The paper deals with an analytical study of various corrected Newtonian potentials. We offer a complete description of the corrected potentials, for the entire range of the parameters involved. These parameters can be fixed for different models in order to obtain a good concordance with known data. Some of the potentials are generated by continued fractions, and another one is derived from the Newtonian potential by adding a logarithmic correction. The zonal potential, which models the motion of a satellite moving in the equatorial plane of the Earth, is also considered. The range of the parameters for which the potentials behave or not similarly to the Newtonian one is pointed out. The shape of the potentials is displayed for all the significant cases, as well as the orbit of Raduga-1M 2 satellite in the field generated by the continued fractional potential U3, and then by the zonal one. For the continued fractional potential U2 we study the basic problem of the existence and linear stability of circular orbits. We prove that such orbits exist and are linearly stable. This qualitative study offers the possibility to choose the adequate potential, either for modeling the motion of planets or satellites, or to explain some phenomena at galactic scale
Authors
Mira Cristiana Anisiu Romanian Academy, Tiberiu Popoviciu Institute of Numerical Analysis, Cluj-Napoca
Iharka Szucs-Csillik Romanian Academy, Institute of Astronomy, Astronomical Observatory Cluj-Napoca
M.-C. Anisiu, I. Szucs-Csillik, Corrected Newtonian potentials in the two-body problem with applications, Astrophys. Space Sci., 361 (2016) 382, 8 pp., https://doi.org/10.1007/s10509-016-2967-x
[1] Abd El-Salam, F.A., Abd El-Bar, S.E., Rasem, M., Alamri, S.Z.: Astrophys. Space Sci.350, 507 (2014) ArticleADSGoogle Scholar
[2] Anisiu, M.-C.: In: Dumitrache, C., Popescu, N.A., Suran, D.M., Mioc, V. (eds.) AIP Conference Proceedings, vol. 895, p. 308 (2007) ChapterGoogle Scholar
[3] Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, Reston (1999) BookMATHGoogle Scholar
[4] Blaga, C.: Rom. Astron. J.25, 233 (2015) Google Scholar
[5] Bozis, G., Anisiu, M.-C., Blaga, C.: Astron. Nachr.318, 313 (1997) ArticleADSGoogle Scholar
[6] Diacu, F., Mioc, V., Stoica, C.: Nonlinear Anal.41, 1029 (2000) ArticleMathSciNetGoogle Scholar
[7] Fabris, J.C., Pereira Campos, J.: Gen. Relativ. Gravit.41, 93 (2009) ArticleADSGoogle Scholar
[8] King-Hele, D.G., Cook, G.E., Rees, J.M.: Geophys. J. Int.8, 119 (1963) ArticleADSGoogle Scholar
[9] Kinney, W.H., Brisudova, M.: Ann. N.Y. Acad. Sci.927, 127 (2001) ArticleADSGoogle Scholar
[10] Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Courier Corporation, North Chelmsford (2000) MATHGoogle Scholar
[11] Maneff, G.: C. R. Acad. Sci. Paris178, 2159 (1924) Google Scholar
[12] Mücket, J.P., Treder, H.-J.: Astron. Nachr.298, 65 (1977) ArticleADSGoogle Scholar
[13] Ragos, O., Haranas, I., Gkigkitzis, I.: Astrophys. Space Sci.345, 67 (2013) ArticleADSGoogle Scholar
[14] Roman, R., Szücs-Csillik, I.: Astrophys. Space Sci.349, 117 (2014) ArticleADSGoogle Scholar
[15] Roy, A.E.: Orbital Motion. CRC, Bristol (2004) BookMATHGoogle Scholar
[16] Seeliger, H.: Astron. Nachr.137, 129 (1895) ArticleADSGoogle Scholar
[18] Szücs-Csillik, I., Roman, R.: In: Workshop on Cosmical Phenomena that Affect Earth and Their Effects, October 18, 2013, Bucharest (2013) Google Scholar
[19] Whittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (1917)
1611.02457v1
Corrected Newtonian potentials in the two-body problem with applications
M.-C. Anisiu ^(1){ }^{1}andI. Szücs-Csillik ^(2){ }^{2}Romanian Academy, Tiberiu Popoviciu Institute of Numerical Analysis, Cluj-NapocaandRomanian Academy, Institute of Astronomy, Astronomical Observatory Cluj-Napoca
Received qquad\qquad ; accepted qquad\qquad
Abstract
The paper deals with an analytical study of various corrected Newtonian potentials. We offer a complete description of the corrected potentials, for the entire range of the parameters involved. These parameters can be fixed for different models in order to obtain a good concordance with known data. Some of the potentials are generated by continued fractions, and another one is derived from the Newtonian potential by adding a logarithmic correction. The zonal potential, which models the motion of a satellite moving in the equatorial plane of the Earth, is also considered. The range of the parameters for which the potentials behave or not similarly to the Newtonian one is pointed out. The shape of the potentials is displayed for all the significant cases, as well as the orbit of Raduga1M21 M 2 satellite in the field generated by the continued fractional potential U_(3)U_{3}, and then by the zonal one. For the continued fractional potential U_(2)U_{2} we study the basic problem of the existence and linear stability of circular orbits. We prove that such orbits exist and are linearly stable. This qualitative study offers the possibility to choose the adequate potential, either for modeling the motion of planets or satellites, or to explain some phenomena at galactic scale.
