Abstract
If a quark-nova occurs inside a collapsar, the interaction between the quark-nova ejecta (relativistic iron-rich chunks) and the collapsar envelope leads to features indicative of those observed in Gamma Ray Bursts. The quark-nova ejecta collides with the stellar envelope creating an outward moving cap ( 1–10) above the polar funnel. Prompt gamma-ray burst emission from internal shocks in relativistic jets (following accretion onto the quark star) becomes visible after the cap becomes optically thin. Model features include (i) precursor activity (optical, X-ray, -ray), (ii) prompt -ray emission, and (iii) afterglow emission. We discuss SN-less long duration GRBs, short hard GRBs (including association and nonassociation with star forming regions), dark GRBs, the energetic X-ray flares detected in Swift GRBs, and the near-simultaneous optical and -ray prompt emission observed in GRBs in the context of our model.
1. Introduction
Recent observations following the launch of the Swift satellite challenge the traditional models of GRBs (e.g., [1, 2]). In particular the traditional afterglow modeling, which has been successful in many ways, appears to have serious limitations (e.g., [3] for a recent review). Here we show how appealing to a quark-nova occurring inside a collapsar can lead to phenomenology reminiscent of that seen by Swift. We start with a brief review of the Quark-Nova explosion.
A quark-nova (QN), [4–6] is the explosion driven by phase transition of the core of a neutron star (NS) to the quark matter phase (i.e., neutron star core collapse) leading to the formation of a quark star (QS). The gravitational potential energy released (plus latent heat of phase transition) during this event is converted partly into internal energy and partly into outward propagating shock waves which impart kinetic energy to the material that forms the ejecta (i.e., the outermost layers of the neutron star crust). The ejection of the outer layers of the NS is driven by the thermal fireball generated as the star cools from its birth temperature down to 7.7 MeV [7, 8]. The fireball expands approximately adiabatically while pushing the overlaying crust and cooling fairly rapidly. The energy needed to eject the crust is less than 1% of fireball energy.
The initial composition of the ejecta is representative of matter in the outer layers of the neutron star crust (with density below 1011 g ), dominated by iron-group elements and neutron-rich large Z nuclei beyond iron [9]. As the ejecta expands, r-process takes effect leading to the formation of even heavier elements. As shown in Jaikumar et al. [10], the QN is effective at turning at most 10% of the ejecta into elements above [10]. In previous work we explored the dynamical and thermal evolution of this ejecta [11, 12]. As the ejecta moves outwards, it expands and cools undergoing a liquid to solid transformation. (For ejecta birth temperature of the order of 10 MeV, the relativistic electrons are only mildly degenerate (see appendix in [11]). Thus additional heat deposition into the ejecta during the expansion (e.g., due to nuclear decays of r-processed elements) could lead to nondegeneracy leaving the ejecta in gaseous form. Here, we assume that the degeneracy is not lifted during the early ejecta expansion.) The relativistic expansion causes rapid breakup into small chunks because of the inability of causal communication laterally in the shell. Whether liquid or solid iron, interionic forces (mediated by the electrons) provide the tension leading to breakup (which does not occur for a gas). The size of the clumps depends on whether the breakup occurs in the liquid or solid phase. In the solid/liquid phase the ejecta breaks up into 107/ chunks with chunk mass of 1019/ gm. Table 1 in Ouyed and Leahy [11] lists the properties of the clumps/chunks.
In this paper, we explore consequences of a quark-nova occurring during the supernova explosion in a rotating massive star. Before the QN has occurred, one has the progenitor collapse—much like in the collapsar picture except that in our case a QS is formed instead of a black hole (BH) (see Figure 1). We note that for low angular momentum progenitors, the combination of a high NS core density at birth and, most likely, fall-back material would drive the protoneutron star to a black hole. High angular momentum progenitors (collapsars) will delay the formation of a black hole for three main reasons: (a) the progenitor's core tends to shed more mass and angular momentum as it shrinks reducing central core mass and fall-back; (b) high spin keeps the core density of the resulting neutron star from crossing the black hole formation limit; (c) high angular momentum in the material around the core reduces the accretion rate onto the central object. The subsequent accretion onto the quark star explains the prompt emission in our model [8]. The conversion from NS to QS depends on the NS central density at birth. As shown by Staff et al. [13], spin-down leads to increase of core density and subsequent conversion. Thus progenitor's angular momentum does not mean that the conversion to QS is unlikely; it only affects the delay between the SN and QN. In summary, collapsars seem to provide favorable conditions for the QN to occur inside them. Furthermore, the high angular momentum in the envelope leads to funnel formation which allows the QS jet to escape the envelope and the GRB to be visible.
