Simulations of a Supernova Imposter

The Fate of the Most Massive Stars ASP Conference Series, Vol. 332, 2005 Roberta M. Humphreys and Krzysztof Z. Stanek Simulations of a Supernova Imposter David Arnett, Casey Meakin, and Patrick A. Young Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson AZ 85721 Abstract. We discuss possible non-supernova eruptions in massive stars like η Carina, using a combination of stellar evolution and radiation hydrodynamics, to ascertain the essential underlying physics of such phenomena. A summary of a combination of numerical hydrodynamical simulations, in one, two, and three dimensions, is used to explore various facets of the problem. We apply insight from multidimensional simulations to both stellar evolution and stellar hydrodynamics in the case of a 120M star of solar abundances. We conclude that such stars should rapidly evolve to the redward of the Humphreys-Davidson limit, where they become vigorously unstable to density clumping and mass ejection, and predict CNO abundance differences in the clumpy and unclumped medium. 1. INTRODUCTION As discussed in a companion paper at this conference (Young 2004), individual eruptions of η Carinae analogues may have kinetic energies of > 10 50 ergs, within an order of magnitude of a core collapse supernova. In this paper we address the hydrodynamics of the onset of catastrophic eruptions of the most massive stars, the supernova imposters. 2. Physics of Massive Star One of the most massive of well-observed stellar objects is η Carinae. It represents an extreme case for stellar physics. The increasing importance of radiation pressure with increasing stellar mass is well known; it is accompanied by an increase in the importance of radiation energy relative to thermal energy in ions and electrons. This causes the adiabatic exponent for the radiation-gas mixture to approach its marginal stability value of γ = 4/3. It also implies that leakage of thermal energy is enhanced (for the sun only about a percent of the internal energy is in the form of radiation). As Cowling (1941) showed, the normal modes of stars in reaction to a perturbation are p-modes (sonic waves) and g-modes (gravity waves). In the compressible case, we have the Brunt-Väisälä frequency N 2 = gδ H P ( ad + ϕ δ µ), (1) from Kippenhahn & Weiget (1990), eq. 6.18, where the symbols have their usual meaning, or Hansen & Kawaler (1994), eq. 5.35 and 10.92. Notice than 75

76 Arnett, Meakin, and Young Figure 1. Density Perturbations from g-modes. For simplicity a 2D simulation is shown. The vertical dimension represents a fractional variation in density. The bottom part of the wedge is convective. At the transition to the convective stability boundary, density variation reach a maximum, due to the interaction of convective plumes with the elastic nonconvective region. Beyond this boundary density variations carried by internal waves may be seen. This simple idea solves much of the old problem in stellar evolution of what to do at the edge of convective zones. The interface deforms into a nonspherical surface, with mixing only occurring due to higher order processes (173). The importance of such effects increases with solar mass, and should be included in models of η Carina. the quantity in parenthesis is the Ledoux condition for convective instability. When this is negative, the linear solutions grow exponentially, giving rise to convection, and for the low viscosity of stellar plasma, turbulence. While the nonconvective case, in the linear limit, gives a discrete spectrum of wave solutions, the turbulence spectrum is continuous. Convection will drive wave motion in nonconvective region. For stellar convection, the temperature gradient is almost adiabatic, so N 2 gδ H P, (2) resulting in an impedence mismatch. This is less pronounced for the gravity modes (the internal waves in the stratified plasma) because of their lower fre-

