Quasi-stars, giants and the Sch\"onberg-Chandrasekhar limit

The Sch\"onberg-Chandrasekhar (SC) limit is a well-established result in the understanding of stellar evolution. It provides an estimate of the point at which an evolved isothermal core embedded in an extended envelope begins to contract. We investigate contours of constant fractional mass in terms of homology invariant variables U and V and find that the SC limit exists because the isothermal core solution does not intersect all the contours for an envelope with polytropic index 3. We find that this analysis also applies to similar limits in the literature including the inner mass limit for polytropic models of quasi-stars. Consequently, any core solution that does not intersect all the fractional mass contours exhibits an associated limit and we identify several relevant cases where this is so. We show that a composite polytrope is at a fractional core mass limit when its core solution touches but does not cross the contour of the corresponding fractional core mass. We apply this test to realistic models of helium stars and find that stars typically expand when their cores are near a mass limit. Furthermore, it appears that stars that evolve into giants have always first exceeded an SC-like limit.Comment: 9 pages, 7 figures, published in MNRAS. Updated to (more closely) match published versio

General Relativistic Stars : Polytropic Equations of State

In this paper, the gravitational field equations for static spherically symmetric perfect fluid models with a polytropic equation of state, $p=k\rho^{1+1/n}$, are recast into two complementary 3-dimensional {\it regular} systems of ordinary differential equations on compact state spaces. The systems are analyzed numerically and qualitatively, using the theory of dynamical systems. Certain key solutions are shown to form building blocks which, to a large extent, determine the remaining solution structure. In one formulation, there exists a monotone function that forces the general relativistic solutions towards a part of the boundary of the state space that corresponds to the low pressure limit. The solutions on this boundary describe Newtonian models and thus the relationship to the Newtonian solution space is clearly displayed. It is numerically demonstrated that general relativistic models have finite radii when the polytropic index $n$ satisfies $0\leq n \lesssim 3.339$ and infinite radii when $n\geq 5$. When $3.339\lesssim n<5$, there exists a 1-parameter set of models with finite radii and a finite number, depending on $n$, with infinite radii.Comment: 31 pages, 10 figure

Nonlinear stability analysis of the Emden-Fowler equation

In this paper we qualitatively study radial solutions of the semilinear elliptic equation $\Delta u + u^n = 0$ with $u(0)=1$ and $u'(0)=0$ on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great importance in Newtonian astrophysics and the constant $n$ is called the polytropic index. By introducing a set of new variables, the Emden-Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of $n$ these different methods yield different results. We identify a parameter range for $n$ where all three methods imply stability.Comment: 12 pages; new reference added; 3 new references added; fully revised versio

Differential rotation in fully convective stars

Under the assumption of thermal wind balance and effective entropy mixing in constant rotation surfaces, the isorotational contours of the solar convective zone may be reproduced with great fidelity. Even at this early stage of development, this helioseismology fit may be used to put a lower bound on the midlatitude {\em radial} solar entropy gradient, which in good accord with standard mixing length theory. In this paper, we generalize this solar calculation to fully convective stars (and potentially planets), retaining the assumptions of thermal wind balance and effective entropy mixing in isorotational surfaces. It is found that each isorotation contour is of the form $R^2 = A+B\Phi(r)$, where $R$ is the radius from the rotation axis, $\Phi(r)$ is the (assumed spherical) gravitational potential, and $A$ and $B$ are constant along the contour. This result is applied to simple models of fully convective stars. Both solar-like surface rotation profiles (angular velocity decreasing toward the poles) as well as "antisolar" profiles (angular velocity increasing toward the poles) are modeled; the latter bear some suggestive resemblance to numerical simulations. We also perform exploratory studies of zonal surface flows similar to those seen in Jupiter and Saturn. In addition to providing a practical framework for understanding the results of large scale numerical simulations, our findings may also prove useful in dynamical calculations for which a simple but viable model for the background rotation profile in a convecting fluid is needed. Finally, our work bears directly on an important goal of the CoRoT program: to elucidate the internal structure of rotating, convecting stars.Comment: 21 pages, 20 figures. Accepted for publication in MNRA

Merger of white dwarf-neutron star binaries: Prelude to hydrodynamic simulations in general relativity

