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In this work, the Hamiltonian of the four-body problem is considered under the effects of solar radiation pressure. The equations of motion of the infinitesimal body are obtained in the Hamiltonian canonical form. The libration points and the corresponding Jacobi constants are obtained with different values of the solar radiation pressure coefficient. The motion and its stability about each point are studied. A family of periodic orbits under the effects of the gravitational forces of the primaries and the solar radiation pressure are obtained depending on the pure numerical method. This purpose is applied to the Sun-Earth-Moon-Space craft system, and the results obtained are in a good agreement with the previous work such as (Kumari and Papadouris, 2013).

Since the restricted four-body problem has an important role in astro-dynamics and space dynamics, therefore many attempts dealing with this problem have been done using numerical and analytical methods. Numerical simulations of the one-dimensional Newtonian four-body problem for the special case were conducted in which the bodies are distributed symmetrically about the centre of mass and were studied the Schubart-like periodic orbit’s stability to perturbation where it is apparently stable in one-dimension but is unstable in three-dimensions [

In this work the Hamiltonian of the restricted four-body problem is constructed under the effect of solar radiation pressure, the location of the libration points are obtained at different values of the solar radiation pressure coefficient, the stability of motion about the collinear libration points is studied and Poincare surface sections is used to illustrates these stabilities.

According to Newton-Lebedev law, the general photo-gravitational force is described as the geometrical sum of two opposite forces, 1) Apart from the

gravitational acceleration of the Sun on the spacecraft F g = G m s m r s 2 , m refers to

the mass of space craft, while m_{s} refers to the mass of Sun; 2) The radiation force

acting on spacecraft is obtained by F r a d = L ⊙ 4 π r s 2 c A , L ⊙ is Luminosity of Sun,

A is the cross-section area of the spacecraft surface, c is the speed of light, and r_{s} is the distance between Sun and the spacecraft. So that

F = F g − F r a d = F g ( 1 − F r a d F g ) = F g ( 1 − β ) (1)

where, β = F r a d F g = L ⊙ 4 π r s 2 c 1 G m s A m

The potential of the force exists by the Sun is

V R P = − ∫ F d r = − ∫ F g ( 1 − β ) d r = − ( 1 − β ) ∫ m s r s 2 d r = ( 1 − β ) m s r s (2)

Then, in the case of restricted four-body problem the effective potential included the effects of solar radiation pressure (SRP) is given by

V ( x , y , z ) = 1 2 ( x 2 + y 2 ) + 1 − μ r 1 + μ r 2 + ( 1 − β ) μ s r 3 (3)

where x, y, z are the coordinates of the fourth body, r 1 , r 2 , r 3 are the dimensionless distances from the fourth body to the primaries, and μ , 1 − μ , μ s are the dimensionless masses for the primaries, defined as

μ = m 2 m 1 + m 2 ; μ s = m s m 1 + m 2 (4)

where m_{1}, m_{2} and m_{s} are the masses of primaries respectively.

To construct the Hamiltonian of the restricted four-body problem using the concept of Lagrange and Hamiltonian principle for the rotating frame ( ξ , η , ζ )

L ( ξ , η , ζ , ξ ˙ , η ˙ , ζ ˙ ) = T − V = 1 2 ( ξ ˙ 2 + η ˙ 2 + ζ ˙ 2 ) + G [ m 1 r 1 + m 2 r 2 + ( 1 − β ) m 3 r 3 ] (5)

Since

ξ = x cos ω + y sin ω (6.1)

η = x sin ω − y cos ω (6.2)

ζ = z (6.3)

where, ω = n t is the angular velocity of the rotating system, ω is the rate at which the primaries rotates about their center of mass and change their position per time (second) and n is the mean motion. Using Equation (3) and Equation (6), after some little algebraic reductions then

L ( x , y , z , x ˙ , y ˙ , z ˙ ) = 1 2 [ ( x ˙ 2 + y ˙ 2 + z ˙ 2 ) + ( x 2 + y 2 ) + 2 ( x y ˙ − x ˙ y ) ] + 1 − μ r 1 + μ r 2 + ( 1 − β ) μ s r 3 (7)

By using the definition of the momentum p i = ∂ L ∂ q ˙ i , then

x ˙ = p x + y (8.1)

y ˙ = p y − x (8.2)

z ˙ = p z (8.3)

the Hamiltonian is defined by [

H = ∑ i p i q ˙ i − L (9)

Substitute from Equation (7) and Equation (8) into Equation (9), this yields the Hamiltonian of the restricted four-body problem with solar radiation pressure in the form

