2.1 Making numerical estimates
In core collapse, estimates of the GW emission from simulations are usually done outside of the actual
hydrodynamics calculation – that is, there is no feedback from the loss of energy from GWs from the
system. Of course, where the GWs drive or damp an unstable mode (e.g., bar modes, Section 4.3), this
approximation will lead to erroneous results. Such an assumption is almost always justified, as the GW
emission is many orders of magnitude less than the total energy in the system. Calculating the GW
emission from a hydrodynamic simulation requires the discretization of Equation (1) and the discretization
used depends upon the numerical technique and the dimension of the calculation. Here we present some
common discretizations.
An example of a discretization in spherical coordinates for a 2-dimensional simulation is that used by
Müller & Janka [208
] (see also [306
, 201
]). Here they define the gravitational quadrupole radiation field
(
):
where
is the distance between the observer and the source,
is the angular coordinate in our
coordinate system, and
is given by:
where
,
are the velocities along the ordinate axes in spherical coordinates and
is
the Newtonian potential. In spherical coordinates, the partial derivatives in
and
are easily
defined
and the integrals simply become summations over the grid space.
In three dimensions, we can simplify this equation:
where
and
, where
where i and j are the coordinate axes.
For smooth particle hydrodynamics (SPH), this formalism becomes [45]:
where
is the distance to the source. The angle brackets in these equations denote averaging over all
source orientations, so that for example
The quantities
are the second time derivatives of the trace-free quadrupole moment of the source. The
diagonal elements are given by, for example,
where
is the mass of an SPH particle, and
is its coordinate location. The other diagonal
elements can be obtained using Equation (18) plus cyclic permutation of the coordinate labels:
via
,
,
;
via
,
,
. The off-diagonal elements are
given by, for example,
the remaining off-diagonal elements can be found via symmetry (
) plus cyclic permutation. This
approach provides only averaged square amplitudes, instead of the specific amplitude of the GW
signal.
For GWs from neutrinos, many groups use [208
] (based on the original analysis of Epstein [76
] and
Turner [317
]):
where
is the gravitational constant,
is the speed of light and
is the distance of the object.
denotes the emergent strain for an observer situated along the source coordinate frame’s z-axis
(or pole) and
is the comparable strain for an observer situated perpendicular to this z-axis (or
equator).
In SPH calculations, these expressions reduce to:
where
is the timestep, the summation is over all particles emitting neutrinos that escape this star
(primarily, the boundary particles). There are some assumptions about the emission of the neutrinos in this
model, but Fryer et al. [107
] argued that these assumptions lead to errors smaller than a factor of
2. The
correspond to observers along the positive/negative directions of each axis: for
instance, in the polar equation, this corresponds to the positive/negative z-axis. In the equatorial
region, the x,y-axis is determined by the choice of x or y position for the (x,y) coordinate in the
equation.
The signal arising from neutrino emission is an example of a GW burst “with memory”, where the
amplitude rises from a zero point and ultimately settles down to a value offset from this initial value. The
noise sources and optimal data analysis techniques for these signals will differ from, for example, signals
from bounce that do not have memory [25, 80].
Other methods have been developed to calculate the GW emission (e.g., Regge variables [328, 121],
Zerilli equations [346], Moncrief metric [202], see [214] for a review). Shibata and collaborators (e.g., [277])
have used the gauge-invariant Moncrief variables in flat spacetime:
where
where
are functions of
and
and are two-sphere integrations with
dependencies on the coordinates
and the Lorentz factor
. Most of these are defined in Shibata &
Nakamura [280]. With these gauge-invariant variables, we can derive the energy emitted in GWs:
Shibata & Sekiguchi [282
] define a number of other useful quantities for determining GW emission using
this formalism.