Research Note

The internal structure of neutron stars and white dwarfs, and the Jacobi virial equation^⋆

Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080 Granada, Spain
e-mail: claret@iaa.es

Received: 6 March 2012
Accepted: 29 May 2012

Abstract

Aims. The internal structure constants k_j and the radius of gyration are useful tools for investigating the apsidal motion and tidal evolution of close binaries and planetary systems. These parameters are available for various evolutionary phases but they are scarce for the late stages of stellar evolution. To cover this gap, we present here the calculations of the apsidal-motion constants, the fractional radius of gyration, and the gravitational potential energy for two grids of cooling evolutionary sequences of white dwarfs and for neutron star models.

Methods. The cooling sequences of white dwarfs were computed with LPCODE. An additional alternative to the white dwarf models was also adopted with the MESA code which allows non-stop calculations from the pre main-sequence (PMS) up to the white dwarf cooling sequences. Neutron star models were acquired from the NSCool/TOV subroutines. The apsidal-motion constants, the moment of inertia, and the gravitational potential energy were computed with a fourth-order Runge-Kutta method.

Results. The parameters are made available for four cooling sequences of white dwarfs (DA and DB types): 0.52, 0.57, 0.837 and 1.0 M_⊙ and for neutron star models covering a mass range from 1.0 up to 2.183 M_⊙, in 0.1 mass step. We show that, contrary to previously established opinion, the product of the form-factors β and α, which are related to the moment of inertia, and gravitational potential energy, is not constant during some evolutionary phases. Regardless of the final products of stellar evolution (white dwarfs, neutron stars and perhaps black holes), we found that they recover the initial value of product αβ at the pre main-sequence phase (≈0.4). These results may have important consequences for the investigation of the Jacobi virial equation.