KERR PARAMETERS FOR STELLAR MASS BLACK HOLES AND THEIR CONSEQUENCES FOR GAMMA-RAY BURSTS AND HYPERNOVAE

Enrique Moreno Méndez; Gerald E. Brown; Chang-Hwan Lee; Frederick M. Walter

doi:10.1088/0004-637X/727/1/29

1. INTRODUCTION

Recent estimates of the Kerr parameters a_⋆ by Shafee et al. (2006) for two soft X-ray transients (SXTs), GRO J1655−40 and 4U 1543−47 (IL Lupi), facilitate a test of stellar evolution. The spins of the black holes in these binaries should be produced during common-envelope evolution which begins with the evolving massive giant and companion donor, and terminates in a helium-star binary, the hydrogen envelope of the massive star having been stripped off and the helium in the core having been burned.

Lee et al. (2002; hereafter denoted as LBW) assumed common-envelope evolution to begin only after He core burning has been completed, i.e., Case C mass transfer (Brown et al. 2001b). Otherwise the He envelope, if laid bare, would be blown away to such an extent that the remaining core would not be sufficiently massive to evolve into a black hole (Brown et al. 2001a). The black hole progenitor, with helium core burning having been completed, is tidally locked with the donor (secondary star) so the spin period of the helium star is equal to the orbital period of the binary. In this process, LBW assumed uniform rotation of He star by postulating that the core and envelope of the He star are strongly connected by a strong internal magnetic field. The core of the helium star drops into a rapidly spinning black hole due to angular momentum conservation. The spin of the black hole depends chiefly on the mass of the donor because the orbital period depends on the donor mass (see Section 3). LBW calculated this Kerr parameter (a_⋆) as a function of the binary orbital period in their Figure 12.

In the LBW calculation, the a_⋆ for GRO J1655−40 (Nova Sco 94) was slightly greater than for 4U 1543−47 (see Figures 11 and 12 of LBW) at the time of the collapse. We also note that LBW predicted a_⋆>0.6 for seven SXT sources with main-sequence companions and a_⋆ ≅ 0.5 for XTE J1550−564 (V381 Normae) and GS 2023+338 (V404 Cygni), both of which have evolved companions. The agreement of these predicted natal Kerr parameters with those measured by Shafee et al. (2006)⁴ means that only a small amount of rotational energy could have been lost after the formation of the black hole. This supports the assumption of Case C mass transfer and tidal locking at the donor-He star stage assumed in the LBW calculations.

We assume that the Blandford–Znajek (Blandford & Znajek 1977) mechanism is responsible for extracting sufficient rotational energy from the pre-black-hole binary to power the explosion. The maximum available energy from this mechanism (see Thorne et al. 1986 and Appendix A) for a_⋆>0.5 is E>3 × 10⁵³ erg, orders of magnitude larger than observed in and hence sufficient to power any GRB/hypernova explosion. Based on this consideration, we interpret the SXTs to be relics of gamma-ray bursts (GRBs) and hypernovae. Generally, GRBs are divided into two groups according to the duration time; the GRBs that are relevant to our model are the long GRBs whose duration time is generally longer than 2 s. There are many GRB models with rapidly rotating neutron stars (e.g., Bucciantini et al. 2009); however, most likely, these would not produce black hole binaries seen as SXTs. So, we focus on the Woosley collapsar model for the development of our model.

It should be noted that the way in which the hypernova explodes can be similar to the Woosley collapsar model (Woosley 1993). In our scenario the H envelope is removed by the donor and the rotational energy is naturally produced in the common envelope phase. The necessity for Case C mass transfer, given galactic metallicity and the measured system velocity (provided by the Blaauw–Boersma kick (Blaauw 1961; Boersma 1961) at the time of the formation of the black hole (Brown et al. 2000)) lock us into the Kerr-parameter values we find.

In Section 2, we show how to determine the Kerr parameters of SXT black hole binaries and the connection of tidal locking to Case C mass transfer. We also discuss the energetics for GRBs and hypernovae based on the black hole spin.

In Section 3, we show that the rotational energy of the black hole binary is determined mainly by the mass of the donor. We discuss 12 galactic transient sources with rotational energies ⩾10⁵³ erg, all of which are likely relics of GRBs and hypernovae. The difference between the energies of the natal rotation of the black hole and the GRB and hypernova explosion provides the observed available rotational energy.

2. CASE C MASS TRANSFER AND TIDAL LOCKING

In Case C, mass transfer takes place during He-shell burning in the progenitor of the black hole. Aside from the fact that we use Case C mass transfer to achieve the tidal coupling of the donor to the black hole progenitor, the rest of our scenario, especially the collapse, is the same as in the Woosley collapsar model. Nevertheless Case C mass transfer and tidal locking are essential in order to acquire the necessary spin during the collapse of the He star into the black hole.⁵

The hypernova explosions associated with GRBs are all of type Ic, in which He lines do not appear. In our Case C mass transfer, as also in the Woosley collapsar model, all the He has not necessarily burned before explosion. The lack of observed He in the spectra may cause a problem in our model. There are two possibilities: (1) the interacting He may fall into the black hole or (2) the He may not mix with the ⁵⁶Ni, so that in either case He lines would not be seen.

In our estimations (Brown et al. 2000), the black hole formation was described by a Blaauw–Boersma explosion and we did not include the effect of natal kick which turned out to be very important for a few systems (Brandt et al. 1995; Nelmans et al. 1999; Gualandris et al. 2005).⁶ If we include the natal kick, the eccentricity after the explosion will be increased. This results in a smaller orbital period which then helps maintain the tidal synchronization before the explosion

2.1. Soft X-ray Transients as Relics of GRBs and Hypernovae

LBW found that there are two classes of SXTs, those with low-mass main-sequence companions⁷ (denoted as AML, for angular momentum loss) and others with more massive evolved companions (denoted as Nu, for nuclear evolved).

2.1.1. AMLs

Due to angular momentum loss via gravitational wave radiation and magnetic braking, after black hole formation the orbital periods of AMLs decrease. LBW used this to trace back the orbital period at the time of black hole formation in their Figure 10. The estimated Kerr parameters for AMLs are a_⋆>0.6 (about half of them are a_⋆>0.8). The maximum available energies for these systems via the Blandford–Znajek formalism are E>3 × 10⁵³ erg. The AMLs are likely the relics of GRBs and hypernovae.

2.1.2. Nu's

The evolution of Nu's after black hole formation is mainly controlled by the donor. As the donor evolves beyond the main-sequence stage, the orbit widens due to the conservative mass transfer from the donor to the black hole. We will discuss the possibility that Nu's are also relics of GRBs and hypernovae.

We briefly reconstruct the case that GRO J1655−40 is a relic of a GRB/hypernova. This was discussed in considerable detail by Brown et al. (2000) who assumed a Kerr parameter of a_⋆ = 0.8. Note that Shafee et al. (2006) give the value for a_⋆ after the GRB and hypernova explosion, whereas Brown et al. (2000) give the pre-explosion value. The two are not significantly different, since the explosion can be powered by the energy from an ∼5% change in a_⋆ when a_⋆ is large. That GRO J1655−40 may have formed in a hypernova explosion is suggested by the existence of α-particle nuclei enrichment in the donor. Israelian et al. (1999) found that the F-star donor has O, Mg, Si, and S abundances 6–10 times solar. These nuclei were presumably absorbed by the donor, which acts as witness to the explosion.⁸

Due to the similarity in the orbital periods of GRO J1655−40 and 4U 1543−47 (see Figure 11 of LBW), we argue that 4U 1543−47 is also a relic of a GRB and hypernova. Although the black hole masses are not known as well in XTE J1550−564 (V383 Normae) and GS 2023+338 (V404 Cygni), it can be seen from Figures 11 and 12 of LBW that using their reconstructed pre-explosion periods they have a Kerr parameter of a_⋆ ≅ 0.5, possibly somewhat less definite than the prediction of the a_⋆ for GRO J1655−40. The latter two binaries have black holes with twice the mass as the first two and smaller a_⋆'s, and, therefore, larger accretion disks (since the Schwarzschild radius is proportional to M_BH and therefore also the Innermost Stable Circular Orbit (ISCO) increases with M_BH; also, slowly rotating BH has a larger radius than a rapidly rotating BH for the same mass). From our arguments in Appendix B, we believe that they may be able to accept more rotational energy.

Brown et al. (2000) remarked that for GRO J1655−40, "After the first second the newly evolved black hole has ∼10⁵³ erg of rotational energy available to power these. The time scale for delivery of this energy depends (inversely quadratically) on the magnitude of the magnetic field in the neighborhood of the black hole, essentially that on the inner accretion disk. The developing supernova explosion disrupts the accretion disk; this removes the magnetic fields anchored in the disk, and self-limits the energy the Blandford–Znajek mechanism can deliver." This, together with the total creation rate of binaries of this type of 3 × 10⁻⁴ galaxy⁻¹ yr⁻¹ estimated by Brown et al. (2000), is sufficient to reproduce the population of subluminous bursts in nearby galaxies (Section 3.4). We posit that the SXTs were subluminous, at least in the cases of GRO J1655−40 and 4U 1543−47, because their Kerr parameters measured by Shafee et al. (2006) are indistinguishable from the predicted natal a_⋆ = 0.8 within observational errors. Note that this large natal Kerr parameter indicates that the system had enough explosion energies to disrupt the disk. The disruption of black hole disk is discussed in detail in Appendix B.

Our evolutionary model for black hole binaries in our Galaxy might appear to be irrelevant for the long (high-luminosity) γ-ray bursts because Fruchter et al. (2006) show that these come chiefly from low metallicity, very massive stars in galaxies of low overall metallicity, quite unlike our Milky Way. However, we can construct a quantitative theory of the rotational energies of the black holes which power the central engine for the GRBs and hypernovae in our Galaxy because we can calculate the black hole Kerr parameters. Having this quantitative theory it is straightforward to apply it to explosions in low metallicity galaxies. There, high luminosities in GRBs result because the donors are more massive than those in our Galaxy, and therefore the black holes are able to supply more rotational energy before disrupting the accretion disks. In high metallicity binaries with low-mass donors, the excessive rotational energy destroys the central engine before it is able to deliver a high-luminosity GRB. However, there may be some overlap of the high metallicity systems with the same mass donor as in the low metallicity systems, as we shall discuss. We predict that this will be the case for XTE J1550−564. We expect that the energy used up in the explosion (the difference between our estimated energy and the measured one) should be nearly that of cosmological (high-luminosity) GRBs.

Recently the eclipsing massive black hole binary M33 X−7 was discovered in this nearby spiral galaxy (Orosz et al. 2007; Bulik 2007). Since the metallicity is ∼0.1 solar, we believe it to mimic low-metallicity stars which are more massive than the galactic ones. The donor has mass ∼70 M_☉ now, and was possibly ∼80 M_☉ earlier. We expect that this system may have gone through a dark explosion due to the high donor mass, which implies a low rotational energy at the time of formation of the black hole, as we discuss in the next section (see also Moreno Méndez et al. 2008).

2.1.3. Nu's: Evolution of Cyg X−1, XTE J1819−254, and GRS 1915+105

A discussion, highly relevant to the evolution of XTE J1819−254 (V4641 Sgr), which is itself a template of an earlier GRS 1915+105 development, was given by Podsiadlowski et al. (2003) for Cyg X−1. They discussed the latter as if the donor and black hole were very nearly equal in mass, which we shall show will happen in the future for Cyg X−1, although its donor is now ∼18 M_☉ and black hole ∼10 M_☉. The donor could have been substantially more massive when the black hole was born after common envelope evolution if it subsequently lost mass through a stellar wind.

