---
abstract: 'The very first stars in the Universe can be very massive, up to $10^3M_\odot$. If born in large numbers, such massive stars can have strong impact on the subsequent star formation, producing strong ionising radiation and contaminating the primordial gas with heavy elements. They would leave behind massive black holes that could act as seeds for growing supermassive black holes of active galactic nuclei. Given the anticipated fast rotation, such stars would end their live as supermassive collapsars and drive powerful magnetically-dominated jets. In this letter we investigate the possibility of observing the bursts of high-energy emission similar to the Long Gamma Ray Bursts associated with normal collapsars. We show that during the collapse of supercollapsars, the Blandford-Znajek mechanism can produce jets as powerful as few$\times10^{52}$erg/s and release up to $10^{54}$erg of the black hole rotational energy. Due to the higher intrinsic time scale and higher redshift the initial bright phase of the burst can last for about $10^4$ seconds, whereas the central engine would remain active for about one day. Due to the high redshift the burst spectrum is expected to be soft, with the spectral energy distribution peaking at around 20-30keV. The peak total flux density is relatively low, $10^{-7}\mbox{erg}\, \mbox{cm}^{-2} \mbox{s}^{-1}$, but not prohibitive. If one supercollapsar is produced per every mini-halo of dark matter arising from the 3-$\sigma$ cosmological fluctuations then the whole sky frequency of such bursts could reach several tens per year.'
author:
- |
    Serguei S. Komissarov$^{1,3}$ [^1] and Maxim V. Barkov$^{1,2,3}$ [^2]\
    $^{1}$Department of Applied Mathematics, The University of Leeds, Leeds, LS2 9GT, UK\
    $^{2}$Space Research Institute, 84/32 Profsoyuznaya Street, Moscow 117997, Russia\
    $^{3}$NORDITA, AlbaNova University Centre, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
date: 'Received/Accepted'
title: 'Supercollapsars and their X-ray Bursts'
---

[ cosmology: early Universe – black hole physics – gamma-rays: bursts – X-rays: bursts – radiation mechanisms: non-thermal – relativity – magnetic fields]{}

Introduction {#intro}
============

According to the modern hierarchical clustering theories of galaxy formation the first stars are born within collapsed haloes of dark matter of $\simeq 10^6M_\odot$ at $z\simeq 20$. The primordial gas falls into the potential well of these haloes and fragments into clamps of $\simeq 10^3M_\odot$ via gravitational instability [@BCL02]. Because this gas is metal-free its cooling is rather slow and further fragmentation into smaller clamps seems to be avoided [cf. @TAS09; @SGB09]. Instead, the clumps contract in a quasi-static fashion as a whole, suggesting that the first stars can be very massive indeed, $M>100M_\odot$. The actual initial mass function (IMF) of first metal-free stars (Population III stars), however, is not known yet as too many factors come into play, making the problem intractable analytically, and rather challenging numerically. In particularly, the initial mass of protostars can be very small, down to $10^{-3}M_\odot$, and the eventual accumulation of mass proceeds via accretion of the surrounding gas. Very high accretion rate, $\dot{M}>\dot{M}_c \simeq 4\times10^{-3} M_\odot\mbox{yr}^{-1}$, may limit the final mass to few hundreds solar masses as the protostellar luminosity reaches the Eddington limit [@OK03]. For lower accretion rate, the accretion may proceed even after the onset of nuclear burning in the stellar core and result in the final mass $M\simeq 10^3M_\odot$. Numerical studies of cosmological gravitational instability suggest that, although in principle the accretion rate can be as high as few$\times 10^{-2}M_\odot\mbox{yr}^{-1}$, in reality the rotational support against gravity often become important and reduces the rate below $\dot{M}_c$ [@G07]. @Ohk09 studied the evolution of accreting Population III stars from the pre-main sequence evolution to the core-collapse and confirmed that the final mass can be as large as $10^3 M_\odot$. Very massive first stars are also predicted in theories involving dark matter annihilation [e.g. @Nat09].

Population III stars with masses $140M_\odot \le M \le 260M_\odot$ most likely end their life as pair-instability supernovae which leave no compact remnant behind [@F01]. If such stars were the main outcome of initial star formation they would overproduce heavy elements in the early Universe, in conflict with the observations of extremely metal pure stars in galactic bulges and the observed abundances of intergalactic and intercluster medium [@UN02; @HW02; @CL02].

