# Gravitational Waveform Catalog

**H. Dimmelmeier, J.A. Font, H.-T. Janka, A. Marek, E. Müller, C.D. Ott**

**General Relativistic Simulations of Rotational Supernova Core Collapse with a Microphysical Equation of State and Deleptonization (Extended Model Set with Variation of the Equation of State and Progenitor Mass) ***(link einfuegen)*

We have simulated a set of 136 rotational supernova core collapse models in axisymmetry. To simulate these models, we numerically solve the fully general relativistic hydrodynamic equations in a flux-conservative formulation [Banyuls, et al., 1997] on a grid using spherical coordinates. For an accurate resolution of shocks, we have implemented a modern high-resolution shock-capturing scheme, which uses characteristic information of relativistic hydrodynamics [Font, 2000]. The exact metric in the ADM spacetime foliation is approximated by assuming conformal flatness for the three-metric, which significantly reduces the complexity of the hydrodynamic and metric equations.

As initial configuration we choose the presupernova stellar models e15a, e15b, e20a, e20b, s11.2, s15, s20, or s40 [Woosley, et al., 2002]. The progenitor models e15a, e15b, e20a, and e20b have an angular momentum distribution from stellar evolution calculations, while on the s11.2, s15, s20, and s40 models we impose rotation [Komatsu, et al., 1989a, Komatsu, et al., 1989b] with different rates and profiles.

During core collapse, we utilize a microphysical equations of state specifically designed for supernova core collapse, either the one by Shen et al. (Shen EoS) [Shen, et al., 1998] or the one by Lattimer and Swesty (LS EoS) [Lattimer and Swesty, 1998]. To approximate the effects of neutrinos in the infall phase, a very efficient parametric deleptonization scheme is used [Liebendörfer, 2005].

A detailed description of the models and other interesting information can be found in a published article [Dimmelmeier, et al., 2008].

Figures of the waveforms for each model in EPS and JPG format:

Figures of the waveforms in EPS format

Figures of the waveforms in JPG format

Raw data of the gravitational wave signal and the maximum density evolution for each model:

Gravitational wave signal data

(gzipped tar archive, 9.8 MByte, including a README file).

Density evolution data

(gzipped tar archive, 6.3 MByte, including a README file).

**General Relativistic Simulations of Rotational Supernova Core Collapse with a Microphysical Equation of State and Deleptonization:**

We have simulated a set of 54 rotational supernova core collapse models in axisymmetry. To simulate these models, we numerically solve the fully general relativistic hydrodynamic equations in a flux-conservative formulation [Banyuls, et al., 1997] on a grid using spherical coordinates. For an accurate resolution of shocks, we have implemented a modern high-resolution shock-capturing scheme, which uses characteristic information of relativistic hydrodynamics [Font, 2000]. The exact metric in the ADM spacetime foliation is approximated by assuming conformal flatness for the three-metric, which significantly reduces the complexity of the hydrodynamic and metric equations.

As initial configuration we choose the presupernova stellar model s20 [Woosley, et al., 2002], on which we impose rotation [Komatsu, et al., 1989a, Komatsu, et al., 1989b] with different rates and profiles.

During core collapse, we utilize a microphysical equation of state [Shen, et al., 1998] specifically designed for supernova core collapse. To approximate the effects of neutrinos in the infall phase, a very efficient parametric deleptonization scheme is used [Liebendörfer, 2005].

A detailed description of the models and other interesting information can be found in a published article [Dimmelmeier, et al., 2007].

Figures of the waveforms for each model in EPS and JPG format:

Figures of the waveforms in EPS format

Figures of the waveforms in JPG format

Raw data of the gravitational wave signal and the maximum density evolution for each model:

Gravitational wave signal data

(gzipped tar archive, 4.2 MByte, including a README file).

Density evolution data

(gzipped tar archive, 1.6 MByte, including a README file).

We have simulated a set of **26 rotational supernova core collapse models** (_link einfuegen_) in axisymmetry. To simulate these models, we numerically solve the fully general relativistic hydrodynamic equations in a flux-conservative formulation [Banyuls, et al., 1997] on a grid using spherical coordinates. For an accurate resolution of shocks, we have implemented a modern high-resolution shock-capturing scheme, which uses characteristic information of relativistic hydrodynamics [Font, 2000]. The exact metric in the ADM spacetime foliation is approximated by assuming conformal flatness for the three-metric, which significantly reduces the complexity of the hydrodynamic and metric equations.

