Mathematical Algorithm for Solving Two–Body Problem

M. H. Alghamdi 1  and A. A. Alshaery 1
  • 1 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
M. H. Alghamdi
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  • Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
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and A. A. Alshaery

Abstract

In this paper, computational algorithm with the aid of Mathematica software is specifically designed for the gravitational two–body problem. Mathematical module is established to find the position and velocity vectors. Application of this module for different kind of orbits (elliptic, parabolic and hyperbolic) leads to accurate results, which proved module efficiency and to be skillful. The classical power series method is to be utilized as the methodology.

1 Introduction

The classical two–body problem is the dynamical system which describes the motion of two objects. Of course, it is the simplest and only integrable system within frame of the classical Newtonian potential between two–bodies is applied. There are many applications in mathematical astronomy or in engineering and physical sciences can be analyzed within the frame work of two–body systems, it can be used in both quantum mechanics (particles motion) and celestial mechanics (Stars motion)

The multivariate of the perturbed forces in inner and outer space change the two–body from a simple and integrable system to one is complex and is not integrable. Thereby the analytical solutions will be invalid in most real applications. The analysis of two–body problem under the effect of many perturbed forces have received a comprehensive an extended study in the literature space dynamics. For example, the perturbed two–body problem by radiation pressure force and many types of drag forces or both together has been investigated [6,9,10,11,12]. Furthermore, the analytical solutions of the satellite motion within frame of non–sphericity and zonal harmonics perturbations effect have been studied by [7, 8]. [4] have studied also the dynamics of anisotropic Kepler problem with small anisotropy. They proved that at every energy level the anisotropic dynamical system has two periodic orbits.

The perturbed two–body system is not limited to the perturbations of either radiation pressure and drag force or the non–sphericity of the bodies. But the other bodies can be taken as a perturbed forces for the motion of two–body system [5]. Another consequence of two–body problem is that the three-body problem can be reduced to the system of two–body, if we consider either the third body has mass equals zero or if it moves to infinity while the other primary does not. There are a considerable work have been constructed to find the periodic solution for many perturbed two–body systems, see for details [1, 2, 13]. In addition the two–body problem within frame of the corrections law of Newtonian inverse square law of gravitation can be studied as a perturbed model [3].

The motion of planets and asteroids is a very important field of interest in astronomy, and space dynamics. It can be formulated as a dynamical system of differential equations based on the Newton’s laws of gravitation [14]. The law states that everybody attracts every other body along a line intersecting the two bodies with a force equals to F = Gm1m2/r2, where F is the force between the two bodies, G is the gravitational constant, m1 is the mass o f the first body, m2 is the mass of the second body, and r the distance between the centres of masses of the bodies [15].

Additionally, the Newtonian gravitation can be extended to N–bodies by simply summing the forces [16,17,18]. Moreover, various analytical and numerical methods have been used to examine such problems of celestial mechanics comprising of the two–body, three–body and the generalized N–body problem, see [19,20,21,22,23,24,25,26]. But we aim in this paper to computationally tackle the gravitational two–body problem with the application of the power series method [14].

2 Vector two–body equation

We start off this section by first introducing the equation of the gravitational N–body problem as [16]
r¨i=Gri3(m0+mi)r¨iGj=1,jiNmj{rirjrij3+rjrj3},i=1,2,...,N
where r¨i is the position vector of mi relative to m0, mi and m0 are masses of ith body and central body, respectively and G is the universal constant of gravitation. Note that if all masses are equal zero except m0, mi, then Eq. (1) becomes
r¨+μr3r=o
where μ = G(m0 + mi). The above equation is called the classical vector two–body equation.

