*4.1. The Continuum*

The apparent radiation temperature of an opaque outflow can be defined in various ways—e.g., based on the photon-energy distribution of the emergen<sup>t</sup> continuum, or its slope at selected wavelengths, or on subsets of absorption features, etc. These alternative *T*'s can differ by 20%, leading to confusion when one attempts to compare values quoted in papers. "Effective temperature" *T*eff used for stellar atmospheres is not appropriate, because an outflow has no fundamental reference radius that is meaningful for that purpose. (Also note that an optical depth value of 2/3 has no significance in this context. Regarding photon escape probabilities, *τ*tot ∼ 1.0 to 1.3 in a diffuse outflow corresponds to *τ*tot ≈ 2/3 in a plane-parallel atmosphere.)

And we must be careful with the word "photosphere." The region with optical depth *<sup>τ</sup>*tot(*r*) ∼ 1 has little effect on an outflow's emergen<sup>t</sup> photon energy distribution, because the dominant opacity is usually Thomson scattering by free electrons. That process has only a weak effect on photon energies. Consider instead a deeper region where absorption and re-emission events are frequent enough to establish *<sup>T</sup>*gas ≈ *T*radiation. Outside some radius *r*esc, the average photon escapes via multiple scattering before it experiences an absorption event. Evidently the emergen<sup>t</sup> photon energy distribution depends mainly on temperatures that exist just inside radius *r*esc. In this overview "photosphere" means that region.

Classical diffusion theory gives the approximate size of *r*esc [70,71]. Suppose that local opacities for absorption, scattering, and their sum are *<sup>κ</sup>*abs, *κ*sc, and *κ*tot, averaged over photon energies in some optimal way. Define a "thermalization opacity"

$$\kappa\_{\rm th}(r) \equiv \left[ \Im \kappa\_{\rm tot}(r) \kappa\_{\rm abs}(r) \right]^{1/2},\tag{1}$$

with associated optical depth

$$
\pi\_{\rm th}(r) = \int\_r^{\infty} \rho(r') \kappa\_{\rm th}(r') \, dr' \, . \tag{2}
$$

Often called thermalization depth or diffusion depth, *<sup>τ</sup>*th(*r*) is typically of the order of 0.6 *<sup>τ</sup>*tot(*r*) in a giant eruption or an LBV event photosphere. Calculations show that *r*esc is approximately the radius where *τ*th = 1 [70,71], and we can regard the photosphere as the region where 1 < *τ*th < 2. This is not a formal statement, but in practice it applies for any reasonable density law *ρ*(*r*) and for large as well as small opacity ratios *<sup>κ</sup>*abs/*<sup>κ</sup>*sc. The emergen<sup>t</sup> continuum is created mostly at *τ*th ≈ 1.5 to 2.0, while absorption and emission lines are formed mainly at *τ*th < 1 or perhaps *τ*th < 1.5. If *T*1 and *T*2 are the temperatures at *τ*th = 1 and 2, then we can liken *T*1 to the *T*eff of a star with a similar spectrum, though their values may disagree because they are defined differently. Caveat: In published models of opaque winds, most authors define the photospheric radius by *τ*tot = 1 or even *τ*tot = 2/3, rather than *τ*th = 1. With those choices, a quoted "photosphere temperature" is cooler than the emergen<sup>t</sup> distribution of photon energies.

In a simple model where opacity depends only on *ρ* and *T*, the temperature at a given location depends approximately on two quantities, *τ*th and *MV*˙ −1*L*−0.67 where *V* is the local outflow velocity [71,72]. Figure 4 shows examples of *T*1 and *T*2 in spherical outflows. Corresponding radii are shown in Figure 5. These sketches are intended only for conceptual purposes; they are based on simplified models that ignore some major details (see below).

**Figure 4.** Photosphere temperatures in simplified opaque outflow models with *L* = 10<sup>6</sup> and 10<sup>8</sup> *L* and *V* = 300 km s<sup>−</sup>1. The curves show temperatures corresponding to thermalization depths of 1 and 2. *<sup>T</sup>*(*<sup>τ</sup>*th) depends approximately on the quantity *MV*˙ −1*L*−0.67. Temperature values here are imprecise and very likely overestimated, because the models are highly idealized; see text.

