The shapes of column density PDFs - The importance of the last closed contour

The probability distribution function of column density (PDF) has become the tool of choice for cloud structure analysis and star formation studies. Its simplicity is attractive, and the PDF could offer access to cloud physical parameters otherwise difficult to measure, but there has been some confusion in the literature on the definition of its completeness limit and shape at the low column density end. In this Letter we use the natural definition of the completeness limit of a column density PDF, the last closed column-density contour inside a surveyed region, and apply it to a set of large-scale maps of nearby molecular clouds. We conclude that there is no observational evidence for log-normal PDFs in these objects. We find that all studied molecular clouds have PDFs well described by power-laws, including the diffuse cloud Polaris. Our results call for a new physical interpretation for the shape of the column density PDFs. We find that the slope of a cloud PDF is invariant to distance but not to the spatial arrangement of cloud material, and as such it is still a useful tool to investigate cloud structure.Comment: A&A Letter, accepted. Comments welcom

Polyominoes with nearly convex columns: An undirected model

Column-convex polyominoes were introduced in 1950's by Temperley, a mathematical physicist working on "lattice gases". By now, column-convex polyominoes are a popular and well-understood model. There exist several generalizations of column-convex polyominoes; an example is a model called multi-directed animals. In this paper, we introduce a new sequence of supersets of column-convex polyominoes. Our model (we call it level m column-subconvex polyominoes) is defined in a simple way. We focus on the case when cells are hexagons and we compute the area generating functions for the levels one and two. Both of those generating functions are complicated q-series, whereas the area generating function of column-convex polyominoes is a rational function. The growth constants of level one and level two column-subconvex polyominoes are 4.319139 and 4.509480, respectively. For comparison, the growth constants of column-convex polyominoes, multi-directed animals and all polyominoes are 3.863131, 4.587894 and 5.183148, respectively.Comment: 26 pages, 14 figure

On column density thresholds and the star formation rate

We present the results of a numerical study designed to address the question of whether there is a column density threshold for star formation within molecular clouds. We have simulated a large number of different clouds, with volume and column densities spanning a wide range of different values, using a state-of-the-art model for the coupled chemical, thermal and dynamical evolution of the gas. We show that star formation is only possible in regions where the mean (area-averaged) column density exceeds $10^{21} \: {\rm cm^{-2}}$. Within the clouds, we also show that there is a good correlation between the mass of gas above a K-band extinction $A_{\rm K} = 0.8$ and the star formation rate (SFR), in agreement with recent observational work. Previously, this relationship has been explained in terms of a correlation between the SFR and the mass in dense gas. However, we find that this correlation is weaker and more time-dependent than that between the SFR and the column density. In support of previous studies, we argue that dust shielding is the key process: the true correlation is one between the SFR and the mass in cold, well-shielded gas, and the latter correlates better with the column density than the volume density.Comment: 21 pages and 12 figures. Accepted for publication in MNRA

Measurement of adsorption of a single component from the liquid phase : modelling investigation and sensitivity analysis

In this work, we consider an alternative approach for the measurement of adsorption from the liquid phase. Consider a mixture consisting of a non-adsorbed component (B) and an adsorbed component (A) present at some low concentration. Initially, a feed of component B only flows through a column packed with an adsorbent. Then, the feed is switched to the mixture of A and B. As soon as the mixture enters the column, there will be a reduction in the outlet flow rate as component A leaves the liquid phase and passes into the adsorbed phase. There are three stages to this work. The first is to develop overall and component balances to show how the amount adsorbed of component A can be determined from the variation in the column outlet flow rate. The second is to determine the actual variation in the column outlet flow rate for both plug flow and axial-dispersed plug flow. The final stage is to consider the suitability of a gravity-fed system to deliver the feed to the column. An analysis of the results shows that the experimental arrangement should be able to accurately monitor adsorption from the liquid phase where the mass fraction of the solute is of the order of 1%: the limiting experimental factor is how constant the volumetric flow rate of the liquid feed can be maintained

Constraint control of distillation processes

There is a growing interest to design and operate chemical processes for reduced energy consumption. As an example a comparison is made between the distillation of binary mixtures in a conventional distillation column, a vapour recompression system and a two column heat integrated system. For all three configurations constraint control schemes are proposed. Utility costs can be reduced with about a factor of two by using a beat integrated system

Mutual Event Observations of Io's Sodium Corona

We have measured the column density profile of Io's sodium corona using 10 mutual eclipses between the Galilean satellites. This approach circumvents the problem of spatially resolving Io's corona directly from Io's bright continuum in the presence of atmospheric seeing and telescopic scattering. The primary goal is to investigate the spatial and temporal variations of Io's corona. Spectra from the Keck Observatory and McDonald Observatory from 1997 reveal a corona that is only approximately spherically symmetric around Io. Comparing the globally averaged radial sodium column density profile in the corona with profiles measured in 1991 and 1985, we find that there has been no significant variation. However, there appears to be a previously undetected asymmetry: the corona above Io's sub-Jupiter hemisphere is consistently more dense than above the anti-Jupiter hemisphere

Small-Scale structure in the Galactic ISM: Implications for Galaxy Cluster Studies

Observations of extragalactic objects need to be corrected for Galactic absorption and this is often accomplished by using the measured 21 cm HI column. However, within the beam of the radio telescope there are variations in the HI column that can have important effects in interpreting absorption line studies and X-ray spectra at the softest energies. We examine the HI and DIRBE/IRAS data for lines of sight out of the Galaxy, which show evidence for HI variations in of up to a factor of three in 1 degree fields. Column density enhancements would preferentially absorb soft X-rays in spatially extended objects and we find evidence for this effect in the ROSAT PSPC observations of two bright clusters of galaxies, Abell 119 and Abell 2142. For clusters of galaxies, the failure to include column density fluctuations will lead to systematically incorrect fits to the X-ray data in the sense that there will appear to be a very soft X-ray excess. This may be one cause of the soft X-ray excess in clusters, since the magnitude of the effect is comparable to the observed values.Comment: 16 pages, 9 figures, to appear in the Astrophysical Journal, vol. 597 (1 Nov 2003

The power spectrum of the Lyman-alpha clouds

We investigate the clustering properties of 13 QSO lines of sight in flat space, with average redshifts from z~2 to 4. We estimate the 1-D power spectrum and the integral density of neighbours, and discuss their variation with respect to redshift and column density. We compare the results with standard CDM models, and estimate the power spectrum of Lyman-alpha clustering as a function both of redshift and column density. We find that a) there is no significant periodicity or characteristic scale; b) the clustering depends both on column density and redshift; c) the clustering increases linearly only if at the same time the HI column density decreases strongly with redshift. The results remain qualitatively the same assuming an open cosmological model.Comment: Accepted for publication in MNRA

On k-Column Sparse Packing Programs

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail