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Mon. Not. R. Astron. Soc. 310, 1039±1052 (1999) The structure of steady detonation waves in Type Ia supernovae: pathological detonations in C±O cores Gary J. Sharpew Department of Applied Mathematics, The University of Leeds, Leeds LS2 9JT Accepted 1999 July 23. Received 1999 July 5; in original form 1999 April 15 A B S T R AC T The structure of steady, one-dimensional detonation waves in C±O is investigated for initial densities in the range 2 107 to 1 109 g cm23 . At these and greater densities, the selfsupporting detonation wave is of the pathological type. For such waves the detonation speed is an eigenvalue of the steady equations, and the reaction zone contains an internal frozen sonic point where the thermicity vanishes. The self-supporting flow downstream of this singular point is supersonic, and is very different from that in supported (overdriven) detonations. A method for determining the structure of pathological detonation waves is described. These waves are examined, and the self-sustaining wave is compared with and contrasted to the supported detonations considered previously by Khokhlov. We show that the thickness of the self-sustaining detonation is a few times the thickness of supported detonations, and that the self-sustaining detonation produces more of the iron-peak and less of the intermediate mass elements than do supported detonations. Implications for the cellular detonation instability are also discussed. Key words: instabilities ± nuclear reactions, nucleosynthesis, abundances ± shock waves ± supernovae: general ± white dwarfs. 1 INTRODUCTION It is generally accepted that Type Ia supernovae are the result of the thermonuclear explosion of C±O white dwarf stars. Whether the explosion occurs as a deflagration (subsonic burning), a detonation (supersonic burning), or a combination of both (delayed detonation) is still a matter of some debate. In the simplest models the white dwarf accretes matter from a companion, which triggers an explosion at the centre when the star is close to the Chandrasekhar mass. However, pure detonation models fail to produce the observed abundances of intermediate mass elements because of the high densities involved (Arnett, Truran & Woosley 1971), while for pure deflagration models the flame speed has to be very nicely tuned in order to reproduce the observed characteristics of the supernova (Nomoto, Sugimoto & Neo 1976; Nomoto, Thielemann & Yokoi 1984; Thielemann, Nomoto & Yokoi 1986). All is not lost for detonations, however, since at lower densities (& 107 g cm23 according to Khokhlov 1989) the thickness of the detonation wave becomes comparable to the size of the star, which can lead to the production of intermediate mass elements as a result of incomplete burning. This is exploited in delayed detonation models, which seem to give the best overall agreement with observations (Khokhlov 1991; Arnett w E-mail: [email protected] q 1999 RAS & Livne 1994; Bychkov & Liberman 1995; Niemeyer & Woosley 1997; Khokhlov, Oran & Wheeler 1997). In these models the explosion is initially a deflagration which burns part of the star and causes it to expand to lower densities, whereupon the rest of the burning proceeds as a detonation. An alternative scenario, in which detonations produce intermediate mass elements because of low densities, concerns helium- or double-detonation models (Livne 1990; Livne & Glasner 1990, 1991; Woosley & Weaver 1994; Livne & Arnett 1995; Wiggins & Falle 1997a; Wiggins, Sharpe & Falle 1998). In these models a detonation in a helium envelope induces a detonation in a low-density, sub-Chandrasekhar C±O core, either at the core±envelope boundary or at the centre of the core. It therefore seems very likely that detonations play a role in Type Ia supernovae. Khokhlov (1989) investigated the steady one-dimensional structures of nuclear detonations in C±O mixtures and in pure He. In C±O the detonation wave consists of three burning layers: a carbon-burning stage, followed by an oxygen-burning or nuclear statistical quasi-equilibrium (NSQE) relaxation layer, and finally a silicon burning, or nuclear statistical equilibrium (NSE) relaxation, stage. Khokhlov (1989) found that the length-scale for silicon burning is very much greater than that for oxygen burning, which is in turn very much greater than that for carbon burning. He also found that, for initial densities greater than about 107 g cm23, the self-sustaining detonation is of the pathological type as a result of 1040 G. J. Sharpe endothermic photo-disintegration reactions. Such detonations travel at the minimum possible speed, which is greater than the Chapman±Jouguet (CJ) speed, and have an internal frozen sonic point (the so-called pathological point) where the thermicity vanishes. Downstream of the sonic point the flow is supersonic. Unlike CJ detonations, for which the detonation speed does not depend on the form of the reaction rates, for pathological detonations the detonation speed is an eigenvalue of the steady equations. This results in the structure of the self-sustaining detonation (the object of interest in the case of Type Ia supernovae) downstream of the sonic point being quite different from that of supported, or overdriven, detonations, which are subsonic throughout. However, because of the singular nature of the sonic pathological point, Khokhlov (1989) could not find the self-sustaining structure by direct integration and considered only supported detonations. No conclusions about the downstream selfsustaining structure can be drawn from such calculations, however. For example, the thickness of the self-sustaining detonation may be much greater because of lower temperatures. Although the governing equations for detonations admit such steady, one-dimensional waves, terrestrial experiments show that detonations usually have a time-dependent, multidimensional structure. In fact the steady detonation waves are prone to both pulsating, or one-dimensional, and cellular, or multidimensional, instabilities. Pulsating detonations are rare in nature, but they are seen for instance when blunt bodies are fired into detonatable gases at supersonic velocities (e.g. Lehr 1972). Usually detonations propagate in the cellular mode (Fickett & Davis 1979). In these cases the leading shock is wrinkled, consisting of alternate weak incident