Commit a4c7fc8c authored by Francesco Zappa's avatar Francesco Zappa

Syncronized with ligo gitlab

parents
Copyright (c) 2018 Bernuzzi, Perego, Zappa
Permission is hereby granted, free of charge, to any person
obtaining a copy of this software and associated documentation
files (the "Software"), to deal in the Software without
restriction, including without limitation the rights to use,
copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following
conditions:
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
OTHER DEALINGS IN THE SOFTWARE.
\ No newline at end of file
# Gravitational-wave energy, peak luminosity and angular momentum of binary neutron star mergers
Fitting formulae for gravitational-wave luminosity, energy and angular momentum at merger and postmerger derived from the [CoRe](http://www.computational-relativity.org/) database of numerical relativity simulations of quasicircular binary neutron star mergers.
All the fitting formulas were developed in the following references to which we refer for a comprehensive presentation. Documentation can be also found on [TDS](https://tds.virgo-gw.eu/?content=3&r=14706) and [DCC](https://dcc.ligo.org/T1800417)
```
@article{Zappa:2017xba,
author = "Zappa, Francesco and Bernuzzi, Sebastiano and Radice,
David and Perego, Albino and Dietrich, Tim",
title = "{Gravitational-wave luminosity of binary neutron stars
mergers}",
journal = "Phys. Rev. Lett.",
volume = "120",
year = "2018",
number = "11",
pages = "111101",
doi = "10.1103/PhysRevLett.120.111101",
eprint = "1712.04267",
archivePrefix = "arXiv",
primaryClass = "gr-qc",
SLACcitation = "%%CITATION = ARXIV:1712.04267;%%"
}
```
```
@article{Bernuzzi:2014kca,
author = "Bernuzzi, Sebastiano and Nagar, Alessandro and Balmelli,
Simone and Dietrich, Tim and Ujevic, Maximiliano",
title = "{Quasiuniversal properties of neutron star mergers}",
journal = "Phys. Rev. Lett.",
volume = "112",
year = "2014",
pages = "201101",
doi = "10.1103/PhysRevLett.112.201101",
eprint = "1402.6244",
archivePrefix = "arXiv",
primaryClass = "gr-qc",
SLACcitation = "%%CITATION = ARXIV:1402.6244;%%"
}
```
## Code requirements
```
Python 2.7.15
Numpy 1.14.3
Scipy 1.1.0
```
\ No newline at end of file
#!/usr/bin/env python
"""
Fit models for BNS peak luminosity, GW energy, and angular momentum based on numerical relativity simulations
"""
__author__ = "F.Zappa (Jena U), S.Bernuzzi (Jena U), D.Radice (Princeton & IAS), A.Perego (INFN & Parma U), T.Dietrich (AEI)"
__copyright__ = "Copyright 2017/18"
import numpy as np
from scipy.misc import factorial2
# ---------- fit coefficients -------------------------------------------
coefs_e_pm_fit_pow = [ 2.44358614, -0.01892881]
slope_e_pm_extrap = -5.12643669729e-05
offset_e_pm_extrap = 0.038079362826
#~ coefs_Lpeak_fit_q_nr = [ 0.0162958980711, 7.87825615e-04, -2.09234890e-07, 2.09277467e-02 ]
coefs_Lpeak_fit_q_nr = [ 0.021782890899735566, 0.00052426885345038126, -9.3658539544098073e-08, 0.027742691256389923 ]
coef0_Lpeak_fit_q_extrap = 0.021782890899735566
high_kappaL_Lpeak_fit_q_extrap = 3581.76666667
slope_Lpeak_fit_q_extrap = -6.08468023236e-06
offset_Lpeak_fit_q_extrap = 0.0167245171163
coefs_e_j_final_fit = [ 0.94665103, -0.43994502, 0.05334342 ]
coefs_j_e_final_fit = [ 4.39506775, -17.211734 , 38.549884 ]
e0_e_mrg_fit = 0.12
a_e_mrg_fit = 1.2e3
coefs_e_mrg_fit = [ 5.09166459e-02, 6.44073902e-05, 9.53252086e-02, 2.64027080e-04 ]
j0_j_mrg_fit = 2.8
a_j_mrg_fit = 1.2e3
coefs_j_mrg_fit = [ 0.07831028, 0.00019308, 0.06631369, 0.00012563 ]
# ------------------------------------------------------------------------
def lam_q_to_kappa(lam, q):
""" Compute kappa_A parameter from Lambda_A parameter and mass ratio """
return 3*(q**4)/((1+q)**5)*lam
def barlamdel_to_kappal(q, barlamAl, barlamBl, ell):
"""$\kappa^{A,B}_\ell(\bar{\lambda}_\ell)$
Assume $q=M_A/M_B>=1$
"""
XA = q/(1.+q);
XB = 1. - XA;
f2l1 = factorial2(2*ell-1);
p = 2*ell + 1;
kappaAl = f2l1 * barlamAl * XA**p / q;
kappaBl = f2l1 * barlamBl * XB**p * q;
#kappaTl = kappaAl + kappaBl;
return kappaAl, kappaBl
def q_to_nu(q):
""" Compute sym mass ratio from mass ratio, assume q>=1 """
if np.any(q<1.):
raise ValueError("q must be >=1")
return q/((1.+q)**2)
def kappalum(kappa, q):
""" Compute effective kappa_A for lluminosity fit from kappa_A and mass ratio """
return 2 * (3 + q) * kappa
def e_pm_fit_const(ktid):
""" Fit energy post merger, prompt collapse
Valid for ktid < 63
"""
cpcol = 0.018
return np.full_like(ktid, cpcol, dtype=np.double)
def e_pm_fit_avg6373(ktid):
""" Fit energy post merger, NS remnant
Average value in region 63 < ktid < 73 """
avg = (0.018 + 0.1)/2.
