 Open Access
 Authors : Girija S.
 Paper ID : IJERTV11IS010014
 Volume & Issue : Volume 11, Issue 01 (January 2022)
 Published (First Online): 10012022
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A CostEffective Mission to Mercury Using Multiple Gravity Assist
Girija S
Research Scholar Department of Aerospace Engineering
IIT Bombay, Mumbai, India
Abstract: Interplanetary missions are required to gather information necessary to ensure the long terms survival of the human race. However, such missions are costly and it is required to reduce the cost. For this purpose, trajectories of an interplanetary mission to the planet Mercury are considered. In this paper, cost reductions of such missions by reducing the change in velocity requirements are discussed. When the change in velocity requirement is small, fuel consumption is reduced. This reduces the overall cost of the mission. The technique involves using multiple gravityassisted maneuvers
Inclination of gravity direction of planet with the local horizontal (rad)
Horizontal distance of satellite from Sun (km)
Vertical distance of satellite from Sun (km)
Horizontal distance of planet from Sun (km)
to achieve the same. The amount by which the change in velocity is reduced depends not only on the type of planet used for the flyby but also on the sequence in which the planets are
Vertical distance of planet from Sun (km)
used. In this paper, different types of multiple gravityassisted missions are designed. Their performances with respect to
Sign of inclination vector of planet
single gravityassisted missions and direct missions are compared. The different planets that were used for flyby are Mars, Earth, Venus and Mercury. Along with this, launch opportunities and mass savings of each mission are also found.
Apogee velocity of the satellite with respect to the Sun (km/s)
NOMENCLATURE
r Radial distance of the secondary body from the primary body (km)
V Velocity of the secondary body with respect to the primary body (km/s)
Phase angle of the secondary body with respect to horizontal (rad)
Flight path angle of the secondary body from the primary body (rad)
The product of universal gravitational constant and the mass of the primary body (32)
a Semimajor axis of the transfer ellipse (km)
Angular velocity of the target planet (rad/s)
Time period required for the transfer (s) e eccentricity of the transfer ellipse
Perigee distance of the transfer ellipse (km)
Initial targeted distance of the spacecraft during the initial mission design (km)
Radial distance of the planet from the primary body (km)
Distance of satellite with respect to Sun (km)
True anomaly that the source planet make in the transfer ellipse (rad)
Phase angle of satellite with respect to Sun (rad)
Velocity of satellite with respect to Sun (km/s)
True anomaly that the target planet makes in the transfer ellipse (rad)
Flight path angle of satellite with respect to Sun (rad)
The product of universal gravitational constant and the mass of the Sun (32)
The product of universal gravitational constant and the mass of the planet (32 )
Eccentric anomaly of source planet (rad)
Eccentric anomaly of target planet (rad)
Time since periapsis passage of source planet (s)
Time since periapsis passage of target planet (s)
Spacecraft velocity with respect to Sun at the arrival of the SOI of the target planet (km/s)
Spacecraft velocity with respect to Sun at departure from the SOI of the target planet (km/s)
Velocity of target planet with respect to Sun (km/s)
Velocity at infinity on the arrival asymptote (km/s)
Velocity at infinity on the departure asymptote (km/s)
Angle through which the spacecraft velocity vector is to be turned (rad)
Hyperbolic asymptote angle (rad)
solar sails or solar electric propulsion techniques in their mission.
Low thrust and multiple gravity assist trajectory have also been designed for Mercury in 2011 [710]. The aero gravity assist (AGA) technique was designed in 2006 Ref. [11]. Packages for the design and optimization of low thrust gravity assist trajectories were developed in 2002. [12, 13], mentions patching different low thrust trajectories together based on the shape of the trajectory.
Packages to develop trajectories for different planetary sequences were developed [14, 15]. This paper is based on our previous work regarding single gravityassisted missions to Mercury [16] in which only one planet is used for the flyby purpose. Based on this, reduced velocity requirement and mass savings are analyzed. Whereas, the current paper proposes the use of more than one planet for gravityassisted interplanetary travel. Same planets in different sequences are also evaluated and the corresponding velocity requirements
Angle between velocity of the planet and the velocity of the spacecraft (rad)
are tabulated. The objective of this paper is to design a low cost mission to Mercury using multiple gravityassisted maneuvers. Different types of flyby sequences are simulated
Angle between and (rad) Angle between and (rad)
G Universal gravitational constant
Mass of sun in kg