shocks and stronger Mach stems, joined at triple points by transverse waves which travel back and forth perpendicular to the front and extend back into the reaction zone. The tracks of these triple points map out the boundaries of the `cells', which can be remarkably regular. Reactions are essentially complete within one cell length, which is one to two orders of magnitude greater than the corresponding steady, onedimensional reaction zone length (Lee 1984). Linear stability analyses of steady detonations show that they are unstable to onedimensional disturbances provided that the reaction rates are sufficiently temperature sensitive, whereas, unless the heat of reaction is unreasonably small, they are always unstable to multidimensional disturbances, even when the reaction rates are independent of temperature (Erpenbeck 1965; Sharpe 1997; Short & Stewart 1998, 1999). Kriminski, Bychkov & Liberman (1998) investigated the onedimensional linear stability (i.e. the pulsational instability) of detonations in carbon white dwarfs. However, they considered only the C+C reaction. They found that the linear stability spectrum contains a purely real (non-oscillatory) mode for initial densities greater than about 2 107 g cm23 , and hence suggest that detonations cannot propagate for such densities. They also suggest that this may be a reason why burning starts as a deflagration at high densities, with a deflagration-to-detonation transition occurring at densities of about 2 107 g cm23 . They base this conclusion on the one-dimensional numerical simulations of He & Lee (1995), in which the detonation dies to a shock followed by a deflagration when the linear spectrum has nonoscillatory modes. However, the results of He & Lee (1995) have been shown to be incorrect as a result of poor numerical resolution (Williams, Bauwens & Oran 1996, Sharpe & Falle 1999). In fact the detonation always re-ignites after an induction time and propagates as a series of failures followed by re-ignitions. Also, Short & Quirk (1997) have shown that the real eigenvalue of linear stability analyses does not correspond to the detonation failure/reignition regime in the non-linear simulations. Most importantly, in multidimensions the detonation propagates in the cellular mode, and failure does not occur (Williams et al. 1996). Numerical simulations were used to investigate the onedimensional instability by Khokhlov (1993) and Koldoba, Tarasova & Chechetkin (1994). Both considered only the instability arising from the carbon burning layer for supported detonations. Khokhlov (1993) found the detonation was unstable to one-dimensional perturbations for initial densities .2 107 g cm23 . Boisseau et al. (1996) carried out two-dimensional numerical simulations of detonations in C±O. They resolved only the carbon-burning stage, which was overdriven because of the assumed energy release of the subsequent burning stages. Not surprisingly, they found that nuclear detonations in C±O are unstable to the cellular instability. Their simulations showed that pockets of unburnt carbon were produced, and the effective burning length was increased by about three times the steady reaction length. However, they considered detonations propagating down narrow tubes. When the widths of such tubes are much smaller than the natural cell size, the cell width is constrained to be the tube width. Hence one cannot make any conclusions about the average natural cell size of unconfined detonations from such calculations (the natural cell widths of terrestrial experiments are of the order of one to two magnitudes greater than the corresponding induction zone length of the steady one-dimensional detonation: see Lee 1984). In order to determine the natural cell sizes, simulations with tube widths much larger than the natural cell size are required. It is very likely that the oxygen- and silicon-burning stages also have their associated instabilities. However, it is not clear that one can separate each burning stage into supported detonations and investigate the instabilities separately. The instability of one burning stage may affect the others. For instance, experiments show that the transverse waves can themselves be detonations and have their own transverse waves moving back and forth across them (see Fickett & Davis 1979 and references therein). This suggests that when there are reactions with disparate length-scales there may be a `cells within cells' structure, such as that seen in detonations in nitromethane mixtures (Presles et al. 1996). Furthermore, Sharpe (1999) carried out a linear stability analysis of pathological detonations, which showed that, unlike the case in which the unsupported detonation is of the CJ type, predictions of the cell size are very sensitive to the detonation speed near the self-sustaining detonation speed. Hence one should be very careful about drawing any conclusions regarding cell sizes, etc., of selfsustaining, pathological detonations by studying even only slightly overdriven detonations. Unfortunately, however, because of the large differences in reaction lengths of the burning stages in C±O, it is currently impossible to resolve more than one stage at a time in numerical simulations. Although such structures are almost always hydrodynamically unstable, the correct underlying steady, one-dimensional structure is an essential starting point for any stability analysis or timedependent numerical simulation. Since the structure of the steady, self-sustaining detonation is quite different from that of steady supported detonations, one would also expect the time-dependent, multidimensional structure of self-sustaining detonations to be quite different from their supported counterparts, especially in light of the linear stability analysis of Sharpe (1999). Also, the structure of the one-dimensional pathological detonation can be q 1999 RAS, MNRAS 310, 1039±1052 Detonations in Type Ia supernovae exploited to estimate the nucleosynthesis caused by large-scale multidimensional effects. The burning up to the sonic pathological point occurs on a very short length-scale (compared to the size of the white dwarf and to the total detonation length) and Khokhlov (1989) found that very little heat is absorbed in the subsequent burning, which occurs on a relatively large length-scale. Hence the burning up to the sonic point can be modelled as an instantaneous `detonation' and, to a first approximation, the subsequent burning can be decoupled from the flow. For instance, Wiggins et al. (1998) used this to estimate the effects of the cusp in the detonation front found by Wiggins & Falle (1997a), and the associated rarefactions at the core±envelope boundary, which occur when the detonation is ignited at a point in the helium envelope. In this paper we investigate the steady, one-dimensional detonation waves in C±O when the self-sustaining detonation is of the pathological type. In Section 2 we give the governing equations for such waves. In Section 3 we determine the pathological detonation speeds, and then in Section 4 we describe a method for determining the solutions of the equations near pathological points. This method is then used to calculate the complete steady, one-dimensional detonation waves, which are examined in Section 5. Conclusions and ideas for future work are given in Section 6. 