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 (1View Equation) 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 [208Jump To The Next Citation Point] (see also  [306Jump To The Next Citation Point, 201Jump To The Next Citation Point]). Here they define the gravitational quadrupole radiation field (TT h):

∘ - 2 E2 h𝜃𝜃 = 1∕8 (15 ∕π)sin 𝜃A 20∕r, (11 )
where r is the distance between the observer and the source, 𝜃 is the angular coordinate in our r − 𝜃 coordinate system, and E2 A20 is given by:
E2 4 3∕2 √ --- ∫ 1 ∫ ∞ 2 2 A 20 = (G∕c )(16π ∕ 15) ρ(r,μ,t)[vrvr(3μ − 1 ) + v𝜃v𝜃(2 − 3μ ) − vϕvϕ (12 ) ∘ ------- −1 0 ∘ ------- − 6vrv𝜃μ 1 − μ2 + r∂ Φ∕∂r (3μ2 − 1) + 3∂Φ âˆ•∂𝜃μ 1 − μ2]r2drdμ,
where μ = cos𝜃, vr,v𝜃,vϕ are the velocities along the ordinate axes in spherical coordinates and Φ is the Newtonian potential. In spherical coordinates, the partial derivatives in 𝜃 and r are easily defined3 and the integrals simply become summations over the grid space.

In three dimensions, we can simplify this equation:

TT 2 2 2 2 2 2 h ij (X, t) = 1∕R [(d Ixx∕dt − d Iyy∕dt )e+ + 2d Ixy∕dt ex] (13 )
where e = e ⊗ e − e ⊗ e + 𝜃 𝜃 ϕ ϕ and e = e ⊗ e + e ⊗ e x 𝜃 𝜃 ϕ ϕ, where
∫ d2Iij∕dt2 = G ∕c4 d3x ρ(2vivj − xi∂jΦ âˆ•∂j − xj∂Φ âˆ•∂i), (14 )
where i and j are the coordinate axes.

For smooth particle hydrodynamics (SPH), this formalism becomes [45]:

c8∕G2 ⟨(rh+ )2⟩ = 4-(I¨xx − ¨Izz)2 +-4-(¨Iyy − I¨zz)2 + 1-(¨Ixx − ¨Iyy)2 15 15 10 14- ¨ 2 4--¨ 2 4--¨ 2 + 15(Ixy) + 15(Ixz) + 15(Iyz) , (15 )
8 2 2 1 2 2 2 4 2 4 2 c ∕G ⟨(rh×) ⟩ = 6(I¨xx − ¨Iyy) + 3(¨Ixy) + 3-(¨Ixz) + 3-(I¨yz) , (16 )
where r is the distance to the source. The angle brackets in these equations denote averaging over all source orientations, so that for example
∫ ⟨(rh+ )2⟩ = 1-- dΩr2h+ (𝜃,ϕ)2. (17 ) 4π
The quantities ¨Iij are the second time derivatives of the trace-free quadrupole moment of the source. The diagonal elements are given by, for example,
2 ∑N ¨Ixx = -- mp(2xp ¨xp − yp¨yp − zpz¨p + 2x˙2p − y˙2p − ˙z2p), (18 ) 3 p=1
where mp is the mass of an SPH particle, and (xp,yp,zp) is its coordinate location. The other diagonal elements can be obtained using Equation (18View Equation) plus cyclic permutation of the coordinate labels: ¨Ixx → I¨yy via x → y, y → z, z → x; I¨xx → ¨Izz via x → z, z → y, y → x. The off-diagonal elements are given by, for example,
∑N ¨Ixy = mp (xp¨xp + yp¨yp + 2x˙py˙p); (19 ) p=1
the remaining off-diagonal elements can be found via symmetry (¨Iji = I¨ij) 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 [208Jump To The Next Citation Point] (based on the original analysis of Epstein [76Jump To The Next Citation Point] and Turner [317Jump To The Next Citation Point]):

2G ∫ t−R∕c (hTTxx )pole =-4- dt′ c∫ r −∞ ′ ′ ′ dLν(Ω-′,t′) d Ω (1 + cos𝜃 )cos(2ϕ ) dΩ ′ , (20 ) 4π
∫ ∫ TT 2G t−R∕c ′ ′ ′ ′ (hxx )equator = -4- dt dΩ (1 + sin 𝜃 cosϕ ) c r −∞ 4π cos2-𝜃′ −-sin2𝜃′sin2ϕ-′dLν(Ω-′,t′) cos2 𝜃′ + sin2𝜃′sin2ϕ ′ dΩ ′ , (21 )
where G is the gravitational constant, c is the speed of light and r is the distance of the object. (hTxTx )pole denotes the emergent strain for an observer situated along the source coordinate frame’s z-axis (or pole) and (hTT)equator xx is the comparable strain for an observer situated perpendicular to this z-axis (or equator).

In SPH calculations, these expressions reduce to:

TT 2G ∑ ∑N 2 2 (hxx )pole =-4-- Δt (1 ± z∕r)(2x ∕r − 1)ΔL ν, (22 ) cR p=1
∘ ---------- 2G ∑ ∑N ∘ ---------- z2 ∕r2 − 1 − z2∕r2(x,y)2∕r2 (hTTxx )equator =-4-- Δt (1 ± 1 − z2∕r2(x,y)∕r)--2--2--∘-------2--2-----2--2ΔL ν, (23 ) c R p=1 z ∕r + 1 − z ∕r (x,y) ∕r
where Δt 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. [107Jump To The Next Citation Point] 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:

E ∘ ----------------- Rlm (t,r) = 2 (l − 2)!∕(l + 2)!(4k2lm + l(l + 1 )k1lm) (24 )
O ∘ ----------------- Rlm(t,r) = 2(l + 2)!∕(l − 2)!(2Clm ∕r + r∂∕∂rDlm ) (25 )
where
k1lm ≡ Klm + l(l + 1)Glm + 2r∂∕∂rGlm − 2h1lm ∕r, (26 )
k2lm ≡ Hlm ∕2 + l(l + 1)Glm + 2r∂∕∂rGlm − 2h1lm ∕r, (27 )
where Clm,Dlm, Klm, Glm, Hlm are functions of r and t and are two-sphere integrations with dependencies on the coordinates r,ϕ, 𝜃 and the Lorentz factor γ. Most of these are defined in Shibata & Nakamura [280]. With these gauge-invariant variables, we can derive the energy emitted in GWs:
dE ∕dt = r2∕32π ∑ [|∂RE ∕∂t |2 + |∂RO ∕ ∂t|2]. (28 ) lm lm l,m
Shibata & Sekiguchi [282Jump To The Next Citation Point] define a number of other useful quantities for determining GW emission using this formalism.


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