Comparison between Cyclers and Stop-over Cyclers
for a Regular Earth-Mars Transportation System
Natacha Linder a, Massimiliano Vasile b
a
Faculté des Sciences Appliquées, Université de Liège, Belgium
b
Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Italy
Abstract
This work consists of a comparison between cyclers and stop-over cyclers, in the
framework of a possible future cyclic human Earth-Mars transportation system. At
first, a comparison among the vast variety of possible cyclers has been carried out
in the attempt of selecting the ones that are interesting for a realistic transportation
system. Then, cyclers are compared to stop-over cyclers in terms of flight times,
waiting times at the Earth and at Mars, ∆Vs and mass budget, and some
considerations are derived on the actual convenience of the two concepts.
Introduction
The advanced robotic and human exploration of the solar system, with Mars as,
most likely primary target (as envisaged by the Aurora program), will require some
sort of infrastructure that allows regular links between the Earth and Mars, in
particular for repeated human missions. Cyclers and stop-over cyclers are two
concepts that have been frequently proposed as promising options to set up this
infrastructure. The actual convenience of one of the two concepts with respect to
the other has still to be clearly assessed.
Cyclers are trajectories that repeat periodically, visiting alternatively two planets
but without stopping at any one of them. They have been studied since the 1960's
[1] at first between Earth and Venus and later between Earth and Mars, at the
beginning of the 1970's [2]. Aldrin, in 1985, was the first to propose a cycler that
repeats every synodic period and uses the gravity of Earth and Mars to correct the
orbit at every planetary encounter [3]. This type of trajectory is interesting because
it provides fast flight times, and thus reduces the problem induced by a long stay in
space. On the other hand the cyclers have high approach and flyby velocities and
require rendezvous and docking manoeuvre on hyperbolic orbits. More recently,
stop-over cyclers have been studied as an alternative concept [6]. They are
trajectories that stop at each encounter with a planet, and wait there the next
opportunity to continue their travel. This waiting time is used to perform several
tasks among which refuelling. Conceptually, the same transportation elements can
be used both for stop-overs and for cyclers but the overall cost for putting in place
the transportation system and for its maintenance could make either of the two more
attractive.
In this work, a high number of solutions for cyclers have been computed using
simplifying assumptions. Then solutions for stop-overs cyclers have been computed
and compared to cyclers in terms of ∆V and propellant budget, time of flight, visit
frequency and time, in order to determine which concept is the cheapest and most
attractive. In order to check the actual consistency of the simplifying assumption
adopted for cyclers, new optimised solutions have been optimised using a more
realistic model.
Cyclers
Simplified Orbit Model
In order to quickly analyse a large number of alternative solutions for cyclers the
simplified model proposed in [7] is also adopted in this work. The Earth-Mars
synodic period S is assumed to be equal to 2 years and 1/7 (783 days, while a more
accurate value is 780 days), and given the fact that the eccentricity of Earth and
Mars are very small (e = 0.0167 for Earth and 0.093 for Mars), both orbits have
been considered to be circular. Furthermore, Earth’s orbit, Mars’ orbit and the
cycler trajectory are supposed to lie in the ecliptic plane. It is thus a 2-dimensional
problem with a cycler’s trajectory conic and prograde. Only the Earth is assumed to
have sufficient mass to provide gravity-assist manoeuvres and these manoeuvres
occur instantaneously. Thus, the spacecraft trajectory is not perturbed when flybying Mars. Each of the computed solutions can be uniquely identified by three
values:
1) n, the number of synodic periods passed before repeating the cycler;
2) whether the solution is long-period (L), short-period (S) or a unique-period
(U) solution;
3) N, the number of revolutions performed rounded down to the nearest
integer. A solution is denoted by an expression of the form nPN where P is
L, S, or U depending on the case.
Further details can be found in [8].
Number of vehicles required
When cycler’s trajectories crosses Mars’ orbit, it does it at two points. By choosing
the correct launch date, the spacecraft can encounter Mars at the first point, or at the
second one and just before returning to Earth. The first type of cycler is called an
outbound cycler because it is used to go from Earth to Mars. The second one, the
inbound cycler, is used to travel from Mars to Earth. There is a cycler of each type
per synodic period and one vehicle per travel is required. In the framework of
human missions to Mars, the goal is to have as many visits as possible of Mars and
of Earth (like regular buses), with a frequency as high as possible, to minimize the
waiting time at both planets. But when n is greater than one, the cycler takes more
than one synodic period to return to Earth and more than 2 vehicles are required. In
general, the number of needed vehicles is equal to 2n. This number has to be kept as
small as possible in order to minimise the cost of the infrastructure. For this reason,
for the present study, a limit for n has been fixed at 6.
