η\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.Comment: Accepted in Ap

Motif counting beyond five nodes

Counting graphlets is a well-studied problem in graph mining and social network analysis. Recently, several papers explored very simple and natural algorithms based on Monte Carlo sampling of Markov Chains (MC), and reported encouraging results. We show, perhaps surprisingly, that such algorithms are outperformed by color coding (CC) [2], a sophisticated algorithmic technique that we extend to the case of graphlet sampling and for which we prove strong statistical guarantees. Our computational experiments on graphs with millions of nodes show CC to be more accurate than MC; furthermore, we formally show that the mixing time of the MC approach is too high in general, even when the input graph has high conductance. All this comes at a price however. While MC is very efficient in terms of space, CC’s memory requirements become demanding when the size of the input graph and that of the graphlets grow. And yet, our experiments show that CC can push the limits of the state-of-the-art, both in terms of the size of the input graph and of that of the graphlets