46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit
25 - 28 July 2010, Nashville, TN
AIAA 2010-6577
L1 to Mars: A Combined Robotic/Piloted Mars Mission
Brice N. Cassenti*
University of Connecticut, Storrs, CT, 06269
Piloted Mars mission have traditionally considered landing on Mars followed by a long
stay time before the Earth-Mars planetary alignment would allow a minimum energy
return. The cost in propellant to land and return to Mars is a major consideration, The
recent Augustine Commission report has considered less expensive exploratory missions to
asteroids and the moons of Mars, where landing, or even orbital insertion, at Mars is
avoided. An option that may eventually prove advantageous would be to park a spacecraft
at the Sun-Mars L1 Lagrange point. Only low thrust electric would be required once the
spacecraft left the Earth for the transit to and from Mars. Electric propulsion would be
beneficial as it could be used for station keeping duties at L1. Previously deployed robotic
spacecraft on the Martian surface could be controlled with a considerably shorter roundtrip communication time (from 5 to 20 minutes becomes less than 10 seconds) allowing for
very efficient exploration. Such missions would benefit greatly through the use of combined
bimodal nuclear thermal propulsion (NTP) and nuclear electric propulsion (NEP) by
shortening mission time and reducing propellant requirements.
Nomenclature
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
Specific energy
Thrust
Standard acceleration of gravity (0.00981 km/s2)
Specific angular momentum
Specific impulse
Mass
Initial spacecraft mass
Final spacecraft mass
Spacecraft mass ratio
Jet power
Surface radius
Heliocentric orbit radius
Time
Inverse of solar radius, 1/r
Velocity with respect to planet
Heliocentric velocity
Exhaust velocity
Hyperbolic excess velocity
β
∆v
µ
θ
ζ
=
=
=
=
=
Angle of thrust
Change in spacecraft speed
Mass of Mars / Mass of Sun
Solar angle
L1 solar radius / Mars solar radius
Subscripts
H
P
=
=
Hohmann
Periapsis
E
F
g0
h
Isp
M
mi
mf
MR
Pjet
R
R
t
u
V
v
ve
v∞
*
Professor in Residence, Mechanical Engineering, 191 University Road, Associate Fellow AIAA
1
American Institute of Aeronautics and Astronautics
Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
I. Introduction
he human exploration of Mars has been a goal for over a century1. Our interest in Mars stems from the fact that
Mars presents an environment that is closer to the Earth’s than any other planet, and is likely to have now or
have had life present. But the interest extends beyond the similarities; the geology of Mars and its atmosphere
present variations that cannot occur on Earth.
Missions to Mars generally use Hohmann transfers to minimize the propellant mass. The missions leave from
low earth orbit and enter a low orbit about Mars and landing craft are deployed for examination of the surface by a
crew. Although this is typical, there are an infinite number of possible variations on these missions including, for
example: a direct entry without going into orbit about Mars, a fly-by of Mars (using a free return trajectory), and
rendezvous with a moon of Mars without a landing2,3. Traditional piloted Mars missions are constrained by a long
stay time before the Earth-Mars planetary alignment would allow a minimum energy Hohmann return1.
The cost in propellant to land on Mars and launch is a major consideration2,3.
The recent Augustine
Commission report4 has considered less expensive exploratory missions to asteroids and the moons of Mars, where
landing, or even orbital insertion, at Mars is avoided. An option that may prove advantageous would be to park a
spacecraft at the Sun-Mars Lagrange point L1. Only low thrust electric propulsion would be required once the
spacecraft left the Earth for the transit to and from Mars resulting in a significant savings in propellant since there
would be no landing. Previously deployed robotic spacecraft on the Martian surface could be controlled with a
considerably shorter round trip communication time (communication times of 25 minutes would be reduced to less
than 10 seconds) allowing for very efficient exploration. Such missions should benefit greatly from using combined
bimodal nuclear thermal propulsion (NTP) and nuclear electric propulsion (NEP) by shortening mission time and
reducing propellant requirements. An illustration of this is presented that shows using L1 is competitive with
nuclear thermal propulsion and significantly better than chemical propulsion.
