Mass-density Relationship Research Articles

Center of mass and anatomical coordinate system definition for sheep head kinematics, with application to ovine models of traumatic brain injury.

Pathological outcomes of traumatic brain injury (TBI), including diffuse axonal injury, are influenced by the direction, magnitude, and duration of head acceleration during the injury exposure. Ovine models have been used to study injury mechanics and pathological outcomes of TBI. To accurately describe the kinematics of the head during an injury exposure, and better facilitate comparison with human head kinematics, anatomical coordinate systems (ACS) with an origin at the head or brain center of mass (CoM), and axes that align with the ovine Frankfort plane equivalent, are required. The aim of this study was to determine the mass properties of the sheep head and brain, and define an ACSvirtual for the head and brain, using anatomical landmarks on the skull with the aforementioned origins and orientation. Three‐dimensional models of 10 merino sheep heads were constructed from computed tomography images, and the coordinates of the head and brain CoMs, relative to a previously reported sheep head coordinate system (ACSphysical), were determined using the Hounsfield unit–mass density relationship. The ACSphysical origin was 34.8 ± 3.1 mm posterosuperior of the head CoM and 43.7 ± 1.7 anteroinferior of the brain CoM. Prominent internal anatomical landmarks were then used to define a new ACS (ACSvirtual) with axes aligned with the Frankfort plane equivalent and an origin 10.4 ± 3.2 mm from the head CoM. The CoM and ACSvirtual defined in this study will increase the potential for comparison of head kinematics between ovine models and humans, in the context of TBI.

Statistical mass function of prestellar cores from the density distribution of their natal clouds

The mass function of clumps observed in molecular clouds raises interesting theoretical issues, especially in its relation to the stellar initial mass function (IMF). We propose a statistical model of the mass function of prestellar cores (CMF), formed in self-gravitating isothermal clouds at a given stage of their evolution. The latter is characterized by the mass-density probability distribution function (ρ-PDF), which is a power-law with slope q. The different molecular clouds are divided into ensembles according to the PDF slope and each ensemble is represented by a single spherical cloud. The cores are considered as elements of self-similar structure typical for fractal clouds and are modeled by spherical objects populating each cloud shell. Our model assumes relations between size, mass, and density of the statistical cores. Out of these, a core mass-density relationship ρ ∝ mx is derived where x = 1∕(1 + q). We find that q determines the existence or nonexistence of a threshold density for core collapse. The derived general CMF is a power law of slope − 1 while the CMF of gravitationally unstable cores has a slope (−1 + x∕2), comparable with the slopes of the high-mass part of the stellar IMF and of observational CMFs.

Statistical link between the structure of molecular clouds and their density distribution

We introduce the concept of a class of equivalence of molecular clouds represented by an abstract spherically symmetric, isotropic object. This object is described by use of abstract scales in respect to a given mass density distribution. Mass and average density are ascribed to each scale and thus are linked to the density distribution: a power-law type and an arbitrary continuous one. In the latter case, we derive a differential relationship between the mean density at a given scale and the structure parameter which defines the mass-density relationship. The two-dimensional (2D) projection of the cloud along the line of sight is also investigated. Scaling relations of mass and mean density are derived in the considered cases of power-law and arbitrary continuous distributions. We obtain relations between scaling exponents in the 2D and 3D cases. The proposed classes of equivalence are representative for the general structure of real clouds with various types of column-density distributions: power law, lognormal or combination of both.

A DEFINITION FOR GIANT PLANETS BASED ON THE MASS–DENSITY RELATIONSHIP

We present the mass-density relationship (log M - log rho) for objects with masses ranging from planets (M ~ 0.01 M_Jup) through stars (M > 0.08 M_Sun). This relationship shows three distinct regions separated by a change in slope in log M -- log rho plane. In particular, objects with masses in the range 0.3 M_Jup to 60 M_Jup follow a tight linear relationship with no distinguishing feature to separate the low mass end (giant planets) from the high mass end (brown dwarfs). The distinction between giant planets and brown dwarfs thus seems arbitrary. We propose a new definition of giant planets based simply on changes in the slope of the log $M$ versus log rho relationship. By this criterion, objects with masses less than ~ 0.3 M_Jup are low mass planets, either icy or rocky. Giant planets cover the mass range 0.3 M_Jup to 60 M_Jup. Analogous to the stellar main sequence, objects on the upper end of the giant planet sequence (brown dwarfs) can simply be referred to as "high mass giant planets", while planets with masses near that of Jupiter can be considered to be "low mass giant planets".