(a)
(b)
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The paper is structured as follows. In Section 2 we investigate the ejecta interaction with the stellar envelope for the two cases of thin and thick envelope. In Section 3 we apply our model to GRBs and explain how the interaction of the chunks with the stellar envelope can lead to precursor, prompt, and afterglow emissions reminiscent of those observed in GRBs. A discussion is given in Section 4 before concluding in Section 5.
2. Ejecta's Interaction with Stellar Envelope
Wolf-Rayet stars can have extended envelopes, the profile of which depends on evolution and metallicity. The evolutionary effects generally result in WN (nitrogen burning) stars evolving into WC (carbon burning) stars with much smaller masses and radii due to mass-loss (e.g., [14, 15]). What is of interest here is the structure of the envelope at the time of stellar collapse, which is not yet fully understood. For simplicity, we take the stellar structure of a Helium Wolf-Rayet star [16] to be representative of the progenitor and consider the low- and high-metallicity cases (see their Figure 2). The main difference is that the high-metallicity star has an extended envelope with density (10−10–10−9) g and a density inversion near the surface (3for a star). In the low-metallicity case, the star envelope cuts off sharply at 1.5.
When the broken pieces of ejecta impact this stellar envelope, they undergo a shock and become heated to a temperature where is the Lorentz factor of the ejecta and the proton mass. Equation above shows that the chunk temperature is insensitive to the presence of heavier elements since does not vary much. Hereafter, and for simplicity, we assume an iron-dominated ejecta (i.e., and ). Here, the shock efficiency was roughly estimated to be , where is the envelope density at the shock radius. Noting that nondegenerate iron will vaporize if heated to 0.3 eV (see CRC tables 2005 [17] for vaporization temperature of iron at normal density), we then define a critical envelope density g above which the chunks will be vaporized and lose considerable momentum.
One might argue that normal core collapse supernovae accompanied by quark-novae should be more energetic than the canonical 1051 ergs observed. However, this depends on the delay between the SN and the QN. If the delay between the QN and SN is long enough, the chunks will not re-energize the SN. The mean density in the envelope ( in the simplest of cases) depends on (i) the progenitor's precollapse profile (which depends on evolution and metallicity) and (ii) on the delay between the QN and SN. The longer the delay, the smaller the envelope density when the chunks hit it. For a typical envelope we find that is reached when the envelope is at cm. In other words, for cases where the delay between the SN and QN exceeds a few days (for SN ejecta velocity 1000 km ), the QN ejecta will encounter a thin envelope yielding weak interaction. For more massive envelopes, delays of the order of weeks are required for the density to drop below critical; shorter delays lead to complete dissipation of the chunks energizing the preceding supernova remnant. As shown in Leahy and Ouyed [18], this can account for superluminous supernovae such as SN 2006gy [18]. The density of the stellar envelope along the rotation axis is also affected by rotation. For a rotating progenitor the collapse proceeds faster along the polar axis leaving a low density path called the funnel ([19] and references therein), with opening half angle . If the QN ejecta propagated into this funnel, then it would encounter negligible resistance (i.e., thin envelope case), while in other directions the QN ejecta would interact with the higher density SN ejecta (see Figure 1). In these equatorial regions most of the QN energy is lost to energizing the SN ejecta, and only a fraction ( is directed into the funnel. The bulk of the ejecta energy not entering the funnel, 1052 , goes into re-energizing the SN ejecta and can result in a hypernova (see discussion in Section 4.4).
2.1. Thick Envelope
If the envelope density is higher than , the chunks will be vaporized upon impact leading to runaway dissipation and total merging of the ejecta with the envelope. A significant fraction, , of the kinetic energy of the QN ejecta goes into heating the envelope. In this case, the thermal energy of the combined ejecta and envelope (hereafter referred to as the cap) is where the resulting thermal energy per nucleus is with the mean mass per nucleus. The maximum nucleus temperature, , occurs when the envelope and ejecta masses are equal; is the ejecta's initial Lorentz factor in units of 10. Note that whenever exceeds 1 MeV due to nuclear dissociation. Thermalization with pair creation places an upper limit on the electron temperature of 1 MeV. Subsequent energy transfer from nuclei to electrons contributes to further pair creation as the nuclei cool to 1 MeV. If MeV, then most of the ejecta's kinetic energy ends up as pairs. The end result would then be a cap rich in pairs.