This paper has three major components: 1) the astrodynamics for trajectory considerations, 2) a comparison of
propulsion options, and 3) a mission scenario that allows a permanent base to be constructed.
T
II. Astrodynamics
Two approaches will be considered for the trajectory calculations: impulse propulsion maneuvers, and low thrust
propulsion. Missions to the Sun-Mars L1 do not require maneuvers in deep gravitational wells. This makes low
thrust trajectories a possibility but does not preclude the use of impulse maneuvers. For impulse maneuvers a liquid
hydrogen-liquid oxygen engine with a specific impulse of 450 seconds will be used in the calculations or a nuclear
thermal rocket with a specific impulse of 950 seconds will be used. For low thrust propulsion the characteristics of
the VASIMR engine will be used5. For VASIMR the jet power6, P jet , is assumed to remain constant, where
P jet =
1 2
m& ve ,
2
(1)
m& is the mass flow rate, and ve is the exhaust velocity. Since the thrust, F , and the specific impulse, I sp , for a
rocket can be approximated from6
F = m& ve
(2)
ve = I sp g 0 ,
(3)
and
where
g 0 is the standard gravitational acceleration. Then
FI sp =
The parameter
2 Pjet
g0
.
2 Pjet / g 0 is constant and hence the product FI sp is also. This assumption is apparent in the
discussion of reference [5]. For VASIMR the jet power was conservatively taken as 7.5 MW5.
2
American Institute of Aeronautics and Astronautics
(4)
The astrodynamics parameters used are given in Table 1. In the table G is the Newton gravitational constant
6.673x10-11 m3/kg-s2.
Table 1. Celestial Mechanics Parameters7,8
Sun
3
Earth
2
GM – km /s
1.33 E+11
3.99E+05
Radius of planet - km 6.36E+03
Orbital radius* - km
1.50E+08
*Saturn and Earth about Sun, Titan about Saturn
Mars
4.27E+04
3.39E+03
2.28E+08
The one important parameter that remains is the location of the Sun-Mars L1 point. In the restricted problem of
three bodies there are five equilibrium (Lagrangian) points as shown in Fig. 1. In the figure L1 is located between
the Sun and Mars and is an unstable equilibrium point9. Each of the five equilibrium points moves with the same
angular velocity as Mars and,hence, each remains at the same location relative to the Sun and Mars. The distance
from the Sun to L1, rL1 , is given by
rL1 = ζrMars
(5)
where r1 is the radius of Mars solar orbit, and
ζ ≈ 1 − (µ / 3)1/ 3 .
(6)
The quantity µ is the ratio of the mass of Mars to the mass
of the Sun. Using the mass values in Table 1, equation (6)
yields ζ as approximately 0.995251. A more accurate
calculation by Szebehely10 yields 0.995249. Note that
Szebehely lists the point as L2. The recent convention is
to list the Lagrangian point between the two main
components in the system as L1.
The two types of trajectories that will be considered
need quite different methods to determine the required
speed changes. Impulse maneuvers are frequently
considered and the methods for this case only need to be
summarized, but low thrust trajectories require more
elaborate methods, and, hence, a more detailed discussion.
A. Impulse Transfer Velocity Change
Hohmann transfers can be used to estimate the
performance requirements. The missions will start from
low earth orbit (LEO) and proceed directly to L1. The
required parameters are summarized in Table 1.
For the Hohmann transfer ellipse, the excess
hyperbolic velocity after launch from low earth orbit can
be readily calculated. Assume the Earth and Mars orbit the Sun in circular orbits with the radii in Table 1. The
specific energy of the Hohmann transfer orbit, EH, is
EH = −
GM Sun
GM Sun
1
= v2 −
rEarth + rL1 2
r
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American Institute of Aeronautics and Astronautics
(7)
where
GM Sun , rEarth and rMars are given in Table 1, v is the velocity with respect to the Sun, and r is the
distance to the Sun. From equation (7), at the orbit of the Earth the velocity becomes 32.7 km/s. The velocity of the
Earth about the Sun GM Sun / rEarth is 29.8 km/s leaving a velocity with respect to Earth of 2.9 km/s. We will take
(
)
this as the speed of the spacecraft at an infinite distance from the Earth. We can use this to estimate the speed
required in low earth orbit to achieve 2.9 km/s by considering the specific energy, E , for orbits about the Earth
GM Earth 1 2
1
E = V p2 −
= V∞
2
REarth
2
(8)
where V p is the velocity at perigee with respect to the Earth of the hyperbolic escape orbit,
respect to the Earth at an infinite distance from the Earth (2.9 km/s), and
V∞ is the speed with
GM Earth and REarth are the gravitational
