Issue 
A&A
Volume 527, March 2011



Article Number  A5  
Number of page(s)  7  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201015895  
Published online  18 January 2011 
Mixing in classical novae: a 2D sensitivity study^{⋆}
^{1}
Departament de Física i Enginyeria Nuclear, EUETIBUniversitat Politècnica
de Catalunya,
c/Comte d’Urgell 187,
08036
Barcelona,
Spain
email: jordi.jose@upc.edu
^{2}
Institut d’Estudis Espacials de Catalunya,
c/Gran Capità 2–4, Ed.
Nexus201, 08034
Barcelona,
Spain
^{3}
Departament de Física Aplicada, Universitat Politècnica de
Catalunya, c/Esteve Terrades
5, 08860
Castelldefels,
Spain
^{4}
Department of Physics & Astronomy, Stony Brook
University, Stony
Brook, NY
117943800,
USA
^{5}
Dipartimento di Fisica “Enrico Fermi”, Università di Pisa and
INFN, Sezione di Pisa, Largo B.
Pontecorvo 3, 56127
Pisa,
Italy
Received:
8
October
2010
Accepted:
24
November
2010
Context. Classical novae are explosive phenomena that take place in stellar binary systems. They are powered by mass transfer from a lowmass, main sequence star onto a white dwarf. The material piles up under degenerate conditions and a thermonuclear runaway ensues. The energy released by the suite of nuclear processes operating at the envelope heats the material up to peak temperatures of ~(1−4) × 10^{8} K. During these events, about 10^{4}−10^{5}M_{⊙}, enriched in CNO and other intermediatemass elements, are ejected into the interstellar medium. To account for the gross observational properties of classical novae (in particular, a metallicity enhancement in the ejecta above solar values), numerical models assume mixing between the (solarlike) material transferred from the companion and the outermost layers (CO or ONerich) of the underlying white dwarf.
Aims. The nature of the mixing mechanism that operates at the coreenvelope interface has puzzled stellar modelers for about 40 years. Here we investigate the role of KelvinHelmholtz instabilities as a natural mechanism for selfenrichment of the accreted envelope with core material.
Methods. The feasibility of this mechanism is studied by means of the multidimensional code FLASH. Here, we present a series of 9 numerical simulations perfomed in two dimensions aimed at testing the possible influence of the initial perturbation (duration, strength, location, and size), the resolution adopted, or the size of the computational domain on the results.
Results. We show that results do not depend substantially on the specific choice of these parameters, demonstrating that KelvinHelmholtz instabilities can naturally lead to selfenrichment of the accreted envelope with core material, at levels that agree with observations.
Key words: novae, cataclysmic variables / nuclear reactions, nucleosynthesis, abundances / convection / hydrodynamics / instabilities / turbulence
Movie is only available in electronic form at http://www.aanda.org
© ESO, 2011
1. Introduction
Classical novae are cataclysmic stellar events. Their thermonuclear origin, theorized by Schatzmann (1949, 1951) and Cameron (1959) – see also Gurevitch & Lebedinsky (1957) and references therein – has been established through multiwavelength observations and numerical simulations pioneered by Sparks (1969), who performed the first 1D, hydrodynamic nova simulation. These efforts helped to establish a basic picture, usually referred to as the thermonuclear runaway model (TNR), which has been successful in reproducing the gross observational properties of novae, namely the peak luminosities achieved, the abundance pattern, and the overall duration of the event; see Starrfield et al. (2008), José & Shore (2008), José & Hernanz (2007) for recent reviews.
Many details of the dynamics of nova explosions remain to be explored. In particular, there are many observed cases of nonspherical ejecta, inferred from line profiles during the early stages of the outburst and from imaging of the resolved ejecta, including multiple shells, emission knots, and chemical inhomogeneities. Although the broad phenomenology of the outburst can be captured by 1D calculations, it is increasingly clear that the full description requires a multidimensional hydrodynamical simulation of such outbursts. To match the energetics, peak luminosities, and the abundance pattern, models of these explosions require mixing of the material accreted from the lowmass stellar companion with the outer layers of the underlying white dwarf. In fact, because of the moderate temperatures achieved during the TNR, a very limited production of elements beyond those from the CNOcycle is expected (Starrfield et al. 1998, 2009; José & Hernanz 1998; Kovetz & Prialnik 1997; Yaron et al. 2005), and the specific chemical abundances derived from observations (with a suite of elements ranging from H to Ca) cannot be explained by thermonuclear processing of solarlike material. Mixing at the coreenvelope interface represents a likely mechanism.