The donor in the Podsiadlowski et al. (2003) scenario has a stellar wind of 3 × 10⁻⁶ M_☉ yr⁻¹ (Herrero et al. 1995) throughout the post-common envelope evolution. Once the mass of the donor decreases to a mass comparable to the mass of the black hole, the donor establishes thermal equilibrium and fills its Roche lobe, transferring mass at the rate of 4 × 10⁻³ M_☉ yr⁻¹. Because of the continuing wind loss the donor shrinks significantly within its Roche lobe and the system widens. The donor starts to expand again after it has exhausted all of the hydrogen in the core and fills its Roche lobe a second time. In this phase the mass transfer reaches a second peak of ∼4 × 10⁻⁴ M_☉ yr⁻¹, where mass transfer is driven by the evolution of the H-burning shell. The most interesting feature of this calculation is that the system becomes detached after the common envelope evolution. Since the donor is close to filling its Roche lobe, the wind may be focused towards the accreting black hole, as is inferred from the tomographic analysis of the mass flow in Cyg X−1 by Sowers et al. (1998).

Sowers et al. (1998) decompose the stellar wind of the supergiant into two moments, one representing the approximately spherically symmetrical part of the wind and the second representing the focused enhancement of wind density in the direction of the black hole. The latter component of the wind transfers mass in an essentially conservative way (although the former would bring about net mass loss). We shall employ a similar wind in both XTE J1819−254 and GRS 1915+105 to transfer matter from the donor to the black hole later on. Note that the wind, transferring matter at a hypercritical (much greater than Eddington) rate, basically shuts off the initial Roche lobe overflow. The second period of Roche lobe overflow transfer is driven by the evolution of the H-burning shell; i.e., by the secondary star becoming a red giant.

Podsiadlowski et al. (2003) say that "irrespective of whether this particular model is applicable to Cyg X−1, the calculation... illustrates that it is generally more likely to observe a high-mass black hole X-ray binary in the relatively long-lived wind mass-transfer phase following the initial thermal timescale phase which only lasts a few 10⁴ yr. In this example, the wind phase lasts a few 10⁵ yr, but it could last as long as a few 10⁶ yr if the secondary were initially less evolved." We believe that the above scenario applies not only to Cyg X−1, XTE J1819−254, and GRS 1915+105, but also to LMC X−1 and M33 X−7—all binaries with donors more massive than the black hole companion. ⁹ As we show below, these all likely had dark explosions because the donor had too high a mass at the time of the explosion to give an energetic GRB. We suggest that the wind in these cases may resemble the tidal stream of Blondin et al. (1991). In the two-dimension system studied by these authors, they show that when D/R_⋆ becomes less than ∼2, where D is the binary separation and R_⋆ is the radius of the donor, the tidally enhanced-wind accretion exceeds Bondi–Hoyle accretion (steady-state, spherical accretion), a factor that increases to several as D/R_⋆ decreases.

None of the three binaries we consider had appreciable natal a_⋆'s because of the high masses of the donors. The high Kerr parameter must come from mass accretion (see Figure 11 of LBW and Figure 6 of Brown et al. 2000) and the accreted mass must be slightly greater than the original black hole mass in order to result in such a high Kerr parameter for GRS 1915+105 (McClintock et al. 2006). Taking the current black hole mass to be 14 M_☉ we infer a natal mass of ∼6 M_☉.

If XTE J1819−254 is to be a template for GRS 1915+105, then its black hole would have had a mass of ∼6 M_☉ when born, and would have evolved to its present value of 9.6 M_☉ by accretion by wind from the donor. Thus, the mass ratio has become inverted through mass exchange from the donor to the black hole. As outlined in LBW, XTE J1819−254 will transfer by wind another 4.6 M_☉ from donor to black hole to match the present GRS 1915+105 with black hole mass ∼14 M_☉. The total mass accreted by GRS 1915+105 is estimated to be ∼8 M_☉. The companion mass evolved in this way is 1.9 M_☉, somewhat larger than the measured mass (Harlaftis & Greiner 2004) of 0.8 ± 0.5 M_☉. The missing mass may have been lost from the system in jets and winds, which is not taken into account in our approximation of conservative mass transfer.

Our evolution of the black hole birth is similar to the second evolution version of Sadakane et al. (2006) which gives the right chemical abundances to the secondary star in XTE J1819−254. They suggested that the black hole mass at birth was 7.2 M_☉, and to obtain the right surface abundances they proposed that the explosion was dark; i.e., of low energy. Mirabel & Rodrigues (2003) suggest that the explosion in Cyg X−1 was also dark, much less energetic than the one in GRO J1655−40. We shall generalize that the explosions are dark because the donors have high masses.

2.2. Energetics for Gamma-ray Bursters and Hypernovae

We will construct estimates of the spin energies for the transient sources in the next section, but we want to make some general comments here. The principal question with GRBs is whether there is enough angular momentum to power the GRB and hypernova explosion. In the case of the widely accepted theory, Woosley's collapsars, this question is unanswered, although one may take the point of view that we observe GRBs and hypernova explosions, so there must be enough angular momentum. Nonetheless, Heger et al. (2002) say that "when recent estimates of magnetic torques (Spruit 2002) are added, however, the evolved cores spin an order of magnitude slower. This is still more angular momentum than observed in young pulsars, but too slow for the collapsar model for gamma-ray bursts." Furthermore, the usual scenario for the interactions of Wolf–Rayets with other stars is that they slow down the rotation.

The hypernova formed in SN1998bw had ∼3 × 10⁵² erg in kinetic energy (Iwamoto et al. 1998). In addition, the jet formation in the GRB requires lifting all of the matter out of the way of the jet. MacFadyen (2000) estimates that this costs ∼10⁵² erg in kinetic energy. At early times the thermal and kinetic energies in supernova explosions are roughly equal, satisfying equipartition. We believe this to be at least roughly true in our explosions here, so that ∼6 × 10⁵² erg would be needed for GRB 980425/SN1998bw and possibly more, because MacFadyen (2000) describes GRB 980425 as a "smothered" explosion.¹⁰

As noted by Brown et al. (2007), the Blandford–Znajek efficiency drops substantially as the Kerr parameter decreases below a_⋆ ∼ 0.5. Thus, the available rotational energy will decrease rapidly with increasing donor mass. In Appendix B we estimate that the most rotational energy that can be accepted by a binary with a 5 M_☉ donor is of ∼6 × 10⁵² erg.