The idea of modifying the original 1//r1 / r Newtonian potential starts with Newton himself. In his Principia he has already proposed a potential of the form A//r+B//r^(2)A / r+B / r^{2} and studied the relative orbit in this case too. Potentials of this type have been physically justified later by Maneff (also spelled Manev), in a series of papers starting with Maneff (1924). A deep insight in the Maneff field can be found in Diacu et al. (2000).
In this paper we deal with an analytical study of some corrected Newtonian potentials. The study is motivated by the fact that nowadays many authors consider various corrected Newtonian potentials without being concerned whether those potentials have or have not the properties of the Newtonian one.
It is known that the Newtonian potential mu//r\mu / r, regarded as a function of rr, satisfies the following conditions:
(i) lim_(r rarr0+)=+oo\lim _{r \rightarrow 0+}=+\infty;
(ii) lim_(r rarr+oo)=0\lim _{r \rightarrow+\infty}=0;
(iii) it decreases from +oo+\infty to 0 as r in(0,+oo)r \in(0,+\infty).
Its graph is the dashed one in Fig. 3.
We offer a complete description of some corrected potentials, for the entire range of the parameters. All the corrected potentials are central and inhomogeneous; their role is important, for example, in the inverse problem of dynamics (Bozis et al. 1997; Anisiu 2007).
We consider potentials derived recently by Abd El-Salam et al. (2014) from continued fractions, as well as the logarithmically corrected Newtonian potential introduced by Mücket & Treder (1977). We identify those which satisfy the conditions (i)-(iii) (at least
for a certain range of the parameters), as the Newtonian potential does. These potentials are suitable to be used to explain some phenomena in the motion of satellites or planets.
As an application we choose to model the orbits of Raduga-1M 2 satellite (launch date 28 January 2010) using the corrected Newtonian potentials. Raduga-1M satellites are military communication satellites, and are the geostationary component of the Integrated Satellite Communication System, where they work in conjuction with the highly eccentric orbit Meridian satellites.
When a relative orbit is designed using a very simple orbit model, then the control station of the formation will need to continuously compensate for the modeling errors and burn fuel. This fuel consumption, depending on the modeling errors, could drastically reduce the lifetime of the spacecraft formation. Using the corrected potentials can reduce the modeling errors (Szücs-Csillik & Roman 2013).
Some potentials, as the logarithmically corrected one, do not satisfy at least one of the conditions (i)-(iii), and they can be used to model the motion at galactic scale. For example, such a potential was considered in cosmologies which avoid to involve dark matter (Kinnev & Brisudova 2001), or to study the rotation curves of spiral galaxies (Fabris & Pereira Campos 2009).
Exponentially corrected potentials (Seeliger 1895) are also of interest.
A further study will be dedicated to the restricted three-body problem, and regularization methods will be applied for a better understanding of the motion, as in Roman & Szücs-Csillik (2014).
Section 2 introduces the potentials generated by continued fractions. It starts with some theoretical results, which allow us to establish a clear difference between the odd and even such potentials. A special attention is given to the fractional potential which includes
the first three terms, namely U_(3)U_{3}, whose graph is similar to the Newtonian one for c_(2) > c_(1)//8c_{2}>c_{1} / 8.
In Section 3 we present a zonal potential, which is of great help in modeling, for example, the motion of a satellite moving in the equatorial plane of the Earth.
Section 4 is dedicated to the logarithmic Newtonian potential.