Departure conditions
The cycler is supposed to be in a highly elliptical orbit (aphelion radius: 2.105 km
and perihelion radius = 200 km) before departure. A taxi, being initially on a 200
km circular orbit, join the cycler just before departure, by making a rendezvous
when the cycler is close to its perihelion.
Arrival at Mars
Since the taxi must rendezvous with the cycler as it passes Mars, the Mars V∞ has to
be as small as possible. This velocity is a characteristic of the cycler and cannot be
changed. What can be chosen is the perihelion radius (rp) of the hyperbolic
trajectory around Mars. To reduce the cost of the taxi, this radius has to be as small
as possible, but on the other hand, the assumption that only Earth, and not Mars, has
a sufficient mass to provide gravity-assist manoeuvres has been made. Doing that,
an error in the trajectory occurs at Mars, all the more important since the perihelion
radius decreases. If β is the semi-angle between the two hyperbola’s asymptotes,
this angle has to be as close as possible of π/2 to make the smallest error in the
chosen model. For each considered cycler, an error of less than 1% (1.8° of
deviation) on the ideal value of β has been considered as acceptable, and rp has been
calculated with this value. This situation is called Case 1.
Return to Earth and swing-by
When the spacecraft fly-bys the Earth again, it has to perform a gravity assist
manoeuvre to rotate its velocity vector and to return to Mars once more on the right
track. If the cycler is relatively slow when it arrives at Earth, the planet, at a certain
rp, performs the gravity turn. Anyway this occurs just for few types of cyclers (only
three cyclers are ballistic for n between 1 and 6, namely the 6S7, 6S8 and 6S9) and
the spacecraft will have to provide a certain ∆Vfly to return on track. If the cycler is
ballistic, the swing-by is performed at the altitude corresponding to the needed
deviation angle of the velocity and if it is not, the altitude is chosen as low as
possible to have the highest natural deviation angle (rp = 200 km) and then the
required additional ∆Vfly is calculated.
Comparison among cyclers
In the goal to compare the cyclers with the stop-overs, a first choice among cyclers
has to be made by keeping only the ones that seem to be realistic in terms of ∆V. To
do this, for each cycler, the needed ∆V for departure, taxi at Mars, taxi at Earth, and
flyby have been added, and the plot is illustrated on Figure 1. Two different regions
can be observed in this plot: one is the group of long period solutions; the other is
the group of short period solutions. Some cyclers are very costly and will be
eliminated in the following study.
Thus, only eight cyclers, whose main characteristics are shown in Table 1, have
been retained. The three n = 6 cyclers are ballistics and fly-by Earth at an altitude of
1404 km, 5412 km and 13844 km, respectively.
Figure 1: Comparison between cyclers from n=1 to n=6
Cycler
1L1
2L3
3L5
4S6
5S7
6S7
6S8
6S9
V∞
at Mars
[km/s]
9.73
3.03
2.94
4.00
3.61
6.63
3.85
1.19
V∞
at Earth
[km/s]
6.54
5.65
7.60
8.51
5.86
4.99
4.02
3.04
∆Vd
[km/s]
∆Vfly
[km/s]
1.97
1.54
2.55
3.08
1.64
1.25
0.89
0.59
1.35
5.06
12.09
15.95
11.2
0
0
0
Shorter
flight time
[days]
146
290
283
163
172
134
181
213
Wait
at Mars
[km/s]
730
543
682
505
526
630
549
526
Wait
at Earth
[km/s]
541
441
317
734
694
666
652
611
Table 1: Characteristics of the eight most promising cyclers
Stop-over cyclers
Direct transfers will now be considered, where the spacecraft travels from Earth to
Mars and stops there, while waiting for the next opportunity to return to Earth. For
these types of trajectories, the actual ephemeris of both Mars and Earth have been
considered. Moreover, highly elliptical orbits for departure and arrival of the stopover have been selected [6]:
• At Earth: perihelion at 6578 km (RT + 200 km) and aphelion at 2 105 km
• At Mars: perihelion at 3593 km (RM + 200 km) and aphelion at 9.5 104 km
By choosing as the initial date the January 1st, 2026, a set of round trips can be
calculated. We proceeded as follows:
1) In the year 2026, the minimum Earth-Mars transfer ∆V is reached for a
flight time of 250 days and this minimum occurs at a known launch date;
2) the arrival date being known, we searched on a period of 18 months, for
which date and which flight time we have a minimum ∆V to return to
Earth. It gave us the waiting time around Mars;
3)
we then calculated the date of arrival at Earth, and the waiting time around
Earth, by searching the optimal date to return to Mars. This can be
repeated several times to construct a set of round trips.