More massive stars, which will be referred to as Very Massive Stars (VMSs) are expected to collapse into black holes with very little mass loss [@F01]. They would leave behind massive black holes (MBHs), which could play the role of seeds for the supermassive black holes (SMBHs) of Active Galactic Nuclei (AGNs). Assuming that MBHs are formed at the rate of one per mini-halo developed from a 3$\sigma$-fluctuation, @MR01 estimated their density to be around 5000 per galaxy like Milky Way and their total mass comparable to the total mass of SMBHs. This suggests that SMBHs could form via mergers of MBHs, the idea that has being actively developed in recent years. Even more massive VMSs, with $M\simeq 10^6M_\odot$, could be formed in more massive dark matter haloes, with total mass $M\simeq 10^8M_\odot$, collapsed at $z\simeq10$ [@BL03; @BVR06]. Although much more rare, such events can provide an alternative way of producing SMBHs.

From the observational perspective it is difficult to distinguish between a VMS and a cluster of less massive Population III stars. This suggests to investigate the potential observational consequences of VMS collapse, which could be quite spectacular because of the very high mass involved. Given their expected fast rotation, it seems likely that supercollapsars (classified as type-III collapsars in @H03) develop accretion disks, drive relativistic jets, and produce bursts of high energy emission in the fashion similar to their less massive relatives [@MW99; @BK08]. If detected, they would become the most distant strong sources of light and provide us with a new way of probing the physical conditions in the epoch close to the “Dark ages”. Even non-detection could be useful, allowing to set constraints on models of star formation in the early Universe and the origin of SMBHs. There has been a number of papers looking into cosmological evolution of GRBs, including those from the Population III stars [e.g. @BL02; @LD07; @NB07]. They assumed that the Population III GRBs are similar to those from the lower redshift Population II stars whose mass is significantly below $100M_\odot$. However, because of their very high mass and redshift, the GRB-like bursts of supercollapsars can be rather special. There are two crucial differences between a normal collapsar and a supercollapsar. One is that instead of a proto-neutron star of solar mass the supercollapsars develop proto-black holes of tens of solar masses, within which the neutrinos from electron capture are trapped [@F01; @Su07]. The other is that the accretion disks of supercollapsars are far too large and cool for the neutrino annihilation mechanism. This has already been seen in the numerical simulations of supercollapsar with mass $M=300M_\odot$ [@F01]. Utilising the study of hyper-accreting disks by @Bel08 we find that at best the rate of heating due to this mechanism is $$\dot{E} \simeq 2\times10^{48} \dot{M}_0^{9/4}  M_{h,3}^{-3/2}  
 \mbox{erg s}^{-1},$$ where $\dot{M}$ is the accretion rate and $M_h$ is the black hole mass. (Here and in other numerical estimates below we use the following notation: $\dot{M}_k$ is the mass accretion rate measured in the units of $10^k M_{\odot} \mbox{s}^{-1}$ and $M_k$ is the mass measured in the units of $10^kM_\odot$.) Such low values have lead @F01 to conclude that the magnetic mechanism is the only candidate for producing GRB jets from supercollapsars. In the following we analyse one particular version of the mechanism, namely the one where the jets are powered by the rotational energy of the black hole via the Blandford-Znajek process [@BZ77; @BK08].

Blandford-Znajek jets from Supercollapsars {#BZ-jets}
==========================================