Section 5 is dedicated to the study of the existence and linear stability of circular orbits. We prove that circular orbits can be traced by a body moving in the field generated by the continued fractional potential U_(2)U_{2}, and these orbits are linearly stable.
In Section 6 we formulate some concluding remarks.
This qualitative study is useful because it offers a complete description of the potentials, for each value of the parameters involved; therefore in the following attempts to explain phenomena on various scales in the universe, the suitable potentials can be chosen knowing in advance their properties.
2. Potentials generated by continued fractions
We remind some definitions and properties concerning the continued fractions. These can be found, for example, in the book of Battin (1999), where the author consider them as basic topics in analytical dynamics, and emphasize their important role in many aspects of Astrodynamics.
Continued fractions were used at first to approximate irrational numbers, the partial numerators and denominators being then integer numbers. A continued fraction is given by the expression
where the partial numerators a_(n)(n inN={1,2,3,dots})a_{n}(n \in \mathbb{N}=\{1,2,3, \ldots\}) and the partial denominators b_(n)(n inN)b_{n} (n \in \mathbb{N}) are real (or complex) numbers.
An infinite sequence (A_(n)//B_(n))_(n inN)\left(A_{n} / B_{n}\right)_{n \in \mathbb{N}} is associated to the continued fraction (1) in the following way:
The fraction A_(n)//B_(n)A_{n} / B_{n} is called a partial convergent or simply a convergent. The expressions A_(n)A_{n} and B_(n)B_{n} satisfy the fundamental recurrence formulas
This can be easily proved by mathematical induction. If the limit lim_(n rarr+oo)A_(n)//B_(n)\lim _{n \rightarrow+\infty} A_{n} / B_{n} exists, it represents the value of the continued fraction; otherwise, the continued fraction is divergent.
Following Battin (1999) we mention some interesting monotonicity properties of the sequence A_(n)//B_(n)A_{n} / B_{n}, for a_(n),b_(n) > 0,n inNa_{n}, b_{n}>0, n \in \mathbb{N}.
Using the idea of continued fractions, Abd El-Salam et al. (2014) considered a perturbation of the Newtonian potential. They started with the continued fraction (11) and put for r,c_(n) > 0,n inNr, c_{n}>0, n \in \mathbb{N},
called continued fractional potential or simply fractional potential. In this formula, when applied to the two-body problem, rr stands for the mutual distance between two punctual bodies and mu\mu for the product of the gravitational constant GG with the sum of masses m_(1)m_{1} and m_(2)m_{2}. In what follows we shall consider c_(1),c_(2),dots > 0c_{1}, c_{2}, \ldots>0.
We remark that it is unnecessary and physically unsustained to put the constant mu\mu everywhere in the continuous fraction. In what follows we shall consider a simplified form by mentaining r=b_(1)=b_(2)=dotsr=b_{1}=b_{2}=\ldots, as in (6), but changing a_(n)a_{n} into
It is known that the Newtonian potential (10), regarded as a function of rr defined on ( 0,+oo0,+\infty ), is decreasing from +oo+\infty to 0 . Therefore, it is natural to study the monotonicity of the other fractional potentials and their limits at 0 and +oo+\infty.
We begin with the behaviour of the continued fractional potential U_(2):[0,+oo)rarr[0,+oo)U_{2}:[0,+\infty) \rightarrow [0,+\infty), which has U_(2)(0)=0U_{2}(0)=0 and lim_(r rarr+oo)U_(2)(r)=0\lim _{r \rightarrow+\infty} U_{2}(r)=0. The first two derivatives of U_(2)U_{2} are respectively
The first derivative of U_(2)U_{2} has a unique positive root sqrt(c_(1))\sqrt{c_{1}}, and the second derivative of U_(2)U_{2} has a root equal to 0 and a positive root sqrt(3c_(1))\sqrt{3 c_{1}}. It follows that the potential U_(2)U_{2} increases from 0 to sqrt(1//c_(1))//2=U_(2)(sqrt(c_(1)))\sqrt{1 / c_{1}} / 2=U_{2}\left(\sqrt{c_{1}}\right) and then it decreases to 0 as r rarr+oor \rightarrow+\infty, having an inflection point at sqrt(3c_(1))\sqrt{3 c_{1}}. So, whatever the positive value of the constant c_(1)c_{1} is, the graph of the continued fractional potential U_(2)U_{2} looks different from that of the Newtonian potential for rr relatively small. The dot graph from Fig. 3 represents U_(2)U_{2} for c_(1)=0.1c_{1}=0.1 and mu=1\mu=1.
