arXiv subjects

Connor Bottrell

Publications and source records attributed to Connor Bottrell.

54 records · Page 3Linked to original sources

arXiv2022

The merger fraction of post-starburst galaxies in UNIONS

Post-starburst (PSB) galaxies are defined as having experienced a recent burst of star formation, followed by a prompt truncation in further activity. Identifying the mechanism(s) causing a galaxy to experience a post-starburst phase therefore provides integral insight into the causes of rapid quenching. Galaxy mergers have long been proposed as a possible post-starburst trigger. Effectively testing this hypothesis requires a large spectroscopic galaxy survey to identify the rare PSBs as well as high quality imaging and robust morphology metrics to identify mergers. We bring together these critical elements by selecting PSBs from the overlap of the Sloan Digital Sky Survey and the Canada-France Imaging Survey and applying a suite of classification methods: non-parametric morphology metrics such as asymmetry and Gini-M20, a convolutional neural network trained to identify post-merger galaxies, and visual classification. This work is therefore the largest and most comprehensive assessment of the merger fraction of PSBs to date. We find that the merger fraction of PSBs ranges from 19% to 42% depending on the merger identification method and details of the PSB sample selection. These merger fractions represent an excess of 3-46x relative to non-PSB control samples. Our results demonstrate that mergers play a significant role in generating PSBs, but that other mechanisms are also required. However, applying our merger identification metrics to known post-mergers in the IllustrisTNG simulation shows that ~70% of recent post-mergers (<200 Myr) would not be detected. Thus, we cannot exclude the possibility that nearly all post-starburst galaxies have undergone a merger in their recent past.

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arXiv2022

Realistic synthetic integral field spectroscopy with RealSim-IFS

The most direct way to confront observed galaxies with those formed in numerical simulations is to forward-model simulated galaxies into synthetic observations. Provided that synthetic galaxy observations include similar constraints and limitations as real observations, they can be used to (1) carry out even-handed comparisons of observation and theory and (2) map the observable characteristics of simulated galaxies to their a priori known origins. In particular, integral field spectroscopy (IFS) expands the scope of such comparisons and mappings to an exceptionally broad set of physical properties. We therefore present RealSim-IFS: a tool for forward-modelling galaxies from hydrodynamical simulations into synthetic IFS observations. The core components of RealSim-IFS model the detailed spatial sampling mechanics of any fibre-bundle, image slicer, or lenslet array IFU and corresponding observing strategy, real or imagined, and support the corresponding propagation of noise adopted by the user. The code is highly generalized and can produce cubes in any light- or mass-weighted quantity (e.g. specific intensity, gas/stellar line-of-sight velocity, stellar age/metallicity, etc.). We show that RealSim-IFS exactly reproduces the spatial reconstruction of specific intensity and variance cubes produced by the MaNGA survey Data Reduction Pipeline using the calibrated fibre spectra as input. We then apply RealSim-IFS by producing a public synthetic MaNGA stellar kinematic survey of 893 galaxies with $\log M_{\star}/M_{\odot}>10$ from the TNG50 cosmological hydrodynamical simulation.

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arXiv2022

The observability of galaxy merger signatures in nearby gas-rich spirals

Galaxy mergers are crucial to understanding galaxy evolution, therefore we must determine their observational signatures to select them from large IFU galaxy samples such as MUSE and SAMI. We employ 24 high-resolution idealised hydrodynamical galaxy merger simulations based on the "Feedback In Realistic Environment" (FIRE-2) model to determine the observability of mergers to various configurations and stages using synthetic images and velocity maps. Our mergers cover a range of orbital configurations at fixed 1:2.5 stellar mass ratio for two gas rich spirals at low redshift. Morphological and kinematic asymmetries are computed for synthetic images and velocity maps spanning each interaction. We divide the interaction sequence into three: (1) the pair phase; (2) the merging phase; and (3) the post-coalescence phase. We correctly identify mergers between first pericentre passage and 500 Myr after coalescence using kinematic asymmetry with 66% completeness, depending upon merger phase and the field-of-view of the observation. We detect fewer mergers in the pair phase (40%) and many more in the merging and post-coalescence phases (97%). We find that merger detectability decreases with field-of-view, except in retrograde mergers, where centrally concentrated asymmetric kinematic features enhances their detectability. Using a cut-off derived from a combination of photometric and kinematic asymmetry, we increase these detections to 89% overall, 79% in pairs, and close to 100% in the merging and post-coalescent phases. By using this combined asymmetry cut-off we mitigate some of the effects caused by smaller fields-of-view subtended by massively multiplexed integral field spectroscopy programmes.

