## 1 Introduction

In deep space missions, a gravity assist trajectory is often used, which uses the gravity of a planet (or other celestial body) to alter the path and speed of a spacecraft. This technique allows to reach destinations which would not be accessible with current technology or to reach targets with significantly reduced propulsion requirements. Many spacecrafts such as Voyager, Galileo, and Cassini use the gravity assist technique to achieve their targets. The two Voyager spacecrafts provide a classic example. Voyager 2 launched in August 1977 took one G. A. from Jupiter, one from Saturn, later from Uranus, and then move up to Neptune and beyond. Galileo passed by Venus then twice by Earth, and finally go up to its path Jupiter. Cassini passed by Venus twice, then Earth, and finally Jupiter on the way to Saturn [1,2,3].

In a gravity assist trajectory, angular momentum is transferred from the orbiting planet to a spacecraft, while the value of it’s speed relative to planet is not changed during a gravity assist flyby, but it’s direction is changed. However, both value and direction of spacecraft’s speed relative to the sun are changed during a gravity assist flyby, due to the planet relative orbital velocity is added to the spacecraft’s velocity on its way out.

The application of a “multi-conic method” with differential correction was explored by Wilson and Howell [4] with applications to the Sun-Earth-Moon environment. Their work is based on the original multi-conic method, which approximates trajectory legs by considering separate perturbing influences. This method is somewhat of a compromise between patched conics and fully integrated trajectories. In another work, Marchand, Howell, and Wilson [6] utilized a multi-step correction process for obtaining trajectories in an n-body ephemeris model. This procedure begins with a “seed” trajectory, divides the trajectory into nodes, and performs differential correction on the states at the nodes to satisfy specified constraints in the n-body model. The design of a transfer trajectory combining Solar Electric Propulsion (SEP) and gravity assist (GA) can be regarded as a general trajectory optimization problem [7]. The dynamics of the spacecraft is governed mainly by the gravity attraction of the Sun, when the spacecraft is outside the sphere of influence of a planet, and by the gravity attraction of the planet during a gravity assist maneuver. Low-thrust propulsion is then used to shape trajectory arcs between two subsequent encounters and to meet the best incoming conditions for a swing-by.

An interesting approach is to choice to direct collocation as demonstrated by Betts [8], who efficiently optimized a transfer trajectory to Mars combining low-thrust with two swing-bys of Venus. In this paper an original direct optimization approach has been used to design an optimal interplanetary trajectory. The proposed approach is characterized by a transcription of both states and controls by Finite Elements in Time (DFET) [9]. A set of additional parameters, not included among states and controls, are allowed and can be used for a combined optimization of both the trajectory and other quantities peculiar to the original optimal control problem (parametric optimization). In particular, in this paper, the orbital elements of each hyperbola are treated as additional parameters and opposite to the work of Betts, swing-by trajectories are not transcribed with collocation but using multiple shooting.

In this work we study the interplanetary trajectory of a spacecraft leaving Earth and making fly by with Mars in it’s destination to Jupiter. We introduced a simple and accessible algorithms for interplanetary trajectory planning that do not require gross simplifications and are able to find the required solution. The algorithms are implemented in Mathematica program, which allows for their straightforward use in an academic setting.

## 2 Description of the Mission

The complete trajectory has been divided into five different segments. Three of them are planetary segments around Earth, Mars and Jupiter, respectively, the other two are heliocentric elliptic orbits Fig. (1). The classical analysis of scales for interplanetary missions is adopted [5]. That is, since planetary radii are significantly smaller than planetary Spheres of influence (SoI), the limit of the (SoI) is considered (from the point of view of the planetary segments) to be located at infinity. On the other hand, from the perspective of the heliocentric trajectories Earth - Mars and Mars - Jupiter, the (SoI) are reduced to a point. Finally, using the method of patched conics, the five segments are joined to compose the complete trajectory.

## 3 Gravity Assists

A gravity assist maneuver is applied in an interplanetary trajectory to use of planet’s gravitational field and momentum in order to increase or decrease the spacecraft’s heliocentric orbital energy. In the planet centered reference frame of the patched conic method, the trajectory (unpowered gravity assist) does not change in orbital energy, but is simply redirected from entering
*V*_{∞} to a heliocentric spacecraft velocity, the change in heliocentric orbital velocity and energy is apparent, particularly when visualized in Figs. (1) and (3).

*V⃗* is the planet heliocentric velocity,

*δ* is the gravitational bend angle (turn angle).