1041 find our results for supported detonations are in very good agreement with those of Khokhlov (1989), who used an extended network. Note that not all the Yi are independent, since 13 X X i 1; 5 i1 where X i Ai Y i is the mass fraction and Ai is the atomic mass of the ith species. Equation (5) can be used to eliminate one of the Yi in terms of the other nuclei abundances, and hence equation (4) reduces to 12 (for the a -network) independent equations. It appears that neither Khokhlov (1989) nor Wiggins et al. (1998) took this into account, so that their partial derivatives with respect to one of the Yi refers to changing that Yi while keeping all of the other nuclei abundances constant, which from (5) we cannot do. In this paper we choose to eliminate the 56Ni nuclei abundance, so that from now on Y refers to (Y1,¼,Y12). Equation (1)±(4) are closed by specifying an equation of state. We use an equation of state which includes contributions from radiation, non-degenerate ions and arbitrarily degenerate and relativistic electrons and electron-positron pairs. We then have p ; p r; T; Y 2 G O V E R N I N G E Q UAT I O N S and A detonation is a supersonic regime of burning in which a strong shock heats the fuel to high temperatures, causing reactions to proceed at a rapid rate behind the shock. The heat released thereby helps to support the detonation. The governing equations for gaseous detonations are the reactive Euler equations. For steady, one-dimensional flow these are dr du r 0; dx dx 1 du dp ru 0; dx dx 2 dE p dr 2 0 dx r2 dx 3 u and dY R ; dx u 4 where x is the distance behind the shock, u the fluid velocity in the shock rest frame, r the density, p the pressure, E the internal energy per unit mass, Yi ni rN A is the nuclei abundance of the ith species, with ni the number density of the ith species, NA is Avogadro's number, and Ri ; Ri r; T; Y the corresponding reaction rates. We define here the mean molecular weight of ions, m , by X m 1= Y i : i We use an a -network consisting of the 13 species 4He, 12C, O, 20Ne, 24Mg, 28Si, 32S, 36Ar, 40Ca, 44Ti, 48Cr, 52Fe and 56Ni, linked by 27 reactions. The reaction rates are taken from Fryxell, MuÈller & Arnett (1989). Although this is a simple network, we 16 q 1999 RAS, MNRAS 310, 1039±1052 E ; e r; T; Y 2 q Y; where e is the internal energy of the ions, photons, electrons and positrons and q NA 13 X Qi Y i i1 the total binding energy of the mixture, Qi the binding energy per nucleus . 0 of the ith species and Y13 (the 56Ni nuclei abundance) is given by (5). Equations (1)±(3) can be integrated to give the usual conservation relations for steady, one-dimensional flow ru r0 D; 6 p r; T; Y ru2 p r0 ; T 0 ; Y 0 r0 D2 7 and E r; T; Y p r; T; Y u2 r 2 E r0 ; T 0 ; Y 0 p r0 ; T 0 ; Y 0 D2 ; r0 2 8 where D is the detonation speed and the 0 subscript denotes quantities in the upstream (initial) state. For a given initial state and detonation speed, eliminating u using (6), equations (7) and (8) give two relations between the 14 quantities r , T and Y. For instance, to find the post-shock density and temperature, we solve equations (7) and (8) iteratively using a Newton±Raphson method, given that the composition across the shock is unchanged (Khokhlov 1989). In this paper we will consider the initial material to be 0:512 C 0:516 O by mass and the initial temperature to be 1 108 K. Note that in the upstream state the electrons are highly degenerate so 1042 G. J. Sharpe that the equation of state is not sensitive to the temperature. From equations (7) and (8), this means that the post-shock temperature and hence the detonation wave structure are themselves insensitive to the initial temperature, and so the only parameter left to vary is the initial density. Alternatively, using the differential relations 12 X p p p dp dr dT dY i 9 r T; Y T r; Y Y i r; T; Y i i1 and de e r T; Y dr e T r;Y dT 12 X e i1 Y i r; T; Y i dY i ; 10 where the subscripts refer to the quantities being kept constant and Y i Y 1 ; ¼; Y i21 ; Y i1 ; ¼; Y 12 ; equations (1)±(3) give dr ra2 s ´ R 2 f dx u h and dT dx p T 21 (" r;Y 11 p u 2 r 2 # T;Y ) 12 dr X p dY i 2 : dx Y i r;T;Y i dx i1 12 Here h a2f 2 u2 is the sonic parameter, where " # 21 p p e p e 2 af 2 r T; Y r2 r T; Y T r; Y T r; Y is the frozen sound speed (i.e. the sound speed evaluated at constant composition), and s ´ R is the thermicity, which is roughly proportional to the rate at which nuclear binding energy is transformed into heat and kinetic energy (Fickett & Davis 1979). The thermicity coefficients are given by ( 1 p si Y i r; T; Y i ra2f " # 21 ) e q p e : 2 2 Y i r;T;Y i Y i Y i T r;Y T r;Y Given the post-shock state as initial conditions, equations (4), (11) and (12) can then be integrated into the reaction zone x . 