**Figure 5.** Radii of the photospheric locations shown in Figure 4. For large *M* ˙ the photosphere becomes geometrically thin (small Δ*r*/*r*) and thus resembles a plane-parallel model.

With Figure 4 in view, imagine a case with constant luminosity *L* while *M* ˙ /*V* gradually increases. Initially the flow is transparent so our model does not apply. But when *M* ˙ /*V* becomes large enough to be opaque, then it determines the apparent temperature. Increasing *M* ˙ /*V* causes the photosphere to move outward, and *T*1 ∝ roughly (*M*˙ /*V*)−0.3 as shown in the upper half of Figure 4. Below 9000 K, however, the proportionality changes to (*M*˙ /*V*)−0.07 because the opacities decline rapidly. *T*1 < 7000 K requires a very large flow density. As noted above, *T*1 is a fair indicator of the absorption and emission lines – except for a caveat in Section 4.3 below.

(This paragraph concerns technicalities that don't affect the main concepts.) Each temperature in Figure 4 refers to a location in the flow, which does not represent any specific observable quantity. For comparison with observations, one would need to calculate emergen<sup>t</sup> radiation in a manner resembling [71] and [73]; but a model with wavelength-dependent opacities would be much better. Figure 4 is based on many simplified models with constant luminosities *L*, mass-loss rates *M*˙ , and flow velocities *V*. It was assumed that *<sup>T</sup>*gas(*r*) = *<sup>T</sup>*rad(*r*), with LTE Rosseland mean opacities which are readily available (http://cdsweb.u-strasbg.fr/topbase/, [74], and refs. therein). Hydrogen and helium mass fractions were *X* = 0.50 and *Y* = 0.48. Integrating the spherical radiative transfer equations [73] with those opacities, we can calculate *<sup>T</sup>*(*<sup>τ</sup>*th). One recent analysis [72] appears at first sight to favor lower temperatures than Figure 4, but the disagreements involve the distinction between *τ*th and *τ*tot, and varying definitions of the observed *T*. Very likely the true ionization in the outer regions is larger than the LTE values; if so then the temperatures in Figure 4 are overestimates. Errors of that type are probably comparable to the differences between alternative definitions of *T* for the emergen<sup>t</sup> radiation.

Figure 4 is fairly consistent with observed giant eruptions, such as *η* Car marked in Figure 1. Exceptionally high flow densities occurred in *η* Car's 1830–1860 Great Eruption, with *M*˙ > 1 *M* y<sup>−</sup><sup>1</sup> and *L* ∼ 3 × 10<sup>7</sup> *L* [57]; so it should have had exceptionally low photospheric temperatures. Reflected light echos provide spectra of that event [75–77]. The light-echo researchers deduced a temperature near 5000 K, but that was an informal value, not well-defined and not based on a quantitative analysis. Judging from the published description and reasons noted in Section 4.3 below, the spectrum most likely indicated *T*1 ∼ 5500 to 6500 K in Figure 4 [78]. (Recall that *T*1 denotes the temperature at a particular location in the outflow, not an emergen<sup>t</sup> radiation temperature.) If we define "apparent temperature" in a different way, its value may have been as low as 5000 K [72]. A second, lesser eruption of *η* Car in the 1890's had *T*1 ∼ 7500 K, according to the earliest spectrogram of this object [62]. Several extragalactic giant eruptions have been observed since 2000, usually showing *T*1 > 8000 K (Section 3.2 above). LBV outbursts have much smaller luminosities, and usually reach 8000–9000 K like AG Car in Figure 1. The assumptions in Figure 4 are not valid for them, because their photospheres are not located in the high-speed part of the flow (Section 4.5 below). However, those observed minimum temperatures of LBV's are very likely determined by the rapid decline of opacity below 9000 K, even though the photospheres resemble static atmospheres.

In summary, there is no known observational reason to doubt the general appearance of Figure 4 for opaque radiation-driven outflows.