VMSs are expected to rotate rapidly, close to the break-up speed and hence produce rapidly rotating MBHs. Moreover, in the absence of strong magnetic field in the prestine primordial gas, VMSs will be weakly magnetized and as the result could develop rapidly rotating cores [@WH06]. This suggests that the spin parameter of MBHs can be very high, $a \simeq 1$, yeilding the enormous rotational energy $E{_{\mbox{\tiny rot}}} \simeq 5\times10^{56} M_{h,3} \mbox{erg}$. In order to estimate the Blandford-Znajek luminosity we need to know the strength of magnetic field accumulated by the hole. The usual approach is to relate it to the gas pressure in the disk and for this we need to know the parameters of the accretion disk itself. Accurate determination of these parameters and their time evolution requires to know the structure of VMS prior to the collapse and the physics of accretion disk. Unfortunately, this information is lacking at the moment. In particular, although the structure of supermassive stars, $M_s \gg 10^3M_\odot$ is very well described by a polytropic model with $n=3$ [@ZN71], we are more interested in stars with $M_s \leq 10^3M_\odot$. @F01 studied the structure and evolution of a $300M_\odot$ star. Prior to the collapse the star entered the red giant phase and expanded from $R_s=4\times10^{12}$cm to $1.5\times10^{14}$cm. However, the initial rotation rate of this star was slow compared to about $\simeq50\%$ of the brake-up speed invoked the current single-star model of GRB progenitors. In this model, the progenitors remain chemically homogeneous and compact all the way up to the collapse [@YL05; @WH06]. Given this lack of information about the progenitor structure we will follow @b90 and assume $\rho \propto R^{-3}$ ($R_c\leq\! R \leq\! R_s$) distribution of mass density in the star prior to collapse. In fact, this distribution agrees reasonably well with the numerical models of rapidly rotating low metallicity stars considered as likely progenitors of normal GRB [see Figure 2 in @KNJ08]. As to the stellar rotation, we will assume that it is uniform ($\Omega$=const) in the stellar envelope, with $50\%$ of break-up speed at the stellar surface.

Due to the slow neutrino cooling, the accretion disks of supercollapsars are expected to be radiatively inefficient, with possible exception only for the very inner region. This suggests to use the ADAF (Advection Dominated Accretion Flow) model [@NY94] to describe these disks. Since the radiation pressure dominates, we can use the ratio of specific heats $\gamma=4/3$ which gives us

$$v{_{\mbox{\tiny in}}} \simeq \frac{3\alpha}{7}v_k, 
\qquad c_s^2 \simeq \frac{2}{7}v^2_k,
\qquad H \simeq R c_s/v_k,
\label{ADAF}$$

where $v{_{\mbox{\tiny in}}}$ is the accretion speed, $c_s$ is the sound speed, $v_k=\sqrt{GM/R}$ is the Keplerian speed, $H$ is the vertical disk scale, and $\alpha$ is the effective viscous stress parameter of the $\alpha$-disk model [@SS73]. The disk density and pressure can be estimated combining the above equations with the expression for the mass accretion rate, $ \dot{M}\simeq2\pi R H\rho v{_{\mbox{\tiny in}}}$. Straightforward calculations yield $$P \simeq \frac{\sqrt{14}}{12\pi\alpha} \frac{\dot{M}(GM)^{1/2}}{R^{5/2}}.
\label{p}$$ The poloidal magnetic field should scale with the thermodynamic pressure, so we write $B^2 = 8\pi P/ \beta$, where $\beta$ is the magnetization parameter. Applying this equation at the radius of the marginally bound orbit $R{_{\mbox{\tiny mb}}} = f_1(a)R_g$, where $f_1(a)=2-a+2(1-a)^{1/2}$ and $R_g=GM_h/c^2$ is the BH’s gravitational radius, we find

$$B{_{\mbox{\tiny mb}}} \simeq 3\times10^{13}  f_1^{-5/4} \beta_{1}^{1/2} 
   \dot{M}_{0}^{1/2}  \mbox{Gauss},
\label{b}$$

where $\beta_1=\beta/10$.[^3] If the magnetic field is generated in the disk then it is likely to change polarity on the scale $\simeq H$. This may lead to significant variation in the strength and polarity of the magnetic field accumulated by the black hole and reduce the Blandford-Znajek luminosity [e.g. @BB09]. We will assume that this effect is accounted for in the value of $\beta$.

The power of jet energized via the Blandford-Znajek mechanism can be estimated using the monopole solution for magnetospheres of rotating black holes [@BZ77], which gives $$L{_{\mbox{\tiny BZ}}} =
\frac{1}{3c}\left(\frac{\Psi_h\Omega_h}{4\pi}\right)^2
\label{bz1}$$ where $\Omega_h= f_2(a)c^3/GM_h$ is the angular velocity of BH, $f_2(a)= a/2(1+\sqrt{1-a^2})$, and $\Psi_h$ is the magnetic flux threading one of the BH’s hemispheres. Inside the marginally bound orbit the disk plasma quickly dives into the BH and the magnetic flux can be roughly estimate as $\Psi=2\pi R{_{\mbox{\tiny mb}}}^2 B{_{\mbox{\tiny mb}}}$ [@RGB06]. Combining this result with Eqs.(\[b\],\[bz1\]), we find $$L{_{\mbox{\tiny BZ}}} \simeq
\frac{\sqrt{14}}{9}\frac{f_1^{3/2}f_2^2}{\alpha\beta} \dot{M}c^2 \simeq 
\frac{0.05}{\alpha_{-1}\beta_1}  \dot{M}c^2, 
\label{bz2}$$ where $\alpha_{-1}=\alpha/0.1$ ( for $0.5\!<a\!<1$ the combination $f_1^{3/2} f_2^2$ depends weakly on $a$ and is approximately $1/4$).

