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At least 73 records · Page 4

Ulysses field and plasma observations of magnetic holes in the solar wind and their relation to mirror-mode structures

The term 'magnetic hole' has been used to denote isolated intervals when the magnitude of the interplanetary magnetic field drops to a few tenths, or less, of its ambient value for a time that corresponds to a linear dimension of tens to a few hundreds of proton gyro-radii. Data obtained by the Ulysses magnetometer and solar wind anlayzer have been combined to study the properties of such magnetic holes in the solar wind between 1 AU and 5.4 AU and to 23 deg south latitude. In order to avoid confusion with decreases in field strength at interplanetary discontinuities, the study has focused on linear holes across which the field direction changed by less than 5 deg. The holes occurred preferentially, but not without exception, in the interaction regions on the leading edges of high-speed solar wind streams. Although the plasma surrounding the holes was generally stable against the mirror instability, there are indications that the holes may have been remnants of mirror-mode structures created upstream of the points of observation. Those indications include the following: (1) For the few holes for which proton of alpha-particle pressure could be measured inside the hole, the ion thermal pressure was always greater than in the plasma adjacent to the holes. (2) The plasma surrounding many of the holes was marginally stable for the mirror mode, while the plasma environment of all holes was significantly closer to mirror instability than was the average solar wind. (3) The plasma containing trains of closely spaced holes was closer to mirror instability than was the plasma containing isolated holes. (4) The near-hole plasma had much higher ion beta (ratio of thermal to magnetic pressure) than did the average solar wind. (5) Near the holes, T(sub perp)/T(sub parallel) tended to be either greater than 1 or larger than in the average wind. (6) The proton and alpha-particle distribution functions measured inside the holes occasionally exhibited the flattened phase-space-density contoures in nu(sub perp)/nu(sub parallel) space found in some numerical simulations of the mirror instability.

Winterhalter, Daniel↗

Observational Evidence for Cosmological Coupling of Black Holes and its Implications for an Astrophysical Source of Dark Energy

Observations have found black holes spanning 10 orders of magnitude in mass across most of cosmic history. The Kerr black hole solution is, however, provisional as its behavior at infinity is incompatible with an expanding universe. Black hole models with realistic behavior at infinity predict that the gravitating mass of a black hole can increase with the expansion of the universe independently of accretion or mergers, in a manner that depends on the black hole's interior solution. We test this prediction by considering the growth of supermassive black holes in elliptical galaxies over 0 < z ≲ 2.5. We find evidence for cosmologically coupled mass growth among these black holes, with zero cosmological coupling excluded at 99.98% confidence. The redshift dependence of the mass growth implies that, at z ≲ 7, black holes contribute an effectively constant cosmological energy density to Friedmann's equations. The continuity equation then requires that black holes contribute cosmologically as vacuum energy. We further show that black hole production from the cosmic star formation history gives the value of Ω Λ measured by Planck while being consistent with constraints from massive compact halo objects. We thus propose that stellar remnant black holes are the astrophysical origin of dark energy, explaining the onset of accelerating expansion at z ~ 0.7.

79 ASTRONOMY AND ASTROPHYSICS↗

Hole Polaron Transport in Bismuth Vanadate BiVO 4 from Hybrid Density Functional Theory

In this paper, we address the chemical character and the mobility of hole polarons in BiVO 4 (BVO). Two distinct structures for hole polarons in BVO have been reported in the literature to date. “Standard” hybrid density functional theory (DFT) calculations have predicted hole polarons to be (BiO 8 ) dodecahedron-centered polarons, while Hubbard U-DFT (DFT + U) calculations have predicted single oxygen atom-centered polarons. Resolving these contrasting findings is critical to validate theory in materials research. We investigated the structure of hole polarons in BVO using hybrid DFT with varying fractions (α) of the exact exchange interaction that enters hybrid functionals. For values of the fraction α, from α = 0.25 to α = 0.45, we obtained both structures, the h + (BiO 8 ) hole structure and the h + (O) hole structure. For the smaller values of α, the h + (BiO 8 ) structure is lower in energy, while for the larger values of α, the h+(O) structure is more stable. The piece-wise linearity of DFT energies E(q) with respect to a point defect charge q is maintained satisfactorily in all cases, although the deviation is the smallest for hybrid DFT with a small exact exchange fraction. For α = 0.25, we determined the diffusivity of h + (BiO 8 ) holes from the Marcus/Holstein two-state model. The activation barrier for Bi-to-Bi hops (ΔG # ~76 meV) and mobility (μ ~0.029 cm2 V –1 s –1 ) are found to be in close agreement with the published THz experimental data (ΔG # ~90 meV and μ exp ~0.02 cm2 V –1 s –1 , respectively). In h + (BiO 8 ) structures, nearly 25% of the hole charge and unpaired spin reside on the Bi atom and the rest is divided among the eight O atoms. The hole is best described as a Bi-centered hole, rather than a Bi 6s hole. In contrast, in h + (O) structures, about 75% of the unpaired spin population resides on the O atom. These findings suggest that if one wishes to resolve this theoretical dichotomy, there remains a need for further experimental characterization of hole polarons in BVO with techniques that could identify the oxidation state of Bi and O atoms in operando conditions and help validate theoretical results of structure and transport.