constant (G) times the mass of the Earth and the surface radius of the Earth respectively and are given in Table 1.
Using equation (2) launch from LEO will require a total speed equal to 11.6 km/s. The velocity change from LEO is
just the difference between the total speed 11.6 km/s) and LEO speed GM Earth / REarth or 3.7 km/s.
(
)
The solar orbital speed of L1 is given by the angular velocity (for a circular orbit) of Mars multiplied by the
radial distance from L1 to the Sun, or
vL1 = rL1
GM Sun
GM Sun
=ζ
.
3
rMars
rMars
(9)
Using Table 1, vL1 is 24.0 km/s. From equation (7), or the conservation of angular momentum, the spacecraft
arrives at L1 with a heliocentric speed of 21.6 km/s. Hence a velocity change of 2.4 km is required for the
spacecraft to match the speed at L1. Adding the speed change to enter the Hohmann transfer ellipse yields a total of
6.1 km/s. This should be compared to the speed change required to enter a low orbit about Mars which yields 5.8
km/s. Hence it is slightly more difficult (by 0.3 km/s) to rendezvous at L1 than it is to enter a low orbit about Mars.
Assuming a chemical rocket with a specific impulse of 450 seconds the mass ratio, MR (i.e., the initial
mass, mi . on leaving earth orbit to remaining mass after the rendezvous at L1, m f ), is approximately given by
MR = mi / m f = exp(∆v / ve ) .
(10)
This yields 3.98 for the mass ratio for chemical rockets. For a nuclear rocket, with a specific impulse of 950
seconds, the mass ratio becomes 1.92. For a mass of 270 tons initially in low earth orbit, the mass arriving at L1 is
68 tons using chemical propulsion and 135 tons using nuclear thermal propulsion.
B. Low Thrust Trajectory Calculation
Optimized low thrust trajectories cannot be readily calculated, typically requiring elaborate computationally
intensive simulations. The current practice is to use complex numerical techniques including control system
concepts to develop optimized trajectories (see for example references [11, 12]). Yet for this study the objective is
only to determine if missions to the Sun-Mars L1 point will be competitive with missions that orbit Mars. The lack
of exact solutions to simplified missions makes it difficult to obtain even rough sizing estimates. A new method has
been adopted that can be further expanded to perform rough optimization studies.
Assume that the planets and spacecraft orbit in the plane of the ecliptic and that the spacecraft is under the
gravitational influence of the Sun (i.e., the spacecraft is not in the sphere of influence of either the Earth or Mars).
Then the equations of motion for the spacecraft orbiting the Sun in polar coordinates with the Sun at the origin
become
r&& − r 2θ& 2 = −
GM sun Fr
+
m
r2
and
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(11)
( )
d 2 & rFθ
r θ =
dt
m
(12)
r is the distance of the spacecraft from the Sun, θ is the angular location of the spacecraft, t is the time, m
d( )
is the mass of the spacecraft, Fr and Fθ are the radial and axial components of the thrust, and (& ) =
. The
dt
where
mass varies according to
m& = −
Fr2 + Fθ2
.
I sp g 0
(13)
Assuming that the spacecraft was boosted to escape velocity using chemical or nuclear thermal propulsion, the
initial conditions can be written as
θ = 0, r = rEarth , θ& =
GM sun
, r& = 0, m = m0 at t = 0 .
3
rEarth
(14)
m0 is the initial spacecraft mass. The thrust must be applied to have the spacecraft arrive at L1 with
where
GM Sun
r& = 0 and θ& =
at r = rL1 .
3
rMars
We must find acceptable trajectories with variable thrust components, Fr and
(15)
Fθ , using solutions to the
differential equations (11) to (13) subject to the initial conditions (14) that will satisfy conditions (15). The best
tf
& (t )dt , where t f is the time at which conditions (15) are satisfied. This is a
approach would be to minimize ∫ m
0
complex variational calculus problem that can be solved numerically13. In this study we will assume functions to
describe the trajectory that satisfy conditions (14) and (15). Equations (11) and (12) can then be used to find the two
components of the thrust. The mass flow rate can then be found, using equations (13) and (4) from the total thrust,
The solution can be simplified if the trajectory is changed from a dependence on time to a dependence on the
angle θ . Assume functional forms for the inverse of the radius, u (θ ) , and the specific angular momentum, h(θ ) ,
where
u = 1/ r
(16)
h = r 2θ& .
(17)
and
The functions
u (θ ) and h(θ ) assumed were
rEarthu (θ ) =
1  rEarth  rEarth   πθ  rEarth
 cos
 =
+ 1 −
1 +
2
rL1
rL1   θ L1 
r

