$\eta$ Carinae's Dusty Homunculus Nebula from Near-Infrared to
  Submillimeter Wavelengths: Mass, Composition, and Evidence for Fading Opacity
  Infrared observations of the dusty, massive Homunculus Nebula around the
luminous blue variable $\eta$ Carinae are crucial to characterize the mass-loss
history and help constrain the mechanisms leading to the Great Eruption. We
present the 2.4 - 670 $\mu$m spectral energy distribution, constructed from
legacy ISO observations and new spectroscopy obtained with the {\em{Herschel
Space Observatory}}. Using radiative transfer modeling, we find that the two
best-fit dust models yield compositions which are consistent with CNO-processed
material, with iron, pyroxene and other metal-rich silicates, corundum, and
magnesium-iron sulfide in common. Spherical corundum grains are supported by
the good match to a narrow 20.2 $\mu$m feature. Our preferred model contains
nitrides AlN and Si$_3$N$_4$ in low abundances. Dust masses range from 0.25 to
0.44 $M_\odot$ but $M_{\rm{tot}} \ge$ 45 $M_\odot$ in both cases due to an
expected high Fe gas-to-dust ratio. The bulk of dust is within a 5$"$ $\times$
7$"$ central region. An additional compact feature is detected at 390 $\mu$m.
We obtain $L_{\rm{IR}}$ = 2.96 $\times$ 10$^6$ $L_\odot$, a 25\% decline from
an average of mid-IR photometric levels observed in 1971-1977. This indicates a
reduction in circumstellar extinction in conjunction with an increase in visual
brightness, allowing 25-40\% of optical and UV radiation to escape from the
central source. We also present an analysis of $^{12}$CO and $^{13}$CO $J =
5-4$ through $9-8$ lines, showing that the abundances are consistent with
expectations for CNO-processed material. The [$^{12}$C~{\sc{ii}}] line is
detected in absorption, which we suspect originates in foreground material at
very low excitation temperatures.
