3. Gravitational wave signature
The non-radial motions and the resulting time-dependence and asphericities of the density stratification due to convection in the unstable layers both inside the proto-neutron star and in the neutrino-heated hot-bubble will produce gravitational radiation. Thus, we have analysed the two- and three-dimensional models discussed in the previous section and have computed the amount and the signature of gravitational radiation that can be expected from convection in Type II supernovae. Moreover, the density and temperature variations in the neutrino-decoupling region (see Fig. 7) and the fast convective overturn below the neutrinosphere lead to anisotropies of the neutrino emission which will also be a source of gravitational waves (Epstein 1978, Turner 1978).
The gravitational wave signal was calculated using a post-Newtonian approach, where numerically troublesome higher order time derivatives of the quadrupole moment are transformed into much better tractable spatial derivatives. In particular, for the gravitational quadrupole radiation field, , we used an expression derived independently by Nakamura & Oohara (1989), and by Blanchet et al. (1990):
where is the distance between the observer and the source, is the Newtonian gravitational potential, is the mass-density and is the velocity. The other quantities have their usual meaning except for (with ) which denotes the transverse-traceless (TT) projection operator onto the plane orthogonal to the outgoing wave direction , acting on symmetrical Cartesian tensors according to
represents the partial derivative with respect to the coordinate. The integrand in Eq. ( 4 ) is defined on a compact manifold and is known to the (2nd order) accuracy level of the numerical algorithm of the hydro-code. Eq. ( 4 ) can be shown to be equivalent to the standard representation
It can easily be shown that evaluating the integral of Eq. ( 4 ) by an integration scheme (of at least 2nd order) is by one order of accuracy superior to twice applying numerical time-differentiation methods to quadrupole data given at discrete points of time (see Mönchmeyer et al. 1991).
3.1.1. Evaluation in two dimensions
The gravitational radiation field gives direct information about the second time derivative of the mass-quadrupole tensor (see Eq. ( 7 )). In case of axisymmetry the quadrupole moment, Q, is the only independent component of the quadrupole tensor. Its relation to the Cartesian components of the radiative mass-quadrupole tensor is
As all our simulations were performed in spherical coordinates, it is natural to represent the (total) radiation field in terms of the "pure-spin tensor harmonics" and with amplitudes and in the following way (Thorne 1980):
For the definitions of , , , and , see Thorne (1980). In spherical coordinates the coefficients and have especially simple integral representations over the source. By symmetry, there is only one nonvanishing quadrupole term in Eq. ( 9 ), namely . Higher-order terms are neglected in the quadrupole approximation of the gravitational radiation field . Transforming Eq. ( 4 ) to spherical coordinates and expressing in terms of unit vectors in the and direction, one obtains by comparison of Eq. ( 4 ) with the lowest-order term of Eq. ( 9 ) for the quadrupole wave amplitude the expression
where , and .
From the definition of (Thorne 1980, Eq. (2.39e)) one derives for the components of the formula
The total energy radiated per unit frequency will be denoted by and was calculated by interpolating the data for the amplitudes that are given at non-equidistant times onto an equidistant temporal grid and using the fast Fourier transform technique with a rectangular window function.
3.1.2. Evaluation in three dimensions
If the source is of genuine three-dimensional nature, it is common to express the gravitational quadrupole radiation field, , in the following tensorial form (see, e.g., Misner et al. 1973)
and being the unit polarization vectors in and -direction of a spherical coordinate system and the tensor product. The amplitudes and represent the only two independent modes of polarization in the TT gauge, and are given by the following expressions for
3.1.3. Gravitational waves from neutrino emission
In order to estimate the gravitational wave signal associated with the anisotropic emission of neutrinos, we use Eq. (16) of Epstein (1978) in the limit of a very distant source, . In addition, we make use of the approximation that the gravitational wave signal measured by an observer at time t is caused only by radiation emitted at time t' = t - R/c . Hence, we take , which means that only a neutrino pulse itself is assumed to cause a gravitational wave signal but memory effects after the pulse has passed the observer are disregarded. With these simplifications, one gets for the dimensionless gravitational wave amplitude
with being the angle between the direction towards the observer and the direction of the radiation emission, and denoting the direction dependent neutrino luminosity, i.e., the energy radiated at time t per unit of time and per unit of solid angle into direction . The angular integral over the radiation source is performed over and thus over all angles and which specify the (beam) direction in the source coordinate frame (x',y',z') that we identify with the coordinate frame used for the hydrodynamical simulations. With the angles and defining the radiation direction in the observer's frame (x,y,z) (where the observer is located at distance R along the z -axis) one has , and the gravitational wave amplitude is given by
Replacing by in Eq. ( 25 ) yields .
