Recycling of Neutron Stars in Common Envelopes and Hypernova Explosions
In this paper we propose a new plausible mechanism of supernova explosions specific to close binary systems. The starting point is the common envelope phase in the evolution of a binary consisting of a red super giant and a neutron star. As the neutron star spirals towards the center of its companion it spins up via disk accretion. Depending on the specific angular momentum of gas captured by the neutron star via the Bondi-Hoyle mechanism, it may reach millisecond periods either when it is still inside the common envelope or after it has merged with the companion core. The high accretion rate may result in strong differential rotation of the neutron star and generation of the magnetar-strength magnetic field. The magnetar wind can blow away the common envelope if its magnetic field is as strong as G, and can destroy the entire companion if it is as strong as G. The total explosion energy can be comparable to the rotational energy of a millisecond pulsar and reach erg. However, only a small amount of Ni is expected to be produced this way. The result is an unusual type-II supernova with very high luminosity during the plateau phase, followed by a sharp drop in brightness and a steep light-curve tail. The remnant is either a solitary magnetar or a close binary involving a Wolf-Rayet star and a magnetar. When this Wolf-Rayet star explodes this will be a third supernovae explosion in the same binary.
Usually, neutron stars (NS) have magnetic field G and rotate with a period of fraction of a second, the Crab pulsar being a typical example. However, we now know that the “zoo” of NS is much more diverse and they can have both much weaker and much stronger magnetic field and rotate with both much longer and much shorter periods.
Around 10% of NS have surface dipolar magnetic field G (Kouveliotou et al., 1998). These “magnetars” are believed to be born in core collapse explosions of rapidly rotating stars. During the collapse, the proto-neutron star naturally develops strong differential rotation, which allows for generation of magnetic field via --dynamo (Duncan & Thompson, 1992; Thompson & Duncan, 1993). The strength of saturated magnetic field strongly depends on the rotational period, with shorter periods leading to stronger magnetic field and more rapid release of rotational energy. In order to generate the magnetar strength magnetic field the rotation period must be around few milliseconds (Duncan & Thompson, 1992). Such a strong field allows to release up to erg of magnetar’s rotational energy in very short period of time. This is sufficient to drive extremely powerful supernova explosions, on the hypernova scale, and to produce powerful Gamma Ray Bursts (GRB, e.g. Usov, 1992; Thompson et al., 2004). Turbulence required for the magnetic dynamo action can also be generated via the magneto-rotational instability (MRI, Balbus & Hawley, 1991). Calculations based on the linear theory show that in the supernova context strong saturation field can be reached very quickly, on the time scale of only several tens of rotational periods (Akiyama et al., 2003; Obergaulinger et al., 2009).
Millisecond pulsars are found in low mass binaries, and it is generally thought that they have been spun up via disk accretion (Alpar et al., 1982; Archibald et al., 2009). This origin implies mass increase by about 0.2 compared to normal radio pulsars, whose masses are narrowly distributed around (Thorsett & Chakrabarty, 1999). It is rather difficult to measure the mass of a millisecond pulsar, but the few available results agree with this prediction of the accretion model (Kaspi et al., 1994; Jacoby et al., 2005; Demorest et al., 2010). The most massive pulsar found to date, almost 2, is a millisecond pulsar (Demorest et al., 2010). The magnetic field of these millisecond pulsars is very low, down to Gauss. Most likely, their initial magnetic field was of similar strength to normal pulsars, but now it is buried under the layers of accreted matter (Bisnovatyi-Kogan & Komberg, 1974; Alpar et al., 1982). The reason why these pulsars could not generate magnetar-strength magnetic field in the same way as in the core-collapse scenario is the very long time scale of spinning up compared to the viscous time-scale. As the result, the differential rotation remains weak and there is not enough energy for effective magnetic dynamo (Spruit, 1999).
The rotational frequency of the fastest X-ray pulsar is Hz (Chakrabarty et al., 2003), whereas the fastest known radio pulsar has the frequency of Hz (Backer et al., 1982). Most likely, it is the gravitational radiation losses what places the upper limit on the rotation rates because at high spin the r-mode oscillations become excited (Shapiro & Teukolsky, 1983; Levin, 1999). Spruit (1999) argued that this instability may also result in magnetic explosion. The idea is that the heating of NS, associated with these oscillations, reduces its viscosity and decouples its interior from the outer layers. Being most disturbed, the outer layers rapidly loose some of their angular momentum via gravitational radiation. This leads to strong differential rotation and generation of magnetar strength magnetic field in the NS interior. This field becomes unstable to buoyancy, emerges on the surface, and magnetically driven pulsar wind rapidly extracts the rotational energy of the NS. Spruit (1999) proposed this as an alternative scenario for long GRBs. It is unlikely that a supernova-like event can accompany a GRB in this scenario. Although the wind energetics is sufficient, only a small fraction of this energy can be deposited into the companion star, simply because of its small geometrical cross-section. Moreover, recent results suggest that the amplitude of r-modes may saturate at a much lower level due to nonlinear interaction with other modes (Arras et al., 2003; Brink et al., 2005; Bondarescu et al., 2007).