return np.full_like(ktid, avg, dtype=np.double)
def e_pm_fit_pow(ktid):
""" Fit energy post merger, NS remnant
Valid for ktid > 73 """
return coefs_e_pm_fit_pow[0] * ktid**(-7./10.)+ coefs_e_pm_fit_pow[1]
def e_pm_extrap(ktid):
""" Extrapolate the fit e_pm,
Valid in 457.9 < ktid < 742.8 """
return slope_e_pm_extrap*ktid + offset_e_pm_extrap
def e_pm_fit(ktid):
""" Fit energy post merger, note that this is EGW/(M nu) """
if np.any(ktid < 0.):
raise ValueError("Invalid values of tidal parameter!")
conds = [ktid <= 63.,
(ktid > 63.) & (ktid < 73.),
(ktid >= 73.) & (ktid <= 457.9),
(ktid > 457.9) & (ktid <= 742.803726537),
ktid > 742.803726537]
funcs = [e_pm_fit_const,
e_pm_fit_avg6373, #lambda x: np.full_like(x,np.nan,dtype=np.double),
e_pm_fit_pow,
e_pm_extrap,
lambda x: np.full_like(x,0.,dtype=np.double)]
return np.piecewise(ktid, conds, funcs)
def Lpeak_fit_q_nr(klum):
""" Fit Lpeak, note this is Lpeak/(nu^2) * q^2
Valid for klum < 3581.76666667 """
return coefs_Lpeak_fit_q_nr[0] * (1 + coefs_Lpeak_fit_q_nr[1] * klum + coefs_Lpeak_fit_q_nr[2] * klum**2)/(1 + coefs_Lpeak_fit_q_nr[3] * klum)
def Lpeak_fit_q_extrap(klum):
""" Extrapolate the fit Lpeak, note this is Lpeak/(nu^2) * q^2
Valid in 3581.76666667 < klum < 6330.39378883 """
return coef0_Lpeak_fit_q_extrap * (slope_Lpeak_fit_q_extrap*(klum - high_kappaL_Lpeak_fit_q_extrap) + offset_Lpeak_fit_q_extrap);
def Lpeak_fit_q(klum):
""" Fit Lpeak, note this is Lpeak/(nu^2) * q^2 """
if np.any(klum < 0.):
raise ValueError("Invalid values of tidal parameter!")
high_kappaL = 3581.76666667
extreme_kappaL = 6330.39378883;
conds = [klum <= high_kappaL,
(klum > high_kappaL) & (klum < extreme_kappaL),
klum >= extreme_kappaL]
funcs = [Lpeak_fit_q_nr,
Lpeak_fit_q_extrap,
lambda x: np.full_like(x,0.,dtype=np.double)]
return np.piecewise(klum, conds, funcs)
def e_j_final_fit(j_fin):
""" Fit final/total radiated energy as a function of final angular momentum e^tot_GW (j_rem) """
return coefs_e_j_final_fit[0] + coefs_e_j_final_fit[1] * j_fin + coefs_e_j_final_fit[2] * j_fin**2
def j_e_final_fit(e_fin):
""" Fit final angular momentum as a function of final/total radiated energy j_rem(e^tot_GW) """
return coefs_j_e_final_fit[0] + coefs_j_e_final_fit[1] * e_fin + coefs_j_e_final_fit[2] * e_fin**2
def e_mrg_fit(ktid, nu):
""" Fit for the energy emitted up to merger with a correction in nu. Note that this EGW/(M nu)(t=tmrg) """
k = ktid + a_e_mrg_fit * (1 - 4 * nu)
if np.any(ktid<0.):
raise ValueError("Invalid values of tidal parameters!")