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arXiv2022

Star formation characteristics of CNN-identified post-mergers in the Ultraviolet Near Infrared Optical Northern Survey (UNIONS)

The importance of the post-merger epoch in galaxy evolution has been well-documented, but post-mergers are notoriously difficult to identify. While the features induced by mergers can sometimes be distinctive, they are frequently missed by visual inspection. In addition, visual classification efforts are highly inefficient because of the inherent rarity of post-mergers (~1% in the low-redshift Universe), and non-parametric statistical merger selection methods do not account for the diversity of post-mergers or the environments in which they appear. To address these issues, we deploy a convolutional neural network (CNN) which has been trained and evaluated on realistic mock observations of simulated galaxies from the IllustrisTNG simulations, to galaxy images from the Canada France Imaging Survey (CFIS), which is part of the Ultraviolet Near Infrared Optical Northern Survey (UNIONS). We present the characteristics of the galaxies with the highest CNN-predicted post-merger certainties, as well as a visually confirmed subset of 699 post-mergers. We find that post-mergers with high CNN merger probabilities (p(x)>0.8) have an average star formation rate that is 0.1 dex higher than a mass- and redshift-matched control sample. The SFR enhancement is even greater in the visually confirmed post-merger sample, a factor of two higher than the control sample.

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arXiv2022

Passive spiral galaxies deeply captured by Subaru Hyper Suprime-Cam

This paper presents a thousand passive spiral galaxy samples at $z=$ 0.01-0.3 based on a combined analysis of the Third Public Data Release of the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP PDR3) and the GALEX-SDSS-WISE Legacy Catalog (GSWLC-2). Among 54871 $gri$ galaxy cutouts taken from the HSC-SSP PDR3 over 1072 deg$^2$, we conducted a search with deep-learning morphological classification for candidates of passive spirals below the star-forming main sequence derived by UV to mid-IR SED fitting in the GSWLC-2. We then classified the candidates into 1100 passive spirals and 1141 secondary samples based on visual inspections. Most of the latter cases are considered to be passive ringed S0 or pseudo-ringed galaxies. The remainder of these secondary samples has ambiguous morphologies, including two peculiar objects with diamond-shaped stellar wings. The selected passive spirals have a similar distribution to the general quiescent galaxies on the EW$_\mathrm{Hδ}$-D$_n$4000 diagram and concentration indices. Moreover, we detected an enhanced passive fraction of spiral galaxies in X-ray clusters. Passive spirals in galaxy clusters are preferentially located in the midterm or late infall phase on the phase-space diagram, supporting the ram pressure scenario, which has been widely advocated in previous studies. The source catalog and $gri$-composite images are available on the HSC-SSP PDR3 website (https://hsc.mtk.nao.ac.jp/ssp/data-release/). Future updates, including integration with a citizen science project dedicated to the HSC data, will achieve more effective and comprehensive classifications.

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arXiv2022

The combined and respective roles of imaging and stellar kinematics in identifying galaxy merger remnants

One of the central challenges to establishing the role of mergers in galaxy evolution is the selection of pure and complete merger samples in observations. In particular, while large and reasonably pure interacting galaxy pair samples can be obtained with relative ease via spectroscopic criteria, automated selection of post-coalescence merger remnants is restricted to the physical characteristics of remnants alone. Furthermore, such selection has predominantly focused on imaging data -- whereas kinematic data may offer a complimentary basis for identifying merger remnants. Therefore, we examine the theoretical utility of both the morphological and kinematic features of merger remnants in distinguishing galaxy merger remnants from other galaxies. Deep classification models are calibrated and evaluated using idealized synthetic images and line-of-sight stellar velocity maps of a heterogeneous population of galaxies and merger remnants from the TNG100 cosmological hydrodynamical simulation. We show that even idealized stellar kinematic data has limited utility compared to imaging and under-performs by $2.1\%\pm0.5\%$ in completeness and $4.7\%\pm0.4\%$ in purity for our fiducial model architecture. Combining imaging and stellar kinematics offers a small boost in completeness (by $1.8\%\pm0.4\%$, compared to $92.7\%\pm0.2\%$ from imaging alone) but no change in purity ($0.1\%\pm0.3\%$ improvement compared to $92.7\%\pm0.2\%$, evaluated with equal numbers of merger remnant and non-remnant control galaxies). Classification accuracy of all models is particularly sensitive to physical companions at separations $\lesssim40$ kpc and to time-since-coalescence. Taken together, our results show that the stellar kinematic data has little to offer in compliment to imaging for merger remnant identification in a heterogeneous galaxy population.