A similar recycling of NS may occur during the common envelop (CE) phase, after the primary becomes a red super giant (RSG; Bisnovatyi-Kogan & Syunyaev, 1971; Paczynski, 1976; Tutukov & Yungelson, 1979; Postnov & Yungelson, 2006). Due to the dynamic friction, the NS then spirals inside the RSG, accreting on its way. Now one can imagine two interesting outcomes of such process. First, the neutron star may accumulate too much mass and collapse into a black hole (BH). This BH is likely to be rapidly rotating and drive a stellar explosion in the collapsar fashion (Fryer & Woosley, 1998; Zhang & Fryer, 2001; Barkov & Komissarov, 2010).
Second, the NS may first spin up to a millisecond period and drive a magnetic explosion of the type proposed by Spruit (1999) but now inside the common envelope. The magnetar wind can keep energising such supernovae, producing a similar effect to radioactive decay (Woosley, 2010; Kasen & Bildsten, 2010). High accretion rates may modify the way the NS is recycled. The accreted gas can form a massive rapidly rotating layer above the NS crust (Inogamov & Sunyaev, 1999, 2010). The strong differential rotation between the layer and the NS core may result in development of the Kelvin-Helmholtz instability when the NS crust melts down under the weight of the layer. This may lead to turbulence and strong amplification of the NS magnetic field.
The accretion onto neutron stars during the in-spiral has been studied by Chevalier (1996), who concluded that the high angular momentum of the gas gravitationally captured by NS prevents it from effective neutrino cooling and keeps the mass accreting rate well below the rate of the Bondi-Hoyle capture. As the result, the NS accumulates very little mass while still inside the common envelope. On the other hand, he suggested that during the merger with the companion core the mass accretion rate rises and the NS collapses into a black hole. While carefully analysing various effects of rotation, Chevalier (1996) did not use any particular model for the primary. Moreover, there is a great deal of uncertainty with the regard to the specific angular momentum of the gravitationally captured gas. In our study we come back to this problem, consider realistic models of RSG stars and allow for the uncertainty.
The paper is organised as follows. In Section 2 we consider the accretion and recycling of NSs during the in-spiral. We conclude that the outcome is very sensitive to the assumed specific angular momentum of the gravitationally captured gas. Given the current uncertainty with the regard to the angular momentum, it seems possible that NSs begin to accrete with Bondi-Hoyle rate while still inside common envelopes. In this case, they rapidly spin up to millisecond periods. In Section 3 we speculate on the possible mechanisms of generating magnetar-strength magnetic field and study the magnetic interaction of millisecond magnetars with accretion flows typical for the in-spiral problem. We find that if the magnetar forms inside the common envelope and its magnetic field is about G, the magnetospheric pressure can overcome the gravity and drive an outflow, with eventual release of up to erg of the magnetar rotational energy. If the magnetar forms only after the merger with the core then higher magnetic field, G, is required to drive stellar explosions. In Section 4 we discuss the properties of such explosions and their observational signatures. Because the RSG envelope is rich in hydrogen the supernova will be classified as type-II. Because of the very high energy release and relatively small mass of the RSG, the speed of the ejecta is expected to be very high, cm/s, and the luminosity at the plateau phase erg/s. However, due to the small amount of Ni generated in the explosion and the rapid spin-down of the magnetar, the plateau is followed by a sharp drop in brightness and a steep light-curve tail. We show that the termination shock of the magnetar wind produces a high energy synchrotron and inverse Compton emission which may energise the tail, but the supernova ejecta soon becomes transparent to this emission, limiting its potential to mimic the effect of radioactive decay. The flux of gamma-ray emission is expected to be rather low and difficult to observe, unless the explosion occurs in the Local Group of galaxies. In the case of off-center explosion, the remnant is a very close binary system consisting of a WR star and a magnetar, but the strong magnetic field prevents magnetar from accreting plasma of the WR wind. Our results are summarized in Section 5.
In this paper, the dimensional estimates are presented using the following notation: the time is measured in s, the distance in cm, the speed in , the mass in , mass accretion rate in , and the magnetic field in G.
2 In-spiral dynamics and recycling of neutron stars
As the neutron spirals inside its giant companion it accretes mass and angular momentum. The accretion can proceed at the Eddington rate or at the much higher Bondi-Hoyle rate. This depends on whether the radiation is trapped in the accretion flow and the neutrino cooling is sufficiently effective (Houck & Chevalier, 1991; Chevalier, 1993, 1996; Fryer et al., 1996). Only if the accretion proceeds at the Bondi-Hoyle rate the recycling of NS is sufficiently fast and the explosion can occur during the inspiral.
In the Bondi-Hoyle problem, an accreting point mass is moving with finite speed through a uniform medium of mass density and sound speed . The Bondi-Hoyle mass accretion rate is well approximated by the equation
is the accretion radius, and
where is the Mach number (Shima et al., 1985). The mean angular momentum of accreted matter is zero.