INTRODUCTION
The cost of an interplanetary mission is dependent on fuel consumption. Now, the amount of fuel consumed is a function change in velocity or the deltaV (V) requirement. The time taken for a particular mission is inversely proportional to its velocity. Hence, a lowcost mission cannot guarantee a minimum time mission. However, in a lowcost mission, when the fuel requirement is reduced, the payload capacity increases. This in turn increases the scope of the mission. Hence lowcost missions are necessary.
To move a spacecraft from one orbit to another, a change in velocity is required which is usually achieved by thrusters. In a lowcost mission, the use of thruster is reduced by employing other means to supply the change in velocity. This can be achieved by using different types of propulsion techniques, such as solar sails [1], beampowered propulsion, laser propulsion [2], solar electric propulsion [3,4], etc. However, this paper focuses on the use of multiplegravity assist maneuvers for achieving a lowcost mission. Hohmann transfer to Mercury requires a net velocity change of about 17 km/s. The high value of V requirement, as well as technological challenges, calls for a reduction in the overall cost of the mission.
Missions that used multiple gravity assists are NASAs Voyager 1 and 2, launched in 1977; MESSENGER [5] launched on 3rd August 2004; etc. The BepiColombo mission of the European Space Agency is supposed to have eight flybys around Earth, Venus and Mercury, [6]. The gravity assist techniques are often accompanied by the use of
and their performances are compared. Launch opportunities are also calculated for the mission.
This paper discusses various concepts in the following order. The mathematical model is described in section II. Multiple gravityassisted missions and their performance are discussed in sections III and IV respectively, along with their simulation results. Launch opportunities are covered in section V. The paper ends with a conclusion in section VI.

MATHEMATICAL MODEL
The spacecraft is to be transferred from the low earth orbit to Mercury by applying a decremental velocity tangential to the spacecraft's motion. Once the spacecraft reaches Mercury, another velocity is povided to the spacecraft tangentially so that it follows the orbit of Mercury. The same technique is done for a direct or a gravity assist transfer. The mathematical model is similar to the one mentioned in [17] and [18]. The following assumptions are made for the mathematical model of the system.

Sun is located at the center of the coordinate system and it is stationary.

The planet is only under the influence of the Sun.

The eccentricity of the Earth's orbit is zero. Hence the orbit is circular.

The orbits are coplanar.

Planet around Sun
The mathematical model of a planet orbiting the sun is discussed in this section. Figure 1 shows a planet (P) orbiting the Sun in a restricted circular orbit.
The coordinate system is Sun Centric Coordinate System (SCS). The motion of the planet P around the Sun is governed by the equations. (1)(4).
= 2
[( ) + ( )] cos1 { }(15)
= sin[( )
( )]
(16)


MULTIPLE GRAVITY ASSISTED MISSION
Figure 1. Illustration of planet around the Sun
= sin
(1)
Multiple gravityassisted mission involves the use of more than one planet for gravityassisted maneuver. Trajectories can be designed initially to reach either Mars or Venus since these planets lie next to Earth. The first gravity assist can take place as soon as the spacecraft reaches the first planet. If the spacecraft passes through the inner circular
= cos
(2)
position, then the velocity of the spacecraft decreases.
However, the velocity of the spacecraft increases if it passes
= sin
(3)
through the outer circular position of the planet. The
2
different planets used in this paper for multiple gravity
=
) cos
(4)
assisted designs are Mars, Earth, Venus and Mercury. The
Where, =GÃ— .
(
2
trajectory must be designed so that the flyby planet is in the right position when the spacecraft passes by it.
Value of G is taken as 6.678 Ã— 1020312

Spacecraft around Sun
In this system, three planets viz., P1, P2 and P3 influence the motion of a satellite around the Sun. If the satellite is orbiting around P1 initially, then the motion of the spacecraft is defined as shown in eqns. (6)(9).
= sin()
(6)
= cos()
(7)
= cos(
2
=1
)
sin()
2
(8)
= ( ) cos
2
+ sin( )
2
=1
(9)
Where,
2 2
rsati=[(xsatxpi) +(ysatypi) ]
(10)
=
(11)
= cos
(12)
= cos
(13))
= cos
(14)
1) Design
The design of a gravityassisted mission can be done by designing a Fixed Time (FT) transfer or a Hohmann transfer to a nearby planet, as discussed in [16]. Figure 2 shows the velocities of the flyby planet and the spacecraft. The velocity of the spacecraft as it approaches the sphere of influence of the flyby is considered as the approach velocity, while the spacecraft velocity as it departs the sphere of influence is called the departure velocity. The velocity of the spacecraft changes near the arrival and departure points according to equations (17)(21).
=
(18)
sin
sin =
(19)
= 180
(20)
=
(18)
sin
sin =
(19)
= 180
(20)
= 180 2 (17)
2 = 2 + 2 2 cos( + ) (21)
Figure 2 Gravity assist
Where, cos() = 1 .
The spacecraft is passed through the outer or inner positions
of the planet so that required velocity change is achieved to reach the next planet for a gravity assist. The process is repeated until the spacecraft reaches Mercury. Trajectory designs of four different types of multiple gravityassisted missions to Mercury are discussed in this paper. The nomenclature of the mission is such that the first and the last planet name initials stand for the starting and the target planet. The intermediate ones are the flyby planets. The different mission names discussed here are EarthVenus MarsMercury (EVMaM), EarthMarsVenusMercury (EMaVM), EarthEarthVenusMercury (EEVM), EarthEarthVenusMercury Bielliptical (EEVM Bi) and EarthMarsVenusVenusMercury (EMaVVM). The trajectory of the missions to the first flyby planet was designed either by using Hohmann transfer or by using Fixed Timed (FT) transfer, discussed in [16]. The parameters used to design fixed time transfers are enclosed in table 2.


