η Carinae's Dusty Homunculus Nebula from Near-infrared to Submillimeter Wavelengths: Mass, Composition, and Evidence for Fading Opacity

2.1.1. SWS 2.4–45.2 $\mu m$

The ISO spectroscopic observations used here have been first published by MWB99, who did not provide data reduction details because of journal brevity constraints. The observations over the 2.4–45.2 μm range were obtained on 1996 January 27 with the SWS (de Graauw et al. 1996) using the Astronomical Observing Template (AOT) S01 grating scan at the slowest scan rate (speed 4), providing a spectral resolution $\lambda /{\rm{\Delta }}\lambda$ in the range of 800–2000. Two complete scans were performed, in forward and reverse over the full grating ranges arranged by the four optically independent detector modules, each with their own throughput aperture, and individual sub-bands corresponding to spectral order separation.

Basic data processing was carried out with the ISO off-line pipeline (OLP) software version 7.0 in the IDL-based SWS Interactive Analysis package IA3.¹⁰ The standard pipeline produces photometrically and wavelength-calibrated spectral segments for each of the 12 pixels in the four detector bands on a grating scan and spectral order basis. Dark currents were inspected for nonlinear behavior that can arise from a cosmic-ray hit on any element of the detector array or amplification electronics, and sky signal was initially masked where discontinuities were noticed above 3σ of the rms noise level. The mean flux level within each resolution bin was then estimated using the stable detector output, and the masked detector signal was reincorporated by fitting the regions between the jumps with a first- or second-order polynomial and rectifying the output to the mean flux. This method results in improvements of 10%–20% in continuum signal-to-noise ratios (S/Ns).

Special care was required for the photometric calibrations, due to the high-input signal from η Carinae. The slower scan speed of the S01-4 AOT results in twice the dwell time per resolution element than the three faster speeds (3 s), and some partial saturation occurred in the 19.5–29.0 μm range of Band 3, where the signal from η Carinae has peaked. However, there are also twice as many readouts (48) along each integrated signal ramp, allowing for a statistically better fit of the unsaturated portions during linearization of the ramps to a calibrated photometric scale. In the 26.5–29.0 μm range of the second scan, ∼60% of the signal ramps are too saturated for reliable slope fitting, due to the accumulated memory (un-reset by voltage biasing) of the preceding scanning. These affected data were removed, and we assign a higher absolute photometric uncertainty to the remaining data, 30%, compared to 23% in the 12.0–26.5 μm range of the rest of Band 3, 25% over 29.5–45.2 μm (Band 4), 8% over 2.4–4.1 μm (Band 1), and 15% over 4.1–12.0 μm (Band 2). These are formal uncertainties at the key wavelengths in each sub-band (see Table 1 in de Graauw et al. 1996). The flux calibration at 3.5 μm (in Band 1) agrees well inside the 8% formal uncertainty with ground-based measurements by Harvey et al. (1978) and L-band photometry by Ghosh et al. (1988). The L-band photometry varies less than 0.1 mag over a 20 yr period through 1994. Our signal nonlinearity corrections were tested qualitatively on S01-4 observations of red hypergiant NML Cyg, the brightest of the SWS secondary stellar calibrators, and Mars, which was observed at opposition in 1997 July (and peaked slightly higher in flux density at 25 μm during this time compared to η Carinae). The ramp linearization and memory corrections resulted in good continuity between bands at the relative and absolute levels.

During observing, the minor axes of the four SWS apertures corresponding to the cross-dispersed grating axes were oriented at a position angle of 49° from celestial north, nearly parallel with the major axis of the Homunculus Nebula (schematically shown in Figure 1). The orientation is retrievable from the spacecraft pointing history files (IIPH) and is a serendipitous result of the roll angle of the telescope during the periods of η Carinae's limited visibility to ISO, constrained by Sun/Earth/Moon/Jupiter avoidances. The apertures for Bands 1 and 2 have the same projected widths, covering $14^{\prime\prime}$ of the projected $\sim 18\buildrel{\prime\prime}\over{.} 5$ infrared nebula. Since the triangular photometric spatial response beam shapes are consistent (Beintema et al. 2003), we do not expect a discontinuity between these two corresponding wavelength ranges regardless of source extent on the sky. The Homunculus is fully contained in the Band 3E and 4 apertures, which also have flattened cross-dispersed beam profiles. Here we would expect a sizable sudden increase in continuum fluxes starting at $\lambda \gt 27.5\,\mu {\rm{m}}$ (Band 3E) and 29.0 μm (Band 4) if emission is extended over these wavelengths, due to the change in aperture size and beam shape differences. This effect is quite obvious in many extended nebulae observed with SWS, such as NGC 7027 (e.g., Bernard-Salas et al. 2001) and the roughly $20^{\prime\prime}$ diameter ring nebula around AG Carinae with significant IR emission away from the central star (Voors et al. 2000). Such effects are not observed in the calibrated η Carinae spectrum, however, leading us to conclude that the primary source of continuum emission at $\lambda \lt 30\,\mu {\rm{m}}$ is no more than $5^{\prime\prime}$ across along the equatorial axis and $7^{\prime\prime}$ along the polar axis. These outside dimensions are in excellent agreement with the earliest measurements of η Carinae's angular size over the 4.3–20 μm range by Westphal & Neugebauer (1969), Gehrz et al. (1973), and Aitken & Jones (1975) and with the roughly $5^{\prime\prime} \times 3^{\prime\prime}$ volume containing the two apparent loops and broken torus-like structure that dominates the emission in mid-IR imaging—physical features that have been numerically modeled by Portegies Zwart & van den Heuvel (2016) as the results of a stellar merger.

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2.1.2. LWS 45.0–196.8 μm

The spectroscopic observations presented here are the portions relevant to our study obtained with SPIRE and HIFI in the OT 1 program The Homunculus: Clues to Massive Ejection from the Most Massive Stars (T. Gull, P.I.). Range spectroscopy with the Photodetector Array Camera and Spectrometer (PACS) instrument was also obtained toward and around the Homunculus and will be presented by T. Gull et al. (in preparation).

2.2.1. SPIRE

The SPIRE observations were taken with its FTS (Griffin et al. 2010) on 2011 September 16 in two separate observations on the same date near apastron (see Table 1), using the nominal and bright detector modes. The nominal mode is intended for source fluxes up to around 500 Jy over any portion of the available 194–671 μm (447–1544 GHz) range and provides maximum sensitivity for faint spectral features using optimal voltage biasing of the bolometer detector arrays. The bright mode operates at 3 to 4 times lower detector sensitivity with much higher biasing and an out-of-phase analog signal amplifier/demodulator. The SED of η Carinae over Herschel wavelengths is on the Rayleigh–Jeans tail, with flux densities from several tens to around 700 Jy. It has been shown by Lu et al. (2014) that the photometric calibrations of the bright and nominal modes generally agree to within 2%, including these η Carinae observations that were included in their analysis, when using latest pipeline processing software and calibration tables (discussed below). Given this agreement, and lacking any noticeable saturation effects or differences between the two observations in the strengths of the spectral features, we use the bright-mode spectrum out of consideration for the higher fluxes at the short-wavelength end.

The SPIRE FTS employs two bolometer detector arrays, corresponding to the long-wavelength or SLW module covering 303–671 μm (446.7–989.4 GHz) and the short-wavelength or SSW module covering 194–313 μm (959.3–1544 GHz). The data have been processed in the final version 14.1 of the Standard Product Generation (SPG) pipeline at the Herschel Science Centre and analyzed in the Herschel Interactive Processing Environment (HIPE; Ott 2010). The FTS absolute and relative photometric calibration is based primarily on Uranus. Detailed analysis of the flux uncertainties shows that the absolute fluxes at the levels of η Carinae are uncertain by up to 6%. (Swinyard et al. 2014; Hopwood et al. 2015).

The beam widths of the FTS vary nonlinearly with wavelength, between $16^{\prime\prime}$ and $22^{\prime\prime}$ FWHM for the SSW section and from $28^{\prime\prime}$ to $43^{\prime\prime}$ in the SLW section. There is a ≈40% difference in beam size in the overlap region between the two sections, $22^{\prime\prime}$ in the SSW band versus $37^{\prime\prime}$ in the SLW band at 310 μm, resulting in discontinuities in flux levels between the two sections. There are two reasons for this: (1) extended background emission around the Homunculus is present and can be characterized and corrected in the off-axis detectors (see Figures 2 and 3), and (2) the Homunculus is neither a point source nor fully extended, whereas the SPIRE spectral data products are provided to users for both limiting cases. SPIRE data processing tools in HIPE include background subtraction using the unvignetted co-aligned off-axis detectors and an interactive semi-extended source correction tool, described by Wu et al. (2013). Details on these last steps are given in Section 2.3.

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2.2.2. HIFI

As shown in Figure 3, the background levels around the Homunculus are easily detected in the SPIRE off-axis detectors. Background subtraction is straightforward using the prescription described by Wu et al. (2013) and available as an interactive script in HIPE. The script allows the user to select (or de-select) any of the off-axis detectors for the correction; we adhere to the standard set of the unvignetted, co-aligned detectors. The uncertainties from this procedure are estimated to be a fraction of 1 Jy (Hopwood et al. 2015) assuming a smooth background, i.e., insignificant compared to the high flux levels of the Homunculus.