Local habitat conditions explain the variation in the strength of self-thinning in a stream salmonid.

Self-thinning patterns are frequently used to describe density dependence in populations on timescales shorter than the organism's life span and have been used to infer carrying capacity of the environment. Among mobile animals, this concept has been used to document density dependence in stream salmonids, which compete over access to food and space. The carrying capacity, growth conditions, and initial cohort sizes often vary between streams and stream sections, which would influence the onset and strength of the density dependence. Despite much effort in describing habitat relationships in stream fishes, few studies have explicitly tested how the physical environment affects the slope of the thinning curves. Here, we investigate the prevalence and strength of self-thinning in juvenile stages of a steelhead (Oncorhynchus mykiss) population in Idaho, USA. Further, we investigate the roles of local physical habitat and metabolic constraints in explaining the variation in thinning curves among study sites in the watershed. Only yearling steelhead exhibited an overall significant thinning trend, but the slope of the mass–density relationship (−0.53) was shallower than predicted by theory and reported from empirical studies. There was no detectable relationship in subyearling steelhead. Certain abiotic factors explained a relatively large portion of the variation in the strength of the self-thinning among the study reaches. For subyearling steelhead, the slopes were negatively associated with the average water depth and flow velocity in the study sites, whereas slopes in yearlings were steeper in sites that incurred a higher metabolic cost. Our results show that the prevalence and strength of density dependence in natural fish populations can vary across heterogeneous watersheds and can be more pronounced during certain stages of a species' life history, and that environmental factors can mediate the extent to which density dependence is manifested in predictable ways.

The role of belowground competition and plastic biomass allocation in altering plant mass–density relationships

Metabolic scaling theory (MST) predicts a ‘universal scaling law’ for plant mass–density relationships, but empirical observations are more variable. Possible explanations of this variability include plasticity in biomass allocation between the above‐ and belowground compartment and different modes of competition, which can be asymmetric or symmetric. Although complex interactions of these factors are likely to occur, so far the majority of modelling and empirical studies has focussed on mono‐factorial explanations. We here present a generic individual‐based model, which allows exploring the plant mass–density relationship in realistic settings by representing plasticity of biomass allocation and different modes of competition in the above‐ and belowground compartment. Plants grew according to an ontogenetic growth model derived from MST. To evaluate the behavior of the simulated plants related to the allocation patterns and to validate model predictions, we conducted greenhouse experiments with tree seedlings. The model reproduced empirical patterns both at the individual and population level. Without belowground resource limitation, aboveground processes dominated and the slopes of mass–density relationships followed the predictions of MST. In contrast, resource limitation led to an increased allocation of biomass to belowground parts of the plants. The subsequent dominance of symmetric belowground competition caused significantly shallower slopes of the mass–density relationship, even though the growth of individual plants followed MST. We conclude that changes in biomass allocation induced by belowground resource limitation explain the deviations from the mass–density relationship predicted by MST. Taking into account the plasticity of biomass allocation and its linkage to the above‐ and belowground competition is critical for fully representing plant communities, in particular for correctly predicting their response of carbon storage and sequestration to changing environmental conditions.

Deviation of Mass-Density Allometry in Ramet Populations of Switchgrass

We studied a ramet population of switchgrass, caespitose in appearance, and proved the absence of density-dependent mortality. We present a deviation of mass-density allometry based on spatially explicit densities along the vertical space of the population. The number of ramets in the mean-ramet-height space explains the variation in the average weight of whole ramets (M) more accurately, with an asymptote towards critical self-thinning trajectory for such a mass-density relationship. With the development of size heterogeneity, a ‘vertical packing’ process appears in the population. We define the process as a similar ‘self-thinning’ trajectory, through the initial crowding of ramets in the mean-ramet-height space and continued transferring into the upper space and the lower space. The process presents a possibly competitive mechanism of self-thinning, local-competition-driving size deviation and vertical space packing. Here, log10 (M) = 2.91 - 1.25 log10 (NL). Local crowding degree (NL) is the number of ramets per unit area (m2) in the mean-ramet-height space class. Similar ‘self-thinning’ occurs in the ramet population, but it just indicates how those ramets escape out of the mean-ramet-height space class, and therefore how the mean weight of whole ramets increases. Self-thinning should be the result of local competition among effective number of individuals in a population, rather than apparent crowding degree presented by whole individuals. The critical mass-density allometry based on whole individuals should be only a special case.