The details of the mixing episodes by which the envelope is enriched in metals have challenged theoreticians for nearly 40 years. Several mechanisms have been proposed, including diffusioninduced mixing (Prialnik & Kovetz 1984; Kovetz & Prialnik 1985; Iben et al. 1991, 1992; Fujimoto & Iben 1992), shear mixing at the diskenvelope interface (Durisen 1977; Kippenhahn & Thomas 1978; MacDonald 1983; Livio & Truran 1987; Kutter & Sparks 1987; Sparks & Kutter 1987), convective overshootinduced flame propagation (Woosley 1986), and mixing by gravity wave breaking on the white dwarf surface (Rosner et al. 2001; Alexakis et al. 2004). The multidimensional nature of mixing has been addressed by Glasner & Livne (1995) and Glasner et al. (1997, 2005, 2007) with 2D simulations of COnovae performed with VULCAN, an arbitrarily Lagrangian Eulerian (ALE) hydrocode capable of handling both explicit and implicit steps. They report an effective mixing triggered by KelvinHelmholtz instabilities that produced metallicity enhancements to levels in agreement with observations. Similar studies (using the same initial model as Glasner et al. 1997) were conducted by Kercek et al. (1998, 1999) in 2D and 3D, respectively. Their results, computed with the Eulerian code PROMETHEUS, displayed mild TNRs with lower peak temperatures and velocities than Glasner et al. (1997) and insufficient mixing. While Glasner et al. (1997) argue that substantial mixing can naturally occur close to peak temperature, when the envelope becomes fully convective and drives a powerful TNR, Kercek et al. (1998) conclude instead that mixing must take place much earlier: if it occurs around peak temperature, it leads to mild explosions or to events that do not resemble a nova.
The differences between these studies have been carefully analyzed by Glasner et al. (2005), who conclude that the early stages of the explosion, before TNR ignition when the evolution is quasistatic, are extremely sensitive to the adopted outer boundary conditions. They show that Lagrangian simulations, in which the envelope is allowed to expand and mass is conserved, lead to consistent explosions. In contrast, in Eulerian schemes with a “free outflow” outer boundary condition, the choice adopted in Kercek et al. (1998), the outburst can be artificially quenched. The scenario was revisited by Casanova et al. (2010), who show that simulations with an Eulerian scheme – the FLASH code – and a proper choice of the outer boundary conditions can produce deepmixing of the solarlike accreted envelopes with core material. The puzzling results reported by Kercek et al. (1998) stress the need for a systematic evaluation of the effect that different choices of model parameters (e.g. the intensity and location of the initial temperature perturbation, resolution, or size of the computational domain) may have on the results. To this end, we performed a series of 9 numerical simulations in 2D aimed at testing the influence of these parameters on the level of metal enhancement of the envelope. Here we report the results of these simulations.
Our paper is organized as follows. In Sect. 2 we explain our input physics and initial conditions. Then Sect. 3 is devoted to studying the mixing at the coreenvelope interface for our fiducial model. In Sect. 4 the effects of the size of the initial perturbation are analyzed, while in Sect. 5 we discuss the effects of the size of the computational domain. In Sect. 6 we quantify the influence of the grid resolution. Finally, in Sect. 7 we discuss the significance of our results and draw our conclusions.