The heliocentric velocity of the spacecraft *V*^{(v)} resulting from the gravity assist is increased or decreased depending on how the maneuver is performed. If the hyperbolic periapsis occurs on the trailing side of the planet with respect to the planet’s heliocentric velocity, then the spacecraft heliocentric velocity will be increased by the gravity assist. If the hyperbolic periapsis occurs on the leading side of the planet with respect to the planet’s heliocentric velocity, then the spacecraft the heliocentric velocity will be decreased by the gravity assist.

## 4 lambert’s problem

Lambert’s problem is characterized by taking two position vectors *r*_{1}, *r*_{2} and the time of flight between them Δ *t* and solving for the incoming and outgoing velocity vectors of the transfer trajectory *V⃗*_{1}, *V⃗*_{2}. In theory, Lambert’s problem will need to be solved twice. The first one for the departure from Earth and the arrival at the flyby planet (Mars), and the second for the departure from the flyby planet and the arrival at the target planet (Jupiter), see Table (1). From these departure and arrival velocities, we can then calculate the Δ*V* requirements that the spacecraft will need to be able to perform those maneuvers. There are many methods that can be used to solve Lambert’s problem. For this design we will be using the Universal Variable method.

Calculations of Lambert problems

lambert earth - Mars | Lambert Mars -Jupiter | ||
---|---|---|---|

r_{0x} (km) | -1.280952970127814 × 10^{8} | r_{2x} = r_{1x} (km) | 1.588109522284044 × 10^{8} |

r_{0y} (km) | -7.873040871488884 × 10^{7} | r_{2y} = r_{1y} (km) | -1.331196049556935 × 10^{8} |

r_{0z} (km) | 4241.23706297353500 | r_{2z} = r_{1z} (km) | -6690470.129802731 |

r_{1x} (km) | 1.588109522284044 × 10^{8} | r_{3x} (km) | 5.331461279416993 × 10^{8} |

r_{1y} (km) | -1.3311960495569350 × 10^{8} | r_{3y} (km) | 5.392020271071861 × 10^{8} |

r (km)_{z} | -6690470.12980273100 | r_{3z} (km) | -9671957.471152349 |

Δ t_{1}(s) | 1.054080811623402 × 10^{7} | Δ t_{2}(s) | 8.25504364820473 × 10^{7} |

μ (_{s}km^{3}/s^{2}) | 1.32712428000 × 10^{11} | μ (_{s}km^{3}/s^{2}) | 1.32712428 × 10^{11} |

V_{0x} (km/s) | 18.61716546638244600 | V_{2x} (km/s) | 30.823073404118684 |

V_{0y} (km/s) | -28.29950136444239000 | V_{2y} (km/s) | 4.934176592416298 |

V_{0z} (km/s) | -1.152116764359739900 | V_{2z} (km/s) | -0.8994633203622276 |

V_{1x} (km/s) | 21.7254264303175400 | V_{3x} (km/s) | -5.888427200446674 |

V_{1y} (km/s) | 13.84469939965363100 | V_{3y} (km/s) | 3.2105733597573787 |

V_{1z} (km/s) | 0.013528429195021072 | V_{3z} (km/s) | 0.22569580208145187 |

### Universal Variable Method

The algorithm that is used is taken from Fundamentals of Astrodynamics and Applications [10]. This algorithm utilizes the bisection method which provide a strong solution for a wide variety of transfer orbits. The formulation of this method begins with the *f* and *g* universal variable, defined by the following formulas:

Where *r*_{1} and *r*_{2} are the magnitudes of the initial and final position vectors,

*χ* is a universal variable, *z* is square of the difference in eccentric anomalies, *E*, at two position (*z* = (Δ*E*)^{2})), *C*(*z*) and *S*(*z*) are define as:

The *f* and *g* expressions in terms of the orbital elements

Equating the corresponding equations in the two groups Eqs. (1) and Eqs. (2), we obtain

We get from Eq. (3)

Substituting with *χ* in Eq. (6) and cancelling

We can write this equation more compactly by defining two auxiliary symbols, A and y as:

Using these definitions of A and y, Eqs. (7) and (8) may be written more compactly as

If we now solve for Δ*t* from Eq. (4), we get

Using the auxiliary symbols A and y to write Eqs. (3 - 6) in the following simplified expressions:

Then the solution of Lambert problems yields the following relations:

## 5 Mission Analysis

### 5.1 Initial impulse (Escape from the Earth at *T*_{0})

In order to escape the gravitational pull of a planet, the spacecraft must travel a hyperbolic trajectory relative to the planet, arriving at it’s sphere of influence with a relative velocity *V*_{∞1} (hyperbolic excess velocity) greater than zero Fig. (2).

The heliocentric velocity of S/C
*V⃗*_{∞1}.