3 Power series solution

3.1 Lagrange’s fundamental invariants

In favour of Eq. (2), the Lagrange’s fundamental invariants ε, λ and ψ are defined by [14, 27, 28]
ε=μr3λ=1r2(rv)ψ=1r2(vv)
where v=r˙ .
Lagrange’s invariants in Eqs.(3) satisfy the following differential equations [14, 27, 28]
ε˙+3ελ=0λ˙+2λ2+(εψ)=0ψ˙+2λψ+2λε=0

3.2 Basic differential equations

Here, to analyze the gravitational two-body problem by formulate and examine the following nonlinear differential equations [14, 27, 28]
q¨+εq=0ε˙+3ελ=0λ˙+2λ2+(εψ)=0ψ˙+2λψ+2λε=0
In analyzing the previous system of nonlinear differential equations, with employ the power series method [16]. Thus, leads to the power series expansion of the four dependent variables in Eqs. (4) as follows:
q(t)=n=0qn(tt0)nε(t)=n=0εn(tt0)nλ(t)=n=0λn(tt0)nψ(t)=n=0ψn(tt0)n
Upon substitution of Eqs. (5) to Eqs. (4) and then solve for the coefficients of qn, εn, λn and ψn, we get the following recurrence relations
qn+2=1(n+1)(n+2)p=0nεpqnpεn+1=3(n+1)p=0nεpλnpλn+1=1(n+1)(ψnεn2p=0nλpλnp)ψn+1=2(n+1)p=0nλp(εnp+ψnp)
Also with the recurrence relations above taking the following starting values
q0q(t0),q1q˙(t0),ε0ε(t0),λ0λ(t0),ψ0ψ(t0)
Furthermore, the values of q0 and q1 are known from the formulated problem, while ε0, λ0 and ψ0 to be determined from r0r(t0) and v0v(t0) as follows
ε0=1r03μλ0=1r02(x0x˙0+y0y˙0+z0z˙0)ψ0=1r02(x˙02+y˙02+z˙02)
with r0=x02+y02+z02 .

In the next section with a help of Eqs. (6, 7), the computational values of qn are determined for n = 9 using a designed algorithm which is called QPS, see Appendix A for algorithm.

3.3 f and g functions

Here, two Lagrangian coefficients functions f and g [14,27] introduced instead of one that led to the following position vector
r=fr0+gv0
where r0 and v0 are the initial values of the position and velocity vectors. Note also that the velocity vector is given by
v=f˙r0+g˙v0
Moreover, it is remarkable that Eqs. (8, 9) satisfies the first equation in Eq. (4) with the following initial conditions [14, 27]
q(t0)={1q=f0q=gdq(t0)dt={0q=f1q=g
As in the above, using the power series method to get the relation recursively. The power series expansion of these functions take the form [27]
f=n=0fn(tt0)ng=n=0gn(tt0)n
together with the starting values [14, 27]
f0=1f1=0g0=0g1=1

Also in the next section, analytical computations determine for the values of fn and gn for n = 10 using a designed algorithm called FGPS, see Appendix B for the algorithm.

4 Results and Discussion

In this chapter, computational values determine for the values qn +2 for n=9, fn and gn for n = 10 using Mathematica software with designed computational algorithms QPS and FGPS are given in Appendix A and Appendix B, respectively.

Furthermore, a mathematical modules establish to find the scalar values of the position and velocity vectors of the two-body motion via Mathematica software with the designed computational algorithm RVPS is given in Appendix C then we apply this module for different kinds of orbits (elliptic, parabolic and hyperbolic).