White dwarf-neutron star binaries generate detectable gravitational radiation. We construct Newtonian equilibrium models of corotational white dwarf-neutron star (WDNS) binaries in circular orbit and find that these models terminate at the Roche limit. At this point the binary will undergo either stable mass transfer (SMT) and evolve on a secular time scale, or unstable mass transfer (UMT), which results in the tidal disruption of the WD. The path a given binary will follow depends primarily on its mass ratio. We analyze the fate of known WDNS binaries and use population synthesis results to estimate the number of LISA-resolved galactic binaries that will undergo either SMT or UMT. We model the quasistationary SMT epoch by solving a set of simple ordinary differential equations and compute the corresponding gravitational waveforms. Finally, we discuss in general terms the possible fate of binaries that undergo UMT and construct approximate Newtonian equilibrium configurations of merged WDNS remnants. We use these configurations to assess plausible outcomes of our future, fully relativistic simulations of these systems. If sufficient WD debris lands on the NS, the remnant may collapse, whereby the gravitational waves from the inspiral, merger, and collapse phases will sweep from LISA through LIGO frequency bands. If the debris forms a disk about the NS, it may fragment and form planets.Comment: 28 pages, 25 figures, 6 table

Inverse approach to Einstein's equations for fluids with vanishing anisotropic stress tensor

We expand previous work on an inverse approach to Einstein Field Equations where we include fluids with energy flux and consider the vanishing of the anisotropic stress tensor. We consider the approach using warped product spacetimes of class $B_1$. Although restricted, these spacetimes include many exact solutions of interest to compact object studies and to cosmological models studies. The question explored here is as follows: given a spacetime metric, what fluid flow (timelike congruence), if any, could generate the spacetime via Einstein's equations. We calculate the flow from the condition of a vanishing anisotropic stress tensor and give results in terms of the metric functions in the three canonical types of coordinates. A condition for perfect fluid sources is also provided. The framework developed is algorithmic and suited for the study and validation of exact solutions using computer algebra systems. The framework can be applied to solutions in comoving and non-comoving frames of reference, and examples in different types of coordinates are worked out.Comment: 15 pages, matches version to appear in Phys.Rev.

Boyle's law and gravitational instability

We have re-examined the classical problem of the macroscopic equation of state for a hydrostatic isothermal self-gravitating gas cloud bounded by an external medium at constant pressure. We have obtained analytical conditions for its equilibrium and stability without imposing any specific shape and symmetry to the cloud density distribution. The equilibrium condition can be stated in the form of an upper limit to the cloud mass; this is found to be inversely proportional to the power 3/2 of a form factor \mu characterizing the shape of the cloud. In this respect, the spherical solution, associated with the maximum value of the form factor, \mu = 1, turns out to correspond to the shape that is most difficult to realize. Surprisingly, the condition that defines the onset of the Bonnor instability (or gravothermal catastrophe) can be cast in the form of an upper limit to the density contrast within the cloud that is independent of the cloud shape. We have then carried out a similar analysis in the two-dimensional case of infinite cylinders, without assuming axisymmetry. The results obtained in this paper generalize well-known results available for spherical or axisymmetric cylindrical isothermal clouds that have had wide astrophysical applications, especially in the study of the interstellar medium.Comment: 9 pages, 2 figures, to appear in A&

The Role of Ejecta in the Small Crater Populations on the Mid-Sized Saturnian Satellites

