Gravitational waves from gamma-ray bursts
Maurice H.P.M. van Putten LIGO Project, NW17-161, 175 Albany St., Cambridge, MA 02139-4307 ABSTRACT We present a mechanism for long bursts of gravitational radiation from Kerr black holes surrounded by a torus. These systems are believed to form in corecollapse of massive stars in association with gamma-ray bursts. The torus catalyzes black hole-spin energy mostly into gravitational radiation, with a minor output in winds, thermal and neutrino emissions. Torus winds impact the remnant envelope of the progenitor star from within, which may account for X-ray emission lines and leaves a supernova remnant. The frequency in gravitational radiation satis?es fgw = 470Hz (ESN R /4×1051 )1/2 (0.1/β)1/2 (7M⊙ /MH )3/2 , where ESN R is the kinetic energy in the SNR, MH is the black hole mass and β ? 0.1 the initial ejection velocity, as detected in GRB 011211. Ultimately, this leaves a black hole binary surrounded by a SNR, which is conceivably illustrated by RX J050736-6847.8.
arXiv:astro-ph/0301607v1 30 Jan 2003
There is increasing evidence for a GRB-supernova association with young massive stars. These events are probably associated with the prompt formation of a Kerr black hole in core-collapse (Woosley 1993). The dissipative timescale of spin-energy in the horizon of the black hole agrees with the durations of tens of seconds of long GRBs, and a small fraction of the baryon-free energy in angular momentum released along the axis of rotation is consistent with the inferred baryon-poor input to the observed (Frail et al., 2001) Eγ ? 3 × 1051 erg in GRB-energies (van Putten, 2001; van Putten & Levinson, 2003). Kerr black holes produce luminous output in gravitational radiation, winds, thermal and neutrino emissions through a surrounding torus. These torus emissions are contemporaneous with forementioned minor output in baryon-poor out?ows along the axis of rotation of the black hole. Gravitational radiation forms a major output of the system, representing approximately 10% of the rotational energy Erot of the black hole. About 1 ?2%Erot is emitted in MeV-neutrinos. Both these emissions do not a?ect the environment. The output in torus
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winds of about 1%Erot impact the remnant stellar envelope from within with continuum radiation and with kinetic energy. We recently suggested that this continuum radiation from within may be the source of X-ray line-emissions, when the envelope has expanded and goes through a transition from optically thick to optically thin. This proposal is similar but not identical to the re?ection model, as discussed by (Lazzati et al., 2002) where line-emission is excited by incidence of continuum radiation on the surface of an optically thick slab. The e?ciency of continuum-to-line emission is generally on the order of a few percent in such re?ection processes. Based on Lazzati et al. (2002), Ghisellini et al. (2002) discuss a model-dependent analysis of required continuum-energies Ec for excitation of line-emissions in GRB 970508 (Piro et al., 1999), GRB 970828 (Yoshida et al., 1999), GRB 991216 (Piro et al., 2000), GRB GRB 000214 (Antonelli 2000) and GRB 011211 (Rees et al. 2002). This indicates energies Ec ≥ 4 × 1052 erg in GRB 991216, pointing towards an energy reservoir in excess of that required for the GRB-energies Eγ , assuming a canonical e?ciency of kinetic energyto-gamma rays of about 15%. This supports the notion that the GRB inner engine is processing other channels of emissions, in addition to and in excess of baryon-poor input to GRB-afterglow emissions. The predicted long burst in gravitational radiation associated with GRB-supernovae (“hypernovae”) may be detected by upcoming gravitational wave-experiments. We mention broad band laser interferometric instruments LIGO (Abramowici1992), VIRGO (Bradaschia 1992), TAMA (Massaki 1991) and GEO (e.g., Schutz & Papa, 1999), and bar or sphere detectors presently under construction. Calorimetry on this energy output provides a method for testing the existence of Kerr black holes as objects in nature. Calorimetry on hypernova remnants may further provide constraints on the angular velocity of the torus and hence its frequency in gravitational radiation. This serves to design and optimize speci?c search strategies for the accompanying bursts of gravitational radiation. Remnants of hypernovae are predicted to be black hole binaries surrounded by a supernova remnant. An example of this morphology is RX J050736-6847.8 (Chu et al. 2001). In Section 2, we review radiation from Kerr black holes surrounded by a torus. In Section 3, we calculate a supernova connection from the the impact of the torus winds on the remnant envelope of the progenitor star. We propose a remnant in Section 4, and conclude with suggestions for future observations in Section 5.
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Number of bursts
Fig. 1.— Shown is the histogram of redshift corrected durations of 27 long bursts with individually determined redshifts from their afterglow emissions. We identify these long durations with the lifetime of rapid spin of a Kerr black hole in a state of suspended accretion. These durations are e?ectively de?ned by the rate of dissipation of black hole-spin energy in the horizon – an unobservable sink of energy – subject to a new magnetic stability criterion for the torus. (Reprinted from M.H.P.M. van Putten, 2002, ApJ, 575, L71.)