After background subtraction, the remaining flux difference between the SSW and SLW sections is very small, ≈6 Jy in the overlap region, or ≈3% of the average flux at 306 μm. In other words, the SPIRE spectrum indicates that this feature causing the jump is very small compared to the beam size, as these spectra have been calibrated as point-like in the SPIRE pipeline. We can treat the remaining discontinuity as arising from semi-extended emission, as follows.

The extended source flux density F_ext in Jy as formulated by Wu et al. (2013) is

$$\begin{eqnarray}&&{F}_{\mathrm{ext}}(\nu )={F}_{p}(\nu )\ {\eta }_{c}(\nu ,{{\rm{\Omega }}}_{\mathrm{source}})\displaystyle \frac{{{\rm{\Omega }}}_{\mathrm{source}}}{{\eta }_{f}(\nu ,{{\rm{\Omega }}}_{\mathrm{source}})\ {{\rm{\Omega }}}_{\mathrm{beam}}(\nu )},\end{eqnarray} \tag{ 1 }$$

where F_p is the point-source flux density, ${\eta }_{f}$ is the signal coupling efficiency between source of angular size ${{\rm{\Omega }}}_{\mathrm{source}}$ and the telescope (i.e., the forward efficiency), and ${\eta }_{c}$ is an empirical correction efficiency to take into account how the beam model couples to the "true" source distribution and thus depends on a geometric source distribution model. ${{\rm{\Omega }}}_{\mathrm{beam}}(\nu )$ is the size of the beam at frequency ν.

Three representative 2D source distribution models are available with the extended source correction algorithm, namely, a flat top hat model, a Gaussian distribution function, and a Sérsic profile that is often applied to galaxies and stellar nebulae with low-brightness central regions and more extended emission away from the core. The Sérsic profile is given by $I{(r)=A\exp [-(r/r0)}^{1/n}]$, where A is a normalization factor such that the area of the profile is 1.0, r is the distance from the center of the profile, r₀ is the scale radius, and n is the Sérsic index. A small value of n corresponds to compact sources (n = 0.5 for a point source), and larger values for more extended emission (n = 10.0 for a fully extended source). Input parameters that define the source shape for all three source distribution models in the algorithm are the diameter ${{\rm{\Omega }}}_{\mathrm{source}}$, eccentricity e, rotation (or position angle), x and y detector coordinate offsets from the central pointing (needed for a suspected mis-pointing of the telescope or an off-center emission region of interest), and the Sérsic index n when applicable. With the diameter set to an initial guess and remaining parameters fixed, the algorithm can perform a search to find the optimum diameter that effectively closes the gap to a mean flux ratio of 1.00 in the overlap region between SSW and SLW spectra. The final spectrum from each model is nearly identical (<0.5% differences at all wavelengths) regardless of input initial guesses, since it is the beam shapes that drive the corrections at the final estimated size for each corresponding model self-consistently.

We have applied all three source distribution models to the SPIRE spectra of η Carinae, adopting an initial conservative guess of 100 for the source size and setting the eccentricities e and rotation angles as shown in Figure 4 and listed in Table 2 to represent simplified geometries for emission from the lobes and the equatorial region. For the Sérsic model, n = 2.0 confines 80% of the power within an ellipse with major axis $a=8^{\prime\prime}$. When the initial guess is more than a factor of two from the optimized diameter, we apply the algorithm again using the result of the first iteration with smaller ${\rm{\Delta }}{{\rm{\Omega }}}_{\mathrm{source}}$ bin sizes, converging to a final optimum value. Again, to a very high degree of accuracy the corrected fluxes are independent of the assumed input model even if that model is not a perfect representation of the source, as long as a consistent source size is used.

Table 2.  975 GHz Source Angular Diameters^a

Source Model		Rot. Angle	e	${{\rm{\Omega }}}_{s}$
		(deg)		(arcsec)
Sérsic	n = 1.5	55.0	0.75	0.875
	n = 2.0	55.0	0.75	0.35
	n = 1.25	140.0	0.90	1.75
	n = 2.0	140.0	0.90	0.55
Gaussian		55.0	0.75	4.75
		140.0	0.90	5.25

Note.

^aAngular diameters ${{\rm{\Omega }}}_{s}$ refer to best-fit major-axis diameters of the emitting source that provides optimum calibration of the background-subtracted SPIRE spectra.

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The results given in Table 2 demonstrate that the largest extent of the emitting source at 306 μm is $5^{\prime\prime}$, as fitted by the Gaussian profile on both the polar and equatorial axes. If we regard the Sérsic model values to be more realistic, then the source is confined to significantly smaller sizes.

Before reaching a final conclusion on the indicated source size, consideration must be made about the background correction; we must acknowledge that the smooth background assumption is not completely valid. As can be seen in Figure 2, there are several SSW detectors closer to the Homunculus (within ≈30'' of the core region) that we have not used in the corrections, lacking co-aligned SLW detectors. The inner SSW detector output indicates higher levels of background emission compared to the exterior co-aligned detectors, by a few Jy; see Figure 5, where we have plotted spectra from two pairs of detectors oriented radially from the core region, showing a clear radial drop-off of the fluxes. An ad hoc application of inferred background levels that would fill the beams at corresponding SLW wavelengths completely closes the gap in the on-source spectrum, removing all indications of continuum source extension in the point-calibrated spectrum of η Carinae. Rather than commit to the conclusion that the far-IR emission is escaping from a point-like region, the indications are for a ∼309 μm source size in the range of 2''–5''. This is not implied to be the size of the cool thermal emission region, regardless of the assumed source geometries, but only a feature that leads to the discontinuity in flux at this wavelength (after background corrections). We will show below that it is most likely associated with a structure observed with higher angular resolution at longer wavelengths.

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There are several important implications for these results concerning the radial decrease of background levels from the Homunculus:

• 1.  
  The HIFI fluxes that are corrected for the backgrounds at positions 3' on either side of the nebula are likely to be undercorrected. However, we examine the continuum levels only for relative changes due to variability (Section 2.4).
• 2.  
  The higher backgrounds within $30^{\prime\prime}$ of η Carinae could indicate lower-density material that has been ejected, or is still being ejected, along the lines proposed by Gomez et al. (2010) from ground-based submillimeter observations.
• 3.  
  While we have no indications of background contamination in the SWS observations (which would otherwise show jumps in the spectrum where the aperture sizes change toward the longer wavelengths; see Section 2.1.1), most certainly the LWS data are affected in a way that must be taken into account in the dust models.

Variability at near-IR wavelengths has been studied by Whitelock et al. (2004) in the JHKL bands during 2000–2004 and addressed by Chesneau et al. (2005) in ${JHKL}^{\prime}$ and narrowband 3.74 μm and 4.05 μm filters at various locations across the core region and the nebula, with the objective of confronting differences between their measurements of the correlated flux from the SED of the primary star and the CMFGEN non-LTE model atmosphere by Hillier et al. (2001). Chesneau et al. (2005) have emphasized a number of challenges to establishing direct comparisons with the model atmosphere, including stellar wind asymmetries and intrinsic variability of the primary star, and influences of the dust-enshrouded companion star as a phase-dependent source of ionization. Gomez et al. (2010) have also studied variability at 870 μm over an extended region around η Carinae with the Atacama Pathfinder Experiment (APEX) telescope shortly after a radio maximum near apastron (cf. Duncan & White 2003) and found indications of significant continuum variability compared with previous measurements, attributing the variation to the ionized stellar wind rather than dust emission. Since we have demonstrated that the bulk of the cool dust observed with SPIRE must be concentrated around the central region, temporal measurements of the far-IR continuum levels may further elucidate the importance of variability versus wind asymmetries.

We examine our HIFI observations for indications of variability at multiple frequencies over 3/4 of an orbital phase in epoch 12 (see Table 1), adopting the SPIRE SED taken at ${\rm{\Delta }}{\rm{\Phi }}$ = 0.4828 as a reference. As a heterodyne instrument, HIFI is designed for line stability rather than high-accuracy broadband or continuum measurements, limited primarily by standing wave residuals compared to the baselines for line sources with zero continuum. We estimate uncertainties of 5% in setting the SPIRE reference levels and compute the errors on the HIFI continuum measurements from the rms noise in the data and the standard deviations between measurements taken on the same day—there are always at least two for the H and V spectra originating from the optically independent mixer assemblies.

We proceed first by converting the HIFI data from the antenna temperature scale to flux densities in order to utilize the SPIRE SED as the reference at corresponding frequencies. In the framework of the intensity calibration of HIFI, the correction algorithm has the form

$$\begin{eqnarray}&&{F}_{\mathrm{ext}}(\nu )={T}_{A}(\nu )\ \displaystyle \frac{2\ k}{({A}_{\mathrm{geom}}\ {\eta }_{A})\ K(x)},\end{eqnarray} \tag{ 2 }$$

where $K(x)=[1-\exp (-{x}^{2})]/{x}^{2}$ is the source distribution function of $x\propto ({{\rm{\Omega }}}_{\mathrm{source}}/{{\rm{\Omega }}}_{\mathrm{beam}})$, with ${{\rm{\Omega }}}_{\mathrm{beam}}\sim \lambda \sqrt{1/({\eta }_{A}\ {A}_{\mathrm{geom}})}$, the effective area of the telescope A_geom = 8.45 m², and with T_A, ${\eta }_{A}$, and ${\eta }_{B}$ as above in Section 2.2.2. The values of ${\eta }_{A}$ and ${\eta }_{B}$ at each observed wavelength λ are derived from beam calibration observations (Roelfsema et al. 2012) and stored in calibration look-up tables with each set of observation products (see footnote 13). The HIFI conversion algorithm used here assumes a Gaussian source profile. However, we are not actually using this algorithm to estimate a source size; we simply adopt an equivalent source size of $5^{\prime\prime}$ consistent with the semi-extended source corrections to the SPIRE data and put HIFI fluxes on a Jy scale.