In Eq. ( 25 ) and need to be expressed in terms of the angles , , , and when and define the orientation of the observer's coordinate system (observer located in z -direction and y -axis lying in the x' - y' -plane) relative to the source coordinate frame. Choosing the y - and y' -axes coincide and the expressions become rather simple. For the z -axis and the z' -axis are identical, too, and the observer is situated along the system's z' - (polar) axis. In that case one obtains
When is axially symmetrical, the gravitational wave signal for an observer on the symmetry axis vanishes and . For the observer is positioned perpendicular to the source's z' -axis in the equatorial plane (z -axis and x' -axis coincide) and becomes
Using the dimensionless gravitational wave amplitude of Eq. ( 29 ) and taking into account that there is only one non-zero component of the quadrupole amplitude, Eqs. ( 11 ) and ( 12 ) allow one to estimate the total energy E(t) that is associated with the gravitational waves produced by the anisotropic emission of neutrinos in the two-dimensional (axially symmetrical) case until time t:
is a numerical factor of order unity, typically around 0.5. Employing Eq. ( 13 ) and Eqs. ( 16 )-( 23 ) and assuming that all amplitudes contribute roughly equally, one finds that Eq. ( 31 ) also holds for the three-dimensional situation but with a slightly different value of .
3.2. Two-dimensional results
The results of our gravitational wave analysis for four different models are displayed in Figs. 18 to 21 which show the quadrupole wave amplitudes, the quadrupole amplitude spectra, the spectral energy densities, and the energies radiated in form of gravitational waves, respectively. One first notices that the signal forms and signal strengths as well as the spectral distributions of the gravitational wave energy depend on whether the gravitational radiation is produced by convection inside the proto-neutron star or by convection in the hot-bubble region. Generally speaking, in case of convective overturn processes in the proto-neutron star, the maximum gravitational wave amplitude is significantly larger ( cm instead of cm), the spectral energy distribution is peaked at higher frequencies (at 500 Hz to 1000 Hz instead of at about 100 Hz), and much more energy is radiated in form of gravitational waves ( instead of ) as compared to multi-dimensional processes in the hot-bubble region.
In model HB2D, which refers to the two-dimensional simulation of convection inside the proto-neutron star starting with Hillebrandt's configuration of a post-bounce stellar core, the strongly unstable lepton and entropy gradients give rise to large convective velocities which exceed cm/s and are hence transonic. As the unstable layer involves a relatively large amount of mass ( ) the angular kinetic energy of the convective flow is substantial ( erg). Moreover, because of the compact and quite massive collapsed stellar core ( ) in model HB2D the non-radial flow takes place deep in a strong gravitational potential with convective elements undershooting the stable layer down to a radius of about 15 km. Consequently, this model produces a relatively strong gravitational wave signal with a maximum amplitude of cm (Fig. 18) and a total emitted gravitational wave energy of (Fig. 21). Corresponding to the characteristic overturn timescales of the convective eddies of about 2-10 ms, the frequency spectrum of the quadrupole amplitude has a broad maximum at about 100-500 Hz (Fig. 19) and the energy spectrum shows most power being radiated between 100 and roughly 1000 Hz (Fig. 20).
