1 Introduction

The classical two-body problem is controlled by the interaction of two point masses moving under mutual gravitational attraction and Newton’s second law, where the massive body is called the primary or the central body, while the smaller body is called the secondary body. But in the framework of general relativity the mechanical laws and the equations of motion, which describe any dynamical system according to Einstein’s theory are much more strenuous and complicated in analysis than under the assumptions of Newtonian mechanics. However, the motions of celestial bodies under conceptions of Einstein mechanics differ little from their Newtonian representation. But for astronomical purposes, relativistic effects may be conveniently treated as a first-order perturbation. A comprehensive and extended study on relativistic celestial mechanics of the solar system, a theoretical development of gravitational physics as it applies to the dynamics of celestial bodies and the analysis of precise astronomical observations, is presented by Kopeikin et al. (2011).

In perturbed Keplerian orbits, the motion of a secondary body is subject to other forces in addition to the gravitational attraction of the central body. In this case, the other forces are called the perturbing forces. These lead to a disturbance in the motion of the secondary body, however, their effect may be very small in comparison with the main forces. There are different kinds of perturbation forces; some of these decrease the velocity of the body as in drag forces, for more details Elshaboury and Mostafa (2014), and other kinds increase the velocity as in thrust forces. There are also some forces causing a loss of orbital energy and angular momentum and coerce particles to slowly spiral to the sun in the case of our solar system, as in the Poynting–Robertson (PR) effect and ion drag from the solar wind; see Burns et al. (1979) for a comprehensive review of the various radiation forces.

The perturbing forces may include: repulsive forces such as radiation pressure which depends on the cross section of the particle and other ones depending on the size, the electrical charge on a dust grain, which are represented by the Lorentz force. A stochastically distributed dust between a planet and the sun can also be considered as a perturbing force, this force generate supplementary random force on the orbiting particle, for instance Jezewski and Mittleman (1983), Mavraganis (1991), Mavraganis and Michalakis (1994), Sharma and Parthasarathy (2007). The non-sphericity effect, if one of the two bodies has irregular shape, the two-body problem becomes insoluble because the lack of sphericity in the body shape produce also an additional force, which can also be treated as a perturbing force; see Jezewski (1983), Martinusi et al. (2015) and Abouelmagd et al. (2015b). Furthermore there are some interest papers which presented significant study on the effect of oblateness in the framework of restricted three-body problem constructed by Abouelmagd (2012, 2013), Abouelmagd et al. (2014a, 2014b, 2015a).

Some of the aforementioned perturbing forces or one of them may be the only considerable perturbing force, if we assume that there are no other bodies outside the dynamical system. But this is not true for all real systems in space. For instant, Earth-Moon motion will suffer from an extra force of the gravitational attraction by the other planets and their satellites; this force is called the interaction force. In most systems that involve multiple gravitational attractions, the dominating effect is produced by the central body. The central body is a star in the case of a stellar system and the other bodies are its planets; or a planet in the case of planetary systems and the other bodies are its satellites. The gravitational effect of the other body or bodies can be treated as a perturbation for the unperturbed motion of the planet or satellite around its central body. But our main significant contribution in the present paper is to find the solution of the perturbed two-body problem in the framework of a relativistic effect without secular terms by the multiple scales method.

This paper is organized as follows: in Sect. 2 we briefly review the \(N\)-body problem in the inertial and heliocentric references frames, as well as the perturbed acceleration of interaction gravitational forces. In Sect. 3 we deduce the equations of motion of the perturbed two-body problem by post-Newtonian relativistic terms. In Sect. 4 we analyze the angular momentum and the energy relations. In Sect. 5 we obtain the solution of the relativistic perturbed two-body problem by a multiple scales method. We end the paper with some comments.

2 Background

2.1 \(N\)-body problem in the inertial reference frame

The classical \(N\)-body problem is defined in the framework of \(N\) points moving under their gravitational attraction without other celestial bodies outside the system. Furthermore, we assume that \(N=n+1\) such that each point mass is fully described by its mass \(m_{i}\) and its position vector \(\underline{x}_{i}(t)\), \((i=0,1,2,\ldots,n+1)\) for all time \(t\). In the case of our planetary system, \(m_{0}\) may denote the mass of the Sun and it could be considered as the unit of mass. We may also define the velocity vector of each point mass by \(\underline{\dot {x}}_{i}(t)\). If the initial state vectors \(\underline{x}_{i}(t_{0})\) and \(\underline{\dot{x}}_{i}(t_{0})\) are known at the initial time \((t=t_{0})\), then the task of finding the trajectories \(\underline{x}_{i}(t)\) for each point mass \(m_{i}\) in the inertial reference frame can be accomplished for all time \(t\).

According to Newton’s universal law of gravitation and his second law, we can write down the equations of motion in the inertial frame for the \(N\)-point masses in the form

$$ \underline{\ddot{x}}_{i}=-G\sum _{j=0,j\neq i}^{n+1} m_{j} \dfrac{\underline {x}_{ji}}{x_{ji}^{3}}, $$
(1)

where \(G\) is the constant of gravitation, \(x_{ij}=|x_{ij}|=|x_{ji}|\), \(\underline{x}_{ij}=\underline{x}_{j}-\underline{x}_{i}\), and \((i=0,1,2,\ldots,n+1)\).

The right hand side of Eq. (1) represents the superposition of the gravitational forces acting on point mass \(m_{i}\) under the condition that there are no masses outside the system of the \(N\)-point masses. It is important to note that Eq. (1) represents an ordinary coupled nonlinear differential equations system of second order in time. The mass \(m_{i}\) does not change with time, which is not perfectly true in most cases.