In order to utilize the SPIRE spectrum as a reference, we next apply an offset of −4.3 Jy to all HIFI continuum measurements so that the HIFI data taken at Φ = 12.5183 (obsID 1342232979) match that of SPIRE at 498 GHz. This removes a systematic offset that may be present from the different locations of the background correction measurements applied in the data processing. It also assumes that no variability has occurred between Φ = 12.4838 and 12.5183. The results are shown in Figure 6, in which the measurements are mirrored in the shaded region in order to check for periodicity.

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Indications of variability at the mean frequency of 500 GHz (600 μm) are very tenuous, due to the size of the error bars (neither HIFI nor the SPIRE FTS is a precision photometer) and nonuniform coverage, but a simple least-squares fit of a sine wave to the data suggests a periodicity in phase with the orbital period, at a peak-to-peak amplitude of ≈15% around the normalized continuum. The indicated increase in continuum levels peaks near ${\rm{\Delta }}{\rm{\Phi }}$ = 0.57, following apastron passage, which might suggest either that dust temperatures are decreasing near the central region during apastron (thus raising the far-IR continuum) or that more dust is being destroyed than created during periastron passage, or possibly from changes in ionization in the stellar wind of the primary. In any case, any large variations (cf. Gomez et al. 2010 at 870 μm) would be more apparent; the indications of variability at the indicated levels here should be regarded with caution and are too low to affect our analysis of the dust grains below.

The 2.4 μm–0.7 mm SED of η Carinae is shown in Figure 7. We include ground-based mid-IR measurements by Robinson et al. (1973), Sutton et al. (1974), and Harvey et al. (1978), showing reasonably good agreement with the SWS and LWS continuum levels at most wavelengths. Some data near the peak of the SED are considerably higher, however, a factor of ≈1.5 between the ground-based flux points at 20 μm, for example. Mid-IR variability between the Robinson et al. (1973) and Sutton et al. (1974) observations at this level is unlikely (but not out of the question), as they were obtained 1–2 yr apart. The filled symbols between 2 and 175 μm were selected by Cox et al. (1995) to fit the IR to submillimeter SED, obtaining ${L}_{\mathrm{IR}}$ = 5 × 10⁶ ${L}_{\odot }$. From our SED we obtain a lower value, ${L}_{\mathrm{IR}}$ = 2.96 × 10⁶ ${L}_{\odot }$. The difference is clearly due to the high-end flux points they selected, especially at 11 and 20 μm. The 10 μm band dominates the IR luminosity, as shown in the inset plot in Figure 7: integrated over 8 to 16 μm, it makes up 75% of the total luminosity. The submillimeter portion of the SED contributes less than 5% to the total over the 2.0–1000 μm baseline. Gomez et al. (2010) obtained L(12–1000 μ) = 1.6 $\times {10}^{6}\,{L}_{\odot }$, while we measure a value just slightly higher, 1.7 $\times {10}^{6}\,{L}_{\odot }$ over the same range (and using the same distance of 2.3 kpc to η Carinae; Allen & Hillier 1993).

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The continuity between the SWS and LWS spectra is quite good (with the caveats about residual photometric nonlinearities in the overlap range; Section 2.1.2). The obvious discontinuity between the LWS and SPIRE spectrum near 200 μm is due to the uncorrected backgrounds in the LWS aperture at these longer wavelengths. We will take this into account in our dust modeling. We also include points from observations compiled by Brooks et al. (2005) and at the APEX Telescope by Gomez et al. (2010). The vertical bars follow Brooks et al. (2005) and Gomez et al. (2010) to represent the range of variability as indicated in their study.

There is very good continuity between the SPIRE spectrum and the ALMA observations (which were obtained late in epoch 12 with beams 045–288 FWHM from short to long wavelengths, thus the lower flux point at 450 μm); both are significantly higher over most submillimeter wavelengths compared to the ranges reproduced from Brooks et al. (2005) . Structures observed in selected H recombination lines and the excess emission peaking near 3 mm have been postulated by Abraham et al. (2014) to be due to the detection of an unresolved source of free–free emission near the core created in the 1941 ejection event that formed the "Little Homunculus" (Ishibashi et al. 2003), dubbing this new thermal feature a "Baby Homunculus." Gull et al. (2016) have since pointed out that this feature is more likely to be an accumulated ionized source of emission, due to the periodic interacting wind structures, mapped in the 3 cm radio continuum by Duncan & White (2003) and extended by [Fe ii] and [Fe iii] structures observed with the Space Telescope Imaging Spectrograph, starting in mid-2009 and followed across the most recent 5.5 yr cycle. The continuity between the SPIRE and ALMA continuum flux levels would suggest that the ≈2'' source measured from the semi-extended source correction of the SPIRE data (Section 2.3) is related to these inner wind–wind structures.

Interestingly, the flux brightening reported by Gomez et al. (2010) at 850 μm is based on observations with APEX/LABoCa (which has a similar beam size to the SPIRE data), taken 1.7 yr after the maximum in the radio and X-ray cycle, which occurs close to apastron (Duncan et al. 1997), whereas the SPIRE and ALMA data have also been acquired close to and just after apastron passage (${\rm{\Delta }}{\rm{\Phi }}$ = 0.48 and 0.69, respectively) and are higher at 850 μm and nearly all upper ranges to the data compiled by Brooks et al. (2005) at similar wavelengths. This could indicate even higher increases in submillimeter emission correlated with the radio cycle, and it is also in phase with the low-amplitude variability indicated (albeit tenuously) in our HIFI continuum measurements (Section 2.4). Variability at a level of ≈25% has also been seen in the 3 mm continuum between the ALMA data by Abraham et al. (2014) and Australia Telescope Compact Array observations by Loinard et al. (2016), indicated to be lower 1.67 yr after the ALMA observations, which were taken near periastron.

Following MWB99, we show an updated three-component modified blackbody fit for reference. We follow MWB99 more closely, consistently avoiding the principle solid-state dust features at 9–15 μm and 23–38 μm for the fitting. The best-fit temperature of the cool modified blackbody remains ${T}_{d,\mathrm{cool}}$ = 110 K, but the total contribution of this component to the total emission is slightly lower. The lower IR luminosity estimate begins to alleviate the problem of energy output from such a small region indicated by the ISO and Herschel/SPIRE beam constraints, but this model is only a coarse representation of the thermal emission in the spatially integrated spectrum. We will address the inferred source sizes below. The total modified blackbody curve remains a difficult or misleading tool for setting a reference baseline that reveals the astonishing collection of solid-state features present in the SWS spectrum, as many of these are blended (i.e., the method tends to overfit the baseline). This is better accomplished with the full dust model. In Figure 8, we show the narrow dust features after subtraction of the broadband contributions from our modeling (presented below) from the total observed SED. We point out the significant, broad emission band over the 17–23 μm range, including substantial emission near 18 μm. Emission at these wavelengths was claimed (incorrectly) by Chesneau et al. (2005) not to be present in this same ISO spectrum as a justification for a dominant contribution by corundum (Al₂O₃) rather than silicates to the 10 μm band. We will address the balance of dust species in our modeling below.

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We also show the CMFGEN model atmosphere for the primary star (Hillier et al. 2001) computed at the time of the SPIRE observation and extrapolated to 1 mm. By comparison to the ${F}_{\nu }\sim {\nu }^{0.6}$ power law for an optically thick free–free wind, the CMFGEN has an obvious downturn beginning already in the mid-IR. This is due to the recombination of H in the outer wind, under the influence of the ionizing companion star at this epoch. At most epochs, the companion will likely ionize the outer wind, leading to an increase in free–free emission, and is probably one reason for variability at these longer wavelengths.

We use the radiative transfer code DUSTY (Ivezic et al. 1999) to model the dust in the Homunculus Nebula. This code solves the 1D radiative transfer equation for a spherically symmetric dust shell centered on a source of radiation. The inputs are an SED for the source, the physical parameters of the dust, a radial density profile, and the optical depth of the dust shell at a reference wavelength. We employ the CMFGEN model SED shown in Figure 7 for the central source. The optical properties of the dust are specified by a set of optical constants (the real and complex indices of refraction n and k), which are derived from reflectance measurements of representative samples in the laboratory. The code then requires a grain size distribution, density profile, optical depth, and the temperature of the dust at the inner radius of the shell. Up to 10 sets of optical constants may be provided at a specific grain size distribution, temperature, density profile, and optical depth.