3. DONOR MASS AND BLACK HOLE SPIN ANTI-CORRELATION

3.1. Mass–Period Relation

Using the relation between the mass of the He core to the star,

$\begin{eqnarray} &&M_{\rm{He}}=0.08(M_{\rm Giant}{/}\mbox{$M_\odot $})^{1.45}\mbox{\,$M_\odot $}, \end{eqnarray} \tag{ 1 }$

LBW found that following common envelope evolution

$\begin{eqnarray} &&a_f=\left(\frac{M_d}{\mbox{\,$M_\odot $}}\right)\left(\frac{M_{\rm Giant}}{\mbox{\,$M_\odot $}} \right)^{-0.55}a_i. \end{eqnarray} \tag{ 2 }$

Here a_f is the final distance between the secondary star and the giant following the stripping of its H envelope, and a_i is its initial distance. The He core has inherited the rotational period of the He star and is tidally locked with the donor. The giant masses found by LBW were all about 30 M_☉. Using Kepler's laws we have the pre-explosion period (in days)

$\begin{eqnarray} &&\frac{1}{P_b}=\left(\frac{4.2\mbox{\,$R_\odot $}}{a_f}\right)^{3/2}\left(\frac{M_d+M_{\rm{He}}}{\mbox{\,$M_\odot $}}\right)^{1/2}. \end{eqnarray} \tag{ 3 }$

From Figure 12 of LBW one sees that the Kerr parameter increases sharply as the period of the binary P_b decreases. From Equation (2) we see that a_f is proportional to M_d, the donor mass, and from Equation (3) we see that P_b is proportional to a^3/2_f. Estimated Kerr parameters for galactic sources are summarized in Table 1, which suggests the dependence of a_⋆ on the donor mass for the galactic sources.

Table 1. Parameters at the Time of Black Hole Formation

Name	M_BH	M_d	Initial	E_BZ
	( M_☉)	( M_☉)	a_⋆	(10⁵¹ erg)
AML: with low-mass companion
J1118+480	∼5	<1	0.8	∼430
Vel 93	∼5	<1	0.8	∼430
J0422+32	6–7	<1	0.8	500–600
1859+226	6–7	<1	0.8	500–600
GS1124	6–7	<1	0.8	500–600
H1705	6–7	<1	0.8	500–600
A0620−003	∼10	<1	0.6	∼440
GS2000+251	∼10	<1	0.6	∼440
Nu: with evolved high-mass companion
GRO J1655−40	∼5	1–2	0.8	∼430
4U 1543−47	∼5	1–2	0.8	∼430
XTE J1550−564	∼10	1–2	0.5	∼300
GS 2023+338	∼10	1–2	0.5	∼300
XTE J1819−254	6–7	∼10	0.2	10–12
GRS 1915+105	6–7	∼10	0.2^a	10–12
Cyg X−1	6–7	≳30	0.15	5–6

Notes. E_BZ is the rotational energy which can be extracted via the Blandford–Znajek mechanism with optimal efficiency epsilon _Ω = 1/2 (see Appendix A). Parameters for Nu are taken from Brown et al. (2007). The AML (Angular Momentum Loss) binaries lose energy by gravitational waves, shortening the orbital period whereas the Nu (Nuclear Evolution) binaries will experience mass loss from the donor star to the higher mass black hole and, therefore, move to longer orbital periods. ^aa_⋆>0.98 is the measured Kerr parameter (McClintock et al. 2006), the difference comes after the accretion of ∼8 M_☉.

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The increased gravitational binding between donor and the He-star remnant must furnish the energy to remove the hydrogen envelope, modulo the product of a shape parameter for the density profile (λ) and the efficiency with which orbital energy is used to expel the envelope (α_ce): λα_ce (see LBW for a discussion of these). The latter decreases inversely with the radius of the giant, so that when the radius is large, the envelope can be removed by a low-mass companion. Combining Equations (2) and (3) we have

$\begin{eqnarray} &&\frac{1}{P_b}= \left(\frac{4.2\mbox{\,$R_\odot $}/a_i}{M_d/\mbox{\,$M_\odot $}}\right)^{3/2} \left(\frac{M_d+M_{\rm{He}}}{\mbox{\,$M_\odot $}}\right)^{1/2} \left(\frac{M_{{\rm giant}}}{\mbox{\,$M_\odot $}}\right)^{0.83} \end{eqnarray} \tag{ 4 }$

so that P_b ∝ M^3/2_d for M_d ≪ M_He and P_b ∝ M_d for higher donor masses. As we develop later, the distances a_i at which mass transfer begins depend very little on donor mass so we can use Equation (4) in order to scale from one donor mass to another. LBW found that all the giants they needed for the transient sources had mass ∼30 M_☉, and, therefore, $M_{\rm{He}}\sim 11 \mbox{\,$M_\odot $}$ , substantially larger than the donor masses for the most energetic GRBs.

For higher-mass donors, mass loss because of the explosion can be neglected, giving pre- and post-explosion periods which are nearly the same.

3.2. Explosion Energies of Galactic Black Hole Binaries

We estimate the explosion energies using the Blandford–Znajek mechanism (Blandford & Znajek 1977; Lee et al. 2000) as summarized in Appendix A, in which the energy is deposited in a fireball in the perturbative region (Blandford & Znajek 1977; Lee et al. 2000).¹¹Paczyński (1986) and Goodman (1986) have shown that in order to power a GRB all that is necessary is for the fireball to have enough energy so that the temperature is well above the pair production temperature; i.e., T>1 MeV. Then the GRB and the afterglow will follow, just from having as source the localized hot fireball.

The estimated explosion energies of galactic sources are summarized in Table 1. This is the first quantitative calculation of the explosions giving rise to GRBs and hypernovae, in the sense that we calculate the energy that is supplied in the form of rotational energy. What the central engine does with this energy is another matter. We cannot calculate how much of the energy is accepted by the accretion disk in detail, because we cannot calculate analytically the Rayleigh–Taylor instability in the magnetic field of the black hole coupling to the accretion disk (but see Appendix B).