SN Imposters 77 quencies, so convection drives internal waves. Recent 2D and 3D hydrodynamic simulations have suggested the deep importance of this coupling for stellar evolution (Young et al. 2001, 2004, Young & Arnett 2004), in addition to its diagnostic importance via stellar seismology. Internal waves become more important for larger stellar masses; their effects are subtle for the Sun. The Sun rings like a bell with sound waves. These, in combination with gravity and rotation determine the shape of stars. We concentrate on slowly rotating stars for our exploratation; this is an over-simplification (see Maeder and Meynet, this conference). On a longer time scale for most stars, radiative diffusion (and sometimes convection) transfer energy within the star. The sonic time scale is τ sound l/s, where l is a characteristic dimension of the region considered, and s is the sound speed. The radiative diffusion time scale is tau rad l 2 /λc = l 2 ρκ/c, where λ is the radiative mean free path, ρ the mass density, κ the opacity, and c the speed of light. While c s, this is generally overcome by l λ. For massive stars, the opacity approaches that for Thomson scattering, which is a minimum for ionized stellar plasma. The lowest densities are found in the envelopes of very extended stars, the red supergiants. Thus, for massive red supergiants, the time scale for sound travel can exceed that for the time scale from radiative heat transfer. This is the regime of the strange modes (see Glatzel, this conference). Traditional stellar pulsation is built on the notion of Eddington that τ sound τ rad, which is not true in the outer layers of such stars. This breakdown results in the development of density inversions (density increasing outward) in the envelopes of red giant models, and as we will argue, favors the formation of density inhomogeneities (clumping). 3. Progenitor Evolution and Initial Model The stellar evolutionary sequences were done with the TYCHO code (Young et al 2001, Young & Arnett 2004). Features of the calculations reported here are: The stars have masses of 120M, with solar abundances. A Kudritzki algorithm is used for blue supergiant mass loss (modelling line-driven radiative winds). The hydrostatic evolution is conducted with hydrodynamically consistent convective boundaries (mixing induced by internal waves) as one option (Young et al. 2004, Young & Arnett 2004). We find excellent agreement with observations of lower mass stars (0.4 < M/M < 23 (see Young et al 2001, Young & Arnett 2004, Young 2004). The hydrodynamic evolution is computed in one dimension (spherical symmetry), with identical microphysics physics as used in the evolutionary stages, but with dynamic convection and flux-limited radiative diffusion, The OPAL (Iglesias & Rogers 1996) and Alexander & Ferguson (1994) opacities, the OPAL, Timmes, and Arnett equations of state, and a 176 nucleus nuclear reaction network are used. Rotation is set to zero.

78 Arnett, Meakin, and Young Figure 2. Within the uncertainty about mixing, qualitatively different types of evolution are possible. The two tracks shown are identical except for mixing; after core hydrogen exhaustion they move to opposite sides of the HR diagram. The model with internal wave mixing goes to the red side. The most massive, well observed wide eclipsing are shown, as well as some Cepheids, to illustrate how far from standard stellar evolution conditions these models are. Figure 2 shows two trajectories for a 120M star in the HR diagram which differ by the treatment of mixing (both withing the range used in recent calculations). The paths diverge after hydrogen enhaustion, giving different mass loss histories. The model with wave induced mixing trepasses into the Humphreys- Davidson forbidden zone, while less mixing allows the star to return to the blue. Also plotted are data for some wide eclipsing binaries and some Cepheid variables; the 120M star lies far from the well-tested regions of the HR diagram. The red supergiant will be taken to be the initial model for further evolution with numerical hydrodynamics. The structure is shown in Figure 3. About 4M were lost in the blue supergiant stage (core hydrogen burning). Almost all the remaining 116M has been processed in the convective core by CNO burning, so by mass there is only a sliver (about 8M ) left which contains hydrogen. However, the radius of the helium core is < 6 10 11 cm, while the photosphere lies at a radius > 10 14 cm. By volume the star is hydrogen rich, even though the core contains 108M of nearly pure helium!

SN Imposters 79 Figure 3. Hydrogen versus mass coordinate, and hydrogen versus logarithm base 10 of the radius. The 108M core occupies almost no volume; the envelope is made of 8M that is hydrogen rich! 4. Hydrodynamic Experiments The characteristics of the hydrodynamic calculations are: The radial hydrodynamics was done on a lagrangian (comoving) grid, with pseudo-viscous smearing ofshocks. The nonradial hydrodynamics (convective mixing of heat) was done by integrating the buoyancy force against a drag term. The drag was chosen to reproduce mixing length theory as used in the hydrostatic evolution. These choices were guided by our multi-dimensional hydrodynamic simulations. Radiation flow was treated using flux-limited diffusion and the same opacities as before. The equation of state was as before, and partial ionization of hydrogen, helium and metals. No significant burning occurred, although our 176 element network was used. While the hydrodynamic treatment is oversimplified, it does allow us to examine some aspects of radial and nonradial flow in the context of a strong radiation field. The evolution of the core gives an increasing luminosity at the base of the hydrogen envelope. Although the increase is small, the hydrogen envelope is within a few percent of its limiting Eddington luminosity. Figure 4 shows the relaxation of the hydrodynamic model which occurs after rezoning in mass to optimize the numerical computation (right panel). We omit the helium core from the computational grid. The model quickly relaxes to a state near the one that the original zoning gave, and then simply remained hydrostatic. In the left panel, the effect of imposing a plausible increase in luminosity at the bottom of the hydrogen layer. We find that a small outburst results, which is shown in Figure 5. Without the increasing luminosity, this does not happen. It appears that a variety of