and
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American Institute of Aeronautics and Astronautics
(18)
 θ
 ζ 2 r 
h(θ ) = α GM Sun rEarth exp 
ln 2 L1  .
 2θ L1  α rEarth 
(19)
Equations (18) and (19) satisfy the conditions in (14) and (15), except, of course, those on the mass and time
(i.e., except m = m0 at θ = 0 and t = 0 at θ = 0 ) which are not involved. The parameter α in equation (19)
is the initial fraction of the Earth’s specific angular momentum, and, for this study, was taken as one. For α = 1
the spacecraft was launched from low earth orbit to escape velocity by either a chemical rocket or a nuclear thermal
rocket. The parameter θ L1 is the angle (in radians) that the spacecraft traversed from launch at the Earth to arrival at
L1.
The time to reach the angle θ can be found from the angular momentum by integrating
3
 GM Sun rEarth
rEarth
dt
=

dθ
GM Sun 
h



1

 .
 (r u )2 

 Earth
(20)
Equation (20) was integrated numerically using the trapezoidal rule.
The thrust-to-mass components can now be found using equations (11) and (12) as
Fr GM Sun
= 2
m
rEarth


 d 2u
 GM Sun rEarth  
2
2
 2 + u 
α
(
r
u
)
1
−
r
 
 Earth 
Earth 

h2
 


 dθ

(21)
and
Fθ GM Sun
= 2
m
rEarth
2 

  ζ 2 rL1 
h2
3 α
 .

 ln 2
(rEarthu ) 





θ
2
GM
r
α
r

Sun Earth  
 L1 
Earth  

(22)
Note that
2
  πθ 
 d 2u
 1  rEarth  rEarth   π 
 
 − 1 cos
 .
rEarth  2 + u  = 1 +
− 1 −
rL1
rL1   θ L1 
θ L1 



 dθ
 2 



(23)
We can now find the mass by integrating
[
2
2

dm
2  ( Fr / m ) + (Fθ / m )
=m 
dθ
Pjet hu 2

] .

