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R-modes and dense matter physics

EU Network discussion meeting, Meudon, France

7th-8th December, 2001


One of the main Networks projects of the Southampton Group concerns the dynamics of superfluid neutron stars. During the meeting Nils Andersson and Reinhard Prix sketched the current status of this effort. The main aim is to study instabilities in superfluid stars and understand the many complex issues that are involved, eg the role of the so-called mutual friction that supposedly suppresses the CFS instability in the f-modes completely. However, in order to be able to address this problem one must first i) develop a suitable framework, ii) construct stationary rotating models allowing for the possibility that the superfluid component(s) rotate at a rate different from that of the charged components, iii) understand the oscillations of a nonrotating superfluid star, and (finally) iv) study the inertial modes of the star.

The group has recently made significant progress on points ii and iii, and Reinhard described very recent results for a Newtonian model including the so-called entrainment effect. The results showed the two classes of co- and countermoving waves that are characteristic of the superfluid problem. There was also evidence of mode crossings as the entrainment parameter was varied. This result seemed to contradict a similar calculation in relativity (by Nils, Greg Comer and David Langlois) which shows so-called avoided crossings. A long debate followed... The general feeling of the meeting was that one would probably expect avoided crossings but since there is no proof that actual crossings cannot occur in this system everybody involved agreed to go back to the drawing board and try to understand if there was, indeed, a reason why the Newtonian modes should disagree with the relativistic ones.

Do the r-modes exist?

During the last year or so, it has become increasingly clear that we do not yet fully understand the slow-rotation problem for r-modes in general relativity. The confusion stems from what is known to the afficionados as the "Kojima equation", an equation that governs a purely axial perturbation of a slowly rotating star at first order of rotation. It has been proved by Lockitch, Andersson and Friedman that this equation is only relevant for non-barotropic stars (where the perturbation is described by an equation of state different from that of the background), and also that the equation leads to sensible r-mode results for uniform density stars. However, once one turns to polytropes or realistic equations of state one runs into difficulties. For a certain range of parameters the eigenvalue problem becomes singular, and the question is how the occurence of this singularity should be dealt with and/or interpreted. Johannes Ruoff presented recent results on this problem. The Thessaloniki group are pushing hard to provide reliable time-evolutions of the relativistic perturbation equations for a rotating star, and serious progress has been made. But an understanding of the fate of the r-modes still seems elusive. From a physical point of view, one can argue that the situation is similar to that of co-rotation in a differentially rotating Newtonian star. This problem is regularised by even a small amount of viscosity. Hence, it is generally felt that a similar thing will happen in the relativistic r-mode problem. For example, second order rotational effects may regularise the equations in a narrow region surrounding the original singularity. However, this has not yet been demonstrated and a reliable method for determining the relativistic modes remains outstanding. Johannes showed some recent time-evolution results that seemed promising, albeit very complex. There should be a lot of progress in this area in the near future.

Horst Beyer discussed a mathematical approach to the problem. He was interested in the nature of the spectrum of purely axial relativistic slow-rotation modes. Using powerful theorems from spectral theory, he has approached the perturbation equations. The results typically indicate the presence of a continuous spectrum. At the end of the discussion Horst indicated that the analysis leads to worrying conclusions regarding the reliability of the slow-rotation approximation in relativity. It seemed as if the approximation will not allow a complete statement regarding the nature of the spectrum without invoking higher order (in rotation) terms. This may be a sign that the slow-rotation expansion is not consistent.

Numerical time-evolutions

There are currently two Network projects aimed at understanding various features of the r-mode instability via Newtonian time-evolutions of the perturbation equations.

Loic Villain presented recent results from the Meudon group. These illustrated nicely the advantage of approaching the problem at the linear level. The numerical code, which uses the spectral method for the spatial problem and finite differencing in time, is very stable and evolves the modes for hundreds of oscillations. The current incarnation of the code is based on the so-called anelastic approximation, includes current multipole radiation reaction and also allows for differential rotation in the background star. The results show how the r-mode "saturates" by concentrating the velocity perturbation in the equatorial region in the case of differential rotation. There was some discussion of singularities that seemed to form during the evolution. These occured near the surface of the star, and a possible interpretation is that there is a "boundary layer" (the origin of which was not understood) that needs to be analysed. It was suggested that the code could be tested with alternative surface boundary conditions.