The in-spiral problem is more complicated due to the density and velocity gradients across the direction of motion of the NS. Provided, the gradients are small on the scale of accretion radius the above expression for is still quite accurate (Ishii et al., 1993; Ruffert, 1997, 1999). However, the accreted matter can now have non-vanishing mean angular momentum. This was first suggested by Illarionov & Sunyaev (1975) in connection with X-ray binaries, where the accretion disk of the black hole is fed by the stellar wind of the primary. Taking into account only the velocity gradients related to the orbital motion, they found that the mean specific angular momentum of accreted matter inside the accretion cylinder is
where is the angular velocity of the orbital motion. Shapiro & Lightman (1976) took into account the density gradient of the wind as well and refined this result,
where the sign minus shows that the disk rotation is now in the opposite sense to the orbital motion. Later, Davies & Pringle (1980) criticised the approach used by the previous authors. They generalised the original calculations of Hoyle & Lyttleton (1939) to include the density and velocity gradients and concluded that to the first order in , where is the length scale of the variations, . The 2D numerical simulations of Ishii et al. (1993) seemed to agree with this conclusion, whereas their 3D simulations still indicated net accretion of angular momentum. Ruffert (1997, 1999) carried out extensive 3D numerical investigation of the problem. Introducing angle-dependent accretion radius, he also derived another version of the analytic expression for
are the dimensionless gradients of density and velocity respectively, with being the transverse Cartesian coordinate. However, the numerical results did not agree with this expression, with varying between 7% and 70% of what is available in the accretion cylinder. Moreover, in the simulations only the cases with either velocity or density gradient were considered, whereas in the in-spiral problem both are present. Although the computational resources and numerical algorithms have improved since 1999, no further attack on this problem has been attempted yet and the issue of remains open.
For the inspiral problem , , and , where is the orbital radius of NS. Assuming power law for density distribution inside the RSG, , we also find . This allows us to write Eq.6 as
where is a free parameter, which reflects our current ignorance.
In our case, is the mass of the NS and is its orbital speed. Assuming that the interaction does not disturb the RSG inside the orbit, we have
where is the orbital radius and is the mass of RSG inside the orbit. Under the same assumption, and can be replaced with the density and sound speed of the undisturbed RSG at radius . The motion of the NS inside RSG is only mildly supersonic, (Chevalier, 1993), allowing us to write
where and only weakly depends on the model of RSG.
Given the specific angular momentum we can estimate the distance from the NS at which the Bondi-Hoyle trapped gas will form an accretion disk as
In the literature this radius is often called the circularisation radius. Since the NS is not a point mass the accretion disk forms only if . When this condition is satisfied, the recycling of NS proceeds at the highest rate.
For the accretion on NS may proceed in more or less spherical fashion and is well described by the Bondi solution, in which the accretion flow becomes supersonic at (for the polytropic index ). Its collision with the NS and/or its accretion disk creates a shock wave which re-heats the flow. This will have no effect on the mass accretion rate only if the following two conditions are satisfied (Chevalier, 1996).
Firstly, the radiation produced by the gas heated at the accretion shock should not be able to escape beyond . If it does the mass accretion rate is limited by the Eddington value
This condition is satisfied when the accretion shock radius
is smaller compared to the radiation trapping radius
For this to occur has to exceed the critical value
(Chevalier, 1996). This result was derived assuming that and ignoring the General Relativity corrections.
Secondly, the shock has to be inside the sonic surface of the Bondi flow111The shock radius in Eq.13 is obtained assuming cold supersonic flow at infinity. Otherwise, this shock cannot be a part of the stationary solution and hence the Bondi-type accretion cannot be realised. This condition is satisfied when the exceeds the critical value
(Chevalier, 1996). As we shell see this condition is more restrictive.
A note of caution has to be made at this point. The analytical expression for the accretion shock radius (13) is based on a number of assumptions and simplifications and its accuracy has not yet been tested via numerical simulations. At the moment, this can only be considered as an order of magnitude estimate. When becomes as small as the NS radius, this expression gives which exceeds the value given by the more reliable spherically symmetric model of Houck & Chevalier (1991) by a factor of ten (see Eqs. 13 and 23). Two reasons explain this disagreement. First, in the spherically symmetric models the gravity is relativistic, whereas in the models with rotation it is Newtonian. Second, the models use different assumptions on the geometry of the neutrino cooling region. It appears that Eq.13 becomes increasingly less accurate when approaches , overestimating . As the result, Eq.16 overestimates the value of . We also note, that the critical rates are quite sensitive to the parameter ,
In this study, we have considered six different models of RSG. In the first two models we assumed that mass distribution in the stellar envelope is the power law , , with (model RSG2) and (model RSG3). In both these models, the total stellar mass , the helium core mass , the inner and the outer radii of the power law envelope are cm and cm respectively. For these models all calculations can be made analytically. Not only this helps to develop a feel for the problem, but also provides useful test cases for numerical subroutines. The next two models are based on stellar evolution calculations and describe RSG in the middle of the He-burning phase (Heger et al., 2004; Woosley et al., 2002). Their Zero Age Main Sequence (AMS) masses are (model RSG20He) and (model RSG25He). These data were kindly provided to us by Alexander Heger. The last two models, RSG20 and RSG25, describe pre-supernova RSG with the same AMS masses222These models can be downloaded from the website http://homepages.spa.umn.edu/alex/stellarevolution/ .. Their have very extended envelopes, increasing the chance of CE phase.
Figure 1 shows the Bondi-Hoyle and the critical accretion rates for these six models and the two extreme values of the parameter , representing the cases of both very efficient () and rather inefficient () accretion of angular momentum (see Eq.9). One can see that the condition of radiation trapping is not very restrictive. For it is satisfied for cm in models RSG20 and RSG25, for cm in models RSG20He and RSG25He, and much earlier for .