The rotational energy that is not initially accepted by the black hole does, however, remain in the disk and appears later in the Kerr parameter of the black hole. Thus far, in GRO J1655−40 and 4U 1543−47 a tiny part of the available rotational energy was accepted by the central engine, so little that the final rotational energy could not be discriminated from our calculated initial energy, because the uncertainty in the measurement of Kerr parameters was the same order of magnitude as the explosion energy.

We can understand why AMLs and GRO J1655−40, 4U 1543−47, XTE J1550−564, and GS 2023+338 among Nu's have such rotational energies from Case C mass transfer. The lower the companion mass, the greater the radius R_sg that the supergiant must reach before its Roche lobe meets the companion. Given giant radii such as those shown in Figure 1 of Brown et al. (2001b), the typical separation distance between giant and companion is ∼1200–1300 R_☉, higher (because of the Roche lobe of the companion star) than the 1000 R_☉ the giant radius reaches. The binding energy of the supergiant envelope goes as R⁻¹_sg and at such a large distance it can be removed by the change in binding energy of an ∼1 M_☉ donor, spiralling in from ∼1500 R_☉ to ∼5 R_☉. A higher mass donor would end up further out, since it would not have to spiral in so close in order to release enough binding energy. Thus low-mass ∼1–2 M_☉ companions can naturally increase their gravitational binding energy by removing the high R_sg envelopes. A detailed discussion of these matters is given in Brown et al. (1999). Thus Case C mass transfer naturally gives the ultra-high rotational energies of the binaries with low-mass donors discussed as relics of GRB and hypernova explosions in our Galaxy.

3.3. Evolution of Cyg X−1

One can infer that there was essentially no mass loss in the Cyg X−1 explosion because the space velocity of Cyg X−1 relative to the Cyg OB3 cluster of O-stars of 9 ± 2 km s⁻¹ is typical of the random velocities of stars in expanding associations (Mirabel & Rodrigues 2003). In comparison, GRO J1655−40 likely underwent a very strong explosion from the fact that its space velocity after explosion was 112 ± 18 km s⁻¹ (although only a small part of the available energy was used up in the system velocity). In our reconstruction of the Cyg X−1 evolution the explosion took place when the black hole mass was about 7 M_☉, 3 M_☉ less than today. The mass transfer from donor to black hole is nonconservative because of the higher mass of the donor. The initial donor would have been substantially more massive than it is today, at least ∼30 M_☉. For a 30 M_☉ donor the maximum available energy is <10⁵² erg.

Mirabel & Rodrigues (2003) argue that Cyg X−1 had a dark explosion. As discussed in Appendix A, the efficiency epsilon _Ω can be taken to be 0.5 for the higher a_⋆, say a_⋆>0.5, but it decreases for small a_⋆, so that for a_⋆ = 0.15 we estimate it to be 0.15 and for a_⋆ = 0.2 we calculate epsilon _Ω = 0.2. The (5–6) × 10⁵¹ erg is clearly not enough for an explosion in Cyg X−1.

3.4. Subluminous Bursts

All of the GRBs in our binary model come about from the same mechanism, but their rotational energy is set by the mass of their donor. Our mechanism predicts the Kerr parameter with which the black hole is born, before powering the GRB and hypernova explosion. There must be a "Goldilocks" scenario for the energy needed to power a high-luminosity GRB, neither too big nor too small. For the Galactic X-ray transients with the 1–2.5 M_☉ low-mass donors, the available energy is clearly too large. We know this because the calculated initial Kerr parameters were essentially the same as those found by Shafee et al. (2006); thus, very little of the energy had been used up in the explosion. On the other hand, M33 X−7, which we will discuss in more detail later, had a donor of zero-age main-sequence (ZAMS) mass ∼80 M_☉, with Kerr parameter a_⋆ ⩽ 0.12 which probably went into a dark explosion, like Cyg X−1. These bracket (but rather widely) the luminous explosions.

Initially in supernova explosions the kinetic energy, the main part of which results from clearing out the matter in the way of the jet that accompanies the GRB, is about equal to the thermal energy of the hypernova. MacFadyen (2000) finds this to be at least approximately true and it would be expected from equipartition of energy. GRB 980425, a subluminous GRB (Brown et al. 2008), must have had an energy of ∼60 bethes (1 bethe ≡10⁵¹ erg). Most of the energy is used up ramming the path through the star for the GRB and in the end only about 1 bethe is visible.

Note that the high-luminosity GRBs turn out to be only a small fraction of the total number, even if we use a beaming factor of 100 for them. Thus, they must be formed in very special circumstances (see Appendix B).

The question is whether there are enough transient sources to supply subluminous GRBs in nearby galaxies. Brown et al. (2000) estimated that in our Galaxy the total creation rate of the transient source binaries is ∼3 × 10⁻⁴ galaxy⁻¹ yr⁻¹. Given 10⁵ galaxies within 200 Mpc this number translates into 3750 Gpc⁻³ yr⁻¹. Liang et al. (2007) find a beaming factor typically less than 14; such a beaming factor would reduce our number¹² to 268 Gpc⁻³ yr⁻¹, in agreement with that of Liang et al. (2007) of ∼325⁺³⁵²₋₁₇₇ Gpc⁻³ yr⁻¹. This is much higher than their estimated rate of high-luminosity GRBs of 1.12^+0.43_−0.20 Gpc⁻³ yr⁻¹ (Liang et al. 2007). The usual beaming factor for the high-luminosity bursts is estimated to be about ∼100 (Piran 2005). Even with such a large factor, the high-luminosity GRBs are estimated to be much rarer, by a factor of ∼40, than the subluminous ones. Even neglecting the Woosley collapsar rate, we have enough binaries to account for all of the bursts.