Planet of
departure
Earth
Mars
Earth
Mars
Earth
Mars
Earth
Mars
Vehicle 1
Launch date ∆V departure Arrival date ∆V break
m/d/y
m/d/y
[km/s]
[km/s]
11/19/2026
0.75
07/27/2027
1.07
10/10/2028
0.87
06/17/2029
1.13
02/18/2031
0.79
09/16/2031
1.45
02/10/2033
0.67
09/18/2033
0.74
06/27/2035
0.64
01/14/2036
0.75
07/24/2037
1.37
03/31/2038
0.72
10/14/2039
1.11
06/20/2040
0.74
09/01/2041
1.01
05/08/2042
0.78
Flight time
[days]
250
250
210
220
200
250
250
250
Planet of
departure
Earth
Mars
Earth
Mars
Earth
Mars
Earth
Mars
Vehicle 2
Launch date ∆V departure Arrival date ∆V break
m/d/y
m/d/y
[km/s]
[km/s]
12/30/2028
0.75
08/17/2029
1.39
11/28/2030
0.74
07/26/2031
1.03
04/17/2033
0.58
11/03/2033
1.11
05/08/2035
0.92
11/24/2035
0.58
09/02/2037
0.99
04/10/2038
0.69
08/28/2039
1.40
05/04/2040
0.62
11/10/2041
0.89
07/18/2042
0.96
09/28/2043
0.91
06/04/2044
1.09
Flight time
[days]
230
240
200
200
220
250
250
250
Table 2: Vehicle 1 and Vehicle 2 trajectory parameters of stop-over round trips
We noted that two vehicles are needed in order to profit from all the launch
opportunities. As with the cyclers, this set of stop-overs round trips is supposed to
build regular links between the two planets. Thus, if the vehicle 1 has not come
back to Earth when another launch opportunity occurs, it is logical to use another
spacecraft, instead of cancelling the flight. A taxi at Earth will cost 3.05 km/s, when
the one at Mars costs only 1.34 km/s. In Table 2 is stored the succession of launch
and arrival dates, for Vehicle 1 and Vehicle 2 and the corresponding needed ∆V.
Comparison between Cyclers and Stop-over Cyclers
For 5 round trips, the average stop-over characteristics are summarized in Table 3.
These values depend of course of the number of considered round trips.
Number of
needed
vehicles
2
Interval of varying
flight time
[day]
[200; 250]
Average
flight time
[day]
225
Average waiting
time at Mars
[day]
513
Average waiting
time at Earth
[day]
619
Table 3: Stop-over characteristics for 5 round trips
For the cyclers, fly-by at Earth is considered only if a new vehicle is not used for
this transfer (in this case, we have considered the ∆V of departure).
Evolution in time
In Figure 2, the needed ∆V have been computed for different numbers of round
trips, from 1 to 15. Cyclers are far more expensive than stop-overs. The principal
reason is the high value of ∆Vfly and/or the high cost of cyclers’ taxis.
Figure 2: Evolution of the needed ∆V for different numbers of round trips
Shorter stop-over transfer time
The stop-over is cheaper than the cyclers, but it has also a longer transfer time.
Thus we can imagine that if the stop-over flight time is shorten, the cost will greatly
increase, until it becomes higher than that of the cyclers. However, as shown in
Figure 3, where the flight time of the stop-over has been imposed at 150 days, the
line of the stop-over hardly reaches the 6S9 cycler.
Figure 3: Evolution of the needed ∆V if the stop-over’s flight time is 150 days
On the other hand, the average waiting time at Mars, for 5 round trips, has increased
from 513 days to 597 days. Waiting time at Earth decreased, but only by 27 days.
Furthermore, it is easier for astronauts to wait for a flight when they are on Earth
than if they are on Mars.
Variation of the fly-by radius at Mars
The principal problem of the cyclers is that the model used to build them requires
that they have to fly-by very far from Mars in order to avoid an important error in
the β angle, which increases a lot taxis’ cost. However, this constraint is not
physical.
If we impose that all cyclers fly-by at the altitude of Phobos (5985 km), it gives the
result shown in Figure 4. This situation for the cyclers is called Case 2.
Figure 4: Comparison for cyclers that fly-by at Mars at rp = 9378 km and a stopover with flight time equal to 150 days
This rp is lower than all the above cyclers’ fly-by radii (included between 28465 km
and 578320 km) but if the cost is indeed reduced, it is not significant and absolute
error on the β angle is already included between 3 and 55 %. Furthermore, it seems
to be clear that 3L5, 4S6 and 5S7 cyclers are not affordable because of their high
cost of fly-by. They can be eliminated from the study.
Comparison in terms of mass budget
The initial payload is known (mp = 15.5 tons for the taxis, and 70 tons for the
spacecraft [5] but this quantity does not take into account the tank mass.
The tank mass is considered here to be an addition to the payload, in order to keep
the initial value for engines and life support systems. If the tank is assumed to
weigh a fraction r of the propellant mass ∆m, we can write:
(1)
m tank = r ∆m
where r is equal to 10%.