On the contrary, the condition can be quite restrictive. In fact, it is never satisfied when . In this case, the circularisation radius is always too far away from the NS and the neutrino cooling is not efficient enough. For the results seem more promising as this condition is satisfied long way before NS merges with the core. For RSGs in the middle of the He-burning phase this occurs almost the the same point where the radiation becomes trapped, and a bit later for the presupernova RSGs. Thus, for NS begins to hyper-accrete already inside the envelope of the RSG primary. However, for such a small value of the accretion disk becomes very compact and may even disappear, in which case the recycling efficiency is reduced.
Figure 2 shows all the characteristic radii of the in-spiral problem for . One can see that for RSG20 and RSG25 models, the disk accretion domain has cm. For the model RSG20He this domain includes the whole stellar envelope and for RSG25He it splits into two zones. What is most important is that in all models the accretion begins to proceed with the Bondi-Hoyle rate while still in the disk regime.
Once the accretion disk is formed the specific angular momentum of the gas settling on the NS surface is that of the last Keplerian orbit. In this case the star will reach the rotational rate after accumulating
where is the final moment of inertia of NS and is the Keplerian angular momentum at the NS radius. Obviously, the same mass is required to recycle the observed millisecond pulsars in low mass binaries and this allows us to conclude that the NS may well reach the gravitationally unstable rotation rate before collapsing into a black hole.
When the accretion proceeds at the Bondi-Hoyle rate, the NS mass and orbit are related approximately as
where . Chevalier (1993) found that for 10 RSG varies slowly between 5 and 7. According to these results, NS increases its mass by after its orbital radius decreases by only a factor of few. This shows that the “window of opportunity”, where the disk accretion can proceed at the Bondi-Hoyle rate, does not have to be particularly wide. In order to obtain more accurate estimate, one could integrate the evolution equation
where is the average density of the primary inside , is the drag coefficientShima et al. (1985), and (Chevalier, 1993). Figures 3 and 4 show the results of such integration for the models RSG20, RSG25, and RSG25He, assuming that the hyper-accretion regime begins near the stellar surface and at cm respectively. One can see that in all these cases the gravitationally unstable rotation rate is reached when decreases by less than a factor of five, in good agreement with the original estimate.
For , the mass accretion rate is low and the NS cannot spin-up significantly inside the CE. What occurs in this case depends on the details of CE evolution. The CE can either be ejected leaving behind a close NS-WR binary or survive, leading to a merger of the NS and the RSG core (Taam & Sandquist, 2000). The configuration after the merger is similar to that of the Thorne-Zydkow object (Thorne & Zytkow, 1977; Barkov et al., 2001). The common view is that such objects can not exist because the effective neutrino cooling leads to hyper-accretion and prompt collapse of the NS into a BH. This conclusion is based on the spherically symmetric model, where the circularisation radius is zero and the flow stagnation surface coincides with the surface of the NS. However, during the inspiral the orbital angular momentum of the NS is converted into the spin of the primary. Thus, one would expect the core to be tidally disrupted and form a massive accretion disk around the NS. As the result, the cooling rate will be lower compared to that found in the spherically symmetric case. It is not obvious for how long this configuration may exist. However, this again opens the possibility of recycling of the NS up to the gravitationally unstable rotation rate and magnetically driven stellar explosion.
In the case of core-collapse, even the magnetic field of magnetar strength cannot prevent accretion and fails to drive stellar explosion (Komissarov & Barkov, 2007). The magnetosphere becomes locked under the pile of accreting matter, and a separated mechanism, e.g. the standard neutrino-driven explosion, is required to release it. However, the density around the Fe core of presupernovae is very high, leading to very high accretion rates. In our case the accretion rates are expected to be significantly lower. In the next section we explore if they are low enough to allow purely magnetically-driven explosions.
3 Magnetic explosion
3.1 Magnetic field generation
The main route to formation of “magnetars” is believed to be the core collapse of rapidly rotating stars. During the collapse, the proto-neutron star naturally develops strong differential rotation, with the angular velocity decreasing outwards. This allows generation of super-strong magnetic field via the --dynamo (Duncan & Thompson, 1992; Thompson & Duncan, 1993) or magneto-rotational instability (Burrows et al., 2007).
In our case, the conditions are rather different. The neutron star is already fully-developed, with a solid crust, when it enters the envelope of its companion. The typical NS crust has the thickness cm and the density at the bottom (see e.g. Gnedin et al., 2001; Lattimer & Prakash, 2007, ). The neutron star may have already been spinned up via accretion from the stellar wind, but but its rotation rate may still be well below the critical for development of the gravitational wave instability. In any case, the accreted gas of companion’s envelope forms a very rapidly rotating layer above the crust. As long as the crust exists, this layer and the NS core are basically decoupled, and interact mainly gravitationally. A strong tangential discontinuity exists at the outer boundary of the crust (Inogamov & Sunyaev, 1999, 2010).
As the mass of the outer layer increases, the crust pressure goes up. Because of the degeneracy of the crust matter, its pressure depends mainly on its density, and only weakly on temperature. When the density reaches the critical value , the crust begins to melt Brown (2000), and the barrier separating the NS core and its outer rotating layer disappears. If we ignore the centrifugal force than this occurs when the NS accumulates mass comparable to the crust mass, . The centrifugal force significantly reduces the effective gravitation acceleration and the above estimate for the layer mass is only a lower limit. Assuming the rotational energy of the layer at the time of melting may well reach ergs. Once the crust has melted, the core and the layer can begin to interact hydromagnetically, with up to ergs of energy in differential rotation to be utilised.