14 SOLAR ENERGY↗

The Effect of Hole Quality on the Bearing Strength of Carbon Fiber Laminates

On programs involving flight hardware for launch vehicles that the author has been involved in over the years, the question always arises as to how to best machine and inspect holes drilled for fasteners. While common sense dictates that ‘well-drilled’ holes are desired over ‘poorly drilled’ holes for bolt bearing applications, the effect of hole quality on the bearing strength of carbon fiber laminates has not been extensively studied in the open literature. If this effect is not quantitatively known, the question of what hole quality needs to be required for the flight hardware cannot be answered. In addition, the function of the part with respect to the holes needs to be taken into account. There is no such thing as a defect-free hole since some chip-out of fibers within the hole will occur regardless of drill bit and backing plate pressure. For most holes machined in a carbon fiber laminate, some extent of delamination and fiber breakout on the entrance and exit of the hole is going to occur. Much time and money can be spent on trying to perfect the drilling technique and subsequent inspection of holes in carbon fiber laminates, including adding extra plies of cloth to the surfaces to help prevent fiber breakout; but is this effort really justified if little-to-no increase in bearing strength is realized by good-quality holes over those of lesser quality?

Nettles, A. T.↗

Printability and Overall Cooling Performance of Additively Manufactured Holes With Inlet and Exit Rounding

Abstract To improve cooling effectiveness of gas turbine hardware, various film cooling hole shapes have previously been researched. Unique design modifications have recently been made possible through the design freedom allotted by additive manufacturing (AM). As one example, creating a rounded inlet for a film-cooling hole can mitigate separation at the inlet. This study explores various geometric features by exploiting the uses of additive manufacturing for shaped film cooling holes at engine scale. Both printability and cooling performance were evaluated. Resulting from this study, additively manufactured holes with hole inlet and exit rounding were printed with some variations from the design intent (DI). The largest deviations from the design intent occurred from dross roughness features located on the leeward side of the hole inlet. The measured overall effectiveness indicated that an as-built inlet fillet decreased in-hole convection as well as decreased jet mixing compared to the as-built sharp inlet. Including an exit fillet, which prevented an overbuilt diffuser exit, was also found to decrease jet mixing. A particular insight gained from this study is the importance of the convective cooling within the hole to the overall cooling performance. In-hole roughness, which is a result of additive manufacturing, increased convective cooling within the holes but also increased jet mixing as the coolant exited the hole. The increased jet mixing caused low overall effectiveness downstream of injection.

Engineering↗

Black Hole Formation Accompanied by the Supernova Explosion of a 40 M ⊙ Progenitor Star

Abstract We have simulated the collapse and evolution of the core of a solar-metallicity 40 M ⊙ star and find that it explodes vigorously by the neutrino mechanism, despite its very high “compactness.” Within ∼1.5 s of explosion, a black hole forms. The explosion is very asymmetrical and has a total explosion energy of ∼1.6 × 10 51 erg. At black hole formation, its baryon mass is ∼2.434 M ⊙ and gravitational mass is 2.286 M ⊙ . Seven seconds after black hole formation, an additional ∼0.2 M ⊙ is accreted, leaving a black hole baryon mass of ∼2.63 M ⊙ . A disk forms around the proto−neutron star, from which a pair of neutrino-driven jets emanates. These jets accelerate some of the matter up to speeds of ∼45,000 km s −1 and contain matter with entropies of ∼50. The large spatial asymmetry in the explosion results in a residual black hole recoil speed of ∼1000 km s −1 . This novel black hole formation channel now joins the other black hole formation channel between ∼12 and ∼15 M ⊙ discovered previously and implies that the black hole/neutron star birth ratio for solar-metallicity stars could be ∼20%. However, one channel leaves black holes in perhaps the ∼5–15 M ⊙ range with low kick speeds, while the other leaves black holes in perhaps the ∼2.5–3.0 M ⊙ mass range with high kick speeds. However, even ∼8.8 s after core bounce the newly formed black hole is still accreting at a rate of ∼2 × 10 −2 M ⊙ s −1 , and whether the black hole eventually achieves a significantly larger mass over time is yet to be determined.

Astronomy & Astrophysics↗

Meterwave observations of a coronal hole

Meter-wave maps are presented showing a coronal hole at 30.9, 50.0, and 73.8 MHz using the Clark Lake Radioheliograph in October 1984. The coronal hole seen against the disk at all three frequencies shows interesting similarities to, and significant differences from its optical signatures in HeI lambda10830 spectroheliograms. The 73.8 MHz coronal hole, when seen near disk center, appears to coincide with the HeI footprint of the hole. At the lower frequencies, the emission comes from higher levels of the corona, and the hole appears to be displaced, probably due to the non-radial structure of the coronal hole. The contrast of the hole relative to the quiet Sun is much greater than reported previously for a coronal hole observed at 80 MHz. The higher contrast is certainly real, due to the superior dynamic range, sensitivity, and calibration of the Clark Lake instrument. Using a coronal hole model, the electron density is derived from radio observations of the brightness temperature. A very large discrepancy is found between the derived density and that determined from Skylab EUV observations of coronal holes. This discrepancy suggests that much of the physics of coronal holes has yet to be elucidated.

Wang, Z.↗

Black Holes Shed Light on Galaxy Formation

This videotape is comprised of several segments of animations on black holes and galaxy formation, and several segments of an interview with Dr. John Kormendy. The animation segments are: (1) a super massive black hole, (2) Centarus A active black hole found in a collision, (3) galaxy NGC-4261 (active black hole and jet model), (4) galaxy M-32 (orbits of stars are effected by the gravity of the black hole), (5) galaxy M-37 (motion of stars increases as mass of black hole increases), (6) Birth of active galactic nuclei, (7) the collision of two galaxy leads to merger of the black holes, (8) Centarus A and simulation of the collision of 2 galaxies. There are also several segments of an interview with John Kormendy. In these segments he discusses the two most important aspects of his recent black hole work: (1) the correlations between galaxies speed and the mass of the black holes, and (2) the existence of black holes and galactic formation. He also discusses the importance of the Hubble Space Telescope and the Space Telescope Imaging Spectrograph to the study of black holes. He also shows the methodology of processing images from the spectrograph in his office.

Source record↗

Analyzing the Subjectivity of Hole-Type Image Quality Indicators for Radiography

Hole-type penetrameters used as image quality indicators (IQIs) for radiography have an inherent degree of subjectivity to their interpretation. The 1T (one times the thickness of the penetrameter) hole is so small, it can be difficult to distinguish from noise. It is suspected that an operator’s knowledge of the true location of the 1T hole may subconsciously influence a false positive identification of the 1T hole when in fact it cannot be discerned. In the case of computed radiography (CR), the size of the phosphor particles may lead to a noise pattern with features on the scale of the 1T hole. Per NASA-STD-5009, the 1T hole must be detected in order to achieve adequate sensitivity. This is based on the historical understanding that this sensitivity will enable detection of the minimum detectable flaw sizes listed in the standard. It’s important to understand if 1T sensitivity is being achieved, and the associated risk if not. This study sought to determine the true detectability of 1T-sized holes in aluminum and Inconel by creating and inspecting a set of penetrameters with randomly placed holes. Enough holes and vacant zones were created to enable a full probability of detection study with 90% detectability, 95% confidence. Testing is ongoing, but preliminary results have shown poor detectability. The detection rate is slightly better for Inconel than aluminum, slightly better using a micro-focus vs. mini-focus tube, and definitively better for film than CR. One of the key questions of this study is whether historical requirements for film are applicable for CR, and these initial findings suggest they may not be. There is also a requirement in the NASA standard for the minimum contrast-to-noise ratio of the hole. The results have shown that this numerical threshold does not correlate well with visual detection. This raises questions about the true nature of detection, in an age of image processing vs. human judgement. As the results indicate that the detection of 1T holes is unreliable, the next challenge will be determining what sensitivity is really achieved, and what is needed.