return e0_e_mrg_fit * (1 + coefs_e_mrg_fit[0] * k + coefs_e_mrg_fit[1] * k**2)/(1 + coefs_e_mrg_fit[2] * k + coefs_e_mrg_fit [3] * k**2)
def j_mrg_fit(ktid, nu):
""" Fit for the angular momentum of the binary at merger with a correction in nu. Note that this J/(M**2 nu) """
k = ktid + a_j_mrg_fit * (1 - 4 * nu)
if np.any(ktid<0.):
raise ValueError("Invalid values of tidal parameters!")
return j0_j_mrg_fit * (1 + coefs_j_mrg_fit[0] * k + coefs_j_mrg_fit[1] * k**2)/(1 + coefs_j_mrg_fit[2] * k + coefs_j_mrg_fit[3] * k**2)
if __name__=='__main__':
""" Usage example """
# Units CGS
Lplanck = 3.628504984913064e59 # Planck luminosity c^5/G [erg/s]
clight = 2.99792458e+10 # [cm/s]
Msun = 1.98892e+33 # [g]
G = 6.674e-8 # [cm^3/(g s^2)]
Energy_cgs = Msun*clight**2
AngMom_cgs = G * Msun**2/clight
# Choose pars for a binary
M = np.array([2.8, 2.4]) # Msun
lam_A = np.array([500., 300.])
lam_B = np.array([500., 200.])
q = np.array([1., 1.2])
# Symmetric mass ratio
nu = q_to_nu(q)
# Compute the kappa's
kappa_A = lam_q_to_kappa(lam_A, q)
kappa_B = lam_q_to_kappa(lam_B, 1./q)
# Compute kappaT2 and kappa effective for luminosity fit
kappaT2 = kappa_A + kappa_B
kapelum = kappalum(kappa_A, q) + kappalum(kappa_B, 1./q)
# Eval fits
Lpeak_q2 = Lpeak_fit_q(kapelum)
e_pm = e_pm_fit(kappaT2)
e_mrg = e_mrg_fit(kappaT2, nu)
j_mrg = j_mrg_fit(kappaT2, nu)
e_fin = e_mrg + e_pm
j = j_e_final_fit(e_fin) # j final requires e_fin !
# Rescale and CGS units
E_cgs = e_fin*nu*M * Energy_cgs
Lpeak_cgs = Lpeak_q2*(nu**2 * 1./(q**2)) * Lplanck
J_cgs = j_mrg*M*nu**2 * AngMom_cgs
for i in range(len(q)):
# Print results of total energy, luminosity peak and angular momentum at merger
print('\n')
print("M = %e q = %e nu = %e lam_A,B = %e, %e"%(M[i], q[i], nu[i], lam_A[i], lam_B[i]))
print("kappa_A = %e kappa_B = %e"%(kappa_A[i], kappa_B[i]))
print("kappaT2 = %e kappa_eff^lum = %e"%(kappaT2[i], kapelum[i]))
print("Natural units (c=G=Msun=1)")
print("\tEGW_tot/(nu M) = %e\n\tLpeak/nu^2*q^2 = %e\n\tJ_merger/(nu M^2) = %e"%(e_fin[i], Lpeak_q2[i], j_mrg[i]))
print("CGS units")
print("\tEGW_tot = %e [erg]\n\tLpeak = %e [erg/s]\n\tJ_merger = %e [g cm^2/s]"%(E_cgs[i], Lpeak_cgs[i], J_cgs[i]))
\documentclass[aps,prl,onecolumn,superscriptaddress,groupedaddress,nofootinbib,floatfix,notitlepage]{revtex4-1}
%% Note for comments
%% use two %
%% use a space after %% write here
%% packages
\usepackage{graphicx,psfrag}
\usepackage{mathrsfs}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage{multirow}
\usepackage{comment}
\usepackage[normalem]{ulem}
\usepackage{hyperref}
\usepackage{makecell}
%% macros
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\bel}{\begin{align}}
\newcommand{\eel}{\end{align}}
\newcommand{\stext}[1]{\text{#1}}
\newcommand{\tento}[1]{\times 10^{#1}}
\def\l{\ell}