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arXiv2021

Towards robust determination of non-parametric morphologies in marginal astronomical data: resolving uncertainties with cosmological hydrodynamical simulations

Quantitative morphologies, such as asymmetry and concentration, have long been used as an effective way to assess the distribution of galaxy starlight in large samples. Application of such quantitative indicators to other data products could provide a tool capable of capturing the 2-dimensional distribution of a range of galactic properties, such as stellar mass or star-formation rate maps. In this work, we utilize galaxies from the Illustris and IllustrisTNG simulations to assess the applicability of concentration and asymmetry indicators to the stellar mass distribution in galaxies. Specifically, we test whether the intrinsic values of concentration and asymmetry (measured directly from the simulation stellar mass particle maps) are recovered after the application of measurement uncertainty and a point spread function (PSF). We find that random noise has a non-negligible systematic effect on asymmetry that scales inversely with signal-to-noise, particularly at signal-to-noise less than 100. We evaluate different methods to correct for the noise contribution to asymmetry at very low signal-to-noise, where previous studies have been unable to explore due to systematics. We present algebraic corrections for noise and resolution to recover the intrinsic morphology parameters. Using Illustris as a comparison dataset, we evaluate the robustness of these fits in the presence of a different physics model, and confirm these correction methods can be applied to other datasets. Lastly, we provide estimations for the uncertainty on different correction methods at varying signal-to-noise and resolution regimes.

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arXiv2021

Convolutional neural network identification of galaxy post-mergers in UNIONS using IllustrisTNG

The Canada-France Imaging Survey (CFIS) will consist of deep, high-resolution r-band imaging over ~5000 square degrees of the sky, representing a first-rate opportunity to identify recently-merged galaxies. Due to the large number of galaxies in CFIS, we investigate the use of a convolutional neural network (CNN) for automated merger classification. Training samples of post-merger and isolated galaxy images are generated from the IllustrisTNG simulation processed with the observational realism code RealSim. The CNN's overall classification accuracy is 88 percent, remaining stable over a wide range of intrinsic and environmental parameters. We generate a mock galaxy survey from IllustrisTNG in order to explore the expected purity of post-merger samples identified by the CNN. Despite the CNN's good performance in training, the intrinsic rarity of post-mergers leads to a sample that is only ~6 percent pure when the default decision threshold is used. We investigate trade-offs in purity and completeness with a variable decision threshold and find that we recover the statistical distribution of merger-induced star formation rate enhancements. Finally, the performance of the CNN is compared with both traditional automated methods and human classifiers. The CNN is shown to outperform Gini-M20 and asymmetry methods by an order of magnitude in post-merger sample purity on the mock survey data. Although the CNN outperforms the human classifiers on sample completeness, the purity of the post-merger sample identified by humans is frequently higher, indicating that a hybrid approach to classifications may be an effective solution to merger classifications in large surveys.

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arXiv2020

The relationship between fine galaxy stellar morphology and star formation activity in cosmological simulations: a deep learning view

Hydrodynamical simulations of galaxy formation and evolution attempt to fully model the physics that shapes galaxies. The agreement between the morphology of simulated and real galaxies, and the way the morphological types are distributed across galaxy scaling relations are important probes of our knowledge of galaxy formation physics. Here we propose an unsupervised deep learning approach to perform a stringent test of the fine morphological structure of galaxies coming from the Illustris and IllustrisTNG (TNG100 and TNG50) simulations against observations from a subsample of the Sloan Digital Sky Survey. Our framework is based on PixelCNN, an autoregressive model for image generation with an explicit likelihood. We adopt a strategy that combines the output of two PixelCNN networks in a metric that isolates the fine morphological details of galaxies from the sky background. We are able to \emph{quantitatively} identify the improvements of IllustrisTNG, particularly in the high-resolution TNG50 run, over the original Illustris. However, we find that the fine details of galaxy structure are still different between observed and simulated galaxies. This difference is driven by small, more spheroidal, and quenched galaxies which are globally less accurate regardless of resolution and which have experienced little improvement between the three simulations explored. We speculate that this disagreement, that is less severe for quenched disky galaxies, may stem from a still too coarse numerical resolution, which struggles to properly capture the inner, dense regions of quenched spheroidal galaxies.