As discussed in Sect. 2.2, convection inside the proto-neutron star is much weaker in those of our models which we particularly used to study convection in the hot-bubble region. There are three reasons for this. Firstly, only the outer part of the convective zone in the proto-neutron star is included in the simulations, i.e., only about of the roughly that are unstable against convection. Secondly, the lepton and entropy gradients in Bruenn's initial model are less unstable than those in Hillebrandt's collapsed stellar core. Thirdly, due to the smaller core mass of only about the gravitational potential is weaker in Bruenn's model and there is less mass in the unstable layer around and below the neutrinosphere. In the hot-bubble simulations the convective velocities inside the proto-neutron star are therefore significantly smaller, only about cm/s, and, correspondingly, the angular kinetic energies reach only about - erg at a time when the convection inside the proto-neutron star is fully developed (at ms, compare Fig. 9 in Janka & Müller 1996). Thus, the resulting gravitational wave signal from the convective overturn inside the proto-neutron star is very weak in these models as compared to model HB2D. In fact, in the models where we followed the neutrino effects in the hot-bubble region, the gravitational wave emission is strongly dominated by the waves produced by mass motions in the neutrino-heated layer between proto-neutron star and supernova shock.
We have analysed three such models, namely models ML2D, LP2D, and EP2D (equivalent to models T2c, T3c, and T4c, respectively, in Janka & Müller 1996), whose gravitational wave signature we discuss in the following. We point out here that the actual gravitational wave signal of these models will be considerably larger if the whole convection zone inside the proto-neutron star, which yields only a minor contribution to the gravitational wave emission in the presented models, is included in the simulations. Models ML2D, LP2D, and EP2D are nevertheless useful to investigate especially the characteristics of the gravitational waves originating from turbulent motions in the hot-bubble region. We find that the structure of the produced waves contains detailed information about duration, strength, and pattern of the accretion and convection processes behind the supernova shock. When these dynamical processes have direct influence on the neutrino emission, e.g., when matter accreted onto the neutron star produces additional neutrino emission, we notice correlations of the neutrino luminosity with the gravitational wave amplitude and luminosity. Moreover, our models suggest that strength and duration of turbulent processes around the proto-neutron star are correlated with the size and temporal decay of the neutrino fluxes from the proto-neutron star and with the explosion energy of the supernova. For this reason we expect, and indeed observe, characteristic differences of the gravitational wave signature of models with different explosion dynamics and different explosion parameters.
In model ML2D the temporal modulations of the gravitational wave signal and the neutrino losses from the hot-bubble region show clear correlations (compare the upper right panel in Fig. 18 with Fig. 18 in Janka & Müller 1996). Neutrino emission and gravitational wave emission from the turbulent layers around the proto-neutron star in this model are produced by downflows of cold material from the postshock region. These narrow downflows reach very high velocities of more than twice the speed of sound and are abruptly decelerated at a radius of about 80 km (see Figs. 15 and 16 in Janka & Müller 1996). The peaks and characteristic features in the neutrino emission are associated with the dissipation of kinetic energy of the gas in the surroundings of the proto-neutron star. The aspherical, dynamical gas motions also act as source of gravitational waves. The large variations of the gravitational wave amplitude around -140 ms (Fig. 18) are directly correlated with spikes in the lepton number loss/gain rate and the energy loss/gain rate of the stellar gas at the same epoch (Fig. 18 in Janka & Müller 1996). Similarly, the variation of the gravitational wave amplitude between 150 ms and 200 ms is mirrored by a correlated activity in the lepton and energy loss/gain rates. Actually, one can observe a slight time-lag of the neutrino emission. This shift can be explained by the fact that the gravitational wave emission traces the dynamical infall of the downflows, whereas the neutrino emission peaks at the moment when the infalling gas reaches highest temperatures and densities, i.e., typically at the moment when it is decelerated and strongly compressed near the proto-neutron star. The characteristic timescale of gas motions and convective overturn in the hot-bubble region is of the order of several ten to about 100 ms. The frequency spectrum of the quadrupole amplitude therefore peaks at frequencies of about 10 to roughly 100 Hz (Fig. 19), and the energy spectrum has a maximum between 50 and 200 Hz with significant power in the frequency range from 10 to 400 Hz (Fig. 20).