The spherical symmetry assumption for our spatially unresolved composite ISO spectrum uses the simplest possible geometry, with the fewest free parameters, and is justified for lack of other constraints on the actual geometry. While we have argued that the bulk of the dust emission is concentrated in the inner ≈5'' based on the ISO aperture and SPIRE beam size constraints (Section 2.3), the geometry of the dust is probably quite complex; MWB99 suggested that the coolest thermal component (${T}_{{\rm{d}}}\approx 110$ K) is primarily located in a configuration related to the disrupted torus-like structure apparent in 18 μm imaging (see Figure 4(f)). Dust may also be forming in regions of interaction between the present-day stellar winds of the companion stars, and with prior ejected material. Hence, the dust distribution does not resemble a typical red supergiant or asymptotic giant branch (AGB) star ejection shell, and we avoid the ambiguities of geometry for an initial analysis of the dust chemistry, temperatures, and masses. Under these conditions, the input dust density profile and optical depths are somewhat arbitrary, and thus we adopt the DUSTY default ${r}^{-2}$ density profile and fix the relative thickness of the shell to less than 100 and ${\tau }_{{\rm{d}}}$(0.55 μm) ≤ 2.0. This bounding helps reduce the propensity for a high number of degenerate solutions because of the large number of free parameters in the modeling, and it will be examined as part of the uncertainties on our results (Section 3.4).

Grain sizes a are represented by the so-called "MRN" (Mathis et al. 1977) power-law distribution $n(a)\propto {a}^{-q}$ for ${a}_{\min }\leqslant a\leqslant {a}_{\max }$. The grains are assumed to be spherical, homogeneous, and nonaligned as given by standard Mie theory. While other grain shape distributions can be found in the literature, namely, the Continuous Distribution of Ellipsoids (CDE; Bohren & Huffman 1983) and the Distribution of Hollow Spheres (DHS; Min et al. 2005), both of which can alter dust band profiles in comparison to spherical grains under the same temperature and density conditions, it is clear when fitting a limited number of observations with many free input parameters that grain shape, size, temperature, and composition cannot be disentangled and must be fit in combination. Adopting the most fundamental and least ambiguous shape possible therefore avoids overfitting the problem. Despite these multiparameter limitations, fitting a dust spectrum across a wide wavelength baseline that covers the main optically active modes of vibration (e.g., Jaeger et al. 1998) provides some constraints, and we may indeed have support for spherical grains in at least one major dust species, as will be shown below.

The optical properties of dust species spanning the widest range of compositions possible have been assembled from the literature into over 50 tables of indices of refraction n and k as a function of wavelength. The compositions include the principle families of silicates, glasses, metal oxides, pure metals, sulfites, carbonaceous grains (carbon and carbides), and nitrates. These materials include crystalline and amorphous grain structures and may include the same species measured at different temperatures in the laboratory. The silicates have varied degrees of metal (Fe, Mg, Ca, Al, Na) inclusions, as well as both O-rich and O-poor species, where the latter are more common in the circumstellar environments of evolved stars (Ossenkopf et al. 1992). C and O are both known to be at very low abundances overall, while N is enhanced in the Homunculus as a consequence of the CNO cycle, but abundance variations are present in the outer ejecta (Smith & Morse 2004) and rotational transitions of the CO molecule have been detected (Loinard et al. 2012 and below); hence, we include the possibility of both extremes of carbonaceous dust and nitrates.

Our modeling emphasizes four key spectral regions: (1) the power law-like continuum 2.4–9 μm, (2) the uniquely broad and strong band peaking at 10 μm, (3) the 15–30 μm region containing a number of narrow to broad dust bands, and (4) the featureless long-wavelength region 35 to ∼250 μm covering the LWS data and portions of the SWS and SPIRE data. The SPIRE spectrum used in the fitting has been corrected for background emission as described above, while the LWS data have not been corrected (they can only be inferred). Fitting is done to all regions simultaneously, and we allow grain sizes to run logarithmically from 0.001 to 100 μm using a power index q of 3.5 (the MRN default) and 0 (fixed sizes) and with ${T}_{{\rm{d}}}$ varying between 50 and 1500 K for most species (particularly at the hot end for pure metal grains).

The mass of dust component i is calculated as

$$\begin{eqnarray}&&{M}_{{\rm{d}},i}=\displaystyle \frac{{D}^{2}{F}_{\nu ,i}(\lambda )}{{\kappa }_{\nu ,i}(\lambda ){B}_{\nu }(\lambda ,{T}_{{\rm{d}},i})},\end{eqnarray} \tag{ 3 }$$

where $D=2.3\,\mathrm{kpc}$ is the distance to η Carinae, and for a dust component i, ${F}_{\nu ,i}(\lambda )$ is the flux density, ${\kappa }_{\nu ,i}$ is the mass absorption coefficient, and ${B}_{\nu }(\lambda ,{T}_{{\rm{d}},i})$ is the Planck function evaluated at the derived dust temperature. The mass absorption coefficient ${\kappa }_{\nu }$ is equal to $3{Q}_{\mathrm{abs}}/4\rho a$, where ρ is the dust bulk density, typically adopted to equal 3.0 g cm⁻³, and ${Q}_{\mathrm{abs}}$ is the grain absorption efficiency. The values of ${Q}_{\mathrm{abs}}$ are calculated from the optical constants and grain size using Mie theory as described by Bohren & Huffman (1983). The total mass is then calculated by summing Equation (3) over all components i of the fit.

Next we present our two sets of models emerging as best fits to the 2.4–250 μm dust spectrum, representing the range of derived dust compositions, temperatures, and masses. The results are summarized in Table 3.

Table 3.  Homunculus Dust Mass by Grain Composition

Species	Typea	Ref.b	${T}_{{\rm{d}}}$	${a}_{\min }$	${M}_{{\rm{d}}}$
			(K)	(μm)	(${M}_{\odot }$)
Model A
Fe (pure)	C	1	95	0.1	0.11
Fe (pure)	C	1	160	1.0	0.09
Ca2Mg0.5AlSi1.5O7	A	6	110	0.01	0.03
Fe (pure)	C	1	300	0.01	0.01
Fe (pure)	C	1	400	0.01	2.71 × 10−3
α-Al2O3 corundum	C	5	120	0.1	2.50 × 10−3
Si3N4	C	8	250	0.1	1.22 × 10−3
Mg0.5Fe0.43Ca0.03Al0.04SiO3	C	3, 4	185	4.0	7.40 × 10−4
Mg0.5Fe0.43Ca0.03Al0.04SiO3	C	3, 4	140	3.5	6.71 × 10−4
Mg0.9Fe0.1S	C	7	170	3.0	6.10 × 10−4
"Cosmic" pyroxene	A	2	350	0.2	1.51 × 10−4
AlN	C	8	250	0.6	1.12 × 10−4
Fe (pure)	C	1	600	0.01	3.87 × 10−5
Fe (pure)	C	1	350	1.0	2.38 × 10−5
Fe (pure)	C	1	800	0.01	1.01 × 10−5
					Total 0.25
Model B
MgFeSiO4 olivine	C	10	115	1.6	0.19
O-poor silicate	A	9	95	70.0	0.11
MgFeSiO4 olivine	C	10	122	1.6	0.06
MgFeSiO4 olivine	C	10	140	1.5	0.05
O-poor silicate	A	9	150	4.5	0.02
Fe (pure)	C	1	300	0.01	5.11 × 10−3
α-Al2O3 corundum	C	5	120	0.1	1.46 × 10−3
O-poor silicate	A	9	170	1.5	1.23 × 10−3
Fe (pure)	C	1	400	0.05	7.64 × 10−4
Fe (pure)	C	1	225	1.0	6.98 × 10−4
Mg0.9Fe0.1S	C	7	180	3.0	6.07 × 10−4
"Cosmic" pyroxene	A	2	350	0.2	1.83 × 10−4
Mg2AlO4 spinel	C	11	385	2.0	4.67 × 10−5
Fe (pure)	C	1	350	1.0	2.90 × 10−5
Fe (pure)	C	1	800	0.05	3.09 × 10−6
Fe (pure)	C	1	600	0.05	2.08 × 10−6
					Total 0.44

Notes.

^aType refers to crystalline (C) or amorphous (A) grain structure. ^bDust optical constants references: (1) Ordal et al. 1985; (2) Henning & Mutschke 1997; (3) Jaeger et al. 1994; (4) Dorschner et al. 1995; (5) Zeidler et al. 2013; (6) Mutschke et al. 1998; (7) Begemann et al. 1994; (8) Kischkat et al. 2012; (9) Ossenkopf et al. 1992; (10) Fabian et al. 2000; (11) Fabian et al. 2001.

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Figure 9 shows the results of the fitting in which the featureless portion of the dust continuum is dominated by pure Fe grains at temperatures between 95 K and at least 800 K (panel (a)) and the aluminosilicate glass (ASG) Ca₂Mg_0.5AlSi_1.5O₇ at 120 K (panel (b)). These all require relatively small grains, which have higher extinction efficiencies compared to large grains, and dominate the dust mass in the three lowest-temperature Fe components and the ASG; see Table 3. The Rayleigh–Jeans portion of the aggregate model is equally weighted from SWS wavelengths to the midpoint of the LWS spectrum, then avoids the uncorrected background contamination at longer LWS wavelengths, and then is weighted to unity over the 190–215 μm portion of the SPIRE spectrum (which is background corrected but contains the onset of the source of excess emission peaking at longer wavelengths; see Figure 7). The ASG at the tabulated grain size and temperature provides a similar broad but somewhat more peaked profile compared to Fe that provides a decent match to the ∼23–45 μm range of the observed SED, particularly in combination with Mg_0.9Fe_0.1S for the 25–30 μm plateau of emission (panel (c)), better than other species in this Fe-dominated model.