The initial configurations are rotating 4/3-polytropes in equilibrium with a central density of *rho*_{c ini} = 10^{10} g/cm^{3} constructed by numerically solving the hydrodynamic equilibrium equations in a general relativistic spacetime [Komatsu, et al., 1989a, Komatsu, et al., 1989b]. The initial state is completely determined by the central density, the rotation rate *beta*_{rot ini} = *E*_{rot ini} / *E*_{pot ini}, and the rotation parameter *A*, which specifies the degree of differential rotation.

The equation of state during core collapse consists of a polytropic and a thermal contribution. The collapse is initiated by reducing the adiabatic exponent *gamma*_{1}. For densities *rho* > 2.0 ^{.} 10^{14} g/cm^{3}, the adiabatic exponent is set to *gamma*_{1} = 2.5 mimicking the stiffening of the equation of state above nuclear matter density. We have also simulated one model with a soft supranuclear equation of state with *gamma*_{1} = 2.0.

A detailed description of the models and other interesting information can be found in a series of published articles [Dimmelmeier, et al., 2001, Dimmelmeier, et al., 2002a, Dimmelmeier, et al., 2002b].

Figures of the waveforms for each model in EPS and JPG format:

Figures of the waveforms in EPS format

Figures of the waveforms in JPG format

Raw data of the gravitational wave signal, the maximum density evolution, and the radiated gravitational wave energy for each model:

Gravitational wave signal data

(gzipped tar archive, 21 MByte, including a README file).

Density evolution data

(gzipped tar archive, 15 MByte, including a README file).

Gravitational wave energy data

(gzipped tar archive, 20 MByte, including a README file).

**References:**

- Banyuls, F., Font, J.A., Ibanez, J.M., Marti, J.M., Miralles, J.A.,

"Numerical {3+1} general relativistic hydrodynamics",

*Astrophys. J.*,**476**, 221-231, (1997),

[Article in PDF format via Astrophys. J.].

- Dimmelmeier, H., Font, J.A., and Müller, E.,

"Gravitational waves from relativistic rotational core collapse",

*Astrophys. J. Lett.*,**560**, L163-L166, (2001),

[Article in astro-ph].

- Dimmelmeier, H., Font, J.A., and Müller, E.,

"Relativistic simulations of rotational core collapse. I. Methods, initial models, and code tests",

*Astron. Astrophys.*,**388**, 917-935, (2002),

[Article in astro-ph].

- Dimmelmeier, H., Font, J.A., and Müller, E.,

"Relativistic simulations of rotational core collapse. II. Collapse dynamics and gravitational radiation",

*Astron. Astrophys.*,**393**, 523-542, (2002),

[Article in astro-ph].

- Dimmelmeier, H., Ott, C.D., Janka, H.-T., Marek, A., and Müller, E.,

"Generic gravitational wave signals from the collapse of rotating stellar cores",

*Phys. Rev. Lett.*,**98**, 251101, (2007),

[Article in astro-ph].

- Dimmelmeier, H., Ott, C.D., Marek, A., and Janka, H.-T.,

"The gravitational wave burst signal from core collapse of rotating stars",

*Phys. Rev. D*,**78**, 064056, (2008),

[Article in astro-ph].

- Font, J.A.,

"Numerical hydrodynamics in general relativity",

*Living Rev. Relativ.*,**3**, (2000),

[Article in Living Rev. Relativ.].

- Komatsu, H., Eriguchi, Y., and Hachisu, I.,

"Rapidly rotating general relativistic stars - I. Numerical method and its application to uniformly rotating polytropes",

*Mon. Not. R. Astron. Soc.*,**237**, 355-379, (1989),

[Article in PDF format via ADS].

- Komatsu, H., Eriguchi, Y., and Hachisu, I.,

"Rapidly rotating general relativistic stars - II. Differentially rotating polytropes",

*Mon. Not. R. Astron. Soc.*,**239**, 153-171, (1989)

[Article in PDF format via ADS].

