Ratio Of Rotational Kinetic Energy Research Articles

Physiological demands in simulated tennis matches and hitting tests take account of the translational and rotational kinetic energy ratio of the ball.

Assessment of fatigue effect on hitting ability in tennis has been controversial in previous studies. The purpose of this study was to determine the relationship between player fatigue and groundstroke type in tennis. We hypothesized that subjects with higher blood lactate concentration during play would apply heavier spin to the ball. We divided players into two groups based on their blood lactate concentration during a pre-measured hitting test (HIGH and LOW). Each group performed a simulated match-play protocol consisting of repeated running and hitting tests, which simulated a three-set match. Heart rate, percent of heart rate reserve, oxygen uptake, pulmonary ventilation, and respiratory exchange were measured. The distance between the ball's landing point and the target, and the ball's kinematics, were recorded during the hitting test between sets. We found no significant difference in ball kinetic energy between groups, but the HIGH group hit the ball with a greater ratio of rotational kinetic energy to total kinetic energy. However, the progression of the simulation protocol did not affect physiological responses (including blood lactate concentration) or hitting ability. Therefore, it is suggested that the type of groundstrokes used by players is one of the factors that should be considered when discussing fatigue in tennis.

Impact of rotation on the multimessenger signatures of a hadron-quark phase transition in core-collapse supernovae

We study the impact of rotation on the multimessenger signals of core-collapse supernovae (CCSNe) with the occurrence of a first-order hadron-quark phase transition (HQPT). We simulate CCSNe with the \texttt{FLASH} code starting from a 20~$M_\odot$ progenitor with different rotation rates, and using the RDF equation of state from \textit{Bastian} 2021 that prescribes the HQPT. Rotation is found to delay the onset of the HQPT and the resulting dynamical collapse of the protocompact star (PCS) due to the centrifugal support. All models with the HQPT experience a second bounce shock which leads to a successful explosion. The oblate PCS as deformed by rotation gives rise to strong gravitational-wave (GW) emission around the second bounce with a peak amplitude larger by a factor of $\sim10$ than that around the first bounce. The breakout of the second bounce shock at the neutrinosphere produces a $\bar{\nu}_e$-rich neutrino burst with a luminosity of serveral 10$^{53}$~erg~s$^{-1}$. In rapidly rotating models the PCS pulsation following the second bounce generates oscillations in the neutrino signal after the burst. In the fastest rotating model with the HQPT, a clear correlation is found between the oscillations in the GW and neutrino signals immediately after the second bounce. In addition, the HQPT-induced collapse leads to a jump in the ratio of rotational kinetic energy to gravitational energy ($\beta$) of the PCS, for which persistent GW emission may arise due to secular nonaxisymmetric instabilities.

Bayesian parameter estimation of core collapse supernovae using gravitational wave simulations

Using the latest numerical simulations of rotating stellar core collapse, we present a Bayesian framework to extract the physical information encoded in noisy gravitational wave signals. We fit Bayesian principal component regression models with known and unknown signal arrival times to reconstruct gravitational wave signals, and subsequently fit known astrophysical parameters on the posterior means of the principal component coefficients using a linear model. We predict the ratio of rotational kinetic energy to gravitational energy of the inner core at bounce by sampling from the posterior predictive distribution, and find that these predictions are generally very close to the true parameter values, with 90% credible intervals and wide for the known and unknown arrival time models respectively. Two supervised machine learning methods are implemented to classify precollapse differential rotation, and we find that these methods discriminate rapidly rotating progenitors particularly well. We also introduce a constrained optimization approach to model selection to find an optimal number of principal components in the signal reconstruction step. Using this approach, we select 14 principal components as the most parsimonious model.

A lower angular momentum limit for self-gravitating protostellar disc fragmentation

We attempt to verify recent claims (made using semi-analytic models) that for the collapse of spherical homogeneous molecular clouds, fragmentation of the self-gravitating disc that subsequently forms can be predicted using the cloud's initial angular momentum alone. In effect, this condition is equivalent to requiring the resulting disc be sufficiently extended in order to fragment, in line with studies of isolated discs. We use smoothed particle hydrodynamics with hybrid radiative transfer to investigate this claim, confirming that in general, homogeneous spherical molecular clouds will produce fragmenting self-gravitating discs if the ratio of rotational kinetic energy to gravitational potential energy is greater than ~ 5e-3, where this result is relatively insensitive to the initial thermal energy. This condition begins to fail at higher cloud masses, suggesting that sufficient mass at large radii governs fragmentation. While these results are based on highly idealised initial conditions, and may not hold if the disc's accretion from the surrounding envelope is sufficiently asymmetric, or if the density structure is perturbed, they provide a sensible lower limit for the minimum angular momentum required to fragment a disc in the absence of significant external turbulence.