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arXiv2020

Spatially resolved star formation and fuelling in galaxy interactions

We investigate the spatial structure and evolution of star formation and the interstellar medium (ISM) in interacting galaxies. We use an extensive suite of parsec-scale galaxy merger simulations (stellar mass ratio = 2.5:1), which employs the "Feedback In Realistic Environments-" model (fire-2). This framework resolves star formation, feedback processes, and the multi-phase structure of the ISM. We focus on the galaxy-pair stages of interaction. We find that close encounters substantially augment cool (HI) and cold-dense (H2) gas budgets, elevating the formation of new stars as a result. This enhancement is centrally-concentrated for the secondary galaxy, and more radially extended for the primary. This behaviour is weakly dependent on orbital geometry. We also find that galaxies with elevated global star formation rate (SFR) experience intense nuclear SFR enhancement, driven by high levels of either star formation efficiency (SFE) or available cold-dense gas fuel. Galaxies with suppressed global SFR also contain a nuclear cold-dense gas reservoir, but low SFE levels diminish SFR in the central region. Concretely, in the majority of cases, SFR-enhancement in the central kiloparsec is fuel-driven (55% for the secondary, 71% for the primary) -- whilst central SFR-suppression is efficiency-driven (91% for the secondary, 97% for the primary). Our numerical predictions underscore the need of substantially larger, and/or merger-dedicated, spatially-resolved galaxy surveys -- capable of examining vast and diverse samples of interacting systems -- coupled with multi-wavelength campaigns aimed to capture their internal ISM structure.

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arXiv2020

Comparison of Multi-Class and Binary Classification Machine Learning Models in Identifying Strong Gravitational Lenses

Typically, binary classification lens-finding schemes are used to discriminate between lens candidates and non-lenses. However, these models often suffer from substantial false-positive classifications. Such false positives frequently occur due to images containing objects such as crowded sources, galaxies with arms, and also images with a central source and smaller surrounding sources. Therefore, a model might confuse the stated circumstances with an Einstein ring. It has been proposed that by allowing such commonly misclassified image types to constitute their own classes, machine learning models will more easily be able to learn the difference between images that contain real lenses, and images that contain lens imposters. Using Hubble Space Telescope (HST) images, in the F814W filter, we compare the usage of binary and multi-class classification models applied to the lens finding task. From our findings, we conclude there is not a significant benefit to using the multi-class model over a binary model. We will also present the results of a simple lens search using a multi-class machine learning model, and potential new lens candidates.

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arXiv2020

Rapid early coeval star formation and assembly of the most massive galaxies in the universe

The current consensus on the formation and evolution of the brightest cluster galaxies is that their stellar mass forms early ($z \gtrsim 4$) in separate galaxies that then eventually assemble the main structure at late times ($z \lesssim 1$). However, advances in observational techniques have led to the discovery of protoclusters out to $z \sim 7$, suggesting that the late-assembly picture may not be fully complete. If these protoclusters assemble rapidly in the early universe, they should form the brightest cluster galaxies much earlier than suspected by the late-assembly picture. Using a combination of observationally constrained hydrodynamical and dark-matter-only simulations, we show that the stellar assembly time of a sub-set of the brightest cluster galaxies occurs at high redshifts ($z > 3$) rather than at low redshifts ($z < 1$), as is commonly thought. We find, using isolated non-cosmological hydrodynamical simulations, that highly overdense protoclusters assemble their stellar mass into brightest cluster galaxies within $\sim 1$ $\mathrm{Gyr}$ of evolution -- producing massive blue elliptical galaxies at high redshifts ($z \gtrsim 1.5$). We argue that there is a downsizing effect on the cluster scale wherein some of the brightest cluster galaxies in the cores of the most-massive clusters assemble earlier than those in lower-mass clusters. In those clusters with $z = 0$ virial mass $\geqslant 5\times10^{14}$ M$_\mathrm{\odot}$, we find that $9.8$% have their cores assembly early, and a higher fraction of $16.4$% in those clusters above $10^{15}$ M$_\mathrm{\odot}$. The James Webb Space Telescope will be able to detect and confirm our prediction in the near future, and we discuss the implications to constraining the value of $σ_\mathrm{8}$.

