Design of Gravity Assist trajectory from Earth to Jupiter

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Abstract

The goal of this paper is to find a combination of conical trajectories, using gravitational assisted maneuvers (swing-by), which perform the transfer from a nearby of the departure planet (Earth) to the vicinity of the arrival planet (Jupiter), making a closest approaches with Mars (flyby) to reduce the fuel consumption for the journey. A detailed description of the mission from Earth— Mars—Jupiter, that used this technique is presented. The table of flyby dates, altitudes of closest approaches is also included. A methodology known as the Patched Conics was used, where the trajectory is divided into three parts:

  1. Departure phase, inside of the sphere of influence of the departure planet,

  2. Heliocentric phase, during the journey between the planets,

  3. Arrival phase, inside the sphere of influence of the arrival planet.

Abstract

The goal of this paper is to find a combination of conical trajectories, using gravitational assisted maneuvers (swing-by), which perform the transfer from a nearby of the departure planet (Earth) to the vicinity of the arrival planet (Jupiter), making a closest approaches with Mars (flyby) to reduce the fuel consumption for the journey. A detailed description of the mission from Earth— Mars—Jupiter, that used this technique is presented. The table of flyby dates, altitudes of closest approaches is also included. A methodology known as the Patched Conics was used, where the trajectory is divided into three parts:

  1. Departure phase, inside of the sphere of influence of the departure planet,

  2. Heliocentric phase, during the journey between the planets,

  3. Arrival phase, inside the sphere of influence of the arrival planet.

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.

Fig. 1

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Fig. 1

Description of the mission from Earth to fly by with Mars

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00011

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(1)to exitingV(2) . When converting the planetocentric entering and departing V to a heliocentric spacecraft velocity, the change in heliocentric orbital velocity and energy is apparent, particularly when visualized in Figs. (1) and (3).

V1(v) is the spacecraft heliocentric velocity pre-encounter,

V2(v) is the spacecraft heliocentric velocity post-encounter,

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 r1, r2 and the time of flight between them Δ t and solving for the incoming and outgoing velocity vectors of the transfer trajectory V⃗1, V⃗2. 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 ΔV 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.

Table 1

Calculations of Lambert problems

lambert earth - MarsLambert Mars -Jupiter
r0x (km)-1.280952970127814 × 108r2x = r1x (km)1.588109522284044 × 108
r0y (km)-7.873040871488884 × 107r2y = r1y (km)-1.331196049556935 × 108
r0z (km)4241.23706297353500r2z = r1z (km)-6690470.129802731
r1x (km)1.588109522284044 × 108r3x (km)5.331461279416993 × 108
r1y (km)-1.3311960495569350 × 108r3y (km)5.392020271071861 × 108
rz (km)-6690470.12980273100r3z (km)-9671957.471152349
Δ t1(s)1.054080811623402 × 107Δ t2(s)8.25504364820473 × 107
μs (km3/s2)1.32712428000 × 1011μs (km3/s2)1.32712428 × 1011
V0x (km/s)18.61716546638244600V2x (km/s)30.823073404118684
V0y (km/s)-28.29950136444239000V2y (km/s)4.934176592416298
V0z (km/s)-1.152116764359739900V2z (km/s)-0.8994633203622276
V1x (km/s)21.7254264303175400V3x (km/s)-5.888427200446674
V1y (km/s)13.84469939965363100V3y (km/s)3.2105733597573787
V1z (km/s)0.013528429195021072V3z (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:

f=1χ2r1C(z)g=Δtχ3μC(z)g˙=1χ3r2C(z)f˙=μr1r2χ(zS(z)1)

Where r1 and r2 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 = (ΔE)2)), C(z) and S(z) are define as:

C(z)=zsinzz3,S(z)=1coszz

The f and g expressions in terms of the orbital elements

g=r1r2μPsinΔνf=1r2P(1cosΔν)g˙=1r1P(1cosΔν)f˙=μPtanΔν2(1cosΔνP1r11r2)

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

f=1r2P(1cosΔν)=1χ2r1C(z)

g=r1r2μPsinΔν=Δtχ3μC(z)

g˙=1r1P(1cosΔν)=1χ3r2C(z)

f˙=μPtanΔν2(1cosΔνP1r11r2)=μr1r2χ(zS(z)1)