Asymmetric supernovae and gamma-ray bursts

Spectropolarimetry of core collapse supernovae has shown that they are asymmetric and often, but not universally, bi-polar. Jet-induced supernova models give a typical jet/torus structure that is reminiscent of some objects like the Crab nebula, SN 1987A and Cas A. Asymmetry in the strength of polar jets is a plausible mechanism to produce substantial pulsar “kick” velocities. Jets may arise from the intrinsic rotation and magnetic fields that are expected to accompany core collapse. We summarize the potential importance of the magneto-rotational instability (MRI) for the core collapse problem in the context of the non-monotonic behavior expected: increasing centrifugal support will lead to a maximum rotation and magnetic field production as a function of the initial rotation of the iron core. Non-axisymmetric instabilities are predicted for differentially rotating proto-neutron stars with values of the ratio of rotational kinetic energy to binding energy, T/∣ W∣≳0.01. The non-axisymmetric instabilities are likely to drive magnetosonic waves into the surrounding time-dependent density structure. These waves represent a mechanism of the dissipation of the rotational energy of the proto-neutron star, and the outward deposition of this energy may play a role in the supernova explosion process. The phase of deleptonization and contraction of the proto-neutron star lasting several seconds is likely to be an important phase of magnetic non-axisymmetric evolution. In the special circumstance that the proto-neutron star is born sufficiently rapidly rotating that it is subject to bar-mode instabilities on secular timescales, a possible outcome is that the deleptonizing neutron star will evolve along the locus T/∣ W∣∼0.14 releasing a significant fraction of its binding energy as MHD power sufficient to account for a GRB. This power will be provided over an extended time, 10 s, that is strongly reminiscent of the timescale of long GRBs and is also comparable to the time for a shock or jet to propagate through the core of a Type Ib/c supernova.

AMR simulations of the low [formula omitted] bar-mode instability of neutron stars

It has been recently argued through numerical work that rotating stars with a high degree of differential rotation are dynamically unstable against bar-mode deformation, even for values of the ratio of rotational kinetic energy to gravitational potential energy as low as O ( 0.01 ) . This may have implications for gravitational wave astronomy in high-frequency sources such as core collapse supernovae. In this paper we present high-resolution simulations, performed with an adaptive mesh refinement hydrodynamics code, of such low T / | W | bar-mode instability. The complex morphological features involved in the nonlinear dynamics of the instability are revealed in our simulations, which show that the excitation of Kelvin–Helmholtz-like fluid modes outside the corotation radius of the star leads to the saturation of the bar-mode deformation. While the overall trends reported in an earlier investigation are confirmed by our work, we also find that numerical resolution plays an important role during the long-term, nonlinear behavior of the instability, which has implications on the dynamics of rotating stars and on the attainable amplitudes of the associated gravitational wave signals.

Deleptonization and Nonaxisymmetric Instabilities in Core‐Collapse Supernovae

The timescale of deleptonization by neutrino loss and associated contraction of a proto-neutron star is short compared to the time it takes to propagate a shock through the helium core of a massive star, and so the deleptonization phase does not occur in the vacuum of space, but within the supernova ambiance, whether or not there has been a successful explosion. Dynamical nonaxisymmetric instabilities (NAXI) are predicted for sufficiently strongly differentially rotating proto-neutron stars. Some modes are unstable for small values of the ratio of rotational kinetic energy to binding energy, T/|W| 0.01. The NAXI are likely to drive magnetoacoustic waves into the surrounding time-dependent density structure. These waves represent a mechanism of the dissipation of the free energy of differential rotation of the proto-neutron star, and the outward deposition of this energy may play a role in the supernova explosion process. We estimate the power produced by this process and the associated timescale and discuss the possible systematics of the deleptonization phase in this context. A likely possibility is that the proto-neutron star will spin down through these effects before deleptonization and produce substantial but not excessive energy input.

