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The dynamic behavior of flood waves on rivers is essential to flood prediction. Natural flood waves are complex due to tributary inputs, rainfall variations, and overbank flows, so this study examines hydropower dam releases, which are simpler to analyze because channel effects are isolated. Successive arrival times and heights of peaks along 9 rivers with multiple stream gauges downstream of hydroelectric dams show that flow peaks typically become exponentially lower and wider with distance. The propagation velocity of peaks increases with water depth and channel slope but decreases with downstream distance and greater channel tortuosity. A rich hierarchy of velocities was found. Hydropower pulses progress at or in slight excess of the theoretical celerity, which is faster than the propagation rate of average natural floods, which in turn exceeds the mean velocity of water in the channel, yet the water moves faster than the peaks of record floods. The progressive changes to the height, shape, and velocity of hydropower flow peaks are simulated by the first analytical solution to the convolution integral for a rectangular source pulse that is based on diffusion-advection theory. Available data support some widely held expectations while refuting others. An expanded definition of "water mining" is proposed. [ABSTRACT FROM AUTHOR]

Available inventories of baryonic mass in the universe are based largely on galactic data and empirical calculations made >20 years ago. Values falling below cosmological estimates underlie proposals that certain rarified gassy regions could have extremely high T, which motivated absorption measurements and hydrodynamic models. Yet, the shortfall remains. We inventory the total baryonic mass, focusing on gravitational interactions and updated measurements. A recent analytical inverse method for analyzing galactic rotation curves quantified how baryon mass and associated volumetric density (ρ) depend on distance (r) from galactic centers. The model is based on the dynamical consequences of the observed oblate shape of galaxies and the Virial Theorem. The parameter-free solution provides ρ(r) ∝ 1/r2 which describes star-rich galactic interiors, gas-rich outer discoids, circumgalactic media, and gradation into intergalactic media. Independent observational determinations of baryonic ρ validate that our 1/r2 result describes baryons alone. This solution shows that total baryonic mass associated with any galaxy is 2.4 to 40 times detectable luminosity, depending on galaxy size and spacing. Luminosity data within 50 Mpc show that Andromeda equivalents separated by ~1 Mpc represent the local universe. Combining the above yields (6 ± 2) × 10−25 kg m−3 for the present-day universe. Three other approaches support this high density: (1) evaluating trends and luminosity data near 1000 Mpc; (2) using a recent estimate for the number of galaxies in the universe; (3) calculating an energy balance. We discuss uncertainties in the critical density. Implications of large baryonic ρ are briefly discussed. [ABSTRACT FROM AUTHOR]

Kepler's discoveries were permitted by his remarkable insight to place the Sun at the focus of an elliptical planetary orbit. This coordinate system reduces a 2-dimensional orbit to a single spatial dimension. We consider an alternative coordinate system centered on the "image focus," which is the symmetrical (mirror) counterpart of the "real focus" occupied by the Sun. Our analytical approach provides new purely geometric formulae and an exact relationship for the dynamic property of orbital time. In addition, considering the mirror symmetry of the ellipse leads to a simple approximation: the radial hand of an orbital clock rotates counterclockwise at a nearly steady angular velocity 2π/T about the "image focus," where T is the orbital period. This approximation is a useful pedagogic tool and has good accuracy for orbits with low to moderate eccentricities, since the deviation from the exact result goes as eccentricity squared. Planetary comparisons are made. In particular, the angular speeds of Mercury and Jupiter are highly variable in the geocentric and heliocentric reference frames, but are nearly constant in the image focus reference frame. Our findings resolve whether the image focus is the location for observing uniform motion of an elliptical orbit, and pertain to their stability. [ABSTRACT FROM AUTHOR]

Extremely slow recession of the Moon from the Earth has been recently proposed and attributed to conversion of Earth's axial spin to lunar orbital momentum. This hypothesis is inconsistent with long-standing recognition that the Moon's orbit involves three-body interactions. This and other short-comings, such as Earth's spin loss being internal, are summarized here. Considering point-masses is justified by theory and observational data on other moons. We deduce that torque in the Earth-Moon-Sun system increases eccentricity of the lunar orbit but decreases its inclination over time. Consequently, the average lunar orbital radius is decreasing. We also show that lunar drift is too small to be constrained through lunar laser ranging measurements, mainly because atmospheric refraction corrections are comparatively large and variations in lunar cycles are under-sampled. Our findings support co-accretion and explain how orbits evolve in many-body point-mass systems. [ABSTRACT FROM AUTHOR]