The gravitational wave signal of model LP2D is characterized by the low-frequency emission from the aspherical expansion of postshock material and by the superimposed high-frequency modes that are produced by small-scale convective processes in the neutrino heated hot-bubble region. The prolate, large-scale deformation of the supernova shock leads to a time-dependent mass quadrupole moment that varies within a typical time of about 100 ms. While the effect of the expansion of supernova shock and ejecta becomes prominent in the gravitational wave signature at times later than about 100 ms after the start of the simulation, the overturn of neutrino-heated material determines the wave amplitude during the first 100 ms where signal variations on timescales of 10-20 ms are visible. The gravitational wave amplitude reaches a maximum value of cm (Fig. 18) and has typical frequencies between about 10 and 100 Hz (Fig. 19). The energy spectrum peaks at about 50-70 Hz ((Fig. 20) and the total energy radiated in gravitational waves is which is almost a factor of 20 smaller than in model ML2D where it is within a similar time of 200-300 ms after supernova shock formation. While the major part of the gravitational wave emission of model ML2D is produced by anisotropic, dynamical downflows of matter from the postshock region to the proto-neutron star, roughly half of the gravitational wave energy of model LP2D results from overturn motions of neutrino-heated gas and the other half from the large-scale expansion.
In model EP2D the explosion happens faster (due to higher core neutrino fluxes) and the convective overturn in the neutrino-heated region is correspondingly shorter. The contribution of the large-scale deformation and expansion of the postshock region to the gravitational wave signal therefore clearly dominates the wave amplitude after about 100 ms (Fig. 18). On a timescale of hundreds of milliseconds the wave amplitude exhibits the slow variation associated with the change of the mass quadrupole moment due to the global dynamical evolution of the explosion. However, it shows only little substructure of higher frequencies because the rapid expansion of model EP2D limits the duration of the phase of convective overturn in the hot-bubble region to about 100 ms. This can be verified in the quadrupole amplitude spectrum (Fig. 19) which confirms the clear dominance of low-frequency modes (around 1-10 Hz) and essentially no pronounced features at higher frequencies. The corresponding energy spectrum (Fig. 20) is very flat and spans the range of frequencies between about 1 Hz and roughly 100 Hz. Even stronger than in case of model LP2D, the global, asymmetrical expansion of model EP2D on a timescale of several 100 ms is reflected in the fact that significant, or even most, power is in a fundamental, low-frequency signal of a few Hz. Model EP2D emits about of gravitational wave energy which is about 50% less than model LP2D. In the lower right panels of Figs. 18-21 the thin solid lines correspond to the gravitational wave signal that originates from the incompletely represented (see above) convective region inside the proto-neutron star for model EP2D. A comparison of the thin and thick lines, the latter representing the total signal, shows that most of the emission is produced by the convection in the hot-bubble region. Only at early times after bounce ( ms, which is when the convective overturn in the neutrino-heated region is not yet fully developed) do convective motions around and inside the neutrinosphere contribute significantly to the total gravitational wave emission (see Fig. 18) at frequencies above 50 Hz (Figs. 19 and 20) and with an integrated energy of about . At later times the signal of convection in the proto-neutron star is minor and the results shown for models ML2D, LP2D, and EP2D in Figs. 18-21 do indeed primarily originate from the mass motions in the hot-bubble region and from the explosive expansion of the supernova.
Comparing the three models we see that with increasing neutrino flux (imposed as an inner boundary condition) and hence with increasing explosion energy the convective activity in the hot-bubble region changes from violent, long-lasting convective overturn associated with accretion processes (model ML2D) to rapid expansion and relatively slowly changing large-scale deformation (model LP2D) which eventually dominate the overall asymmetry and the quadrupole moment of the exploding star (model EP2D). This change of the characteristics of non-radial motions in the hot-bubble and postshock regions is directly reflected in the dominant frequencies of the gravitational wave signal, which drop from about 200 Hz (model ML2D) down to less than 10 Hz (model EP2D). Thus, a measurement of the frequency of the wave signal provides important insights into the explosion dynamics. Moreover, since the signal produced by the convection inside the proto-neutron star is typically of much higher frequency (500-1000 Hz), such a measurement would also allow to discriminate the contributions from both convection zones. Unfortunately, the calculated maximum dimensionless amplitudes (Eq. ( 11 )) are too small to be detected for a supernova outside our own Galaxy, because they lie in the range for a source at a distance of 10 kpc.