\def\lm{\ell m}
\def\p{\partial}
\def\Lie{{\cal L}}
\def\non{\nonumber}
\def\half{\frac{1}{2}}
\def\e{{\rm e}}
\def\i{{\rm i}}
\def\ergsec{{\rm erg\,s^{-1}}}
\def\gccm{{\rm g\,cm^{-3}}}
\def\Msun{{\rm M_{\odot}}}
\def\GMc2{{\rm G M_{\odot} c^{-2}}}
\def\Mpc{{\rm Mpc}}
\def\eps{\epsilon}
\def\veps{\varepsilon}
\def\vrho{\varrho}
\def\hatOmega{\hat{\Omega}}
\def\hatomega{\hat{\omega}}
\def\O{\mathcal{O}}
\def\Mpr{M_\text{pc}}
\def\Cpr{c_\text{pc}}
\def\egw{e_\text{GW}}
\def\Egw{E_\text{GW}}
%% macros for comments
\usepackage{color}
%% colors
\definecolor{cyan}{rgb}{0,0.9,0.9}
\definecolor{orange}{rgb}{0.9,0.5,0}
\definecolor{magenta}{rgb}{1,0,1}
\definecolor{purple}{rgb}{0.8,0.4,0.8}
\definecolor{gray}{rgb}{0.8242,0.8242,0.8242}
%%
\newcommand{\zp}[1]{{\textcolor{red}{\texttt{FZ: #1}} }}
\newcommand{\bs}[1]{{\textcolor{green}{\texttt{SB: #1}} }}
\newcommand{\dr}[1]{{\textcolor{purple}{\texttt{DR: #1}} }}
\newcommand{\pa}[1]{{\textcolor{blue}{\texttt{AP: #1}} }}
\newcommand{\td}[1]{{\textcolor{orange}{\texttt{TD: #1}} }}
%% ...
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\cor}[2]{\sout{#1}\textcolor{red}{#2}}
%% +++++++++++++++++++++++++++++++++++++++++++++++++++++++
\begin{document}
\title{Gravitational-wave energy, luminosity and angular momentum from
numerical relativity simulations of binary neutron stars' mergers}
%% http://www.computational-relativity.org/database.html
%%
%
\author{Francesco \surname{Zappa}$^{1}$}
\author{Sebastiano \surname{Bernuzzi}$^{1}$}
%\author{David \surname{Radice}$^{3,4}$}
\author{Albino \surname{Perego}$^{2}$}
%\author{Tim \surname{Dietrich}$^{6,7}$}
%
%\affiliation{${}^1$ Dipartimento di Scienze Matematiche Fisiche ed Informatiche, Universit\`a di Parma, I-43124 Parma, Italy}
\affiliation{${}^1$ Theoretical Physics Institute, University of Jena, 07743 Jena, Germany}
\affiliation{${}^2$ Istituto Nazionale di Fisica Nucleare, Sezione Milano Bicocca, Italia}
%\affiliation{${}^3$ Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA}
%\affiliation{${}^4$ Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA}
%\affiliation{${}^5$ Dipartimento di Fisica, Universit\`{a} degli Studi di Milano Bicocca, Piazza della Scienza 3, 20126 Milano, Italia}
%\affiliation{${}^6$ Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M\"uhlenberg 1, Potsdam-Golm, 14476, Germany}
%\affiliation{${}^7$ Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands}
\date{\today}
\begin{abstract}
We collect fitting formulae for gravitational-wave (GW) luminosity, energy and angular momentum
derived from the {\tt CoRe} database of numerical relativity simulations of quasicircular binary neutron star mergers.
All the fitting formula were developed in \cite{Zappa:2017xba}, to which we refer for a comprehensive presentation.
