Axial instability of rotating relativistic stars

Perturbations of rotating relativistic stars can be classified by their behavior under parity. For axial perturbations (r-modes), initial data with negative canonical energy is found with angular dependence $e^{im\phi}$ for all values of $m\geq 2$ and for arbitrarily slow rotation. This implies instability (or marginal stability) of such perturbations for rotating perfect fluids. This low $m$-instability is strikingly different from the instability to polar perturbations, which sets in first for large values of $m$. The timescale for the axial instability appears, for small angular velocity $\Omega$, to be proportional to a high power of $\Omega$. As in the case of polar modes, viscosity will again presumably enforce stability except for hot, rapidly rotating neutron stars. This work complements Andersson's numerical investigation of axial modes in slowly rotating stars.Comment: Latex, 18 pages. Equations 84 and 85 are corrected. Discussion of timescales is corrected and update

Constraining phases of quark matter with studies of r-mode damping in neutron stars

The r-mode instability in rotating compact stars is used to constrain the phase of matter at high density. The color-flavor-locked phase with kaon condensation (CFL-K0) and without (CFL) is considered in the temperature range 10^8K < T <10^{11} K. While the bulk viscosity in either phase is only effective at damping the r-mode at temperatures T > 10^{11} K, the shear viscosity in the CFL-K0 phase is the only effective damping agent all the way down to temperatures T > 10^8 K characteristic of cooling neutron stars. However, it cannot keep the star from becoming unstable to gravitational wave emission for rotation frequencies f ~ 56-11 Hz at T ~ 10^8-10^9 K. Stars composed almost entirely of CFL or CFL-K0 matter are ruled out by observation of rapidly rotating neutron stars, indicating that dissipation at the quark-hadron interface or nuclear crust interface must play a key role in damping the instability.Comment: 8 pages, 2 figure

A simple derivation of Kepler's laws without solving differential equations

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to en explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by beginners at university (or even before) can be considered as a first application of mechanical concepts to a physical problem of great historical and pedagogical interest

Generalized r-Modes of the Maclaurin Spheroids

Analytical solutions are presented for a class of generalized r-modes of rigidly rotating uniform density stars---the Maclaurin spheroids---with arbitrary values of the angular velocity. Our analysis is based on the work of Bryan; however, we derive the solutions using slightly different coordinates that give purely real representations of the r-modes. The class of generalized r-modes is much larger than the previously studied `classical' r-modes. In particular, for each l and m we find l-m (or l-1 for the m=0 case) distinct r-modes. Many of these previously unstudied r-modes (about 30% of those examined) are subject to a secular instability driven by gravitational radiation. The eigenfunctions of the `classical' r-modes, the l=m+1 case here, are found to have particularly simple analytical representations. These r-modes provide an interesting mathematical example of solutions to a hyperbolic eigenvalue problem.Comment: 12 pages, 3 figures; minor changes and additions as will appear in the version to be published in Physical Review D, January 199

Had the planet mars not existed: Kepler's equant model and its physical consequences

We examine the equant model for the motion of planets, which has been the starting point of Kepler's investigations before he modified it because of Mars observations. We show that, up to first order in eccentricity, this model implies for each orbit a velocity which satisfies Kepler's second law and Hamilton's hodograph, and a centripetal acceleration with an inverse square dependence on the distance to the sun. If this dependence is assumed to be universal, Kepler's third law follows immediately. This elementary execice in kinematics for undergraduates emphasizes the proximity of the equant model coming from Ancient Greece with our present knowledge. It adds to its historical interest a didactical relevance concerning, in particular, the discussion of the Aristotelian or Newtonian conception of motion

The R-Mode Oscillations in Relativistic Rotating Stars

The axial mode oscillations are examined for relativistic rotating stars with uniform angular velocity. Using the slow rotation formalism and the Cowling approximation, we have derived the equations governing the r-mode oscillations up to the second order with respect to the rotation. In the lowest order, the allowed range of the frequencies is determined, but corresponding spatial function is arbitrary. The spatial function can be decomposed in non-barotropic region by a set of functions associated with the differential equation of the second-order corrections. The equation however becomes singular in barotropic region, and a single function can be selected to describe the spatial perturbation of the lowest order. The frame dragging effect among the relativistic effects may be significant, as it results in rather broad spectrum of the r-mode frequency unlike in the Newtonian first-order calculation.Comment: 19 pages, 4 figures, AAS LaTeX, Accepted for publication in The Astrophysical Journa

The meaning of variation to healthcare managers, clinical and health-services researchers, and individual patients

Healthcare managers, clinical researchers and individual patients (and their physicians) manage variation differently to achieve different ends. First, managers are primarily concerned with the performance of care processes over time. Their time horizon is relatively short, and the improvements they are concerned with are pragmatic and 'holistic.' Their goal is to create processes that are stable and effective. The analytical techniques of statistical process control effectively reflect these concerns. Second, clinical and health-services researchers are interested in the effectiveness of care and the generalisability of findings. They seek to control variation by their study design methods. Their primary question is: 'Does A cause B, everything else being equal?' Consequently, randomised controlled trials and regression models are the research methods of choice. The focus of this reductionist approach is on the 'average patient' in the group being observed rather than the individual patient working with the individual care provider. Third, individual patients are primarily concerned with the nature and quality of their own care and clinical outcomes. They and their care providers are not primarily seeking to generalise beyond the unique individual. We propose that the gold standard for helping individual patients with chronic conditions should be longitudinal factorial design of trials with individual patients. Understanding how these three groups deal differently with variation can help appreciate these three approaches