0 to determine the steady, one-dimensional solutions, with u given by (6). Note that it is also possible to define an equilibrium sound speed ae, which is the sound speed evaluated with the composition shifting in order to maintain equilibrium (i.e. R 0), with ae # af (Fickett & Davis 1979). The Chapman±Jouguet condition is then that the flow is equilibrium sonic at the end of the reaction zone, i.e. ae u when R 0. However, for pathological detonations this state is unattainable and the reaction zone has an interior frozen sonic point where the thermicity vanishes. For a detailed discussion of the properties of pathological detonations based on the p 2 1=r plane see, for example, Fickett & Davis (1979), Khokhlov (1989) or Sharpe (1999). For the sake of clarity we briefly summarize. Immediately behind the shock the flow is frozen subsonic. If the detonation speed is too low, then a frozen sonic point is reached inside the wave at which s ´ R ± 0, so that from (11) the derivatives of the thermodynamic quantities diverge there (but note that the derivatives of the nuclei abundances do not), and no physically valid solution exists (Wood & Salsburg 1960). The minimum, or self-sustaining, detonation speed is then given by the value of D for which s ´ R 0 at the frozen sonic point. Note that the derivatives of the thermodynamic variables are indeterminate at such a pathological point. The thermicity can become zero with R ± 0 due to endothermic or dissapative effects. In the case of detonations in C±O the thermicity passes through zero as a result of an endothermic stage of the reactions. Fig. 1 shows a schematic diagram of the possible pathological wave structures. The flow downstream of the pathological point can either be frozen supersonic (dashed line in Fig. 1), corresponding to the selfsustaining detonation, or frozen subsonic with a discontinuity in the thermodynamic derivatives at the pathological point for a supported pathological detonation (dotted line in Fig. 1), which is the limit of supported detonations as the detonation speed tends to the self-sustaining speed from above. The structure of the detonation downstream of the pathological point is then quite different for the self-sustaining and supported pathological detonations. Detonations with speeds greater than this minimum, pathological speed, corresponding to piston-supported detonations, are frozen subsonic throughout and the thermodynamic quantities have an interior turning point where the thermicity passes through zero. 3 D E T O N AT I O N S P E E D The Chapman±Jouguet detonation speed can be found by solving equations (6)±(8) together with the conditions u ae r; T; Y and R r; T; Y 0: The pathological detonation speed, however cannot be found by solving a set of algebraic equations. In order to determine the selfsustaining speed we follow Khokhlov (1989). Given a value of the initial density and an initial guess for the corresponding pathological detonation speed we integrate the equations into the reaction zone. If the detonation speed is too low then the solution will terminate at a frozen sonic point, whereas if it is too high the thermicity will vanish, corresponding to a supported detonation wave. We can thus iterate using bisection to obtain the Figure 1. Diagram of pathological detonation wave structures. q 1999 RAS, MNRAS 310, 1039±1052 Detonations in Type Ia supernovae 1043 pathological detonation speed to within any desired degree of accuracy. Fig. 2 shows the detonation speed obtained by this method as a function of the initial density. We find that the selfsustaining detonation becomes Chapman±Jouguet below initial densities of about 2 107 g cm23 . 4 S O L U T I O N S N E A R PAT H O L O G I C A L POINTS Since we are interested in the self-sustaining detonation, we require the solution at, and downstream of, the frozen sonic pathological point. We cannot obtain the downstream selfsustaining structure by direct integration from the shock because of the singular nature of (11) and (12) at such points. It is therefore necessary to analyse the structure of the solutions in the neighbourhood of the pathological points. We note first that equations (6)±(8) can be viewed as giving the thermodynamic variables as functions of the composition, Y, for a given initial state and detonation speed. Where they are defined, these functions are double valued for a given composition, with a frozen subsonic and a frozen supersonic value, and can only change continuously from one solution branch to the other at a frozen sonic point (Wood & Salsburg 1960). For N independent reactions there is then an N 2 2 surface or locus of `pathological points' (note for N 2 this locus reduces to a single point, cf. Sharpe 1999), which can be determined by solving h0 Figure 2. Detonation speed versus the initial density. 13 and s ´ R 0; 14 together with equations (6)±(8). For the pathological detonation speed, the solution which starts at the post-shock state reaches a point on the corresponding pathological locus, which we denote by Yp, with corresponding density and temperature r p and Tp and detonation speed Dp. Note that for N . 2, knowledge of the pathological detonation speed does not allow one to determine the state at the pathological point algebraically. Note also that the pathological point is a frozen sonic point, but from equations (4), unlike the derivatives of the thermodynamic variables, the reaction rates are well defined there. Hence the nuclear composition can be determined in the neighbourhood of such a point by a linear expansion. Since the composition is thus determined and the thermodynamic variables are double-valued functions of the composition, there are two solutions entering or leaving the pathological point, a frozen subsonic one and a frozen supersonic one. Fig. 3 shows the solutions obtained by integrating from the shock for detonation speeds very near the pathological speed when the initial density is 1 109 g cm23 , from which it can be seen that the pathological point has a `saddle-like' nature. Note that any thermodynamic variable plotted against any nuclei abundance has a similar behaviour. Away from the pathological point, the solutions lie very close to one another, i.e. they form a `separatrix'. For D , Dp, the solution eventually leaves the separatrix from below and the gradients of the thermodynamic variables become infinite as a non-pathological frozen sonic point is reached, while for D . Dp the solution leaves the separatrix from above as the thermodynamic variables pass through a turning point where the thermicity is zero (note that downstream of this turning point, these supported detonation solutions again converge on another separatrix). The closer the detonation speed is to the q 1999 RAS, MNRAS 310, 1039±1052 Figure 3. Solutions integrated from the shock for detonation speeds very close to the pathological speed, solid lines D 1 ^ 10215 Dp , dotted lines D 1 ^ 10210 Dp , and dashed lines D 1 ^ 1025 Dp . (a) Density versus distance behind shock, and (b) density versus 4He