2. Input physics and initial conditions
The simulations reported here were performed with FLASH, a parallelized explicit Eulerian code, based on the piecewise parabolic interpolation of physical quantities for solving the hydrodynamical equations, and with adaptive mesh refinement (see Fryxell et al. 2000). As in Casanova et al. (2010), we used the same initial model as Glasner et al. (1997) and Kercek et al. (1998): a 1M_{⊙} CO white dwarf that accretes solar composition matter (Z = 0.02) at a rate of 5 × 10^{9}M_{⊙} yr^{1}. The model was evolved spherically (1D) and mapped onto a 2D cartesian grid, when the temperature at the base of the envelope reached ≈ 10^{8} K. It initially comprised 112 radial layers – including the outermost part of the CO core – and 512 horizontal layers. The mass of the accreted envelope was about 2 × 10^{5}M_{⊙}. Nuclear energy generation is handled through a network of 13 species (^{1}H, ^{4}He, ^{12,13}C, ^{13,14,15}N, ^{14,15,16,17}O, and ^{17,18}F), and connected through 18 nuclear reactions. We adopted the conductive and radiative opacities from Timmes (2000) and an equation of state based on Timmes & Swesty (2000). Periodic boundary conditions were imposed on both sides of the computational domain with vertical hydrostatic equilibrium with an outflow constraint at the top and a reflecting constraint at the bottom on the velocity (see Zingale et al. 2002). A summary of the main characteristics of the 9 models computed in this work is given in Table 1, where H is the distance from the perturbation to the initial coreenvelope interface, R_{x} and R_{y}^{1}, δT, and δt are the size, strength, and duration of the temperature perturbation, and Z the massaveraged metallicity of the envelope at the end of the calculations.
3. 2D simulations of mixing at the coreenvelope interface
In this section, we describe the basic features of our fiducial model A, as a framework for further discussion of the effect of the parameter choices on our results. A movie, showing the development of KelvinHelmholtz instabilities, in terms of the ^{12}C content, up to the time when the convective front hits the upper computational boundary, ModelA2D.wmv, is available online or at http://www.fen.upc.edu/users/jjose/Downloads.html. The simulation was performed for the conditions of model A, as summarized in Table 1.
For all sequences reported in this work, the relaxation of the initial model to guarantee hydrostatic equilibrium, together with the small amount of numerical viscosity – in contrast with the simulations performed by Glasner et al. (1997) – requires an initial perturbation close to the coreenvelope interface to trigger the onset of instabilities early in the calculations. The initial perturbation is applied to the temperature using four parameters: strength, location, size and duration. Model A assumes a tophat temperature perturbation wherever ((x − x_{0})/R_{x})^{2} + ((y − y_{0})/R_{y})^{2} ≤ 1, where x and y are the space coordinates measured from the center of the perturbation, (x_{0}, y_{0}), and R_{x} and R_{y} indicate its spatial extent. We fixed x_{0} = 5 × 10^{7} cm in all sequences. The strength of the perturbation is 5% in temperature in all cases but one (see Table 1). It is 1 km wide, applied only during the initial timestep (that is, the temperature is fixed only during 10^{10} s), and imposed on the coreenvelope interface (y_{0} = 5.51 × 10^{8} cm). The resolution adopted in model A is 1.56 × 1.56 km, and the size of the computational domain is 800 × 800 km.
Fig. 1 Snapshot of the development of early instabilities, which later spawn KelvinHelmholtz instabilities, shown in terms of the ^{12}C mass fraction (in logarithmic scale) for model A, 158 s from the start of the simulation when the coreenvelope interface temperature is T_{base} ~ 1.36 × 10^{8} K. 
The initial perturbation drives a shear flow that triggers the formation of instabilities (Fig. 1), about 150 s after the start of the simulation. As soon as material from the core is mixed into the envelope, small convective cells develop. At this early stage, the fluid has a large Reynolds number, with a characteristic eddy length of 50 km, fluid velocities ranging between v = 10^{5} − 10^{6} cm s^{1}, and a dynamic viscosity^{2} of 2 × 10^{4} P. The fluid velocity v remains below the speed of sound c_{s} (that is, the Mach number Ma = v/c_{s} is always less than unity, see Fig. 2), hence, the fluid displays a deflagration rather than a detonation – see Williams (1985), for a thorough analysis of the differences between flame propagation under detonation and deflagration conditions. At t = 235 s, the characteristic eddy turnover time is l_{conv}/v_{conv} ~ 10 s. The merging of the small convective cells into large eddies, characteristic of 2D simulations, with a size comparable to the height of the envelope, reinforces the injection of COrich material into the envelope. Convection becomes more turbulent. At this stage (t = 450 s), the nuclear energy generation rate exceeds 10^{15} erg g^{1} s^{1}, while the characteristic convective timescale decreases to ~5 s. The convection front propagates progressively upwards (Fig. 3, left panel), with a velocity of ~ 10 km s^{1}, and eventually reaches the top of our computational domain. The envelope base reaches a peak temperature of 1.64 × 10^{8} K. At this time (t = 496 s), when matter starts to cross the outer boundary of the computational domain, we stop the calculations because of the Eulerian nature of the FLASH code. At this final stage, the mean massaveraged metallicity in the envelope reaches Z ~ 0.21. It is worth noting, however, that the convective eddies are still pumping metalrich matter through the coreenvelope interface. Hence, it is likely that the final metallicity in the envelope will be larger. The simulation shows that the induced KelvinHelmholtz vortices can naturally lead to selfenrichment of the accreted envelope with core material to levels that agree with observations and that the expansion and progress of the runaway is almost spherically symmetric for nova conditions even for a pointlike TNR ignition.