Available data on insulating, semiconducting, and metallic solids verify our new model that incorporates steady-state heat flow into a macroscopic, thermodynamic description of solids, with agreement being best for isotropic examples. Our model is based on: (1) mass and energy conservation; (2) Fourier's law; (3) Stefan–Boltzmann's law; and (4) rigidity, which is a large, yet heretofore neglected, energy reservoir with no counterpart in gases. To account for rigidity while neglecting dissipation, we consider the ideal, limiting case of a perfectly frictionless elastic solid (PFES) which does not generate heat from stress. Its equation-of-state is independent of the energetics, as in the historic model. We show that pressure-volume work (PdV) in a PFES arises from internal interatomic forces, which are linked to Young's modulus (Ξ) and a constant (n) accounting for cation coordination. Steady-state conditions are adiabatic since heat content (Q) is constant. Because average temperature is also constant and the thermal gradient is fixed in space, conditions are simultaneously isothermal: Under these dual restrictions, thermal transport properties do not enter into our analysis. We find that adiabatic and isothermal bulk moduli (B) are equal. Moreover, Q/V depends on temperature only. Distinguishing deformation from volume changes elucidates how solids thermally expand. These findings lead to simple descriptions of the two specific heats in solids: ∂ln(cP)/∂P = −1/B; cP = nΞ times thermal expansivity divided by density; cP = cVnΞ/B. Implications of our validated formulae are briefly covered. [ABSTRACT FROM AUTHOR]

Galactic mass consistent with luminous mass is obtained by fitting rotation curves (RC = tangential velocities vs. equatorial radius r) using Newtonian force models, or can be unambiguously calculated from RC data using a model based on spin. In contrast, mass exceeding luminous mass is obtained from multi-parameter fits using potentials associated with test particles orbiting in a disk around a central mass. To understand this disparity, we explore the premises of these mainstream disk potential models utilizing the theorem of Gauss, thermodynamic concepts of Gibbs, the findings of Newton and Maclaurin, and well-established techniques and results from analytical mathematics. Mainstream models assume that galactic density in the axial (z) and r directions varies independently: we show that this is untrue for self-gravitating objects. Mathematics and thermodynamic principles each show that modifying Poisson’s equation by summing densities is in error. Neither do mainstream models differentiate between interior and exterior potentials, which is required by potential theory and has been recognized in seminal astronomical literature. The theorem of Gauss shows that: (1) density in Poisson’s equation must be averaged over the interior volume; (2) logarithmic gravitational potentials implicitly assume that mass forms a long, line source along the z axis, unlike any astronomical object; and (3) gravitational stability for three-dimensional shapes is limited to oblate spheroids or extremely tall cylinders, whereas other shapes are prone to collapse. Our findings suggest a mechanism for the formation of the flattened Solar System and of spiral galaxies from gas clouds. The theorem of Gauss offers many advantages over Poisson’s equation in analyzing astronomical problems because mass, not density, is the key parameter. [ABSTRACT FROM AUTHOR]

Empirical laws proposed for the decline in star spin with time have heretofore been tested using ambiguous fitting models. We develop an analytical inverse model that uses histogram data to unequivocally determine the physical law governing how dwarf star spin depends on time (t) and mass (M). We analyze shapes of paired histograms of axial rotation period (П) and angular velocity (ω = 2π/П) to utilize the fact that a variable and its reciprocal are governed by the same physics. Copious data on open clusters are used to test the formula ∂ω/∂t ∝ − ωn where n is unrestricted, and thus covers diverse possibilities. Histogram conjugates for each of 15 clusters with 120 to 812 measurements provide n = 1.13 ± 0.19. Results are independent of initial spin rate, bin size, cluster parameters, and star mass. Notably, 11 large clusters with mostly M-types yield fits with n = 1.07 ± 0.12. Associations behave similarly. Only exponential decay (n = 1) explains the similar shapes of the conjugate histograms for the spin period and angular velocity, despite the asymmetric (inverse) relationship of these variables. This rate law is consistent with viscous dissipation. Forward modeling confirms that n is near unity and further shows that coeval formation of all stars in a cluster does not occur. We therefore explore a constant rate of star production, which is reasonable for tiny stars. Inverse models show that episodic production increases with mass, but is unimportant below ~0.55 MSun. We infer star and cluster ages, and find that star production becomes less regular with time, as interstellar gas and dust are progressively depleted. Our new analytical approach of extracting a physical law from conjugate histograms is general and widely applicable. [ABSTRACT FROM AUTHOR]

