Advances in Astronomy

Volume 2016, Article ID 9193627, 10 pages

http://dx.doi.org/10.1155/2016/9193627

## Effective Perihelion Advance and Potentials in a Conformastatic Background with Magnetic Field

^{1}Federal University for Latin American Integration, Apartado Postal 2123, 85867-970 Foz do Iguaçu, PR, Brazil^{2}Casimiro Montenegro Filho Astronomy Center, Itaipu Technological Park, 85867-970 Foz do Iguaçu, PR, Brazil^{3}Facultad de Ciencias Básicas, Universidad Tecnológica de Bolívar, CP 131001, Cartagena, Colombia^{4}Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70543, 04510 Mexico City, Mexico

Received 8 August 2016; Accepted 18 October 2016

Academic Editor: Josep M. Trigo-Rodríguez

Copyright © 2016 Abraão J. S. Capistrano and Antonio C. Gutiérrez-Piñeres. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Exact solutions of the Einstein-Maxwell field equations for a conformastatic metric with magnetized sources are investigated. In this context, effective potentials are studied in order to understand the dynamics of the magnetic field in galaxies. We derive the equations of motion for neutral and charged particles in a spacetime background characterized by this class of solutions. In this particular case, we investigate the main physical properties of the equatorial circular orbits and related effective potentials. In addition, we obtain an effective analytic expression for the perihelion advance of test particles. Our theoretical predictions are compared with the observational data calibrated with the ephemerides of the planets of the solar system and the Moon (EPM2011). In general, we show that the magnetic punctual mass predicts values that are in better agreement with observations than the values predicted in Einstein’s gravity alone.

#### 1. Introduction

Magnetics fields are extensively studied in literature and their influence on the galactic dynamics are currently subject of active research, for example, on the understanding of the galactic jets and inner process of “active” galaxy core, neutron stars dynamics [1], and/or movement of charged particles in spacetimes [2–4] or neutral particles in charged galactic halo [5–7]. In summary, they are present in almost every celestial object from stars, pulsars, and nearby galaxies to clusters of galaxies. An interesting review can be found in [8–12]. To this matter, the Einstein-Maxwell equations have been revealed to be an important tool to deal with this problem and help us on the understanding of the dynamics of magnetic fields in galaxies. Important approaches are the relativistic models with disk like configurations and relativistic disk accretion models as proposed in recent years [13–18] and references therein. In a recent publication [15], we studied the behavior of a test particle submitted to a magnetic field in a relativistic galaxy disk model and how its influence may affect the dynamics. In a different approach, in this paper we investigate effective potentials using Einstein-Maxwell equations motivated by the necessity to understand how the dynamics of a galaxy respond to flattening and how the magnetic field may be related to this process. This may be a basis for futures advances in both galactic and stellar formations. We also explore the possibility of an influence of magnetic field in the apsidal precession in solar system scale, in other words, how it might affect the movement of test particles embedded in solar gravitational field. To this matter, we use data calibrated with the ephemerides of the planets of the solar system and the Moon (EPM2011) [19, 20].

The present paper is divided into sections. In Section 2, we study the basic framework of a conformastatic background and investigate some applications using the isothermal-sphere logarithm potential and Toomre-Kuzmin-like potential, which are compatible with axisymmetric systems. In Section 3, we obtain an expression for the perihelion advance of a charged test particle in a generic conformastatic spacetime in the presence of a magnetic field and perform a comparison between our results, the results from Einstein’s gravity alone, and the values observed for the secular perihelion precession of some inner planets and minor objects of the solar system. In the conclusion section, we make the final considerations.

#### 2. The Conformastatic Background

General relativistic scenaries described by using a conformally flat space of orbits are very alluring from mathematical point of view and in physical applications. Using the definition proposed by Synge [21], we start our analysis with the use of conformally flat space, which is the main characteristic of a conformastatic spacetime (e.g., the Schwarzschild metric). Considering the background of a conformastatic gravitational source in presence of a magnetic field described by the line element in standard cylindrical coordinates, one can write [14]where the metric potential depends only on the variables and . The vacuum Einstein-Maxwell equations in geometrized units, such that , are given bywhere and is the electromagnetic energy-momentum tensor. The Greek indices run from 1 to 4.

With the electromagnetic potential and the line element in (1) the Einstein-Maxwell equations in (2a) and (2b) are equivalent to the system of equationsBy using the procedure to obtain solutions of the Einstein-Maxwell equations presented in [14], suitable solutions of the system in (4a), (4b), (4c), (4d), and (4e) can be displayed aswhere is a solution of Laplace’s equation.

##### 2.1. Motion of Test Charged Particles

The motion of a test particle of charge and mass moving in a magnetized background is described by the Lagrangianwhere , being an arbitrary parameter. The corresponding Hamiltonian of the particle iswhere the canonical momentum is given by . The motion equations are given bywhere . Accordingly, by introducing (7) into (8a) and (8b) we obtainFrom (7) and the normalization condition (with for space-like, null, and time-like curves) we have the conditionOn the another hand, from (9a) and (9b), we haveand alsorespectively, whereas, from (9c) and (9d), we obtainwhere

We can write the last system in the formwhere is called the “effective potential” (see equations (3.68) pg. 160 in [16]). In terms of the solution in (5a), (5b), and (5c) one obtainswhereThus the three-dimensional motion of the particle in an axis-symmetric potential can be reduced to the two-dimensional motion of the particle in a “Newtonian potential” .

##### 2.2. Circular Motion in the Plane

To study the circular motion of the test charged particle we start with the conditionsThen, from the first of these equations, (7) and (10), we have the energy of the particle as follows:From the second condition in (20) we haveNotice that if , from (21) and (14), we obtainThus, by introducing the corresponding metric coefficients of the line element in (1), such aswhich lacks physical meaning, hence the condition is equivalent to

The minimum radius for stable circular orbit occurs in the inflection points of the effective potential. Thus we must solve the equationor, equivalently, solve the equationOn the other hand, by calculating the derivative with respect to in both sides of (21), we obtain for the angular momentwhere and we have used the Einstein-Maxwell equation . By substituting this value for in (21) we obtain the energy of the particle as follows:Since the Lagrangian in (6) does not depend explicitly on the variables and , one can obtain the following two conserved quantities:and alsowhere and are, respectively, the energy and the angular momentum of the particle as measured by an observer at rest at infinity. Furthermore, the momentum of the particle can be normalized so that . Accordingly, for the metric in (1), we havewhere, with , the notation , 0, denotes space-like, null, and time-like curves, respectively.

As an application of (18), we use an axial bidimensional isothermal potential, which has the form and straightforwardly we get the expression Hence, integrating the former expression, it is necessary to obtain a convergence of the integral away from origin; we use a Laurent expansion . Finally, after long algebra, we can write the form of the effective potential felt by charged particle with mass moving with velocity and total angular momentum as follows: where we denote and . In Figure 1, we notice that a small value of the velocity induces outgoing lines from the center as expected, as noted in the three panels. In (b), we notice that time-like curves suggest that the magnetic lines distort the path of a test charged particle away from the center of the galaxy.