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arXiv2019

Deep learning predictions of galaxy merger stage and the importance of observational realism

Machine learning is becoming a popular tool to quantify galaxy morphologies and identify mergers. However, this technique relies on using an appropriate set of training data to be successful. By combining hydrodynamical simulations, synthetic observations and convolutional neural networks (CNNs), we quantitatively assess how realistic simulated galaxy images must be in order to reliably classify mergers. Specifically, we compare the performance of CNNs trained with two types of galaxy images, stellar maps and dust-inclusive radiatively transferred images, each with three levels of observational realism: (1) no observational effects (idealized images), (2) realistic sky and point spread function (semi-realistic images), (3) insertion into a real sky image (fully realistic images). We find that networks trained on either idealized or semi-real images have poor performance when applied to survey-realistic images. In contrast, networks trained on fully realistic images achieve 87.1% classification performance. Importantly, the level of realism in the training images is much more important than whether the images included radiative transfer, or simply used the stellar maps (87.1% compared to 79.6% accuracy, respectively). Therefore, one can avoid the large computational and storage cost of running radiative transfer with a relatively modest compromise in classification performance. Making photometry-based networks insensitive to colour incurs a very mild penalty to performance with survey-realistic data (86.0% with r-only compared to 87.1% with gri). This result demonstrates that while colour can be exploited by colour-sensitive networks, it is not necessary to achieve high accuracy and so can be avoided if desired. We provide the public release of our statistical observational realism suite, RealSim, as a companion to this paper.

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arXiv2019

A definitive merger-AGN connection at z~0 with CFIS: mergers have an excess of AGN and AGN hosts are more frequently disturbed

The question of whether galaxy mergers are linked to the triggering of active galactic nuclei (AGN) continues to be a topic of considerable debate. The issue can be broken down into two distinct questions: 1) Can galaxy mergers trigger AGN? 2) Are galaxy mergers the dominant AGN triggering mechanism? A complete picture of the AGN-merger connection requires that both of these questions are addressed with the same dataset. In previous work, we have shown that galaxy mergers selected from the Sloan Digital Sky Survey (SDSS) show an excess of both optically-selected, and mid-IR colour-selected AGN, demonstrating that the answer to the first of the above questions is affirmative. Here, we use the same optical and mid-IR AGN selection to address the second question, by quantifying the frequency of morphological disturbances in low surface brightness r-band images from the Canada France Imaging Survey (CFIS). Only ~30 per cent of optical AGN host galaxies are morphologically disturbed, indicating that recent interactions are not the dominant trigger. However, almost 60 per cent of mid-IR AGN hosts show signs of visual disturbance, indicating that interactions play a more significant role in nuclear feeding. Both mid-IR and optically selected AGN have interacting fractions that are a factor of two greater than a mass and redshift matched non-AGN control sample, an excess that increases with both AGN luminosity and host galaxy stellar mass.

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arXiv2019

Bulge plus disc and Sérsic decomposition catalogues for 16,908 galaxies in the SDSS Stripe 82 co-adds: A detailed study of the $ugriz$ structural measurements

Quantitative characterization of galaxy morphology is vital in enabling comparison of observations to predictions from galaxy formation theory. However, without significant overlap between the observational footprints of deep and shallow galaxy surveys, the extent to which structural measurements for large galaxy samples are robust to image quality (e.g., depth, spatial resolution) cannot be established. Deep images from the Sloan Digital Sky Survey (SDSS) Stripe 82 co-adds provide a unique solution to this problem - offering $1.6-1.8$ magnitudes improvement in depth with respect to SDSS Legacy images. Having similar spatial resolution to Legacy, the co-adds make it possible to examine the sensitivity of parametric morphologies to depth alone. Using the Gim2D surface-brightness decomposition software, we provide public morphology catalogs for 16,908 galaxies in the Stripe 82 $ugriz$ co-adds. Our methods and selection are completely consistent with the Simard et al. (2011) and Mendel et al. (2014) photometric decompositions. We rigorously compare measurements in the deep and shallow images. We find no systematics in total magnitudes and sizes except for faint galaxies in the $u$-band and the brightest galaxies in each band. However, characterization of bulge-to-total fractions is significantly improved in the deep images. Furthermore, statistics used to determine whether single-Sérsic or two-component (e.g., bulge+disc) models are required become more bimodal in the deep images. Lastly, we show that asymmetries are enhanced in the deep images and that the enhancement is positively correlated with the asymmetries measured in Legacy images.