(24)
Again the trapezoidal rule was used to numerically integrate equation (24).
A sample calculation illustrates the probable performance of VASIMR used for an L1 mission. The parameter
θ1 (the solar angle traversed by the spacecraft form Earth escape to L1 rendezvous) was chosen to be 120 degrees,
which is slightly better than a Hohmann transfer. For the initial mass, 188 tons was used. A total of 7.5 MW was
assumed for the ion propulsion system, and the parameter alpha was chosen as one. A nuclear thermal rocket was
used to boost the initial mass from low earth orbit to escape velocity. For a specific impulse of 950 seconds the
mass ratio is 1.42 and the initial mass in low earth orbit is 267 tons. For the transfer to L1 the numerical integrations
result in a mass ratio of 1.34 and a mass of 140 tons at L1, which is slightly larger than a pure nuclear thermal
rocket. Of course, with the VASIMR engine station keeping at L1 will be considerably less difficult than with
nuclear thermal rockets.
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American Institute of Aeronautics and Astronautics
The trajectory for the transfer is shown in
Figure 2. The trajectory and the orbits of the
Earth and Mars are marked every 10 days. The
final locations are also shown. The trip takes
168 days which is less than a Hohmann
transfer to Mars of 259 days using nuclear
thermal propulsion.
Figures 3 illustrate the engine performance
as a function of the elapsed mission time. Both
the thrust and the specific impulse vary by a
factor of five over the mission. The thrust
peaks initially at about 0.25x10-3 of the initial
weight and drops to a minimum just before the
midpoint of the mission. The specific impulse
peaks at 25,000 seconds when the thrust is a
minimum.
The angle of the thrust will be defined as
Tanβ =
Fr
.
Fθ
(25)
Figure 4 illustrates the time history for the
thrust angle. Initially the thrust is nearly
directed away from the Sun. At the time of maximum specific impulse the thrust points along the direction of
positive orbital velocities. The spacecraft arrives at L1 with the thrust nearly pointed toward the Sun. These thrust
directions, of course, are not optimal.
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American Institute of Aeronautics and Astronautics
III. Propulsion System Comparison
The choice for the propulsion system is critical for any interplanetary mission.
Higher performance systems can greatly increase the mass arriving at L1. Tables 2
and 3 summarize the results for an initial mass in low earth orbit (IMLEO) of 267
tons14. For the low thrust mission discussed the total mission times are not
significantly different, but the time at L1 is about thirty percent greater providing
more time for science at essentially the same cost. We also must consider the fact
that the Hohmann transfer is optimal for impulse maneuvers, but the low thrust trajectory
considered was not optimized. Hence the low thrust trajectory shows more promise and
further investigations will probably show a significant improvement over the current analysis.
Table 3 compares the mass arriving at L1 for chemical, nuclear thermal and VASIMR.
The same general conclusion applies as for the mission times. Nuclear thermal propulsion
and VASIMR are comparable, but chemical propulsion is not competitive, and, as in the case
of the mission times, an optimized low thrust trajectory would improve the low thrust
trajectory mass arriving at L1.
IV. Mission Scenario
Using reference [14] the first mission can consist of three launches from LEO. The first would be the launch of
the robotic landing craft with return supplies and a return vehicle before the launch of the crew. The return vehicle
would be placed on a trajectory for rendezvous at L1. The robotic landing craft would be diverted and directly enter
the atmosphere of Mars. The robotic landers and the return vehicle would be monitored to be sure they were
operating correctly before the crew was launched. The launch of the crew in two separate vehicles would then
proceed. Each of the vehicles would be able to support the entire crew for the entire journey. At L1 the crew can
use all or just one return vehicle to conduct the robotic exploration. The advantage in conducting robotic
investigations is the significant reduction in the round-trip communications time. A round-trip signal from L1 to
Mars takes about 7 seconds. This should be compared to the time it takes a round-trip signal from the Earth to
Mars. Depending on the relative locations, the signal will take between 9 minutes and 42 minutes (typically the
round-trip time can be taken as 25 minutes). This would obviously significantly reduce the difficulties now
encountered in the robotic exploration of Mars. The use of three spacecraft provides the crew with sufficient