One possibility is viscous heating and neutrino emission. We can estimate the neutrino luminosity as , where is the neutrino diffusion time. Following the analysis of Lattimer & Prakash (2007) and Barkov (2008), this time can be estimated as
where is the optical depth of the hot envelope for the neutrino emission. Assuming equilibrium with the electron fraction , the mean neutrino cross-section is about , where Bahcall (1964) and is mean energy of neutrino (Thompson et al., 2001). The temperature can be estimated as
where is the radiation constant. The corresponding diffusion time is quite short,
Another process is development of the Kelvin-Helmholtz instability, production of turbulence and amplification of magnetic field. For the Kelvin-Helmholtz instability to develop the Richardson number has to be below than 1/4 (Chandrasekhar, 1961). In our case
where is the gravitational acceleration and we use . Thus, the instability condition is marginally satisfied. Once the turbulence has developed the magnetic field amplification is expected to proceed as discussed in Duncan & Thompson (1992). The e-folding time is given by the eddy turn-over time, which one would expect to be below the rotation period of the outer layer. Thus, strong magnetic field can be generated on the time below the viscous and the neutrino diffusion time scale333The viscous dissipation timescale is very long s (Shapiro, 2000). This conclusion is supported by the recent numerical simulations of similar problems, which show that the time as short as 0.03 s can be sufficient to generate dynamically strong magnetic field (Obergaulinger et al., 2009; Rezzolla et al., 2011). The strong toroidal magnetic field can suppers the instability in the case if (Chandrasekhar, 1961) or if G. This is likely to determine the saturation strength of magnetic field. Such a strong magnetic field is also buoyant (Spruit, 1999) and will emerge from under the NS surface into the surrounding accretion flow.
Another possibility of generating super-strong magnetic in recycled NS was proposed by Spruit (1999) with application to X-ray binaries. In this model, the NS spins up to the critical rotation rate where it becomes unstable to exciting r-modes and gravitational radiation (Levin, 1999; Spruit, 1999). This may lead to effective braking of the outer layers of NS and developing of strong differential rotation. Recent studies, however, suggest that non-linear coupling with other modes leads to the saturation of the r-mode at a much lower amplitude (Arras et al., 2003; Brink et al., 2005; Bondarescu et al., 2007), though the mass accretion rates assumed in these studies are much lower than those encountered in common envelopes.
3.2 Criteria for explosion
Depending on the mass accretion rate, this may just modify the properties of the accretion flow near the star, the “accretor” regime, or drive an outflow (e.g. Illarionov & Sunyaev, 1975). In the latter case, two different regimes are normally discussed in the literature, the “propeller” regime and the “ejector” regime. In the ejector regime, the magnetosphere expands beyond the light cylinder and develops a pulsar wind. In the propeller regime, the magnetosphere remains confined within the light cylinder. If the magnetic axis is not aligned with the rotational axis it may act in a similar fashion to an aircraft propeller, driving shock waves into the surrounding gas, heating it and spinning it up. If the energy supplied by the propeller is efficiently radiated away, this may lead to a quasi-static configuration (e.g. Mineshige et al., 1991). If not, the most likely outcome is an outflow, followed by expansion of the magnetosphere, and transition to the ejector regime.
This classification is applied both in the case of quasi-spherical accretion and the case of disk accretion. In the disk case, the external gas above and below the disk is usually not considered as important dynamical element. However in our case this is not true because the accretion disk is rather compact and feeds from a more or less spherical component of the accretion flow. If the expanding magnetosphere removes this component, the disk will no longer be supplied with mass and quickly disappear, no matter what its regime is. For this reason, we will not consider the complicated interaction between the magnetosphere and the disk444An interested reader may find out more about the disk-magnetosphere interaction from Ustyugova et al. (2006) and references therein., but will focus on the interaction with the low angular momentum flow of the polar regions which accretes directly onto the NS avoiding the disk. In order to simplify the problem, we will treat this flow as spherically symmetric, keeping in mind that its mass accretion rate is only a fraction of the Bondi-Hoyle rate.
The extent of the magnetosphere is often estimated by the balance between the magnetic pressure and the ram pressure of accreting flow. This is fine when the accretion shock is close to the NS. If not, the thermal pressure at the magnetospheric radius can be significantly higher compared both to the actual ram pressure at the shock and to the ram pressure that would be found at the magnetospheric radius if the free-fall flow could continue down to this radius. In this case, the initial radius of emerging magnetosphere is rather determined by the balance of the magnetic pressure and the thermal pressure of the quasi-hydrostatic “settling” flow downstream of the accretion shock (Mineshige et al., 1991).
