*5.3. The Iron Opacity Peak*

The "iron opacity peak" locale in a star, described below, is probably crucial; but its instabilities are too complex for simple analysis, math expressions, and predictions. A decisive analysis will require numerous 3-D simulations which have not been feasible so far.

Opacity has a dramatic maximum at temperatures around 180,000 K, for reasons concerning ionization stages of iron. In a typical LBV-like very massive star, *κ*tot > 2 *κ*sc throughout a temperature range such as 100,000 to 300,000 K, though the actual limits depend on mass density. Vigorous convection occurs there because *<sup>κ</sup>*tot*L* obviously exceeds 4 *<sup>π</sup>cGM*, if *L* signifies the total energy flow. Such a region offers a zoo of instabilities, and dynamically it decouples the outer layers from the stellar interior. Since the associated mass and energy greatly exceed the photosphere, this region is the most promising part of the star for eruption mechanisms. Its usual name, the iron opacity peak zone, might be confused with the iron peak of cosmic abundances; and "iron opacity bump zone" is both inelegant and cumbersome. For convenience, an ugly acronym will be used here: OPR = iron opacity peak region in the star. Although it usually occurs in the outer 1% of the star's mass distribution, its spatial radius may be considerably smaller than the stellar radius *R*. A second opacity peak will also be mentioned, involving helium at lower temperatures.

The OPR mass and energy are difficult to estimate from observational data. If *μ*(*T*) is the mass column density of layers cooler than *T*, and radiation pressure dominates, then *μ* ≈ *<sup>P</sup>*/*g* ∼ *aT*4/3*g* so

$$m(T) \sim 4\pi R\_T^2 \mu(T) \sim \frac{4\pi R\_T^4 a T^4}{3GM\_\*},\tag{4}$$

where *RT* is a radius that has temperature ≈ 0.8 *T*. But the *R*<sup>4</sup> *T* factor is quite uncertain, because *RT* may lie deep within an extended envelope. Consider, for instance, *η* Car before its giant eruption. If we know only that *M*∗ ≈ 150 *M*, *L* ≈ 4 × 10<sup>6</sup> *L*, and *T*eff ≈ 20,000 to 25,000 K [58], then the mass in the temperature range 100,000–300,000 K may have been anywhere in the range 0.002 to 0.1 *M*. The thermal timescale for this OPR might have any value ranging from a few days to a few months, depending partly on how we define it. Given these strong dependences, the effects of OPR instabilities may be very sensitive to the evolutionary state and structure of the star—and thus consistent with observed facts about giant eruptions and LBV's.

A standard LBV eruption might expel no more than the OPR mass, and the two amounts may even be related. But a giant eruption rooted in that region must be geyser-like (Section 5.2). Some forms of instability cannot easily function like geysers, for reasons involving timescales – see a remark later below.

Since 1993, almost every stability analysis of very massive stars has emphasized "strange modes" of pulsation [36,40,97,111–113]. Apart from mathematical details, they have the following attributes.


These characteristics are almost perfectly suited to the OPR in a star near the Eddington Limit. Item 3 causes the local mass density to be relatively low, thereby enabling item 2. For a very brief account of strange modes, see [112].

Altogether, then, in a star near the Eddington Limit, the OPR forms a queasy sort of cavity between the stellar interior and the outer layers—with strong consequences for pulsation modes. Even if we consider only 1-D radial motions, gas-dynamical simulations reveal phenomena that appear crucial for LBV's and giant eruptions [39,85,113–115]. An essential factor is the time dependence of convection. Normally a massive stellar interior obeys the Eddington Limit by shifting some of the energy flux to convection where necessary [104]. But this assumption fails in a structure that changes rapidly, e.g., in pulsating layers. Convection needs some time to develop, and the dominant convective cells have finite turnover times. Hence the convective energy flux lags behind the total energy flux, especially in the circumstances listed above for strange modes. As explained in the papers cited above, this fact causes the radiative flux to exceed the Eddington Limit at some times and places in a pulsation cycle. No actual runaway outburst occurred in the simulations, but their boundary conditions and lack of non-radial modes may have inhibited such a development.

Three-dimensional simulations show the spatial fluctuations of convection, and reveal some opacity-related phenomena that cannot appear in the 1-D models [34,110,116]. For instance, helium opacity can become large within clumps of gas that have been lifted to regions with *T* < 70,000 K [34]. The result is a second opacity-peak region, indirectly caused by the iron opacity bump. Local regions in and below the photosphere can thus have large radiative accelerations. The outer layers become supersonically turbulent, and local parcels of mass can be ejected in a chaotic way. In this manner we begin to graduate from "pulsations" to "stellar activity" or even "weather"— see Figure 2 in ref. [34]. Unfortunately, the 3-D calculations are so expensive in CPU time that only a few have been attempted.