H = 1 2 ( p x 2 + p y 2 + p z 2 ) + y p x − x p y − [ 1 − μ r 1 + μ r 2 + ( 1 − β ) μ s r 3 ] (10)

where

p x , p y , p z are the components of momenta in Cartesian coordinates. And the canonical form is given by

x ˙ = ∂ H ∂ p x = p x + y (11.1)

y ˙ = ∂ H ∂ p y = p y − x (11.2)

z ˙ = ∂ H ∂ p z = p z (11.3)

p ˙ x = − ∂ H ∂ x = y ˙ + x − ( 1 − μ ) ( x + μ ) r 1 3 − μ ( x + μ − 1 ) r 2 3 − μ s ( 1 − β ) ( x − R s cos θ ) r 3 3 (11.4)

p ˙ y = − ∂ H ∂ y = − x ˙ + y − ( 1 − μ ) ⋅ y r 1 3 − μ y r 2 3 − μ s ( 1 − β ) ( y − R s sin θ ) r 3 3 (11.5)

p ˙ z = − ∂ H ∂ z = − ( 1 − μ ) ⋅ z r 1 3 − μ z r 2 3 − μ s ( 1 − β ) z r 3 3 (11.6)

where

θ = ω s t , ω s is the angular velocity of the center of mass of the two primaries about the Sun, and R s is the distance between the Sun and the center of mass of the two primaries. Equation (11) represents the equations of motion of the fourth body under the effect of gravitational forces and the solar radiation pressure.

Now, the Jacobi constant is defined as

p ˙ x 2 + p ˙ y 2 + p ˙ z 2 = − 2 V + C (12)

when the velocity tends to zero, then Equation (12) becomes

C = 2 V (13)

Equation (13) enables to obtain the zero velocity curves and is used to apply the Poincare surface sections (PSS) to study the stability of motion about each libration point.

One of the special solutions of the restricted four-body problem is the equilibrium points at which the components of the velocity of the fourth body are zero. The subject of equilibrium points is to find the location of the points where a fourth body could be placed. To obtain the location of libration points, the conditions x ˙ = y ˙ = z ˙ = p ˙ x = p ˙ y = p ˙ z = 0 , are applied on Equation (11), then

p x + y = 0 (14.1)

p y − x = 0 (14.2)

p z = 0 (14.3)

Then

p x = − y , p y = x , and p z = 0 , then

x − ( 1 − μ ) ( x + μ ) r 1 3 − μ ( x + μ − 1 ) r 2 3 − μ s ( 1 − β ) ( x − R s cos θ ) r 3 3 = 0 (15.1)

y − ( 1 − μ ) ⋅ y r 1 3 − μ y r 2 3 − μ s ( 1 − β ) ( y − R s cos θ ) r 3 3 = 0 (15.2)

− ( 1 − μ ) ⋅ z r 1 3 − μ z r 2 3 − μ s ( 1 − β ) z r 3 3 = 0 (15.3)

Since the motion lies in the x-y plane then Equation (15.3) will be vanished. Now, the collinear points can be determined from Equation (15.1), with θ = 0

x = ( 1 − μ ) ( x + μ ) r 1 3 − μ ( x + μ − 1 ) r 2 3 − μ s ( 1 − β ) ( x − R s ) r 3 3 (16)

Let X L denotes the coordinates of the libration points, then applying the binomial function on Equation (16), then

X L 5 ( R s 3 − μ S ( 1 − β ) ) + X L 4 ( − 2 R s 3 + 2 μ S ( 1 − β ) + μ S R s ( 1 − β ) + 4 R s 3 μ − 4 μ S ( 1 − β ) μ ) + X L 3 ( R s 3 − μ S ( 1 − β ) − 2 μ S R s ( 1 − β ) − 6 R s 3 μ + 6 μ S ( 1 − β ) μ + 4 μ S R s ( 1 − β ) μ + 6 R s 3 μ 2 − 6 μ S ( 1 − β ) μ 2 ) + X L 2 ( − R s 3 + μ S R s ( 1 − β ) + 2 R s 3 μ − 2 μ S ( 1 − β ) μ − 6 μ S R s ( 1 − β ) μ

− 6 R s 3 μ 2 + 6 μ S ( 1 − β ) μ 2 + 6 μ S R s ( 1 − β ) μ 2 + 4 R s 3 μ 3 − 4 μ S ( 1 − β ) μ 3 ) + X L ( 2 R s 3 − 4 R s 3 μ + 2 μ S R s ( 1 − β ) μ + R s 3 μ 2 − μ S ( 1 − β ) μ 2 − 6 μ S R s ( 1 − β ) μ 2 − 2 R s 3 μ 3 + 2 μ S ( 1 − β ) μ 3 + 4 μ S R s ( 1 − β ) μ 3 + R s 3 μ 4 − μ S ( 1 − β ) μ 4 ) + 3 R s 3 μ − 3 R s 3 μ 2 + μ S R s ( 1 − β ) μ 2 − 2 μ S R s ( 1 − β ) μ 3 + μ S R s ( 1 − β ) μ 4 − R s 3 = 0 (17)

This is a quantic equation and its solution has five real parts depends on the parameters μ S , μ , R s and β these roots give the positions of the collinear libration points.