3.5. Binaries in Low-metallicity Galaxies

The effect of cutting down the angular momentum losses via winds in galactic stars gave a hint about how the rotational energy in the binaries could be decreased so as to be in the ballpark needed for high-luminosity GRBs. The wind losses are sensitive to galactic metallicity, with lower metallicity stars having smaller $\dot{m}$ . In general the low metallicity stars are more massive than galactic ones, which we believe has the effect of scaling up all of the galactic masses. We pursue the question of cosmological GRBs and their abundances in Appendix B.

As discussed earlier, Orosz et al. (2007) have recently measured the Kerr parameter of the black hole in M33 X−7 where the metallicity is ∼0.1 solar. Consistent with Brown et al. (2008), the donors in low-metallicity galaxies are generally more massive than those of our Galaxy; the donor in M33 X−7 is now ∼70 M_☉ (∼80 M_☉ at the time of common envelope evolution). In fact, it is so massive that we estimate a_⋆ to be <0.12 at the time of the explosion (see Moreno Méndez et al. 2008 for further discussion on this system); it probably went through a dark explosion. We also should be able to connect GRB 980425 with our galactic GRBs because of its nearly solar metallicity (Sollerman et al. 2005).¹³

Far from being irrelevant, as the results of Fruchter et al. (2006) imply, the measurements of Shafee et al. (2006) for galactic sources teach us how to calculate the energies of GRB and hypernova explosions. We can easily extend our theory to low-metallicity galaxies by increasing the donor masses.

3.6. A General Discussion of Black Hole Masses

If one accepts the Schaller et al. (1992) numbers literally, then Case C mass transfer is actually limited to a narrow interval of ZAMS masses ∼19–22 M_☉ as found by Portegies Zwart et al. (1997). This is because the binary orbit widens with mass loss from the supergiant so that in order to initiate mass transfer only after helium burning the supergiant has to expand sufficiently to compensate for this widening of the orbit. A graphic display of this is shown in Figure 1 of Brown et al. (2001b).

LBW realized that in order to produce black holes from the wider range of ZAMS masses 18–35 M_☉ necessary for their evolution of transient sources, they had to reduce the wind losses in the red giant stage (see Figure 3 of LBW). This was necessary because Brown et al. (2001a) had shown that high-mass black hole binaries could be evolved with the black holes coming from ZAMS masses 18–35 M_☉ provided Case C mass transfer was used.

LBW argued that the progenitors of the black hole binaries GRO J1655−40, 4U 1543−47, and GRS 1915+105 all came from 30–33 M_☉ giants. Note that the black holes produced from lower mass progenitors should be more common than the higher mass 30–33 M_☉ progenitors because of the mass function. In the case of GRO J1655−40 and 4U 1543−47 (which have low-mass donors; see Table 1) the explosion was so energetic that the 5.5 M_☉ black hole is only about half of the progenitor He star mass, i.e., the explosion was so violent that nearly half of the mass of the system was lost in the explosion; a loss of half or more would result in system breakup. Presumably many if not most binaries with lower-mass black holes did lose more than half of their system mass and did not survive the explosion.

GRS 1915+105, as well as XTE J1819−254 and Cyg X−1 (which have high-mass donors; see Table 1), have black holes of 6–7 M_☉. Little mass was lost in the explosion, which came from 20–22 M_☉ progenitors (our evolution of GRS 1915+105 in the present paper is an improvement over that in LBW). Thus, the black holes in galactic SXT sources seem to come from a wide range of ZAMS mass progenitors.

The donor mass in high-luminosity GRBs might be larger than the <2 M_☉ donors of most galactic SXTs (in which the disk is disrupted in the energetic explosion that was caused by the high natal black hole spin) but smaller than the ≳10 M_☉ ZAMS mass companions of GRS 1915+105, XTE J1819−254, and Cyg X−1. However, we do not yet know how rapidly the binaries are left rotating after powering the explosion. Measurements of black hole binaries with donor masses in the 10–20 M_☉ range would be very helpful.

4. CONCLUSIONS

Our theory of GRBs and hypernova explosions is essentially unchanged from the earlier treatment by Brown et al. (2000). Lee et al. (2002) showed how to calculate the Kerr parameters of the black holes through an understanding of the tidal locking. Our predicted Kerr parameters have been checked by the measurement of the Kerr parameters of GRO J1655−40 and 4U 1543−47 by Shafee et al. (2006). Both Paczyński (1986) and Goodman (1986) have shown that when sufficient energy has been delivered to the fireball (so that the temperature is above the pair-production threshold) the GRB and hypernova explosions follow and the afterglow is as observed. Therefore the production of energy and the explosion decouple, but the latter follows from the former once sufficient energy is furnished. In this sense, we have a complete and predictive theory of GRBs and hypernovae.

The GRB and hypernova explosion model is based on that of Woosley's collapsar model, but with an important improvement: namely, that the required amount of rotational energy is obtained from the tidal spin up of the black hole progenitor by the donor. The donor then, after furnishing the angular momentum, acts as a passive witness to the explosion, but can record some details of the latter in the chiefly alpha-particle nuclei which it accretes. The magnetic field lines threading the disk of the black hole power the central engine in the Blandford–Znajek mechanism. The jet formation and hypernova explosion are powered as in the MacFadyen & Woosley (1999) collapsar. Population synthesis shows that there are enough binaries to reproduce all GRBs.

The properties of the SXT sources in our Galaxy can be studied in detail. While they are higher in metallicity than the high-luminosity GRBs, it is easy to extend our description to the low-metallicity case, because the rotational energy is determined by the mass of the donor. Donors in low-metallicity galaxies tend to be more massive than in high-metallicity ones, furnishing a lower rotational energy.

We find that the subluminous GRBs come from two sources. (1) High-metallicity systems with low-mass donors, where the magnetic field coupling to the black hole disk is so high that it dismantles the central engine before much rotational energy can be delivered as summarized in Appendix B. GRO J1655−40 and 4U 1543−47 are excellent examples of these in that only a tiny part of the available rotational energy was used up in the explosion.¹⁴ (2) Binaries with massive, low-metallicity donors, such as the 80 M_☉ donor in M33 X−7. Somewhere in between these extremes the binaries will have the rotational energies of cosmological GRBs.