The mass accretion rate can be estimated following the procedure described in @BK09. The total accretion time includes the travel time of the rarefaction wave send into the stellar envelope by the core collapse, the time of the envelope collapse, and the disk accretion time, which gives the largest contribution. Accounting only for the disk contribution, the accretion time scale for the stellar matter located in the progenitor at radius $R$ is $t \propto l^3/M^2$, where $l=\Omega R^2$ and $M(R)$ is the stellar mass enclosed within radius $R$. Then $\dot{M} = dM/dt \simeq (dM/dR)/(dt/dR)$, where for the Bethe’s model we have $dM/dR \simeq M_s/(R\ln(R_s/R_c))$ and $dt/dR \simeq 6t/R$, where $R_c$ is the stellar core radius. Collecting the results we obtain $$\dot{M} \simeq \frac{1}{6}\frac{M_s}{\ln(R_s/R_c)} \frac{1}{t} 
\simeq 36 \frac{M_{s,3}}{t} M_\odot \mbox{s}^{-1},
\label{m-dot}$$ where $t$ is measured in seconds and we used $R_s/R_c=100$.[^4] Here, we assumed that the whole of the disk is accreted by BH, following the original ADAF model. However, it has been argued that this model has to be modified via including disk wind [Advection Dominated Inflow Outflow Solution (ADIOS), @BB99], which implies a mass loss from the disk and a smaller accretion rate compared to Eq.(\[m-dot\]). While the arguments for disk wind are very convincing, the actual value of mass loss is not well constrained and can be rather low. Given Eq.(\[m-dot\]) the power of BZ-jet is $$L{_{\mbox{\tiny BZ}}} \simeq
\frac{3.2\times 10^{52}\epsilon_m} {\alpha_{-1}\beta_1} \frac{M_{s,3}}{t_2} 
\mbox{erg}\,\mbox{s}^{-1},
\label{bz3}$$ where $t_2=t/100$ and $\epsilon_m<1$ is the fraction of the disk mass reaching the BH. Given the jet propagation speed inside the star, $v_j\simeq 0.2c$, deduced from axisymmetric numerical simulations [@BK08], the jet breakout time is expected to be around of few hundred seconds and, thus, the numerical factor in Eq.(\[bz3\]) gives us the optimistic jet power at the time when it becomes observable. In fact, the initial influx of mass through the polar column is very large and activation of the Blandford-Znajek mechanism can be delayed [@KB09]. The very latest time for the activation is given by the free-fall time of the whole star, $$t{_{\mbox{\tiny ff}}} \simeq 1000 R_{s,12}^{3/2} M_{s,3}^{-1/2} \mbox{s},
\label{free-fall}$$ as by this time the polar column becomes completely empty.

The total duration of the jet production phase has to be similar to the disk lifetime. If VMS is rotating at half of the break-up speed then the initial outer edge of the disk is at $R_d \simeq R_s/4$. Ignoring the edge expansion due to accumulation of angular momentum, the disk life time is given by its “viscous” time scale

$$t{_{\mbox{\tiny ce}}} \simeq \frac{2R_s}{3v{_{\mbox{\tiny in}}}(R_s)} 
          \simeq 5000 \alpha_{-1}^{-1}
     R_{s,12}^{3/2} 
     M_{s,3}^{-1/2} \mbox{s}, 
\label{cet}$$

where $R_{s,12}$ is the stellar radius measured in $10^{12}$cm. By this time the BZ power will be significantly reduced but could still play a role in shaping the light curve of afterglow emission [@BK09].

Using the mass accretion rate given by Eq.(\[m-dot\]) we can check if the neutrino cooling needs to be included in the model. Under the conditions of the supercollapsar’s disk its cooling is dominated by pairs. Using the well known equation for this cooling rate [e.g. @YKGH01] we can compare the cooling time with the accretion time at a given disk radius. The result is

$$\frac{t_d}{t{_{\mbox{\tiny cool}}}} \simeq 0.3 
  \alpha_{-1}^{-9/4} (R/R_g)^{-13/8}
   \dot{M}_{-1}^{5/4}  M_{s,3}^{\!-3/2}.
\label{cooling}$$

Thus, except for the very inner part of the disk, the neutrino cooling is indeed inefficient.