3.3. Three-dimensional results
The analysis of the three-dimensional model caused some problems because our simulation volume involved only a 60 by 60 degree sector of a full sphere. Simply extending the data from the computational volume to the whole sphere by making use of the periodic boundary conditions imposed in angular direction during the simulation did not make sense because a spherical potential was used in the simulation and because the resulting configuration was highly symmetrical in angular direction. The computed sector fits into the full sphere exactly three times in -direction and six times in -direction which gives rise to a 60, 120, and 180 degree rotational symmetry around the z -axis.
In order to compute the gravitational radiation we instead proceeded as follows. First we divided the simulated sector into 36 subsectors of 10 by 10 degree each. Then these 36 subsectors were randomly distributed over the full sphere. This process is justified, because the angular distribution of the dynamical variables varies on scales smaller than or, at least, not much larger than 10 degrees (see Sect. 2.1.2 and Fig. 12). However, one has to take into account that in -direction the linear extent of the subsectors decreases like when approaching the poles at and . We therefore mapped one 10 by 10 degree subsector as constructed in the simulated degree wedge over several pole-near subsectors such that the length scales of the structures were approximately conserved. This mapping procedure of the data was repeated for all radial shells and for all times while keeping the association of subsectors of the computational cone with their randomly chosen mapping locations on the sphere fixed.
The resulting angular distribution is displayed in Fig. 22 showing the density on a sphere of radius 65 km (which is right inside the convective layer in the proto-neutron star) at different moments of the evolution. Fig. 22 shows that the structures are indeed small enough and the mapping process is random enough to produce statistically homogeneous angular distributions. The result is so perfect that if we omitted the white solid line that marks the boundary of the computed sector, it would be impossible to locate the wedge of the simulation by eye inspection.
Fig. 22 further shows that the angular scale of the fluctuations is time-dependent and grows steadily with time from the onset of the instability at about 9 ms (upper left panel in Fig. 22). The level of the fluctuations reaches a maximum at about 13.6 ms (lower left panel in Fig. 22), when the ratio of maximum to minimum density is 2.32 at 65 km. Within the next 3 ms the ratio drops to a value of 1.72 (lower right panel in Fig. 22) which reflects the increasing homogenization of the mixing layer.
From the density distribution on the full sphere as constructed by the mapping procedure we computed the corresponding three-dimensional gravitational potential taking into account the central point mass (see Sect. 2.1). The three-dimensional Poisson solver employed in the calculation of the potential is an extension of the two-dimensional solver of Müller & Steinmetz (1995) and was provided by Zwerger (personal communication). Using the computed three-dimensional gravitational potential and the mapped density and velocity distributions, we were able to derive the gravitational wave signature of model HB3D from Eqs. ( 13 - 23 ).
According to Fig. 23 model HB3D, which is the three-dimensional analogue of model HB2D, emits significantly less energy in form of gravitational waves. There are several reasons for that. Firstly, the convective elements are smaller, only about half of the typical size found in two dimensions ( ), and the mass motions are therefore less coherent and do not cause the strong large-scale deformations seen in 2D. Secondly, the rising and sinking convective elements move with smaller velocities in three spatial dimensions, , which leads to reduced over- and undershooting (only about 0.8 instead of 1.2 pressure scale heights, see Sect. 2.1) and is another reason for the weaker large-scale deformation of the outer layers of the proto-neutron star in the 3D model HB3D. While model HB2D emits a gravitational wave energy of , model HB3D radiates an energy of only during the same time interval of 32 ms (compare Figs. 18 and 23). The quadrupole amplitudes of both polarizations ( and ) are also shown in Fig. 23 for an observer at the pole and at the equator, respectively. The maximum absolute values of the amplitudes never exceed 4 cm which is about a factor of 100 smaller than in the two-dimensional model HB2D (see Fig. 18). This corresponds to a maximum dimensionless gravitational wave amplitude (Eq. ( 13 )) of for a source at distance 10 kpc, about a factor of 15 smaller than in HB2D. A comparison of Figs. 19 and 20 with Fig. 23 (upper and lower right panels, respectively) reveals that the frequency spectrum of the quadrupole amplitude and the spectral energy density show more relative power, respectively are more peaked, towards lower frequencies in case of model HB3D. In HB3D the spectral maxima are around 100-200 Hz and the spectra drop rapidly towards higher frequencies, whereas the amplitude spectrum of HB2D has a broad region of highest power between 100 and 700 Hz (Fig. 18) and the spectral energy density of HB2D peaks between about 200 and 600 Hz. This difference of the wave frequencies is a result of the smaller convective velocities and the correspondingly longer overturn timescales and less violent convection in the three-dimensional situation.