SIMULATION RESULTS

EVMaM
In EarthVenusMarsMercury, the satellite is initially designed to reach Venus using fixed time transfer. The spacecraft, on reaching Venus does a flyby so that its velocity is minimally increased so as to reach Mars. This is done by adjusting the flyby distance. The Mars flyby decreases the spacecraft velocity, due to which it reaches Mercury. The result of this mission is shown in figure 3.
Figure 3 Multiple gravity assist EVMaM
The velocity changes in the spacecraft after each flyby are mentioned in Table 1.
Table 1 Change in velocity EVMaM
Planet
Altitude (km)
(km/s)
(km/s)
days
Venus
8
37.4436
41.6136
132
Mars
10
21.3949
20.8235
270
Total change in velocity =16.8 km/s Time taken to reach Mercury = 375 days.
The total change in velocity required to reach Mercury is less than EarthMercury Hohmann transfer velocity change.
Table 2 Design parameters for fixed time transfer
Mission
Ã— 108
(km)
AÃ— 108
(km)
E
1
(deg.)
(deg.)
EVMaM
1.08
1.28
0.17
195
3.05
EEVM
1.49
8
1.426
0.04
300
0.017
EEVM bi
elliptical
1.49
6
1.4867
0.12
160
2.752

EMaVM
Another mission is designed so that the spacecraft initially possesses EarthMars Hohmann velocity, which is the minimum velocity to reach Mars. On reaching Mars, the spacecraft does a flyby around it. As a result, the spacecraft moves towards Venus whee it performs another flyby and heads towards Mercury. The simulation result is shown in figure 4. The velocity reductions of Mars and Venus are mentioned in table 3.
Total change in velocity = 13.2147 km/s Total time taken to reach Mercury = 290 days.
Figure 4: Multiple gravity assist EMaVM
Table 3 Change in velocity EMaVM
Planet
Altitude (km)
(km/s)
(km/s)
days
Earth
10
21.476
20.152
85
Venus
8
40.989
35.125
253

EEVM
The initial trajectory of this mission is designed so as to meet the Earth again using fixed time transfer. The spacecraft intercepts Earth once again for its first flyby, after departing from Earth. It does its second flyby around Venus and heads towards Mercury. The simulation result for this mission is shown in Figure 5. The change in spacecraft velocity after each flyby is enclosed in table 4.
Total change in velocity = 7.216 km/s Total time taken to reach Mercury = 220 days.
Figure 5: Multiple gravity assist EEVM
Table 4 Change in velocity EEVM
Planet
Altitude (km)
(km/s)
(km/s)
days
Earth
10
28.745
26.466
89
Venus
8
36.751
29.535
166

EEVM Bi
A bielliptical transfer is a mission in which the spacecraft traverses only of the half parts of two different elliptical orbits. This mission is designed in such a way that the spacecraft initially does a half Hohmann transfer to an orbit with an apoapsis of 1.6875 Ã— 108km. The second elliptical orbit through which the spacecraft moves is designed based on the fixed time transfer to the next target planet. Once it reaches the apoapsis of Hohmann transfer, a second impulse is provided so that the spacecraft is transferred to the target point. Hence with the help of two transfer ellipses, the spacecraft is made to reach its destination.
Table 5 change in velocity EEVM bielliptical
Planet
Altitude (km)
(km/s)
(km/s)
days
Earth
1500
29.471
26.746
488
Venus
8
36.480
31.064
650
The first change in velocity given for Hohmann transfer is
1=0.8831 km/s
The second change in velocity given for FT transfer is
2=1.2044 km/s
The third change in velocity given on reaching Mercury is
3=5.78 km/s
Total change in velocity =7.8714 km/s
Time taken to reach Mercury = 700 days.