We find evidence that crater ejecta play an important role in the small crater populations on the Saturnian satellites, and more broadly, on cratered surfaces throughout the Solar System. We measure crater populations in Cassini images of Enceladus, Rhea, and Mimas, focusing on image data with scales less than 500 m/pixel. We use recent updates to crater scaling laws and their constants to estimate the amount of mass ejected in three different velocity ranges: (i) greater than escape velocity, (ii) less than escape velocity and faster than the minimum velocity required to make a secondary crater (v_min), and (iii) velocities less than v_min. Although the vast majority of mass on each satellite is ejected at speeds less than v_min, our calculations demonstrate that the differences in mass available in the other two categories should lead to observable differences in the small crater populations; the predictions are borne out by the measurements we have made to date. Rhea, Tethys, and Dione have sufficient surface gravities to retain ejecta moving fast enough to make secondary crater populations. The smaller satellites, such as Enceladus but especially Mimas, are expected to have little or no traditional secondary populations because their escape velocities are near the threshold velocity necessary to make a secondary crater. Our work clarifies why the Galilean satellites have extensive secondary crater populations relative to the Saturnian satellites. The presence, extent, and sizes of sesquinary craters (craters formed by ejecta that escape into temporary orbits around Saturn before re-impacting the surface) is not yet well understood. Finally, our work provides further evidence for a "shallow" size-frequency distribution (slope index of ~2 for a differential power-law) for comets a few km diameter and smaller. [slightly abbreviated]Comment: Submitted to Icarus. 77 double-spaced pages, including 25 figures and 5 table

An improved algorithm of second order to construct consistent theories of equilibrium figures of rotating celestial bodies

One of the main problems in celestial mechanics is the study of the figure adopted by large deformable bodies in slow rotation around an axis with a constant angular velocity W when they reach their equilibrium configuration. This figure corresponds to the lowest equipotential surface containing the entire mass and, in order to determine it, calculating its potential at an arbitrary point is required. Classical methods address this problem generally by performing a series development of the inverse-distance by using Clairaut’s coordinates. These methods show convergence problems, already in first order in W2, so that to avoid them they must assume no demonstrated hypotheses. The authors of this paper warned and proved this fact in a previous work, for which they used two methods: 1. Taking into account the asymptotic properties of numeric quadrature formulas. 2. By a process similar to that used by Laplace to develop the inverse of the distance between two planets. Thus, the authors demonstrated that, although the intermediate formulas obtained by the classical methods are wrong, the self-gravitational potential, up to first order in w2, obtained with them, were coincident with those obtained by other methods. Now, in this paper an extension up to second order in w2 of the first method mentioned above is proposed. It shows rigorously that up to second order in , the intermediate formulas obtained by the classical methods are erroneous. In addition, the correct developments, up to second order in W2, of those intermediate formulas are obtained. In spite of these discrepancies, the auto-gravitational potential, up to second order in , obtained by the classical methods and the one obtained in this work are coincident, consequently it is proved that the classical theory is correct until second order in W2

Light curves and colours of the faint Uranian irregular satellites Sycorax, Prospero, Stephano, Setebos and Trinculo

After the work of Gladman et al. (1998), it is now assessed that many irregular satellites are orbiting around Uranus. Despite many studies have been performed in past years, very few is know for the light-curves of these objects and inconsistencies are present between colours derived by different authors. This situation motivated our effort to improve both the knowledge of colours and light curves. We present and discuss time series observations of Sycorax, Prospero, Stephano, Setebos and Trinculo, five faint irregular satellites of Uranus, carried out at VLT, ESO Paranal (Chile) in the nights between 29 and 30 July, 2005 and 25 and 30 November, 2005. We derive light curves for Sycorax and Prospero and colours for all of these these bodies. For Sycorax we obtain colours B-V =0.839 +/- 0.014, V-R = 0.531 +/- 0.005 and a light curve which is suggestive of a periodical variation with period about 3.6 hours and amplitude about 0.067 +/- 0.004 mag. The periods and colours we derive for Sycorax are in agreement with our previous determination in 1999 using NTT. We derive also a light-curve for Prospero which suggests an amplitude of about 0.2 mag and a periodicity of about 4 hours. However, the sparseness of our data, prevents a more precise characterization of the light-curves, and we can not determine wether they are one-peaked or two-peaked. Hence, these periods and amplitudes have to be considered preliminary estimates. As for Setebos, Stephano and Trinculo the present data do not allow to derive any unambiguous periodicity, despite Setebos displays a significant variability with amplitude about as large as that of Prospero. Colours for Prospero, Setebos, Stephano and Trinculo are in marginal agreement with the literature.Comment: Submitted to A&A 13 Dec 2006, Accepted 17 Apr 2007. 18 pages, 8 colours figures BW printable, 6 tables. LaTeX 2.09, with packages: natbib, graphicx, longtable, aa4babbage included in the submission file (tar gzipped of 349 KBytes