Assuming that the accretion flow in the polar region is approximately spherical, we can describe it using the analytical model developed by Houck & Chevalier (1991). Repeating their calculations with the more accurate cooling function, adopted later in Chevalier (1996),
and is the gravitational radius of the NS (In the calculations we assume the Schwarzschild space-time.). For reasonable masses and radii of NSs, . In particular for and km. Downstream of the shock we have an adiabatic subsonic flow with the mass density
is the mass density at the shock, the pressure
is the ram pressure of the free falling flow at the shock, the temperature
where is the radiation constant, and the radial component of 4-velocity
Assuming that the magnetosphere expands until its magnetic pressure equals to the gas pressure555 As the result of its oscillations the NS can become quite hot and fill the magnetosphere with substantial amount of plasma. If this case the centrifugal force will need to be included in the force balance, yielding even higher value for . we find the radius of dipolar magnetosphere with the surface strength to be
For this to be above the NS radius the mass accretion rate should not exceed
The traditional criterion for the propeller regime is that the magnetospheric radius is below the light cylinder radius
and above the co-rotation radius
at which the magnetosphere rotates with the local Keplerian velocity.
The ejector regime is realised when . In terms of the mass accretion rate the propeller regime criterion can be written as
and the ejector regime criterion as
The latter is much stronger, and even if only a fraction of the total mass accretion rate is shared by the quasi-spherical component, it could be satisfied only during the early phase of the inspiral (see Figure 1). More likely, the emerged magnetosphere will operate in the propeller regime.
The sound speed in the Houck & Chevalier (1991) solution depends only of the NS mass and the distance from the NS,
The linear rotational velocity of the magnetosphere,
is generally higher and thus we have the so-called supersonic propeller. The power generated by such a propeller is roughly
(Mineshige et al., 1991), exceeding by several orders of magnitude the accretion power
This suggests that the magnetosphere easily blows away the accreting matter and expands beyond the light cylinder, thus creating the conditions for proper pulsar wind from the NS.
One may argue that the accretion flow could re-adjust to the new conditions, with the accretion shock moving further out, the temperature at the magnetospheric radius rising, and the enhanced neutrino cooling compensating for the propeller heating, (A similar problem has been analysed by Mineshige et al. (1991).). Given the strong dependence of the neutrino emissivity on temperature, the cooling is expected to be significant only in the close vicinity of the magnetosphere. In the non-magnetic case, most of the cooling occurs in the volume comparable to the NS volume (Houck & Chevalier, 1991). This suggests that in our case the cooling volume will be approximately the same as the volume of the magnetosphere. Then the energy balance can be written as
where is given by Eq.22. Solving this equation for , simultaneously with the pressure balance
where is given by Eq.27, we find that
independently on the mass accretion rate. Since in this solution , we conclude that the accretion flow cannot re-adjust itself and will be terminated by the emerged magnetosphere.
In the case of cold neutron stars, the pulsar wind power
where is the star magnetic moment and is the angle between the magnetic moment and the rotational axis of the NS (Spitkovsky, 2006). For a hot NS the mass loading of the wind can be substantial, leading to even higher luminosity (Bucciantini et al., 2006; Metzger et al., 2007). The energy released in the magnetically driven explosion is the rotational energy of the NS
where is the NS moment of inertia.
Inside the RSG core the accretion rate is expected to be much higher. For example, in the RSG20He model the core density and the pressure , yielding the Bondi accretion rate
For such a high accretion rate, the propeller regime requires the magnetic field to be well in excess of G, similar to what is required in the magnetar models of GRBs. Whether such a strong dipolar magnetic field can be generated is not clear at present, as the observations of known magnetars suggest lower values. However, this could be a result of the short decay time expected for such a strong field (Heyl & Kulkarni, 1998).
4 Supernova properties
4.1 The plateau.
Since the NS has to reach the maximum allowed rotation rate before it explodes, the model predicts the explosion energy around erg, putting it on the level of hypernovae. The corresponding ejecta speed will be very high
where is the ejected mass.
Since the progenitor is a RSG the supernova spectra will show hydrogen lines and would be classified as type-II. As the NS spirals through the weakly bound outer envelope of its RSG companion, it drives an outflow with the speed about the local escape speed and the mass loss rate about yr (Taam & Sandquist, 2000). The total in-spiral time is about few initial orbital periods. For a RSG with mass and radius cm this yields and s. Thus, we expect a hydrogen rich circumstellar shell of few solar masses to exist at a distance of cm at the time of the supernova explosion. The presence of such a shell will delay the transition to adiabatic expansion phase and increase the supernova brightness similar to how this occurs type-IIn supernovae (Chugai et al., 2004; Woosley et al., 2007; Smith et al., 2007; Smith et al., 2008).
Following Kasen & Bildsten (2010), the supernova luminosity during the phase of adiabatic expansion can be estimated as
where is the expansion time scale and is the progenitor radius, and
is the radiative diffusion time scale, is the opacity (In the estimates below we use , the value for Thomson scattering in fully ionised plasma.). It is assumed that the initial magnetar spindown time scale is smaller compared to , which is certainly satisfied in our case, given the large required magnetic field, G. For the characteristic parameters of the model we have
This high luminosity will be sustained for up to , after which the trapped radiation, generated by the blast wave when it was crossing the star, escapes and the luminosity drops. It most normal type-II supernovae the luminosity after this point is sustained via radioactive decay of isotopes, mainly Ni, produced by the supernova shock in the high density environment around the collapsed core. However, in our model the stellar structure is significantly different from normal pre-supernovae.
4.2 Ni production
If the explosion occurs after the NS is settled into the center of RSG20He the core density is only and it temperature K. where is the Bondi accretion radius.