2.2 \(N\)-body problem in the heliocentric reference frame

The construction of Eq. (1) is set up under the assumption that there are no masses outside of the system of the \(N\)-point masses, which will never be \(100~\%\) true. We have to think of the gravitational attraction which the solar system experiences from our galaxy. If time periods of hundreds of millions of years are considered such effects must be taken into account (the revolution period of the solar system around the galactic center is estimated to be about 250 million years); see for more details Beutler et al. (2005). But if the mass \(m_{0}\) dominates all other masses, it makes sense to rewrite Eq. (1) to describe the motion of the system relative to the point mass \(m_{0}\). For that purpose, we define

$$ \underline{r}_{i}(t)=\underline{x}_{i}(t)- \underline{x}_{0}(t), $$
(2)

where \(i=1,2,\ldots,n+1\) and the vector \(\underline{r}_{i}\) is called the heliocentric position vector in our planetary system.

Starting from Eq. (1) with the help of Eq. (2), we may easily set up the equation of motion for \(N\) point masses in terms of the heliocentric position vector \(\underline{r}_{i}\) in the form

$$ \underline{\ddot{r}}_{i}=-G(m_{0}+m_{i}) \dfrac{\underline {r}_{i}}{r_{i}^{3}}-G\sum_{j=1,j\neq i}^{n+1} m_{j} \biggl[\dfrac{\underline {r}_{ji}}{r_{ji}^{3}}+\dfrac{\underline{r}_{j}}{r_{j}^{3}} \biggr] ; $$
(3)

Eq. (3) represents the equation of motion for \(N\) point masses in the heliocentric coordinates system such that its origin follows the trajectory of the point mass \(m_{0}\). This equation shows that the acceleration may vanish in the heliocentric coordinates but that this is not possible for all time in inertial space according to Eq. (1). In addition, the initial state vectors can be defined as \(r_{i}(t_{0})=r_{i0}\) and \(\dot{r}_{i}(t_{0})=\dot{r}_{i0}\) where \((i=1,2,\ldots,n+1)\).

Remark 1

It is important to state that we are able to analyze the development of the planetary system dynamics without having defined the origin in the inertial space system relative to the central mass \(m_{0}\), using Eq. (3) when the initial state vectors in the heliocentric system are given.

2.3 Equation of motion with the effect of interaction gravitational forces

Starting from Eq. (3) with the set up \(m_{n+1}\equiv m\), \(\underline{r}_{n+1}\equiv\underline{r}\), and \(p\) denotes \(n+1\); without loss of the generality this equation can be written in the form

$$ \underline{\ddot{r}}=-\mu\dfrac{\underline{r}}{r^{3}}+\underline{a}_{p}, $$
(4)

where \(\mu=G(m_{0}+m)\) and \(\underline{a}_{p}\) is the perturbed acceleration of the interaction force, which will be controlled by

$$ \underline{a}_{p}=-G\sum_{j=1}^{n} m_{j} \biggl[\dfrac{\underline {r}_{jp}}{r_{jp}^{3}}+\dfrac{\underline{r}_{j}}{r_{j}^{3}} \biggr] . $$
(5)

In Eq. (4) the first term on the right hand side represents the main term of the force which acts on the point mass \(m\), and, while the sum may be called the perturbation term, this property is correct especially in our planetary system, because the ratio of planetary or satellite masses to the mass of central body \((m_{i}/m_{0}\ll1)\) is a small quantity. The most massive planet, Jupiter, has a mass of \(0.1~\%\) of the solar mass. Furthermore, one of the significant features of Eq. (4), the first term, is a result of the Kepler force, while the second term emerges from the interaction force. Actually, the last force is realistic and exists because there are no encounters between the bodies for any real system. Thereby, the perturbation due to the interaction force is uniquely due to the transformation from the inertial reference frame to the heliocentric coordinates system.

Remark 2

In the case of a point mass \(m\) of negligible mass with respect to all other masses of the system, \(\mu=Gm_{0}\) then Eq. (4) enables us to describe the trajectory of a minor planet or a comet in the heliocentric system.

3 Equations of motion with a relativistic effect

The fixed body of a spherical structure produces a spherically symmetric gravitational field with a metric in terms of rectangular coordinates,

$$\begin{aligned} \mathrm{d}s^{2}&= \biggl[p(r)c_{l}^{2}\, \mathrm{d}t^{2} +2b(r)\dfrac{x^{i}}{r}c_{l}\, \mathrm{d}t\,\mathrm{d}x^{i} \biggr] \\ &\quad {}- \dfrac{1}{r^{2}} \biggl[a^{2}(r) \delta_{ik}+ \biggl(q(r)-\dfrac {a^{2}(r)}{r^{2}} \biggr)x^{i} x^{k} \biggr]\,\mathrm{d}x^{i}\,\mathrm{d}x^{k}, \end{aligned}$$
(6)

where \(c_{l}\) is the speed of light, and in this setting \(a, b, p\), and \(q\) are functions of \(r\) to be determined from the field equations, \(a\) and \(b\) are arbitrary, while \(p\) and \(q\) depend on \(a\) and \(b\). See Adler et al. (1965), Brumberg (1991) and Kopeikin et al. (2011) for complete details.