The latter is assumed to be equal to the spacecraft velocity relative to the Earth. In general it is

The impulse required to be given at the perigee of the hyperbolic orbit to transfer the spacecraft from the parking orbit to the escape hyperbolic orbit is given by

Clearly the direction of

We can obtain
*T*_{0} to find *V⃗*_{∞1} and it’s magnitude (*V*_{∞1}) which by it and a given perigee (*r*_{p1} = *R _{E}* + 300) can calculate the hyperbolic trajectory elements. the angular momentum and eccentricity of the hyperbolic orbit can be obtained from the following relations [5]:

The velocity at the perigee of the hyperbolic orbit is:

The speed of S/c in its circular parking orbit is given by
*V*_{1} required to put the S/C onto the hyperbolic departure trajectory is:

The orientation of the apse line of the hyperbola to the asymptotes of the hyperbolic trajectory measured by the angle, which can be obtained from the relation [5]

The results are summarized in the Table (2).

Escape from Earth at *T*_{0}

*V⃗*_{E0} = {15.113446216468128, -25.49053048920682, 0.0008860019561318039}

*V⃗*_{0} = {18.617165466382446, -28.29950136444239, -1.1521167643597399}

*V⃗*_{∞1} = {3.50372, -2.80897, -1.153}

JD (Julian day No) of *T*_{0} = 2458230.5

*V*_{(parking orbit)} = 11.8689(km/s)

V_{∞1} (km/s) | μ (_{E}km^{3}/s^{2}) | r_{p1} (km) | e_{1} | h_{1} | β(red) | V_{p1} (km/s) | Δ V_{1} (km/s) |
---|---|---|---|---|---|---|---|

4.63635 | 3.986004415 × 10^{5} | 6678.1363 | 1.36014 | 79262.1 | .744807 | 7.72576 | 4.14313 |

### 5.2 Gravity assist maneuver (fly by Mars at *T*_{1} = *T*_{0} + Δ *t*_{1})

Now after solving Lambert’s problem for earth - Mars trajectory and Mars, Jupiter trajectory we have *V⃗*_{1} and *V⃗*_{2}, see Table (1). Such that:

*V⃗*_{1} is the heliocentric S/C velocity at final position for Earth-Mars Lambert’s algorithm,

*V⃗*_{2} is the heliocentric S/C velocity at initial position for Mars- Jupiter Lambert’s algorithm,

*V⃗ _{M}* is the velocity of Mars at time

*T*

_{1}.

Then the heliocentric velocity of spacecraft at the SOI of Mars is *V⃗*_{1} and it need to out from the SOI by *V⃗*_{2}, to complete the trajectory and finally reach to the SOI of Jupiter, on the other hand the spacecraft enter the SOI velocity relative to mars is
*V⃗ _{out}* Fig. (3).

We take the direction of
*V⃗ _{out}* then the turn angle of the hyperbolic orbit flyby can be calculate from this relation [5],

We know that

Now we can calculate the hyperbolic orbital elements using the relation [5],

Then we can calculate the perigee of the hyperbolic fly by Mars, now the spacecraft out the SOI of Mars with the velocity
*V*_{∞2} and in the same direction of *V⃗ _{out}*, to make the velocity of spacecraft relative to Mars is

*V⃗*we give it Δ

_{out}*V*

_{2}in the same direction of

After that the spacecraft out from the SOI of Mars with heliocentric velocity is *V⃗*_{2} = *V⃗ _{M}* +

*V⃗*

_{out}which by it can complete it’s trajectory to reach the SOI of Jupiter. The results are summarized in Table (3).

Fly by Mars at *T*_{1}

*V⃗ _{in}* =

*V⃗*

_{1}= {21.72542643031754, 13.844699399653631, 0.013528429195021072}

*V⃗ _{out}* =

*V⃗*

_{2}= {30.823073404118684, 4.934176592416298, -0.8994633203622276}

*V⃗ _{M}* = {16.48592538172737, 20.64382791494634, 0.02774276866812586}

JD (Julian day No) of *T*_{1} = 2458230.5

V_{∞2} (km/s) | μ (_{M}km^{3}/s^{2}) | r_{p1} (km) | e_{2} | Δ (km) | V (_{out/M}km/s) | δ (red) | Δ V_{2} (km/s) |
---|---|---|---|---|---|---|---|

8.58375 | 4.305 × 10^{4} | 11968.6 | 21.4844 | 12539.3 | 21.2887 | 0.0931244 | 12.7049 |

### 5.3 Capture by Jupiter at (*T*_{2} = *T*_{1} + Δ *t*_{2})

A spacecraft arrives at the sphere of influence of the Jupiter with a hyperbolic excess velocity *V⃗*_{∞3} relative to Jupiter, where