Momentum conservation arguments in this case show that the cap slows quickly, reaching a final speed () of where . We note that if the envelope mass is less than , then the mixed ejecta is moving radially outwards at relativistic speeds ( or ; see Figure 2). Thus separates two regimes within the thick envelope case which is of relevance when applying our model to GRBs.
2.2. Thin Envelope
If the stellar envelope density following the collapse is below the critical density, , the chunks will not be vaporized nor do they expand significantly; rather they pass through the envelope effectively puncturing it. During this interaction, the temperature of a piece of broken ejecta, , is determined by shock heating (1) and will not exceed the eV range; thus any emission would be in the optical band (see Section 3.1.1). As discussed above, high-metallicity stars can have extended thin envelopes. However, inhomogeneities and asphericity in the thick envelope case could lead to low-density regions where the chunks can survive destruction.
3. Application to GRBs
The advent of the Swift mission has enabled a much more intensive sampling of GRB light curves, particularly during its early phases but also extending out to late times. These data allow for a more stringent comparison with the standard blast wave model. In addition to the suggested extended engine activity, the observed X-ray flares (e.g., [20]) appear to be a distinct emission component, which suggests a sporadic late time activity of the central source. Another interesting finding by Swift is that the early optical emission, which has been attributed in some cases before Swift to the reverse shock, is typically dimmer than expected. The chromatic break in the afterglow lightcurves is puzzling as it suggests that the X-ray and optical emission may arise in separate physical components, which would then naturally account for their seemingly decoupled lightcurves. There are other features that seem difficult to explain within the framework of the standard engine and afterglow (we refer the interested reader to [3] for more on this).
Here we show how appealing to a quark-nova following the SN can help alleviate at least some of the issues mentioned above. The interaction of the QN ejecta with the stellar envelope yields precursors and postcursors in the optical and X-ray range as shown next.
3.1. Precursors
3.1.1. Thin Envelope (Optical Precursor)
In our model, the mechanism for production of an optical flash is the heating of the chunks of the QN ejecta in the thin envelope case. In this case, emission is directly related to shock efficiency with emitted energy: where and are the mean weight per electron. We expect the precursor optical emission to be thermal-like despite such a relativistic beaming; equation above takes into account beaming correction ().
The precursor time is governed by 3 timescales:
(a)time to traverse the envelope, if the envelope is optically thin to the radiation emitted at temperature ; (b)geometrical time delay (); (c)cooling time of the chunk: with ; the chunk's area is .The number of particles in the chunk is with (if we take one ion and one free electron in the metal per Fe nucleus). Then and depends strongly on the temperature that the chunks are heated to. If optical (eV) then (c) is longer than (a), so the longer of (b) and (c) would give the observed precursor duration. If the chunks are heated to keV temperatures, (c) is so short that unless the chunk is continuously heated by interaction with the envelope, the longer of (a) or (b) would give observed precursor duration. The observed cooling time is shorter by a factor of ) due to relativistic motion of the chunk toward the observer. We note that for liquid clumps and the area are both larger somewhat lengthening . But is still dominated by the value of leading to the same conclusions.
Near-simultaneous optical and -ray emissions have been observed in a few cases (e.g., [21]). This has led to open debates on the association or nonassociation between the two emissions (e.g., [22]). This is further discussed in Section 4.5. Let us simply mention that, in our model, any observation of optical precursor means that the envelope density must be close to with chunks heated to 0.3 eV (i.e., eV). In those sources, the observed optical precursor could yield crucial information about the delay between the SN and QN.
3.1.2. Thin Envelope with Density Inversion (Optical and X-Ray Precursors)
After the chunks have freely propagated outside of the main envelope, they can interact with a higher density shell further out at a radius of (i.e., the density inversion in the envelope at a few times cm; e.g., Figure 2 in Petrovic et al. [16]). In order for the chunks to dissipate their energy, the density of the outer shell must exceed g (from Section 2). Once the chunks collide with matter possessing this critical density, they spread, resulting in their density decreasing, initiating a runaway dissipation process, ionization and heating. The emission from the shocked material is optically thin, so the observer sees radiation at the shock temperature as given in (1). For example, if the density at the inversion is 100, then the X-ray emission will peak at 3 keV.