APS Conf. Ser. Style 2. Black hole-beauty
A torus around a rotating black hole becomes radiant by three-fold equivalence to pulsars: in poloidal topology, causality and ms rotation periods. In accord with Mach’s principle, the inner face of the torus hereby receives input from the black hole – a nearby compact in?nity with non-zero angular velocity – and the outer face radiates to asymptotic in?nity. In response, conservation of energy and angular momentum causes the black hole to spin down, i.e.: the black hole becomes luminous. See (van Putten & Levinson, 2003) for a theory on tori surrounding rapidly spinning black holes. The torus develops a state of suspended accretion, while the black hole spins rapidly. The equations of suspended accretion permit solutions with a positive gravitational waveluminosity, produced by multipole mass-moments in the torus. These emissions in gravitational radiation and winds are further accompanied by thermal and neutrino emissions, as dissipation in the torus develops MeV-temperatures. Most of the spin-energy of the black hole is dissipated in the horizon. This dissipation is rate-limited, and sets a lower bound of tens of seconds on the lifetime of its spin and, hence, the emissions by the torus. Most of the black hole-luminosity is incident onto the torus, whereby the torus’ emissions form the major energy output of the system. Gravitational radiation, in turn, de?nes most of the output from the torus, and represents about 10% of the spin-energy of the black hole. A small fraction of black hole-spin energy is released along the axis of rotation in the form of baryon-poor out?ows along an open magnetic ?ux-tube. This magnetic ?ux-tube extends from the horizon to in?nity. A horizon half-opening angle corresponding to the curvature in poloidal topology of the inner torus magnetosphere de?nes an energy output in baryon-poor out?ows of about 0.1% of the rotational energy of the black hole. This is in quantitative agreement with the observed value of Eγ ? 3 × 1051 erg in GRB-energies, assuming a canonical e?ciency of kinetic energy-to-gamma rays of about 15%. We stress that our model for GRBs from rotating black holes accounts quantitatively for both the long durations relative to the Keplerian timescale (order parameter ? 104 ), as well as the small output in GRB-energies relative to the rotational energy of the black hole (order parameter ? 10?3 ). The model predicts a correlated output in gravitational radiation and torus winds. In particular, their energies satisfy (van Putten (2003), corrected and simpli?ed) Egw αη ?η ? 2 Erot α(1 + δ) + fw (1)
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in the limit of strong viscosity (large α) and small slenderness (small δ). Here fw denotes the fraction of open magnetic ?ux supported by the torus which connects to in?nity, δ = b/2a is a ratio of minor-to-major radius of the torus and η denotes the ratio of the angular velocity of the torus to that of the black hole, which satis?es η ? 1/4α for large α. The energy emissions in winds satis?es
2 Ew ηfw (1 ? δ)2 ? η2 ? 2 Erot α(1 + δ) + fw
in the same limit considered in (1), applied to the case of a symmetric ?ux-distribution with 1 fw = 2 . In dimensionful form, these energy emissions are Egw ? 4×1053 erg (η/0.1) (MH /7M⊙ ) and Ew ? 4 × 1052 erg (η/0.1)2 (MH /7M⊙ ) , The frequency of quadrupole gravitational radiation fgw and the energy in winds are related by fgw Ew ? 470Hz 4 × 1052 erg
For the purpose of calorimetry on Ew , we set out to identify observable signatures of the impact of the torus winds on its surroundings.
Prompt core-collapse in young massive stars in binaries may produce Kerr black holes surrounded by a disk or torus, formed from matter stalled against an angular momentum barrier (Brown et al., 2000). Prompt collapse takes place on a free-fall time, or less through the agency of magnetic ?elds (see, e.g., van Putten & Ostriker 2001). As a result, the newly formed black hole-torus system is initially surrounded by a remnant stellar envelope. Spinenergy of the black hole released in the form of torus winds promptly impacts this stellar envelope from within. The kinetic energy and radial momentum of the torus winds provides a powerful mechanism to eject matter, producing a supernova associated with the underlying GRB. GRB 991216 and GRB 011211 both show initial ejection velocities β ? 0.1, relative to the velocity of light, as observed in blue-shifted X-ray line-emissions. This blue-shift de?nes the e?ciency β/2 of conversion of an energy Ew in essentially luminal torus winds to a kinetic energy ESN R of matter ejecta. This leaves ESN R ? (1/2)βEw ? few ×1051 erg from Ew = few ×1052 erg. These values are remarkably similar to the kinetic energies in non-GRB supernova remnants. Because the e?ciency β of Ew to ESN R is somewhat larger than the e?ciency of continuum-to-line emissions, it is a preferred method for calorimetry on torus winds.