Given the facts outlined above, the OPR is very likely the root of the LBV phenomenon. It is especially dramatic in stars with LBV-like *L*/*M* ratios, and it is rich in phenomena that appear relevant to the questions in Section 5.2 above. Moreover, effects found in numerical simulations can help to accelerate the ejecta. Therefore, contrary to most papers in this topic, we should not assume that LBV outflows are merely line-driven winds—especially during a major outburst (cf. [117]).

But can the OPR incite a giant eruption? No simulation has ye<sup>t</sup> produced an outright eruption. Maybe this is so because the "weather" analogy is apt! A terrestrial atmosphere simulation would usually go for a long time before it produces a typhoon. By analogy, perhaps a stellar eruption results from an infrequent coincidence of several chaotic processes—a Perfect Storm. Note that the inflated

LBV model in [34] was still expanding when the calculations ended after 700 dynamical timescales, only a few percent of a typical event duration.

As mentioned earlier, if a giant eruption can originate in the OPR layers of the star, then it must be a geyser-style process with instability propagating downward through the stellar layers—or rather, the successive layers move outward past the instability zone. The energy budget thereby becomes complicated, because inner regions tend to contract in order to compensate for the lost energy. As noted by [117], the resulting small increase in local temperature can increase nuclear reaction rates; so the overall event may be indirectly powered by hydrogen burning. Nearly all of the mass is close to dynamical equilibrium throughout this process, but thermal equilibrium fails in the outer regions. This story may lend itself to additional instabilities deep within the star.

Unfortunately the geyser analogy may fail for some types of OPR pulsational instability [118,119]. When a pulse of material has been expelled, the driving mechanism needs time to re-establish itself, and that time may be much longer than the dynamical timescale. In that case the instability cannot easily propagate through deeper layers.

At first sight, a supernova precursor eruption (Section 3.2) cannot originate in the OPR, because such events happen only a few years before core collapse, and the outer layers evolve much slower than that. In the outer layers, there is nothing special about the core's last few years. But this view may be too naive, for reasons noted in [120]. During those final years, turbulence in the core can generate unsteady burning and outward waves, which tend to expand the outer layers—"an early warning system for core collapse." The OPR is so sensitive that it may respond violently to even a small change in the outer-layer structure. Thus it seems conceivable that the opacity peak might play a role in every class of eruption from LBV events to pre-SN outbursts.

### *5.4. Instabilities in and Near the Stellar Core*

Some giant eruptions probably originate near the centers of massive stars, rather than in the OPR. But the definite examples concern true supernovae in special circumstances, and the nature of SN impostors (i.e., giant eruptions that are not related to SN events) remains murky.

A supernova can produce a radiation-driven eruption instead of a visible blast wave. Suppose that a star produces an opaque mass outflow in the years preceding its SN explosion. In that case, when the SN blast wave emerges from the star and moves into the surrounding opaque ejecta, photons may diffuse outward faster than the shock speed [121–123]. Radiation thus reaches the *τ* ∼ 1 radius substantially before the shock does; indeed the shock may emerge long after the time of maximum light. The visible event represents "photon breakout" rather than "shock breakout." Maximum luminosity is far above the Eddington Limit.

The photon diffusion rate can be described in terms of a random walk, but the familiar version of that concept doesn't give a unique diffusion speed for comparison with the SN shock speed. Instead, here's a formal example with an constant diffusion speed. Consider pure scattering in a spherical configuration; absorption and re-emission are equivalent to scattering so far as the total energy flux is concerned. Suppose that the scattering coefficient is *k*(*r*) = *ζ*/*<sup>r</sup>*, with a constant parameter *ζ*. (In the notation of Section 4 above, *k* = *ρκ*.) In this case the time-dependent diffusion equation has a similarity solution that represents an expanding pulse of radiation density:

$$\mathcal{U}(r,t) = \left(\frac{E}{8\pi}\right) \left(\frac{3\mathcal{J}}{ct}\right)^3 \exp\left(-\frac{3\mathcal{J}}{ct}\right) \,,\tag{5}$$

which has total energy *E*. Because of the choice *k* ∝ *r*<sup>−</sup>1, this expression contains a velocity-like ratio *<sup>r</sup>*/*t*. At any given location *r*, the maximum radiation flux occurs at *t* = 0.75*ζr*/*c* when about 24% of the energy has passed. At any given time, half of the radiation is located outside radius *<sup>r</sup>*1/2 ≈ 0.9*ct*/*ζ*; so the median diffusion speed is approximately 0.9*c*/*ζ*. About 10% of the radiation energy moves outward faster than 1.8*c*/*ζ*. If *ζ* is small enough for this speed to outrun the SN blast wave, but large

enough to make the pre-SN outflow opaque – say 1 < *ζ* < 40—then a radiation-driven eruption rapidly develops.