4. Effect of the initial perturbation
To quantify the influence of the initial perturbation on our results, we have performed a series of 2D hydrodynamic tests for a set of different durations, strengths (intensities), locations and sizes of the perturbation. For simplicity, a tophat perturbation, centered at x_{0} = 5 × 10^{7} cm, has been adopted in all models reported in this work.
The effect of the duration of the perturbation was checked by means of a test case (model B), identical to model A but with a perturbation lasting for 10 s. As shown in Table 1, the characteristic timescales for model B, such as the time required for the first instabilities to show up, T_{KH}, or the time needed by the convective front to hit the outer boundary, t_{Y}, become shorter. The role played by a temperature perturbation can be understood in terms of the energy injected into the envelope: the longer the duration of the perturbation, the larger the energy injected, and thus, the shorter the characteristic timescales of the TNR. This has little effect, however, on the overall metallicity enhancement in the envelope since a final CNO mass fraction of ~0.212 was found in model B, whereas ~0.224 resulted in model A.
Both models A and B assumed temperature perturbations of δT ~ 5% during the initial timestep (~10^{10} s). To test the possible influence of the strength of the perturbation, a test case with δT ~ 0.5% (model C) has also been computed. As shown in Table 1 and Fig. 4, the time evolution of models A and C is very similar, and hence, similar final mean CNO mass fractions at the end of the simulations were found (with Z = 0.209 in model C).
Models computed.
Fig. 2 Mach number at two different moments of the simulation, t = 230 s (left panel) and 496 s (right panel), for model A. 
Fig. 3 Left panel: propagation of the convective front as a function of time, for models A, H, and I. Right panel: temperature profile versus radius at two different times, t = 0 s (solid line; T_{base} = 9.84 × 10^{7} K) and t = 496 s (dashed line; T_{base} = 1.64 × 10^{8} K), for model A. 
The effect of the location of the perturbation along the vertical axis has also been studied: whereas model A assumed a temperature perturbation of ~5%, applied at the innermost envelope shell (y_{0} = 5.51 × 10^{8} cm), in model D, a similar perturbation was placed ~5 km above the coreenvelope interface (y_{0} = 5.515 × 10^{8} cm). Both models exhibit a very similar temporal evolution, with almost identical times for the appearance of the first instabilities and for the time required to reach the outer boundary. Similar envelope mean CNO mass fractions (0.224 and 0.235, respectively) were also found.
Finally, the influence of the size of the perturbation has also been analyzed. Whereas model D was evolved with an initial temperature perturbation of size R_{x} = 1 km and R_{y} = 1 km, model E assumed R_{x} = 5 km and R_{y} = 5 km. As before, very similar characteristic timescales (see Table 1) and final mean CNO mass fractions (0.235 and 0.209, respectively) were found.
To summarize, the specific choice of the parameters that define the initial temperature perturbation has a negligible effect on metallicity enhancement of the envelope.
5. Effect of the size of the computational domain
The choice of the computational domain represents a compromise between computational time requirements and numerical accuracy. Several considerations constrain its minimum size. On one hand, the merger of large convective eddies often found in 2D simulations may be severely affected by the adoption of a small computational domain. On the other hand, nova outbursts eventually result in mass ejection. With an Eulerian code such as FLASH, it is not possible to track the material that flows off the grid, and hence, it is important to use domains that are as large as possible along the radial direction (while being sufficiently wide along the horizontal axis). Unfortunately, when the initial 1D model is mapped into the 2D grid, and relaxed to guarantee hydrostatic equilibrium, densities quickly underflow values for large heights (Zingale et al. 2002).