Proposed explanations of galactic rotation curves (RC = tangential velocity vs. equatorial radius, determined from Doppler measurements) involve dramatically different assumptions. A dominant, original camp invoked huge amounts of unknown, non-baryonic dark matter (NBDM) in surrounding haloes to reconcile RC simulated using their Newtonian orbital models (NOMs) for billions of stars in spiral galaxies with the familiar Keplerian orbital patterns of the few, tiny planets in our Solar System. A competing minority proposed that hypothetical, non-relativistic, non-Newtonian forces govern the internal motions of galaxies. More than 40 years of controversy has followed. Other smaller groups, unsatisfied by explanations rooted in unknown matter or undocumented forces, have variously employed force summations, spin models, or relativistic adaptations to explain galactic rotation curves. Some small groups have pursued inverse models and found no need for NBDM. The successes, failures, and underlying assumptions of the above models are reviewed in this paper, focusing on their mathematical underpinnings. We also show that extractions of RC from Doppler measurements need revising to account for the effect of galaxy shape on flux-velocity profiles and for the possible presence of a secondary spin axis. The latter is indicated by complex Doppler shift patterns. Our findings, combined with independent evidence such as hadron collider experiments failing to produce non-baryonic matter, suggest that a paradigm shift is unfolding. [ABSTRACT FROM AUTHOR]

Spiral galaxies and their rotation curves have key characteristics of differentially spinning objects. Oblate spheroid shapes are a consequence of spin and reasonably describe galaxies, indicating that their matter is distributed in gravitationally interacting homeoidal shells. Here, previously published equations describing differentially spinning oblate spheroids with radially varying density are applied to 51 galaxies, mostly spirals. A constant volumetric density (ρ, kg m−3) is assumed for each thin homeoid in these formulae, after Newton, which is consistent with RCs being reported simply as a function of equatorial radius r. We construct parameter-free inverse models that uniquely specify mass inside any given r, and thus directly constrain ρ vs. r solely from velocity v (r) and galactic aspect ratios (assumed as 1:10 for spirals when data are unavailable). Except for their innermost zones, ρ is proven to be closely proportional to rn, where the statistical average of n for all 36 spirals studied is −1.80 ± 0.40. Our values for interior densities compare closely with independently measured baryon density in appropriate astronomical environments: for example, calculated ρ at galactic edges agrees with independently estimated ρ of intergalactic media (IGM). Our finding that central densities increase with galaxy size is consistent with behavior exhibited by diverse self-gravitating entities. Our calculated mass distributions are consistent with visible luminosity and require no non-baryonic component. [ABSTRACT FROM AUTHOR]

Spiral galaxies are spinning, internally densified objects. The Virial Theorem explains galactic rotation curves via its linkage of the rotation rate to the gravitational self-potential (Ug) and the moment of inertia of oblate spheroids. We devise a new analytical solution that allows galactic mass and volumetric density (kg m−3) profiles to be extracted from velocity and its derivative as functions of equatorial radius. This inverse model of rotation curves is direct, unambiguous, and parameter-free. To probe galactic evolution, we combine energy minimization, angular momentum conservation, and the Virial Theorem. The characteristic flat shape of spiral galaxies results from an initial vertical collapse of a spinning, colossal molecular cloud, which reduces Ug while conserving angular momentum. Subsequent inward densification further lowers Ug, producing bulges, but conserving angular momentum requires mass loss, achieved by the outward movement of the distal parts of the spiral arms. Many of the evolutionary patterns of spiral galaxies are exhibited by the changing shapes of hurricanes during formation and dissipation. In contrast, elliptical galaxies evolve from a cloud with roughly random orbits into progressively rounder, internally denser objects, with angular momentum conserved by the development of vertically oriented jets. Galactic evolution is governed by the initial inventory of mass and angular momentum, resulting in separate paths for elliptical and spiral galaxies, as is codified in Hubble's tuning fork diagram. [ABSTRACT FROM AUTHOR]