In a more realistic case with *k*(*r*) ∝ *r*<sup>−</sup><sup>2</sup> rather than *r*<sup>−</sup>1, the diffusion speed accelerates outward. The light curve can resemble Figure 2, with a sudden decline after most of the radiation has passed through the photosphere. Meanwhile, of course, the radiation accelerates the mass outflow. Later the SN blast wave may emerge after the brightness has declined, with only a modest display. Thus SN 2011ht, for instance, may have been either a true supernova with a hidden shock, or an impostor with no shock [42].

One point about shrouded supernovae is so obvious that it is often underemphasized: *the required circumstellar material was probably ejected in one or more giant eruptions* with *M*˙ > 10−<sup>3</sup> *M* y<sup>−</sup>1, years or decades before the core-collapse events (Section 3.2 above). Many researchers assume that the pre-SN stars were LBV's, because LBV's are the best-advertised eruptors. But this surmise is not entirely consistent, because the deduced amount of ejecta usually surpasses the familiar type of major LBV eruption [42]. A giant LBV event (Sections 3.1 and 5.2 above) would be needed—i.e., much stronger than any LBV outburst observed in the past few decades. If such large eruptions really do occur as part of the general LBV story, they must be very infrequent. Thus we should be very surprised if several known SN events were closely preceded by random LBV episodes on that scale. It seems far more likely that the pre-SN outbursts were somehow related to the imminent core collapse, i.e., related to the core structure. Hence the deduced pre-SN mass ejection probably had nothing to do with standard LBV behavior. Those stars may have been LBV's, but there is no good reason to assume that they were. The precursor events may have resembled the outbursts of SN 2009ip (Section 3.2 above), but with longer time scales.

Pulsational pair instability attracted attention a decade ago with reference to supernova impostors [124–127], because it can produce repeated eruptions. A star with initial mass around 150 *M* eventually becomes a pair-production supernova, wherein core temperatures rise high enough to produce a significant rate of *γ* + *γ* → *e*<sup>−</sup> + *<sup>e</sup>*+. This transfer of energy to rest mass causes a pressure deficit, while the adiabatic index falls well below 4/3 which implies dynamical instability. Hence the core begins to collapse, raising the temperature so the pair creation accelerates, and runaway nuclear reactions unbind the whole star. But if the star's mass is somewhat smaller, then the central region stabilizes before it is entirely disrupted, and the episode can repeat. This repetition motivates the term "pulsational" instability. It must be very rare because it occurs only in near-terminal stages of very massive stars. The phenomenon seems too indeterminate to be really satisfying; the time interval between events is extremely sensitive to obscure details, and the first such event probably expels all the hydrogen. For the latter reason, supernova impostors such as *η* Car presumably did not involve this type of event. Apart from having too many syllables, the main fault of pulsational pair instability is the difficulty of making definite statements about it.

Parallel to the computational developments noted in Section 5.3, 3-D simulations have revealed new phenomena in the star's core region. An important fact is that some numerical techniques, especially in 1-D models, entail artificial (i.e., illusory) damping of fluctuations. 3-D convection and turbulence become particularly vigorous during a massive star's final years [120,128], with dynamic effects that cannot be represented in 1-D calculations. Turbulence generates gasdynamic waves, which carry energy outward. Consequently the outer layers, feebly bound because they are close to the Eddington Limit, expand or perhaps even erupt. Mass ejection may occur [120,129,130], while the turbulence also causes the nuclear burning to be unsteady or even explosive. The outer layers are quite vulnerable because their binding energy is much smaller than the nuclear energy being processed in the central region. As mentioned earlier, the opacity-peak region may produce enhanced instabilities because of the waves flowing through it. Given these circumstances, perhaps we should not be surprised that paroxysms occur just before core collapse.

What can we say about core-based eruptions that are *not* related to a SN event? The processes mentioned above would not be suitable. Eta Carinae, for instance, still has considerable hydrogen even after its Great Eruption. Evidently it has not ye<sup>t</sup> evolved far enough to have an exotic core region. It probably has a very capable opacity peak region, but doubts about the geyser process (see above) may require a core-region instability instead. One credible possibility has been suggested in refs. [113,118,119]. In a very massive, moderately evolved star, gravity pulsation modes (like ocean waves rather than pressure waves) may become numerous and strong at the lower boundary of the region that still has some hydrogen. Suppose that they grow enough to mix some hydrogen into the hot dense zones below that boundary. The resulting burst of hydrogen-burning would rapidly lift some material, possibly ejecting a set of outer layers, and then the remaining material would settle down. Events of this type may recur on a thermal timescale, reasonable for an object like *η* Car. Some remarks in [117], concerning enhanced reaction rates when a star's total energy has been reduced by mass ejection, may be relevant to this idea.

Explorations of core instabilities have naturally concentrated on the final pre-SN state, because the structure is highly complex then and because SN-related processes are most fashionable. With the development of 3-D computation, however, unpredicted phenomena may appear at earlier stages of evolution; anyway that's what we need for giant eruptions if the opacity-peak region turns out to be inadequate.
