Secular instability in rapidly rotating Neutron star

The stability of Maclaurin spheroid is of great interests in the study of the shape of celestial objects and has been extensively studied over the past decades. It has been well established that an initially rapidly rotating axis-symmetric Maclaurin spheroid with |T/W|> 0.27, where T and W denote the rotational and gravitaional potential energy of the object respectively, undergoes dynamical instability toward m=2 fundamental mode(barmode), so the initially axisymmetric structure will be distorted into a bar-like shape as time goes on, on the other hand, the bar mode can also grow in another kind of instability, namely, secular instability which occurs at |T/W|~0.14 while there exists some kind of energy dissipative mechanism such as viscosity or gravitaional radiation. The similiar instabilities stand in the cases of compressible polytropes with n > 0.
In the recent developments on gravitational wave theory, large effort is applied on the normal mode analysis of neutron stars in order to work out the wave form template that can be used by LIGO to detect gravitational wave. We are specially interested in the effect of gravitational radiation on the fundamental \($$m=2$$\) mode: what kind of role does the gravitational radiation play in the evolution of a rapidly rotating neutron star? Will it drives the initially axisymmetric star toward the secular instability as predicted by theory? Here, we performed fully 3D simulations in purpose of following the secular instability development of \($$n=0.5$$\) neutron stars and calculate the actual gravitatinal wave from this scenario.
Movies for the f-mode simulation performed on LSU's Beuwolf cluster Super Mike
Evolution Stage Resolution # of processors computer plots movies comments
Non-rotating nuetron star 162*322*128 128 Pentium 4 LSU Super Mike diagnostic data maya movies The m=2 f-mode decays in the non-rotating neutron star
Non-rotating nuetron star 162*322*128 128 Pentium 4 LSU Super Mike diagnostic data maya movies We flipped the sign of the radiation force to see a growing f-mode in a non-rotating neutron star.
rotating nuetron star 162*322*128 128 Pentium 4 LSU Super Mike diagnostic data maya movies The m=2 f-mode should become unstable in the rotating neutron star having |T/W|=0.154, but in our simulation the mode is actually decaying, we concluded that it is not the correct mode we want to kick, it looks to be a plus wave, by CFS instability criterion, only negative mode which is prograding in inertial frame and retrograding in the corotating frame is unstable to GR.
quadrupole radiation wave form , note that the simulation is done in inertial frame.
initial model without perturbation and radiation force.
rotating nuetron star 162*322*128 128 Pentium 4 LSU Super Mike diagnostic data maya movies We kicked a mode which we thought is a negative mode prograding in inertial frame, but this can only be proved if it grows as simulation goes on,
quadrupole radiation wave form , note that the simulation is done in rotating frame.
initial model without perturbation and radiation force.
non-rotating nuetron star 162*322*128 128 Pentium 4 LSU Super Mike diagnostic data maya movies After discussion with Lee, we modified the perturbation routine in the hope that it kicks a pure forward moving mode(minus sign in the v_phi component, the mode is moving clockwise in the movie because of the rendering mismatch), but the current curve still shows a mixture of two modes.
non-rotating nuetron star 162*322*128 128 Pentium 4 LSU Super Mike diagnostic data none The result of this run shows that the initial model is in good equilibrium, the measured D22 amplitude is in the order of 10^-18.
nonrotating star with growing plus wave 162*322*128 Pentium 4 LSU Super Mike diagnostic plot, maya movie In this simulation, we got plus wave by flipping the sign of vphi, the star exploded after some time .
initial perturbations in the nonrotating case 162*322*128 Pentium 4 LSU Super Mike density perturbation, plus vel perturbation, minus vel perturbation The density plot shows the equatorial density perturbations, red is high positive region, black is negative region. The plus wave velocity is yielded by simply flipping the sign of vphi of minus wave.
initial perturbations in the nonrotating case 162*322*128 Pentium 4 LSU Super Mike density perturbation, new plus vel perturbation, The velocity patterns of plus and minus waves in the above row look quite different, I flipped the signs of all three vel components and got new flow pattern for plus wave which is closer to that of minus wave.
nonrotating star with decaying plus wave 162*322*128 Pentium 4 LSU Super Mike diagnostic plot, maya movie , vel field In this simulation, we start with a plus wave which should decay as time goes on, the star looks to be stable now.
nonrotating star with decaying minus wave 162*322*128 Pentium 4 LSU Super Mike diagnostic plot, maya movie , velocity field In this simulation, we flipped the sign of \delta \rho, the density movie and velocity movie suggest it is a minus wave.
nonrotating star with growing minus wave 162*322*128 Pentium 4 LSU Super Mike diagnostic plot, maya movie , velocity field In this simulation, we flipped the sign of the radiation force so that the mode is acutally growing

Some stuffs about the initial model:

As shown in literature, the rigidly rotating polytrope can't has |T/W|>0.14 when n > 0.808, to verify this, I created series of initial models from spherical case to flatter model for n=0.5, 1.0, 1.5, only n = 0.5 case yields models with |T/W|>0.14. Here are the plots.