In the parameterized post-Newtonian approximation (PPN), the function \(a(r)\) for most practically employed quasi-Galilean reference system may be controlled by

$$ a(r)=r \biggl(1+(1-\alpha)\dfrac{m_{c}}{r}+\epsilon \dfrac {m_{c}^{2}}{r^{2}}+\cdots \biggr), $$
(7)

where \((\alpha, \epsilon, \ldots)\) are the parameters which characterize the type of coordinate, \(m_{c}=\mathcal{G}M/c^{2}_{l}\), and \(\mathcal{G}\) is the parameter which may differ from the universal gravitational constant by a constant factor such that \(\mathcal{G}=\mathcal{A}G\), \(M\) is the total mass of the system. For instance, the values \(\alpha=1, \epsilon=0\) in the standard coordinates, while harmonic coordinates are associated with \(\alpha=0, \epsilon=0\); see Brumberg (1991) for more classifications.

After inserting Eq. (7) into Eq. (6), the generalized Schwarzschild metric in the PPN approximation takes the following form in terms of rectangular coordinates:

$$\begin{aligned} \mathrm{d}s^{2}&= \biggl[1-\dfrac{2m_{c}}{r} +2(\beta- \alpha)\dfrac{ m_{c}^{2}}{r}+\cdots \biggr]c_{l}^{2}\, \mathrm{d}t^{2} \\ &\quad{} - \biggl[\delta_{ij}-\dfrac{2 m_{c}}{r} \biggl[(\gamma-\alpha) \delta_{ij} +\alpha\dfrac {x^{i}x^{j}}{r^{2}} \biggr] +\cdots \biggr]\,\mathrm{d}x^{i}\,\mathrm{d}x^{j}; \end{aligned}$$
(8)

the parameters \(\beta\) and \(\gamma\) determine the features of the PPN formalism. For general relativity \(\beta=\gamma=1\). Nevertheless, considering \(\beta\) and \(\gamma\), the value of \(\alpha\) defines specific coordinates conditions; with \(\alpha=0\), the formula in Eq. (8) reduces to the well-known Eddington–Robertson metric.

The rectangular coordinates \(x^{1}, x^{2}\) and \(x^{3}\) are related to the spherical coordinates \((r,\theta, \varphi)\) by

$$\begin{aligned} x^{1}&=r\cos\theta\sin\varphi, \end{aligned}$$
(9a)
$$\begin{aligned} x^{2}&=r\sin\theta\sin\varphi, \end{aligned}$$
(9b)
$$\begin{aligned} x^{3}&=r\cos\varphi. \end{aligned}$$
(9c)

Taking into account the motion of the two-body problem with a perturbation given by a relativistic effect, then the equation of relative motion in the type of vector form as described in Brumberg (1991) and Kopeikin et al. (2011) will be governed by

$$\begin{aligned} &\underline{\ddot{r}}= -\frac{\mathcal{G}M}{r^{3}}\underline{r} + \underline{F}, \end{aligned}$$
(10a)
$$\begin{aligned} &\underline{F}=\frac{\mathcal{G}M}{c^{2}_{l}r^{3}} \biggl[ \biggl(2 \sigma\frac {\mathcal{G }M}{r} -2 \epsilon\, \underline{\dot{r}}.\underline{\dot {r}} + 3 \alpha\frac{(\underline{r}.\underline{\dot{r}})^{2}}{r^{2}} \biggr)\underline{r} +2\mu(\underline{r}.\underline{\dot{r}})\underline{\dot {r}} \biggr], \end{aligned}$$
(10b)

where \(\underline{r}=\underline{r}_{2}-\underline{r}_{1}\) is a relative vector, \(\underline{r}_{1}\), \(\underline{r}_{2}\) are the position vectors of the masses \(M_{1}\) and \(M_{2}\), \(M=M_{1}+M_{2}\). \(\alpha, \epsilon, \mu\), and \(\sigma\) are arbitrary numerical parameters. \(\underline{F}\) is called the Chazy distributing force. The above equation is an important tool in the investigation of the motion of a test particle in a Schwarzschild gravitational field. The general relativity equations of the generalized Schwarzschild problem is a result for the setting:

$$\begin{aligned} &{\sigma=\gamma+\beta-\alpha,} \end{aligned}$$
(11a)
$$\begin{aligned} &{2\epsilon=\gamma+\alpha,} \end{aligned}$$
(11b)
$$\begin{aligned} &{\mu=\gamma+1-\alpha.} \end{aligned}$$
(11c)

Substituting Eqs. (11a), (11b), (11c) into Eqs. (10a), (10b), one obtains

$$\begin{aligned} \underline{\ddot{r}} =& -\frac{\mathcal{G}M}{r^{3}}\underline{r}+ \frac {\mathcal{G}M}{c^{2}_{l}r^{3}} \biggl[ \biggl(2 (\gamma+\beta-\alpha)\frac {\mathcal{G }M}{r} -( \gamma+\alpha)\, \underline{\dot{r}}.\underline {\dot{r}} \\ &{}+ 3 \alpha \frac{(\underline{r}.\underline{\dot{r}})^{2}}{r^{2}} \biggr)\underline{r} +2(\gamma+1-\alpha) (\underline{r}. \underline{\dot {r}})\underline{\dot{r}} \biggr] \end{aligned}$$
(12)

or

$$\begin{aligned} &\ddot{r} - r \dot{\theta}^{2} = -\frac{\mathcal{G}M}{r^{2}} + \frac{\mathcal {G}M}{c^{2}_{l}r^{2}} \biggl[2(\gamma+\beta-\alpha)\frac{\mathcal{G}M}{r} \\ &\phantom{\ddot{r} - r \dot{\theta}^{2} =}{} - (\gamma+ \alpha)r^{2} \dot{\theta}^{2} + (\gamma+2)\dot{r}^{2} \biggr] , \end{aligned}$$
(13a)
$$\begin{aligned} &2 \dot{r} \dot{\theta}+ r \ddot{\theta}= \frac{2\mathcal{G}M(\gamma +1-\alpha)}{c^{2}_{l}r} \,\dot{r}\, \dot{\theta}. \end{aligned}$$
(13b)

