5.2 Studies of stellar QNM excitation

The discussion in the previous sections has shown that we have an acceptable knowledge of stellar pulsations and that we can extract information from the detection of such oscillations. But as was clear from the discussion earlier in this section, the energy released as gravitational radiation during the stellar collapse is basically unknown. Additionally, it is not known how the energy is distributed in the various fluid and spacetime modes. Both uncertainties depend strongly on the details of gravitational collapse, or in general the mechanism that excites the modes. Unless full 3D general relativistic codes are generated for the gravitational collapse or the final stages of binary coalescence we will never be able to give definite answers to the above questions. In the meantime, perturbation theory is a reliable way to get some first hints and indications. A survey of the literature reveals several indications that the pulsation modes are present in the gravitational wave signals from coalescing stars. Waveforms obtained by Nakamura and Oohara [155] show clear mode presence. Ruffert et al. [175] have also obtained gravitational wave signals from coalescing stars that show late-time oscillations. Their waveforms and spectra show oscillations at frequencies between 1.5 and 2 kHz.

The situation is similar for rotating core collapse. Most available studies use Newtonian hydrodynamics and account for gravitational wave emission through the quadrupole formula. The collapse of a non-rotating star is expected to bounce at nuclear densities, but if the star is rotating the collapse can also bounce at subnuclear densities because of the centrifugal force. In each case, the emerging gravitational waves are dominated by a burst associated with the bounce. But the waves that follow a centrifugal bounce can also show large amplitude oscillations that may be associated with pulsations in the collapsed core. Such results have been obtained by Mönchmeyer et al. [151]. Some of their models show the presence of modes with different angular dependence superimposed. Typically, these oscillations have a period of a few ms and damp out in 20 ms. The calculations also show that the energy in the higher multipoles is roughly three orders of magnitude smaller than that of the quadrupole. More recent simulations by Yamada and Sato [210] and Zwerger and Müller [215] also show post-bounce oscillations. The cited examples are encouraging, and it seems reasonable that besides the fluid modes the spacetime modes should also be excited in a generic case. To show that this is the case one must incorporate general relativity in the simulations of collapse and coalescence. As yet there have been few attempts to do this, but an interesting example is provided by the core collapse studies of Seidel et al. [187Jump To The Next Citation Point186182183]. They considered axial (odd parity) or polar (even parity) perturbations of a time-dependent background (that evolves according to a specified collapse scenario). The extracted gravitational waves are dominated by a sharp burst associated with the bounce at nuclear densities. But there are also features that may be related to the fluid and the w-modes. Especially for the axial case, since there are no axial fluid modes for a non-rotating star, it is plausible that this mode corresponds to one of the axial w-modes of the core. Furthermore, the power spectrum for one of the simulations discussed in [187] (cf. their Fig. 2) shows some enhancement around 7 kHz.

Recently, Andersson and Kokkotas [18Jump To The Next Citation Point] have studied the excitation of axial modes by sending gravitational wave pulses to hit the star (see Figure 5View Image). The results of this study are encouraging because they provide the first indication that the w-modes can be excited in a dynamical scenario. Similar results have recently been obtained by Borelli [48] for particles falling onto a neutron star. The excitation of the axial QNMs by test particles scattered by a neutron star have been recently studied in detail [203Jump To The Next Citation Point84Jump To The Next Citation Point28Jump To The Next Citation Point] and some interesting conclusions can be drawn. For example, that the degree of excitation depends on the details of the particle’s orbit, or that in the cases that we have strong excitation of the quadrapole oscillations there is a comparable excitation of higher multipole modes.

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Figure 5: Time evolution of axial perturbations of a neutron star, here only axial w-modes are excited. It is apparent in panel D that the late time behavior is dominated by a time tail. In the left panels (A and C) the star is ultra compact (M ∕R = 0.44) and one can see not only the curvature modes but also the trapped modes which damp out much slower. The stellar model for the panels (B and D) is a typical neutron star (M ∕R = 0.2) and we can see only the first curvature mode being excited.

These results have prompted the study of an astrophysically more relevant problem, the excitation of even parity (or polar) stellar oscillations. In recent work Allen et al. [6Jump To The Next Citation Point] have studied the excitation of the polar modes using two sets of initial data. First, as previously for the axial case, they excited the modes by an incoming gravitational wave pulse. In the second case, the initial data described an initial deformation of both the fluid and the spacetime. In the first case the picture was similar to that of the axial modes i.e. the star was excited and emitted gravitational waves in both spacetime and the fluid modes. Nearly all of the energy was radiated away in the w-modes.

This should be expected since the incoming pulse does not have enough time to couple to the fluid (which has a much lower speed of propagation of information). The infalling gravitational waves are simply affected by the spacetime curvature associated with the star, and the outgoing radiation contains an unmistakable w-mode signature. In the second case one has considerable freedom in choosing the degree of initial excitation of the fluid and the spacetime. In the study the choice was some “plausible” initial data inspired by the treatment of the problem in the Newtonian limit. Then the energy emitted as gravitational waves was more evenly shared between the fluid and the spacetime modes, and one could see the f, p, and w modes in the signal. The characteristic signal was (as expected) a short burst (w-modes) followed by a slowly damped sinusoidal wave (fluid modes).

The previous picture emerged as well in a recent study in the close limit of head-on collision of two neutron stars [5Jump To The Next Citation Point]. There the excitation of both fluid and spacetime modes is apparent but most of the energy is radiated via the fluid modes. These new results show the importance of general relativity for the calculation of the energy emission in violent processes. Up to now the general way of calculating the energy emission has been the quadrupole formula. But this formula only accounts for the fluid deformations and motions, and as we have seen this accounts only for a part of the total energy.

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Figure 6: Excitation of polar modes due to an initial deformation of the star. The initial burst which is dominated mainly by the first w-mode is followed by a sinusoidal waveform dominated by the f and the first p-mode. In the upper panel the actual waveform is shown, while in the lower panel is its power spectrum. The wide peak in the power spectrum corresponds to the w-mode, while the sharp peaks correspond to the various fluid modes.

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