On the Use of the Self‐consistent–Field Method in the Construction of Models for Rapidly Rotating Main‐Sequence Stars

A new formulation of the self-consistent-field (SCF) method for computing models of rapidly, differentially rotating stars is described. The angular velocity is assumed to depend only on the distance from the axis of rotation. In the modified SCF iterative scheme, normalized distributions of two thermodynamic variables—pressure and temperature—are used as trial functions, while the central values of the pressure and temperature are adjusted by a Newton-Raphson iteration. A two-dimensional trial density distribution, which is needed to compute the gravitational potential, is readily obtained from the pressure and temperature through the equation of state in conjunction with a third trial function specifying the two-dimensional shape of the constant-density surfaces. Rotating models of chemically homogeneous main-sequence stars have been computed as necessary in order to illustrate the algorithm and to make comparisons with existing models. Unlike previous implementations of the SCF method, the method described here is not limited to the upper main sequence: it converges for all main-sequence masses, including those well below 9 M⊙. Moreover, the method converges for values of the parameter t = T/|W| (the ratio of rotational kinetic energy to gravitational potential energy) that are at least as high as those obtained by Clement's relaxation technique. The method is also capable of producing models with deep concavities about the poles as well as models with extreme oblateness (far greater than that possible in uniformly rotating stars). For cases with moderate degrees of differential rotation (say for Ω0/Ωe < 10, where Ω0 and Ωe denote the angular velocity at the center and at the equator, respectively), the method has been found to be remarkably robust. For higher degrees of differential rotation, models are restricted to a portion of parameter space away from two regions of nonconvergence, inside which some of the models evidently develop toroidal level surfaces.

Gravitational Waves from Axisymmetric, Rotating Stellar Core Collapse

We have carried out an extensive set of two-dimensional, axisymmetric, purely hydrodynamic calculations of rotational stellar core collapse with a realistic, finite-temperature nuclear equation of state and realistic massive star progenitor models. For each of the total number of 72 different simulations we performed, the gravitational wave signature was extracted via the quadrupole formula in the slow-motion, weak-field approximation. We investigate the consequences of variation in the initial ratio of rotational kinetic energy to gravitational potential energy and in the initial degree of differential rotation. Furthermore, we include in our model suite progenitors from recent evolutionary calculations that take into account the effects of rotation and magnetic torques. For each model, we calculate gravitational radiation waveforms, characteristic wave strain spectra, energy spectra, final rotational profiles, and total radiated energy. In addition, we compare our model signals with the anticipated sensitivities of the first- and second-generation LIGO detectors coming on line. We find that most of our models are detectable by LIGO from anywhere in the Milky Way.

Dynamical instability of new-born neutron stars as sources of gravitational radiation

The dynamical instability of new-born neutron stars is studied by evolving the linearized hydrodynamical equations. The neutron stars considered in this paper are those produced by the accretion induced collapse of rigidly rotating white dwarfs. A dynamical bar-mode (m=2) instability is observed when the ratio of rotational kinetic energy to gravitational potential energy $\beta$ of the neutron star is greater than the critical value $\beta_d \approx 0.25$. This bar-mode instability leads to the emission of gravitational radiation that could be detected by gravitational wave detectors. However, these sources are unlikely to be detected by LIGO II interferometers if the event rate is less than $10^{-6}$ per year per galaxy. Nevertheless, if a significant fraction of the pre-supernova cores are rapidly rotating, there would be a substantial number of neutron stars produced by the core collapse undergoing bar-mode instability. This would greatly increase the chance of detecting the gravitational radiation.

Nonaxisymmetric secular instabilities driven by star/disk coupling

We determine conditions for the onset of nonaxisymmetric secular instabilities in polytropes with a wide range of angular momentum distributions using Lagrangian techniques, and then calculate the growth rate of such instabilities when driven by the coupling of the perturbed star to a circumstellar disk. We use Langrangian displacement vectors with azimuthal dependence proportional to exp (im phi), where m is an integer and phi is the azimuthal coordinate. The onset of secular instability in terms of the quantity T/absolute value of W, the ratio of rotational kinetic energy to gravitational potential energy, is affected by both the compressibility and the angular momentum distribution of the polytrope. The largest effects occur when the angular momentum distribution is varied. For polytropic index n = 3/2, the onset of secular instability for the m = 2 mode (the bar mode), as determined by its neutral point, shifts from T/absolute value of W = 0.141 to 0.093, while the m = 5 mode neutral point shifts from T/absolute value of W = 0.088 to 0.031 over the range of angular momentum distributions we consider. The smallest critical T/absolute value of W-values occur for the angular momentum distributions which are the most peaked toward the equator. For the angular momentum distribution of a Maclaurin spheroid, as the polytropic index n is increased from 3/2 to 5/2, the neutral point for m = 2 shifts from T/absolute value of W = 0.141 to 0.144 and the netural point for m = 5 shifts from T/absolute value of W = 0.069 to 0.078. The netural points for m = 2 and 5 for the Maclaurin sequence (n = 0) are 0.137 and 0.0629, respectively. As the angular momentum distribution becomes more peaked toward the equatorial radius of the polytropes, the critical T/absolute value of W-values generally become less sensitive to the compressibility of the polytrope. Star/disk coupling can drive the secular instability in systems where the star is surrounded by a massive disk and, if the instability can grow to moderate amplitude, then the coupling can transport significant amounts of angular momentum from the star into the circumstellar disk. We find, for the particular case of rotating protostars during the accretion phase, that the instability growth time can be shorter than the accretion time. Further, if the instability can grow to amplitudes on the order of several percent, the star/disk coupling can remove angular momentum from the forming star faster than it is added by accretion.