Now we can simplify the equations by fixing \(\mathcal{G}M=\mathcal {G}(M_{1}+M_{2})\) equal to unity and \(\varepsilon=1/c^{2}_{l}\). For the general relativity, one of the choices is \(\gamma=\beta=1\); furthermore \(\alpha =0\) in harmonic coordinates. In this setting the parameter \(\varepsilon \) represents the relativistic effect. Its acceleration due to the theory of general relativity at maximum is of order \(10^{-9}\) of the main term in the close to Earth satellite. Although this effect is very small, it is mandatory to take such an effect into account for precise orbit determination. Hence Eq. (12) and Eqs. (13a), (13b) will be reduced to the form

$$ \underline{\ddot{r}}= -\frac{1}{r^{3}}\underline{r}+ \frac{\varepsilon }{r^{3}} \biggl[ \biggl(\frac{4}{r} -\, \underline{\dot{r}}. \underline{\dot {r}} \biggr)\underline{r} + 4(\underline{r}.\underline{\dot {r}}) \underline{\dot{r}} \biggr] $$
(14)

or

$$\begin{aligned} &\ddot{r} - r \dot{\theta}^{2} = -\frac{1}{r^{2}} + \frac{\varepsilon}{r^{2}} \biggl[\frac{4}{r} - r^{2} \dot{\theta}^{2} + 3\dot{r}^{2} \biggr] , \end{aligned}$$
(15a)
$$\begin{aligned} &2 \dot{r} \dot{\theta}+ r \ddot{\theta}= \frac{4\varepsilon}{r} \,\dot{r}\, \dot{\theta}. \end{aligned}$$
(15b)

The above equations are necessary and sufficient to investigate the structure of the motion for a test particle in the Schwarzschild gravitational force in the framework of the parameterized post-Newtonian formalism.

4 The constants of perturbed motion

In this section we investigate how the perturbation modifies the angular momentum and the total energy.

4.1 A modified conservation of the angular momentum and total energy

Theorem 1

The angular momentum of the dynamical system for the perturbed two-body problem via the relativistic effect is not conserved, however, its direction is a constant, while the total energy is conserved. Therefore the motion is planar.

Proof of Theorem 1

Let \(\underline{h}_{t}\) be the vector of total orbital angular momentum per unit mass of the body \(m\) according to the heliocentric reference frame; therefore,

$$ \underline{h}_{t} = \underline{r} \wedge\underline{ \dot{r}}=\mathopen{|}\underline {r} \wedge\underline{\dot{r}}\mathclose{|}\underline{ \hat{h}}_{t}, $$
(16)

where \(h_{t}=\mathopen{|}\underline{r} \wedge\underline{\dot{r}}\mathclose{|}\) and \(\underline {\hat{h}}_{t}\) is the unit normal vector of the orbital plane.

Multiply the equation of motion, Eq. (14) with the vectorial product by \(\underline{r}\), and one obtains

$$ \underline{r} \wedge\underline{\ddot{r}}=\dfrac{4\varepsilon\dot {r}}{ r^{2}} \underline{r} \wedge\underline{\dot{r}} ; $$
(17)

now we can rewrite Eq. (17) in the form

$$ \dot{\underline{h}}_{t} =\dfrac{4\varepsilon\dot{r}}{r^{2}} \underline{h}_{t} . $$
(18)

Taking the product of \(\underline{h}_{t}\) with Eq. (18), we have

$$ \dot{\underline{h}}_{t} \wedge \underline{h}_{t}=0 . $$
(19)

Equation (19) shows that the direction of the angular momentum is a constant; therefore the motion is planar.

In the framework of the first parameterized post-Newtonian approximation the Lagrangian will be written in the following form; see Damour and Deruelle (1985) and Blanchet (2001):

$$ L = L_{\mathrm{N}} + \dfrac{1}{c^{2}_{l}}L_{1\mathrm{PN}} ; $$
(20)

therefore, in the aforementioned setting in Sect. 3, \(L_{\mathrm{N}}\) and \(L_{1\mathrm{PN}}\) are given by

$$\begin{aligned} &L_{\mathrm{N}}=\dfrac{1}{2} \underline{\dot{r}}^{2} + \dfrac{1}{r}, \end{aligned}$$
(21a)
$$\begin{aligned} &L_{1\mathrm{PN}}= \dfrac{1}{2r} \biggl[\dfrac{r}{4} \bigl( \underline{\dot{r}}^{2} \bigr)^{2} +3 \underline{ \dot{r}}^{2}-\dfrac{1}{r} \biggr], \end{aligned}$$
(21b)

substituting Eqs. (21a), (21b) int Eq. (20) the \(1\mathrm{PN}\) Lagrangian in the harmonic coordinates is of the type

$$ L=\dfrac{1}{2} v^{2} +\dfrac{1}{r}+ \dfrac{\varepsilon}{2r} \biggl[ \dfrac {r}{4} v^{4}+ 3v^{2}-\dfrac{1}{r} \biggr], $$
(22)

where \(v^{2}= \underline{\dot{r}}^{2}\)

