The internal structure of neutron stars and white dwarfs, and the Jacobi virial equation. II.

Abstract

Context. The Jacobi virial equation is a very powerful tool for exploring several aspects of the stellar internal structure and evolution. In a previous paper we have shown that the function αβ GR /Λ 0.9 (R) is constant (≈0.4) for pre main-sequence stars (PMS), white dwarfs (WD) and for some neutron star (NS) models, where α GR and β GR are the form-factors of the gravitational potential energy and of the moment of inertia. Aims. To investigate the structural evolution of another type of celestial bodies, we extend these calculations to gaseous planets. We also analyse the cases for which this function is not conserved during some stellar evolutionary phases. Concerning NS, we study the influence of the equation of state (EOS) on this function and refine the exponent of the auxiliary function Λ(R). We also present a macroscopic criterion of stability for these stars. Methods. Non-stop calculations from the PMS to the white dwarf cooling sequences were performed with the MESA code. The covered mass range was 0.1-1.7 M . We used the same code to compute models for gaseous planets with masses between 0.1-50 M J . Neutron star models were computed using two codes. The first one is a modified version of the NSCool/TOV subroutines. The second code is a plain TOV solver that allows one to use seven previously described EOS. The relativistic moment of inertia and gravitational potential energy were computed through a fourth-order Runge-Kutta method. Results. By analysing the internal structure of gaseous planets we show that the function αβ GR /Λ 0.8 (R) ≡ Γ(M, EOS) is conserved for all models during the whole planetary evolution and is independent of the planet mass. For the PMS to the white dwarf cooling sequences, we have found a connection between the strong variations of Γ(M, EOS) during the intermediary evolutionary phases and the specific nuclear power. A threshold for the specific nuclear power was found. Below this limit this function is invariant (≈0.4) for these models, i.e., at the initial and final stages (PMS and WD). For NS, we showed that the function Γ(M, EOS) is also invariant (≈0.4) and is independent of the EOS and of the stellar mass. Therefore, we confirm that regardless of the final products of the stellar evolution, NS or WD, they recover the initial value of Γ(M, EOS) ≈ 0.4 acquired at the PMS. Finally, we have introduced a macroscopic stability criterion for NS models based on the properties of the relativistic product αβ GR .

Neutron, quark, and proto-neutron stars at the onset of formation of black-holes: the memory effect

“…In previous papers (Claret 2012;Claret & Hempel 2013) we showed the invariance of the function Γ(M, EOS) for NS models computed with seven relativistic EOS. The model from Hempel & Schaffner-Bielich 2010 was adopted in the EOS tables available on their webpage, using the following relativistic meanfield nucleon interactions: DD2 (Typel et al 2010), FSUgold (Todd-Rutel & Piekarewicz 2005, NL3 (Lalazissis et al 1997), SFHo and SFHx (Steiner et al 2013), TM1 (Sugahara & Toki 1994), and TMA (Toki et al 1995).…”

“…We used the NS models generated with the eight abovementioned EOS to expand our previous test to see whether the function Γ(M, EOS) is independent of the EOS, of the mass, and of the relativistic and non-relativistic framework. As in Claret & Hempel (2013), first we analyse each EOS separately and found the best exponent to the function Λ(R) by means of the leastsquares method (Fig. 1).…”

“…We performed some tests to verify if this is indeed a characteristic of models without crusts. We completed the table for pure quark matter (Bag model with The values of the exponent of Λ(R) listed in Table 1, as well as that considering all EOS together (and the seven ones derived in Claret & Hempel 2013), allow us to consider that Γ(M, EOS) is invariant for neutron/quarks stars and is independent of the adopted EOS, of the stellar mass, and of the adopted relativistic or non-relativistic approach. In previous papers we showed that Γ(M, EOS) is invariant (≈0.40) for PMS and WD models.…”

“…Table 1 are available in electronic form at http://www.aanda.org However, the polytropic models or other simple density distributions adopted by Ferronski et al only cover a limited branch of the Herzsprung-Russel (HR) diagram. In two recent papers that deal with compact stars (Claret 2012;Claret & Hempel 2013) we showed that the function Γ(M, EOS) ≡ αβ GR /Λ 0.8 (R) is invariant (≈0.40) for stars in the pre-mainsequence (PMS) and also at the last stages of evolution, neutron stars (NS) or white dwarfs (WD). In the above definition, Λ(R) is given by 1 − 2GM(R) A&A 562, A31 (2014) for stellar models evolving from the PMS up to the WD phases.…”

“…The invariance of the mentioned function is, in this way, extended to gaseous planets too. Moreover, our investigations concerning the relativistic form-factors α and β allowed us to derive a heuristic macroscopic criterion of stability for NS (Claret & Hempel 2013).…”