80 Arnett, Meakin, and Young Figure 4. Left panel: A slowly increasing luminosity is imposed at the base of the envelope. Right panel: Numerical transients quickly relax (in less that a day) to a state similar to the initial hydrostatic one and remain static for centuries of star time. Figure 5. Left panel: The onset of a small outburst, showing surface luminosity versus time. Right panel: Fine structure of the outburst shown on an expanded scale. The width of the pulse is about one day. modes happen to combine to give the outburst. The real situation will be more complex because we allow only radial modes. Is this the great outburst of η Car? Probably not; it is too small. However, it could be an example of what triggers a large outburst (Young 2004). Such nonlinear mixing of modes may be an inportant feature of massive red supergiants. Figure 6 shows snapshots of density and temperature as the envelope lifts off the helium core, which is indicated by the steep rise in both temperature and density at small radius. The density inversion at the outer edge of the envelope maintains its shape as a subsonic expansion occurs. The temperature maintains its photpspheric structure as well. At late time the region around the coreenvelope interface is poorly resolved; this is a worry, as this region may be important for driving by super-eddington radiative acceleration (Young 2004), as well as strange mode effects (see Glatzel, this conference).

SN Imposters 81 Figure 6. Left panel: Density versus log radius snapshots. The initial model shows the pronounced density inversion commonly found in models of giant stars. Right panel: Temperature versus log radius snapshots. The helium core lies inside the steep increase at small radius. Figure 7. Left panel: Pressure versus log radius snapshots. Right panel: Fluid velocity versus log radius snapshots. The flows are subsonic. Figure 7 shows snapshots of the pressure profile (left panel) which slowly changes its shape as the envelope lifts off, characteristic of subsonic flow. This is also indicated by the radial velocity snapshots seen in the right panel. However, the picture is more interesting if we examine the convective flow velocities, shown in Figure 8. The velocity scale is a factor of ten larger than in the left panel of Figure 7 for the radial velocities, and are supersonic. Our separation of radial and nonradial flows was predicated on the nonradial flows being a perturbation on the radial; this is not so. The system has pronounced nonradial flow at a epoch at which the thermal relaxation is faster than sonic effects. This is a certain recipe for clumping. Converging flows, from collisions of nonradial waves, will give compression, but efficient radiative flow will resist pressure increase. Similarly, shocks will radiate efficiently. Note that this occurs while still in the opaque limit, so that radiative diffusion is adequate. The convective velocities are higher inward, suggesting the onset of a stellar wind, which will poke through the lower density regions in the inhomogeneous shell. This is suggestive of a multiphase medium, with a higher entropy, faster,

82 Arnett, Meakin, and Young Figure 8. wind?). Convective velocity versus log radius snapshots (transition to lower density wind in one phase, and a slower, wind of dense clumps in another. Figure 9 shows the abundance structure. The outer shell is less process by CNO burning, but will clump and be overtaken by the N 14 enriched underlying material. The whole 8M envelope escapes in this simulation, leaving the 108M helium core, which is compact, and highly unstable to the epsilon mechanism (nuclear energized radial pulsations by core helium burning). We recall that 4M were shed by the star in its blue phase; this matter was still less processed by CNO burning. 5. Implications While these calculations suffer from two serious flaws, namely a lack of coupling between radial and nonradial flows and a developing lack of resolution at the core-envelope interface, they provide some clues for proceding to the next step and some robust implications. A significant amount of mass is lost, and there are abundance diagnostics from CNO burning.

