Nested spheroidal figures of equilibrium III. Connection with the gravitational moments J2nJ_{2n}

We establish, in the framework of the theory of nested figures, the expressions for the gravitational moments $J_{2n}$ of a systems made of ${\cal L}$ homogeneous layers separated by spheroidal surfaces and in relative rotational motion. We then discuss how to solve the inverse problem, which consists in finding the equilibrium configurations (i.e. internal structures) that reproduce ``exactly'' a set of observables, namely the equatorial radius, the total mass, the shape and the first gravitational moments. Two coefficients $J_{2n}$ being constrained per surface, ${\cal L}=1+\frac{n}{2}$ layers ($n$ even) are required to fix $J_2$ to $J_{2n}$. As shown, this problem already suffers from a severe degeneracy, inherent in the fact that two spheroidal surfaces in the system confocal with each other leave unchanged all the moments. The complexity, which increases with the number of layers involved, can be reduced by considering the rotation rate of each layer. Jupiter is used as a test-bed to illustrate the method, concretely for ${\cal L}=2,3$ and $4$. For this planet, the number of possible internal structures is infinite for ${\cal L} > 2$. Intermediate layers can have smaller or larger oblateness, and can rotate slower or faster than the surroundings. Configurations with large and massive cores are always present. Low-mass cores (of the order a few Earth masses) are predicted for ${\cal L} \ge 4$. The results are in good agreement with the numerical solutions obtained from the Self-Consistent-Field method.Comment: Accepted for publication in MNRAS, 24 page

Fabric dependence of wave propagation in anisotropic porous media

Current diagnosis of bone loss and osteoporosis is based on the measurement of the Bone Mineral Density (BMD) or the apparent mass density. Unfortunately, in most clinical ultrasound densitometers: 1) measurements are often performed in a single anatomical direction, 2) only the first wave arriving to the ultrasound probe is characterized, and 3) the analysis of bone status is based on empirical relationships between measurable quantities such as Speed of Sound (SOS) and Broadband Ultrasound Attenuation (BUA) and the density of the porous medium. However, the existence of a second wave in cancellous bone has been reported, which is an unequivocal signature of poroelastic media, as predicted by Biot’s poroelastic wave propagation theory. In this paper the governing equations for wave motion in the linear theory of anisotropic poroelastic materials are developed and extended to include the dependence of the constitutive relations upon fabric - a quantitative stereological measure of the degree of structural anisotropy in the pore architecture of a porous medium. This fabric-dependent anisotropic poroelastic approach is a theoretical framework to describe the microarchitectural-dependent relationship between measurable wave properties and the elastic constants of trabecular bone, and thus represents an alternative for bone quality assessment beyond BMD alone

A computational method for rotating, multi-layer spheroids with internal jumps

Abstract We discuss the structure of differentially rotating, multilayer spheroids containing mass-density jumps and rotational discontinuities at the interfaces. The study is based upon a scale-free, numerical method. Polytropic equations of state and cylindrical rotation profiles are assumed. The Bernoulli equation and the Poisson equation for the gravitational potential are solved for each layer separately on a common computational grid. The 2-layer (core-envelope) case is first investigated in detail. We find that the core and the envelope are not, in general, homothetical in shape (cores are more than spherical than the envelope). Besides, the occurence of a mass-density jump all along the interface is prone to a rotational discontinuity (unless the polytropic indices are the same). In particular, for given rotation laws, the mass-density jump is not uniform along the interface. Tests, trends and examples (e.g. false bipolytrope, critical rotation, degenerate configurations) are given. Next, we consider the general case of systems made of {\cal L}&gt;2 layers. This includes the full equation set, the virial equation, a comprehensive step-by-step procedure and two examples of tripolytropic systems. The properties observed in the 2-layer case hold for any pairs of adjacent layers. In spite of a different internal structure, two multilayer configurations can share the same mass, same axis ratio, same size and same surface velocity (which is measured through a degeneracy parameter). Applications concern the determination of the interior of planets, exoplanets, stars and compact objects.</jats:p

Nested spheroidal figures of equilibrium – III. Connection with the gravitational moments <i>J</i>2<i>n</i>

ABSTRACT We establish, in the framework of the theory of nested figures, the expressions for the gravitational moments J2n of a systems made of ${\cal L}$ homogeneous layers separated by spheroidal surfaces and in relative rotational motion. We then discuss how to solve the inverse problem, which consists in finding the equilibrium configurations (i.e. internal structures) that reproduce ‘exactly’ a set of observables, namely the equatorial radius, the total mass, the shape, and the first gravitational moments. Two coefficients J2n being constrained per surface, ${\cal L}=1+\frac{n}{2}$ layers (n even) are required to fix J2 to J2n. As shown, this problem already suffers from a severe degeneracy, inherent in the fact that two spheroidal surfaces in the system confocal with each other leave unchanged all the moments. The complexity, which increases with the number of layers involved, can be reduced by considering the rotation rate of each layer. Jupiter is used as a test-bed to illustrate the method, concretely for ${\cal L}=2,3$, and 4. For this planet, the number of possible internal structures is infinite for {\cal L} &gt; 2. Intermediate layers can have smaller or larger oblateness, and can rotate slower or faster than the surroundings. Configurations with large and massive cores are always present. Low-mass cores (of the order of a few Earth masses) are predicted for ${\cal L} \ge 4$. The results are in good agreement with the numerical solutions obtained from the self-consistent field method.</jats:p

Rigidly rotating, incompressible spheroid-ring systems: new bifurcations, critical rotations and degenerate states