EMaVVM
The initial trajectory of the mission is the same as the one discussed in EMaVM. Mars and Venus provide the spacecraft with first and second gravity assist. After the second gravity assist, the spacecraft takes a path that is similar to the orbit of Venus. The concept of mean motion resonance orbit is used to calculate the instant at which spacecraft and Venus meet again at the same point. Generally, resonant orbit gives an idea about when a spacecraft and a planet will meet again at the same position because their orbits are resonant according to [19]. Suppose the orbital period of the spacecraft has a time period q and that of a planet is b. Then the spacecraft and the planet will encounter at the same place when the spacecraft has performed b revolutions while the planet performs q revolutions around the Sun.
The initial change in spacecraft velocity, 1=2.9438 km/s. The velocity change in the spacecraft due to the gravity assist of Mars and Venus are given in Table 6.
Table 6 change in velocity EMaVVM
Planet
Altitude (km)
(km/s)
(km/s)
days
Mars
10
19.440
18.233
129
Venus
8
40.053
33.85
300
Venus
8
33.856
28.09
18.8yrs
Mercury
3400
54.206
47.88
19yrs
Once the spacecraft perform Venus gravity assists, its trajectory changes. The period of the new trajectory is found to be 207 days.
=
207 23
=
225 25
Figure 6 : Multiple gravity assist EEVM bielliptical
The result of the Bielliptical EarthEarthVenusMercury transfer is shown in Figure 6. The change in velocity after each flyby is enclosed in table 5.
Figure 7: Multiple gravity assist EMaVVM
The spacecraft should meet the planet, Venus, again at the same position by the end of its 23rd revolution around the Sun. However, results show that the spacecraft completes 24 revolutions around the Sun to meet Venus again at the position where the first Venus flyby occurred. This is due to the error in the measurement of the orbital period and integration errors. The simulation result for multiple gravity assist using resonant orbit concept is shown in figure 7.


COMPARISON OF PERFORMANCE OF MULTIPLE GRAVITY ASSISTED MISSION AND
LAUNCH OPPORTUNITIES
It gives better performance than the single gravityassisted transfer for 59 days mentioned in [16].
When compared to minimum V for a single gravity assisted mission, the performance of EVMaM and EMa VM are similar. Hence there are no mass savings for these missions. Mass savings can be observed for EEVM, EE VM (bielliptical) and EMaVVM. However, EMaV VM takes 19 years to complete. The launch opportunities for the various multiple gravityassisted missions are enclosed in table 8. It is assumed that
= 0 in all the cases while calculating the launch opportunities.
Missions using multiple gravityassisted techniques are feasible only if the planets are present at the required position and time. Accordingly, the mission duration also varies. Due to this, the comparison with respect to the direct missions to the same planet, based on mission duration is not significant. The values of direct mission, as well as single gravityassisted mission to Mercury, were obtained from [16]. Using these results, a comparative study of the performance with a different sequence of planets considered for multiple gravityassisted missions is done. In this paper, performance is considered as a measure of change in velocity. Table 7 gives the performance of multiple gravity assisted missions for different planetary sequences when compared to the direct mission. A particular planetary sequence is of better performance if the total change in velocity is less. The maximum V obtained for multiple gravityassisted techniques is 16.8 km/s which is less than that required for a direct mission. Hence compared to direct missions, multiple gravityassisted missions are economical.
Table 7 Mass savings
Planetary Sequence
(km/s)
Mass savings1 (kg)
Mass savings2 (kg)
EVMaM
16.8
66
–
EMaVM
13.21
1266.66
–
EEVM
bielliptical
7.871
3041.99
928.67
EEVM
7.216
3261.33
1148.00
EMaVV M
2.943
4685.66
2572.3
Mass savings1 in Table 7 represents the savings in the propellant mass of a gravityassisted mission compared to a direct Hohmann transfer to Mercury. This evaluation is based on the total change in velocity obtained for the missions. The total change in velocity for direct missions was obtained from [16]. Mass savings2 of Table 7 gives the savings in the propellant mass of a multiple gravityassisted mission when compared to the minimum change in velocity obtained for a single gravityassisted mission to Mercury. This velocity was mentioned as 10.6 km/s in [16]. The performance of EMaVM is similar to that of EVMaM.
Table 8 Launch opportunities
Planet Sequence
.
Launch Opportunity
EVMaM
300
330
143
12052038,
22062068
EMaVM
220
260
11
15012032
EEVM
bielliptical
66
145
–
13102023
EEVM
90
270
–
30092020
EMaVVM
75
85
7
26072054