Such that:

*V⃗*_{3} is the heliocentric S/C velocity at final position for Mars- Jupiter lambert algorithm. see Table (1), or it is the heliocentric S/C velocity at SOI of Jupiter at *T*_{2}

*V⃗ _{J}* is the heliocentric velocity of Jupiter at

*T*

_{2}.

the goal’s mission is landing on Jupiter To achieve this goal we make the perigee of the hyperbola equal to the Jupiter radius (*r*_{p3} = *R _{J}* = 69911 km) and the velocity of S/C relative Jupiter equal zero at perigee. we give S/C the third impulse in the perigee of the hyperbola equal to the it’s velocity and in the opposite direction. The results are summarized in Table (4),

Capture from Jupiter at *T*_{2}

*V⃗*_{3} = {-5.888427200446674, 3.2105733597573787, 0.22569580208145187}

*V⃗ _{J}* = {9.130213923868334, 9.803076409849977, -0.2456201552966279}

JD (Julian day No) of *T*_{2} = 0.21253784951961086

*V⃗*_{∞3} = {-15.0186, -6.5925, 0.471316}

V_{∞2} (km/s) | μ (_{J}km^{3}/s^{2}) | r_{p1} (km) | e_{2} | Δ (km) | V_{p3} (km/s) | Δ V_{3} (km/s) |
---|---|---|---|---|---|---|

16.4086 | 1.26675 × 10^{8} | 69911 | 1.14859 | 265842 | 62.3951 | -62.3951 |

## 6 Conculsion

The problem of preliminary interplanetary design to outer planets has been studied using Gravity-assisted maneuvers techniques which have been introduced as a resource to get the required energy to reach far planets. Deep Space Maneuvers and impulses at the flyby periapsis have also been described as means to increase the degrees of freedom in the global trajectory design process. The method is applied to transfer trajectory from the Earth to planet Jupiter making flyby with Mars to gain an extra energy to reach to the target planet ( Jupiter). Lambert problem were used to find a solution for the position vectors from initial orbits in each transfer.

*S*=

Index for Sun

*E*=

Index for Jupiter

*M*=

Index for Mars

*μ*=

Gravitational parameter

*R*=

Planet radius

*r*=

_{p}Periapsis radius

Input hyperbolic excess velocity

Output hyperbolic excess velocity

*t*

_{1}=

The time of flight between Earth and Mars

*t*

_{2}=

The time of flight between Mars and Jupiter

*T*

_{0}=

The spacecraft departure date from the Earth (22/4/2018 0^{h} 0^{m} 0^{s})

*T*

_{1}=

The spacecraft departure and arrival date to Mars (22/8/2018 0^{h} 0^{m} 8^{s})

*T*

_{2}=

The spacecraft departure and arrival date to Jupiter (3/4/2021 10^{h} 40^{m} 44^{s})

*r*

_{0k}=

the component of position vector of Earth and the S/C at *T*_{0} relative to the sun in k - direction (k = x, y, z)

*r*

_{1k}=

the component of position vector of Mars and the S/C at *T*_{1} relative to the sun in k - direction (k = x, y, z)

*r*

_{3k}=

the component of position vector of Jupiter and the S/C at *T*_{2} relative to the sun in k - direction (k = x, y, z)

## References

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Wilson R. S. and Howell K. C. Trajectory Design in the Sun Earth Moon System Using Lunar Gravity Assists. Journal of Spacecraft and Rockets. (35): 191–198 (1998).

- [6]↑
Marchand B. G. Howell K. C. and Wilson R. S. Improved Corrections Process for Constrained Trajectory Design in the n-Body Problem. Journal of Spacecraft and Rockets. (44): 884 – 897 (2007).

- [7]↑
Vasile M. Bernelli-Zazzera F. Jehn R. Janin G. Optimal Interplanetary Trajectories Using a Combination of Low- Thrust and Gravity Assist Manoeuvres. Rio de Janeiro Brazil IAF-00-A.5.07 51st IAF Congress (2000).

- [8]↑
Betts J. T. Optimal Interplanetary Orbit Transfers by Direct Transcription. The Journal of the Astronautical Sciences. (40): 247–268 (1994).

- [9]↑
Vasile M. Direct Transcription by FET for Optimal Space Trajectory Design. Internal Report DIASR 99–02 Politecnico di Milano Dipartimento di Ingegneria Aerospaziale (1999).

- [10]↑
Bate R. R. Mueller D. D. and White J. E. Fundamentals of Astrodynamics. Dover Publications Mineola (1971).