The high BZ power given by Eq.(\[bz3\]) suggests that the GRB-like burst emission from such jets could be seen even from $z\simeq 20$ and in the next section we discuss the properties of such bursts in more details.

Observational signatures {#signatures}
========================

Assuming that the radiation mechanism of the supercollapsar jets is similar to that of normal GRB jets, we expect the peak in the spectral energy distribution of the prompt emission in the source frame to be around $0.5$MeV. However, the cosmological redshift effect reduces the peak down to $$E{_{\mbox{\tiny max}}} \simeq 25\mbox{keV} {\left(\!\frac{1+z}{20}\!\right)}^{-1}, 
\label{peak freq}$$ which is still inside the energy window of [*Swift*]{}’s BAT. For the same reason the observed total duration of the burst increases up to $$t_b \simeq 1 {\left(\!\frac{1+z}{20}\!\right)} \alpha_{-1}^{-1}
     R_{s,12}^{3/2} M_{s,3}^{-1/2} \mbox{day}.
\label{duration}$$ The characteristic source frame time scale for the decay of BZ luminosity in the model presented above is given by the time since the onset of the collapse. Thus, the initial time scale for the burst decay will be of order of the jet break out time, few$\times 10^2$ seconds, or a bit longer if the activation of the BZ-mechanism is significantly delayed. In the observers frame this translates into few$\times 10^3 - 10^4$ seconds. Thus, these bursts would appear not only unusually soft but also unusually long-lasting.

The total flux density of the burst emission received on Earth and the isotropic luminosity are related via $F = L/(4\pi r{_{\mbox{\tiny L}}}^2)$, where $r{_{\mbox{\tiny L}}}$ is the luminosity distance to the source [e.g. @P93]. However, the emission from GRB jets is highly anisotropic due to the relativistic beaming. Moreover, not all of the Blandford-Znajek power is converted into the radiation within the energy window of the receiver. This leads to

$$F = \epsilon_c\frac{L{_{\mbox{\tiny BZ}}}}{4\pi r{_{\mbox{\tiny L}}}^2 {\cal A}}, 
\label{flux1}$$

where ${\cal A}\ll 1$ is the solid angle of the radiation beam and and $\epsilon_c<1$ is the conversion efficiency. In flat Universe

$$r{_{\mbox{\tiny L}}} = \frac{c}{H_0}(1+z) \int\limits_{0}^{z} 
   (\Omega_m(1+z)^3+\Omega_{\Lambda})^{-1/2} dz.
\label{lum dist}$$

For $z=20$ and the density parameters $\Omega_{\Lambda}=0.72$, $\Omega_{m}=0.28$ [@k09] this gives us

$$F \simeq  2 \times 10^{-7} 
         \frac{\epsilon_{c,-1} \epsilon_m}{\alpha_{-1}\beta_1 {\cal A}_{-3}} 
         M_{s,3} t_3^{-1}  
    \mbox{erg}\, \mbox{cm}^{-2} \mbox{s}^{-1},
\label{flux}$$

where $\epsilon_{c,-1}=\epsilon_c/0.1$, ${\cal A}_{-3}={\cal A}/10^{-3}$, and $t_3=t/10^3$. One can see that for the first $10^4$s this is above the sensitivity of BAT, $10^{-8} \mbox{erg}\, \mbox{cm}^{-2} \mbox{s}^{-1}$, and thus such a burst could trigger BAT. Having said this, we keep in mind that there is a great deal of uncertainty with respect to the values of various parameters appearing in Eq.(\[flux\]).

The time dependence in Eq.(\[flux\]) gives the evolution of mean bolometric flux. It is not clear if the supercollapsar bursts will also exhibit the fine substructure characteristic of normal GRBs. If the variability of normal GRBs is due to internal shocks in baryon dominated flow, as this is proposed in the currently most popular model of prompt gamma-ray emission [@MR94], then the supercollapsar burst produced by magnetically-dominated BZ jet may well be smooth and featureless. However, there are models of normal GRBs that attribute the observed variability to unsteady magnetic dissipation [@LB03; @GUB09; @KN09]. If they are correct then the supercollapsar bursts will also show fine substructure.