4.1 Values of qn

With the application of QPS algorithm for n = 9, the following symbolic expression of the q’s coefficients are
q0=q0q1=q1q2=12(q0)ε0q3=16(ε0)(q1(3q0)λ0)
q4=124ε0((6q1)λ0+q0(3(ψ05λ02)2ε0))q5=1120ε0(((15q0)λ0)(7λ02+2ε03ψ0)+q1(9(ψ05λ02)8ε0))q6=1720ε0(((30q1)λ0)(14λ02+5ε06ψ0)q0(22ε02+(6ε0)(70λ0211ψ0)+45(21λ04+ψ02(14λ02)ψ0)))
q7=15040(ε0(((63q0)λ0)(165λ04+(100ε0)λ02+12ε02+25ψ02(6(25λ02+6ε0))ψ0)q1(172ε02+(36ε0)(70λ0211ψ0)+225(21λ04+ψ02(14λ02)ψ0))))q8=140320(ε0(((126q1)λ0)(495λ04+(350ε0)λ02+52ε02+75ψ02(18(25λ02+7ε0))ψ0)+q0(584ε03+(36ε02)(73ψ0560λ02)(54ε0)(1925λ04+67ψ02(1120λ02)ψ0)315(429λ065ψ03+(135λ02)ψ02(495λ04)ψ0))))q9=1362880(ε0(((15q0)λ0)(2368ε03+(444ε02)(77λ0224ψ0)+(18ε0)(7007λ04+827ψ02(5698λ02)ψ0)+189(715λ0635ψ03+(385λ02)ψ02(1001λ04)ψ0))q1(7136ε03+(108ε02)(1785λ02232ψ0)+(432ε0)(1925λ04+67ψ02(1120λ02)ψ0)+2205(429λ065ψ03+(135λ02)ψ02(495λ04)ψ0))))

4.2 Values of fn

With the application of FGPS for n = 10; T = 1, 2, the following symbolic expressions of the f’s are
f0=1f1=0f2=ε02f3=ε0λ02
f4=124(ε0)(15λ02+2ε03ψ0)f5=18(ε0λ0)(7λ02+2ε03ψ0)f6=1720(ε0)(22ε02+(6ε0)(70λ0211ψ0)+45(21λ04+ψ02(14λ02)ψ0))
f7=180(ε0λ0)(165λ04+(100ε0)λ02+12ε02+25ψ02(6(25λ02+6ε0))ψ0)f8=140320(ε0(584ε03+(36ε02)(560λ0273ψ0)+(54ε0)(1925λ04+67ψ02(1120λ02)ψ0)+315(429λ065ψ03+(135λ02)ψ02(495λ04)ψ0)))f9=124192((ε0λ0)(2368ε03+(444ε02)(77λ0224ψ0)+(18ε0)(7007λ04+827ψ02(5698λ02)ψ0)+189(715λ0635ψ03+(385λ02)ψ02(1001λ04)ψ0)))f10=13628800(ε0(28384ε04+(48ε03)(31735λ023548ψ0)+(54ε02)(245245λ04+6559ψ02(126940λ02)ψ0)+(90ε0)(420420λ063461ψ03+(107514λ02)ψ02(441441λ04)ψ0)+14175(2431λ08+7ψ04(308λ02)ψ03+(2002λ04)ψ02(4004λ06)ψ0)))

4.3 Values of gn

Again with the application of FGPS for n = 10; T = 1, 2, the following symbolic expressions of g’s coefficients are
g0=0g1=1g2=0g3=ε06g4=ε0λ04
g5=1120(ε0)(45λ02+8ε09ψ0)g6=124(ε0λ0)(14λ02+5ε06ψ0)g7=ε0(172ε02+(36ε0)(70λ0211ψ0)+225(21λ04+ψ02(14λ02)ψ0))5040
g8=1320(ε0λ0)(495λ04+(350ε0)λ02+52ε02+75ψ02(18(25λ02+7ε0))ψ0)g9=1362880(ε0(7136ε03+(108ε02)(1785λ02232ψ0)+(432ε0)(1925λ04+67ψ02(1120λ02)ψ0)+2205(429λ065ψ03+(135λ02)ψ02(495λ04)ψ0)))g10=1120960((ε0λ0)(15220ε03+(12ε02)(14938λ024647ψ0)+(81ε0)(7007λ04+827ψ02(5698λ02)ψ0)+756(715λ0635ψ03+(385λ02)ψ02(1001λ04)ψ0)))

4.4 Applications on orbits

With the application of RVPS algorithm on different kinds of orbits with the initial values components for the position and velocity vectors is given by then, we get the final scalar values of the position and velocity vectors

5 Conclusion

The dynamical system of the two–body problem is addressed, by applying the mathematical modules, which are established to find the scalar values of the position and velocity vectors of the two–body motion. The Application of this module for different kinds of orbits (elliptic, parabolic and hyperbolic) leads to accurate results, that prove module efficiency.