Consequently Eq. (22) shows the constants of motion:

$$\begin{aligned} &\underline{h}=\underline{r}\wedge \dfrac{\partial L}{\partial \underline{\dot{r}}} , \end{aligned}$$
(23a)
$$\begin{aligned} &E= \underline{\dot{r}}.\dfrac{\partial L}{\partial\underline{\dot {r}}} -L, \end{aligned}$$
(23b)

or

$$\begin{aligned} &\underline{h}=\underline{r} \wedge\underline{\dot{r}} \biggl[1+ \dfrac {1}{2}\varepsilon \biggl(v^{2}+ \dfrac{6}{r} \biggr) \biggr] , \end{aligned}$$
(24)
$$\begin{aligned} &E=\dfrac{1}{2} v^{2} -\dfrac{1}{r}+ \dfrac{\varepsilon}{2r} \biggl[ \dfrac {3r}{4} v^{4}+ 3v^{2}+ \dfrac{1}{r} \biggr] . \end{aligned}$$
(25)

Since the total angular momentum \(h_{t}=|\underline{r} \wedge\underline {\dot{r}}|\), we can rewrite Eq. (24) in the form

$$ h_{t}=h \biggl[1- \dfrac{1}{2}\varepsilon \biggl(v^{2}+ \dfrac{6}{r} \biggr) \biggr] ; $$
(26)

from Eq. (25), we can rewrite Eq. (26) in the form

$$ h_{t}=h \biggl[1- \varepsilon \biggl( \dfrac{4}{r}+E \biggr) \biggr] . $$
(27)

We would like to mention that Eqs. (25)–(27) are considered to be direct results for the relations provided in the book by Brumberg (1991), for the case of general relativity (\(\gamma= \beta =1\) and \(\alpha=0\)). Specifically, Eqs. (24) and (25) correspond to the expressions (3.1.52) and (3.1.53) in Brumberg (1991), accordingly.

Equation (27) shows that the magnitude of \(h_{t}\) is a function in the radial vector \(r\), and this implies that the total angular momentum is not conserved. Since \(\underline{h}_{t}=\underline{r}\wedge {\underline{\dot{r}}}\), we conclude that the angular momentum vector \(\underline{h}_{t}\) is always perpendicular to the plane (the plane of the motion) which includes both vectors \(\underline{r}\) and \(\underline {\dot{r}}\). This proves that the position and velocity vectors lie in the plane of motion and the total angular momentum \(h_{t}\) is not conserved as in the classical case.

Finally, we state that \(E\) represents the total energy, which is conserved, while the angular momentum is not. □

Remark 3

The change in the magnitude of the angular momentum is very small due to the perturbation of the general relativity, because the term of the relativistic perturbation depends on the factor \(1/c^{2}\). So if this term is neglected, the total angular momentum is conserved and the structure of the orbital plane of motion coincides with the one of the unperturbed model.

In this context, the mechanical energy-like invariant of motion is given by

$$ \dfrac{1}{2}\bar{v}^{2}-\dfrac{1}{r}+ \dfrac{7\varepsilon}{ r^{2}}=E_{l} , $$
(28)

where \(\bar{v}^{2}= (1-\frac{6 \varepsilon}{r} )v^{2}\) and \(E_{l}\) is a constant representing the total energy; see the Appendix for details.

Theorem 2

In neglecting of \(O(\varepsilon e)\) in Eq. (28), the energy-like approximate integral of motion \(E_{l}\) becomes an exact integral of motion as in expression of Eq. (29). We have

$$ \dfrac{1}{2}{v}^{2}-\dfrac{1}{r}=E ; $$
(29)

here \(E=E_{l}-4\varepsilon/h^{4}\).

The proof of this statement is available in the Appendix.

Theorem 2 establishes that the energy-like approximate integral of motion coincides with the appropriate energy emerging in the unperturbed motion. The difference between the approximate integral of motion and the exact one is of the order of \(O(\varepsilon)\).

Theorem 3

For low-eccentricity orbits, the total angular momentum of the relativistically perturbed two-body problem is conserved.

Proof of Theorem 3

Since

$$ \underline{r}.\underline{\dot{r}}=r\dot{r} $$
(30)

the scalar product \(\underline{r}.\underline{\dot{r}}\) can be approximated in the form

$$ \underline{r}.\underline{\dot{r}}=-h \bigl[e-e^{2} \cos f +e^{3} \cos^{2}f +O \bigl(e^{4} \bigr) \bigr] $$
(31)

for low eccentricity, and the scalar product \(\underline{r}.\underline {\dot{r}}\) is of the order of \(O(e)\). Then we may state that \(\underline {r}.\underline{\dot{r}}\approx0\). Furthermore the energy theorem of the two-body problem may be reduced for small values of the eccentricity orbit to \(\underline{\dot{r}}.\underline{\dot{r}}\approx 1/r\). In this context, we can write Eq. (14) in the form

$$ \underline{\ddot{r}}=- \biggl[\dfrac{1}{r^{3}}- \dfrac{3 \varepsilon}{ r^{4}} \biggr]\underline{r} . $$
(32)

Now we take the vector product of \(\underline{r}\) with Eq. (32), and we obtain

$$ \ddot{\underline{r}}\wedge\underline{r}=0 . $$
(33)

After integration, the above equation can be written as

$$ r^{2} \dot{\theta}=h, $$
(34)

where \(h\) is a constant. This shows that the total angular momentum is conserved as in the classical case without any changes in its magnitude. □

5 Solution provided by multiple scales method

We start by rewriting Eqs. (15a), (15b) in the form

$$ \ddot{r} - \dfrac{h^{2}_{t}}{r^{3}} = - \dfrac{1}{r^{2}} + \varepsilon \biggl(\dfrac{4}{r^{3}} - \dfrac{h^{2}_{t}}{r^{4}} +\dfrac{3 \dot{r}^{2}}{r^{2}} \biggr) . $$
(35)

Letting \(r = \dfrac{1}{u}\), we obtain

$$ \dfrac{\mathrm{d}^{2}u}{\mathrm{d}\theta^{2}} + u = \dfrac{1}{h^{2}} + \varepsilon \biggl(a_{0}+a_{1}u+u^{2}+ \biggl( \dfrac{\mathrm{d}u}{\mathrm{d}\theta} \biggr)^{2} \biggr), $$
(36)

where \(a_{0}=2E/h^{2}\) and \(a_{1}=4/h^{2}\).