nuclei abundance. r0 1 109 g cm23 . pathological speed, the later the solution leaves the separatrix. One can see from Fig. 3 that although we can obtain the pathological detonation speed to very high accuracy by simply shooting at the pathological point from the shock, we cannot actually reach the singular pathological point because of its saddle-like nature, although we can get very close to it. One idea for overcoming this problem is to start at a 1044 G. J. Sharpe pathological point, shoot towards the shock and try to satisfy the shock conditions. Unfortunately, equations (4), (11) and (12) are unstable to numerical integration in the upstream direction. Besides which, the pathological point has N 2 2 (i.e. 10 for the a -network) degrees of freedom so it would be very difficult to determine the correct values in this way. We thus need to find a very good guess for the state at the pathological point given the initial conditions and the pathological detonation speed. Fig. 4 shows values of the state obtained at the frozen sonic point (for D , Dp ) or the zero thermicity point (for D . Dp ) against the detonation speed when the initial density is 1 109 g cm23 . Note that the state at such points becomes increasingly sensitive to the detonations speed as D approaches Dp, which is a result of the saddle-like nature of the pathological point. However, from Fig. 4, it can be seen that the sonic value of the density (and also the pressure) is not very sensitive to the detonation speed as D tends to Dp from below (the reason for this is clear if one considers Rayleigh lines and Hugoniot curves in a p 2 1=r plane). Hence we can determine the value of the density at the sonic pathological point to very good accuracy. We can now determine a good guess for the composition at the pathological point, and for the distance between the shock and the pathological point. The procedure is to integrate the solutions from the shock for detonation speeds very close to the pathological speed, so that we approach as near to the pathological point as possible (cf. Fig. 3). We can then interpolate the separatrix to the known pathological density, in the space of the density and each of the nuclei abundances or the distance from the shock, in order to determine the values at the pathological point. We make a brief aside here. From Fig. 4 it can be seen that the state at the turning point in the thermodynamic variables where the thermicity vanishes is extremely sensitive to the detonation speed for supported detonations with detonation speeds near the pathological speed, and becomes increasingly sensitive as D ! Dp from above. In particular we find that the minimum in h a2f 2 u2 , which occurs at the zero thermicity point, is itself extremely sensitive to the detonation speed. We believe it is this sensitivity of the sonic nature of the flow which is, in turn, the cause of the sensitivity in the linear stability spectrum of such detonations on the detonation speed, as found by Sharpe (1999). We therefore expect the stability of overdriven waves in C±O to be extremely sensitive to the detonation speed near the pathological speed, and we reiterate that one must be very careful in drawing any conclusions about the stability of the self-sustaining detonation by using even slightly overdriven detonation. Now, we note that (for an a -network) the pressure and the frozen sound speed depend on the composition only through the mean molecular weight of ions, m . Hence, given the initial conditions, pathological detonation speed and sonic density, we can solve the momentum equation, (7), together with condition (13), for the frozen sonic values of the temperature and the mean molecular weight (which gives one of the nuclei abundances, Yj say, as a function of the other abundances). We can then determine a point on the pathological locus by solving the energy equation, (8) together with condition (14) for two of the nuclei abundances, Yk and Yl say, with the remaining abundances given by the interpolated pathological values, found as described above. One can see from Fig. 4 that the sonic values of the temperature and the composition are quite sensitive to the detonation speed near the pathological speed, and so also on the sonic density. Hence by fine tuning the sonic density (by which we mean changes in the density by the order of only about 103 g cm23) we find we can Figure 4. State at sonic point (dashed line) or zero thermicity point (solid line) against detonation speed: (a) density, (b) temperature, (c) distance behind shock and (d) mean molecular weight. r0 1 109 g cm23 . q 1999 RAS, MNRAS 310, 1039±1052 Detonations in Type Ia supernovae 1045 make the values of Yj, Yk and Yl simultaneously very close to their interpolated values, which lends credence to the interpolation procedure. We can thus find a point on the pathological locus which is very close to the desired pathological point. We use the nuclei abundances with the highest interpolated values for Yj, Yk and Yl, usually 4He, 28Si and 32S, since these are the most important. Figs 5 and 6 show the dependence of the state at the pathological point, as determined by this method, on the initial density. The temperature and the log of the density at the pathological point depend almost linearly on the log of the initial density. The distance between the pathological point and the shock increases quite dramatically as the initial density decreases. Also, as the initial density decreases, the state at the pathological point becomes more completely burnt; the abundances of carbon and oxygen decrease, while the abundances of the products of silicon burning and the iron-peak elements increase. Given the state at the pathological point, it remains to determine the downstream solutions. Since equations (11) and (12) are indeterminate at such points it is necessary to linearize about the pathological values in order to determine the behaviour of the solutions in the neighbourhood of the pathological point. We follow partly the analysis of Wood & Salsburg (1960), who considered linear expansions about such points for a very general system with an arbitrary number of reactions. Near the pathological point we write r rp 1 dr; T T p 1 dT and Y Y p 1 dY; where the p superscript refers to quantities evaluated at the pathological point and dr , d T and d Yi i 1; ¼; 12 are small. We note first