CONCLUSION
The costeffectiveness of multiple gravityassisted interplanetary missions has been studied in this paper and the results are compared with the direct mission as well as single gravityassisted mission. In general, the cost of the interplanetary mission is proportional to the change in velocity requirement of the spacecraft. This requirement is reduced in this paper with the help of multiple gravity assisted flyby techniques. From the analysis carried out, it can be concluded that the change in velocity produced by each planet depends upon the planet and the flyby altitude. It also depends upon the entry flight path angle of the spacecraft. However, the entry flight angle is dependent on the initial mission design. Hence the performance of multiple gravity assists depends on the design of the initial phase of the mission. The EEVM planetary sequence performs better when compared to the rest of the missions analyzed in this paper. Launch opportunities for multiple gravityassisted maneuvers have also been found out.
REFERENCES
[1.] Yarnoz DG, Jehn R, Croon M, Interplanetary navigation along the lowthrust trajectory of BepiColombo, Acta Astronautica, 2006 Jul 1; 59(15):28493. [2.] Patel MR., Automated design of DeltaV gravityassist trajectories for solar system exploration , Doctoral dissertation, Purdue University. [3.] Williams SN, CoverstoneCarroll V., Mars missions using solar electric propulsion, Journal of Spacecraft and Rockets. 2000 Jan;37(1):717. [4.] Landis GA, Oleson S, McGuire M, Fincannon J, Bury K., Solar electric propulsion for advanced planetary missions, Photovoltaic Specialists Conference (PVSC), 2011 37th IEEE 2011 Jun 19 (pp. 001589001594), IEEE. [5.] Longuski JM, Williams SN., Automated design of gravityassist trajectories to Mars and the outer planets, Celestial Mechanics and Dynamical Astronomy, 1991 Sep 1;52(3):20720. [6.] Longuski JM, Williams SN., Automated design of multiple encounter gravityassist trajectories, Astrodynamics 19891990 (pp. 985994). [7.] Casalino L, Colasurdo G, Pastrone D. Optimal lowthrust escape trajectories using gravity assist, Journal of Guidance, Control, and Dynamics, 1999 Sep;22(5):63742. [8.] McAdams JV, Dunham DW, Farquhar RW, Taylor AH, Williams BG., Trajectory design and maneuver strategy for the MESSENGER mission to Mercury, Journal of spacecraft and rockets, 2006 Sep;43(5):105464. [9.] Petropoulos AE, Longuski JM, Bonfiglio EP., Trajectories to Jupiter via gravity assists from Venus, Earth, and Mars, Journal of Spacecraft and Rockets, 2000 Nov;37(6):77683. [10.] Papkov OV., Multiple gravity assist interplanetary trajectories, Routledge; 2017 Nov 1. [11.] Shang H, Cui P, Luan E, Qiao D., Study of design and optimization of lowthrust transfer trajectory with planetary aero gravity assist, Computational Engineering in Systems Applications, IMACS Multiconference on 2006 Oct 4 (Vol. 1, pp. 690695). IEEE. [12.] Casalino L, Colasurdo G, Rosa Sentinella M., Lowthrust trajectories to Mercury with multiple gravity assists, 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 2007 (p. 5233). [13.] Hollenbeck GR., New flight techniques for outer planet missions, AIAA Conference on the Exploration of the Outer Planets 1975 Jul. [14.] Ma X, Fang J, Ning X., An overview of the autonomous navigation for a gravityassist interplanetary spacecraft, Progress in Aerospace Sciences, 2013 Nov 1; 63:5666. [15.] Bonfiglio EP. Automated design of gravityassist and aerogravity assist trajectories, M. S. Thesis, School of Aeronautics and Astronautics, Purdue Univ., West Lafayette, IN, Aug. 1999. [16.] Girija S, R Gifty Ernestina Benjamin and Mayadevi N., Interplanetary mission to mercury using gravity assist maneuver, 15nth National Conference on Technical Trends. No.221,India,2014
[17.] Miele A, Wang T., Optimal trajectories and mirror properties for roundtrip Mars missions, Acta Astronautica, 1999 Dec 1;45(11):65568. [18.] Miele A, Wang T., Nearoptimal guidance scheme for a Mars trajectory, Acta Astronautica, 2002 Jul 1;51(19):35178. [19.] Malhotra R., Orbital resonances and chaos in the solar system, Solar System Formation and Evolution 1998 (Vol. 149, p. 37).