3.4. Gravitational waves from anisotropic neutrino emission
The gravitational wave emission associated with the anisotropic radiation of neutrinos can only be estimated for the presented models because no multi-dimensional neutrino transport was used in the simulations, in fact some of the models were computed without including neutrino effects at all.
In order to estimate the anisotropy parameter defined in Eq. ( 29 ) we used the following procedure. Assuming that the neutrino flux can be approximated by black-body emission and thus scales with the fourth power of the temperature T and with the area of the emitting surface region, we employed to evaluate Eq. ( 29 ). Here denotes a mass-weighted radial average of in the neutrinospheric layer that is considered to encompass densities of to . The mean is evaluated by adding up the mass-weighted contributions of all grid cells along a specified radial beam direction . Repeating this for all angular directions and for all given time levels of a model yields the needed input into Eq. ( 29 ).
Evaluating the two-dimensional model HB2D we find that the anisotropic neutrino emission associated with the convective processes inside the proto-neutron star leads to an equatorial anisotropy parameter of (Eq. ( 29 ) with the angular factor of Eq. ( 27 )) at a time when the convective overturn is fully developed ( ms after the start of the simulation). We obtain numerical values that are rather close to those of the relative quadrupole moment of . A test integration, moreover, was in agreement with the analytical result for the axially symmetrical case, namely that the corresponding polar anisotropy parameter (Eq. ( 29 ) with the angular factor of Eq. ( 26 )) is negligibly small. Using the temperature distribution from the three-dimensional simulation HB3D we get for ms, again fairly similar to what is obtained from the quadrupole formula.
The anisotropy parameters therefore turn out to be roughly a factor of 10 smaller in the 3D case than in the 2D situation. Correspondingly, when going from 2D to 3D, the gravitational radiation field (Eq. ( 28 )) becomes about one order of magnitude smaller and the gravitational wave energy (Eq. ( 31 )) approximately two orders of magnitude, if the neutrino luminosities of both cases are similar. Assuming constant luminosity and anisotropy for an emission time , one gets
For erg/s and a typical emission time of one second - which assumes that convective overturn and anisotropic neutrino emission continue during most of the time when the gravitational binding energy of the proto-neutron star (a few erg) is lost by neutrino emission - one finds to be of the order of several 100 cm, i.e., for a source at a distance of 10 kpc. The gravitational wave energy is . For the three-dimensional case these numbers are and .
The gravitational wave signal from anisotropic mass motions due to convection inside the proto-neutron star was found (Sects. 3.2 and 3.3) to be and for model HB2D and and for the three-dimensional model HB3D. A comparison shows that the gravitational wave amplitude associated with the neutrino emission is somewhat larger than the wave amplitude due to convective motions. In two dimensions the neutrino gravitational wave amplitude is about 5 times larger, in three dimensions the factor can become even 10. The total energy radiated in gravitational waves, however, is dominated by the contributions from the mass quadrupole moment. Neutrino gravitational waves account only for a minor fraction (for a few per cent at most) of the total gravitational wave energy in two-dimensional models, while in 3D they contribute up to several 10% of the gravitational wave energy.
Like the gravitational waves from convective motions inside the newly formed neutron star, the neutrino gravitational waves are predominantly emitted in the frequency band between several 10 Hz and a few 100 Hz because of the common origin and thus similar timescales of the anisotropic processes. In case of the neutrino gravitational waves there is also a superimposed low-frequency component ( Hz to a few Hz) caused by the long-time variation of the neutrino emission from the cooling neutron star on a timescale of about one second.
© European Southern Observatory (ESO) 1997