The Sun-Earth-Moon system is considered with the effect of SRP. Then the perturbed motion around collinear libration points is obtained with initial conditions of small displacement from x 0 , y 0 and x ˙ 0 = y ˙ 0 = 0 . To obtain the periodic orbits family about each of collinear libration points for different values of β the following steps are used.

1) The values of potentials V x x , V y y are determined for the value of β and the corresponding coordinate of libration point.

2) The Eigen values are obtained from the characteristic equation.

3) The Eigen values will be four values (2 real and 2 imaginaries).

4) The two imaginary values responding to give the stable periodic orbits about the libration point.

Now, to apply these steps the linear equations for the motion about the collinear libration points for the fourth body are written as follows,

p ˙ x − 2 p y = x V x x + y V x y + z V x z , (18.1)

p ˙ y + 2 p x = x V x y + y V y y + z V y z , (18.2)

p ˙ z = z V z z . (18.3)

where the partial derivatives of the effective potential for the four-body problem with SRP are obtained as

V x x = 1 + 3 m s ( − R s + x ) 2 ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 5 / 2 − m s ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 3 / 2 + 3 μ ( − 1 + x + μ ) 2 ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 5 / 2 − μ ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 3 / 2 + 3 ( 1 − μ ) ( x + μ ) 2 ( y 2 + z 2 + ( x + μ ) 2 ) 5 / 2 − 1 − μ ( y 2 + z 2 + ( x + μ ) 2 ) 3 / 2 (19.1)

V y y = 1 + 3 m s y 2 ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 5 / 2 − m s ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 3 / 2 + 3 y 2 μ ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 5 / 2 − μ ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 3 / 2 + 3 y 2 ( 1 − μ ) ( y 2 + z 2 + ( x + μ ) 2 ) 5 / 2 − 1 − μ ( y 2 + z 2 + ( x + μ ) 2 ) 3 / 2 (19.2)

V z z = 1 + 3 m s z 2 ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 5 / 2 − m s ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 3 / 2 + 3 z 2 μ ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 5 / 2 − μ ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 3 / 2 + 3 z 2 ( 1 − μ ) ( y 2 + z 2 + ( x + μ ) 2 ) 5 / 2 − 1 − μ ( y 2 + z 2 + ( x + μ ) 2 ) 3 / 2 (19.3)

V x y = V y x = 3 m s ( − R s + x ) y ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 5 / 2 + 3 y μ ( − 1 + x + μ ) ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 5 / 2 + 3 y ( 1 − μ ) ( x + μ ) ( y 2 + z 2 + ( x + μ ) 2 ) 5 / 2 (19.4)

V z y = V y z = 3 m s y z ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 5 / 2 + 3 y z μ ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 5 / 2 + 3 y z ( 1 − μ ) ( y 2 + z 2 + ( x + μ ) 2 ) 5 / 2 (19.5)

V z x = V x z = λ 3 m s ( − R s + x ) z ( 1 − β ) ( ( − R s + x ) 2 + y 2 + z 2 ) 5 / 2 + 3 z μ ( − 1 + x + μ ) ( y 2 + z 2 + ( − 1 + x + μ ) 2 ) 5 / 2 + 3 z ( 1 − μ ) ( x + μ ) ( y 2 + z 2 + ( x + μ ) 2 ) 5 / 2 (19.6)

Then characteristic equation can be rewritten as

λ 4 + ( 4 − V x x − V y y ) λ 2 + V x x V y y = 0 (20)

The solutions of the linear Equation (18) can be written as

x = A i ∑ i = 1 4 e λ i t (21.1)

y = B i ∑ i = 1 4 e λ i t (21.2)

where, A i and B i represent constant coefficient. Equation (20) has four roots λ i , i = 1 , 2 , 3 , 4 the real roots of λ i _{ }give unstable motion, while the imaginary roots represent the stable motion.

e 2 = d 2 − 1 d 2 (22.1)

T = 2 π | s | (22.2)

where

d = λ i 2 − V x x 2 λ i − V x y , ands is the coefficient of the imaginary Eigen value.

