LOW THRUST TRANSFER TO SUN-EARTH L1 AND L2 POINTS
WITH A CONSTRAINT ON THE THRUST DIRECTION
Alexander A. Sukhanov and Natan A. Eismont
Space Recearch Institute (IKI) of the Russian Academy of Sciences
84/32 Profsoyuznaya St., Moscow 117997, Russia
e-mails: [email protected], [email protected]
Low-thrust transfers from a low Earth circular orbit (LEO) to the
Sun-Earth L1 and then to L2 points are analyzed. A spin-stabilized
spacecraft with the spin axis orthogonal to the Sun direction is
considered. The thrusters provide jet acceleration along the spin axis
in both directions. Thus, the thrust is always orthogonal to the Sun
direction. The spiral spacecraft ascent from the LEO is considered
first. Each orbit of the spiral has two thrust arcs and two coast ones.
Then the spacecraft is inserted into an L1 halo orbit in the ecliptic
plane. After the operations in the halo orbit are completed the
spacecraft is transferred to an L2 halo orbit. The transfers containing
zero, one, or two complete orbits around the Earth are considered.
Introduction
A low-thrust mission to one or two of the Sun-Earth collinear libration points was being
developed in Russia a few years ago. Main mission goals were solar wind exploration and
magnetic storm prediction; however, perhaps most important goal was testing of new
technologies. The mission concept was the following:
– a light spacecraft equipped with solar electric propulsion (SEP) was to be assembled at
the International Space Station1;
– the spacecraft was to ascend from LEO in a spiral orbit, to be transferred to the L1 point,
and to be inserted into a halo orbit using SEP;
– after the operations in the L1 halo orbit are completed the spacecraft could transfer to an
L2 halo orbit.
The mission has not been adopted, nevertheless some of its features can be of a certain
interest. For instance, the following: a spin-stabilized spacecraft was considered with
immovable solar arrays and thrusters what simplified the spacecraft design and control and
lowered the mission cost. The spin axis is orthogonal to the Sun direction and the thrusters
provide the thrust along the spin axis in both directions. Thus, the thrust is always orthogonal
to the Sun direction and all the transfers and maneuvers were to be performed under this
constraint on the thrust direction. This paper presents main results of the mission analysis,
namely the following:
– Earth to L1 halo transfer including the spiral ascent near Earth;
– L1 halo to L2 halo transfers with different numbers of complete orbits around the Earth.
Some of the important problems of the mission (such as communication, parameters of the
film solar arrays, a prolonged being in the van Allen belts etc.) are not discussed in the paper
because they were outside authors’ competence.
1
Mir space station was considered for this purpose at an early phase of the project.
The spacecraft concept
The electrically propelled spacecraft design is subject to the following requirements:
– the continuous thrust direction must be close to the spacecraft velocity vector for a long
time of the thrust run to provide maximum efficiency of the thrust;
– the solar panels must have a big area and be directed to Sun for all time of the SEP run to
provide the power-consuming thrusters with maximum electric power.
These requirements often contradict each other, especially in the spiral orbit where the
spacecraft performs hundreds of orbits and its thrust must follow the velocity vector in each
of them. This would lead to complicated both the spacecraft construction and control.
A simple and elegant solution of the problem has been proposed for the considered
mission. The spin-stabilized spacecraft reminds a bicycle’s wheel with the spacecraft body in
the middle and the solar arrays along the rim (see Fig. 1).
Figure 1. Overall view of the spacecraft
The one-sided arrays form a cylindrical surface of 9-m diameter and 2-m height. Thus, the
total area of the arrays is about 55 m2. Assuming 85% coverage of the arrays by photocells,
maximum effective area of the arrays (i.e. cross-sectional area of the photocells orthogonal to
the solar radiation flux) is about 15 m2. Film solar arrays providing about 1.5 kW of the
electric power assumed to be used. Wet initial mass of the spacecraft was estimated as of
about 290 kg.
The spacecraft has 8 thrusters D-38 developed in the Energia Rocket and Space
Corporation. Four of the thrusters are installed on one side of the spacecraft and another four
on the opposite one; thus, their thrust was to be directed along the spacecraft spin axis in two
opposite directions. However, only two of four co-directed thrusters run simultaneously;
another pair is auxiliary. Parameters of the D-38 thruster are given in Table 1.
