The dynamics of the radiative envelope of rapidly rotating stars. I. A spherical Boussinesq model

Context: The observations of rapidly rotating stars are increasingly detailed and precise thanks to interferometry and asteroseismology; two-dimensional models taking into account the hydrodynamics of these stars are very much needed. Aims: A model for studying the dynamics of baroclinic stellar envelope is presented. Methods: This models treats the stellar fluid at the Boussinesq approximation and assumes that it is contained in a rigid spherical domain. The temperature field along with the rotation of the system generate the baroclinic flow. Results: We manage to give an analytical solution to the asymptotic problem at small Ekman and Prandtl numbers. We show that, provided the Brunt-Vaisala frequency profile is smooth enough, differential rotation of a stably stratified envelope takes the form a fast rotating pole and a slow equator while it is the opposite in a convective envelope. We also show that at low Prandtl numbers and without $\mu$-barriers, the jump in viscosity at the core-envelope boundary generates a shear layer staying along the tangential cylinder of the core. Its role in mixing processes is discussed. Conclusions: Such a model provides an interesting tool for investigating the fluid dynamics of rotating stars in particular for the study of the various instabilities affecting baroclinic flows or, even more, of a dynamo effect.Comment: 17 pages, accepted in Astronomy and Astrophysic

Frequency ratio method for seismic modeling of Gamma Doradus stars

A method for obtaining asteroseismological information of a Gamma Doradus oscillating star showing at least three pulsation frequencies is presented. This method is based on a first-order asymptotic g-mode expression, in agreement with the internal structure of Gamma Doradus stars. The information obtained is twofold: 1) a possible identification of the radial order n and degree l of observed frequencies (assuming that these have the same l), and 2) an estimate of the integral of the buoyancy frequency (Brunt-Vaisala) weighted over the stellar radius along the radiative zone. The accuracy of the method as well as its theoretical consistency are also discussed for a typical Gamma Doradus stellar model. Finally, the frequency ratios method has been tested with observed frequencies of the Gamma Doradus star HD 12901. The number of representative models verifying the complete set of constraints (the location in the HR diagram, the Brunt-Vaisala frequency integral, the observed metallicity and frequencies and a reliable identification of n and l) is drastically reduced to six.Comment: 11 pages, 12 figure

Finite-amplitude disturbances in self-gravitating media. II

AbstractThis paper deals with finite-amplitude axisymmetric disturbances in a self-gravitating fluid column of finite radius R. It is shown that the cutoff wavelength λnl above which gravitational breakup occurs now depends on the relative amplitude ϵR of the initial perturbation. Actually, for small-but finite-amplitude disturbances, λnl = λl (1 − 0.34368 ϵ2R2), where λl ( = 5.8898R) designates the cutoff wavelength predicted in the linear approximation

Do structured methods help eco-innovation: An evaluation of the product ideas tree diagram

This paper reports on the first test of the Product Ideas Tree diagram (PIT): a structured method aimed to help Eco-innovation. The PIT diagram structures ideas output from chaotic idea generating sessions. This study compared four ways of conducting an Eco-innovation workshop. The results show that structured methods help Eco-innovation by improving the constructive communication between the participants. Further development of the PIT diagram promises to contribute several new approaches to sustainable product and process design

Breaking the core-envelope symmetry in p-mode pulsating stars

It has been shown that there is a potential ambiguity in the asteroseismic determination of the location of internal structures in a pulsating star. We show how, in the case of high-order non-radial acoustic modes, it is possible to remove this ambiguity by considering modes of different degree. To support our conclusions we have investigated the seismic signatures of sharp density variations in the structure of quasi-homogeneous models.Comment: 3 pages, 3 figures, accepted for publication in Astronomy and Astrophysic

Self-gravitating Newtonian disks revisited

Recent analytic results concerning stationary, self-gravitating fluids in Newtonian theory are discussed. We give a theorem that forbids infinitely extended fluids, depending on the assumed equation of state and the rotation law. This part extends previous results that have been obtained for static configurations. The second part discusses a Sobolev bound on the mass of the fluid and a rigorous Jeans-type inequality that is valid in the stationary case.Comment: A talk given at the Spanish Relativity Meeting in Portugal 2012. To appear in Progress in Mathematical Relativity, Gravitation and Cosmology, Proceedings of the Spanish Relativity Meeting ERE2012, University of Minho, Guimaraes, Portugal, 3-7 September 2012, Springer Proceedings in Mathematics & Statistics, Vol. 6