\end{abstract}
\maketitle
%% \tableofcontents
%% Template for figures
%% \begin{figure}[t]
%% \centering
%% \includegraphics[width=0.49\textwidth]{en_fit_powerlaw.png}
%% \caption{ }
%% \label{fig: }
%% \end{figure}
%% Template for tables
%% \begin{table}[t]
%% \centering
%% \caption{ }
%% \begin{tabular}{ccc}
%% \hline
%% col$1$ & col$2$ & col$3$ \\
%% \hline
%% & & \\
%% \hline
%% \end{tabular}
%% \label{tab: }
%% \end{table}
\begin{table}[t]
\centering
\caption{ Binary parameters. }
\begin{tabular}{ccc}
\hline
Quantity & Definition & \\
\hline
\hline
\makecell{Gravitational mass of \\ star A in isolation} &
$ M_{\rm A} $ & \\
\hline
\makecell{Total gravitational \\ mass of the binary} &
$ M = M_{\rm A} + M_{\rm B} $ & \\
\hline
Mass ratio &
$ q = M_{\rm A} / M_{\rm B} \geq 1$ & \\
\hline
Symmetric mass ratio &
$ \nu = q/(1+q)^2 $&\\
\hline
Compactness of star A &
$ C_{\rm A} = \frac{G M_{\rm A}}{R_{\rm A} c^2} $ &\\
\hline
\makecell{Gravito-Electric Quadrupolar \\ Love number~\cite{Damour:2009wj} of star A} &
$ k_2^{\rm A} $ &\\
\hline
\makecell{Neutron Star's Gravito-Electric \\ Quadrupolar Tidal polarizability~\cite{Damour:2009wj} of star A
\footnote{The relation between this parameter and the more common $\Lambda_2$ is given by
$\Lambda_2^\stext{A} = \frac{1}{3} \left(\frac{M}{M_{\rm A}}\right)^5 \frac{M_{\rm A}}{M_\stext{B}} \kappa_2^\stext{A}$}} &
$ \kappa_2^{\rm A} = 2\left(\frac{M_{\rm A}}{M}\right)^5 \frac{M_{\rm B}}{M_\stext{A}}
\frac{k_2^{\rm A}}{(C_{\rm A})^5}$ &\\
\hline
\makecell{Binary's Quadrupolar \\ Tidal polarizability} &
$ \kappa_2^{\rm T} = \kappa_2^{\rm A} + \kappa_2^{\rm B} $ &\\
\hline
\makecell{Effective Binary's Quadrupolar \\ Tidal polarizability for Luminosity} &
$ \kappa_2^{\rm L} = 2 \left[\left( 3 + \frac{M_{\rm A}}{M_{\rm B}} \right)\kappa_2^{\rm A} + ({\rm A} \leftrightarrow {\rm B}) \right] $ &\\
\hline
&&\\
\end{tabular}
\label{tab:bin_par}
\end{table}
\begin{table}[t]
\centering
\caption{Fitting formulae. Note that all the quantities shown are dimensionless.}
\begin{tabular}{ccc}
\hline
Quantity & Fitting formula & Parameters\\
\hline
\hline
GW Luminosity peak &
$ L_\text{peak}(\kappa^\text{L}_2,\nu) = \begin{cases} \kappa^\text{L}_2\lesssim 3582 & L_0 \frac{\nu^2}{q^2(\nu)}
\frac{ 1 + n_1 \kappa^\text{L}_2 + n_2 ({\kappa^\text{L}_2})^2 }
{ 1 + d_1 \kappa_2^\text{L}}\\
\kappa^\text{L}_2\gtrsim 3582 &L_0\left[a \left(\kappa^\text{L}_2 -b\right) + c \right]
\end{cases}$
& \makecell{$L_0 = 2.178 \times 10^{-2}$ \\ $ n_1 = 5.24 \times 10^{-4}$ \\
$n_2 = -9.36 \times 10^{-8}$ \\ $d_1 = 2.77 \times 10^{-2}$ \\ $a = -6.1\times 10^{-6}$ \\ $b = 3582$ \\ $c = 1.7 \times 10^{-2}$} \\
\hline
GW Energy at merger
& \makecell{$e^{\rm mrg}_{\rm GW}(\kappa^\text{T}_2,\nu) = e_0
\frac{ 1 + n_1 \hat{\kappa}^\text{T}_2 + n_2 ({\hat{\kappa}^\text{T}_2})^2 }
{ 1 + d_1 \hat{\kappa}_2^\text{T} + d_2 ({\hat{\kappa}^\text{T}_2})^2 } $ \\
\\$ \hat{\kappa}_2^\text{T} = \kappa_2^\text{T} + a (1 - 4 \nu) $}
& \makecell{$ a = 1.2\times 10^3$ \\ $e_0 = 0.12$ \\ $ n_1 = 5.09\times 10^{-2} $ \\
$n_2 = 6.44 \times 10^{-5}$ \\ $d_1 = 9.53\tento{-2} $ \\ $d_2 = 2.64 \tento{-4}$} \\
\hline
Binary's angular momentum at merger &