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Hot Fe grains provide the best fit to the power-law-like range of the SED to ∼9 μm, for ${T}_{{\rm{d}}}\geqslant 300$ K. As pointed out by MWB99, the grains are likely to be distributed over a continuous range of temperatures up to ∼1500 K and are probably not in thermal equilibrium. Our fitting has used temperature intervals of 50 K, and as shown in Figure 9, the 300, 350, 400, 600, and 800 K components provide the measurable contributions in this wavelength range, none contributing more than ∼1% to the total dust mass.

Panels (b) and (c) in Figure 9 show the contributions of the remaining dust species emerging from the fitting that fills in the observed band structure. As shown in Figure 10, the fit to the strong 10 μm band with the unusually broad red wing is well fit by a simple combination of amorphous pyroxene at 350 K and two nitrides at 250 K. All three are only minor contributors to the total dust mass, but they provide enormous amounts of scattered radiation needed to match the high luminosity ($\approx 52\,{L}_{\odot }$) of this band. The shape of the band at $\lambda \gtrsim 14.5$ μm is also influenced by a similarly low concentration of the metal-rich crystalline silicate Mg_0.5Fe_0.43Ca_0.03Al_0.04SiO₃ (essentially an olivine or mafic silicate with Ca and Al inclusions, known in terrestrial settings as augite) at 185 K, which is a more important contributor to the 20 μm band discussed below. The composition we derive in this model for the 10 μm band deviates significantly from the olivine and corundum mix proposed by Chesneau et al. (2005), most certainly due to the limited wavelength coverage (7.5–13.5 μm) in their ground-based observations and exclusion of nitrides as potential carriers in their model. The implications for nitrides as key carriers of the 10 μm feature will be discussed in Section 5.

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Another interesting feature in Model A is the good match between corundum (Al₂O₃) at 120 K and the narrow Lorentzian-like profile at 20.2 μm, as well as the surrounding plateau in combination with the crystalline metal-rich silicate augite; see Figure 10. Only spherical grains of corundum calculated in Mie theory produce such narrowband structure compared to the CDE and DHS grain shapes; see Fabian et al. (2001), Mutschke et al. (2009), Imai et al. (2009), and Koike et al. (2010) for studies of grain shape effects on the optical properties of various silicate grains. We would otherwise need to add an ad hoc synthetic Lorentzian profile to account for the contribution at 20.2 μm, but this would fail to add any opacity to the SED at 16 μm, where a secondary band from corundum contributes in our model. The 20.2 μm match gives us some confidence that spherical grains not only are the least ambiguous shape to assume but also may indeed be observationally consistent with at least some dust carriers.

The results for Model B are shown in Figure 11. This model is similar to Model A in terms of a near-equal goodness of fit and uses Fe carriers at comparable grain sizes and temperatures to fit the region to ∼9.0 μm. However, the remaining "featureless" part of the SED now requires substantial contributions from olivine and O-poor silicates at temperatures between 95 and 150 K in order to match the Rayleigh–Jeans side of the composite spectrum. The absorption efficiencies of these carrier grains are markedly lower compared to the astronomical silicates adopted in the simple modes by MWB99 and Smith et al. (2003), driving up the dust masses.

A contribution from warm pyroxene is similarly needed to produce the blue portion of the 10 μm band, but the rest is a somewhat complicated mix of 115–140 K olivine, magnesium oxide (spinel), and a stronger contribution from Mg_0.9Fe_0.1S. Corundum also remains the only carrier that can provide the narrow 20.2 μm feature. Corundum has also been shown to be a component in the dust modeling of the 10 μm band from VLTI/MIDI spectra (7.5–13.5 μm) by Chesneau et al. (2005). In their models, however, this is present at higher temperatures (>250 K) and proportions, emitting as a broad band (a result of the CDE approximation to the grain shapes), in combination with olivine to fit the observed 10 μm feature. The full profile in our spatially integrated spectrum exhibits a substantial red wing to ≈15.4 μm, where cooler corundum along with the metal-rich silicate augite does contribute in our models (including Model B below, also in combination with olivine) in minor proportions and relies on spherical grains in order to reproduce the narrow 20.2 μm band. A close-up comparison of the main contributors in Model B to the 10 and 20 μm bands is shown in Figure 12.

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A main difference in how the SED was fit for Model B compared to Model A is that we left the entire LWS range unweighted. This gives a comparative and conservative sense of the extent of possible background contamination in these uncorrected observations, which this model indicates to start at around 60 μm. If we force a similar match to the midpoint of the LWS range, a stronger cool contribution from either the Fe or O-poor silicate (or a combination both) would result, driving up the dust mass even further. As it is, we already regard ${M}_{{\rm{d}}}$(A) = 0.25 ${M}_{\odot }$ to be a very high amount of dust mass, while ${M}_{{\rm{d}}}$(B) = 0.45 ${M}_{\odot }$ is astonishing.

The formal fitting uncertainties on both models are very similar (taking into account the difference in weighting of the LWS spectrum as just described)—the derived dust temperatures have a maximum uncertainty of ±5 K, affecting the dust masses though the predicted values of the emergent fluxes ${F}_{\nu ,i}(\lambda )$ and in the Planck function in Equation (3) most strongly at low ${T}_{{\rm{d}}}$. The highest-mass component in Model B, namely, the olivine at 115 K, has a conservative uncertainty range of −0.03 to +0.05 ${M}_{\odot }$. Grain sizes a are derived together with ${T}_{{\rm{d}}}$, and their uncertainties over the same range of temperature uncertainties have very little effect on derived masses because of the invariance of a with ${Q}_{\mathrm{abs}}/a$ for $2\pi | k| a/\lambda \ll 1$.

As previously alluded, the main source of uncertainty on the dust masses is due to the lack of constraints on the dust-emitting geometry. For large, easily observed ejection shells around AGB and red supergiant (RSG) stars and the LBV AG Carinae, the number of shells and their inner and outer radii can be constrained with IR imaging, and appropriate density profiles can be assumed with knowledge of the outflow type. As stressed already for η Carinae's environment, however, we have only an inferred size of the dust-emitting region from aperture constraints. The 12 and 18 μm imaging from MWB99 suggests that the bulk of the dust is likely to be present in the equatorial region in toroidal structures, possibly also in an inner unresolved torus or disk, and/or in pinwheel-like structures created by colliding stellar winds in the orbital plane similar to those observed in some WC binaries (e.g., Monnier et al. 2007), and to an unsettled extent in the bipolar lobes. Thus, in the representation of a single shell model we bounded the shell radii and optical depths to reduce some of the degeneracies of solutions from the large number of free parameters (a common issue even for well-constrained geometries and relatively simple dust chemistries), and in doing so, we applied Equation (3), which assumes that dust is optically thin and therefore yields lower limits on the values of M_d.

While we cannot improve on our assumptions for the dust geometry without higher angular resolution IR observations, which may then justify 2D or 3D modeling, we can give a sense of the effects on M_d when shell radii and optical depth limits are relaxed for example dust species. Using the highest-mass silicate component in Model A, Ca₂Mg_0.5AlSi_1.5O₇, we can obtain a similar output flux distribution for a shell with inner radius r₁ = 1.85 × 10¹⁶ cm, a shell thickness Δr = 15 × r₁, and an optical depth at 0.55 μm of ${\tau }_{0.55}$ = 1.2. The dust density has been assumed to vary as ${r}^{-2}$. The dust mass can then be calculated as

$$\begin{eqnarray}&&{M}_{{\rm{d}},i}=\displaystyle \frac{16{\pi }^{2}a\rho {\tau }_{\lambda }r^{\prime} }{3{Q}_{\mathrm{abs},\lambda }},\end{eqnarray} \tag{ 4 }$$

where $r^{\prime}$ is the radial center of mass and other parameters are the same as defined above for Equation (3). The value of $r^{\prime}$ for an ${r}^{-2}$ density profile is simply the average of the inner and outer radii. Equation (4) yields M_d = 0.04 ${M}_{\odot }$ in this example, which is close to that estimated from the optically thin case mainly because τ has not increased very much above unity. Interestingly, the shell's derived angular size in this geometry is found to be 05 and 80 at the inner and outer radii, respectively, which is not inconsistent with the size of the emitting region inferred by aperture constraints (Section 2.1.1) where the cool dust peaks. Higher optical depths can be offset by a thinner shell size, compared to the initial limit ${\rm{\Delta }}r\leqslant 100$ for Models A and B, but there is no straightforward way to estimate this for all species in simultaneous fitting without descending into a very high number of degenerate solutions involving all dust parameters. With these factors taken together, the dust masses given in Table 3 should be regarded as lower limits. An additional uncertainty on the gas-to-dust ratio will be discussed when we address the implied total mass of the Homunculus from each model (Section 5).