A Simple Model for Solar Isorotational Contours

The solar convective zone, or SCZ, is nearly adiabatic and marginally convectively unstable. But the SCZ is also in a state of differential rotation, and its dynamical stability properties are those of a weakly magnetized gas. This renders it far more prone to rapidly growing rotational baroclinic instabilities than a hydrodynamical system would be. These instabilities should be treated on the same footing as convective instabilites. If isentropic and isorotational surfaces coincide in the SCZ, the gas is marginally (un)stable to {\em both} convective and rotational disturbances. This is a plausible resolution for the instabilities associated with these more general rotating convective systems. This motivates an analysis of the thermal wind equation in which isentropes and isorotational surfaces are identical. The characteristics of this partial differential equation correspond to isorotation contours, and their form may be deduced even without precise knowledge of how the entropy and rotation are functionally related. Although the exact solution of the global SCZ problem in principle requires this knowledge, even the simplest models produce striking results in broad agreement with helioseismology data. This includes horizontal (i.e. quasi-spherical) isorotational contours at the poles, axial contours at the equator, and approximately radial contours at midlatitudes. The theory does not apply directly to the tachocline, where a simple thermal wind balance is not expected to be valid. The work presented here is subject to tests of self-consistency, among them the prediction that there should be good agreement between isentropes and isorotational contours in sufficiently well-resolved large scale numerical MHD simulations.Comment: Final version: 21 pages, 4 figures, to appear in MNRAS; thorough revision, typos and minor errors corrected, expanded development and reordering of the material. Conclusions unchanged from origina

The rotational broadening and the mass of the donor star of GRS 1915+105

The binary parameters of the microquasar GRS 1915+105 have been determined by the detection of Doppler-shifted 12CO and 13CO lines in its K-band spectrum (Greiner et al., 2001, Nature, 414, 522). Here, we present further analysis of the same K-band VLT spectra and we derive a rotational broadening of the donor star of V sin i=26+-3 km/s from the 12CO/13CO lines. Assuming that the K-type star is tidally locked to the black hole and is filling its Roche-lobe surface, then the implied mass ratio is q = M_d/M_x = 0.058+-0.033. This result, combined with (P, K, i)=(33.5 d, 140 km/s, 66 deg) gives a more refined mass estimate for the black hole, $M_x=14.0+-4.4 M_{\odot}$, than previously estimated, using an inclination of i=66+-2 deg (Fender et al. 1999) as derived from the orientation of the radio jets and a more accurate distance. The mass for the early K-type giant star is $M_d=0.81\pm0.53 M_{\odot}$, consistent with a more evolved stripped-giant donor star in GRS 1915+105 than, for example, the donor star of the prototype black-hole X-ray transient, V404 Cyg which has the longest binary period after GRS 1915+105.Comment: 4 pages, 1 figure, A&A Lette

Theoretical study of γ\gamma Doradus pulsations in pre-main sequence stars

The question of the existence of pre-main sequence (PMS) $\gamma$~Doradus ($\gamma$~Dor) has been raised by the observations of young clusters such as NGC~884 hosting $\gamma$~Dor members. We have explored the properties of $\gamma$~Dor type pulsations in a grid of PMS models covering the mass range $1.2 M_\odot < M_* < 2.5 M_\odot$ and we derive the theoretical instability strip (IS) for the PMS $\gamma$~Dor pulsators. We explore the possibility of distinguishing between PMS and MS $\gamma$~Dor by the behaviour of the period spacing of their high order $gravity$-modes ($g$-modes).Comment: 5 pages, 6 figures, Proc. HELAS IV Conference, Lanzarote, February 2010. Eds T. Roca Cort\'es, P. Pall\'e and S. Jim\'enez Reyes. Accepted in Astron. Nac

On general relativistic uniformly rotating white dwarfs

The properties of uniformly rotating white dwarfs (RWDs) are analyzed within the framework of general relativity. Hartle's formalism is applied to construct the internal and external solutions to the Einstein equations. The WD matter is described by the relativistic Feynman-Metropolis-Teller equation of state which generalizes the Salpeter's one by taking into account the finite size of the nuclei, the Coulomb interactions as well as electroweak equilibrium in a self-consistent relativistic fashion. The mass $M$, radius $R$, angular momentum $J$, eccentricity $\epsilon$, and quadrupole moment $Q$ of RWDs are calculated as a function of the central density $\rho_c$ and rotation angular velocity $\Omega$. We construct the region of stability of RWDs ($J$-$M$ plane) taking into account the mass-shedding limit, inverse $\beta$-decay instability, and the boundary established by the turning-points of constant $J$ sequences which separates stable from secularly unstable configurations. We found the minimum rotation periods $\sim 0.3$, 0.5, 0.7 and 2.2 seconds and maximum masses $\sim 1.500$, 1.474, 1.467, 1.202 $M_\odot$ for $^{4}$He, $^{12}$C, $^{16}$O, and $^{56}$Fe WDs respectively. By using the turning-point method we found that RWDs can indeed be axisymmetrically unstable and we give the range of WD parameters where it occurs. We also construct constant rest-mass evolution tracks of RWDs at fixed chemical composition and show that, by loosing angular momentum, sub-Chandrasekhar RWDs (mass smaller than maximum static one) can experience both spin-up and spin-down epochs depending on their initial mass and rotation period while, super-Chandrasekhar RWDs (mass larger than maximum static one), only spin-up.Comment: The Astrophysical Journal; in pres