Appendix A: QPS

  1. *PurposeTo generate n symbolic expressions of the q’s coefficients of the time power series solution i=0qi(tt0)i of the single harmonic oscillator q¨+εq=0 .
  2. *Inputq0, q1, ε0, λ0, ψ0, n
  3. *Outputn symbolic expressions of the q’s coefficients.
  4. *Module list.
    Module[{},Do[{qn+2=i=0nεiqni(n+1)(n+2),εn+1=3i=0nεiλnin+1,λn+1=2i=0nλiλni+ψnεnn+1,ψn+1=2i=0nλi(ψni+εni)n+1,{n,0,m}]]
Table 1

The initial values components for the position and velocity vectors

Orbit Typex0 (km)y0 (km)z0 (km)0 (km/s)y˙0(km/s)ż0 (km/s)
Elliptic5096.5306253997.328251−1767.351714.6830160850.6023868474.217758697
0.4625816700.0633660530.94365569−0.3861046700.4854997700.538650150
0.4675859500.0604306950.94778613−0.3435505000.4642865800.582594050
Parabolic−1616.9409947756.699643−7712.188395−0.67303031378.43493095700.705548375
0.201227080−0.449587910−0.7966663201.10390070000.4558421100−0.841963950
0.000786356−0.105595880−1.1217328000.97159778000.4937057600−0.766396930
Hyperbolic0000100000.00000000.000000000.000000000.000000009.20000000
−1.61740150−1.0018533−0.627945830.563250920.19817751−1.37540280
−0.66846076−2.0580722−1.964201000.788746240.74895884−0.78257152
Table 2

The final scalar values of the position and velocity vectors

Orbit Typer(km)v(km/s)
Elliptic1.197020000.64264300
1.199896700.66492594
0.422982991.87403310
Parabolic2.17063000.95989400
55.8733770.18921011
55.81316100.18930981
Hyperbolic1.65262001.1352500
226.741561.1219743
213.825591.0585222

Appendix B: FGPS

  1. *PurposeTo generate n symbolic expressions of the f’s or g’s coefficients of the time power series of the Lagrange functions f or g.
  2. *InputT:A positive integer takes the value 1 or 2, such that:
    • T = 1 if it is required to find the f’s coefficients,
    • T = 2 if it is required to find the g’s coefficients, ε0, λ0, ψ0, n
  3. *Outputn symbolic expressions of the f’s or g’s coefficients according to the value of T.
  4. *Module list.
    Module[{},Which[T==1,Goto[1],T==2,Goto[2]];Label[1];q0=1;q1=0;Goto[3];Label[2];q0=0;q1=1;Label[3];Evaluate[QPS(q0,q1,ε0,λ0,ψ0,m)]]

Appendix C: RVPS

  1. *PurposeTo generate the position and velocity values for any orbit.
  2. *Inputx0, y0, z0, 0, y˙0 , ż0, μ, m, t and t0.
  3. *OutputThe position and velocity values for any orbit.
  4. *Module list.
    Module[{},r0=x2+y2+z2;ε0=μr03;λ0=xxd+yyd+zzdr02;ψ0=xd2+yd2+zd2r02;τ=tt0;F=j=0mqjτj;Fd=j=0mjqjτj1;G=j=0mqjτj;Gd=j=0mjqjτj1;xx=Fx+Gxd;yy=Fy+Gyd;zz=Fz+Gzd;r=xx2+yy2+zz2;xxd=Fdx+Gdxd;yyd=Fdy+Gdyd;zzd=Fdz+Gdzd;v=xxd2+yyd2+zzd2]]

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  • [1]

    Abouelmagd E I, Mortari D, Selim H H (2015). Analytical study of periodic solutions on perturbed equatorial two-body problem. International Journal of Bifurcation and Chaos. 25 (14): 1540040. https://doi.org/10.1142/S0218127415400404.