Since the mass of the envelope at the inversion radius is much less than the ejecta mass, the shock propagates at . Since , an observer would see the emission from all of the chunks, and so the overall precursor pulse would be due to the sequential viewing of different individual pulses from each chunk along the curved surface. The precursor duration is then due to a geometrical delay:
3.1.3. Thick Envelope Case (-Ray Precursor)
If significant portions of the stellar envelope are above the critical density, then one would expect the kinetic energy of the chunks to be deposited in a thin dissipation zone at the base of the envelope. This will effectively spread the ejecta, forming a piston with a strong shock ahead of it. This piston should remain relativistic until it has swept up approximately of envelope mass, at which point it slows, reaching a final velocity given by (4). Although the shock heats up and dissociate the nuclei, however, as noted above, the actual temperature will be limited to 1 MeV due to thermal pair creation. (The details of the shock heating of the envelope and its subsequent cooling are complex. Using blackbody cooling as an upper limit leads to an extremely rapid cooling time (mainly due to the large emitting area) of , where the envelope temperature is in units of 1 MeV. We note that the actual cooling time is defined by the shock propagation time through the optically thin outer parts of the envelope (105 cm), which yields timescales 10−4 s.)
The precursor consists of a short burst of radiation when the shock reaches the outer edge of the envelope. The precursor would have a typical temperature of a pair plasma, keV with a duration also defined by geometrical delays: The precursor brightness in the thick envelope case depends on the final speed of the combined ejecta: (i) if it is relativistic (), the usual blueshift and beaming applies yielding higher brightness (); (ii) if it is not relativistic, we expect the precursor to be dimmer and harder to detect.
We approximate the precursor fluence by assuming that all of the thermal energy is radiated. In the nonrelativistic case and for , while the resulting fluence in the relativistic case (i.e., ) is written as and is shown in Figure 3. It peaks at , or
Note that when , the thermal energy per nucleus is in the keV range leading to an X-ray precursor instead of a -ray precursor.
3.2. Prompt GRB Emission
As shown in Figure 4, the phase following the precursor phase consists of the quark star accreting the disk material. As shown in Ouyed et al. [8] whenever the quark star is heated above 7.7 MeV, it will release a burst of photons with energy 3 which can momentarily impede accretion, until the burst has faded at which point another accretion episode ensues leading to another burst. In its simplest form, this episodic process [8] can be responsible for creating intermittent fireballs (loaded shells with Lorentz factor in the hundreds) eventually leading to internal shocks as described by Kobayashi et al. [23]. Compared to any other jet launching mechanism (e.g., from a black hole), the QS is able to emit far more energy for a given amount of accreted material [7, 8]. Part of the effectiveness of our model can be attributed to the high efficiency in which the QS converts accreted matter to radiation.
The column density of the cap is with a corresponding optical depth where is the Thompson cross-section. This implies that the cap is initially Compton optically thick to the photons from the internal shocks occurring underneath. Thus the prompt GRB phase can only be observed as optically thin once the cap is somehow destroyed or pushed to a higher radius by the QS shells.
3.2.1. Cap Acceleration and Removal
In the thin envelope case, the first few shells from the QS accretion phase could easily remove the opaque envelope material, making subsequent bursts detectable by the observer. Alternatively, in the case of a thick envelope, the cap will be bombarded by many QS shells before it starts accelerating. In general the number of collisions with the QS shells needed to dissipate or remove the cap to distances large enough to become transparent to radiation is , that is, about collisions using our fiducial values. Equation (11) above indicates that at a radius 3 which occurs at time 1000.
3.2.2. Cap Temperature and Spectrum
An approximate equilibrium temperature for the cap can be found from the relation, , which yields where and are the QS temperature and radius in units of 10 MeV and 10 km, respectively; is the radiative efficiency of the internal shocks which we take to be 10 [23]. This equilibrium temperature is actually the peak temperature because the QS heating is episodic (see [8]), and the temperature is only lower in-between episodes. This quasicontinuous supply of photons by the QS will keep the fireball spectra close to thermal during its evolution. The spectra should thus consist of a blackbody in the early phase which would eventually evolve into an optical thin emission. Interestingly, it has been suggested in literature that a hybrid model with a thermal and nonthermal components can explain all types of spectral evolution and shapes of the observed prompt GRB emissions (e.g., [24] and references therein). This is further discussed in Section 4.7.