Equation (36) is considered more general than the well-known Binet equation; see for instance Navickas and Ragulskis (2013).

5.1 Solution of unperturbed problem

Equation (36) represents the trajectory of the second body around the primary. It is important to note that this trajectory will follow a Kepler orbit when the effect of general relativity is switched off (\(\varepsilon= 0\)). Therefore, the solution is given by

$$ u(f)= \dfrac{1}{h^{2}} \, (1 + e \cos f), $$
(37)

where \(f\) is the true anomaly, \(e= \kappa\,h^{2}\) is the orbit eccentricity, and \(\kappa\) is a constant to be determined from the initial conditions

$$ u (0) = u_{0} = \dfrac{1}{h^{2}}(1 + e)\quad \text{and} \quad \dfrac {\mathrm{d}}{\mathrm{d}f}u(0)= 0 . $$
(38)

Furthermore, Eq. (37) represents a periodic solution where \(u(f + 2\pi) = u(f)\).

Now we want to find a solution of Eq. (36) with the effect of perturbation parameter \(\varepsilon\). So we have to use some perturbation methods. For instance, we have the KBM method, the Lindstedt–Poincaré technique, and the method of multiple scales or the classical theory of perturbation. The first two methods provide a way to obtain asymptotic approximations of periodic solutions; see for more details Celletti (2010) and Abouelmagd et al. (2014a). But they cannot be used to find solutions that form aperiodically on a slow variable scale. The method of multiple scales is a more general approach in which we construct a solution including one or more new slow variables for each parameter scale of interest in the problem. It does not require this solution to depend periodically on the slow variable.

5.2 Multiple scales method

A solution obtained by the classical perturbation approach will include a secular term leading to unbounded growth. The multiple scales method eliminates such unwanted secularities. The analysis procedure of multiple scales method will be established in the following four steps:

  • The first step

    We look for a solution \(u(f)\equiv y_{n}(f,\tau)\) where \(\tau :=\varepsilon f\). Here \(y_{n}\) depends on two variables scales, namely \(f\) is a fast variable and \(\tau\) is a slow variable such that \(\tau\) is not negligible when \(f\) is of order \(O(\varepsilon^{-1})\). We can extend this procedure to as many variable scales as we like, but in this case the other variable scales will be of order \(O(\varepsilon^{n})\) \((n=2,3,\ldots)\). Consequently we will obtain more equations, which add more difficulty without any further insight into the method, especially for a very small value for \(\varepsilon\). So here we consider two variable scales only.

  • The second step

    We write \(u\) as a perturbation series:

    $$ u (f) = \sum\limits _{n = 0}^{\infty}\epsilon^{n} y_{n}(f,\tau) . $$
    (39)

    Thereby, the solution of Eq. (36) can be given by \(u(f)\), which depends only on the variable \(f\). Nevertheless the multiple scales method seeks a solution that is a function of both the fast true variable scale \(f\) and the slow variable \(\tau\). Even though in the actual solution \(f\) and \(\tau\) are in correlation to each other, this method treats them as independent variables. This strategy enables us to eliminate the secular effect in an elegant way. We want to emphasize that \(f\) and \(\tau\) are ultimately not independent. Now we expand the notation of the derivative with respect to \(f\) by a differential operator \(D_{f}\):

    $$ \dfrac{\mathrm{d}}{\mathrm{d}f} u (f) = D_{f}\sum \limits_{n = 0}^{\infty}\epsilon^{n} y_{n}(f,\tau) $$
    (40)

    for all \(f\), \(\dfrac{\mathrm{d}}{\mathrm{d}f}\tau=\varepsilon\), and the formula of the differential operator \(D_{f}\) is given by

    $$ D_{f}:= \biggl(\dfrac{\partial}{\partial f}+ \varepsilon \dfrac{\partial }{\partial\tau} \biggr). $$
    (41)

    We now insert Eq. (41) into Eq. (40) and assume that \(y_{n}\) is continuously differentiable with respect to \(f\) and \(\tau\). The first and second derivatives of \(u\) are given by