The Sun-Earth-Moon-spacecraft system is used to illustrate this work. The Earth’s mass m 1 = 5.98 × 10 24 kg , the mass of Moon m 2 = 7.35 × 10 22 kg , the mass of Sun m s = 1.99 × 10 30 kg , Reena Kumari (2013). The canonical units of masses and distances are used in which the mass of the Earth

= μ E = 1 − μ = m 1 m 1 + m 2 = 0.9878715 ; mass of the Moon

= μ M = μ = m 2 m 1 + m 2 = 0.0121506683 ; mass of the Sun

= μ s = m s m 1 + m 2 = 328900.48 , and the distance between the Sun and the center of

the system = R_{s} = 389.1723985.

Now, the solution of Equation (17) numerically will gives the locations of the collinear libration points, the corresponding Jacobi constant is obtained from Equation (13),

β | L1 | L2 | L3 | L4 | L5 | |||||
---|---|---|---|---|---|---|---|---|---|---|

X | C | X | C | X | C | X | C | X | C | |

0 | −1.86027 | 7.87523 | −0.89726 | 4.09822 | 0.59344 | 2.49354 | 0.90838 | 4.40506 | 1.05927 | 8.82768 |

0.01 | −1.87369 | 7.87823 | −0.90095 | 4.05882 | 0.59680 | 2.46279 | 0.90755 | 4.34356 | 1.05984 | 8.77038 |

0.03 | −1.80859 | 7.49129 | −0.92646 | 4.05485 | 0.60150 | 2.38657 | 0.90653 | 4.22586 | 1.06049 | 8.65799 |

0.05 | −1.73955 | 7.09377 | 0.95601 | 4.06534 | 0.60631 | 2.31073 | 0.90547 | 4.10788 | 1.06116 | 8.54558 |

0.07 | −1.66483 | 6.67933 | −0.99133 | 4.09659 | 0.61124 | 2.23536 | 0.90436 | 3.98963 | 1.06184 | 8.43319 |

0.09 | −1.58098 | 6.23529 | −1.0359 | 4.16138 | 0.61629 | 2.16043 | 0.90320 | 3.87109 | 1.06254 | 8.32077 |

0.1 | −1.533 | 5.99366 | −1.06394 | 4.21476 | 0.61886 | 2.12312 | 0.90260 | 3.81169 | 1.0629 | 8.26455 |

areas are allowed at which the fourth body is moving and never moves from one allowed region to another.

Also, the effects of SRP separate the asymptotic circles.

they are more closed around the libration point (the center). Here, the regular islands are expanded gradually because of radiation pressure. Again, the island centered about x = 1.06 shows that the trajectory is regular, which mean that the region in the neighborhood of x = 1.06 is stable, and this region shrinks towards center. This is more clear at

The family of periodic orbits about L5 which related to different values of solar radiation pressure coefficient β is shown in

the calculations of d = λ i 2 − V x x 2 λ i − V x y , which specify the eccentricity of the orbit, it is

clear from Equation (19) that the values of β has a great effects on the eccentricity of the orbit.

In this work, the study is concentrated on the motion about the collinear libration points. The restricted four-body problem is studied by assuming the effect of radiation pressure. The boundaries of allowed regions for the motions of the infinitesimal mass are determined using zero velocity surfaces at different values of the radiation pressure coefficient. It is found that allowed possible regions of the motions decrease with the increase in the value of Jacobi constant C. With the help of PSS, it is observed that the stability region gets expanded in presence of radiation pressure and at the point x = 1.0629 orbits are stable. The effect of solar radiation pressure controls the positions of the libration points and the stability of motion about these libration points. This work enables the maneuvers to be done in the spacecraft missions.

The authors declare no conflicts of interest regarding the publication of this paper.

Ismail, M.N., Ibrahim, A.H., Zaghrout, A.S., Younis, S.H., Elmalky, F.S. and Elmasry, L.E. (2019) Solar Radiation Pressure Effects on Stability of Periodic Orbits in Restricted Four-Body Problem. World Journal of Mechanics, 9, 191-204. https://doi.org/10.4236/wjm.2019.98013

F g : Gravitational force,

F r a d : Radiation force,

m: Mass of spacecraft,

m_{s}: Mass of the Sun,

m 1 : Mass of the Earth,

m 2 : Mass of the Moon,

r s : The dimensionless distance between Sun and the spacecraft,

r 1 : The dimensionless distance between Earth and the spacecraft,

r 2 : The dimensionless distance between Moon and the spacecraft,

L ⊙ : Luminosity of Sun,

A: The cross-section area of the spacecraft surface,

c: The speed of light,

β : The solar radiation pressure parameter,

V R P : The potential of solar radiation pressure force,

μ : The dimensionless mass of Moon,

1 − μ : The dimensionless mass of Earth,

μ s : The dimensionless mass of Sun,

ω : The angular velocity of the rotating system,

n: The mean motion,

p x , p y , p z : The components of momentum in Cartesian coordinates,

ω s : The angular velocity of the center of mass of the two primaries about the Sun.