Table 1. D-38 thruster parameters
Parameter name
Parameter
value
Power, W
Specific impulse, s
Efficiency (including losses in PPU)
Thrust force, N
Mass flow rate, kg/s
Resource, hours
Propellant
750
2200
0.5
0.035
1.6⋅ 10-6
3000
xenon
The spacecraft spin axis is orthogonal to the Sun direction during the SEP run. Thus, the
thrust always lies in the plane orthogonal to the Sun direction. It is assumed that any thrust
direction in the plane can be chosen. The spacecraft contains 85 kg of xenon what provides
about 7.5 km/s of the spacecraft characteristic velocity.
Spacecraft ascent from LEO
Spiral ascent strategy. Typical space station orbital parameters were taken for the initial
spacecraft orbit: a circular orbit of the 400-km altitude and 51.6-degree inclination.
After the separation from the space station and starting its ascent from LEO by means of
SEP the spacecraft moves in an expanding spiral orbit. While the spacecraft jet acceleration is
much lower than the gravitational one the instantaneous spacecraft orbit remains very close to
the circular one of the growing radius. Optimal thrust direction in this case is always close to
the spacecraft orbital velocity vector. However this optimal direction cannot be provided for
the spacecraft described above (except two points in each orbit where the spacecraft velocity
is orthogonal to the Sun direction). The following strategy of the SEP control taking into
account the spacecraft concept has been selected for the mission: the SEP runs along two 120degree arcs, ±60 degree from the projection of the Sun direction onto the orbit plane; two
thrusters providing proper thrust direction run in each of the arcs (see Fig. 2).
Figure 2. The SEP runs in the Earth vicinity
This strategy leads to a loss of 17 percent of the SEP effectiveness (and respectively to a
higher propellant consumption) and to longer with factor 1.7 time of flight comparing to the
Shadowed arc and inclination to ecliptic, deg.
permanent tangential thrust. This is the payment for the simplified spacecraft design and
control.
The 120-degree arc has been chosen as a compromise: a shorter arc would provide higher
effectiveness of the SEP but the flight time would increase; a longer thrust arc would lead to a
less effectiveness of the propulsion.
Shadowing. The thrust arc behind the Earth (with respect to Sun) can be entirely or partly
shadowed by the Earth for a long time (see Fig. 2). It is impossible to avoid the shadowing
completely; the only way to diminish it is an appropriate selection of both the Sun position
and the longitude of the ascending node of the spacecraft orbit at the launch time. Analysis
shows that the spacecraft launch in June-July or December-January with the longitude of the
ascending node of about 280 to 300 degrees minimizes the average shadowing down to 7.5
percent (i.e. in average about 7.5 percent of the whole thrust arc per one orbit is shadowed).
However this optimal solution would lead to a high (higher than 50 degrees) final inclination
of the spacecraft orbit to the ecliptic plane what is not good for the further insertion into the
halo orbit. Therefore a compromise has been selected: launch in May or November with the
longitude of the ascending node around 260 degrees. This gives the average shadowing of
about 8.5 percent and the final inclination to the ecliptic plane of about 35 degrees (it is clear
that the minimal possible inclination is about 28 degrees if the inclination to the equator is
51.6 degrees). The thrust arc shadowing and inclination to the ecliptic versus time are shown
in Fig. 3; as is seen in the Figure one of the thrust arcs is completely shadowed at the
beginning of flight.
120
Shadowed arc
80
Inclination
to ecliptic
40
0
0
100
200
300
Time of flight, days
Figure 3. Shadowed arc and inclination to the ecliptic versus time
Parameters of the spiral orbit. Radius of the spiral orbit versus time is given in Fig. 4;
Table 2 presents parameters of the spiral orbit in the Earth vicinity.
Orbit radius, 10 3 km
100
10
1
0
100
200
300
Time of flight, days
Figure 4. Orbit radius versus time
Table 2. Parameters of the spiral orbit
Parameter name
Time of flight, days
Number of orbits
Consumed characteristic velocity, m/s
Propellant consumption, kg
Spacecraft mass, kg
Parameter
value
280
1330
6850
78.9
211.1
Notice that the final inclination to the ecliptic is sensitive to the initial longitude of the
ascending node: 10-degree variation of the longitude changes the inclination in 3 degree.