We postulate that high-luminosity, cosmological GRBs are produced only in binaries with donor masses ∼5 M_☉, but this is uncertain until the Kerr parameters or the system velocity of binaries such as XTE J1550−564 is measured. With such a Kerr parameter (or system velocity) in hand, we can subtract the rotational energy left in the binary from the calculable pre-explosion energy (see Table 1). The difference is the energy of the explosion. In the cases of GRO J1655−40 and 4U 1543−47, the energy used up in the explosion was tiny compared with the initial rotational energy, but this must change as the initial rotational energy decreases, and black hole mass increases.

We have shown that there are 12 relics of GRBs and hypernova explosions in the Galaxy. Cyg X−1 likely went through a low-energy dark explosion. XTE J1819−254 and GRS 1915+105 are likely remnants of dark explosions as well; however, they might have instead been subluminous GRBs and hypernova explosions. The soft X-ray black hole binaries are the remnants of the subluminous GRBs.

We thank Jeff McClintock for many useful discussions. The Smithsonian-Harvard group and collaborators have opened up an exciting new field of activity. We also thank the referee for the very helpful suggestions. C.H.L. thanks APCTP where a part of this work has been discussed during the APCTP International School on Numerical Relativity and Gravitational Waves. G.E.B. and E.M.M. were supported in part by the US Department of Energy under grant No. DE-FG02-88ER40388. C.H.L. was supported by the BAERI Nuclear R & D program (M20808740002) of MEST/KOSEF and the Mid-career Researcher Program through an NRG grant funded by the MEST (No. 2009-0083826).

APPENDIX A: THE BLANDFORD–ZNAJEK MECHANISM

The rotational energy of a black hole with angular momentum J is a fraction of the black hole mass energy (Blandford & Znajek 1977; Lee et al. 2000),

$\begin{eqnarray} &&E_{\rm rot} = f(a_\star) M_{\rm BH} c^2, \end{eqnarray} \tag{ A1 }$

where

$\begin{eqnarray} &&f(a_\star) = 1 - \sqrt{\frac{1}{2} \left(1+\sqrt{1-a_\star ^2}\right)}. \end{eqnarray} \tag{ A2 }$

For a maximally rotating black hole (a_⋆ = 1) f(1) = 0.29. In the Blandford–Znajek mechanism (Blandford & Znajek 1977; Lee et al. 2000), the efficiency of extracting the rotational energy is determined by the ratio between the angular velocities of the black hole Ω_H and the magnetic field velocity Ω_F,

$\begin{eqnarray} &&\epsilon _\Omega = \Omega _F/\Omega _H. \end{eqnarray} \tag{ A3 }$

One can deduce the analytical expression for the energy extracted:

$\begin{eqnarray} &&E_{\rm{BZ}} = 1.8\times 10^{54}\epsilon _{\Omega }f(a_{\star })\frac{M_{\rm{BH}}}{\mbox{\,$M_\odot $}}{\rm erg}. \end{eqnarray} \tag{ A4 }$

The rest of the rotational energy is dissipated into the black hole, increasing the entropy or equivalently, the irreducible mass.

For optimal energy extraction epsilon _Ω ≃ 0.5 (Blandford & Znajek 1977; Lee et al. 2000). Although the use of epsilon _Ω = 0.5 is close to the actual efficiency for high a_⋆, it decreases with a_⋆ from 0.69 at a_⋆ ∼ 0.8 to 0.46 at a_⋆ ∼ 0.4 (Brown et al. 2000). There would be a further decrease of ≳50% to the a_⋆ of 0.15 of Cyg X−1 and M33 X−7. This low efficiency virtually ensures that these two binaries will have gone through dark explosions.

The hypernova results from the magnetic field lines anchored in the black hole and extending through the accretion disk, which is highly ionized so the lines are frozen in it. When the He star falls into a black hole, the latter is so much smaller in radius that it has to rotate much faster than the progenitor He star so as to conserve angular momentum.

Initially, large amounts of energy, up to 10⁵² erg, were attributed to GRBs. However, when the correction is made for beaming, the actual GRB energy is distributed about a "mere" ∼10⁵¹ erg (Piran 2002). However, ∼10⁵² erg are required to clear the way for the jet. Hypernova explosions are usually modeled after the nearby supernova 1998bw. The hypernova by Nomoto et al. (2001) that Israelian et al. (1999) compared with GRO J1655−40 had an energy of 3 × 10⁵² erg.

APPENDIX B: DISMANTLING THE ACCRETION DISK BY HIGH-ENERGY INPUT

The amount of energy deposited into the accretion disk of the black hole is almost unfathomable, the 4.3 × 10⁵³ erg $\simeq \frac{1}{4}\mbox{\,$M_\odot $}\, c^2$ being 430 times the luminous energy of a strong supernova explosion. Near the horizon of the black hole, the physical situation might become quite complicated (Thorne et al. 1986). Field-line reconstruction might be common and lead to serious breakdowns in the freezing of the field to the plasma; the field on the black hole sometimes might become so strong as to push it back off the black hole and into the disk (Rayleigh–Taylor instability) concentrating the energy even more. During the instability the magnetic field lines will be distributed randomly in "globs," the large ones having eaten the small ones. It seems reasonable that the Blandford–Znajek mechanism is dismantled. Later, however, conservation laws demand that the angular momentum not used up in the GRB and hypernova explosion be reconstituted in the Kerr parameter of the black hole. The radius of the (Kerr) black hole is

$\begin{eqnarray*} &&R=\frac{GM}{c^2}= 1.48\times 10^{6}\frac{M}{10\mbox{\,$M_\odot $}}{\rm cm}. \end{eqnarray*}$

Given the above scenario of the very high magnetic couplings dismantling the disk by Rayleigh–Taylor instability (e.g., in GRO J1655−40), we wish to make a "guesstimate" of the same effect for XTE J1550−564.