Erin Lanigan↗

Analyzing the Subjectivity of Hole-Type Image Quality Indicators for Radiography

Hole-type penetrameters used as image quality indicators (IQIs) for radiography have an inherent degree of subjectivity to their interpretation. The 1T (one times the thickness of the penetrameter) hole is so small, it can be difficult to distinguish from noise. It is suspected that an operator’s knowledge of the true location of the 1T hole may subconsciously influence a false positive identification of the 1T hole when in fact it cannot be discerned. In the case of computed radiography (CR), the size of the phosphor particles may lead to a noise pattern with features on the scale of the 1T hole. Per NASA-STD-5009, the 1T hole must be detected in order to achieve adequate sensitivity. This is based on the historical understanding that this sensitivity will enable detection of the minimum detectable flaw sizes listed in the standard. It’s important to understand if 1T sensitivity is being achieved, and the associated risk if not. This study sought to determine the true detectability of 1T-sized holes in aluminum and Inconel by creating and inspecting a set of penetrameters with randomly placed holes. Enough holes and vacant zones were created to enable a full probability of detection study with 90% detectability, 95% confidence. Testing is ongoing, but preliminary results have shown poor detectability. The detection rate is slightly better for Inconel than aluminum, slightly better using a micro-focus vs. mini-focus tube, and definitively better for film than CR. One of the key questions of this study is whether historical requirements for film are applicable for CR, and these initial findings suggest they may not be. There is also a requirement in the NASA standard for the minimum contrast-to-noise ratio of the hole. The results have shown that this numerical threshold does not correlate well with visual detection. This raises questions about the true nature of detection, in an age of image processing vs. human judgement. As the results indicate that the detection of 1T holes is unreliable, the next challenge will be determining what sensitivity is really achieved, and what is needed.

Erin Lanigan↗

Experimental and Computational Investigation of Shaped Film Cooling Holes Designed to Minimize Inlet Separation

Abstract Film cooling is used to protect turbine components from the extreme temperatures by ejecting coolant through arrays of holes to create an air buffer from the hot combustion gases. Limitations in traditional machining meant film cooling holes universally have sharp inlets, which create separation regions at the hole entrance. The present study uses experimental and computational data to show that these inlet separation are a major cause of performance variation in crossflow fed film cooling holes. Three-hole designs were experimentally tested by independently varying the coolant velocity ratio (VR) and the coolant channel velocity ratio (VRc) to isolate the effects of crossflow on hole performance. Leveraging additive manufacturing (AM) technologies, the addition of a 0.25D radius fillet to the inlet of a 7-7-7 shaped hole is shown to significantly improve diffuser usage and significantly reduce variation in performance with VRc. A second AM design used a very large radius of curvature inlet to reduce biasing caused by the inlet crossflow. Experiments showed that this “swept” hole design did minimize biasing of the coolant flow to one side of the shaped hole, and it significantly reduced variations due to varying VRc. RANS simulations at six VR and three VRc conditions were made for each geometry to better understand how the new geometries changed the velocity field within the hole. The sharp and rounded inlets were seen to have very similar tangential velocity fields and jet biasing. Both AM inlets created more uniform, slower velocity fields entering the diffuser. The results of this article indicate that large improvements in film cooling performance can be found by leveraging AM technology.

Engineering↗

Effects of load proportioning on the capacity of multiple-hole composite joints

This study addresses the issue of adjusting the proportion of load transmitted by each hole in a multiple-hole joint so that the joint capacity is a maximum. Specifically two-hole-in-series joints are examined. The results indicate that when each hole reacts 50% of the total load, the joint capacity is not a maximum. One hole generally is understressed at joint failure. The algorithm developed to determine the load proportion at each hole which results in maximum capacity is discussed. The algorithm includes two-dimensional finite-element stress analysis and failure criteria. The algorithm is used to study the effects of joint width, hole spacing, and hole to joint-end distance on load proportioning and capacity. To study hole size effects, two hole diameters are considered. Three laminates are considered: a quasi-isotropic laminate; a cross-ply laminate; and a 45 degree angle-ply laminate. By proportioning the load, capacity can be increased generally from 5 to 10%. In some cases a greater increase is possible.