We consider now the continued fractional potential U_(3):(0,+oo)rarr(0,+oo)U_{3}:(0,+\infty) \rightarrow(0,+\infty), for which lim_(r rarr0+)U_(3)(r)=+oo\lim _{r \rightarrow 0+} U_{3}(r)=+\infty and lim_(r rarr+oo)U_(3)(r)=0\lim _{r \rightarrow+\infty} U_{3}(r)=0. The first two derivatives of U_(3)U_{3} are respectively
has the discriminant Delta=c_(1)(c_(1)-8c_(2))\Delta=c_{1}\left(c_{1}-8 c_{2}\right). For c_(2) > c_(1)//8c_{2}>c_{1} / 8, the equation (16) has no real roots, hence U_(3)^(')(r) < 0U_{3}^{\prime}(r)<0; then U_(3)(r)U_{3}(r) strictly decreases from +oo+\infty to 0 . For c_(2)=c_(1)//8c_{2}=c_{1} / 8, the equation (16) has a double real root u_(1,2)=3c_(1)//8u_{1,2}=3 c_{1} / 8, hence U_(3)^(')(r) <= 0U_{3}^{\prime}(r) \leq 0 and U_(3)(r)U_{3}(r) strictly decreases from +oo+\infty to 0 , and its graph has an inflection point at r=sqrt(3c_(1)//8)(:}r=\sqrt{3 c_{1} / 8}\left(\right. where {:U_(3)^('')(sqrt(3c_(1)//8))=0)\left.U_{3}^{\prime \prime}\left(\sqrt{3 c_{1} / 8}\right)=0\right). The last case is c_(2) < c_(1)//8c_{2}<c_{1} / 8, when (16), hence U_(3)^(')(r)U_{3}^{\prime}(r) also, has two distinct positive roots.
Fig. 1. -U_(3)-U_{3} for c_(1)=0.3;0.8;1c_{1}=0.3 ; 0.8 ; 1 and c_(2)=0.1,mu=1c_{2}=0.1, \mu=1
In this situation, U_(3)(r)U_{3}(r) strictly decreases from +oo+\infty to a local minimum, then it strictly increases to a local maximum, and finally decreases to 0 .
We illustrate in Fig. 1 these cases for U_(3)U_{3} with mu=1\mu=1 and
а) c_(1)=0.3,c_(2)=0.1c_{1}=0.3, c_{2}=0.1 (dash);
b) c_(1)=0.8,c_(2)=0.1c_{1}=0.8, c_{2}=0.1 (dot);
c) c_(1)=1,c_(2)=0.1(c_{1}=1, c_{2}=0.1( solid )).
In conclusion, the continued fractional potential U_(3)U_{3} has three kinds of graphs, depending on the values of the coefficients c_(1)c_{1} and c_(2)c_{2}. The graph of U_(3)U_{3} is similar to that of the Newtonian potential for c_(2) > c_(1)//8c_{2}>c_{1} / 8. We emphasize that the behaviour of U_(3)U_{3} in the vicinity of 0 , for any c_(1),c_(2) > 0c_{1}, c_{2}>0, is similar to that of the Newtonian potential, in contrast with that of U_(2)U_{2} given in (12).
It follows that a good choice for a continued fractional potential, which is close to the Newtonian one and still easy to handle, is U_(3)U_{3} with c_(2) > c_(1)//8c_{2}>c_{1} / 8.
Fig. 2.-Raduga-1M 2 satellite orbit using continued fractional potential U_(3),r=sqrt(q_(1)^(2)+q_(2)^(2))U_{3}, r=\sqrt{q_{1}^{2}+q_{2}^{2}}
We consider the continued fractional potential U_(3)U_{3} with c_(1)=0.0001c_{1}=0.0001 and c_(2)=0.00002c_{2}=0.00002, and we apply it for Raduga-1M 2 GEO satellite with semi-major axis 42164 km and the period of revolution 1436 minutes. The orbit is displayed in Fig. 2. This choice of the small parameters c_(1)c_{1} and c_(2)c_{2} is situated in the range c_(2) > c_(1)//8c_{2}>c_{1} / 8, so that the potential U_(3)U_{3} is very similar to the Newtonian one.