The black hole radius is proportional to M_BH. The ratio of $M_{\rm{BH}}(1550-564)/M_{\rm{BH}}(1655-40)\simeq 2$ , but we use a ratio of 2.5 because GRS 1655−40 has a Kerr black hole and, in XTE J1550−564, the black hole is about halfway between Kerr and Schwarzschild. Thus, the area of the circle inside the last stable circular orbit is ∼6 times larger for XTE J1550−564. Taking the field strength for Rayleigh–Taylor instability to go as the inverse of the area means that its effect would be reduced by a factor of 6 in going from GRS 1655−40 to XTE J1550−564. If our scenario that the magnetic field coupling is correct for GRO J1655−40, then not much of the effect would be left in XTE J1550−564 which should accept most of the energy for a cosmological (highly luminous) GRB.

The question then is, what is the latter? We know that SN1998bw had a hypernova energy of ∼30 bethe. From the equipartition of energy, we would estimate the kinetic energy to clear out a path for the jet in the GRB to be about equal to the thermal energy, which is roughly true in MacFadyen (2000). So, the total energy would be ∼60 bethes, which we know can be accepted by the binary. MacFadyen (2000) suggested that the accompanying GRB 980425 was "smothered," so that may be a lower limit, although it is larger than estimates we have seen to date, so we choose it as the energy of a cosmological GRB.

Now, 60 bethes is ∼20% of our estimated rotational energy for XTE J1550−564. However, the decrease in the Kerr parameter necessary to go from 300 to 240 bethes is only 0.06 of our natal a_⋆ ∼ 0.5, or ∼12% (note that it is neither linear with a_⋆ nor with the rotational energy), which requires an accurate measurement in the Kerr parameter, but this is nonetheless much larger than the ∼6% decrease estimated for GRO J1655−40. (Note that for Schwarzschild black holes, the rotational energy goes approximately as a²_⋆.) On the other hand, this may be the minimal difference between natal and present rotational energies because no other model leaves the system spinning so rapidly. Thus, if no difference is conclusively seen because observational errors are ∼12% then this is also very interesting.

Suppose a decrease of more than 12% is seen between the estimated natal a_⋆ and the observed one. Then this means that we have underestimated the energy of the explosion, but our above estimates are as large as any proposed ones. In any case, the possibility that the system, following the explosion, is left rotating is a new and interesting one.

We thus see that we can fit the Fruchter et al. (2006) condition for no high-luminosity GRBs in our high metallicity stars in it, because all of the donor masses of the GRBs that received the highest rotational energies had companion masses of ∼1–2 M_☉ and the rotational energy furnished to them was so great that the accretion disks were dismantled. The result is that the GRBs were subluminous, like the vast majority of GRBs. The ∼6 times larger surface area should be helpful in allowing XTE 1550−564 to accept the rotational energy, but the amount is still tremendous. We do not have any donors of ∼5 M_☉ in our Galaxy, but such a donor would bring the energy down by ∼1/5, since it goes inversely with donor mass, to the ∼60 bethes that we estimate for GRB 980425, which as noted has nearly galactic metallicity. Although GRB 980425 is essentially "smothered" (MacFadyen 2000), 60 bethes is the highest energy anyone has attributed to the GRB and hypernova explosion. Thus, our estimates would say that donors of ∼5 M_☉ would give the most luminous GRBs and that XTE 1550−564 may or may not have been able to accept ∼60 bethes, but since we view this as an upper limit, this black hole should still be spinning rapidly.

We should enter a proviso here. Our main thesis is that the black hole binary must have a donor sufficiently massive to slow it down enough so that the black hole can accept the strong magnetic coupling through its accretion disk without the Rayleigh–Taylor instability entering. By increasing the area of the accretion disk by a factor ∼6 in going from the ∼5 M_☉ black hole in GRO J1655−40, to the ∼10 M_☉ black hole in XTE 1550−564 the density of magnetic coupling is decreased by a factor ∼6. However, the donor in XTE 1550−564 is only ∼1.3 M_☉, about the same size as in GRO J1655−40. Therefore, the GRB in XTE 1550−564 may still have been subluminous, but less so than GRO J1655−40 because of the larger disk area.

The donor masses at the time of the common envelope evolution of XTE 1819−254 and GRS 1915+105 were ∼10 M_☉. However, they have only ∼1/3 of the energy of the binary with 5 M_☉ donor, ∼10–12 bethes. These black holes accrete a lot of matter, more than doubling the black hole mass in the case of GRS 1915+105 and, ultimately will double that mass in XTE 1819−254. Such binaries with substantial mass exchange do not obey our simple scaling which is designed for natal angular momentum. They must be evolved in detail.

We believe that similar evolutionary arguments will apply to binaries with higher mass donors, and, anyway, that the rotational energy will be cut down by the higher donor masses. Thus, we expect that only binaries with donor masses ∼5 M_☉ will give highly luminous GRBs with rotational energy ∼60 bethes.

In a very rough estimate, using a flat distribution of donors with mass up to 80 M_☉ we can estimate that the number of cosmological GRBs, those with high luminosity, will be ∼5/80 of the total, not far from the ratio of GRBs with high luminosity to subluminous ones found by Liang et al. (2007).

In summary, the commonly accepted estimates of the explosion energies in GRBs are orders of magnitude less than the natal rotational energy in GRO J1655−40. The measured Kerr parameter of a_⋆ ∼ 0.8 (Shafee et al. 2006) has a present rotational energy indistinguishable from the natal one, within error bars. However, the dismantling of the accretion disk by the very strong magnetic couplings should be slower in XTE J1550−564 and the rotational energy is somewhat less, the donor being about the mass of the He star in the Woosley collapsar model. We propose that XTE J1550−564 can accept substantial rotational energy from the central engine but probably less than the energy of a cosmological GRB.

If our suggested scenario is confirmed (by a precise measurement of a_⋆ and/or peculiar velocity), then this should be strong support for the Brown et al. (2000) binary scenario for GRBs.

KERR PARAMETERS FOR STELLAR MASS BLACK HOLES AND THEIR CONSEQUENCES FOR GAMMA-RAY BURSTS AND HYPERNOVAE

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ABSTRACT

1. INTRODUCTION

2. CASE C MASS TRANSFER AND TIDAL LOCKING