The specific size adopted for most of the models computed in this work, i.e. 800 × 800 km, is a bit smaller than those used in Glasner et al. (1997) – 0.1π^{rad}, in sphericalpolar coordinates – and in Kercek et al. (1998) – 1800 × 1100 km, in a cartesian, planeparallel geometry. In this section, we analyze possible dependences of the results on the adopted size of the computational domain. To this end, two additional simulations were performed. In the first one (model F), a wider computational domain has been adopted (i.e., 1600 × 800 km). In the second (model G), aimed at testing the influence of the vertical (radial) length, a domain of 800 × 1000 km has been used (where the choice of 1000 km results from numerical restrictions that limit the vertical extent of our computational domain).
As shown in Table 1, the horizontal width (model F) has no noticeable effect on the timescales of the simulations, either for the time required for the onset of the first instabilities or for the time required for the convective front to reach the outer boundary. The massaveraged CNO abundance in the envelope reached ~0.206 at the end of this simulation, close to the value found for model A. These results confirm that 800 km is an appropriate choice for the width of the computational domain, stressing that above a threshold value the course of the TNR is insensitive to the adopted width, in agreement with the sensitivity study performed by Glasner et al. (2007).
Fig. 4
Left panel: time evolution of the nuclear energy generation rate (in erg s^{1}) for the 9 models computed in this work. Right panel: final CNO mass fraction versus radius. 
The specific length adopted along the vertical direction (see model G), while unimportant for the time of appearance of the instabilities (around 155 s after the start of the simulation, as in model A), affects the time required to reach the outer boundary, located 200 km above the value adopted for model A. Moreover, the larger extension of the computational domain along the radial (vertical) direction allows the convective eddies to pump additional metalrich core material into the envelope compared with all the simulations reported previously in this paper. Indeed, the mean, massaveraged metallicity in model G achieves the largest value of all the simulations reported, ~0.291. This result suggests that the likely mean massaveraged metallicity driven by KelvinHelmholtz instabilities should be Z ≈ 0.3. In summary, we conclude that the size of the computational domain, above a certain threshold value, has little influence on the physical quantities that are more directly related with the mixing process at the coreenvelope interface.
6. Effect of the grid resolution
Fig. 5
Snapshots of the ^{1}H (upper panels) and ^{12}C (middle panels) mass fractions at t ~ 395 s (model A; left panels), and 688 s (model I; right panels). Lower panels: the number of blocks administered, at this stage, is 3184 for model A, and 43 800 for model I. In both simulations, FLASH divides each block in 8 cells. Structures such as vortexs are better resolved in the finer resolution model I. 
All simulations discussed so far (e.g., models A to G) were performed with a resolution of 1.56 × 1.56 km, a value similar to the minimum resolutions adopted in Glasner et al. (2007) which is roughly ~1.4 × 1.4 km, and in Kercek et al. (1998), 1 × 2 km. To quantitatively assess the possible effect of the resolution, two additional test cases were computed with exactly the same input parameters as in model A but with two different resolutions: 1 × 1 km (model H) and 0.39 × 0.39 km (model I)^{3}.
As shown in Table 1, the increase in resolution produces a delay in the time required for the first instabilities to develop, t_{KH}. This seems to be a numerical artifact. In models with a coarser resolution, the larger size of the blocks artificially generates a larger numerical diffusion compared to models with a finer resolution (a similar resolution dependence is clearly seen as well in the Kercek et al. 1998, simulations). Actually, the ratio of differences in the initial build up times (i.e., (model Imodel A)/(model Hmodel A)) scales approximately as the zone size dimensions to the power of two. This is a purely numerical perturbation that forces the development of instabilities. To test this hypothesis, we computed an additional test case (not included in Table 1), identical to model A but without any initial perturbation. The onset of the instabilities in such an extremely low numerical diffusion regime is substantially delayed. The simulations reported by Glasner et al. (1997) also show the early appearance of instabilities in a model with substantial numerical noise: within a very short time (about 10 s), thenumerical noise (roundoff) seeds an intense convective flow in the envelope without any artificial perturbations.
