An all-purpose metric for the exterior of any kind of rotating neutron star
George Pappas and Theocharis A. Apostolatos
Section of Astrophysics, Astronomy, and Mechanics, Department of Physics, University of Athens, Panepistimiopolis Zografos GR15783, Athens, Greece
Theoretical Astrophysics, IAAT, Eberhard Karls University of Tuebingen, Tuebingen 72076, Germany
Abstract:
We have tested the appropriateness of two-soliton analytic metric to describe the exterior of all types of neutron stars, no matter what their equation of state or rotation rate is. The particular analytic solution of the vaccuum Einstein equations proved quite adjustable to mimic the metric functions of all numerically constructed neutron-star models that we used as a testbed. The neutron-star models covered a wide range of stiffness, with regard to the equation of state of their interior, and all rotation rates up to the maximum possible rotation rate allowed for each such star. Apart of the metric functions themselves, we have compared the radius of the innermost stable circular orbit $$R_{\rm{ISCO}}$$, the orbital frequency $$\Omega\equiv\frac{d\phi}{dt}$$ of circular geodesics, and their epicyclic frequencies $$\Omega_{\rho}, \Omega_z$$, as well as the change of the energy of circular orbits per logarithmic change of orbital frequency $$\Delta\tilde{E}$$. All these quantities, calculated by means of the two-soliton analytic metric, fitted with good accuracy the corresponding numerical ones as in previous analogous comparisons (although previous attempts were restricted to neutron star models with either high or low rotation rates). We believe that this particular analytic solution could be considered as an analytic faithful representation of the gravitation field of any rotating neutron star with such accuracy, that one could explore the interior structure of a neutron star by using this space-time to interpret observations of astrophysical processes that take place around it.
In this webpage one can find the expressions that define the multipole moments of the two-soliton spacetime, as well as the expressions that define the metric functions of the various spacetimes that were used, i.e., the two-soliton spacetime, the Manko et al. spacetime and the Hartle-Thorne spacetime.
One will also find the demonstration for the calculation of the first multipole moments for the Hartle-Thorne metric.
These same expressions are given also in the form of a Mathematic 5.2 notebook for further use in calculations.
In this additional webpage one can find a demonstration containing the results of the comparison between the two-soliton spacetime and a specific numerical neutron star moddel (a Case IIa spacetime corresponding to a neutron star model of fast rotation, constructed with L EOS).
This webpage is produced from this Mathematica 5.2 notebook, which one can download and use. For running the notebook one should also have the data files for the specific neutron star. These data files can be downloaded from this link and should be placed in an empty folder, the path of which should be specified in the notebook at the definition:
pathread = "D:\mathematica notebook\NS-model";
replacing the path mentioned there.
The two-soliton spacetime was produced by Manko V. S., Martin J., Ruiz E., 1995, J. Math. Phys., 36, 3063. We should note that there are some notation differences between the cited work and our work presented here and in the notebooks. The reason is that the final expresions for the two-soliton presented in that paper had several typographical errors, so they were rederived from the begining.
Bibliography:
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[2] Manko V. S., Mielke E. W., Sanabria-Gomez J. D., 2000, Phys. Rev. D, 61, 081501
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[5] Manko V. S., Martin J., Ruiz E., 1995, J. Math. Phys., 36, 3063
[6] T P Sotiriou and G Pappas, 2005 J. Phys.: Conf. Ser., 8, 23, (arXiv:gr-qc/0504122)
[7] G Pappas and T A Apostolatos, 2008, Class. Quantum Grav., 25, 228002, (extended version at arXiv:0803.0602v1 [gr-qc])
[8] George Pappas, 2009, J. Phys.: Conf. Ser., 189, 012028, (arXiv:1201.6055 [gr-qc])
update: the correct M_4 moment should have been,
$$\small M_4=\left( a^4 - (3a^2 -2ab + b^2)k + k^2 -\frac{1}{7}kM^2\right) M $$
[9] G Pappas and T A Apostolatos, 2012 Phys. Rev. Lett. 108, 231104, (arXiv:1201.6067 [gr-qc]).
[10] G Pappas and T A Apostolatos, 2013, MNRAS 429, 3007 (arXiv:1209.6148 [gr-qc]).