Ian Jones described the Southampton project, the current status of which is that the numerical code (which allows for rotation up to the break-up velocity) nicely reproduces all known mode-results for simple stellar models. He outlined the next steps that are planned, the main one of which concerns the implementation of a local radiation reaction force. This issue had been discussed during a visit by Guillaume Faye to Southampton in the previous week, and Ian sketched the conclusions of those discussions. Basically, all the required PN expressions for both mass- and current multipoles are available, and it is only a matter of implementing them and testing the code. This is of course far from trivial, but it seemed as if this effort is going well.

Silvano Bonazzola discussed (in his unimitable way) another facet of the Meudon project. He focussed on the nonlinear effects that may lead to r-mode saturation. Of particular relevance here is non-linear coupling that leads to velocity perturbations of arbitrarily small scales, and the associated turbulent viscosity. Given the ease with which one can isolate the small length-scales in a spectral code, Silvano had introduced what he called "evanescent viscosity" to damp out the relevant perturbations. The numerical tests seem to indicate that this leads to a channel for saturating an unstable mode at much lower amplitudes than fully 3D hydrosimulations seem to suggest. The "reliability" of the recent 3D simulations of Tohline et al was discussed, and there was a general feeling that much more work was needed before we can claim to understand this problem well.

Covariant hydrodynamics

In a brief presentation, Brandon Carter told the meeting about his recent insights into Newtonian hydrodynamics. He argued that it was natural, and often much easier, to do Newtonian calculations using a fully covariant framework. This idea may seem obvious to a relativist, but it is typically not at all appreciated by a fluid dynamicist. Brandon outlined the theory and showed how a theorem for the conservation of vorticity followed in a very simple way. In fact, the covariant calculation also provided a conservation law that may not be known from the standard description. It was commented that the covariant description provides an ideal framework since it allows a straightforward comparison between Newtonian equations and relativistic ones. This could be particularly important for the understanding of oscillations of stars, where one would often like to show that a relativistic study reproduces familiar Newtonian results.

Bulk viscosity in dense matter

Pawel Haensel provided a useful introduction to bulk viscosity in dense matter. He outlined the necessary steps needed to estimate the viscosity coefficients (eg the relaxation rates associated with the relevant nuclear reactions), and gave an overview of recent work on bulk viscosity in stars with superfluid components. Superfluidity suppresses the reactions and so leads to a significantly weaker bulk viscosity. Pawel generally argued that one had to be very careful before drawing strong conclusions in this game - the uncertainties are simply too great. He suggested that people should base their models on relatively simple approximate formulas from their series of papers, in which physical quantities (like the superfluid transition temperature) appear as free parameters. He said that there are several reasons why one should be sceptical about results from relativistic mean-field calculations, because these models (as used by for example Glendenning) still do not reproduce nuclear scattering data very well. It was noted that all current indications for the presence of eg hyperons come from mean-field models. A considerable discussion concerned the viscosity due to the presence of hyperons. It has recently been argued by Lindblom and Owen that this provides a mechanism that can wipe out the r-mode instability in most astrophysical scenarios. It seemed to the meeting that this conclusion was a little bit too severe, and Nils Andersson sketched some recent thoughts on this issue that had come out of discussions with John Miller and Kostas Kokkotas.

In order to emphasize the relevance of the bulk viscosity, Ian Jones briefly presented recent work on strange stars. This study shows nicely how, essentially because the bulk viscosity is much stronger than in a simple neutron star model, an unstable r-mode may evolve in a distinct way. In fact, this could provide a persistent source of gravitational waves that may be observable with LIGO.

PN radiation reaction

In a short, but interesting talk, Luc Blanchet outlined a general PN expression for radiation reaction, including all PN orders. This formal result is enlightening because it shows the nature of radiation reaction, and can be used as an indicator of when non-local effects will seriously affect the calculations.

Nils Andersson

This work has been supported by the EU Programme 'Improving the Human Research Potential and the Socio-Economic Knowledge Base' (Research Training Network Contract HPRN-CT-2000-00137).