) to HPCen by up to 2 GPa, similar to previous quench experiments on the OEn to HPCen phase transition, where about 30-45 km (1.0-1.5 GPa) of deflection could occur per 0.1 wt% H{sub 2}O. If the mantle X-discontinuity results from pyroxene transitions in a depleted harzburgite layer, because of the strong influence of minor amounts of water on the transformation boundary, the depth of the mantle X-discontinuity could serve as a potentially sensitive indicator of water content in the uppermantle.

58 GEOSCIENCES↗

Cosmological simulations of Milky Way-sized galaxies

We introduce a new set of eight Milky Way-sized cosmological simulations performed using the AMR code Adaptive Refinement Tree + Hydrodynamics in a ΛCDM cosmology. The set of zoom-in simulations covers present-day virial masses that range from 8.3×10{sup 11} M{sub ⊙} to 1.56×10{sup 12} M{sub ⊙} and is carried out with our simple but effective prescriptions for deterministic star formation (SF) and “explosive” stellar feedback. The work focuses on showing the goodness of the simulated set of “field” Milky Way-sized galaxies. To this end, we compare some of the predicted physical quantities with the corresponding observed ones. Our results are as follows. (a) In agreement with some previous works, we found curves of circular velocity that are flat or slightly peaked. (b) All simulated galaxies with a significant disk component are consistent with the observed Tully–Fisher, radius–mass, and cold gas–stellar mass correlations of late-type galaxies. (c) The disk-dominated galaxies have stellar specific angular momenta in agreement with those of late-type galaxies, with values around 10{sup 3} km s{sup −1} kpc{sup −1}. (d) The SF rates at z = 0 of all runs but one are comparable to those estimated for the star-forming galaxies. (e) The two most spheroid-dominated galaxies formed in halos with late active merger histories and late bursts of SF, but the other run that ends also dominated by a spheroid never had major mergers. (f) The simulated galaxies lie in the semi-empirical correlation of stellar to halo mass of local central galaxies, and those that end up as disk-dominated evolve mostly along the low-mass branch of this correlation. Moreover, the growth histories of baryonic and stellar mass of these galaxies are proportional to their growth histories of halo mass over the last 6.5–10 Gyr. (g) Within the virial radii of the simulations, ≈25%–50% of the baryons are missed; the amount of gas in the halo is similar to the amount in stars in the galaxy, and most of this gas is in the warm–hot phase. (h) The profiles of z∼0 vertical gas velocity dispersion, σ{sub z}(r), are nearly flat and can be mostly explained by the kinetic energy injected by stars. The average values of σ{sub z} increase at higher redshifts, following roughly the shape of the SF history.

79 ASTRONOMY AND ASTROPHYSICS↗

THE M31 VELOCITY VECTOR. III. FUTURE MILKY WAY M31-M33 ORBITAL EVOLUTION, MERGING, AND FATE OF THE SUN

We study the future orbital evolution and merging of the Milky Way (MW)-M31-M33 system, using a combination of collisionless N-body simulations and semi-analytic orbit integrations. Monte Carlo simulations are used to explore the consequences of varying all relevant initial phase-space and mass parameters within their observational uncertainties. The observed M31 transverse velocity from Papers I and II implies that the MW and M31 will merge t = 5.86{sup +1.61}{sub -0.72} Gyr from now. The first pericenter occurs at t = 3.87{sup +0.42}{sub -0.32} Gyr, at a pericenter distance of r = 31.0{sup +38.0}{sub -19.8} kpc. In 41% of Monte Carlo orbits, M31 makes a direct hit with the MW, defined here as a first-pericenter distance less than 25 kpc. For the M31-M33 system, the first-pericenter time and distance are t = 0.85{sup +0.18}{sub -0.13} Gyr and r = 80.8{sup +42.2}{sub -31.7} kpc. By the time M31 gets to its first pericenter with the MW, M33 is close to its second pericenter with M31. For the MW-M33 system, the first-pericenter time and distance are t = 3.70{sup +0.74}{sub -0.46} Gyr and r = 176.0{sup +239.0}{sub -136.9} kpc. The most likely outcome is for the MW and M31 to merge first, with M33 settling onto an orbit around them that may decay toward a merger later. However, there is a 9% probability that M33 makes a direct hit with the MW at its first pericenter, before M31 gets to or collides with the MW. Also, there is a 7% probability that M33 gets ejected from the Local Group, temporarily or permanently. The radial mass profile of the MW-M31 merger remnant is significantly more extended than the original profiles of either the MW or M31, and suggests that the merger remnant will resemble an elliptical galaxy. The Sun will most likely ({approx}85% probability) end up at a larger radius from the center of the MW-M31 merger remnant than its current distance from the MW center, possibly further than 50 kpc ({approx}10% probability). There is a {approx}20% probability that the Sun will at some time in the next 10 Gyr find itself moving through M33 (within 10 kpc), but while dynamically still bound to the MW-M31 merger remnant. The arrival and possible collision of M31 (and possibly M33) with the MW is the next major cosmic event affecting the environment of our Sun and solar system that can be predicted with some certainty.

79 ASTRONOMY AND ASTROPHYSICS↗