In Fig. 3 we plot the graphs of the first four continued fractional potentials U_(1)U_{1} (dash), U_(2)U_{2} (dot), U_(3)U_{3} (dashdot), and U_(4)U_{4} (solid) for mu=1,c_(1)=c_(2)=c_(3)=0.1\mu=1, c_{1}=c_{2}=c_{3}=0.1.
Fig. 3.- U_(1),U_(2),U_(3),U_(4)U_{1}, U_{2}, U_{3}, U_{4} for c_(1)=c_(2)=c_(3)=0.1c_{1}=c_{2}=c_{3}=0.1 and mu=1\mu=1
3. On the zonal potential of Earth
Until now we considered the motion of two punctual masses m_(1)m_{1} and m_(2)m_{2}. In the case of the very important problem of the motion of artificial satellites around the Earth, the Earth cannot be approximated by a point in order to obtain a good model of a satellite's motion; a good enough approximation is that of a spheroid.
It is known (King-Hele et al. 1963; Roy 2005) that if it is assumed that the Earth is a spheroid (i. e. we neglect the tesseral and sectorial harmonics), then its potential may be written as a series of zonal harmonics of the form
where mu=GM\mu=G M is the product of the gravitational constant GG with the mass of the Earth M,RM, R is the equatorial radius of the Earth and J_(n)J_{n} are the zonal harmonic coefficients due to the oblateness of the Earth. The coordinates of the satellite are the distance to the center of the Earth rr and its latitude phi\phi, and P_(n)(*)P_{n}(\cdot) are the Legendre polynomials of degree nn. Equation (19) does not take into account the small variation of U_(zon)U_{z o n} with longitude.
Because the coefficients J_(n),n > 2J_{n}, n>2, are much smaller than J_(2)J_{2}, a good approximation of the zonal potential of Earth is
{:(20)U_(zonal)=(mu )/(r)[1-(1)/(2)J_(2)((R)/(r))^(2)(3sin^(2)phi-1)]:}\begin{equation*}
U_{z o n a l}=\frac{\mu}{r}\left[1-\frac{1}{2} J_{2}\left(\frac{R}{r}\right)^{2}\left(3 \sin ^{2} \phi-1\right)\right] \tag{20}
\end{equation*}
In order to compare it with the Newtonian and continued fractional potentials, we consider equatorial orbits, with phi=0\phi=0. The potential is then of the type
with c=R^(2)J_(2)//2c=R^{2} J_{2} / 2.
We remark that UU is an inhomogeneous potential. Such potentials, in relation with the families of orbits generated by them, are studied by Bozis et al. (1997).
The potential UU has lim_(r rarr0+)U(r)=+oo\lim _{r \rightarrow 0+} U(r)=+\infty and lim_(r rarr+oo)U(r)=0\lim _{r \rightarrow+\infty} U(r)=0. The first two derivatives are respectively
The first derivative of UU is negative and its second derivative is positive on ( 0,+oo0,+\infty ), hence the potential UU decreases from its limit in 0 , which is +oo+\infty, to its limit 0 as r rarr+oor \rightarrow+\infty. Therefore, the zonal potential UU has a graph similar to the Newtonian one, as it may be seen in Fig. 4, for mu=1\mu=1 and c=1c=1. For the sake of completeness, we study also the case of c < 0c<0. The first derivative of UU has now the positive root sqrt(-3c)\sqrt{-3 c}, and the second derivative of VV has the positive root sqrt(-6c)\sqrt{-6 c}. It follows that the potential UU increases from its limit in 0 , which is equal with -oo-\infty to U(sqrt(-3c)) > 0U(\sqrt{-3 c})>0 and then it decreases to 0 as r rarr+oor \rightarrow+\infty, having an inflection point at sqrt(-6c)\sqrt{-6 c}. So, whatever the negative value of the constant cc is, the graph of the zonal-type potential VV looks different from that of the Newtonian potential.
The fact that the zonal potential UU, for positive cc, has the graph similar to that of the Newtonian potential, while the fractional potential U_(2)U_{2} has the dotted graph from Fig.
Fig. 4.- The zonal potential UU for c=1c=1 and mu=1\mu=1
3, explains difference of the corresponding plotted graphs for low and medium altitudes in Figs. 4-5 of Abd El-Salam et al. (2014).
Applying the zonal potential UU for the Raduga-1M 2 satellite motion, we obtain the trajectory displayed in Fig. 5.
