We present J-band spectroscopy of passive galaxies focusing on the Na I doublet at 1.14 µm. Like the Na I 0.82 µm doublet, this feature is strong in low-mass stars and hence may provide a useful probe of the initial mass function (IMF). From high signal-to-noise composite spectra, we find that Na I 1.14 µm increases steeply with increasing velocity dispersion, σ, and for the most massive galaxies (σ > ∼ 300 km s −1 ) is much stronger than predicted from synthetic spectra with Milky-Way-like IMFs and solar abundances. Reproducing Na I 1.14 µm at highσ likely requires either a very high [Na/H], or a bottom-heavy IMF, or a combination of both. Using the Na D line to break the degeneracy between IMF and abundance, we infer [Na/H] ≈ +0.5 and a steep IMF (single-slope-equivalent x ≈ 3.2, where x = 2.35 for Salpeter), for the high-σ galaxies. At lower mass (σ = 50-100 km s −1 ), the line strengths are compatible with MW-like IMFs and near-solar [Na/H]. We highlight two galaxies in our sample where strong gravitational lensing masses favour MW-like IMFs. Like the high-σ sample on average, these galaxies have strong Na I 1.14 µm; taken in isolation their sodium indices imply bottomheavy IMFs which are hard to reconcile with the lensing masses. An alternative full-spectrumfitting approach, applied to the high-σ sample, recovers an IMF less heavy than Salpeter, but under-predicts the Na I 1.14 µm line at the 5σ level. We conclude that current models struggle to reproduce this feature in the most massive galaxies without breaking other constraints, and caution against over-reliance on the sodium lines in spectroscopic IMF studies.
We investigate the spatially resolved stellar populations of a sample of seven nearby massive Early-type galaxies (ETGs), using optical and near infrared data, including K-band spectroscopy. This data offers good prospects for mitigating the uncertainties inherent in stellar population modelling by making a wide variety of strong spectroscopic features available. We report new VLT-KMOS measurements of the average empirical radial gradients out to the effective radius in the strengths of the Ca I 1.98 µm and 2.26 µm features, the Na I 2.21 µm line, and the CO 2.30 µm bandhead. Following previous work, which has indicated an excess of dwarf stars in the cores of massive ETGs, we pay specific attention to radial variations in the stellar initial mass function (IMF) as well as modelling the chemical abundance patterns and stellar population ages in our sample. Using state-of-the-art stellar population models we infer an [Fe/H] gradient of -0.16±0.05 per dex in fractional radius and an average [Na/Fe] gradient of -0.35±0.09. We find a large but radially-constant enhancement to [Mg/Fe] of ∼ 0.4 and a much lower [Ca/Fe] enhancement of ∼ 0.1. Finally, we find no significant IMF radial gradient in our sample on average and find that most galaxies in our sample are consistent with having a Milky Way-like IMF, or at most a modestly bottom heavy IMF (e.g. less dwarf enriched than a single power law IMF with the Salpeter slope).
The heavyweight stellar initial mass function (IMF) observed in the cores of massive early-type galaxies (ETGs) has been linked to formation of their cores in an initial swiftlyquenched rapid starburst. However, the outskirts of ETGs are thought to be assembled via the slow accumulation of smaller systems in which the star formation is less extreme; this suggests the form of the IMF should exhibit a radial trend in ETGs. Here we report radial stellar population gradients out to the half-light radii of a sample of eight nearby ETGs. Spatially resolved spectroscopy at 0.8-1.35µm from the VLT's KMOS instrument was used to measure radial trends in the strengths of a variety of IMF-sensitive absorption features (including some which are previously unexplored). We find weak or no radial variation in some of these which, given a radial IMF trend, ought to vary measurably, e.g. for the Wing-Ford band we measure a gradient of +0.06 ± 0.04 per decade in radius.Using stellar population models to fit stacked and individual spectra, we infer that the measured radial changes in absorption feature strengths are primarily accounted for by abundance gradients which are fairly consistent across our sample (e.g. we derive an average [Na/H] gradient of -0.53±0.07). The inferred contribution of dwarf stars to the total light typically corresponds to a bottom heavy IMF, but we find no evidence for radial IMF variations in the majority of our sample galaxies.
A bond pull test is used to determine the strength of the bond of an electronic interconnect to a circuit board. A standard test consists of clamping and pulling the interconnect with a pair of microscopic jaws. In a successful test, the maximum pulling force registered by the jaws will be the failure load of the interconnect to circuit board bond. However, if the interconnect itself deforms before the bond has failed, then this would constitute an unsuccessful test. This paper reports on a theoretical analysis of the optimal geometry for gripping of a cylindrical interconnect. Upper and lower-bound plasticity models have been used to determine the jaw proportions that will maximize the load for the deformation of the interconnect and that should, therefore, be most likely to allow successful measurement of the bond strength. This theoretical analysis is compared to 2D and 3D non-linear finite element calculations. The 2D finite element models are axi-symmetric approximations of a pull test on a cylindrical interconnect. 3D finite element models take into account the actual jaw geometry and allow simulation of both clamping and pulling stages. The maximum calculated pull forces for both 2D and 3D simulations are in good agreement with the plasticity theory. Preliminary validation of the theory and finite element results has been accomplished through experimental clamping and pulling tests on cylindrical metal rods.
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