\makecell{$j^{\rm mrg}(\kappa^\text{T}_2,\nu) = j_0
\frac{ 1 + n_1 \hat{\kappa}^\text{T}_2 + n_2 ({\hat{\kappa}^\text{T}_2})^2 }
{ 1 + d_1 \hat{\kappa}_2^\text{T} + d_2 ({\hat{\kappa}^\text{T}_2})^2 }$ \\
\\$\hat{\kappa}_2^\text{T} = \kappa_2^\text{T} + a (1 - 4 \nu)$}
& \makecell{$ a = 1.2 \tento{3}$ \\ $j_0 = 2.8$ \\ $ n_1 = 7.83\tento{-2}$ \\ $n_2 = 1.93\tento{-4} $
\\ $d_1 = 6.63 \tento{-2}$ \\ $d_2 = 1.26 \tento{-4}$} \\
\hline
\makecell{GW post merger energy \\ (as a function of $\kappa^\text{T}_2$)}&
$e^{\rm pm}_{\rm GW}( \kappa^\text{T}_2 ) = \begin{cases} 0.02 & \kappa^\text{T}_2 \lesssim 63 \\ - & 63 \lesssim \kappa_2^\stext{T} \lesssim 73\\ a(\kappa^\text{T}_2)^{-\frac{7}{10}} + b & 73\lesssim \kappa^\text{T}_2 \lesssim 458 \\ c \kappa^\text{T}_2 +d & \kappa^\text{T}_2 \gtrsim 458 \end{cases} $
& \makecell{$ a = 2.44 $ \\ $ b = -0.019 $\\ $ c =-5.1\times 10^{-5} $\\ $ d = 0.038$}\\
\hline
\makecell{GW total energy \\ (as a function of $j_{\rm rem}$)} &
$ e^{\rm tot}_{\rm GW}(j_{\rm rem}) = c_0 + c_1 j_{\rm rem} + c_2 (j_{\rm rem})^2$ &
\makecell{$ c_0 = 0.94 $ \\ $c_1 = -0.43 $ \\ $c_2 = 0.053$}
\\
\hline
Remnant angular momentum &
$ j_{\rm rem}(e^{\rm tot}_{\rm GW}) = c_0 + c_1 e^{\rm tot}_{\rm GW} + c_2 (e^{\rm tot}_{\rm GW})^2 $ &
\makecell{$ c_0 = 4.39 $ \\ $c_1 = -17.2 $ \\ $c_2 = 38.5 $} \\
\hline
\end{tabular}
\label{tab:fit}
\end{table}
\paragraph*{\bf Simulations.}
In \cite{Zappa:2017xba} we use data from about 100 simulations of
quasi-circular non-spinnng binaries with the {\tt BAM} and {\tt THC}
code. All data are now public at
\begin{center}
\url{http://www.computational-relativity.org/}
\end{center}
A summary of the database is presented in~\cite{Dietrich:2018phi}.
We do not use the simulations {\tt BAM:0023}-{\tt BAM:0034}.
The data employed span the parameter ranges
\begin{align}
&1 < q < 2\\
& 40 < \kappa^T_2 < 500
\end{align}
and refer to 8 EOS and different input physics, cf. discussion
in \cite{Zappa:2017xba}. Following results robustly describe also
binaries with dimensionless spins up to $\chi\sim0.1$, \cite{Zappa:2017xba}.
\paragraph*{\bf Definitions.} Main quantities
\be
e_\stext{GW} \equiv \frac{E_\stext{GW}}{M \nu} = - \frac{M-M_\stext{ADM}(t=0) - {\cal E}_\stext{rad}(t)}{M\nu}
\ee
The conversion factor of $E_\stext{GW}$ to physical units is $\Msun c^2$.
\be
j_\stext{binary} \equiv \frac{J_\stext{binary}}{M^2\nu} = \frac{J_\stext{ADM}(t=0) - {\cal J}_\stext{rad}(t)}{M^2\nu}
\ee
The conversion factor to physical units of $J_\stext{binary}$ is $\frac{G\Msun^2}{c}$.
\be
L_\stext{peak} \equiv \max_t \left( \frac{dE_\stext{GW}}{dt} \right)
\ee
The conversion factor to physical units is $L_\stext{Planck} = c^5/G$.
$M_\stext{ADM}(t=0)$ and $J_\stext{ADM}(t=0)$ are the mass and angular momentum of Arnowitt-Deser-Misner, calculated for
the initial binary configuration.
${\cal E}_\stext{rad}(t)$,~${\cal J}_\stext{rad}(t)$ are the energy and angular momentum radiated through GW during the simulation~\cite{Damour:2011fu}.
\paragraph*{\bf Comments on fits.}
The behaviour of the quantities above is captured by the symmetric mass ratio $\nu$
and the tidal parameters of the binary.