Hyer, M. W.↗

Primordial black holes in globular clusters

It has recently been recognized that significant numbers of medium-mass back holes (of order 10 solar masses) should form in globular clusters during the early stages of their evolution. Here we explore the dynamical and observational consequences of the presence of such a primordial black-hole population in a globular cluster. The holes initially segregate to the cluster cores, where they form binary and multiple black-hole systems. The subsequent dynamical evolution of the black-hole population ejects most of the holes on a relatively short timescale: a typical cluster will retain between zero and four black holes in its core, and possibly a few black holes in its halo. The presence of binary, triple, and quadruple black-hole systems in cluster cores will disrupt main-sequence and giant stellar binaries; this may account for the observed anomalies in the distribution of binaries in globular clusters. Furthermore, tidal interactions between a multiple black-hole system and a red giant star can remove much of the red giant's stellar envelope, which may explain the puzzling absence of larger red giants in the cores of some very dense clusters.

Sigurdsson, Steinn↗

Heating of coronal holes by the resonant absorption and dissipation of Alfven waves and its relation to solar wind acceleration

Coronal hole regions are well known sources of high-speed solar wind, however to account for the observed properties of the solar wind a source of momentum and heat must be included. Alfven waves were suggested as the possible source of heating that accelerates the solar wind. We investigate the propagation of the Alfven waves in coronal holes via numerical solution of the linearized 2-D resistive MHD equations in slab geometry. The Alfven waves are driven at the lower boundary of the coronal hole and propagate into the corona. The waves are reflected at the coronal hole boundary and part of the wave energy leaks out of the coronal hole. We compare the calculated wavelengths and the attenuation rate of the fast mode Alfven waves in the leaky waveguide formed by the coronal hole with the analytical ideal MHD solutions. The formation of resonance heating layers is found to occur when shear Alfven waves propagate in an inhomogeneous coronal hole. The heating is enhanced when fast mode waves couple to the shear Alfven waves. The narrow heating layers are formed near the location of the ideal resonance, which might occur near the coronal hole boundary for a nearly constant density coronal hole, surrounded by a higher density plasma. We investigate the dependence of the heating on the driver frequency, the Lundquist number, and on the heliocentric distance. and find that the low frequency Alfven waves can be an efficient source of heating at large distances from the Sun. We discuss the relation of our results to the observed properties of high-speed solar wind and coronal holes.

Ofman, L.↗

High-Time-Resolution Study of Magnetic Holes in the Solar Wind

The objectives of this investigation are to determine the internal plasma structure of kinetic-scale and larger scale magnetic holes, and to determine their stability, their source mechanism(s), and their spatial extent. It is also of importance to determine the relationship between kinetic-scale holes and long-duration holes. As smaller and smaller magnetic depressions are investigated in order to make this a complete study, a robust criterion is necessary for distinguishing magnetic holes from random or unresolvable fluctuations in the interplanetary magnetic field. In order to resolve this ambiguity, we obtained from the MFI experiments magnetic field measurements from the WIND spacecraft at a time resolution of 46 to 184 ms over certain periods. We have also devised a measure of certainty for magnetic hole detections. The certainty factor, q, is defined as the difference between the mean magnetic field in the hole and the local magnetic field, in units of the local standard deviation of the field strength. For fullest generality, it is necessary to calculate this q over the range of available scales of interest, from 60 ms up to 300 s. This technique establishes a two dimensional matrix of relative probabilities that a hole of some duration (d) might exist in the data set at a given time (t). In identifying q-peaks in time and duration, we also come upon a natural method for distinguishing holes with internal structure from multiple holes in close proximity or holes nested inside of others. If two q-peaks are more than a half-width apart, they are simply said to be separate events.