4. Logarithmically corrected potential
Fabris & Pereira Campos (2009) have analyzed the rotation curves of some spiral galaxies, using a disc modelization, with a Newtonian potential corrected with an extra logarithmic term. More recently, Ragos et al. (2013) have taken into account the effects in the anomalistic period of celestial bodies due to the same logarithmic correction to the Newtonian gravitational potential. We shall compare this corrected potential with the Newtonian one. The corrected potential V:(0,+oo)rarr(0,+oo)V:(0,+\infty) \rightarrow(0,+\infty) is given by
Let us consider that the coefficient alpha\alpha is positive. Then lim_(r rarr0+)V(r)=+oo\lim _{r \rightarrow 0+} V(r)=+\infty and lim_(r rarr+oo)V(r)=+oo\lim _{r \rightarrow+\infty} V(r)=+\infty. The first derivative of VV has the positive root 1//alpha1 / \alpha, and the second derivative of VV has the positive root 2//alpha2 / \alpha. It follows that the potential VV decreases from its limit in 0 , which is equal with +oo+\infty to alpha(1-log(alpha))=V(1//alpha)\alpha(1-\log (\alpha))=V(1 / \alpha) and then it increases to +oo+\infty as r rarr+oor \rightarrow+\infty, having an inflection point at 2//alpha2 / \alpha. So, whatever the positive value of the constant alpha\alpha is, the graph of the logarithmically corrected potential VV looks different from that of the Newtonian potential. We illustrate in Fig. 6 the shape of VV for mu=alpha=1\mu=\alpha=1. For a negative alpha\alpha, we have lim_(r rarr0+)V(r)=+oo\lim _{r \rightarrow 0+} V(r)=+\infty, but lim_(r rarr+oo)V(r)=-oo\lim _{r \rightarrow+\infty} V(r)=-\infty. The first derivative of VV is negative and its second derivative is positive on ( 0,+oo0,+\infty ), hence the potential VV decreases from its limit in 0 , which is also equal with +oo+\infty, to its limit -oo-\infty as r rarr+oor \rightarrow+\infty. We represent in Fig. 7 the potential VV for mu=1\mu=1 and alpha=-1\alpha=-1. We consider the logaritmically corrected potential VV with alpha=-0.0001\alpha=-0.0001, and we apply it for the motion of a star from the dynamical system like "Milky-way" spiral galaxy with initial position at 8 kpc , initial
Fig. 6. - The logarithmically corrected potential VV for mu=alpha=1\mu=\alpha=1
velocity 572kpc//Gyr572 \mathrm{kpc} / \mathrm{Gyr} and the period of revolution 0.04 Gyr. The orbit is displayed in Fig. 8.
5. Circular orbits in the continued fractional potential U_(2)U_{2}
Circular orbits appear in the motion of the equatorial satellites, of some planets in various planetary systems, or of stars in galaxies. Therefore it is important to study if they can be traced in the continued fractional fields. We shall prove that such orbits can be traced by a body moving in the field produced by the continued fractional potential U_(2)U_{2}, which is a generalization of the Newtonian potential. The existence and the stability of circular orbits in a Maneff field was studied recently by Blaga (2015).
We consider a two-body problem with a primary body of mass MM and a secondary one of mass mm, under the influence of the continued fractional potential U_(2)U_{2}. The potential is central, so the two-body problem may be reduced to a central force one, and we shall study the relative motion of the secondary body. The motion is planar and it is governed, in polar
Fig. 7. - The logarithmically corrected potential VV for mu=1\mu=1 and alpha=-1\alpha=-1
coordinates r=sqrt(q_(1)^(2)+q_(2)^(2)),theta=tan^(-1)(q_(2)//q_(1))r=\sqrt{q_{1}^{2}+q_{2}^{2}}, \theta=\tan ^{-1}\left(q_{2} / q_{1}\right), by the equations
For h!=0h \neq 0, we try for a constant solution r=r_(0),r_(0)!=0r=r_{0}, r_{0} \neq 0 for the equations (25). For such a solution, equation (26) gives theta^(˙)=h//r_(0)^(2)=omega_(0)\dot{\theta}=h / r_{0}^{2}=\omega_{0}. The first equation of (25) reads for r(t)=r_(0)r(t)=r_{0} :
If this equation admits a solution r_(0) > 0r_{0}>0, it means that a circular orbit r(t)=r_(0)r(t)=r_{0} is possible, the secondary pursuing the orbit with constant angular velocity omega_(0)=h//r_(0)^(2)\omega_{0}=h / r_{0}^{2}.