Since the ascending node precession is about 5 degrees per day for the space station, the
launch window providing necessary inclination to the ecliptic plane is narrow. Therefore the
spacecraft should be separated in advance in order to start operations exactly at a proper time.
Flight to L1 and insertion into halo orbit
Rather small halo orbit with amplitude Ay ≈ 60 thousand km has been selected for the
mission. Fig. 5 gives two projections of the spacecraft transfer trajectory: projection onto the
ecliptic plane (xy) and the orthogonal one (xz); the spiral shown in Fig. 5 begins from 50,000km radius.
y
L1
x
z
x
L1
Figure 5. The spacecraft trajectory to L1
The transfer trajectory shown in Fig. 5 corresponds to launch in November; for the May
launch the xy projection will not change and the xz one will be mirrored with respect to the x
axis. The trajectory includes the spiral part, the flight to L1, and the halo orbit. Bold arc at the
end of the spiral orbit is the last thrust arc injecting the spacecraft into the transfer trajectory
to L1. This arc is shorter than the typical 120-degree arcs and asymmetric; this is to provide
necessary halo amplitude and z component close to zero during the insertion into the halo.
The thrust arc lasts 4.5 days and consumes 268 m/s of the spacecraft characteristic velocity
(corresponding 2.6 kg of xenon have been included in the spiral orbit propellant
consumption). The bold arc near L1 shows the break maneuver inserting the spacecraft into
the halo orbit. The axes ticks correspond to 200,000-km distance, crosses on the curves mark
10-day time intervals after the injection into the transfer trajectory.
Parameters of the transfer to and insertion into the halo orbit are given in Table 3.
Table 3. Parameters of the flight to halo orbit
Parameter name
Time of flight (after the spiral), days
Characteristic velocity of the insertion
into halo, m/s
Propellant consumption, kg
Spacecraft mass in the halo, kg
Parameter
value
140
290
2.8
208.3
Amplitude Ay of the halo orbit, 103 km
62
As is seen in Fig. 5 a planar halo orbit has been chosen for the mission analysis. It is not
necessary for the mission purposes; however the cost of this insertion is just a little higher
than of the insertion into a 3D halo with a reasonable Az amplitude. So the cost, 290 m/s (see
Table 3), is an upper limit for the insertion into a halo with 60,000-km Ay amplitude.
A tiny variation of the ∆V transferring the spacecraft to the libration point can dramatically
change the halo orbit amplitude. The approximate dependencies are the following: the ∆V
increment in 5 cm/s increases the Ay amplitude in 100 thousand km and reduces the
characteristic velocity and the propellant consumption of the insertion into halo in 20 m/s and
0.2 kg respectively.
On the Moon gravity assist
Moon gravity assist for the transfer to L1 has been analyzed only for the spacecraft with 3axis stabilization. The main advantage of this maneuver is that it can put the spacecraft
trajectory very close to the ecliptic plane and hence lower delta-V of the insertion into the
halo almost in 200 m/s (2 kg of xenon). However the Moon gravity assist may require waiting
in a parking orbit for providing the Moon encounter conditions what can increase the total
flight time in a few weeks.
Transfer from L1 to L2
Introduction. Planar transfers from the L1 halo orbit to an L2 one are considered in this
section. This part of the mission was considered rather as a mission extension, its profile was
completely uncertain. In particular parameters of the L2 halo orbit were not defined. Therefore
different options have been considered for the transfer trajectory design.
There is a great amount of possible halo-to-halo transfers. Even if both of the halos are
given the transfers differ by the number and location of the active maneuvers, their values,
number of complete orbits around the Earth, transfer duration, use of the Moon gravity assist
etc. Some of the transfers are described below.
Models and methods used. Since the specific impulse is given (see Table 1), we have the
constant exhaust velocity (CEV) case. In this case the thrust value is given by
α=
m& p c
(1)
m
where mp is consumed propellant mass, c is exhaust velocity, m is the spacecraft mass.