SN Imposters 83 Figure 9. Detailed abundances versus log radius in the envelope. The outermost matter is most unstable to clumping, and has a different composition: less nitrogen production. Abundance differences are expected between the regions of strongest clumping and those of fastest expansion. There is a strong clumping instability, so radiation spikes are expected. Most of the action occurs in a full 3D hydrodynamics plus radiation diffusion regime. While stellar wind methods will be useful for diagnostics, the driving physics occurs in a non-steady state regime with optical depths greater than unity. The outburst is a hydrodynamic stellar interiors problem, which transitions later to a wind phase. Rotational geometry and a binary companion will have effects which modify the picture, perhaps in important ways. There is no SN imposter yet! However we do get an indication of outburst behavior, and the lack of resolution at the core-envelope interface may weaken this effect (Young (2004) and Glatzel, this conference). What will the He star do? While η Carina has not yet reached this phase, it seems likely to be dramatic, with the unveiling of a very hot and luminous

84 Arnett, Meakin, and Young helium star, unstable to nuclearly driven pulsations, within a fairly massive (12M ) nebula. This work was supported in part by a DOE grant to the University of Arizona, and a subcontract to the ASCI Flash Center at the University of Chicago. References Alexander, D. R. & Ferguson, J. W. 1994, ApJ,, 437, 879 Cowling, T. G., 1941, MNRAS, 101, 367 Hansen, C. J., & Kawaler, S. D., 1994, Stellar Interiors, Springer-Verlag Iglesias, C. & Rogers, F. J. 1996, ApJ,, 464, 943 Kippenhahn, R. & Weigert, A. 1990, Stellar Structure and Evolution, Springer-Verlag This conference. Young, P. A. and Arnett, D. 2004, ApJ,, accepted for publication Young, P. A., Knierman, K. A., Rigby, J. R., and Arnett, D., 2003, ApJ,, 595, 1114 Young, P. A., Mamajek, E. E., Arnett, D., & Liebert, J. 2001, ApJ,, 556, 230 Discussion Guzik: What is the temperature of the base of the envelope you re working with? Approximately? 1 Billion degrees? Arnett:It s in an interesting region and I don t really remember. It s of order 10 6. Maeder: At the beginning of your talk, you mentioned the gravity waves that you consider in the model. Could you comment a bit on the contribution of these gravitiy waves to the transport of chemical elements and to the transport of angular momentum is it significant or not? Arnett: I think we probably agree with you about the transport of angular momentum. Patrick and I wrote a paper where we looked at the lithium and beryllium depletion and we found that with the combination of what we were doing and what Tolonge (?) and Charbonell (?) were doing, we could get a complete picture. But it was important that the rotation both aided and inhibited convection and so we re in agreement there. I think that what happens with the gravity waves is even in the absence of rotation they enhance the mixing, particularily in semi-convection regions, because what s happening, is that you have this marginally stable part of the star which has a H gradient and you re pushing it at the bottom. The more massive the star, the less resistance to mixing it has, because the radiation pressure is going up and when you get to the big guys the effect is more important. Conversely, if we go to low mass stars then this prescription makes almost no difference. Patrick will talk about it later. Davidson: One of the many things we can see in η and can t see in other objects for just practical reasons is the existence of a lot of absurdly slow material. Speeds of 50 km s 1 which is not a natural velocity scale that I can think of in

SN Imposters 85 the the object. I don t know whether this is pertinent to your comment about slow ejected stuff but it might be. Arnett: That s what I m thinking. I m pretty sure that there is a dichotomy in the flow velocity, against low speeds and vice versa. What I don t know is what the number is. On the other hand, there is Ishibashi s little homonculus. It isn t quite that slow but it s pretty slow, 200 km s 1 or something like that? And that is also fairly slow. Most of what we ve discussed observationally today is material outside where this is going on. Townsend: Two questions. First, I might have missed the crucial part of your talk but, I m wondering are your simulations 1D, 2D, 3D? Arnett: All the above. What I m doing is piecing together stuff. So, we ve done 2 and 3 D and we have short parts of the stellar evolution cycle, analyzed that, determine that the gravity waves were a dominant mechanism for additional transport of mass and angular momentum. We put that back in the 1D code, evolved it and lo-and-behold it goes over to the rim. Very quickly. Then we look at that and say, well you know it s above 95% of the Eddington limit and what happens if we just let it go? If we let it go and we don t change the luminosity, it just sits there. We let it go and we raise the luminosity, this is what you get. In 1 D. And then I speculate on what I guess would happen in 2 D. Townsend: You already answered my second question which was on whether gravity waves play an important part for transport of angular momentum. You said yes. Is this from your hydro simulations in the 80 s? Arnett: No this is from our hydro simulations. What we re doing is still angular momentum. When we do it without rotation we still have angular momentum in the radial plane so, we re right down to the same equations.