It has been decided that in the case of cyclers, taxis and spacecrafts have cryogenic
propulsion, which is a necessity for such high taxis' velocities.
On the other hand, if stop-over’s spacecraft has also cryogenic propulsion, low taxis
∆V values allow to consider three types of propellant:
1) Solid propellants, for which Isp is included between 230 and 260 s. In
the following calculations, we will take a value of 250 s.
2) Liquid propellants, for which Isp is included between 270 and 300 s.
We chose the value of 300 s.
3) Cryogenic propellants, for which Isp is above 470 s. Isp = 470 s has been
kept in the algorithms.
Figure 5: Comparison between cyclers in case 2 and stop-overs with a flight
time fixed to 150 days
Figure 5 compares the budget of the most expensive stop-overs (for which the flight
time has been fixed at 150 days) and the budget of the cheapest cyclers (in case 2).
What can be seen on this picture is that it is possible to find a case for which some
cyclers have a lower budget than the stop-overs.
The budget of the 1L1 cycler (not represented in the figure) is quite expensive
compared to that of the stop-over, for a flight time that is more or less equal and a
longer waiting time at Mars. A long stay at a planet is not automatically a
disadvantage since manoeuvres in orbit, such as ascent from and descent to the
surface, take time. In addition, astronauts will also need time for operations on the
ground. The important factor is thus the flight time. As for the 6S8 cycler, it takes
only forty days more than stop-overs to reach its goal and its budget is always
below the lowest stop-over's curve. However, the problem that is most important
not to forget, is the need of 12 vehicles that require these two cyclers, as compared
to only two needed by the stop-overs. If the goal is to make savings in propellant
they are interesting, but the cost in money to build 12 spacecrafts will take a long
time to be written off, without counting launches’ costs and assemblies in orbit.
Finally, cyclers that have been plotted in Figure 5 are the ones in case 2, where the
cyclers fly-by Mars at an altitude of 5985 km. But an error in the model, which can
be big, is made in this case. To move away from this constraint and be free to fly-by
at any altitude from Mars or Earth, a new model is needed.
Optimised Cycler trajectory
In this section, a new model will be developed where both Earth and Mars have
elliptical orbits and where Mars can also provide gravity assist manoeuvres. It will
also be considered that Mars does not lie in the ecliptic plane. To do that, deep
space manoeuvres have to be managed during the travel between the planets. The
principal question of this technique is to know when manage the ∆V. An
optimisation work has thus to be done to discover the combination that gives the
minimum total ∆V needed for the manoeuvres. The results stored on Table 4
compare the theoretical Aldrin's cycler with the optimised one, for seven successive
cycles, as shown in Figure 6.
Transfer
time
[day]
146.2
Theoretical cycler
Fly-by’s
∆Vs sum
[km/s]
16.2
Taxis’
∆Vs sum
[km/s]
194.4
Optimised cycler
Mean
Manoeuvres’
transfer time
∆Vs sum
[day]
[km/s]
165.9
16.1
Taxis’
∆Vs sum
[km/s]
171.5
Table 4: Aldrin's cycler results for 7 round trips (7 outbounds and 7 inbounds)
Fly-by altitudes are made between 200 and 216 km above the surface of the planets.
Lower altitudes should give a better solution but this solution would not be realistic.
Figure 6: Optimisation of 7 cycles of the Aldrin’s cycler
Figure 7: Comparison between the optimised cycler and the theoretical cycler
On Figure 7 has been plotted the evolution of both cyclers' cost. Rapidly, the cost of
the optimised cycler becomes much lower than the other one, until to be 18.26 km/s
more advantageous after 7 round trips, which is not negligible.
Initially, the goal of adding deep space manoeuvres was to improve the model, but
observation shows that when times of burn are correctly chosen, it is possible to
find cheaper solution than in the previous cases.
Conclusion
Cyclers and stop-over cyclers have been compared in terms of ∆V, propellant mass,
flight time and revisit time. In terms of ∆V only, it seems that stop-overs are
generally more advantageous even because they do not require a docking
manoeuvre on hyperbolic orbit. On the other hand when the mass budget is
calculated, 6S8 and 6S9 cyclers become competitive. However these cyclers require
a total of 12 vehicles to take full advantage of all launch opportunities. Aldrin’s
cycler (which offers the lowest number of needed vehicles) is generally too
expensive, nevertheless its optimised version presents an appreciable gain in ∆V
especially for the taxi. This suggests that an optimised version of the 2L3 cycler
(which requires only 4 vehicles) could become competitive compared to stop-overs.
The revisit time and transit time are dominated by the geometry of the two planets
and by its periodicity and does not seem to be dependent on the particular
transportation system.
Acknowledgments
We wish to acknowledge the ESTEC Advanced Concept Team for its help, and we
are also grateful to Pr. Jean-Pierre Swings from the University of Liege to have
permitted us to perform this work.
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