As the magnetically driven explosion develops a shock wave propagates through the body of RSG. Where the post-shock temperature exceeds K, Ni is produced in abundance (Woosley et al., 2002). In order to capture the shock dynamics, we need to know the structure of the accretion flow onto the NS, which has been established before the explosion. This flow is rather complicated because of its rotation and without computer simulations we can only make very rough estimates.
its rotation and near the NS there is an accretion disk. First consider the polar region where the flow is more of less spherically symmetric. Equation 13 shows that the accretion shock is quite close to the NS, particularly when it is near the RSG core, and hence most of the flow can be described using the free fall model. The ram pressure of the accretion flow in this model is given by
where is the free fall speed. Close to the NS the speed of explosion shock is much less compared to and can be found from the pressure balance equation
where is the shock radius and is the energy injected by the magnetar. This yields
and thus the shock speed increases with the distance. This approximation breaks down at the radius
where . The gas temperature exceeds inside the radius determined by the equation
and the total mass of produced Ni can be estimated as
where is the time required for the shock to reach according to Eq.54. From these equations we find that
Thus only a relatively small amount of Ni can be produced even for the highest mass accretion rates allowed in this model.
In the equatorial plane, the accretion shock can be pushed all the way towards (see Section 2). Downstream of the shock the flow is highly subsonic with
The highest density is achieved when the accretion shock is infinitely weak and this law is applied up to the sonic point . In this case, we can put and , the mass density of the RSG. The equation for shock speed is now
which yields the shock radius
is now determined by the equation
where the mass of produced Ni is still given by Eq.57. Combining these equations we find
The highest values for are obtained inside the RSG core. For the RSG20He, where cm, (see Figures 1 and 2), and , we find . Although significantly higher, this is still small compared to the mass deduced from the observations of most supernovae.
If the density behind the shock wave is lower than g cm, which corresponds to , then the photon dissociation of Ni becomes important as well (Magkotsios et al., 2010). Thus, our estimation gives only an upper limit on the amount of produced Ni.
Another potential source of Ni is the wind from the accretion disk itself, like in the collapsar model of GRB. Without a robust disk model, it is difficult to figure out how effective this mechanism is. For example, in the numerical simulations of MacFadyen & Woosley (1999), the outcome was dependent of the rate of introduced “viscous” dissipation. In any case, the much lower accretion rates encountered in our model suggest that the amount of Ni will be lower too. Thus, we conclude that our model predicts significant deficit of Ni.
4.3 The long term effects of magnetar wind
It has been suggested recently that the long-term activity of magnetars born during the core-collapse can power the supernova light curves, mimicking the effect of radioactive decay (Woosley, 2010; Kasen & Bildsten, 2010). In these studies, it was assumed that the rotational energy of the magnetar, released at the rate , is converted into heat at the base of the supernova ejecta. One mechanism of such heating is purely hydrodynamical, via the forward shock wave involved in the collision of the fast magnetar wind with the slower supernova ejecta. In our case, however, this shock is the supernova shock and it is not present since the break-out. The other mechanism, heating by the radiation produced at the termination shock of the magnetar wind, remains in force and here we explore its efficiency.
(In this Section we use cm and .) The last equation shows that unless G the supernova luminosity will drop dramatically after the plateau phase. Such evolution has been observed in some supernovae, for example SN1994W (Tsvetkov, 1995; Sollerman et al., 1998).
In order to see how exactly this energy can be used to heat the supernova ejecta we can appeal to the observed properties of pulsar wind nebulae. For example, the study of the Crab Nebula by Kennel & Coroniti (1984) indicates that immediately behind the termination shock most of the energy is in the form of ultra-relativistic electrons and only a fraction of it is in the magnetic field and there is no good reason to expect this to be different for magnetar winds. However, the magnetic field of a one year old magnetar wind nebula (MWN) is expected to be very strong, making the synchrotron cooling time very short.
One can estimate the magnetic field strength via the pressure balance at the termination shock
where is the radius of termination shock. Assuming that , this yields
The lowest energy electrons accelerated at the termination shock of the Crab Nebula have the Lorentz factor (Kennel & Coroniti, 1984). In the magnetic field of the MWN, they produce synchrotron photons with energy
and their synchrotron cooling time
is shorter compared to the dynamical time scale for few years after the explosion.
The observations of the Crab Nebula also show that the electrons are accelerated up to the radiation reaction limit (de Jager et al., 1996; Komissarov & Lyutikov, 2010). Assuming that the same is true in our case we expect the synchrotron spectrum to continue up to MeV and the highest energy of the electrons to be
These electrons also cool via the Inverse Compton (IC) scattering on the soft photons produced by the supernova shell. The ratio of the IC to the synchrotron energy losses , where is the energy density of the soft radiation and is the magnetic energy density. During the plateau phase
where is the radius of photosphere. Combining this result with Eq. 68 we find
As we have seen, can be much higher than but on the other hand can be much lower than . Thus, it is hard to tell whether the IC emission can compete with the synchrotron one without detailed numerical simulations666 Should the IC losses dominate, and are to be reduced by the factors and respectively.. After the plateau phase drops dramatically and we expect the synchrotron losses to be dominant.
