The Gravitational Wave Signatures of Core Collapse SupernovaeMikhail BelyaevPrinceton [email protected], October 1, 2009Presentation Overview•Gravitational wave physics•Processes in core-collapse supernovae that produce gravitational waves•Using gravitational wave signatures to experimentally probe interior regions of the explosion2Thursday, October 1, 2009Gravitational Waves in Linear Theory•Start with Einstein’s field equations: •Assume a form for the metric which is •In the field equations, only keep terms that are linear in•Applying gauge conditions in vacuum yields a wave equation•Plane waves are solutions to this equation•Properties of gravitational waves in the TT gauge•Waves are transverse to their direction of propagation•Waves propagate at the speed of light•Waves have two polarizationsgµν= ηµν+ hµνhµνGµν=8πGc4Tµν(1c2∂2∂t2−∇2)hTTµν=0hTTµν= Aµνei(k·x−ωt)3Thursday, October 1, 2009The Effect of a Gravitational Wave on an Inertial Ring of Test Particles•Consider two inertial test particles A and B separated by a vector x. The motion of B relative to A in the proper reference frame of A is given byxj(τ )=xk(0)!δjk+12hTTjk"Figure 35.2 of Misner, Thorne, and Wheeler h×= A×0 0 0 00 0 1 00 1 0 00 0 0 0eikz−ωth+= A+0 0 0 00 1 0 000−100 0 0 0eikz− ωt4Thursday, October 1, 2009Quadrupole Formula for GW Strain•In the limit •Ex: radiation from a rotating dumbbell •Consider two point masses of mass m separated by a distance l rotating around their center of mass with angular frequency omega. The dimensionless strain at a distance D is given by•Often the dimensionless strain will be multiplied the distance to the source to give a length which is independent of source distance.2πRλ! 1 ⇐⇒ v ! c,GMRc2!2πRλ,!!Tjk!!T00!2πRλh+=3.2 × 10−21,D= 10 kpc,h+D = 100 cmhTTjk∼2ml2ω2GDc4=!1.7 × 10−49s2cm g"ml2ω2DhTTjk=2GDc4¨ITTjk(t − D/c),Ijk≡!AmA"xAjxAk−13δjkr2A#5Thursday, October 1, 2009Energy Carried by Gravitational Waves•Energy in electromagnetic waves cannot be localized, but it is possible to define a stress energy density averaged over several wavelengths.•For a plane wave propagating along z:•The time-averaged energy flux at the detector is then simply •It is possible to rearrange this expression to find a “characteristic strain” based on the spectral energy density in the gravitational waves. A rigorous treatment (Flanagan et. al. 1998) yieldsT(GW )µν=132π!hTTjk,µhTTjk,ν"T(GW )tt= T(GW )zz= −T(GW )tz=c232πGω2!|A+|2+ |A×|2"dEdAdt=c332πGω2!|A+|2+ |A×|2"hchar(f) ≡1D!2Gπ2c3dEGWdf(SNR)2optimal=!∞0d ln fh2char(f)fS(f)6Thursday, October 1, 2009Gravitational Wave Luminosity•Formula for power emitted as gravitational quadrupole radiation: M : mass involved in motions that change the quadrupole moment R : characteristic radius for the motions T : characteristic timescale or period for the motions Lint : characteristic power in motions that change the quadrupole moment L0 = 3.6x1059 erg/sec : characteristic luminosity in general relativity.•This means that energy losses from gravitational radiation will be important over the course of one period if •In supernovae, , so even taking into account the fact that processes producing gravitational radiation occur at frequencies between 100-1000 Hz and the time to explosion is approximately a second, gravitational radiation carries away a negligible fraction of the energy.•The energy carried away by gravitational waves varies significantly by process but is of the order . This is much less than the 3x1053 ergs of binding energy or even the 1051 ergs of kinetic energy.LGWQuad=G5c5!...Ijk...Ijk"∼Gc5!MR2T3"2∼LintL0LintLint∼ L0Lint! L010−8M"c2=1.8 × 1046erg7Thursday, October 1, 2009Gravitational Waves from Core Collapse Supernovae8Thursday, October 1, 2009Processes in Core Collapse SupernovaeTime after bounce (ms)Radius (km)0408012016020024060120 180 240 300 360 420480 540SASINeutrino Driven ConvectionPrompt ConvectionPNS Convectiong-modesDifferential Rotation?secular/bar instabilities9Thursday, October 1, 2009Rotating Core Collapse and Bounce•A rotating core has an oblate l=2 deformation. Upon bounce this leads to rapid variations in the reduced quadrupole moment and GW emission. Class. Quantum Grav. 26 (2009) 063001 Topical Review01020-300-250-200-150-100-50050100150200t - tbounce(ms)h+D (cm)s11A2O09s15A1O07s20A2O09s20A3O12s40A2O13E20AFigure 2. GW signals (h+D in units of cm, where D is the distance of the source) for a fewexamples from the 2D GR model set of Dimmelmeier et al [108]. The models shown here werecomputed with the Shen EOS [135, 136] and employ 1D presupernova models of [137], spanningthe progenitor mass range from 11.2 M (s11) to 40 M (s40). The models were set up withprecollapse central angular velocities !c,ifrom 1.5 rad s− 1to 11 rad s− 1. For details of therotational setup, see [108]. Model E20A uses a 20-M presupernova model that was evolved by[138] with a 1D prescription for rotation. Note the generic shape of the waveforms, exhibitingone pronounced spike at core bounce and a subsequent ring down. Very rapid precollapse rotation(!c,i! 6 rad s− 1; models s20A3O12 and s40A2O13 in this plot) results in a significant slow-down of core bounce, leading to a lower-amplitude and lower-frequency GW burst. The GW signaldata are available for download from [126].Table 1. Summary of the GW signal characteristics of rotating iron core collapse and corebounce based on the waveforms of Dimmelmeier et al [108]. All models exhibit type-I dynamicsand waveform morphology and can be organized into three distinct groups based primarily ontheir precollapse central angular velocity !c,i. |hmax| is the maximum gravitational-wave strainamplitude (scaled to 10 kpc) at bounce, EGWis the energy radiated away in gravitational waves,fpeakis the frequency at which the GW energy spectrum dEGW/df peaks, and "f50is the frequencyinterval centered about fpeakthat contains 50% of EGW. Note that fmaxused by Dimmelmeier et alis the peak of the GW signal