An additional source of uncertainty in Equations 3 and 4 is on the bulk density of the grains, particularly the Fe grains, for which we adopted the same canonical value of 3.0 g cm⁻³ as for the silicates. The bulk densities are not quoted by Ordal et al. (1985), whose optical constants we employed owing to the continuous IR wavelength coverage of the measurements. Values of ${\rho }_{\mathrm{Fe}}$ quoted in the literature range from 2.8 g cm⁻³ for iron powder and filings (which are probably oxidized) to as high as 7.9 g cm⁻³ (Pollack et al. 1994). The upper value would have the effect of more than doubling the estimated dust masses of all fitted Fe components.

Finally, the fact that two models emerge from our analysis as preferred best fits should underscore that a chemically unique breakdown of the spectral signatures in the observed SED is not possible to derive. This may be partially a consequence of unquantifiable limitations in our library of dust analogs for representing the true dust properties, but it also tied in with the lack of spatial discrimination of the features. In fact, the indications we have that the majority of the warm and cool dust is within the central $5^{\prime\prime}$ and not in the lobes only compounds the degeneracy problem. Nonetheless, our models are chemically sensible for the C- and O-poor and N-rich abundance pattern of the Homunculus: we do not detect any meaningful contributions from carbonaceous compounds (graphite, carbon grains, SiC, etc.) or minerals that are precipitated with water (e.g., montmorillonite, carbonates, etc.), while nitrides, sulfites, and O-poor and metal-rich silicates are strongly favored. From this standpoint, the two models should be taken together as reasonable representations; the actual chemistry may be a mix of both models, and it is clear that the Homunculus is extremely massive, ${M}_{{\rm{d}}}\gt 0.25\,{M}_{\odot }$.

The abundance pattern supporting our derived dust chemistries is not without some recent ambiguities. C and O deficiencies in the Homunculus are consistent with core evolution for a massive star and have been used to explain the lack of detection of the CO molecule in searches at submillimeter and UV wavelengths (Cox & Bronfman 1995; Verner et al. 2005; Nielsen et al. 2005)—that is, until an analysis of recently detected ¹²CO and ¹³CO emission by Loinard et al. (2012), using the APEX telescope. Their study indicates that CO relative to H₂ is at cosmic levels in the Homunculus, while a value for [¹²C/¹³C] ≈ 5 derived from the CO lines is consistent with isotopic enhancement of ¹³C for material that has been lifted (or erupted in η Carinae's case) from a star showing surface abundances consistent with CNO processing in the core. We address these apparent inconsistencies next.

A crucial parameter for understanding the η Carinae system is its luminosity. Typically it has been assumed that this can be obtained from the infrared spectral distribution under the assumption that the majority of visible and UV light is absorbed by circumstellar dust and reradiated in the infrared. Below we examine this assumption and discuss the variability of infrared fluxes. For consistency, all previous estimates of the luminosity are scaled to a distance of 2.3 kpc.

Early observations of η Carinae's infrared flux were made by Neugebauer & Westphal (1968) and Westphal & Neugebauer (1969); their measurements are shown in Figure 7. The latter estimated $L=4.5\times {10}^{6}$ ${L}_{\odot }$ for the energy emitted between 0.4 and 19.5 μm. This is a lower limit for the luminosity—the current data indicate that ∼30% of the energy is emitted longward of 20 μm. This would increase the luminosity at the time of Westphal & Neugebauer's 1968 measurements to $L=6.4\times {10}^{6}$ ${L}_{\odot }$. Later measurements made by Gehrz et al. (1973) and Robinson et al. (1973) from 1 to 20 μm are broadly consistent with those of Neugebauer & Westphal (1968) and Westphal & Neugebauer (1969). The fluxes were in agreement with the earlier measurements except at 3.6 μm. The estimated errors were 0.1 mag at 4.8, 8.4, and 11.2 μm, but with a larger uncertainty of 30% at 20 μm.

Cox et al. (1995) derived ${L}_{\mathrm{IR}}=5\times {10}^{6}$ ${L}_{\odot }$, where ≈85% of the luminosity can be attributed to emission from heated dust in two main thermal components at 210 and 430 K. They used measurements of Robinson et al. (1973) that were obtained in 1971 and 1972, 34 μm data from Sutton et al. (1974), and longer-wavelength data taken in 1977 with the Kuiper Airborne Observatory by Harvey et al. (1978). Using the data selection by Cox et al. (1995), Davidson & Humphreys (1997) estimated $L=5\times {10}^{6}$ ${L}_{\odot }$ after making a 10% allowance for energy not absorbed by dust in the Homunculus, and estimated a 20% error in L.

Our integration of the SED from 1 to 10⁶ μm yields a luminosity of $3.0\times {10}^{6}$ ${L}_{\odot }$, which is significantly lower than the earlier estimates and raises several issues. Is η Carinae variable in the infrared? If the source is variable, is this due to intrinsic changes in luminosity, or due to changes in circumstellar extinction? What is the luminosity of η Carinae? This is of crucial importance since estimates for the mass of the primary star are based primarily on its luminosity via the Eddington limit.

Smith et al. (1995) provide strong evidence that η Carinae is variable at 10 μm. In July 1993 they measured a 12.5 μm flux of $1.02\times {10}^{-13}$ W cm² μm⁻¹, while 20 yr earlier they had measured a flux of $1.63\times {10}^{-13}$ W cm² μm⁻¹ using the same beam. By assuming that the shape of the SED had not changed, they showed that this change was consistent with a measurement of the 10 μm flux made in 1986 by Russell et al. (1987). The 1993 flux measurement of Smith et al. (1995) compares favorably with our value of $1.14\times {10}^{-13}$ W cm² μm⁻¹, which was obtained in 1996.

In the near-IR and optical, the variability of η Carinae is well documented. In the early 1940s η Carinae brightened very rapidly by about 1 mag (Fernández-Lajús et al. 2009), and since then it has been getting steadily brighter. Today it is about 2 mag brighter than it was in the 1970s. In addition to the overall increase in its brightness, the central source is now much brighter than in the past and is increasingly contributing a greater fraction of the observed optical flux. In 8 yr, HST observations reveal that the central region became brighter by about 2 mag (Martin & Koppelman 2004). Superimposed on the longer-term trend is the imprint of the binary with its 5.5 yr period (e.g., Whitelock et al. 2004; van Genderen et al. 2006; Fernández-Lajús et al. 2010; Mehner et al. 2014), as well as more random variations. Recent ground-based photometry is provided by Observatorio Astronómico de La Plata.¹⁶

The most likely scenario to explain the gross behavior of all observations is that the extinction caused by circumstellar dust is declining. This can explain both the fading at infrared wavelengths and the brightening at visible wavelengths. If we assume that the bolometric luminosity of the system has remained constant, a significant amount of UV and visible flux must have been escaping from the system in 1996 (the time of the ISO measurements). Because our line of sight to the system has unusually high extinction (Hillier & Allen 1992), it is difficult to estimate this fraction. However, under the assumption of a constant bolometric luminosity, and by comparison with the earlier measurements, at least 30% of the flux emitted by the two stars must have been escaping in the visible/IR in 1996. Today it could be much larger, since both the system and the central star are much brighter than in 1996. This would be a consequence of more dust being destroyed in the shocks created in the colliding stellar winds during periastron passage than is created during the subsequent cooling phase, leading to a relative brightening at visual wavelengths after each passage. Unfortunately, the temporal resolution of the referenced ground-based and ISO observations with respect to orbit phase is rather poor for demonstrating this, as shown in Figure 15. There is an obvious paucity of measurements from ϕ = 0.75 to periastron and then into the immediate cooling phase of the orbit, emphasizing the need to pursue mid-IR observations to the next periastron event in 2020 and then over the next year (presuming that the system survives) focused on the balance of dust creation and destruction in the central region and lobes.

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Taking all of the above-referenced 1970s photometry into account in an average sense, we conclude that a reasonable estimate for L is $4.1\times {10}^{6}$ ${L}_{\odot }$, and this is within the 20% uncertainty placed on L by Davidson & Humphreys (1997). This may be a lower limit—were we lucky in the 1970s when the first IR measurements were made? Did these measurements provide us with an accurate estimate of the bolometric flux? It is known from HST observations (and also ground-based observations) that the Weigelt blobs suffer very little circumstellar extinction. By contrast, using HST data obtained in 1998, Hillier et al. (2001) estimated that the total extinction to the star along our line of sight was close to 7 mag (but is now much less). The difference in extinction between the Weigelt blobs and the primary star is not a new phenomenon—it must have also been present in the 1940s/1950s since narrow nebula lines, presumably arising in the Weigelt blobs, were very prominent in spectra taken at that time (e.g., Gaviola 1953).

The dust modeling of our combined ISO and Herschel spectral observations has been the main focus of this paper, and superseding the simple blackbody approach to estimating dust temperatures and masses (MWB99; Smith et al. 2003) with a full treatment of the interaction between the radiation field of the erupting star and its dusty environment covered by a wide range of potential optical properties has resulted in dust compositions that are not unexpected for the CNO-processed nebula. Indications of some species have been made in previous studies. Corundum, for example, has been pointed out by Chesneau et al. (2005) to be one of the first species to condense starting at high temperatures (∼1800 K) and high densities (e.g., Tielens 1990). Following Mitchell & Robinson (1978), Chesneau et al. (2005) use warm corundum to fit the broad 10 μm feature, but in higher proportions compared to our models, mainly a consequence of the additional emission in this band unobservable from the ground beyond 13.5 μm and of our approach with fitting the full SED.

