Instituto Federal de Educação, Ciência e Tecnologia de São Paulo, IFSP, São José dos Campos 12.223-201, Brazil
Physics and Astronomy Laboratory, Universidade do Vale do Paraíba, UNIVAP, São José dos Campos 72840, Brazil
Instituto Nacional de Astrofísica, Óptica y Electrónica, INAOE, Puebla, Mexico
Author to whom correspondence should be addressed.
Received: 3 November 2017 / Accepted: 5 December 2017 / Published: 15 December 2017
One of the algebraic potentials most commonly used to represent a galactic bar in the stellar orbits integration is the Ferrers potential. Some researchers may be inclined to implement a numerical differentiation for it in the motion or variational equations, since it can be very laborious to calculate such derivatives algebraically, despite a possible polynomial form, and there are no publications showing the second partial explicit derivatives. The purpose of this work is to present the explicit algebraic form of the partial derivatives of the Ferrers potential using the simplifications suggested by Pfenniger.
It is estimated that 65% of known disk galaxies has a bar substructure [1,2]. This justifies the importance of studying galaxies of this type. It is common to use the integration of stellar orbits immersed in gravitational potentials to conduct a dynamic study of galaxies, including barred galaxies. In this context, the galaxy to be studied is modeled as a composite of several models of gravitational potential that represent each galactic substructure.
In dealing with the bar substructure, there are several analytical potentials that can be chosen to compose the total analytical potential model. The Ferrers potential has a great fidelity in the representation of the galactic bar and it has been extensively used in many works (e.g., [3,4,5,6]). However it does not have a friendly mathematical constitution to work analytically. Some researchers implement a Ferrers numerical differentiation in the motion or variational equations, since it can be very laborious to calculate such derivatives algebraically and there are no publications showing the second partial explicit derivatives.
Therefore, the purpose of this work is to present the explicit algebraic form of the partial derivatives of the Ferrers potential using the simplifications to polynomial form suggested by Pfenniger .
2. The Ferrers Potential
This potential model was proposed in 1877 by Ferrers .
Bringing to the bars context, in this model, the density is given by
where the central density is , is the bar mass and , where are the semi-axes of the ellipsoid which represents the bar.
The potential created by the galactic bar is:
where and is the positive solution of for the region outside the bar () and for the region inside the bar ().
In 1984, Pfenniger , using the Generalized Multinomial Theorem in , simplified the Ferrers potential expression.
where depends on . We can write the potential1 given by the Equation (2) as
The calculations to obtain the values depends on first and second order elliptic integrals and a extensive mathematics. Such calculations, which can be checked in , will not be exposed here.
Finally, after all the algebraic manipulation presented, we have that the Ferrers potential can be compacted in the following polynomial way:
3. The Partial Derivatives
The values depend on and depend on and z. Thus, to calculate the partial derivatives of this potential, it is necessary the derivatives , and .
To this end, the quantities , , and are:
Also note that .
Taking into account the facts described in the previous lines, the partial derivatives of this potential are presented below. The first derivatives were introduced by Pfenniger  and will be presented here for completeness. The main contribution of this paper is to present the second derivatives.
We invite the reader to know our work  where we use all the equations shown in this text to study the influence of galactic bars on the stability of the orbits supported by them. In order to write this study, all these equations were exhaustively tested.
In this text, we have studied some of the algebraic manipulations made by Pfenniger in the Ferrers potential, in order to make this potential more accessible in a polynomial form.
We also calculated the partial derivatives of this potential in an algebraic and explicit way. Such derivatives are laborious despite the polynomial form. Thus, those who wish to implement these derivatives in the motion or variational equations to study the stellar orbits dynamics in analytical potentials, for example, can use the above analytical form instead of numerical methods.
We acknowledge the Brazilian agencies FAPESP (2013/17247-9), CNPq (200906-2015-1) and CAPES, as well as the Mexican agency CONACyT (CB-2014-240426) for supporting this work.
The authors contributed equally to this paper.
Conflicts of Interest
The authors declare no conflict of interest.
Eskridge, P.B.; Frogel, J.A.; Pogge, R.W.; Quillen, A.C.; Davies, R.L.; DePoy, D.L.; Houdashelt, M.L.; Kuchinski, L.E.; Ramírez, S.V.; Sellgren, K.; et al. The Frequency of Barred Spiral Galaxies in the Near-Infrared. Astron. J.2000, 119, 536–544. [Google Scholar] [CrossRef]
Sheth, K.; Regan, M.W.; Scoville, N.Z.; Strubbe, L.E. Barred Galaxies at z > 0.7: NICMOS Hubble Deep Field-North Observations. Astron. J.2003, 592, L13–L16. [Google Scholar] [CrossRef]
Caritá, L.A.; Rodrigues, I.; Puerari, I.; Schiavo, L.E.C.A. The Smaller Alignment Index (SALI) applied in a study of stellar orbits in barred galaxies potential models using the LP-VIcode. New Astron.2018, 60, 48–60. [Google Scholar] [CrossRef]
Patsis, P.A. Orbital Morphology of 3D Bars. In Disks of Galaxies: Kinematics, Dynamics and Peturbations; Astronomical Society of the Pacific: San Francisco, CA, USA, 2002; Volume 275, pp. 161–168. [Google Scholar]