A similar behavior is also found for the time required for the convective front to reach the outer boundary, t_{Y}, and for the history of the nuclear energy generation rate (Fig. 4). As expected, filamentary structures and convective cells are better resolved in the finer resolution model I, compared to those computed with somewhat coarser grids (models A and H; see Fig. 5). These minor differences do not, however, show significant variations in the final, mean CNO abundances achieved in the envelope: while Z ~ 0.224 in model A, models H and I yield 0.201 and 0.205, by mass, respectively. Similar agreement is found in the peak temperatures achieved and in the overall nuclear energy generation rates (Fig. 4).
Thus, the adopted resolution has not a critical effect for the mixing models presented in this work. The variation in the final mean CNO abundance in the envelope, under the range of resolutions adopted, is only about 12% (when comparing results for models A, H, and I), a quite reasonable value.
7. Discussion and conclusions
In this paper we have reported results for a series of nine 2D numerical simulations that test the influence of the initial perturbation (duration, strength, location, and size), the resolution of the grid, and the size of the computational domain on the results. We have shown that mixing at the coreenvelope interface proceeds almost independently of the specific choice of such initial parameters, above threshold values.
The study confirms that the metallicity enhancement inferred from observations of the ejecta of classical novae can be explained by KelvinHelmholtz instabilities, powered by an effective mesoscopic shearing resulting from the initial buoyancy. Fresh core material is efficiently transported from the outermost layers of the white dwarf core and mixed with the approximately solar composition material of the accreted envelope. As soon as ^{12}C and ^{16}O are dredged up, convection sets in and small convective cells appear, accompanied by an increased nuclear energy generation rate. The size of these convective cells increases in time. Eventually, these cells merge into large convective eddies with a size comparable to the envelope height. The range of mean massaveraged envelope metallicities obtained in our simulations at the time when the convective front hits the outer boundary, 0.21 − 0.29, matches the values obtained for classical novae hosting CO white dwarfs.
It is, however, worth noting that the convective pattern is actually produced by the adopted geometry (e.g., 2D), forcing the fluid motion to behave very differently than 3D convection (Shore 2007; Meakin & Arnett 2007). Nevertheless, the levels of metallicity enhancement found in our 2D simulations will likely remain unaffected by the limitations imposed by the 2D geometry (Arnett, private communication). Fully 3D simulations aimed at testing this hypothesis are currently underway.
Acknowledgments
We thank the referee for a careful reading of the manuscript and for constructive comments. The software used in this work was in part developed by the DOEsupported ASC/Alliances Center for Astrophysical Thermonuclear Flashes at the University of Chicago. This work has been partially supported by the Spanish MEC grants AYA201015685 and AYA200804211C0201, by the E.U. FEDER funds, and by the ESF EUROCORES Program EuroGENESIS. We also acknowledge the Barcelona Supercomputing Center for a generous allocation of time at the MareNostrum supercomputer.
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Online material
Movie (Access here)
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All Figures
Fig. 1 Snapshot of the development of early instabilities, which later spawn KelvinHelmholtz instabilities, shown in terms of the ^{12}C mass fraction (in logarithmic scale) for model A, 158 s from the start of the simulation when the coreenvelope interface temperature is T_{base} ~ 1.36 × 10^{8} K. 

In the text 
Fig. 2 Mach number at two different moments of the simulation, t = 230 s (left panel) and 496 s (right panel), for model A. 

In the text 
Fig. 3 Left panel: propagation of the convective front as a function of time, for models A, H, and I. Right panel: temperature profile versus radius at two different times, t = 0 s (solid line; T_{base} = 9.84 × 10^{7} K) and t = 496 s (dashed line; T_{base} = 1.64 × 10^{8} K), for model A. 

In the text 
Fig. 4
Left panel: time evolution of the nuclear energy generation rate (in erg s^{1}) for the 9 models computed in this work. Right panel: final CNO mass fraction versus radius. 

In the text 
Fig. 5
Snapshots of the ^{1}H (upper panels) and ^{12}C (middle panels) mass fractions at t ~ 395 s (model A; left panels), and 688 s (model I; right panels). Lower panels: the number of blocks administered, at this stage, is 3184 for model A, and 43 800 for model I. In both simulations, FLASH divides each block in 8 cells. Structures such as vortexs are better resolved in the finer resolution model I. 

In the text 
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