Pure iron grains, on the other hand, may be much more difficult to condense in most astronomical environments. New sounding rocket experiments under microgravity conditions carried out by Kimura et al. (2017) indicate that the sticking probability for the formation of Fe grains is quite small and that only a few atoms will stick per hundred thousand collisions. This contrasts with significantly higher probabilities determined in ground-based laboratory measurements (e.g., Giesen et al. 2004) for proposed reasons such as thermal convection that can create a buffer in the nucleation site of the grains and complicate interpretations, but otherwise is not entirely understood (Kimura et al. 2017).

Observational evidence for pure Fe dust has been shown in dust models for, e.g., the O-rich shell of the AGB star OH 127.8 + 0.0 (Kemper et al. 2002), in the Type IIb supernova remnant Cas A (Rho et al. 2008), and in the LBV R17 in the LMC (Guha Niyogi et al. 2014). In fact, Mitchell & Robinson (1978) proposed Fe grains along with corundum and forsterite (Mg₂SiO₄) as the primary constituents of the 10 μm band of η Carinae, observed at 8−13 μm. Fe is thermodynamically favored in condensation models suited to LBV envelopes, η Carinae in particular, by Gail et al. (2005). Kemper et al. (2002) have pointed out that while metallic Fe can condense at fairly low temperatures 50–100 K in gas of solar composition, it condenses into FeO or FeS (cf. Jones 1990) below 700 K in O-rich environments, unless it is protected from an oxidizing environment by inclusion onto a grain. The nucleation and survival of Fe grains in the Homunculus are promoted by high Fe abundances (e.g., Hillier et al. 2001) and O deficiency where most of the available O (and Si) is likely to have been incorporated into silicates and gas-phase molecules (CO, OH, HCO, etc.), which are surviving the harsh environment of the hot star radiation field (T. Gull et al., in preparation). While some of the indicated silicates in our model fits do contain Fe inclusions, Mg atoms are thermodynamically more favorable during silicate formation (Gail & Sedlmayr 1999). We found no evidence of FeO, but we do see the presence of magnesium sulfide with a minority inclusion of iron (Mg_0.9Fe_0.1S).

Nitrides in astrophysical environments have also been elusive. Si₃N₄ has been identified in meteorites and attributed to pristine presolar grains deposited into the ISM by Type II supernovae via analysis of the elemental isotopes (Nittler et al. 1995; Lin et al. 2010). Nitrides have not been unambiguously detected in a circumstellar environment without challenge (e.g., Pitman et al. 2006). Small amounts of nitrides are predicted to occur around C-rich AGB and post-AGB stars after enhancement of N from subsolar values in the dredge-up of CNO products from the stellar interior and pulsational shocks propagating into the shells where dust will form (Fokin et al. 2001; Schirrmacher et al. 2003; Pitman et al. 2006). Even in the N-rich environment of the Homunculus, AlN and Si₃N₄ in our preferred Model A are in minor abundances, ≈0.5% of the total dust mass. Their main effect is in supplying opacity to the broad 10 μm feature.

In Section 3 we did not tabulate shell radii of the various dust species for the following reasons. The thermal balance of spherical grains exposed to a stellar radiation field at temperature ${T}_{\star }$ and distance r from a star with radius ${R}_{\star }$ can be derived from Kirchhoff's law,

$$\begin{eqnarray}&&W{\int }_{0}^{\infty }\overline{{C}_{\mathrm{abs}}}(\lambda )\pi {B}_{\lambda }({T}_{\star }){\rm{d}}\lambda =W{\int }_{0}^{\infty }\overline{{C}_{\mathrm{em}}}(\lambda )\pi {B}_{\lambda }({T}_{d}){\rm{d}}\lambda ,\end{eqnarray} \tag{ 5 }$$

where $\overline{{C}_{\mathrm{abs}}}(\lambda )$ and $\overline{{C}_{\mathrm{em}}}(\lambda )$ are the orientation-averaged absorption and emission cross sections that are derived from the absorption and emission efficiencies ${Q}_{\mathrm{em}}$ and ${Q}_{\mathrm{abs}}$ (also rotationally averaged), and $W\equiv {({R}_{\star }/{R}_{{\rm{d}}})}^{2}$ is the dilution factor. If the grains are unpolarized, randomly oriented in a spherical geometry, and the radiation field is isotropic, $\overline{{C}_{\mathrm{abs}}}(\lambda )\approx \overline{{C}_{\mathrm{em}}}(\lambda )/4$ for blackbody radiation. Then using Stefan–Boltzmann's law, Equation (5) simplifies to an expression for the blackbody dust temperature as

$$\begin{eqnarray}&&{T}_{{\rm{d}}}^{\mathrm{BB}}={T}_{\star }{W}^{1/4},\end{eqnarray} \tag{ 6 }$$

and from this one gets a simple approximation ${R}_{{\rm{d}}}$ (au) $\approx 1.72\times {10}^{9}\,{T}_{{\rm{d}}}^{-2}$ for η Carinae. The approximation has been used by Smith et al. (2003) and Chesneau et al. (2005) to estimate the radial distances of the equivalent blackbody thermal components, finding a distance of 900 au (03) for the hot 400 K component. This estimate is useful only as a "blackbody equivalent radius," since even in an environment where approximations on the geometry and isotropy of the radiation field have to be made (as we have done here), the clear non-equivalence between dust absorption and emission efficiencies for different grain compositions and sizes can lead to very different results. These non-equivalences are taken into account in the derivation of the dust temperatures and estimated masses, but for radial distances they are numerically difficult to solve and still subject to the other approximations. Moreover, the hottest dust, i.e., the small Fe grains above 400 K and probably reaching 1500 K to fill in the 2–8 μm opacity, are unlikely to be in thermal equilibrium (e.g., Fischera 2004), so that Equation (6) is inappropriate. That said, it is correct to point out (as done by Smith et al. 2003) that there must be significant shielding in the central region to allow the dust and molecules to survive. Gull et al. (2016) have mapped [Fe ii] wind–wind structures, which may protect the region close to the central source where we suspect much of the molecular gas resides, absorbing UV radiation at $\lambda \lt 2200$ Å and greatly reducing the radiation field to around 3000 Å. The geometry is roughly that of a hemisphere with an opening angle to the east, and we would expect to see a shell of molecules slightly offset to the west with both positive and negative velocities extending from south through west to north. The spatial morphology of these structures changes dramatically with orbit phase as the companion approaches from ≈30 au at apastron to ≈1.6 au, plunging deep into the primary star's wind at periastron (see Figure 1 in Gull et al. 2016).

The presence of iron grains as a major component of the dust in the Homunculus has a strong impact on estimation of the total mass. As a consequence of the low probability of Fe grain formation, most Fe in the envelope is left in the gas phase (e.g., Gail et al. 2005), perhaps as much as ∼80%–85%, accounting for minority inclusions in silicates. In this case, the gas-to-dust ratio f_gd is much higher than 100. If we adopt f_gd = 200 as a very conservative lower limit for the Fe dust and 100 for all other species, then the total mass of the Homunculus is ${M}_{\mathrm{tot}}\approx$ 45 ${M}_{\odot }$ in both models. This mass value is more of a lower limit in Model A, due to the stronger contribution of cool Fe dust. The predicted dust masses are comparable to those in the surrounding outer ejecta as estimated and proposed by Gomez et al. (2006) to originate in an even earlier eruption, leading to some dust formation by collision of ejected material with the local ISM.

What can lead to such large mass loss and not annihilate the system? There are at least some global consistencies between our results and the stellar merger model by Portegies Zwart & van den Heuvel (2016). In their model, the great eruption is the result of one of the stars in the triple system plunging into the dense envelope of the primary star, a few decades after formation of the Homunculus bipolar nebula of 20 ${M}_{\odot }$ that is the result of massive single-star evolution at the Eddington limit (cf. Owocki et al. 2004). The triple-star system orbits become partially re-aligned into the equatorial plane of the primary star, leading to the merger event over two orbital passages that forms the massive loops and torus-like structures in the mechanical dumping of angular momentum. The observed size of that set of structures is consistent with our inferred constraints on the principle dust volume, while the mass is predicted to be $\approx 22\,{M}_{\odot }$, with ≈10% of that in a nitrogen-enriched inner ring of material. Thus, their predictions for the total mass of the Homunculus and the merger are consistent with our dust model results, the main difference being in the distribution of the mass, which is inferred in our modeled observations to be concentrated in the central structures, and far less in the lobes. Portegies Zwart & van den Heuvel do not address observational constraints on the age of the Homunculus with respect to the great eruption, which has a literature range of about one orbital period, not several decades.¹⁷ Also, to what extent the model has assumed that there must be $\approx 20\,{M}_{\odot }$ lost into the Homunculus from the primary several decades prior to the merger is not clear, i.e., whether based on observational claims or an outcome of the modeling. Nonetheless, the general attributes are encouraging, and the parameter space for this modeling may possibly allow for some closer agreement of the mass predictions, or reveal additional paths to significant mass loss.