Lazarus, Alan↗

Method for Estimating Thread Strength Reduction of Damaged Parent Holes with Inserts

During normal assembly and disassembly of bolted-joint components, thread damage and/or deformation may occur. If threads are overloaded, thread damage/deformation can also be anticipated. Typical inspection techniques (e.g. using GO-NO GO gages) may not provide adequate visibility of the extent of thread damage. More detailed inspection techniques have provided actual pitch-diameter profiles of damaged-hardware holes. A method to predict the reduction in thread shear-out capacity of damaged threaded holes has been developed. This method was based on testing and analytical modeling. Test samples were machined to simulate damaged holes in the hardware of interest. Test samples containing pristine parent-holes were also manufactured from the same bar-stock material to provide baseline results for comparison purposes. After the particular parent-hole thread profile was machined into each sample a helical insert was installed into the threaded hole. These samples were tested in a specially designed fixture to determine the maximum load required to shear out the parent threads. It was determined from the pristine-hole samples that, for the specific material tested, each individual thread could resist an average load of 3980 pounds. The shear-out loads of the holes having modified pitch diameters were compared to the ultimate loads of the specimens with pristine holes. An equivalent number of missing helical coil threads was then determined based on the ratio of shear-out loads for each thread configuration. These data were compared with the results from a finite element model (FEM). The model gave insights into the ability of the thread loads to redistribute for both pristine and simulated damage configurations. In this case, it was determined that the overall potential reduction in thread load-carrying capability in the hardware of interest was equal to having up to three fewer threads in the hole that bolt threads could engage. One- half of this potential reduction was due to local pitch-diameter variations and the other half was due to overall pitch-diameter enlargement beyond Class 2 fit. This result was important in that the thread shear capacity for this particular hardware design was the limiting structural capability. The details of the method development, including the supporting testing, data reduction and analytical model results comparison will be discussed hereafter.

Johnson, David L.↗

Generation of Electron Whistler Waves at the Mirror Mode Magnetic Holes: MMS Observations and PIC Simulation

The Magnetospheric Multiscale mission has observed electron whistler waves at the center and at the edges of magnetic holes in the dayside magnetosheath. The magnetic holes are nonlinear mirror structures since their magnitude is anticorrelated with particle density. In this article, we examine the growth mechanisms of these whistler waves and their interaction with the host magnetic hole. In the observations, as magnetic holes develop and get deeper, an electron population gets trapped and develops a temperature anisotropy favorable for whistler waves to be generated. In addition, the decrease in magnetic field magnitude and the increase in density reduce the electron resonance energy, which promotes the electron cyclotron resonance. To investigate this process, we used expanding box particle-in-cell simulations to produce the mirror instability, which then evolve into magnetic holes. The simulation shows that whistler waves can be generated at the center and edges of magnetic holes, which reproduces the primary features of the MMS observations. The simulation shows that the electron temperature anisotropy develops in the center of the magnetic hole once the mirror instability reaches its nonlinear stage of evolution. The plasma is then unstable to whistler waves at the minimum of the magnetic field structures. In the saturation regime of mirror instability, when magnetic holes are developed, the electron temperature anisotropy appears at the edges of the holes and electron distributions become more isotropic at the magnetic field minimum. At the edges, the expansion of magnetic holes decelerates the electrons, which leads to temperature anisotropies.

Magnetospheric Multiscale↗

How the Universe postpones the evaporation and curtails the quantum spreading of black holes

Black holes are expected to evaporate through the process of Hawking radiation. This process is expected to cause the uncertainty in a black hole’s position to grow to ~ $M^4/M^3_{pl}$ over the course of it’s lifetime, even as its momentum spreads only by ~ $M_{Pl}$ For the black holes that have been observed, which have M ≥ M ⊙ this greatly exceeds the Hubble volume. However, the decay of black holes and their quantum spreading, are delayed in the Universe while the influx of energy into the black holes exceeds their Hawking luminosity. We show that for these M ≥ M ⊙ black holes, their decay outside galaxies and clusters is prevented far longer than it takes the black holes to be dragged well beyond the Hubble horizon, where their eventual decay occurs away from the prying eyes of any observer who has not hitched a ride with them. Meanwhile, black holes in an observer’s galaxy or cluster are themselves prevented from decaying long past the extinction of the last stars, and at least until their galaxy/cluster is swept clean of dark matter, in $\gg$ 10 25 y. Even then, if the black holes become unbound, then they are dragged beyond the Hubble radius before undergoing significant decay; if not, they remain in bound orbits, spreading at most over a subvolume of the galaxy/cluster, and long localized by scatterings to much smaller volumes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