Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System

Sawsan Alhowaity

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Dettagli della rivista

Formato: Rivista
eISSN: 2444-8656
Prima pubblicazione: 01 Jan 2016
Frequenza di pubblicazione: 2 volte all'anno
Lingue: Inglese

The N-body problem has a great significant in celestial mechanics, because there are many dynamical systems consist of N-bodies, like the solar system, constellation of satellites, and etc. N-body problem can be used to describe the motion or dynamic of N–point masses, which are moving under Newton’s laws of motion under the mutual gravitational forces.

So, it has many applications in scientific researches, where N ≤ 2, where the problem is solved for N = 2 because it can be reduced to perturbed two–body problem or the perturbed Kepler problem, which is a system of ordinary differential equations that describe the motion of two particles moving under their mutual gravitational attraction force. Some considerable work on the solution of the Kepler problem are studied in [1–8].

Furthermore, there are some considerable contributions to find the periodic solutions of perturbed two–body problem using some perturbation methods are developed in [9], the authors have presented an analytical study to find periodic solutions of the perturbed two–body problem using the classic perturbation theory, the Lindstedt–Poincaré technique, and the Krylov–Bogoliubov–Mitropolsky method, as well as, a numerical integration by Runge–Kutta algorithm is applied to compare the obtained results by the aforemention methods. Also, some work to find also periodic solutions by others powerful methods, such as multiple scales technique, the averaging theory and KB averaging method [10–12]

It is exceeding important to estimate the space bodies positions and velocities vectors at any time. The equations of motion for the N-body trajectories are derived, while the solution is developed (especially N = 2) using a recurrent power series technique [13–19].

In this paper, a universal computational algorithm will be constructed to find the solution of two–body problem using F and G series. Then, an example for the geocentric system motion of the planet Mercury in the solar system will be considered, to find the solution.

Equations of motion

In this section, the practical form of the equations of motion will be presented. The gravitational parameter μ could be written as $μ = k^{2} (1 + m),$ where m is the body mass in terms of the central body mass, while k is the Gaussian gravitational constant, its value depends on the adopted system. The following units will be used in the calculation within frame of Geocentric System:

The Earth at the centre of the system.

The length is measured in terms of the Earth’s radius (gr).

The mass is measured in terms of the Earth’s mass (gm).

The time is measured in minutes (min).

The value of k in this system equals 07436680 (gr)^3/2/min.

The modified time Τ is defined by the equation $τ = k (t - t_{0})$

Using the modified time, the vector equation of motion becomes $\frac{d^{2} r}{d τ^{2}} + \frac{μ}{r^{3}} r = 0,$ while the constant μ becomes μ = 1 + m. Consequently, the fundamental equations become

Power series $r = r_{0} + τ {\dot{r}}_{0} + \frac{1}{2} τ^{2} {\ddot{r}}_{0} + \dots .$

Representations of r and ${\dot{r}}_{0}$ $\begin{aligned} r & =F(τ)r_{0}+G(τ){\dot{r}}_{0}, \\ {\dot{r}}_{0} & =F^{'}(τ)r_{0}+G^{'}(τ){\dot{r}}_{0}, \end{aligned}$ where $\begin{aligned} F^{'} (τ) & =\frac{d F}{d τ}, \\ G^{'} (τ) & =\frac{d G}{d τ}, \end{aligned}$ $F^{'} (τ)$ and $G^{'} (τ)$ are given by the following power series 1 $\begin{aligned} F (τ) & =\sum_{j = 0}^{\infty}f_{j}τ^{j}, \end{aligned}$ 2 $\begin{aligned} G (τ) & =\sum_{j = 0}^{\infty}g_{j}τ^{j}, \end{aligned}$ Finally, the power series coefficients are calculated recursively as illustrated in the following section.

Computational Algorithm

Purpose

Using power series (1) and (2), to compute for a body of mass m (given in terms of the central body mass), the following:

The coordinates of positions vector (x₀, y₀, z₀) and also the velocities components vector $(\dot{x_{0}}, \dot{y_{0}}, \dot{z_{0}})$ at time t, from the corresponding components (x₀, y₀, z₀) and $(\dot{x_{0}}, \dot{y_{0}}, \dot{z_{0}})$ at time t₀.

To compute the distance r and the velocity value v at t (In the computational steps, we shall consider 10 terms only of the power series).