In order to estimate the observed rate of such burst we assume, following @MR01, that the dark matter mini-haloes that host supercollapsars arise from 3-$\sigma$ fluctuations that constitute only $\simeq 0.3\%$ of the dark matter matter of the Universe and that only one supercollapsar per mini-halo is produced. The total mass per Mpc$^3$ at $z=20$ is $M{_{\mbox{\tiny Mpc}}} \simeq 1.5\times 10^{15} M_\odot$. The number density of 3-$\sigma$ mini-haloes is then $$n{_{\mbox{\tiny mh}}} \simeq 0.003 \frac{\Omega{_{\mbox{\tiny dm}}}M{_{\mbox{\tiny Mpc}}}}{10^6 M_\odot} 
       \simeq 10^6 \mbox{Mpc}^{-3}.$$ Let us assume, for the sake of simplicity, that all supercollapsars go off simultaneously at cosmological time $t_e$ corresponding to $z=20$ (a moderate spread around this redshift will not significantly change the result). In flat Universe the observed time separation between events occurring simultaneously at $r_0$ and $r_0+dr_0$, where $r_0$ is the comoving radial coordinate, is $dt_o=cdr_0$. The corresponding physical volume within one steradian of the BAT’s field of view, is $$dV=a^3(t_e) r_0^2 dr_0,$$ where $a(t_e)=(1+z)^{-1}$ is the scaling factor of the Universe at $t=t_e$ (in the calculations we fix the scaling factor via the condition $a(t_o)=1$). $r_0$ and $t_e$ are related via $r_0=r{_{\mbox{\tiny L}}}(1+z)^{-1}$. Putting all this together we find the rate to be $$f_c = {\cal A}\frac{ c n{_{\mbox{\tiny mh}}} r{_{\mbox{\tiny L}}}^2}{(1+z)^5} 
   \simeq 4\, {\cal A}_{-3} {\left(\!\frac{n{_{\mbox{\tiny mh}}}}{10^6}\!\right)} \mbox{yr}^{-1} \mbox{sr}^{-1}.$$ Recent high-resolution simulations of cosmological star formation indicate the possibility of further fragmentation of gas clumps in minihaloes, resulting in formation of binary or even multiple protostars in some realizations [@TAS09; @SGB09]. Thus, the theoretical rate of VMS formation can be significantly smaller compared to the one used in our calculations, making the supercollapsar bursts rare events. This may explain why such bursts have not been seen so far.

Conclusions
===========

In spite of the significant progress in the astrophysics of Gamma Ray Bursts, both observational and theoretical, it may still take quite a while before we fully understand both the physics of the bursts and the nature of their progenitors. At the moment there are several competing theories and too many unknowns. Similarly, we know very little about the star formation in the early Universe. For this reason, the analysis presented above is rather speculative and the numbers it yields are not very reliable. Further efforts are required to develop a proper theory of supercollapsars and to make firm conclusions on their observational impact. On the other hand, our estimates suggest that if we are on the right track then the X-ray bursts of supercollapsars may be detectable already with [*Swift*]{}. The expected very long duration of bursts and their relatively low brightness imply that a dedicated search program using the image trigger may be required. Such search would be useful even in the case of non-detection as this would put important constraints on models of star formation in the early Universe, models of the GRB progenitors, and the origin of SMBHs.

Acknowledgments {#acknowledgments .unnumbered}
===============

This research was funded by STFC under the rolling grant “Theoretical Astrophysics in Leeds” (SSK and MVB). The work on this project was started during the NORDITA program on Physics of Relativistic Flows (Stockholm 2009) and we are grateful to the hosts and organises of this program for their generosity and hospitality. We are also thankful to Felix Aharonyan, Boris Stern, and Alexei Pozanenko for useful discussions.

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[^1]: serguei@maths.leeds.ac.uk (SSK)

[^2]: E-Mail: bmv@maths.leeds.ac.uk (MVB)

[^3]: The inner edge of the disk varies between the marginally bound and the marginally stable orbits, depending of its thickness. In our calculations this does not make much difference.

[^4]: For a $300M_\odot$ star at $t\simeq 8$s this gives $\dot{M}\simeq 1.3M_\odot \mbox{s}^{-1} $, which agrees reasonably well with numbers given in @F01.