%%Range of validity, errors, refs.
\begin{itemize}
\item The merger time $t_{\rm mrg}$ is defined as the peak of the amplitude of the $(2,2)$ mode of the GW.
\item The total energy and final angular momentum at taken at the end
of the simulation, $t_{\rm pm}\sim t_{\rm mrg} + 20-30$~ms. On this
timescale the remnant radiate most of the GW energy
\cite{Bernuzzi:2015opx}. At the end of our simulations the GW
radiation timescale for angular momentum loss is $\dot{{\cal
J}}_\stext{rad}/{\cal J}_\stext{rad}\sim 0.5$~s and rapidly increasing.
\item Luminosity peak~\cite{Zappa:2017xba}.
$L_0$ is the average of the luminosity peaks for binary black hole
(BBH)
mergers with equivalent parameters~\cite{Keitel:2016krm}.
%The fitting formula is obtained for values of $\Lambda^\stext{A,~B}$
%so that $\red{X} \lesssim \kappa^\text{L}_2 \lesssim 3600 $.
For $\kappa^\text{L}_2 = 0 $, it matches nonspinning BBH
with $q\sim1$ in the above sense ($L_0$).
For $\kappa^\text{L}_2 \gtrsim 3600 $, it is linearly extended in
such a way that the luminosity approaches to 0 for large values
of the tidal parameters.
The coefficient of determination of the fit is $R^2 = 0.943$. The
fit errors are below $30 \%$.
\item GW Energy at merger and binary angular momentum at merger \cite{Bernuzzi:2014kca,Zappa:2017xba}. The parameter which appears in this formulae is a tidal parameter
with a correction depending on $\nu$. $a$ is an empirical parameter needed to amplify this correction.
These formulae extend previous results presented in~\cite{Bernuzzi:2014kca} but they are still unpublished in this form.
For the energy, $R^2 = 0.992$ and the maximum errors are below $3\%$; for the angular momentum $R^2 = 0.993$ and the errors are
below $1\%$. This fit can be used to properly estimate the minimum
energy emitted by the binary. The fit for the GW energy at merger is shown in Fig.~\ref{fig:emrg_fit}, note that
for $\kappa_2^\stext{T}=0$ the fit returns the BBH $q=1$ value.
\item GW total energy as a function of $\kappa^\text{T}_2$~\cite{Zappa:2017xba}. The given formula has the form of a piecewise function
and this reflect the two possible outcome of the merger: a prompt BH
formation or a massive NS.
In the case of prompt collapses ($\kappa_2^\stext{T} \lesssim 63 $)the emission of GWs stops almost immediately
after the moment of merger and the amount of energy does not depend much on the EOS. If the remnant object is a neutron star ($\kappa_2^\stext{T} \gtrsim 73 $) the
emission of GWs continues after the merger and quantitatively
depends on the EOS. For $63 \lesssim \kappa_2^\stext{T} \lesssim 73$ the
behaviour is too uncertain and is not possible to give a prediction; the fit returns an average value in those cases.
A possible application of this formula is an estimate on the maximum energy emitted through GWs in the post merger phase. For an equal-mass binary system with $M = 2.8$,
the fit and data predict for the post-merger and total energy resp.
\begin{align}
E_\stext{GW}^\stext{pm} &\lesssim 0.072 \frac{M}{2.8} \Msun c^2 \\
E_\stext{GW}^\stext{tot} &\lesssim 0.126 \frac{M}{2.8} \Msun c^2.
\end{align}
Note that this fit has large uncertainties and errors, as shown Fig.~\ref{fig:etot_fit}. Hence, it should be used to provide an estimate or the upper limits. This relation is still unpublished.
\item GW total energy as a function of $j_{\rm rem}$~\cite{Zappa:2017xba}. For this formula, $R^2 = 0.986$ and errors are below
$8\%$.
\item Remnant angular momentum: For this formula, $R^2 = 0.982$ and the maximum error is around $2\%$. This relation is
still unpublished.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{egw_mrg_fit.png}
\caption{Fit of the energy per unit mass radiated up to merger.}
\label{fig:emrg_fit}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{egw_pm_fitpowerlaw.png}
\caption{Fit of the energy per unit mass radiated in the post-merger phase.}
\label{fig:etot_fit}
\end{figure}
%\bibliography{references}
\begin{thebibliography}{1}
%\cite{Zappa:2017xba}
\bibitem{Zappa:2017xba}
F.~Zappa, S.~Bernuzzi, D.~Radice, A.~Perego and T.~Dietrich,
%``Gravitational-wave luminosity of binary neutron stars mergers,''
Phys.\ Rev.\ Lett.\ {\bf 120}, no. 11, 111101 (2018)
doi:10.1103/PhysRevLett.120.111101
[arXiv:1712.04267 [gr-qc]].