that, since the reaction rates are well defined at the pathological point, from equations (4) we have, to first order in d Ym, dY i p Rp Y pi dY i Y pm dY m Y pm dY m pi ; i 1; ¼; 12 dY m Rm for any m, provided Rpm ± 0 (since the pathological point is a not an equilibrium point, certainly not all the reaction rates can be zero there). Then from equations (6), (7) and (9) we have, to first order in the small quantities, " #21 (" p # p p r0 Dp 2 p p T dT 2 rp dr p T r;Y r r T; Y ) p 12 X Y pm p p : 15 Ri 2 p dY m Rm Y i r;T;Y i i1 Hence we have two unknowns, dr and d Ym. We next define a pseudo-time variable, t , by dx dt : h Then, from (11) and (4) we obtain dr ra2 2 f s´R dt u 16 and dY m Rm : h dt u q 1999 RAS, MNRAS 310, 1039±1052 17 Figure 5. Values of (a) the density, (b) the temperature and (c) the distance behind shock, at the pathological point versus the initial density. Clearly, the pathological point is a critical point of these equations (in fact the pathological locus is a locus of critical points). Linearizing in the small quantities about the pathological point we get s ´ R p p s ´ R p p r dr T dT r T p # 12 X Y pm p s ´ R p dY m Ri Rm Y i i1 ddr ap 2 rp f p dt r0 D 1046 G. J. Sharpe where the matrix Ap depends only on the state at the pathological point. As expected we find that Ap has real eigenvalues, one positive and one negative, so that the pathological point is a saddle point of equations (16) and (17). There are thus two solutions, the separatrices of the saddle, which enter or leave the pathological point, corresponding to the frozen subsonic and supersonic branches. Note that, for subsonic h . 0 solutions, t -stability implies x-stability, whereas, for supersonic solutions h , 0 t -stability implies x-instability and vice versa. Hence two solutions leave the pathological point as x increases (i.e. in the downstream direction), corresponding to the self-sustaining (supersonic) and supported pathological (subsonic) solutions. Of the two solutions which enter the pathological point from upstream, the subsonic one is the branch of the solution between the shock and the pathological point, while the supersonic one corresponds to a non-physical unshocked wave. We can thus use the eigenvectors of Ap very near the pathological point to determine initial conditions for these solutions. We then integrate towards the strong or weak equilibrium points to determine the downstream solutions. We find that these solutions are not very sensitive to the exact location of the pathological point, or the nuclei abundances used for Yj, Yk and Yl. Also, as expected, the supported pathological detonation quickly tends to, and eventually becomes indistinguishable from, the (downstream) separatrix formed by the solutions for overdriven detonations with D very slightly greater than Dp. One should also integrate back upstream to the shock along the subsonic solution to ensure that the original initial conditions are met. However, as pointed out earlier, the equations are numerically unstable to integration in this direction, but since the subsonic solution downstream quickly tends to the subsonic separatrix, we expect the upstream solution to also quickly tend to the upstream separatrix, i.e. to pick up the solutions found by integrating from the shock with D . Dp . Indeed, the subsonic eigenvector of Ap is directed back towards this upstream separatrix. However, this is not too important since we are interested in the difference between the self-sustaining and supported downstream solutions. Hence we can find the complete structure by integrating from the shock with D Dp , as far as possible along the separatrix towards the pathological point, continuing this solution by interpolation to the pathological point, then linearizing about the pathological point to integrate to equilibrium. Fig. 7 shows the structure of the different solution branches determined by this method near the pathological point for an initial density of 1 109 g cm23 (compare with Fig. 3). Figure 6. Composition at the pathological point versus the initial density, (a) nuclei abundances of 4He to 20Ne, (b) nuclei abundances of 24Mg to 36 Ar and (c) nuclei abundances of 40Ca to 56Ni. and ddY m dt rp Rpm r0 Dp Y pm " h r p p # p 12 X h Yp h ; rp dr T p dT pm dY m Rpi T Rm Yi i1 with d T given by (15), or ! ! dr dr d p A ; dt dY m dY m 5 PAT H O L O G I C A L D E T O N AT I O N WAV E STRUCTURES In this section we examine both the self-sustaining and the supported pathological detonation wave structures for three representative initial densities, log r0 ; g cm23 9:0, 8.0 and 7.6. Figs 8±10 show the density, temperature, nuclear energy release and mean molecular weight throughout the waves for these initial densities. Fig. 11 shows the abundances of He, C, O, Si, Ca, Fe and Ni for both the unsupported and supported waves, while Fig. 12 shows the detailed nuclear composition for the unsupported detonation, for the case log r0 8:0. Table 1 gives the detonation lengths (defined as the distance behind the shock at which the differences between all the nuclei abundances and their equilibrium values become less than 1 per cent) and the q 1999 RAS, MNRAS 310, 1039±1052 Detonations in Type Ia supernovae 1047 Figure 7. Pathological detonation wave structure near the pathological point. The solid line is the branch of the solution between the shock and the pathological point, the dashed line is the self-sustaining downstream solution and the dotted line is the supported downstream solution. (a) Density versus distance behind shock, and (b) density versus 4He nuclei abundance. r0 1 109 g cm23 . equilibrium density, temperature and composition. Note that downstream of the pathological point there can be additional points where the thermicity again passes through zero. Indeed, from Fig. 9 it is clear that, for the self-sustaining solution, the density, temperature and heat release pass through a minimum and then increase slightly as equilibrium is approached. As the initial density decreases this effect seems to become more pronounced. For the log r0 7:6 case it is so great that the solution again encounters, and terminates at, a frozen sonic point before equilibrium is reached. Hence for lower initial densities there seems to be no completely steady self-sustaining wave. However, since the flow is frozen sonic at the pathological point, it is also possible