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The energetic impact on the envelope provides a source of continuum emission for excitation of X-ray lines, and deposits kinetic energy. The above shows that both of these processes are remarkably ine?cient. Excitation of X-ray lines by continuum emissions has an estimated e?ciency of less than one percent (Ghisellini 2002). Deposition of kinetic energy by approximately luminal torus winds has an e?ciency of β, denoting the ejection velocity relative to the velocity of light. Matter ejecta in both GRB 991216 (Piro et al., 2000) and GRB 011211 (Reeves, 2002) show an expansion velocity of β ? 0.1. The e?ciency of kinetic energy deposition of the torus wind onto this remnant matter is hereby β/2 = 5%. With Ew as given in (2), this 1 predicts a supernova remnant with ESN R ? 2 βEw ? 2 × 1051 erg, which is very similar to energies of non-GRB supernovae remnants. We emphasize that ultimately, this connection is to be applied the other way around: obtaining estimates for Ew from kinetic energies in a sample of supernova remnants around black hole binaries, by assuming that β ? 0.1 holds as a representative value for the initial ejection velocity obtained from Ew . This assumption may be eliminated by averaging over observed values of β in a sample of GRB-supernova events with identi?ed line-doppler shifts.
Remnants of beauty
The angular momentum in rapidly spinning black holes produced in core-collapse probably derives from orbital angular momentum, following prior interaction of the young massive star with a binary companion (Paczynski 1998; Brown 2000). Therefore, the end product of the black hole-supernova is a black hole binary (a black hole and a low mass companion) surrounded by a supernova remnant, accompanied by a burst of gravitational radiation as an echo in eternity. It becomes of interest, therefore, to search for black hole binaries in supernova remnants. A particularly striking example of an X-ray binary surrounded by a supernova remnant is RX J050736-6847.8. The above suggests that this X-ray binary may harbor a black hole.
We present a mechanism for long-duration bursts (of tens of seconds) of gravitational radiation powered by Kerr black holes. This emission is catalyzed by a surrounding torus, in association with minor outputs in winds, thermal and MeV neutrino emissions. A small fraction of the black hole-spin energy is released contemporaneously along the axis of rotation.
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We propose this as a model for GRB-supernovae (“hypernovae”), following Woosley’s (1993) scenario of black hole formation in core-collapse of massive stars (Paczynski 1998, Brown 2000). The proposed model accounts quantitatively for the secular timescale of long GRBs and the relatively small GRB-energies. The torus emissions in various energy channels are correlated with the Keplerian angular velocity, which establishes a relation between windenergies and the frequency of quadrupole gravitational radiation. This provides an avenue to use calorimetry on torus winds and its remnants in support of predicting frequencies of gravitational radiation. This is intended for designing optimal detection and search strategies in upcoming gravitational wave-experiments. The interaction of the torus wind with the remnant evelope may further produce a burst in radio emission, which could be a source of interest for LOFAR/SKA. Some constraints might be placed on the statistics of these bursts by the recent analysis of Levinson et al. (2002). The model predicts hypernova remnants in the form of black hole binaries surrounded by a supernova remnant. A candidate of particular interest is RX J050736-6847.8. It becomes of interest to pursue searches for a sample of such X-ray binaries surrounded by supernova remnants, of which some may harbor black hole binaries. The author thanks G. Mendell for useful comments, and S. Kim for drawing attention to RX J050736-6847.8. This research is supported by NASA Grant 5-7012, a NATO Collaborative Linkage Grant and an MIT C.E. Reed Fund. The LIGO Observatories were constructed by the California Institue of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation under cooperative agreement PHY 9210038. The LIGO Laboratory operates under cooperative agreement PHY-0107417. This paper has been assigned LIGO Document Number LIGO-P030003-00-R.
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Black hole-spin energy
Baryon poor outflows
Collimating Irradiation of winds remnant envelope
Chemical enhancement of companion star
X-ray emission lines
Soft X-ray transient
Fig. 2.— A tree of black hole-spin energy, catalyzed by a torus: most energy is dissipated in the horizon, most of the output is in gravitational radiation, accompanied by a minor output in winds, thermal and neutrino emissions. A small fraction is released in baryon poor out?ows as input to GRB-afterglow emissions. Direct measurement of gravitational radiation by upcoming gravitational wave-experiments provides a calorimetric test for Kerr black holes (dark connections). Calorimetry on torus winds (below the dashed line, incomplete or unknown) provides an avenue for constraining the angular velocity of the torus, and hence its frequency of gravitational radiation. Ultimately, this leaves a black hole binary, possibly an SXT, surrounded by an SNR. (Reprinted from van Putten & Levinson, ApJ, 2003, in press.)
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A This preprint was prepared with the AAS L TEX macros v5.0.