Hill model was used for the motion analysis. The model is given by the equations
r& = v
v& = n 2 Nr + 2nMv −
µ
r
3
r+α
(2)
where µ is the Earth gravity constant, r, v are the spacecraft position and velocity, r = |r|, n is
the Earth mean motion, α is the thrust vector,
3 0 0 
 0 1 0


N = 0 0 0 , M =  − 1 0 0




0 0 − 1
 0 0 0
Thrust arcs in the transfer are short comparing to the whole transfer time. This is why at
the first step the arcs can be approximated by the junction points (i.e. points of the impulsive
thrust application). In order to obtain the junction points positions a combination of the
Pontryagin’s maximum principle and Lawden’s primer vector was used [1, 2]. The
Hamiltonian is
µ 

H = p 0 m& p + p Tr v + p Tv  n 2 Nr + 2nMv − 3 r  + p Tv α
r 

(3)
where p0, pr, pv are costate variables (pv is the Lawden’s primer vector), superscript T means
transposition. Vectors pr, pv satisfy the adjoint variational equations
T
(
)
 ∂H 
2
p& r = −
 = − G + n N pv
 ∂r 
(4)
T
 ∂H 
T
p& v = −
 = −p r − nM p v
 ∂v 
where
G=
µ  rr T


 ,
−
I
3

r 3  r 2

I is the unit matrix of 3rd order. The constraint on the thrust direction described above can be
written as follows:
x 0T α = 0
(5)
where x0 is unit vector of the x axis. Constraint on the thrust value is
αT α = α 2
(6)
where α is given by (1). The optimal control α should provide maximum of function (3)
under the constraints (5, 6), i.e. the following function is to be maximized:
L = H + λ1 x 0 T α +
λ2
2
(α
T
α −α 2
)
(7)
where λ1, λ2 are indeterminate multipliers. Then
∂L
= p Tv + λ1 x 0T − λ 2 α T = 0 T
∂α
(8)
Vector α can be easily found from (8, 5, 6) as follows:
pA
pA
(9)
pA = pv – (x0Tpv)x0
(10)
α =α
where
is a projection of the pv vector onto the yz plane, pA = |pA|. Also it can be shown that the
switching function κ = κ(t) [2] in this case is given by
κ& =
p& A c
m
(11)
The junction points correspond to the maximum values of pA which can be found by means of
numerical integration of equations (2, 4) and using (10); transfer between the junction points
is ballistic, i.e. α = 0 in (2) when integrating.
After the junction points were found three-thrust-arc transfers were analyzed. The thrust
arcs were the following: launch from the initial L1 halo; a midcourse maneuver applied at a
junction point; insertion into the L2 halo orbit. Since planar halo-to-halo transfers were
considered, the thrust was always directed along the y axis. Thus, it was sufficient to find only
three values of the three maneuvers (i.e. propellant consumptions or characteristic velocities
or the spacecraft velocity changes). The characteristic velocity values ∆v1, ∆v2, ∆v3
corresponding to the thrust arcs were varied in order to perform the transfer and to minimize
the sum ∆v = ∆v1 + ∆v2 + ∆v3. The condition ∆v ≤ ∆vr was also used where ∆vr is the rest of
the available characteristic velocity after the insertion into the L1 halo has been performed.
Zero complete orbits around Earth. The transfer trajectory for this case is shown in Fig. 6.
The thrust is directed toward +y axis for ∆v1,3 maneuvers and −y for ∆v2 one.
y
∆ v1 L
1
Earth
L2
x
∆ v2
Figure 6. Transfer with zero complete orbits
The transfer parameters are given in Table 4.
Table 4. Transfer with zero complete orbits
Parameter name
Parameter
value
∆v3
Consumed characteristic velocity, m/s
∆v1
∆v2
∆v3
Time between ∆v1 and ∆v2, days
The transfer duration, days
Propellant consumption, kg
Final spacecraft mass, kg
Ay amplitude of the L2 halo, 103 km
306
50
195.6
60.5
70
181
2.9
205.4
800
Duration of the transfer is relatively short in this case, just 6 months. However the
available propellant allows only this large halo orbit around L2.
One complete orbit around the Earth. Fig. 7 shows one of the possible transfers with final
Ay = 300 thousand km. The thrust is directed toward +y axis for all three ∆V maneuvers in
this case.
y
∆ v1 L
1
Earth
x
L2 ∆ v
3
∆v2
Figure 7. Transfer with one complete orbit
Parameters of the transfer are given in Table 5.
