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

^{1}
Instituto de Astrofísica de Andalucía,
CSIC, Apartado 3004,
18080
Granada,
Spain

e-mail:
claret@iaa.es

^{2}
Department of Physics, University of Basel,
Klingelbergstr. 82,
4056
Basel,
Switzerland

Received:
15
October
2012

Accepted:
30
January
2013

*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} *Lambd**a*^{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}.

Key words: stars: evolution / stars: interiors / stars: pre-main sequence / stars: neutron / white dwarfs

*© ESO, 2013*