to have a time-dependent continuation of the solution downstream of the pathological point (Wood & Salsburg 1960). Indeed, for any initial density, the steady continuation downstream of the pathological point of the self-sustaining solution is not unique; time-dependent continuations are also possible, depending on the downstream conditions. For a given initial density, the temperature and the density downstream of the pathological point are consistently lower for the self-sustaining wave than for the supported wave. Even though the difference in the temperature is not large, because the reaction rates are extremely temperature sensitive, this results in the structures of the self-sustaining and supported waves being quite q 1999 RAS, MNRAS 310, 1039±1052 Figure 8. Pathological detonation wave structure for r0 1 109 g cm23. The solid line is the branch of the solution between the shock and the pathological point, the dashed line is the self-sustaining downstream solution and the dotted line is the supported downstream solution. (a) Density, (b) temperature, (c) nuclear energy release and (d) mean molecular weight versus distance behind shock. 1048 G. J. Sharpe Figure 9. Pathological detonation wave structure for r0 1 108 g cm23. The solid line is the branch of the solution between the shock and the pathological point, the dashed line is the self-sustaining downstream solution and the dotted line is the supported downstream solution. (a) Density, (b) temperature, (c) nuclear energy release and (d) mean molecular weight versus distance behind shock. Figure 10. Pathological detonation wave structure for log r0 7:6. The solid line is the branch of the solution between the shock and the pathological point, the dashed line is the self-sustaining downstream solution and the dotted line is the supported downstream solution. (a) Density, (b) temperature, (c) nuclear energy release and (d) mean molecular weight versus distance behind shock. q 1999 RAS, MNRAS 310, 1039±1052 Detonations in Type Ia supernovae 1049 Figure 11. Pathological detonation wave structure for r0 1 108 g cm23. The solid line is the branch of the solution between the shock and the pathological point, the dashed line is the self-sustaining downstream solution and the dotted line is the supported downstream solution. (a) 4He, (b) 12C, (c) 16 O and (d) 28Si, (e) 40Ca, (f) 52Fe and (g) 56Ni nuclei abundances versus distance behind shock. different. First, the lower temperature of the self-sustaining detonation means that the fuel takes much longer to burn to equilibrium than for the supported detonation. From Table 1, the self-sustaining detonation lengths are about 2.4 and 3.0 times that q 1999 RAS, MNRAS 310, 1039±1052 of the corresponding supported detonation wave lengths for log r0 9:0 and 8.0, respectively. Hence one would expect the detonation length to become comparable with the length-scale of the star at slightly higher initial densities than the supported 1050 G. J. Sharpe Table 1. Detonation length and equilibrium density, temperature and composition by mass for the self-sustaining and supported pathological detonation waves. log(r 0, g cm23) Figure 12. Nuclear composition in the unsupported detonation wave for r0 1 108 . detonation wave structures would predict. Secondly, since the more tightly bound nuclei are favoured in equilibrium for lower temperatures (Wiggins et al. 1998), the self-sustaining detonation produces more 56Ni and slightly less of the intermediate mass elements. This also results in much less energy being absorbed downstream of the pathological point in the self-sustaining detonation than in the supported wave. From Figs 8±10 it can be seen that very little heat indeed is absorbed for the selfsustaining case. Although the difference in equilibrium compositions is not large, one can see from Fig. 11 that the composition in the wave at a given distance behind the pathological point is quite different. Hence if the detonation was quenched for some reason, e.g. as a result of the expansion of the star, the resulting composition would be quite different for the self-sustaining detonation than for the supported detonation. Finally, from Fig. 11 it is clear that the pathological point occurs near the start of silicon burning; both carbon and oxygen are virtually completely burnt by the time the pathological point is reached. Note the very different length-scales of carbon, oxygen and silicon burning, which can also be seen from Fig. 12. 6 CONCLUSIONS In this paper we have investigated steady, one-dimensional detonation waves in C±O for initial densities greater than 2 107 g cm23 . For such densities the self-sustaining detonation is of the pathological type. We have developed a method for determining the self-sustaining and supported pathological detonations. The structure for the self-sustaining detonation downstream of the pathological point is quite different from that of the corresponding supported detonation. The lower temperatures of the self-sustaining detonation waves mean that the detonation length is significantly larger than that in the supported wave. More of the iron-peak and less of the intermediate elements are produced, and hence there is a greater total nuclear energy release by the self-sustaining wave than by the supported detonation. In this paper we have used a simple a -network for the burning. This should give the correct qualitative results, including (importantly) the energy release. Large differences in the energy release, compared to a complete network, would only be important near the end of the burning, because of the neglect of xdet, cm r , 109 g cm23 T, 109 K XHe, % XC, % XO, % XNe, % XMg, % XSi, % XS, % XAr, % XCa, % XTi, % XCr, % XFe, % XNi, % xdet, cm r , 109 g cm23 T, 109 K XHe, % XC, % XO, % XNe, % XMg, % XSi, % XS, % XAr, % XCa, % XTi, % XCr, % XFe, % XNi, % xdet, cm r , 109 g cm23 T, 109 K XHe, % XC, % XO, % XNe, % XMg, % XSi, % XS, % XAr, % XCa, % XTi, % XCr, % XFe, % XNi, % 9.0 8.0 7.6 Self-sustaining Supported 1:9 103 1.270 8.145 26.6 0.006 0.017 0.001 0.054 5.5 8.5 7.0 8.7 1.0 3.6 12.0 27.0 8:0 102 1.543 8.344 28.5 0.007 0.020 0.002 0.061 5.6 8.6 7.1 8.7 1.1 3.6 11.6 25.1 2:2 105 0.1415 6.658 19.7 0.002 0.006 0.0002 0.018 4.9 7.6 5.8 7.4 0.49 2.4 11.6 40.1 7:3 104 0.1783 6.862 23.0 0.003 0.008 0.0003 0.023 5.3 8.1 6.1 7.6 0.55 2.5 11.3 35.4 2:9 105 * 0.06178* 6.139* 12.9* 0.003* 