We get from Eq. (3)

χ=r1r2(1cosΔν)PC(z)

Substituting with χ in Eq. (6) and cancelling h=μp from both sides, we obtain after simplification;

r1r2(1cosΔν)P=r1+r2r1r2sinΔν1cosΔν(1zS(z))C(z)

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

A=r1r2sinΔν1cosΔν,y=r1r2(1cosΔν)P

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

χ=yC(z)

y=r1+r2A(1zS(z))C(z)

If we now solve for Δt from Eq. (4), we get

μΔt=χ3S(z)+Ay

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

f=1yr1,g=Ayμ,g˙=1yr2

Then the solution of Lambert problems yields the following relations:

V1=1g(r2fr1),V2=1g(g˙r2r1)

5 Mission Analysis

5.1 Initial impulse (Escape from the Earth at T0)

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).

Fig. 2

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Fig. 2

Escape from Earth

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00011

The heliocentric velocity of S/C V0 at departure from the Earth is the sum of the Earth velocity VE and the hyperbolic excess velocity V⃗∞1.

V0=VE+V1

The latter is assumed to be equal to the spacecraft velocity relative to the Earth. In general it is VEV1 , due to the modest capabilities of present space propulsion, so that the maximum angle between VEandV0 is quite small. In particular the heliocentric leg will lie in a plane that can assume only a modest inclination away from the ecliptic plane.

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

ΔV1=VpVc

Vp is the velocity of the spacecraft at perigee of hyperbolic orbit,

Vc is the velocity of the spacecraft in the parking orbit.

Clearly the direction of VpandVc are the same, then ΔV1 is also in the same direction.

We can obtain V0 from solving Lambert’s problem earth - Mars, see Table (1), we calculate VE at T0 to find V⃗∞1 and it’s magnitude (V∞1) which by it and a given perigee (rp1 = RE + 300) can calculate the hyperbolic trajectory elements. the angular momentum and eccentricity of the hyperbolic orbit can be obtained from the following relations [5]:

h1=rp1V12+2μrp1,e1=1+rp1V12μ1

The velocity at the perigee of the hyperbolic orbit is:

VP1=V12+2μErp1

The speed of S/c in its circular parking orbit is given by Vc=μ/rp1 Then the Δ V1 required to put the S/C onto the hyperbolic departure trajectory is:

ΔV1=Vp1Vc=V12+2μErp1μErp1

ΔV1=Vc2+V1Vc21

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]

β=cos11e1=cos111+rp1V12μE

The results are summarized in the Table (2).

Table 2

Escape from Earth at T0

V⃗E0 = {15.113446216468128, -25.49053048920682, 0.0008860019561318039}

V⃗0 = {18.617165466382446, -28.29950136444239, -1.1521167643597399}

V⃗∞1 = {3.50372, -2.80897, -1.153}

JD (Julian day No) of T0 = 2458230.5

V(parking orbit) = 11.8689(km/s)

V∞1 (km/s)μE (km3/s2)rp1 (km)e1h1β(red)Vp1 (km/s)Δ V1 (km/s)
4.636353.986004415 × 1056678.13631.3601479262.1.7448077.725764.14313

5.2 Gravity assist maneuver (fly by Mars at T1 = T0 + Δ t1)

Now after solving Lambert’s problem for earth - Mars trajectory and Mars, Jupiter trajectory we have V⃗1 and V⃗2, see Table (1). Such that:

V⃗1 is the heliocentric S/C velocity at final position for Earth-Mars Lambert’s algorithm,

V⃗2 is the heliocentric S/C velocity at initial position for Mars- Jupiter Lambert’s algorithm,

V⃗M is the velocity of Mars at time T1.

Then the heliocentric velocity of spacecraft at the SOI of Mars is V⃗1 and it need to out from the SOI by V⃗2, 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 V(1) and it need to out with velocity relative to Mars is V⃗out Fig. (3).