    $$\begin{aligned} &\dfrac{\mathrm{d}u}{\mathrm{d}f}=\dfrac{\partial y_{0}}{\partial f} + \varepsilon \biggl(\dfrac{\partial y_{0}}{\partial\tau}+ \dfrac{\partial y_{1}}{\partial f} \biggr) \\ &\phantom{\dfrac{du}{df}=}{} + \varepsilon^{2} \biggl(\dfrac{\partial y_{1}}{\partial\tau}+ \dfrac{\partial y_{2}}{\partial f} \biggr)+O \bigl(\varepsilon^{3} \bigr), \end{aligned}$$
    (42a)
    $$\begin{aligned} &\dfrac{\mathrm{d}^{2}u}{\mathrm{d}f^{2}}=\dfrac{\partial^{2} y_{0}}{\partial f^{2}} + \varepsilon \biggl(\dfrac{\partial^{2} y_{1}}{\partial f^{2}}+ 2 \dfrac{\partial^{2} y_{0}}{\partial\tau\partial f} \biggr) \\ &\phantom{\dfrac{\mathrm{d}^{2}u}{\mathrm{d}f^{2}}=}{} + \varepsilon^{2} \biggl( \dfrac {\partial^{2} y_{2}}{\partial f^{2}}+ 2\dfrac{\partial^{2} y_{1}}{\partial\tau \partial f} + \dfrac{\partial^{2} y_{0}}{\partial\tau^{2}} \biggr) +O \bigl( \varepsilon^{3} \bigr), \end{aligned}$$
    (42b)

    where\(\frac{\mathrm{d}}{\mathrm{d}\theta}=\frac{\mathrm{d}}{\mathrm{d}f}\).

  • The third step

    In the third step we find \(y_{n}\) with the condition in Eq. (38). Therefore we insert Eqs. (42a), (42b) and Eq. (39) into Eq. (36) and comparing the terms in orders of \(\varepsilon\), we obtain a sequence of linear partial differential equations where the first three of this sequence are

    $$\begin{aligned} &\begin{aligned} &\dfrac{\partial^{2}y_{0}}{\partial f^{2}}+y_{0}= \dfrac{1}{h^{2}} , \\ &y_{0}(0,0) =\dfrac{1}{h^{2}} (1+e ) , \qquad \dfrac{\partial }{\partial f}y_{0}(0,0)=0, \end{aligned} \end{aligned}$$
    (43)
    $$\begin{aligned} &\begin{aligned} &\dfrac{\partial^{2}y_{1}}{\partial f^{2}} + y_{1} =a_{0}+a_{1} y_{0}+y_{0}^{2}+ \biggl(\dfrac{\partial y_{0}}{\partial f} \biggr)^{2} -\dfrac{\partial ^{2}y_{0}}{\partial\tau\partial f}, \\ &y_{1}(0,0) =0 , \qquad \dfrac{\partial}{\partial f}y_{1}(0,0)=- \dfrac {\partial}{\partial\tau}y_{0}(0,0), \end{aligned} \end{aligned}$$
    (44)
    $$\begin{aligned} &\begin{aligned} &\dfrac{\partial^{2}y_{2}}{\partial f^{2}} + y_{2} =a_{1}y_{1}+2y_{0} y_{1} +2 \dfrac {\partial y_{0}}{\partial f} \biggl(\dfrac{\partial y_{0}}{\partial\tau }+\dfrac{\partial y_{1}}{\partial f} \biggr)\\ &\phantom{\dfrac{\partial^{2}y_{2}}{\partial f^{2}} + y_{2} =}{}-2 \dfrac{\partial ^{2}y_{1}}{\partial\tau\partial f} -\dfrac{\partial^{2}y_{0}}{\partial\tau ^{2}}, \\ &y_{2}(0,0) =0 , \qquad \dfrac{\partial}{\partial f}y_{2}(0,0)=- \dfrac {\partial}{\partial\tau}y_{1}(0,0). \end{aligned} \end{aligned}$$
    (45)

  • The fourth step

    In this step we calculate the solutions of the previous equations by eliminating the secular terms. It is convenient to assume that the solution of Eq. (43) can be written in the following form to obtain the dependence of the solutions due to the previous partial differential equations in the independently treated variables \(f\) and \(\tau\):

    $$ y_{0} (f,\tau) = \dfrac{1}{h^{2}}+A( \tau)\mathrm{e}^{\mathrm{i}f}+\bar{A}(\tau)\mathrm{e}^{-\mathrm{i}f}, $$
    (46)

    where \(A(\tau)\) shall be a yet arbitrary complex function of \(\tau\), and \(\bar{A}(\tau)\) denotes its complex conjugate. Furthermore these functions will be determined with the conditions that the solutions of \(y_{1}(f,\tau)\) have no secular terms.

    Now substituting Eq. (46) and its partial derivative with respect to \(f\) into Eq. (44), one obtains

    $$ \begin{aligned} &\dfrac{\partial^{2}y_{1}}{\partial f^{2}} + y_{1} =b_{0} +4 A(\tau)\bar{A}(\tau) +s_{1}(\tau)\mathrm{e}^{\mathrm{i}f}+ \bar{s}_{1}\mathrm{e}^{-\mathrm{i}f} , \\ &y_{1}(0,0) =0 , \qquad \dfrac{\partial}{\partial f}y_{1}(0,0)=- \dfrac {\partial}{\partial\tau}y_{0}(0,0), \end{aligned} $$
    (47)

    where \(b_{0}=\frac{1}{h^{2}}(2E+\frac{5}{h^{2}})\) and

    $$\begin{aligned} &s_{1}(\tau)=b_{1} A -2\dfrac{\mathrm{d}A}{\mathrm{d}\tau}i, \end{aligned}$$
    (48)
    $$\begin{aligned} &\bar{s}_{1}(\tau)=b_{1} \bar{A}+ \dfrac{\mathrm{d}\bar{A}}{\mathrm{d}\tau}i, \end{aligned}$$
    (49)

    where \(b_{1}=6/h^{2}\)

    Notice that \(\mathrm{e}^{\pm \mathrm{i}f}\) is a solution of the homogeneous equation associated to Eq. (44). Therefore if the coefficients \(s_{1}(\tau)\) and \(\bar{s}_{1}(\tau)\) are nonzero, then the solution of \(y_{1}(f,\tau)\) will include secular terms in the variable \(f\). But that is exactly what we want to avoid. Hence we set \(s_{1}\) and \(\bar {s}_{1}\) equal to zero. In addition Eq. (49) is just the complex conjugate of Eq. (48); then it can be omitted. If \(A(\tau)\) satisfies the conditions \(s_{1}(\tau)=0\) and \(\bar{s}_{1}(\tau )=0\), the solution of \(y_{1}\) will not contain secular terms and at least no secularities appear in the first two terms in the perturbation series.