0.009* 0.0002* 0.026* 10.1* 14.0* 9.0* 10.4* 0.47* 2.0* 8.9* 32.2* 9:3 105 0.07155 6.274 17.4 0.001 0.003 0.00009 0.011 4.2 6.7 5.0 6.6 0.37 2.0 11.3 46.4 * State at frozen sonic termination point. such processes as electron capture in the a -network. Since the pathological point occurs near the start of silicon burning, this should have little qualitative effect on the structure near such points, but may produce rather different predictions for the equilibrium abundances. However, it is difficult to assess the differences that would occur if a complete network was employed instead of an a -network, without determining the complete detonation wave. While the method described in this paper could be used to determine the complete pathological detonation wave structures, the large number of species (a few hundred, with a few q 1999 RAS, MNRAS 310, 1039±1052 Detonations in Type Ia supernovae thousand reactions) would make it rather difficult to handle the determination of the structure around the pathological points. Secondly, in any time-dependent numerical simulation, one really is restricted to a simple network like the a -network. The detonation wave structures presented here are for detonations propagating into material with a constant pre-shock density. Of course, in a real supernova the density drops with the radius of the white dwarf. However, for the density range investigated here the detonation thickness, or more importantly the distance between the shock and the pathological point (since the flow downstream of this point cannot affect the front), is very much smaller than the size of the white dwarf and also much smaller than the scale of density variations in the star (,107 cm, Khokhlov 1989). Hence such density gradients will have little effect on the propagation of the detonation wave, and the structures presented here can be viewed as snapshots of the detonation wave when it has reached a given density. For lower densities, when the self-sustaining detonation is Chapman± Jouguet, the detonation wave thickness can become comparable with the scale of density variations (Khokhlov 1989). To investigate the effect of the gradient at such densities, one could perform one-dimensional (i.e. spherically symmetric), timedependent simulations in a portion of the star corresponding to the low-density outer layers. Fortunately, for these densities the detonation is hydrodynamically stable to one-dimensional perturbations (Khokhlov 1993). An adaptive gridding technique would be required to resolve the different length-scales. However, at such densities, other effects, such as the curvature of the front, are also likely to become important. A small amount of curvature can cause the detonation to become pathological, even when the planar detonation is CJ, because of the divergence of the flow (Fickett & Davis 1979). In this case, it is the distance between the shock and the sonic pathological point which is the relevant length-scale. In the slightly curved detonation, this distance may be much shorter than the detonation length as a whole (as in the cases studied in this paper), and also much shorter than the scale of the density variations, in which case such variations in the density would be unimportant to the propagation of the detonation. We intend to use the information presented here in several ways. First, using the method of Wiggins & Falle (1997b), we intend to perform two-dimensional simulations of double detonations in Type Ia supernovae, where a detonation is ignited at a point in the He envelope and triggers a detonation in the C±O core at the core±envelope boundary. We have shown in this paper that the burning up to the pathological point occurs on a very short length-scale, and only a small amount of heat is absorbed downstream of the frozen sonic pathological point. Hence we can model the burning up to the frozen sonic point as a thin detonation, with the downstream state given by the onedimensional pathological point values. We can then evaluate the burning beyond the pathological point from the subsequent flow calculated from the hydrodynamics. This can be used to investigate the effects of two-dimensional phenomena, such as asymmetrical expansion of the star and the cusps in the wave front found by Wiggins & Falle (1997a), on the nucleosynthesis and the range of expansion velocities of the intermediate mass elements. Secondly, we intend to perform a linear stability analysis of both CJ and pathological C±O detonations. Sharpe (1997, 1999) has developed methods for dealing with both cases. Preliminary two-dimensional numerical simulations show that such analyses give very good predictions of the natural cell size, at least when q 1999 RAS, MNRAS 310, 1039±1052 1051 the steady detonation is stable to one-dimensional perturbations (Sharpe & Falle, in preparation). It should also reveal the multiscale instabilities of each of the burning layers. We then intend to perform two-dimensional numerical simulations to see how these instabilities on different length-scales affect each other. However, because of the vastly disparate length-scales it is not possible to resolve more than one burning layer at a time. The only way of overcoming this difficulty that occurs to us is to perform successive simulations with the length-scale of the carbon-burning layer artificially increased, and see if we can scale these to the correct value. It would, of course, be easier to first perform one-dimensional simulations. However, the one-dimensional, or pulsating, instability is a very different instability and propagation mechanism than the cellular instability in multidimensions. As mentioned in the introduction, such pulsating detonations are seen only very rarely in terrestrial experiments (when blunt bodies are fired into detonatable gases). Almost always the wave propagates as a cellular detonation, even when the detonation is stable to onedimensional perturbations. Hence the pulsating instability has no relevance in the multidimensional progenitors of Type Ia supernovae. Indeed the one-dimensional instability leads to erroneous predictions about how the detonation would propagate in multidimensions, such as the detonation failure seen in one dimension, which does not occur in multidimensions because of the constant re-ignition by the transverse waves. 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