3.3. Afterglow Emission
The shells from the QS jet (following accretion onto the QS) colliding with the cap produce events similar to internal shocks between shells themselves. The cap provides a buffer for the intermittent shells to be absorbed and subsequently form a heavy, slowly moving “giant” shock (reminiscent of an external shock) that might be of relevance to the afterglow activity. This buffer, of minimum mass , will lead to different types of afterglows depending on whether it is relativistic or not.
The slowly moving wall resulting from the merging of the cap and the multiple QS shells should absorb and emit radiation as it interacts with the surrounding or when it is bombarded by the energetic photons from late internal shocks (e.g., [13]). Iron emission lines have been detected in the X-ray afterglow of few GRBs [25–29]. This might be indicative of heavy elements in the cap which survive nuclear disintegration due to shocks.
4. Discussion
4.1. SN-Less GRBs in Our Model
If the supernova fails to explode, then the consequences are twofold in our model (see Figure 5). First, the QN ejecta will be subject to larger densities due to an infalling stellar envelope, which leads to higher shock efficiency and a harder spectrum than if the SN had occurred.
Second, the infalling material will in most cases force the QS to turn into a black hole. Whereas in the SN case the outcome could be a QS or a black hole depending on the disk and QS's initial mass, in the SN-less case the GRB phase is likely due to jet activity from accretion onto a black hole. In our QS model, as opposed to models with just a black hole, the black hole jet will catch up faster with the mixed ejecta/envelope since the QN ejecta will have cleared out the more compact, dense envelope.
4.2. Short Duration GRBs in Our Model
The short duration GRBs we first discuss here are necessarily related to star forming regions. A discussion on the second class (i.e., those not associated with star forming regions) in our model will be presented elsewhere. A short duration GRB in our model corresponds to the case of a low angular momentum progenitor. In this case the infalling progenitor's envelope will not form a disk and will fall entirely onto the star, resulting in a black hole with no surrounding material to accrete (see Figure 5). Simply put, in our model short GRBs are dominated by the precursor phase which will emit in the -ray frequency band due to the high envelope densities.
We expect that the funnel's opening angle in this case will be wider than in cases involving disks (from angular momentum arguments). This implies that (i) some short GRBs with no SN association should be found in star forming regions; (ii) they might be less numerous than long ones if low angular momentum progenitors are spars; (iii) they are less luminous and thus only the nearby one will be detectable; (iv) their spectrum should be harder since the QN ejecta will interact with a more dense SN ejecta; (v) X-ray precursors of SN-less GRBs or the early phase of the prompt GRB emission in SN-less GRBs should resemble emission from short GRBs.
4.3. Dark GRBs in Our Model
Dark GRBs are defined as those that are not associated with an optical afterglow (e.g., [30]) or any afterglow emission regardless of the frequency band (e.g., [31]). In our model the cap as we have said provides a buffer for the episodic shocks (from accretion onto the QS) to be absorbed and subsequently form an external shock that could in principle explain the observed afterglows.
One possible explanation using our model is that Dark GRBs would correspond to the situations where the interactions between the cap and the upcoming QS shells are reduced or nonexistent. This would be the case if the envelope is thin in which case there is no cap or buffer, or if the cap is moving at relativistic speeds in which case the heating from the colliding shocks is diminished.
4.4. Hypernovae as QNe Signature?
Hypernovae are energetic SNe associated with GRBs and are observed in the late afterfglows of long GRBs. In our model, outside the funnel where the density is above the critical density, the chunks will dissipate their energy entirely into heat. As shown in Leahy and Ouyed [18], this results in a superluminous supernova reminiscent of a hypernova. Furthermore, conditions in the expanding QN ejecta are favorable for the r-process to take effect (as discussed in details in [10]). Hence, additional heavy elements will be deposited in the expanding envelope as the QN ejecta reaches and mixes with the envelope.
4.5. Optical Flashes and X-Ray Precursors
Traditionally, it has been suggested that the optical emission could be produced by the reverse shock emission or could arise from the internal shock emission (e.g., [32]). Observationally, there appears to be two cases of prompt optical emission.