The equilibrium of incompressible spheroid-ring systems in rigid rotation is investigated by numerical means for a unity density contrast. A great diversity of binary configurations is obtained, with no limit neither in the mass ratio, nor in the orbital separation. We found only detached binaries, meaning that the end-point of the ɛ2-sequence is the single binary state in strict contact, easily prone to mass-exchange. The solutions show a remarkable confinement in the rotation frequency-angular momentum diagram, with a total absence of equilibrium for Ω2/πGρ ≳ 0.21. A short band of degeneracy is present next to the one-ring sequence. We unveil a continuum of bifurcations all along the ascending side of the Maclaurin sequence for eccentricities of the ellipsoid less than ≈0.612 and which involves a gradually expanding, initially massless loop

Multi-body figures of equilibrium in axial symmetry

We present an efficient multi-body code devoted to self-gravitating polytropic stars and rings in mutual gravitational interaction. The code implements the Self-Consistent Field method which captures solutions in an iterative manner. It works for any positive polytropic index, rotation law and configuration (axis ratios and relative separations). The number of bodies is free. We have investigated a wide range of equilibria involving 2 up to 8 bodies. A model for the disk around HL Tau is currently under progress

The exterior gravitational potential of toroids

ABSTRACT We perform a bivariate Taylor expansion of the axisymmetric Green function in order to determine the exterior potential of a static thin toroidal shell having a circular section, as given by the Laplace equation. This expansion, performed at the centre of the section, consists in an infinite series in the powers of the minor-to-major radius ratio e of the shell. It is appropriate for a solid, homogeneous torus, as well as for inhomogeneous bodies (the case of a core stratification is considered). We show that the leading term is identical to the potential of a loop having the same main radius and the same mass – this ‘similarity’ is shown to hold in the ${\cal O}(e^2)$ order. The series converges very well, especially close to the surface of the toroid where the average relative precision is ∼10−3 for e = 0.1 at order zero, and as low as a few 10−6 at second order. The Laplace equation is satisfied exactly in every order, so no extra density is induced by truncation. The gravitational acceleration, important in dynamical studies, is reproduced with the same accuracy. The technique also applies to the magnetic potential and field generated by azimuthal currents as met in terrestrial and astrophysical plasmas.</jats:p

The exterior gravitational potential of toroids

International audienceWe perform a bivariate Taylor expansion of the axisymmetric Green function in order to determine the exterior potential of a static thin toroidal shell having a circular section, as given by the Laplace equation. This expansion, performed at the centre of the section, consists in an infinite series in the powers of the minor-to-major radius ratio $e$ of the shell. It is appropriate for a solid, homogeneous torus, as well as for inhomogeneous bodies (the case of a core stratification is considered). We show that the leading term is identical to the potential of a loop having the same main radius and the same mass | this "similarity" is shown to hold in the ${\cal O}(e^2)$ order. The series converges very well, especially close to the surface of the toroid where the average relative precision is $\sim 10^{-3}$ for $e\! = \!0.1$ at order zero, and as low as a few $10^{-6}$ at second order. The Laplace equation is satisfied {\em exactly} in every order, so no extra density is induced by truncation. The gravitational acceleration, important in dynamical studies, is reproduced with the same accuracy. The technique also applies to the magnetic potential and field generated by azimuthal currents as met in terrestrial and astrophysical plasmas

Comparison of synchrotron radiation and conventional X-ray microcomputed tomography for assessing trabecular bone microarchitecture of human femoral heads

articleMicrocomputed tomography (microCT) produces three-dimensional (3D) images of trabecular bone. We compared conventional microCT (CmicroCT) with a polychromatic x-ray cone beam to synchrotron radiation (SR) microCT with a monochromatic parallel beam for assessing trabecular bone microarchitecture of 14 subchondral femoral head specimens from patients with osteoarthritis (n=10) or osteoporosis (n=4). SRmicroCT images with a voxel size of 10.13 microm were reconstructed from 900 2D radiographic projections (angular step, 0.2 degrees). CmicroCT images with a voxel size of 10.77 microm were reconstructed from 205, 413, and 825 projections obtained using angular steps of 0.9 degrees, 0.45 degrees, and 0.23 degrees, respectively. A single threshold was used to binarize the images. We computed bone volume/ tissue volume (BV/TV), bone surface/bone volume (BS/BV), trabecular number (Tb.N), trabecular thickness (Tb.Th and Tb.Th*), trabecular spacing (Tb.Sp), degree of anisotropy (DA), and Euler density. With the 0.9 degrees angular step, all CmicroCT values were significantly different from SRmicroCT values. With the 0.23 degrees and 0.45 degrees rotation steps, BV/TV, Tb.Th, and BS/BV by CmicroCT differed significantly from the values by SRmicroCT. The error due to slice matching (visual site matching +/- 10 slices) was within 1% for most parameters. Compared to SRmicroCT, BV/TV, Tb.Sp, and Tb.Th by CmicroCT were underestimated, whereas Tb.N and Tb. Th* were overestimated. A Bland and Altman plot showed no bias for Tb.N or DA. Bias was -0.8 +/- 1.0%, +5.0 +/- 1.1 microm, -5.9 +/- 6.3 microm, and -5.7 +/- 29.1 microm for BV/TV, Tb.Th*, Tb.Th, and Tb.Sp, respectively, and the differences did not vary over the range of values. Although systematic differences were noted between SRmicroCT and CmicroCT values, correlations between the techniques were high and the differences would probably not change the discrimination between study groups. CmicroCT provides a reliable 3D assessment of human defatted bone when working at the 0.23 degrees or 0.45 degrees rotation step; the 0.9 degrees rotation step may be insufficiently accurate for morphological bone analysis