Input

t, t₀, (x₀, y₀, z₀), $(\dot{x_{0}}, \dot{y_{0}}, \dot{z_{0}})$ , m and k.

Computational steps

$τ = k (t - t_{0})$

μ = 1 + m

$r_{0} = ({x_{0}}^{2} + {y_{0}}^{2} + {z_{0}}^{2})^{1 / 2}$

$ϵ_{0} = \frac{μ}{r_{0}^{3}}$ , $λ_{0} = \frac{x_{0} \dot{x_{0}} + y_{0} \dot{y_{0}} + z_{0} \dot{z_{0}}}{r_{0}^{2}}$ , $ψ_{0} = \frac{{\dot{x_{0}}}^{2} + {\dot{y_{0}}}^{2} + {\dot{z_{0}}}^{2}}{r_{0}^{2}}$

$f_{j}; j = 0, 1, . . ., 10$ (from Appendices A)

$g_{j}; j = 0, 1, . . ., 10$ (from Appendices B)

$F = \sum_{j = 0}^{10} f_{j} τ^{j}$ , and $G = \sum_{j = 0}^{10} g_{j} τ^{j}$

$F^{'} = \sum_{j = 0}^{10} j f_{j} τ^{j - 1}$ , and $G^{'} = \sum_{j = 0}^{10} j g_{j} τ^{j - 1}$

$x = F (τ) x_{0} + G (τ) \dot{x_{0}}$ ,

$y = F (τ) y_{0} + G (τ) \dot{y_{0}}$ , and

$z = F (τ) z_{0} + G (τ) \dot{z_{0}}$ .

10)

$r = (x^{2} + y^{2} + z^{2})^{1 / 2}$

11)

$\dot{x} = F^{'} (τ) x_{0} + G^{'} (τ) \dot{x_{0}}$ ,

$\dot{y} = F^{'} (τ) y_{0} + G^{'} (τ) \dot{y_{0}}$ , and

$\dot{z} = F^{'} (τ) z_{0} + G^{'} (τ) \dot{z_{0}}$ .

12)

$v = ({\dot{x}}^{2} + {\dot{y}}^{2} + {\dot{z}}^{2})^{1 / 2}$

13)

End

Finally, we would like to mention that: It enough to find only ten coefficients from the functions f_j and g_j, that is is due to the fundamental applications of orbit determination depends on the interval of time, where the value of (t – t₀) must be small. But in the case of the interval of time is large, we will follow the strategy of repeating decrements of time interval many times over, thus the desired time interval must be very small in this procedures. At end, it should remark that the application of the aforementioned algorithm of F and G series for the initial value problem (the calculation of r and v at required time t from r₀ and v₀ at the initial time t₀ is effective for some factors), such as the algorithm are satisfied the following properties and advantages:

Its simple reparation provide us with powerful method to find any required number from the coefficients for precise predictions of r and $\dot{v}$ .

There is no neediness to find the Kepler’s equation solution and its variants for parabolic and hyperbolic orbits to evaluate the position vector r and velocity vector $\dot{v}$ .

The algorithm is considered a global procedure, where it can be applied for any of conic orbit.

The algorithm can be applied whatever the length of interval time $(t - t_{0})$ .

Numerical results

In this section, we will take as an example the heliocentric motion of the planet Mercury. We will calculate its position for 20 days. The numerical values are

Sun’s mass = 1MU,

Mass of Mercury in terms of Sun’s mass = 0.000000166,

κ = 0.01720209895,

The initial time t₀ = 3/8/1985,

r₀ = +0.169342, −0.355991, −0.207717 AU,

${\dot{r}}_{0} = + 1.18376, + 0.669777, + 0.234931$ AU/day.

The out put of the application of the above algorithm is listed in Table 1

Table 1.