%%CITATION = doi:10.1103/PhysRevLett.120.111101;%%
%9 citations counted in INSPIRE as of 20 Sep 2018
%\cite{Damour:2009wj}
\bibitem{Damour:2009wj}
T.~Damour and A.~Nagar,
%``Effective One Body description of tidal effects in inspiralling compact binaries,''
Phys.\ Rev.\ D {\bf 81}, 084016 (2010)
doi:10.1103/PhysRevD.81.084016
[arXiv:0911.5041 [gr-qc]].
%%CITATION = doi:10.1103/PhysRevD.81.084016;%%
%105 citations counted in INSPIRE as of 20 Sep 2018
%\cite{Dietrich:2018phi}
\bibitem{Dietrich:2018phi}
T.~Dietrich {\it et al.},
%``CoRe database of binary neutron star merger waveforms and its application in waveform development,''
arXiv:1806.01625 [gr-qc].
%%CITATION = ARXIV:1806.01625;%%
%4 citations counted in INSPIRE as of 20 Sep 2018
%\cite{Damour:2011fu}
\bibitem{Damour:2011fu}
T.~Damour, A.~Nagar, D.~Pollney and C.~Reisswig,
%``Energy versus Angular Momentum in Black Hole Binaries,''
Phys.\ Rev.\ Lett.\ {\bf 108}, 131101 (2012)
doi:10.1103/PhysRevLett.108.131101
[arXiv:1110.2938 [gr-qc]].
%%CITATION = doi:10.1103/PhysRevLett.108.131101;%%
%48 citations counted in INSPIRE as of 20 Sep 2018
%\cite{Bernuzzi:2015opx}
\bibitem{Bernuzzi:2015opx}
S.~Bernuzzi, D.~Radice, C.~D.~Ott, L.~F.~Roberts, P.~Moesta and F.~Galeazzi,
%``How loud are neutron star mergers?,''
Phys.\ Rev.\ D {\bf 94}, no. 2, 024023 (2016)
doi:10.1103/PhysRevD.94.024023
[arXiv:1512.06397 [gr-qc]].
%%CITATION = doi:10.1103/PhysRevD.94.024023;%%
%29 citations counted in INSPIRE as of 20 Sep 2018
%\cite{Keitel:2016krm}
\bibitem{Keitel:2016krm}
D.~Keitel {\it et al.},
%``The most powerful astrophysical events: Gravitational-wave peak luminosity of binary black holes as predicted by numerical relativity,''
Phys.\ Rev.\ D {\bf 96}, no. 2, 024006 (2017)
doi:10.1103/PhysRevD.96.024006
[arXiv:1612.09566 [gr-qc]].
%%CITATION = doi:10.1103/PhysRevD.96.024006;%%
%12 citations counted in INSPIRE as of 13 Dec 2018
%% %\cite{Jimenez-Forteza:2016oae}
%% \bibitem{Jimenez-Forteza:2016oae}
%% X.~Jiménez-Forteza, D.~Keitel, S.~Husa, M.~Hannam, S.~Khan and M.~Pürrer,
%% %``Hierarchical data-driven approach to fitting numerical relativity data for nonprecessing binary black holes with an application to final spin and radiated energy,''
%% Phys.\ Rev.\ D {\bf 95}, no. 6, 064024 (2017)
%% doi:10.1103/PhysRevD.95.064024
%% [arXiv:1611.00332 [gr-qc]].
%% %%CITATION = doi:10.1103/PhysRevD.95.064024;%%
%% %23 citations counted in INSPIRE as of 20 Sep 2018
%\cite{Bernuzzi:2014kca}
\bibitem{Bernuzzi:2014kca}
S.~Bernuzzi, A.~Nagar, S.~Balmelli, T.~Dietrich and M.~Ujevic,
%``Quasiuniversal properties of neutron star mergers,''
Phys.\ Rev.\ Lett.\ {\bf 112}, 201101 (2014)
doi:10.1103/PhysRevLett.112.201101
[arXiv:1402.6244 [gr-qc]].
%%CITATION = doi:10.1103/PhysRevLett.112.201101;%%
%55 citations counted in INSPIRE as of 20 Sep 2018
\end{thebibliography}
\end{document}
Markdown is supported
0% or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment