Figure 1:
2D surface of equilibrium models for EOS L. The surface is bounded by the nonrotating () and massshedding () limits and formed by constant and constant sequences (solid lines). The projection of these sequences in the – plane are shown as longdashed lines. Also shown are the axisymmetric instability sequence (shortdashed line). The projection of the 2D surface in the – plane shows an overlapping (see dotted lines). (Figure 7 of Stergioulas and Friedman [294].) 

Figure 2:
Isoenergy density lines of a differentially rotating protoneutron star at the massshedding limit, of rest mass . (Figure 5a of Goussard, Haensel and Zdunik [135]; used with permission.) 

Figure 3:
Meridional plane cross section of a rapidly rotating strange star at the massshedding limit, obtained with a multidomain spectral code. The various lines are isocontours of the logenthalpy H, as defined in [133]. Solid lines indicate a positive value of H and dashed lines a negative value (vacuum). The thick solid line denotes the stellar surface. The thick dotdashed line denotes the boundary between the two computational domains. (Figure 4 of Gourgoulhon, Haensel, Livine, Paluch, Bonazzola, and Marck [133]; used with permission.) 

Figure 4:
Apparent intersection (due to avoided crossing) of the axisymmetric first quasiradial overtone () and the first overtone of the pmode (in the Cowling approximation). Frequencies are normalized by , where is the central energy density of the star. The rotational frequency at the massshedding limit is 0.597 (in the above units). Along continuous sequences of computed frequencies, mode eigenfunctions are exchanged at the avoided crossing. Defining quasinormal mode sequences by the shape of their eigenfunction, the sequence (filled boxes) appears to intersect with the sequence (triangle), but each sequence shows a discontinuity, when the region of apparent intersection is well resolved. (Figure 3 of Yoshida and Eriguchi [330]; used with permission.) 

Figure 5:
Frequencies of several axisymmetric modes along a sequence of rapidly rotating relativistic polytropes of , in the Cowling approximation. On the horizontal axis, the angular velocity of each model is scaled to the angular velocity of the model at the massshedding limit. Lower order modes are weakly affected by rapid rotation, while higher order modes show apparent mode intersections. (Figure 10 of Font, Dimmelmeier, Gupta, and Stergioulas [104].) 

Figure 6:
Eigenfrequencies (in the Cowling approximation) of f modes along a sequence of models, constructed with the WFF3NV EOS. The vertical line corresponds to the frequency of rotation of the model at the massshedding limit of the sequence. (Figure 1 of Yoshida and Eriguchi [329]; used with permission.) 

Figure 7:
The neutral f mode sequences for EOS A. Shown are the ratio of rotational to gravitational energy (upper panel) and the ratio of the critical angular velocity to the angular velocity at the massshedding limit for uniform rotation (lower panel) as a function of gravitational mass. The solid curves are the neutral mode sequences for , and (from top to bottom), while the dashed curve in the upper panel corresponds to the massshedding limit for uniform rotation. The f mode becomes CFSunstable even at 85% of the massshedding limit, for models constructed with this EOS. (Figure 2 of Morsink, Stergioulas, and Blattning [230].) 

Figure 8:
Eigenfrequencies (in the Cowling approximation) of the mode as a function of the parameter for three different sequences of differentially rotating neutron stars (the line corresponding to uniform rotation). The filled circle indicates the neutral stability point of a uniformly rotating star computed in full general relativity (Stergioulas and Friedman [295]). Differential rotation shifts the neutral point to higher rotation rates. (Figure 1 of Yoshida, Rezzolla, Karino, and Eriguchi [333]; used with permission.) 

Figure 9:
The rmode instability window for a strange star of and R =10 km (solid line). Dashed curves show the corresponding instability windows for normal npe fluid and neutron stars with a crust. The instability window is compared to i) the inferred spinperiods for accreting stars in LMBXs [shaded box], and ii) the fastest known millisecond pulsars (for which observational upper limits on the temperature are available) [horizontal lines]. (Figure 1 of Andersson, Jones, and Kokkotas [11]; used with permission.) 

Figure 10:
Relativistic rmode frequencies for a range of the compactness ratio . The coupling of polar and axial terms, even in the order slow rotation approximation has a dramatic impact on the continuous frequency bands (shaded areas), allowing the rmode to exist even in highly compact stars. The Newtonian value of the rmode frequency is plotted as a dasheddotted line. (Figure 3 of Ruoff, Stavridis, and Kokkotas [257]; used with permission.) 

Figure 11:
Projected trajectories of several fiducial fluid elements (as seen in the corotating frame) for an Newtonian rmode. All of the fluid elements are initially positioned on the meridian at different latitudes (indicated with stars). Blue dots indicate the position of the fluid elements after each full oscillation period. The rmode induces a kinematical, differential drift. (Figure 2c of Rezzolla, Lamb, Marković, and Shapiro [252]; used with permission.) 

Figure 12:
Evolution of the axial velocity in the equatorial plane for a relativistic rmode in a rapidly rotating polytrope (in the Cowling approximation). Since the initial data used to excite the mode are not exact, the evolution is a superposition of (mainly) the rmode and several inertial modes. The amplitude of the oscillation decreases due to numerical (finitedifferencing) viscosity of the code. A beating between the rmode and another inertial mode can also be seen. (Figure 2 of Stergioulas and Font [293].) 

Figure 13:
Time evolution of the rotational velocity profile for a stationary, rapidly rotating relativistic star (in the Cowling approximation), using the 3rd order PPM scheme and a 116^{3} grid. The initial rotational profile is preserved to a high degree of accuracy, even after 20 rotational periods. (Figure 1 of Stergioulas and Font [293].) 

Figure 14:
Movie Simulation of a stationary, rapidly rotating neutron star model in full general relativity, for 3 rotational periods (shown are isodensity contours, in dimensionless units). The stationary shape is well preserved at a resolution of 129^{3}. Simulation by Font, Goodale, Iyer, Miller, Rezzolla, Seidel, Stergioulas, Suen, and Tobias. Visualization by W. Benger and L. Rezzolla at the Albert Einstein Institute, Golm [1]. 

Figure 15:
Density contours and velocity flow for a neutron star model that has developed spiral arms, due to the dynamical barmode instability. The computation was done in full General Relativity. (Figure 4 of Shibata, Baumgarte, and Shapiro [273]; used with permission). 

Figure 16:
Movie Simulation of the development of the dynamical barmode instability in a rapidly rotating relativistic star. Spiral arms form within a few rotational periods. The different colors correspond to different values of the density, while the computation was done in full general relativity. Movie produced at the University of Illinois by T.W. Baumgarte, S.L. Shapiro, and M. Shibata, with the assistance of the Illinois Undergraduate Research Team [31]; used with permission. 

Figure 17:
Movie Gravitational wave emission during the development of the dynamical barmode instability in a rapidly rotating relativistic star. The gravitational wave amplitude in a plane containing the rotation axis is shown. At large distances, the waves assume a quadrupolelike angular dependence. Movie produced at the University of Illinois by T.W. Baumgarte, S.L. Shapiro, and M. Shibata, with the assistance of the Illinois Undergraduate Research Team [31]; used with permission. 

Figure 18:
The first fully relativistic, quasiradial pulsation frequencies for a sequence of rapidly rotating stars (solid lines). The frequencies of the fundamental mode (filled squares) and of the first overtone (filled circles) are obtained through coupled hydrodynamical and spacetime evolutions. The corresponding frequencies obtained from computations in the relativistic Cowling approximation [104] are shown as dashed lines. (Figure 16 of Font, Goodale, Iyer, Miller, Rezzolla, Seidel, Stergioulas, Suen, and Tobias [105].) 
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