Fig. 8.- Motion of a star in a dynamical system like "Milky way" spiral galaxy using the logarithmically corrected potential VV for alpha=-0.0001\alpha=-0.0001
this condition being obtained by developing U_(2)U_{2} around r_(0)r_{0} up to the first-order terms (Whittaker 1917; Roy 2005).
By applying this reasoning for the Newtonian potential given by (10), with mu > 0\mu>0, we obtain easily that the circular orbit r_(N)=h^(2)//mur_{N}=h^{2} / \mu obtained from the equation corresponding to (27) is linearly stable, since
We study now the case of the continued fractional potential U_(2)U_{2} given by (11), with mu,c_(1) > 0\mu, c_{1}>0. Equation (27), where we denote shortly r=r_(0)r=r_{0}, reads
and this fifth degree equation cannot be solved in general. Nevertheless, we remark that all the coefficients, excepting that of r^(5)r^{5}, are positive. It follows that there is precisely one change of sign in the row of the coefficients. Applying the Descartes rule of signs (Korn & Korn 2000) it follows that equation (29) has at most one positive solution. But for r=0r=0 the left hand side of (29) is equal to h^(2)c_(1)^(2) > 0h^{2} c_{1}^{2}>0, and its limit to infinity is -oo-\infty, hence equation (29) has a unique solution r=r_(0)r=r_{0}. Moreover, for r_(N)=h^(2)//mur_{N}=h^{2} / \mu the left hand side of (29) is positive, which means that r_(0) > r_(N)r_{0}>r_{N}, i. e. the radius of the circular orbit in the fractional potential U_(2)U_{2} is greater than the similar one traced in the Newtonian potential.
To get information on the stability of the unique circular orbit r=r_(0)r=r_{0}, we calculate the left hand side of (28), using the expressions of the derivatives of U_(2)U_{2} from (14):
hence it has positive values on the interval ( 0,r_(1)0, r_{1} ) and negative ones on ( r_(1),+oor_{1},+\infty ).
We have proved that the fifth degree polynomial in (29) is positive on ( 0,r_(0)0, r_{0} ) and negative on (r_(0),+oo)\left(r_{0},+\infty\right). We calculate the value of that polynomial for r_(1)r_{1} given by (31) and get:
The main properties of the Newtonian potential are preserved by some of its corrected potentials: the continued fractional potential U_(3)U_{3} given by (121), for c_(2) > c_(1)//8 > 0c_{2}>c_{1} / 8>0; the zonal potential of Earth UU given by (21). It is worth noting that in the case of the continued fraction potential U_(3)U_{3} we have at our disposal two parameters c_(1)c_{1} and c_(2)c_{2}. These can be used to adjust the potential when we have information on the motion of the satellite. The figures illustrate the possible situations which have been proved analytically.
For the continued fractional potential U_(2)U_{2} it is proved that circular orbits exist and are linearly stable.
We remark a strong feature of the continued fractional potentials: they have a simple analytical form, being rational functions, hence they can be easily handled in further applications.
The authors are deeply indebted to the reviewers and to the editor for their valuable comments and suggestions.
The work of the second author was partially supported by a grant of the Romanian National Authority for Scientific Research, CNDI-UEFISCDI, project number PN-II-PT-PCCA-2011-3.2-0651 (AMHEOS).
REFERENCES
Abd El-Salam, F. A., Abd El-Bar, S. E., Rasem, M., & Alamri, S. Z. 2014, Ap&SS, 350, 507
Anisiu, M.-C. 2007, In Dumitrache, C., Popescu, N. A., Suran, D. M., & Mioc, V. (eds.). AIP Conference Proceedings 895, 308
Battin, R. H. 1999, An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, Reston
Fabris, J. C., & Pereira Campos, J. 2009, General Relativity and Gravitation, 41, 93
King-Hele, D. G., Cook, G. E., & Rees, J. M. 1963, Geophys. J. Int., 8, 119
Kinney, W. H., & Brisudova, M. 2001, Annals of the New York Academy of Science, 927, 127
Korn, G. A., & Korn, T. M. 2000, Mathematical handbook for scientists and engineers: Definitions, theorems, and formulas for reference and review. Courier Corporation