*4.2. Distinctive Emission Line Profiles*

The brightest emission lines from an opaque outflow generally have a certain type of profile, illustrated in Figure 6. Smooth broad line wings extend beyond ±2000 km s<sup>−</sup><sup>1</sup> even though the wind speed is less than 700 km s<sup>−</sup>1; and the longer-wavelength side is stronger. These are classic signs of Thomson scattering by free electrons [42,44,79–81]. Since the electrons have r.m.s. thermal speeds of the order of 600 km s<sup>−</sup>1, some of the photons acquire large Doppler shifts in multiple scattering events before they escape. Meanwhile, expansion of the outflowing material favors shifts toward longer wavelengths. Obviously the resulting profile depends on *<sup>τ</sup>*scem, the line-emitting region's average optical depth for Thomson scattering. The shape in Figure 6 indicates *<sup>τ</sup>*scem ∼ 0.5 to 2, and appears to be generic. It specifically represents SN 2011ht [42], but SN 1994w exhibited a similar H*α* profile, and so do other giant eruptions and *η* Car's dense wind [44,63,82].

The moderate size of *<sup>τ</sup>*scem has a simple explanation. In typical eruptions with *T*1 > 7500 K, *<sup>κ</sup>*sc/*<sup>κ</sup>*th ∼ 1 to 2—a consequence of atomic physics. Hence the continuum photosphere boundary *τ*th ≈ 1 automatically has *τ*sc ∼ 1 to 2. Since the emission lines are formed outside the photosphere, we therefore expect *<sup>τ</sup>*scem ∼ 1, or perhaps a little smaller, for bright emission lines. The main point here is that an observed profile like Figure 6 is good evidence for an opaque or semi-opaque outflow. It recognizably differs from emission lines seen in normal stellar winds, expanding shells, nebulae, and supernova remnants.

**Figure 6.** Emission line profile with moderate Thomson scattering. The dashed curve on the left side is a mirror image of the right side. Note that the line wings extend far beyond the velocity indicated by P Cyg absorption. This example is H*α* in the radiation-driven outflow of SN 2011ht [42], but other giant eruptions produce similar line shapes.

(Caveat: In some papers a Thomson-scattered line profile is called "Lorentzian," often without recognizing its significance. That usage gives a flatly wrong impression in two respects. First, in physics the word "Lorentzian" has very specific connotations: 1/(1 + *x*<sup>2</sup>) = the Fourier transform of an exponential decay, the natural shape of an idealized spectral line, closely related to the uncertainty principle. None of these applies to the shape in Figure 6. Second, the wings of a true Lorentz profile are like *x*<sup>−</sup><sup>2</sup> but the wings of a Thomson-scattered profile are like *<sup>e</sup>*<sup>−</sup>*<sup>α</sup>*|*x*|. This difference is fundamental, not just a matter of opinion.)

### *4.3. Cautionary Remarks about Absorption Features*

An opaque outflow also produces absorption lines, but they cannot safely be compared with stellar spectral types. The most dramatic example concerns *η* Car's Great Eruption. Spectra of that event have been obtained via light echos, leading to an estimate *T* ∼ 5000 K which seemed to contradict an expected value of 7500 K [75–77]. But that conclusion had two disabling flaws. (1) In fact the expected value was far below 7500 K [71,72,78]. (2) More pertinent here, spectral classification standards for stars do not apply to a mass outflow. For instance the light-echo spectra of *η* Car's Great Eruption showed absorption features of CN, which would indicate *T*eff < 5000 K in a star—but they may occur in an outflow with *T*1 ∼ 6000 K.

Figure 7 shows why. Each curve represents the column density *ρ dr* of material cooler than *T*. A stellar amosphere with *T*eff ≈ 6000 K has almost no material below 5000 K, but a diffuse outflow with *T*1 ≈ 6000 K can have an appreciable amount of cooler gas at large radii. This difference is a consequence of two facts: (1) the mass distribution in an outflow resembles a power law *ρ*(*r*) ∝ *r*<sup>−</sup>*<sup>n</sup>* instead of the exponential *ρ*(*z*) ∝ *e*<sup>−</sup>*<sup>z</sup>*/*<sup>h</sup>* that roughly describes a stellar atmosphere, and (2) Radiation density in an outflow has a 1/*r*<sup>2</sup> "dilution factor." Figure 7 is merely schematic, but it suggests that an eruption with *T*1 ∼ 6000 K can form cool spectral features such as CN in its outer regions. Absorption lines formed at smaller radii may be good indicators of *T*1, but they must be chosen carefully. Since LTE is a poor approximation for *T* < *T*1, and there are other complications, a realistic model of the absorption line spectrum will be extremely complex (see below). Meanwhile, so far as available information allows us to judge, the light-echo spectra of *η* Car's eruption appear consistent with standard portrayals of that event.

**Figure 7.** Sketch of the mass column density at temperatures below *T*, in a stellar atmosphere and in a dense outflow. These curves are based merely on idealized textbook-style models of *ρ*(*r*) and *<sup>T</sup>*(*r*), but the difference between them is qualitatively valid.

### *4.4. Why a Real Outflow Spectrum is Exceedingly Difficult to Calculate*

The dependence of *T*1 on *M* ˙ was first described long ago [71]. Figure 4 and ref. [72] employ modernized opacities, but the resulting quantities are highly imprecise because the models are highly simplified. A truly realistic calculation must include all of the following complications.


Omitting any of these complications may cause the results to be almost as inaccurate as the simplified models in Figure 4. Some existing codes employ elaborate radiative transfer with some NLTE effects, but there are reasons for skepticism. Items 3, 4, and 5 have multiple undetermined free parameters, not advertised in most research papers. Items 3 and 4 acting together may invalidate the radiative transfer methods for spectral lines. Most important, the realism of a calculation is very difficult to test. Approximately matching an observed spectrum does not prove correctness, since there are enough free parameters to compensate for omitted effects. In summary, existing codes illustrate most of the chief processes, but they must not be regarded as decisive or authoritative. Their uncertainties may be far worse than most authors assume.