V(1)=V1VMVout=V0VM

We take the direction of V(2) (the out velocity of spacecraft in hyperbolic orbit fly by) in the same direction of V⃗out then the turn angle of the hyperbolic orbit flyby can be calculate from this relation [5],

cosδ=V(1).V(2)V2=V(1).VoutVVout

Fig. 3

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Fig. 3

fly by with Mars

Citation: Applied Mathematics and Nonlinear Sciences 3, 1; 10.21042/AMNS.2018.1.00011

We know that

V(1)=V(2)

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

e2=1sin(δ2),rp2=μMV22(e21),Δ=rp1+2μMrpV22

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) relative to Mars with magnitude is V∞2 and in the same direction of V⃗out, to make the velocity of spacecraft relative to Mars is V⃗out we give it Δ V2 in the same direction of V(2),

ΔV2=VoutV2

After that the spacecraft out from the SOI of Mars with heliocentric velocity is V⃗2 = V⃗M + V⃗out

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

Table 3

Fly by Mars at T1

V⃗in = V⃗1 = {21.72542643031754, 13.844699399653631, 0.013528429195021072}

V⃗out = V⃗2 = {30.823073404118684, 4.934176592416298, -0.8994633203622276}

V⃗M = {16.48592538172737, 20.64382791494634, 0.02774276866812586}

JD (Julian day No) of T1 = 2458230.5

V∞2 (km/s)μM (km3/s2)rp1 (km)e2Δ (km)Vout/M (km/s)δ (red)Δ V2 (km/s)
8.583754.305 × 10411968.621.484412539.321.28870.093124412.7049

5.3 Capture by Jupiter at (T2 = T1 + Δ t2)

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

V3=V3VJ

Such that:

V⃗3 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 T2

V⃗J is the heliocentric velocity of Jupiter at T2.

the goal’s mission is landing on Jupiter To achieve this goal we make the perigee of the hyperbola equal to the Jupiter radius (rp3 = RJ = 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),

Table 4

Capture from Jupiter at T2

V⃗3 = {-5.888427200446674, 3.2105733597573787, 0.22569580208145187}

V⃗J = {9.130213923868334, 9.803076409849977, -0.2456201552966279}

JD (Julian day No) of T2 = 0.21253784951961086

V⃗∞3 = {-15.0186, -6.5925, 0.471316}

V∞2 (km/s)μJ (km3/s2)rp1 (km)e2Δ (km)Vp3 (km/s)Δ V3 (km/s)
16.40861.26675 × 108699111.1485926584262.3951-62.3951

ΔV3=Vp3

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.

Communicated by Elbaz I. Abouelmagd

NomenclatureS =

Index for Sun

E =

Index for Jupiter

M =

Index for Mars

μ =

Gravitational parameter

R =

Planet radius

rp =

Periapsis radius

V(1) =

Input hyperbolic excess velocity

V(2) =

Output hyperbolic excess velocity

Δ t1 =

The time of flight between Earth and Mars

Δ t2 =

The time of flight between Mars and Jupiter

T0 =

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

T1 =

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

T2 =

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

r0k =

the component of position vector of Earth and the S/C at T0 relative to the sun in k - direction (k = x, y, z)

r1k =

the component of position vector of Mars and the S/C at T1 relative to the sun in k - direction (k = x, y, z)

r3k =

the component of position vector of Jupiter and the S/C at T2 relative to the sun in k - direction (k = x, y, z)

References

  • [1]

    JPL, Voyager Mission. http://voyager.jpl.nasa.gov/

  • [2]

    NASA, Cassini, Solstice Mission. http://cassini-2.jpl.nasa.gov/

  • [3]

    JPL, Galileo Mission. http://Galileo.jpl.nasa.gov/

  • [4]

    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).

  • [5]

    Curtis H. D. Orbital Mechanics for Engineering Students. (2005) http://Elsevier.com/

  • [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).

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    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).

[1]

JPL, Voyager Mission. http://voyager.jpl.nasa.gov/

[2]

NASA, Cassini, Solstice Mission. http://cassini-2.jpl.nasa.gov/

[3]

JPL, Galileo Mission. http://Galileo.jpl.nasa.gov/

[4]

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).

[5]

Curtis H. D. Orbital Mechanics for Engineering Students. (2005) http://Elsevier.com/

[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).

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