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arXiv2019

What shapes a galaxy? - Unraveling the role of mass, environment and star formation in forming galactic structure

We investigate the dependence of galaxy structure on a variety of galactic and environmental parameters for ~500,000 galaxies at z<0.2, taken from the Sloan Digital Sky Survey data release 7 (SDSS-DR7). We utilise bulge-to-total stellar mass ratio, (B/T)_*, as the primary indicator of galactic structure, which circumvents issues of morphological dependence on waveband. We rank galaxy and environmental parameters in terms of how predictive they are of galaxy structure, using an artificial neural network approach. We find that distance from the star forming main sequence (Delta_SFR), followed by stellar mass (M_*), are the most closely connected parameters to (B/T)_*, and are significantly more predictive of galaxy structure than global star formation rate (SFR), or any environmental metric considered (for both central and satellite galaxies). Additionally, we make a detailed comparison to the Illustris hydrodynamical simulation and the LGalaxies semi-analytic model. In both simulations, we find a significant lack of bulge-dominated galaxies at a fixed stellar mass, compared to the SDSS. This result highlights a potentially serious problem in contemporary models of galaxy evolution.

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arXiv2017

Galaxies in the Illustris simulation as seen by the Sloan Digital Sky Survey - II: Size-luminosity relations and the deficit of bulge-dominated galaxies in Illustris at low mass

The interpretive power of the newest generation of large-volume hydrodynamical simulations of galaxy formation rests upon their ability to reproduce the observed properties of galaxies. In this second paper in a series, we employ bulge+disc decompositions of realistic dust-free galaxy images from the Illustris simulation in a consistent comparison with galaxies from the Sloan Digital Sky Survey (SDSS). Examining the size-luminosity relations of each sample, we find that galaxies in Illustris are roughly twice as large and $0.7$ magnitudes brighter on average than galaxies in the SDSS. The trend of increasing slope and decreasing normalization of size-luminosity as a function of bulge-fraction is qualitatively similar to observations. However, the size-luminosity relations of Illustris galaxies are quantitatively distinguished by higher normalizations and smaller slopes than for real galaxies. We show that this result is linked to a significant deficit of bulge-dominated galaxies in Illustris relative to the SDSS at stellar masses $\log\mathrm{M}_{\star}/\mathrm{M}_{\odot}\lesssim11$. We investigate this deficit by comparing bulge fraction estimates derived from photometry \emph{and} internal kinematics. We show that photometric bulge fractions are systematically lower than the kinematic fractions at low masses, but with increasingly good agreement as the stellar mass increases.

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arXiv2017

Galaxies in the Illustris simulation as seen by the Sloan Digital Sky Survey - I: Bulge+disc decompositions, methods, and biases

We present an image-based method for comparing the structural properties of galaxies produced in hydrodynamical simulations to real galaxies in the Sloan Digital Sky Survey. The key feature of our work is the introduction of extensive observational realism, such as object crowding, noise and viewing angle, to the synthetic images of simulated galaxies, so that they can be fairly compared to real galaxy catalogs. We apply our methodology to the dust-free synthetic image catalog of galaxies from the Illustris simulation at $z=0$, which are then fit with bulge+disc models to obtain morphological parameters. In this first paper in a series, we detail our methods, quantify observational biases, and present publicly available bulge+disc decomposition catalogs. We find that our bulge+disc decompositions are largely robust to the observational biases that affect decompositions of real galaxies. However, we identify a significant population of galaxies (roughly 30\% of the full sample) in Illustris that are prone to internal segmentation, leading to systematically reduced flux estimates by up to a factor of 6, smaller half-light radii by up to a factor of $\sim$ 2, and generally erroneous bulge-to-total fractions of (B/T)=0.

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