(i)For GRB990123 and recently discovered GRB060111b, the optical flashes were uncorrelated with the prompt gamma-ray emission, which suggests that the optical emission and gamma-ray emission should have different origin (e.g., [33]).(ii)For GRB041219a, its optical flash was correlated with the gamma-ray emission [34, 35], and for GRB050904, a very bright optical flare was temporally coincident with an X-ray flare ([36]; see also [37]), which implies that for these two GRBs there should be a common origin for the optical and high-energy emission.More recently, the near-simultaneous optical and -ray emissions in GRB 080319B have reopened this debate. Kumar and Panaitescu [22] argue for a Synchrotron Self-Compton (SSC) origin while recent work [21] argues for physically separated emission regions (e.g., -rays from internal shocks and optical flash from external shock emission).
In our model, the optical emission originates when the chunks are heated during their passage through the thin envelope (see Section 3.1.1) and X-ray emission when the chunks dissipate further out at the density inversion (high-metallicity stars; [16]). The optical and X-ray emissions are produced at different radii with durations dominated by geometry as given in (6) but both composed of short pulses from the chunks. The prompt GRB emission is from internal shocks in the outflow launched during accretion onto the QS (see Section 3.2), physically separated from the optical emission. The delay of the optical from the QN event can be shorter or longer than the corresponding GRB delay depending on the Lorentz factors and emission distances.
4.6. X-Ray Flares
X-ray flares are frequently observed in the early X-ray afterglow of GRBs (e.g., [38, 39]). In Staff, Ouyed, and Bagchi [40] a possible explanation for these X-ray flares with an accreting quark star as the GRB inner engine is given. A quark star can accrete a maximum of about before collapsing to a black hole. Hence the quark star as GRB inner engine can last for about a thousand seconds. If there is still matter left to accrete once the black hole is formed, a new jet is formed from the accretion onto the black hole [41]. Staff et al. [40] proposed that the interaction between the jet from the black hole and the jet from the quark star could make internal shocks and thereby produce X-ray flares. Furthermore, when the black hole jet (or more massive parts of the quark star jet) interacts with the external shock, the shock will be reenergized and a “bump” can result. Internal shocks within the black hole jet itself can also occur, giving rise to more flares; see Figure 5 in Staff et al. [40] for the resulting lightcurves.
The accretion rate onto a black hole is likely very high (), meaning that the black hole phase will be rather short. The flares created by interactions between the black hole jet and the quark star jet therefore have to occur within about a thousand seconds, whereas the activity caused by interaction between the jets and the external shock can occur later. Flares can occur earlier if the quark star collapses to a black hole sooner than after a thousand seconds.
4.7. The Two Components
From (10) we estimate the expanding cap to become optically thin to photons from the internal shocks, at a radius of about at which point the temperature of the thermal component becomes . The emission would thus remain thermal for duration of . The synchrotron radiation from the subsequent internal shocks will then dominate the spectrum in the later stages, , of the prompt GRBs; the QS will continue to accrete until the accretion disk is consumed or the QS turns into a black hole [8, 13]. Whether the two components presumably inherent to some GRBs [24] are an indication of the envelope-shells interaction as described in our model remains to be confirmed. For completeness, however, we should note that accreting quark stars could in principle result, as we have said above, from QNe going off in isolation in which case the standard internal shocks scenario, involving no intervening envelope, applies [8].
5. Conclusion
In this paper, we explore the case of a QN going off inside a collapsar. We find that the interaction between the iron-rich chunks from the QN ejecta and the collapsar envelope leads to features indicative of those observed in Gamma Ray Bursts. These features include (i) precursor activity (optical, X-ray, -ray), (ii) prompt -ray emission, and (iii) afterglow emission. Although the presented model is based on physical arguments, most of these are in reality more complicated and so would require more detailed studies. For example, the launching of the outer layers of the neutron star during the QN is a challenging process to study and involves energy transfer, core-bounce, and generation of a shock wave, including cooling processes and subsequent ejection. We have assumed simple conditions for the ejecta immediately after the QN such as a single Lorentz factor. A range of Lorentz factors would still result in the outermost shell of the ejected material interacting with the progenitor envelope as we have described here. Shells with lower Lorentz factors would interact later in a similar manner and would lead to more complex interaction with the envelope. Another important aspect of our model that requires further studies is the process of clumping, crystallization, and breakup of the ejecta, which would require better knowledge of the ambient conditions surrounding the ejecta. Despite our simplifying assumptions, we feel that our model captures the basic envelope interaction physics and provides interesting features with possible applications to GRBs.
Acknowledgment
This work is supported by an operating grant from the Natural Research Council of Canada (NSERC).