For photons with energy MeV the main source of opacity is the Compton scattering in Thomson regime. The corresponding optical depth is
where is the Thomson cross-section. Thus, the shell becomes transparent in about one year after the explosion. The typical energy of photons from radioactive decays is around MeV and hence the ejecta opacity to these photons is similar. As the result, the synchrotron photons and the photons from radioactive decays are utilised with similar efficiency.
When the scattering is in the Klein-Nishina regime and the cross section is smaller, (Lang, 1980). For MeV synchrotron photons this yields
and these photons begin to escape already after ten days or so. However, provided the spectrum of the shock accelerated electrons is a power-law with , which is expected to be the case, the contribution of such highly energetic photons to the energy transport is lower.
The IC photons with energy TeV will also interact with the soft supernova photons via the two photon pair-production reaction (here we used eV). The corresponding opacity can be estimated as
Thus, we expect the ejecta to become transparent to the IC emission soon after the end of the plateau phase.
The above calculations show that during the plateau phase the high energy emission from the termination shock of the magnetar wind is deposited in the ejected. However this additional heating has little effect on the supernova luminosity. Indeed, the increase of the luminosity due to this heating can be estimated as
For the energy deposition rate does no longer scale as (c.f. Woosley, 2010) as the ejecta becomes transparent to the high energy emission. For example, if the synchrotron emission is the main channel of heating the rate scales as when , and the rate due to IC emission is likely to decline even faster.
Out of the well studied supernova known to the authors none fits the above description. The closest example is SN1994W (Cortini et al., 1994; Sollerman et al., 1998). This is one of the brightest type-II supernovae, with the peak luminosity erg/s. After around 120 days its luminosity drops dramatically down to erg/s. The tail of its light curve is very steep showing very small mass of ejected Ni (Sollerman et al., 1998). In fact the data can be approximated by . All these properties agree nicely with our expectations. On the negative side, the spectral data indicate the lower ejecta speed and the presence of a very slow component with km/s. This supernova is classified as type-IIn and seems to be explained very well in the model involving a collision between the SN ejecta and a massive circumstellar shell ejected few years earlier in giant stellar eruption (Chugai et al., 2004; Woosley et al., 2007; Smith et al., 2007; Smith et al., 2008).
The power of escaped high energy emission can be estimated as . As usual in such cases, it will peak when , which may occur within the first year after the explosion. For a source at the distance Mpc, the corresponding total flux will be
where is the time when the peak is reached. For the synchrotron component the spectral energy distribution is expected to peak around keV (see Eq.70).
What is left behind after the explosion depends on where inside the RSG it occurs. As we have demonstrated, this strongly depends on the mean specific angular momentum of gas captured via the Bondi-Hoyle mechanism. If it is as high as proposed in the early investigations (Illarionov & Sunyaev, 1975; Shapiro & Lightman, 1976) then the NS will accrete only at the Eddington rate until it settles into the core of its companion. What would happens after this is not clear. The current view is that the neutrino cooling will be sufficiently high for the accretion to proceed at the Bondi rate and that the NS collapses into a black hole (Zel’dovich et al., 1972). However, this conclusion is based on spherically symmetric models. The high rotation rate developed in the core of the primary during the in-spiral means that instead of directly accreting onto the NS the core will form a torus, whose neutrino cooling may not be very effective, in which case such a configuration can be relatively long-lived. Moreover, the mass accretion will be accompanied by efficient recycling of the NS, so one can still expect the NS to turn into a millisecond magnetar before its mass becomes high enough for gravitational collapse. Even if the magnetar-driven explosion fails in the dense environment of the core, and the NS eventually collapses into a black hole, this will be a rapidly rotating black hole accompanied by a massive accretion disk – the configuration characteristic of the collapsar model for GRB. However, in contrast to the GRB progenitors the envelope of RSG star is not sufficiently compact for the relativistic jet to penetrate it during the typical life-time of the GRB central engine. Instead, the jet energy will be deposited in the RSG envelope and most likely result in a supernova explosion, leaving behind a black hole remnant. Interestingly, the mass of the presupernova could be below the limit for black hole formation in the normal course of its evolution as a solitary star.
If the angular momentum is much lower, as suggested by Davies & Pringle (1980) then the explosion occurs when the NS is still inside the RSG envelope then the remnant is likely to be a close binary consisting of a WR star and a magnetar777The 2D numerical simulations by Fryer et al. (1996) have indicated the possibility of a neutrino-driven explosion during the CE phase, also leaving behind a close binary system.. In order to show this, we first note that the pressure inside the core of RSG at the He-burning phase is and much higher at the presupernova phase (Woosley et al., 2002). This has to be compared with the pressure inside the high entropy bubble, created by the magnetar
where is the bubble radius, is its expansion speed, and is the time since the explosion. We are interested in its value at the time when the bubble radius is comparable with the separation between the NS and the RSG core. Taking cm and we find that
If the explosion occurs while the NS is still well inside the RSG envelope then and . Thus, the magnetar wind cannot destroy the core. Another point is whether the gravitational coupling between the NS and the core is strong enough for the binary to survive the high mass loss during the explosion. Such a destruction is unlikely because for a sufficiently small separation at the time of the explosion most of the mass loss will come from the envelope, which has no effect on the gravitational attraction between the NS and the core.
The magnetar wind will interact with the WR wind and will terminate when the radius of magnetar magnetosphere drops below the light cylinder radius. The magnetospheric radius is determined by the pressure balance