We have emphasized that the chemistry of our dust models is consistent with observed abundances in the CNO-processed nebula, supported by analysis of ¹²CO and ¹³CO lines that indicates a subsolar abundance ratio of CO/H₂ and an enhancement of ¹³C. Some caution is needed about the isotopic abundance ratio, however, since chemical fractionation of carbon driven by the reaction ¹³C⁺ + CO $\rightleftharpoons$ ¹²C⁺ + ¹³CO + 34.8 K can occur in dense photodissociation regions (PDRs; Langer et al. 1984; Gerin et al. 1998; Röllig & Ossenkopf 2013) and would decrease the ¹²C/¹³C abundance ratio. To check for this most directly requires observations of C i or [C ii] and their isotopes. Neutral C at 492.2 GHz (³P₁–³P₀) and 809.3 GHz (³P₂–³P₁) has been searched but is not detected in our Herschel observations.

The [¹²C ii] ²P_3/2–²P_1/2 line has been detected with PACS (T. Gull et al., in preparation) and HIFI. The HIFI spectrum shown in Figure 16 contains two obvious absorption lines at ${v}_{\mathrm{lsr}}=-1.7$ and −29.3 km s⁻¹, with widths of 2.1 and 4.5 km s⁻¹, respectively. Emission may be present at velocities near the weaker of the two absorption lines, but this is clearly a complex profile also containing a broad apparent "trough" of absorption centered between the two main features. In fact, the entire baseline around these features is composed of 1900 GHz (158 μm) continuum emission and much broader [C ii] emission detected with PACS. The spectral resolution of the PACS grating is 230 km s⁻¹ at this wavelength, and the main emission line may just be barely resolved, so that the HIFI [¹²C ii] spectrum is only showing the structure at the profile peak. The narrow absorption lines are unresolved to PACS. Absorption by C⁺ is a signature of very low excitation temperatures in a foreground cloud, ${T}_{\mathrm{ex}}$(C⁺) ≃ 25–40 K (Goldsmith et al. 2012), along the line of sight to η Carinae's strong continuum and the line emission, which provide disparate sources of background radiation. None of the ${[}^{13}$C ii] hyperfine lines are detected, contrary to what we would expect from [¹²C/¹³C] $\approxeq \,3$ in the Homunculus. We have two possible explanations for this.

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Scenario 1: The inferred C⁺ emission is not part of the Homunculus. [¹²C ii] 158 μm emission is widespread across the Carina Nebula as observed with ISO, arising primarily in the PDR (Mizutani et al. 2004) formed at the interface between the H ii region and molecular cloud mapped out in CO (Brooks et al. 1998; Zhang et al. 2001; see also Cox 1995; Smith et al. 2000; Brooks et al. 2003, for reviews of the large-scale structure of the Carina cloud). Our PACS spectroscopy shows significant emission in the vicinity of the Homunculus, the strongest being in the direct sightline. HIFI spectra at two positions, Carina North and Carina South (Ossenkopf et al. 2013) marked in Figure 2, are shown in Figure 16 in a comparison with η Carinae for the line excitation and kinematics.¹⁸

The strongest absorption feature at all three positions is consistent with H i absorption at −29.4 km s⁻¹, interpreted to originate from neutral hydrogen associated with molecular concentrations and absorbing dust in an expanding shell (Dickel 1974). Additional absorption is present at −43.0 and −31.6 km s⁻¹ in Carina S and at +2.8 km s⁻¹ in Carina N, which may be similar to the −1.7 km s⁻¹ feature in η Carinae (given the similar shift of the main absorption feature). The emission in Carina N may be composed of three or four main velocity components, or it is being self-absorbed at velocities similar to the absorption trough in the η Carinae spectrum. We show a Gaussian line profile in the η Carinae spectrum for the emission that is hinted at in the wings, at a strength that would be needed in combination with the observed absorption features to match the total spectrum. In this composite model, the optical depth at line center ${\tau }_{0}$ = 3.9 for the absorption line at −29.5 km s⁻¹. The absorption is expected when a significant column of low-density gas is observed toward a strong source of background emission at temperature ${T}_{\mathrm{bg}}$, and the excitation temperature ${T}_{\mathrm{ex}}$ is less than the brightness temperature of the background, i.e.,

$$\begin{eqnarray}&&{T}_{\mathrm{ex}}({{\rm{C}}}^{+})\lt \displaystyle \frac{91.2}{\mathrm{ln}(1+91.2/{T}_{{\rm{c}}}(1900.5))}{\rm{K}},\end{eqnarray} \tag{ 7 }$$

where the continuum brightness temperature at 1900.5 GHz ${T}_{{\rm{c}}}$(K)$\,=\,91.2/({e}^{91.2/{T}_{\mathrm{bg}}}-1)$. The value 91.2 K is the equivalent temperature ${T}_{* }\equiv h\nu /k$ at 1900.5 GHz. At the position of η Carinae, where we observe absorption, the net continuum levels peak at around 4.1 K on a main-beam temperature scale (single-sideband continuum plus inferred background emission), and if we include 2.7 K for the cosmic-ray background temperature, giving ${T}_{\mathrm{bg}}$ = 6.8 K, then T_c is very small (negligible) and ${T}_{\mathrm{ex}}^{\mathrm{abs}}\lesssim 7$ K for the absorption component. This is very low compared to most sightlines to dense molecular clouds in star-forming regions, and in fact we require the background emission component of the composite model to reach this temperature or it would be even lower, less than 5 K. These thermal excitation conditions on the absorption are independent of the source of the inferred emission, whether in the extended background or part of the Homunculus.

We can also estimate the excitation temperature for the emission component when it arises in the background and fills the HIFI beam. Following Goldsmith & Langer (1999), ${T}_{\mathrm{ex}}$ is related to the measured radiation temperature ${T}_{{\rm{R}}}$ through the equation

$$\begin{eqnarray}&&{T}_{{\rm{R}}}=\left(\displaystyle \frac{{T}_{* }}{\exp ({T}_{* }/{T}_{\mathrm{ex}})-1}-\displaystyle \frac{{T}_{* }}{\exp ({T}_{* }/{T}_{\mathrm{bg}})-1}\right)[1-{e}^{-\tau }]({\rm{K}}).\end{eqnarray} \tag{ 8 }$$

For optically thick C⁺ line emission that fills the HIFI beam,

$$\begin{eqnarray}&&{T}_{\mathrm{ex}}({{\rm{C}}}^{+})=\displaystyle \frac{91.2}{\mathrm{ln}[1+91.2/({T}_{\mathrm{MB}}{(}^{12}{{\rm{C}}}^{+})+{T}_{{\rm{c}}}(1900.5))]}\ ({\rm{K}}),\end{eqnarray} \tag{ 9 }$$

where ${T}_{\mathrm{MB}}{(}^{12}{{\rm{C}}}^{+})$ ≈ 2.7 K is the line peak on a main-beam temperature scale, yielding ${T}_{\mathrm{ex}}^{\mathrm{em}}({{\rm{C}}}^{+})\simeq 27$ K. To match the inferred emission profile, the C⁺-emitting gas has a column density N(C⁺) of several times 10¹⁸ cm⁻², which is not atypical of dense PDR clouds.

Given the similarity of this empirically derived profile to the observed Carina N emission spectrum, and to some extent the more blueshifted Carina S emission, it is likely that the C⁺ observed with HIFI (and PACS) originates in foreground clouds. In this case, [¹²C/¹³C] ≈ 70, and thus the ¹³C⁺ is too weak to detect with HIFI. Caution with this interpretation is warranted, however, since the emission toward η Carinae could include more than one simple component, as well as higher-velocity components as indicated in the wings of the PACS spectra (T. Gull et al., in preparation), outside the IF range of HIFI. In this case, we should consider a second possibility.

Scenario 2: C⁺ emission is being produced locally in or near the central region. We consider this scenario since [¹²C ii] is observed to be strongest toward the Homunculus in our PACS spectral mapping (T. Gull et al., in preparation). While the absorption line at −29.5 km s⁻¹ is almost certainly associated with the expanding H i shell as noted above, the blueshifted velocities of the other observed absorption lines and the inferred emission are consistent with those of the partially ionized Weigelt structures located in the central $1^{\prime\prime}$. Teodoro et al. (2017) show in an analysis of HST/STIS observations of the two most optically prominent condensations, Weigelt C and D, that the former is brighter in emission lines originating in intermediate- or high-ionization potential ions, particularly near apastron, which happens to be when our HIFI observations of C⁺ were obtained. Because of the large HIFI beam, however, we cannot make a spatial distinction between structures in the central region. The main problem that we have with this scenario is the lack of detection of nebular [¹³C ii], if we expect the Weigelt structures to have the same [¹²C/¹³C] abundance ratio as the rest of the Homunculus. If [¹²C ii] emission does originate from the Weigelt blobs, then the [¹²C/¹³C] ratio must instead be more ISM-like in order for the strongest $F=2-1$ hyperfine transition of ¹³C⁺ to be undetected by HIFI (or else we have grossly overestimated the strength of the ¹²C⁺ emission). From proper-motion analysis, Teodoro et al. (2017) estimate that the Weigelt structures are located at a distance of 1240 au from the central source, on the near side of the Homunculus close to the equatorial plane.

The nature of the C⁺ detected toward the Homunculus favors an origin in the foreground clouds with low ${T}_{\mathrm{ex}}$(C⁺) in the Carina PDR, but it cannot be decisively confirmed without further mapping of the region at high enough spectral resolution to trace the absorption lines detected with HIFI.