*5.1. The Modified Eddington Limit*

Giant eruptions exceed the Eddington Limit. Thus it seems natural to guess that the progenitor stars were very massive, close to the Limit, and vulnerable to the same instabilities as LBV's. This connection between LBV's and giant eruptions may be illusory, but thinking about it leads to some useful ideas.

Observed *L*/*M* ratios are clues to the LBV phenomenon in two different ways. First, of course, their proximity to the Eddington Limit (Section 3.1 above) implies a large role for radiation pressure. Second and less obvious, the distinction between two classes of LBV's favors an instability in the outer layers, not the stellar core. Types LBV1 and LBV2 have different central structures because they represent different stages of evolution (Section 3.1 above and [12]). The stars' outer 3% of mass, however, have similar radiation/gas ratios for both classes, and similar variations of opacity.

In the classical Eddington Limit

$$\left(\frac{L}{M}\right)\_{\text{Edd}} \equiv \frac{4\pi \text{cG}}{\kappa\_{\text{sc}}}\,\text{.}\tag{3}$$

the opacity *κ*sc includes only Thomson scattering by free electrons. Absorption opacity *<sup>κ</sup>*abs practically vanishes as a static photosphere approaches the Limit, because density *ρ* becomes very low. The classical Eddington parameter ΓEdd ≡ *<sup>κ</sup>*sc*L*/4*πcGM* can thereby approach 1 in an old-fashioned radiative atmosphere model. However, the photospheric *<sup>κ</sup>*abs may be appreciable in a model with ΓEdd < 0.95, and deeper layers may have larger opacities in any case. During the 1980's this thought inspired the idea of a "Modified Eddington Limit" that takes *<sup>κ</sup>*abs into account [1,98–103]. It was an empirical hypothesis, not a theoretical prediction, motivated by *η* Car and observed LBV behavior.

Strictly speaking, there are two forms of Modified Eddington Limit. (1) It might be a well-defined limit to the allowed values of *L*/*M* in a static stellar model, like the classical limit but including realistic convection, incipient porosity, etc. (2) Or, more likely, it may signify an instability that arises when *L*/*M* exceeds some value, see Section 5.2 in [1]. In either case the critical *L*/*M* depends on opacities in the outer 3% of the star's mass, or maybe the outer 1%.

Rapid rotation reduces the effective gravity mass, thereby altering any form of Modified Eddington Limit. The terms " Ω limit" and/or " Ω*L* limit" allude to this obvious fact, but the implications are often oversimplified. Two decidedly non-trivial subtleties occur: (1) Rotation causes a star's subsurface temperatures to depend on latitude. Resulting alterations of opacity affect the topics of Section 5.2 and 5.3 below. (2) The specific angular momentum expelled in a giant eruption may be either larger or smaller than in the underlying layers. Consequently the effects of rotation may evolve during the eruption [58].

### *5.2. The Photosphere, Bistability, and Surface Activity*

The easiest place to start is the photosphere. Traditionally, the total energy flow in a stellar interior can exceed 4 *<sup>π</sup>cGM*/*<sup>κ</sup>*tot by inciting convection [104]. But convection becomes inefficient in the photosphere, so radiation must carry nearly all of the energy flux there. Imagine a model wherein *κ*tot increases inward, and radiative forces are less than gravity outside some radius *rc*. Inside *rc*, convection carries the excess energy flux. An increase in *L*/*M* presumably causes *rc* to move outward relative to the stellar material. For some value of *L*/*M*, *rc* moves into the photosphere; so there may be a practical limit, somewhat smaller than (*L*/*M*)Edd. By extrapolating normal atmosphere models, one can identify a limiting *L*/*M* around 0.9 (*L*/*M*)Edd [105–107]. But the reasoning is dangerously subtle, and a different approach suggests that a star may become unstable at a smaller value of *L*/*M*, see below.

Now suppose that "Modified Eddington Limit" connotes an instability that arises somewhere in the range 0.5 < Γ < 0.9, rather than a well-defined static limit. At relevant photospheric densities, *κ*tot has a maximum in the vicinity of *T* ∼ 13000 K, involving the ionization ratio Fe++/Fe<sup>+</sup>. Consequently a high-Γ atmosphere may be very unstable in a particular range of *T* around that maximum, and might act as a relaxation oscillator jumping back and forth across the unstable range [99,102].

Behavior like that is observed at a somewhat higher temperature [15,18,108,109]. Consider a standard hot line-driven wind model wherein the star gradually becomes cooler. As *T*eff declines below 20000 K, Fe++ and other suitable ion species become numerous enough to drastically increase *κ*tot; so the wind becomes slower and much denser. The transition occurs across a narrow range of *T*, hence the term "bistability jump." It was noted around 1990 as a likely cause of LBV events [18,108]. That idea originally meant a difference between two outflow states, but it rapidly evolved into a bistability between two quasi-static states of the star's outer layers [25,88–91]. One state corresponds to a quiescent LBV, the other occurs during an LBV eruption, and intermediate states are more unstable due to the opacity maximum mentioned earlier. In the eruptive state, the outer layers are greatly expanded or "inflated."

But a set of inflated and non-inflated models does not constitute a theory of LBV variability; instead it plays a role more like an existence theorem in mathematics. A proper theory must acknowledge the following questions.


The last item pertains to giant eruptions. In order to expel a mass which greatly exceeds that of the unstable region, the process must be like a geyser: instability begins at the top and moves downward (relative to the material) until some factor stops it. In this way a photospheric instability might even cause a giant eruption. As outer layers depart, a large reservoir of radiative energy is progressively uncovered. At any given time the configuration resembles a steady-state model, since the observed timescale is much longer than the dynamical timescale. Presumably the eruption ends when conditions change at the base of the flow – perhaps when it reaches some particular feature in the pre-eruption interior structure.

Another form of Modified Eddington Limit relates to dynamical processes rather than the temperature dependence of opacity. A static atmosphere dominated by radiation pressure tends to develop inhomogeneities, granulation, and porosity like an outflow; see [84,86] and many refs. therein. Resulting turbulence can engender MHD effects, even though the photosphere is well above the temperatures traditionally associated with stellar activity. These phenomena may influence the outflow rate, and might even determine it. Conceivably, *η* Car's dense wind a century ago [62] may have involved stellar activity analogous to a red supergiant! [110].

Most of the above possibilities are not mutually exclusive.