Table 5. Transfer with one complete orbit
Parameter name
Consumed characteristic velocity, m/s
∆v1
∆v2
∆v3
Time between ∆v1 and ∆v2, days
The transfer duration, days
Propellant consumption, kg
Final spacecraft mass, kg
Ay amplitude of the L2 halo, 103 km
Parameter
value
224
65
18.1
141
82
259
2.2
206.1
300
This transfer has a longer duration (8.6 months) than the zero-orbit one, but can provide
lower amplitude of the final halo orbit for lower propellant consumption. The available
propellant could permit even lower amplitude than one indicated in Table 5.
Two complete orbits around the Earth. One of the options for the transfer with two
complete orbits around the Earth is shown in Fig. 8; the thrust is also directed toward +y axis
for all maneuvers.
y
∆ v1 L
1
Earth
∆v2
L2
∆v3
x
Figure 8. Transfer with two complete orbits
Table 5 gives the transfer parameters.
Table 5. Transfer with two complete orbits
Parameter name
Consumed characteristic velocity, m/s
∆v1
∆v2
∆v3
Time between ∆v1 and ∆v2, days
The transfer duration, days
Propellant consumption, kg
Final spacecraft mass, kg
Ay amplitude of the L2 halo, 103 km
Parameter
value
70
35
1.6
33.2
70
319
0.7
207.6
150
This is the longest transfer (10.5 months), but it allows any amplitude of the L2 halo orbit
for a very low cost. Note that in the case of two complete orbits a two-thrust-arc transfer is
also possible. A symmetric two-thrust-arc transfer is shown in Fig. 9; here ∆v1 = ∆v2 = 43.3
m/s (0.8 kg of the propellant for both) and the transfer duration is 307 days.
y
∆v1 L
1
Earth
Figure 9. Symmetric two-impulse transfer
x
L2 ∆v2
Common notes. In all three transfer cases considered both the L2 halo amplitude and the
transfer duration can be varied by means of changing the ∆V maneuvers values and positions.
Lowering the amplitude in 30 thousand km costs about 10 m/s of the spacecraft characteristic
velocity (~0.1 kg of xenon); lowering the transfer time in 10 days takes 5–7 m/s.
The Moon gravity assist can be easily performed in the considered planar (or near-planar)
transfer. Phasing of the spacecraft trajectory necessary to provide the encounter with Moon
can be obtained by a very small variation of the launch maneuver in the L1 halo orbit. The
gravity assist certainly could either lower the xenon consumption for the transfer or lower the
L2 halo amplitude or the transfer duration. However this maneuver has not been analyzed yet.
Conclusion
Table 6 summarizes characteristics of all the spacecraft movements; the duration of its stay
in the L1 halo orbit is excluded from the total flight duration because it is still undefined.
Table 6. Summary of the spacecraft transfers
Operation
Launch
Ascent in the spiral orbit
Transfer to and insertion
into L1 halo
Transfer to and insertion
into L2 halo
Rest for the correction
maneuvers
Flight time,
month
Total ∆v,
km/s
0
9.3
0
6.85
Total xenon
consumption,
kg
0
78.9
14.0
7.14
81.7
208.3
20–24.5
7.21–7.45
82.4–84.6
205.4–207.6
–
0.05–0.29
0.4–2.6
–
S/C mass,
kg
290
211.1
The run time of each thruster is within 2000 hours what is covered by the thruster resource
time (see Table 1). Nevertheless an installation of a pair of spare thrusters is also possible.
Thus, the spacecraft concept having been taken for the mission provides fulfillment of all
operations necessary for the transfer to the L1 halo orbit and then to the L2 one for reasonable
time and propellant consumption. This is mainly due to the fact that the thrust orthogonal to
the Sun direction is very effective for changing the orbital parameters in the libration points
neighborhood. Although this is also true for planetary missions [3], so this concept can be
applied to them as well.
References
1. L. S. Pontryagin et al. Mathematical Theory of Optimal Processes, Moscow, Nauka, 1969
(in Russian).
2. D. F. Lawden. Optimal Trajectories for Space Navigation, Butterworths, London, 1963.
3. Williams, S.N.; Coverstone-Caroll, V. Benefits of Solar Electric Propulsion for the Next
Generation of Planetary Exploration Missions. Journal of Astronaut. Sci., Vol. 45, No. 2,
April-June 1997, pp. 143-159.