redundancy to assure the safe return of the crew.
Two of the vehicles would be used for the return which will again significantly reduce the risk to the crew.
Either vehicle alone would be capable of returning the crew the entire distance. The crew would transfer to an earth
entry vehicle and enter the atmosphere directly while the two return vehicles would spiral into low earth orbit. One
of these could be outfitted as a future return vehicle carrying robotic landers. The other would be used for
transferring a crew to Mars. A new crew transport vehicle would be constructed for use so that two vehicles return
to Mars. Once one vehicle is located permanently located at Mars it would be a much less complex task to complete
sample return missions from Mars. Samples would not have to be returned to Earth for study, but could be returned
to L1 and examined by later expeditions. When sufficient vehicles were present at L1, it could serve as a
continuously crewed station. Missions could then be outfitted for piloted landings.
V.
Conclusion
The Sun-Mars L1 Lagrange point can provide low cost, low risk, human exploration missions to Mars. Missions
to L1 can effectively use low thrust mission technology and when combined with bimodal nuclear thermal
propulsion becomes the method of choice to gradually build a permanent human presence at Mars.
The mission scenario presented not only gradually builds a permanent presence at the Sun-Mars L1 point, but
also presents minimal risk to the crew. The first mission allows the crew to perform a very effective robotic
investigation of the surface. After the first, sample return mission to L1 can be performed. When a permanent
presence is finally achieved human landings on Mars can be readily completed again with minimum risk to the crew.
Much work still needs to be done. The key to determining the usefulness of using L1 as a base for the
exploration of Mars is the low thrust trajectory. The example shown was not optimized, but the results are still
somewhat better than the same mission performed with nuclear thermal propulsion and are comparable with nuclear
thermal propulsion missions to low Mars orbit, but without a landing. The thrust angles of approximately 90
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American Institute of Aeronautics and Astronautics
degrees in Figure 5 strongly suggest that a better functional form for either the specific angular momentum, or the
inverse of the radius to the Sun, or both will greatly improve the mass arriving at L1. A better choice for the
heliocentric angle traversed by the spacecraft can significantly reduce the time to arrive at Mars and the stay time
required to return. It should be possible to obtain better choices for the functions by applying the principles of
variational calculus.
Acknowledgment
The author wishes to acknowledge the support of University of Connecticut for their help in
preparation and presentation of this paper.
References
1
Von Braun, W., The Mars Project, University of Illinois Press, Urbana, 1953 (reprinted in 1991), pp. 38-64.
Donahue, B. “Nuclear and Thermal Propulsion Vehicle Design for the Mars Flyby with Surface Exploration,”
AIAA/SAE/ASME 27th Joint Propulsion Conference, Sacramento, Paper AIAA 91-2561, June,1991.
3
Zubrin, R.M., The Case for Mars, Simon and Schuster, Inc., New York, 1996, pp.75-84.
4
Review of Human Spaceflight Plans Committee, Seeking a Human Spaceflight Program Worthy of a Great Nation,
http://www.nasa.gov/pdf/396093main_HSF_Cmte_FinalReport.pdf.
5
Petro, A., VASIMR Plasma Rocket Technology, NASA JSC, http://dma.ing.uniroma1.it/users/bruno/Petro.prn.pdf,
May, 2002.
6
Sutton, G. and O. Biblarz, Rocket Propulsion Elements, 7th edition, John Wiley & Sons, Inc., New York, 2001.
7
http://Nssdc.gsfc.nasa.gov/planetary/factsheet/saturnfact.html
8
http://Nssdc.gsfc.nasa.gov/planetary/factsheet/saturniansatfact.html
9
Battin, R., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education Series, American
Institute of Aeronautics and Astronautics, Inc. New York, 1987.
10
Szebehely, V. Theory of Orbits – The restricted problem of three bodies, Academic Press, New York, 1967.
11
Betts, J.T., Survey of Numerical Methods for Trajectory Optimization, Journal of Guidance, Control and
Dynamics, Vol., 21, No. 2, pp. 193-207, 1998.
12
Betts, J.T., Very low-thrust trajectory optimization using a direct SQP method, Journal of Computational and
Applied Mathematics, Vol. 120, pp. 27-40, 2000.
13
Cassenti, B.N. "Optimization of Relativistic Antimatter Rockets", Journal of the British Interplanetary Society,
Vol. 37, pp. 483-490, 1984.
14
Cassenti, B.N., Trajectory Options for Manned Mars Missions, Journal of Spacecraft and Rockets, Vol. 42, No.5,
pp.,890-895, 2005.
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