The components of position and velocity vectors r and $\dot{r}$

t	x	y	z	$\dot{x}$	$\dot{y}$	$\dot{z}$
3/8/1985	0.169342	–0.355991	0.207717	1.183760	0.669777	0.234931
5/8/1985	0.208826	–0.330584	–0.198243	1.108300	0.806612	0.315848
7/8/1985	0.245372	–0.300538	–0.185986	1.012790	0.939173	0.396564
9/8/1985	0.278269	–0.266032	–0.170969	0.895868	1.065510	0.476178
11/8/1985	0.306754	–0.227321	–0.153247	0.756154	1.183160	0.553512
13/8/1985	0.330022	–0.184756	–0.132926	0.592352	1.288970	0.627024
15/8/1985	0.347225	–0.138810	–0.110169	0.403457	1.379000	0.694714
17/8/1985	0.357491	–0.090106	–0.085220	0.189079	1.448410	0.754027
19/8/1985	0.359950	–0.039451	–0.058419	–0.050101	1.491330	0.801771
21/8/1985	0.353785	0.0121299	–0.0302278	–0.311690	1.501050	0.834100
23/8/1985	0.338301	0.0633692	–0.0012525	–0.590801	1.470260	0.846615
25/8/1985	0.313025	0.1127500	0.0277459	–0.879227	1.391890	0.834681

Results and Conclusion

In Table 2, the comparison between the computed values from the above algorithm with the corresponding values of Mercury’s radius vectors given in Astronomical Almanac.

Table 2.

The comparison between the computed values and the given in Astronomical Almanac

Date	Computed	Reference	Residuals
13/8/1985	0.4008980	0.4008978	–0.0000002
23/8/1985	0.3441870	0.3441872	+0.0000002

It can be seen from the residuals that the two–body motion aggress. Finally, we demonstrate that a universal computational algorithm is constructed by using F and G series, which can be applied for any of conic orbit. In particular, to find the solution of the two–body problem. In this context,the solution of geocentric system motion of the Mercury planet in the solar system is found using the obtained computational algorithm.

Figure e tabelle

The comparison between the computed values and the given in Astronomical Almanac

Date	Computed	Reference	Residuals
13/8/1985	0.4008980	0.4008978	–0.0000002
23/8/1985	0.3441870	0.3441872	+0.0000002

The components of position and velocity vectors r and r˙

t	x	y	z	x˙	y˙	z˙
3/8/1985	0.169342	–0.355991	0.207717	1.183760	0.669777	0.234931
5/8/1985	0.208826	–0.330584	–0.198243	1.108300	0.806612	0.315848
7/8/1985	0.245372	–0.300538	–0.185986	1.012790	0.939173	0.396564
9/8/1985	0.278269	–0.266032	–0.170969	0.895868	1.065510	0.476178
11/8/1985	0.306754	–0.227321	–0.153247	0.756154	1.183160	0.553512
13/8/1985	0.330022	–0.184756	–0.132926	0.592352	1.288970	0.627024
15/8/1985	0.347225	–0.138810	–0.110169	0.403457	1.379000	0.694714
17/8/1985	0.357491	–0.090106	–0.085220	0.189079	1.448410	0.754027
19/8/1985	0.359950	–0.039451	–0.058419	–0.050101	1.491330	0.801771
21/8/1985	0.353785	0.0121299	–0.0302278	–0.311690	1.501050	0.834100
23/8/1985	0.338301	0.0633692	–0.0012525	–0.590801	1.470260	0.846615
25/8/1985	0.313025	0.1127500	0.0277459	–0.879227	1.391890	0.834681

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Applied Mathematics and Nonlinear Sciences

Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System

Sawsan Alhowaity
Sawsan Alhowaity

Pubblicato online: 23 Jul 2022

Volume & Edizione: AHEAD OF PRINT

Pagine: -

Ricevuto: 01 Apr 2022

Accettato: 15 Jul 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Anteprima PDF

Articolo

Figure e tabelle

The comparison between the computed values and the given in Astronomical Almanac

The components of position and velocity vectors r and r˙

Riferimenti

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Pianifica la tua conferenza remota con Sciendo

Applied Mathematics and Nonlinear Sciences

Computational Algorithm to Solve Two–Body Problem Using Power Series in Geocentric System

Sawsan AlhowaitySawsan Alhowaity

Pubblicato online: 23 Jul 2022

Volume & Edizione: AHEAD OF PRINT

Pagine: -

Ricevuto: 01 Apr 2022

Accettato: 15 Jul 2022

Dettagli rivista e numero

Applied Mathematics and Nonlinear Sciences

Anteprima PDF

Articolo

Figure e tabelle

The comparison between the computed values and the given in Astronomical Almanac

The components of position and velocity vectors r and r˙

Riferimenti

Articoli recenti

Pianifica la tua conferenza remota con Sciendo

Sawsan Alhowaity
Sawsan Alhowaity