- Lattimer, J.M. and Swesty, F.D.,

"A generalized equation of state for hot, dense matter",

*Nucl. Phys. A*,**535**, 331-376, (1991)

[Article via journal webpage].

- Liebendörfer, M.,

"A simple parameterization of the consequences of deleptonization for simulations of stellar core collapse",

*Astrophys. J.*,**633**, 1042-1051, (2005)

[Article in astro-ph].

- Shen, H., Toki, H., Oyamatsu, K., and Sumiyoshi, K.,

"Relativistic Equation of State of Nuclear Matter for Supernova Explosion",

*Prog. Theor. Phys.*,**100**, 1013-1031, (1998)

[Article in PDF format via Prog. Theor. Phys.].

- Woosley, S.A, Heger, A., and Weaver, T.A.,

"The evolution and explosion of massive stars",

*Rev. Mod. Phys.*,**74**, 1015-1071, (2002)

[Article in PDF format via Rev. Mod. Phys.].

## Important Note

We note that the results for the axisymmetric core collapse models presented here slightly differ from those published in the previous version of the waveform catalog and related webpages (before June 23, 2004) and in some of our papers [Dimmelmeier, et al., 2001, Dimmelmeier, et al., 2002a, Dimmelmeier, et al., 2002b].

This is partly due to improvements related to evaluating the Euler equation source terms in compact form and using exact numerical conservation in the new code, as discussed in [Dimmelmeier, et al., 2004]. However, the main reason for the small discrepancy is that in the simulations in [Dimmelmeier, et al., 2002b] a *symmetric* boundary condition for the shift vector component *β*^{2} across the equatorial plane was chosen. This leads to a nonzero value for *β*^{2} at *θ* = *π* / 2 close to and after core bounce, i.e. when meridional motions set in. As a consequence of this, the deviation is strongest for models where rotation plays a significant role in the collapse dynamics.

The physically accurate *antisymmetric* equatorial boundary condition for *β*^{2} which is used in the results shown in [Dimmelmeier, et al., 2004] systematically yields lower values for *ρ*_{c} after core compared to the models presented in [Dimmelmeier, et al., 2002b]. The differences are negligible for the central density at bounce *ρ*_{c b}, while the post-bounce central density *ρ*_{c f} decreases on average by about 11% in the new simulations. As a result of the modifications in the post-bounce dynamics, the waveform amplitudes and frequencies of the gravitational radiation are altered by a small amount: *h*^{TT} decreases on average by 11%, and the gravitational wave frequency decreases on average by 18% compared to the old (and now obsolete) results.

The comparison of the gravitational wave amplitudes and frequencies obtained in simulations using relativistic gravity with simulations using Newtonian gravity changes as follows:

for the average of the multiple bounce models, and

for the average of all models.

Despite of these small quantitative changes, the qualitative statements related to the influence of general relativistic effects in rotational core collapse made in [Dimmelmeier, et al., 2002b], even when the antisymmetric boundary condition is used.

Note also that due to an error in the model setup, all data for model A4B5G4 in Newtonian gravity in the old version of the waveform catalog and related webpages were inaccurate. This error has been corrected in this version. [back]

- Dimmelmeier, H., Font, J. A., and Müller, E.,

"Gravitational waves from relativistic rotational core collapse",

*Astrophys. J. Lett.*,**560**, L163-L166, (2001),

[Article in astro-ph].

- Dimmelmeier, H., Font, J. A., and Müller, E.,

"Relativistic simulations of rotational core collapse. I. Methods, initial models, and code tests",

*Astron. Astrophys.*,**388**, 917-935, (2002),

[Article in astro-ph]. - Dimmelmeier, H., Font, J. A., and Müller, E.,

"Relativistic simulations of rotational core collapse. II. Collapse dynamics and gravitational radiation",

*Astron. Astrophys.*,**393**, 523-542, (2002),

[Article in astro-ph].

- Dimmelmeier, H., Novak, J., Font, J. A., Ibáñez, J. M., and Müller, E.,

"Combining spectral and shock-capturing methods: A new numerical approach for 3D relativistic core collapse simulations",

*Phys. Rev. D*,**71**, 064023, (2005),

[Article in astro-ph].