    To achieve our objective, let us try to solve Eq. (48) in the framework of the polar coordinates \((R,f)\), so we assume that

    $$ \begin{aligned} & A(\tau)=R(\tau)\mathrm{e}^{\mathrm{i}f} , \\ &R , f:\mathbb{R}\rightarrow\mathbb{R}, \end{aligned} $$
    (50)

    and substituting Eq. (50) into Eq. (48) with the condition \(s_{1}(\tau)=0\), with simple calculations one gets

    $$\begin{aligned} &A(\tau)=R_{0}\mathrm{e}^{\mathrm{i}(f_{0}-c\tau)}, \end{aligned}$$
    (51a)
    $$\begin{aligned} &\bar{A}(\tau)=R_{0}\mathrm{e}^{-\mathrm{i}(f_{0}-c\tau)}, \end{aligned}$$
    (51b)

    where \(c=3/h^{2}\); \(R_{0}\) and \(f_{0}\) are arbitrary constants.

    Substituting Eqs. (51a), (51b) into Eq. (46) and using the initial conditions in Eq. (43), one obtains \(R_{0}= e/2h^{2}\) and \(f_{0}=0\). Hence Eq. (46) can be rewritten in the form

    $$ y_{0} (f,\tau) = \dfrac{1}{h^{2}}+R_{0} \bigl(\mathrm{e}^{\mathrm{i}(f-c\tau)}+\mathrm{e}^{-\mathrm{i}(f-c\tau )} \bigr) . $$
    (52)

The steps are completed with the construction of Eq. (52), but we have to note that Eqs. (51a), (51b) and Eq. (52) investigate that Eq. (47) is in resonance with the solution of its homogeneous part and its general solution will contain secular terms. To avoid these secularities we must find the general solution of \(y_{1}(f,\tau)\) with the condition that the partial differential equation due to \(y_{2}(f,\tau)\) has no secular terms. Consequently we can do the same procedure to determine the other functions of the perturbed series.

Given \(y_{0}\) as in (52), the solution \(u (f,\tau) \) can be written in the form

$$ u (f,\tau) = \dfrac{1}{h^{2}} \bigl[1+e \cos(f-c\tau) \bigr], $$
(53)

or

$$ u (f) = \dfrac{1}{h^{2}} \bigl[1+ e\cos(1-\varepsilon c)f \bigr], $$
(54)

where \(e\) is the eccentricity when \(\varepsilon=0\).

In this setting the error for fixed \(f\) is given by

$$ \mbox{err} (u) := \bigl|u(f)-y_{0}(f)\bigr| $$
(55)

and it is at most of order \(\textit{ O }(\varepsilon)\). But this is only true as long as \(f< \varepsilon^{-1}\). Therefore \(y_{1}\) stays bounded for all \(f, \tau\), hence \(\varepsilon y_{1}\in\textit {O}(\varepsilon)\) and we only have to worry about \(\varepsilon^{2} y_{2} \), which we did not investigate. This term may contain a secular term that grows with \(\textit{O}(f)\). That is why we have to set the validity interval to \(f \in[0,f_{0}/\varepsilon]\) for some \(f_{0} >0\), if the term of \(y_{1}\) is included. According to Eq. (55) the relative error may be \(\in [0,2e/(1+e) )\).

We have to compare the solution obtained by the multiple scales method directly with analytical or semi-analytical known solutions, of course with a predefined accuracy. For this purpose we will rewrite Eq. (54) in the following form:

$$ r = \dfrac{a (1-e^{2} )}{ [1+ e\cos (1-\frac{3 \, \varepsilon}{a (1-e^{2} )} )f ]}. $$
(56)

In the framework of the parameterized post-Newtonian approximation (when \(\gamma=\beta=1\) and \(\alpha=0\)) in the case of general relativity, Eq. (56) is deduced from the relations provided by Brumberg’s book, corresponding to the expressions (3.1.60)–(3.1.65) in Brumberg (1991).

Although the expression of Eq. (56) is considered a zeroth-order approximation of the multiple scales method, we have revealed the time scale separation underlying the old Brumberg result. On the other hand, Brumberg’s solution was based on a template borrowed from the unperturbed two-body problem—an ansatz that we do not use. So we can conclude that the multiple scales method is more general than the Brumberg method. In fact, the latter method provides an approximation to the former.

6 Conclusions

In this work, we investigate the two-body problem perturbed by a post-Newtonian relativistic term. In Theorem 1, we show that the motion is planar and the total energy is conserved, while the angular momentum is not. In Theorem 2, we prove that the invariant-like motion can be reduced to an invariant motion. In Theorem 3, we offer the conservation law of the angular momentum for low-eccentricity orbits. We also offer a method based on multiple time scales, to obtain an approximated solution to the relativistic perturbed two-body problem. We demonstrate that our approach is more general than the one developed in the book by Brumberg (1991). The latter method furnishes an approximation to our solution. Thereby, we reveal the time scale separation underlying Brumberg’s old result.