The stability of thick, self-gravitating disks in protostellar systems

Using numerical hydrodynamics techniques, we perform a nonlinear stability analysis of accretion disk systems that contain thick, self-gravitating disks. The systems are initially represented by a point mass M(sub c) at the center and a geometrically thick, axisymmetric disk of mass M(sub d) that supports uniform specific angular momentum and obeys an n = 3/2, polytropic equation of state. The equilibrium disk structure is uniquely defined upon the specification of two key dimensionless system parameters: M(sub d)/M(sub c) and T/absolute value of W (the ratio of rotational kinetic energy if the disk to the gravitational potential energy of the system). The focus of this work is on the identification of systems within this two-dimensional parameter space that are marginally unstable toward the development of nonaxisymmetric distortions. The geometric form and relative pattern speed of the disk's distortion as well as the likelihood of disk fragmentation as a result of such instabilities is examined, particularly in the context of protostellar systems. The value of T/absolute value of W at which thick disks first become dynamically unstable to nonaxisymmetric distortions is found to vary significantly with the mass ratio of the accertion disk system. Nonaxisymmetric eigenmodes with four distinctly different characters are identified in systems with mass ratios in the range 0.2 less than or equal to (M(sub d)/M(sub c)) less than or equal to 5.

The breakup of self-gravitating rings, tori, and thick accretion disks

view Abstract Citations (66) References (46) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Breakup of Self-gravitating Rings, Tori, and Thick Accretion Disks Tohline, Joel E. ; Hachisu, Izumi Abstract Using Hachisu's self-consistent-field technique, we have constructed equilibrium sequences of self-gravitating, axisymmetric rings (or, tori) having n = 3/2 polytropic structures and radial angular velocity profiles of the form {OMEGA} is proportional to R^-1/2^, with l = 3.0, 3.5, and 4.0. Then, using Tohline's three-dimensional hydrodynamic computer code, we have identified the position along the equilibrium sequences where the rings first become dynamically unstable toward the development of an m = 2 (ellipsoidal), nonaxisymmetric distortion. (Principally for computational convenience, we have limited this study to nonaxisymmetric distortions with even azimuthal mode numbers.) In terms of the ratio of rotational kinetic energy T to gravitational potential energy W, the instability arises in systems having T/|W| >= 0.16. Although the dynamics of odd azimuthal modes has not been examined here, we suspect that tori having smaller values of T/|W| are dynamically stable to all nonaxisymmetric modes because the only odd mode having a longer azimuthal wavelength--the m = 1 mode---cannot develop without requiring a shift in the center of mass of the torus. We conclude that the onset of a dynamical instability in self- gravitating rings occurs at a value of T/|W| that is substantially lower than the value (T/|W|~0.27) at which an analogous instability arises in centrally condensed, self-gravitating configurations. We have quantitatively measured the growth rate and the pattern speed of the m = 2 mode in eight models having values of T/|W| in the range 0.167 <= T/|W| <= 0.271. The character of the unstable eigenmode in these models is clearly different from the unstable, nonaxisymmetric eigenmode that was first shown by Papaloizou and Pringle to arise in zero mass, accretion tori. We are therefore convinced that the instability we have identified here is driven by the self-gravity of the torus and is distinctly different from the Papaloizou-Pringle instability. Although the early nonlinear development of this instability causes the torus to deform into an ellipsoidal configuration with noticeable density enhancements arising on opposite sides of the ring, further evolution shows that the instability does not lead to fragmentation of the torus. We therefore call into question the perceived notion that self-gravitating rings are generally unstable toward fragmentation along their length. Publication: The Astrophysical Journal Pub Date: October 1990 DOI: 10.1086/169205 Bibcode: 1990ApJ...361..394T Keywords: Accretion Disks; Galactic Evolution; Galactic Structure; Star Formation; Angular Velocity; Self Consistent Fields; Astrophysics; ACCRETION; GALAXIES: FORMATION; GALAXIES: STRUCTURE; HYDRODYNAMICS; STARS: FORMATION full text sources ADS |