THE EFFECT OF POPULATION-WIDE MASS-TO-RADIUS RELATIONSHIPS ON THE INTERPRETATION OFKEPLERAND HARPS SUPER-EARTH OCCURRENCE RATES

The ongoing High Accuracy Radial velocity Planet Search (HARPS) has found that 30%-50% of G and K dwarfs in the solar neighborhood host planets with M{sub pl} {approx}< M{sub Nep} in orbits of P {<=} 50 days. At first glance, this high overall occurrence rate seems at best to be marginally consistent with the planet frequency measured during Q0-Q2 of the Kepler mission, whose 1235 detected planetary candidates naively imply that {approx}15% of main-sequence dwarfs harbor a short-period planet with R{sub pl} < 4 R{sub Circled-Plus }. A rigorous comparison between the two surveys is difficult, however, as they observe different stellar populations, measure different planetary physical properties, and are subject to radically different sampling plans. In this article, we report the results of a Monte Carlo study which seeks to partially overcome this apparent discrepancy by identifying plausible planetary population distributions which can jointly conform to the results of the two surveys. We find that, given the HARPS occurrence rate, either a population subject to a mass-density relationship extrapolated from our solar system or a population concurrently consisting of dense silicate-iron planets and low-density gaseous worlds can produce total numbers of planet candidates consistent with those actually detected bymore » Kepler. However, these two mass-to-radius relationships (M-Rs) resolve the apparent occurrence rate discrepancy with different mass and period distributions, enabling future observations to rule out M-Rs that do not fully describe the observed planet population in a global sense. Extracting information of this nature from the transit-radial-velocity comparison has significant implications for the interpretation of planet occurrence rates: if a multi-valued M-R, which allows planets of similar mass to have significantly different radii, emerges from observational data, then multiple formation mechanisms are likely at work. In this event, HARPS may be detecting a large population of dense low-mass planets, while Kepler detects a large population of gaseous sub-Neptunes.« less

Clump mass function at an early stage of molecular cloud evolution - I. A statistical approach

We derive the mass function of condensations (clumps) which were formed through a turbulent cascade over a range of spatial scales $L\le20$ pc during early, predominantly turbulent evolution of a molecular cloud. The approach rests upon the assumption of a statistical clump mass-density relationship $n\propto m^x$ with a scale dependence of the exponent $x$ obtained from equipartition relations between various forms of energy of clumps. The derived clump mass function (ClMF) could be represented by series of 2 or 3 power laws, depending on the chosen equipartition relation, the velocity scaling index and the type of turbulent forcing. The high-mass ClMF exhibits an average slope $\Gamma\simeq-1$, typical for fractal clouds, whereas its intermediate-mass part is shallower or flattened, in agreement with some observational studies.

Mass-density relationship in molecular cloud clumps

We study the mass-density relationship n ~ m^x in molecular cloud condensations (clumps), considering various equipartition relations between their gravitational, kinetic, internal and magnetic energies. Clumps are described statistically, with a density distribution that reflects a lognormal probability density function (pdf) in turbulent cold interstellar medium. The clump mass-density exponent $x$ derived at different scales $L$ varies in most of the cases within the range $-2.5\lesssim x \lesssim-0.2$, with a pronounced scale dependence and in consistency with observations. When derived from the global size-mass relationship m ~ l^{\gamma_{glob}} for set of clumps, generated at all scales, the clump mass-density exponent has typical values $-3.0\lesssim x(\gamma_{glob}) \lesssim -0.3$ that depend on the forms of energy, included in the equipartition relations and on the velocity scaling law whereas the description of clump geometry is important when magnetic energy is taken into account.

Mass–density relationship changes along salinity gradient in Suaeda salsa L.

Whether and why the biomass–density (M–N) scaling relationship varies along environmental gradients were continuously debated in theoretical ecology. In this study, how soil salinity stress affects on the M–N scaling relationship was investigated by using Suaeda salsa L. in beach of Dongtai, Jiangsu Province, China. The results showed that the exponent of the scaling relationship (b) of low salinity level (−1.259) was smaller than that of middle salinity level (−1.025), which in turn was smaller than that high salinity level (−0.698). The plant height–crown radius (H–r) scaling exponents (∂) decreased with increasing salinity stress, while the canopy coverage–density (C–N) scaling exponents (β) showed an inverse trend. The predict data (b) based on ∂ and β by using the geometric model were statistically indistinguishable from their observed values for the three salinity levels. Moreover, two resources utilization parameters (l mean from root to leaf, a total area of leaves) of metabolic theory, photosynthetic rate, and water-use efficiency were more advantageous to Suaeda salsa L. of high stress than to those of low salinity. Therefore, it was implied that the changes of M–N relationship along salinity gradients may be resulted from their different geometric morphologies and resource utilization in response to salinity stress.