  • [2]

    Abouelmagd E I, Elshaboury S M, Selim H H (2016). Numerical integration of a relativistic two-body problem via a multiple scales method. Astrophys. Space Sci. 361 (1): 38. https://doi.org/10.1007/s10509-015-2625-8.

    • Crossref
    • Export Citation
  • [3]

    Abouelmagd E I (2018). Periodic Solution of Two–body problem by KB Averaging Method within Frame of the modified Newtonian Potential The Journal of the Astronautical Sciences 65(3): 291–306. https://doi.org/10.1007/s40295-018-0128-x.

    • Crossref
    • Export Citation
  • [4]

    Abouelmagd E I, Llibre J, Guirao J L G (2017). Periodic Orbits of the Planar Anisotropic Kepler Problem. International Journal of Bifurcation and Chaos 27 (3): 1750039. https://doi.org/10.1142/S0218127417500390.

  • [5]

    Abouelmagd E I, Alhothuali M S, Guirao J L G, Malaikah H M (2015). On the periodic structure of planar photogravitational Hill problem. Applied Mathematics & Information Sciences 9 (5): 2409–2416. http://dx.doi.org/10.12785/amis/090524.

  • [6]

    Elshaboury S M, Mostafa A (2014). The motion of axisymmetric satellite with drag and radiation pressure. Astrophys. Space Sci. 352: 515–519. https://doi.org/10.1007/s10509-014-1975-y.

    • Crossref
    • Export Citation
  • [7]

    Engels R C, Junkins J L (1981). The gravity-perturbed Lambert problem: A KS variation of parameters approach. Celest. Mech. Dyn. Astron. 24 (1): 3–21. https://doi.org/10.1007/BF01228790.

    • Crossref
    • Export Citation
  • [8]

    Jezewski D J (1983). A noncanonical analytic solution to the J2 perturbed two-body problem. Celest. Mech. Dyn. Astron. 30 (4): 343–361. https://doi.org/10.1007/BF01375505.

    • Crossref
    • Export Citation
  • [9]

    Leach P G L (1987). The first integrals and orbit equation for the Kepler problem with drag. J. Phys. A 20 (8): 1997–2002. https://doi.org/10.1088/0305-4470/20/8/019.

    • Crossref
    • Export Citation
  • [10]

    Margheri A, Ortega R, Rebelo C (2012). Some analytical results about periodic orbits in the restricted three body problem with dissipation. Celest. Mech. Dyn. Astron. 113: 279–290. https://doi.org/10.1007/s10569-012-9415-1.

    • Crossref
    • Export Citation
  • [11]

    Mavraganis A G, Michalakis D G (1994). The two-body problem with drag and radiation pressure. Celest. Mech. Dyn. Astron. 58 (4): 393–403. https://doi.org/10.1007/BF00692013.

    • Crossref
    • Export Citation
  • [12]

    Mittleman D, Jezewski D (1982). An analytic solution to the classical two-body problem with drag. Celest. Mech. Dyn. Astron. 28 (4): 401–413. https://doi.org/10.1007/BF01372122.

    • Crossref
    • Export Citation
  • [13]

    Russell R P (2006). Global Search for Planar and Three-Dimensional Periodic Orbits Near Europa. The Journal of the Astronautical Sciences 54 (2): 199–266. https://doi.org/10.1007/BF03256483.

    • Crossref
    • Export Citation
  • [14]

    Battin, R.H. (1999), An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, New York. https://doi.org/10.2514/4.861543.

  • [15]

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