Plant mass–density relationship along a moisture gradient in north‐west China

Summary Metabolic scaling theory predicts that the rate of resource use per unit area is independent of the average mass per individual and that the slope of the log mass–log density relationship should be −4/3. Data were obtained from plant communities along a natural gradient of moisture and latitude in north‐west China to test the generality of this theory. The allometric exponents (slopes of the log mass–log density relationship) for above‐ground biomass decreased with natural moisture levels and plant cover, deviating from the predictions of the energy equivalence theory. Allometric exponents for below‐ground and total biomass were similar among the three sites and were much closer to the predicted value of −4/3. Metabolic scaling theory may be applicable under many growth conditions, but not when restricted to above‐ground biomass under drought stress. The rate of supply of the limiting resource per unit area determines which plant parts behave according to theory.

The interspecific mass-density relationship and plant geometry.

We present an a priori theoretical framework for the interspecific allometric relationship between stand mass and plant population density. Our model predicts a slope of -1/3 between the logarithm of stand mass and the logarithm of stand density, thus conflicting with a previously assumed slope of -1/2. Our model rests on a heuristic separation of resource-limited living mass and structural mass in the plant body. We point out that because of similar resource requirements among plants of different sizes, a nonzero plant mass-density slope is primarily defined by structural mass. Specifically, the slope is a result of (i) the physical size-dependent relationship between stem width and height, (ii) foliage-dependent demands of conductance, and (iii) the cumulative nature of structural mass. The data support our model, both when the potential sampling bias of taxonomic relatedness is accounted for and when it is not. Independent contrasts analyses show that observed relationships among variables are not significantly different from the assumptions made to build the model or from its a priori predictions. We note that the dependence of the plant mass-density slope on the functions of structural mass provides a cause for the difference from the zero slope found in the animal population mass-density relationship; for the most part, animals do not have a comparable cumulative tissue type.

Population Density, Body Mass and Niche Overlap in Australian Birds

1. There is a negative relationship between body mass and ecological population density across 200 species of Australian birds, with a slope of -0.82 on logarithmic axes. This is not significantly different from the slope predicted by the energetic equivalence rule. 2. Contrary to previous studies, there is no tendency for the phylogenetic position of an individual tribe to correlate with the strength of the mass-density relationship across its species. 3. However, in line with recent theory, there is a relationship between the degree of niche overlap among species in a tribe and the abundance pattern of the species. Tribes in which the species appear to share few resources are those in which the relationship between mass and density is most strongly negative

What Shape Is the Relationship between Body Size and Population Density?

On average, small-bodied organisms are more abundant than large ones (Elton 1927). This relationship between individual body mass and population density (hereafter referred to as the mass-density relationship) has been extensively studied, and the literature presents two strikingly different sets of observations on it. The purpose of the present study is to re-examine these apparently conflicting observed patterns and to suggest hypotheses to reconcile them. One set of authors has quantified the mass-density relationship by assembling estimates of population densities from the published literature (Damuth 1981, Peters and Wassenberg 1983, Peters and Raelson 1984, Currie and Fritz 1993). Such studies typically draw data from detailed studies of individual species. These species generally derive from many phyla, drawn from a wide geographic area. This process yields relationships such as the one shown in Fig. 1, which are usually described by the statistical model:

Self-Thinning in Even-Aged Natural Stands of Betula pubescens

The mass-density relationship of the shade intolerant species Betula pubescens Ehrh. does not conform to the self-thinning rule (which states that f = K.N, where y is equal to -1.5), the exponent (-1.37) differing significantly from -1.5 (p < 0.05). According to an extended allometric model, which predicts the relationships of the thinning exponent to allometric exponents derived from stand dimensions, this deviation from -1.5 can be attributed to the relatively high allocation of resources to radial growth and poor ability to add mass into the already occupied space. Crown depth growth in B. pubescens declines with increasing tree height, implying that surviving trees mainly increase their leaf area by radial growth (assuming that the leaf area index remains constant during thinning). Average mass density, estimated on the basis of stand weight and stand volume, decreases as a population develops. The performance of B. pubescens stands during self-thinning is in close agreement with the prediction of the extended allometric model.

The Place of the Self-Thinning Rule in Population Dynamics

The Place of the Self-Thinning Rule in Population Dynamics